ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
ﻃﺮﻕ ﺍﻟﺘﻨﺒﺆ ﺍﻹﺣﺼﺎﺋﻲ
) ا
ء اول (
ﺗﺄﻟﻴﻒ ﺩ .ﻋﺪﻧﺎﻥ ﻣﺎﺟﺪ ﻋﺒﺪﺍﻟﺮﺣﻤﻦ ﺑﺮﻱ
ﺃﺳﺘﺎﺫ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﻤﺸﺎﺭﻙ
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
!%& '(! رب ا
"
! وا
ة وا
م اف ا و%
ا
.)(* و%+
* و, و
.". &أ
.س12ر1
3(
ا45& ب6
7859 ا:(';
;ب =ق ا3
دة او1!
ا7@ هAه
@216;. م1Bف أ1 "C ذن اF. و7 دة إ
&ء ا1& HI ;ب3
ا اAه
م1"
ان اJ; أ7 اA ه7;و3
9* ا3K. HI &;! وH3K. *'%C@ و2Cو
73C; &L آ;ب7N O"P ان وQ%. &;رعH3K.& و12 ر16;C 4'J;
وا
.S;9&ت وا1"!
رة ا1U 7N 4+ع و1P1!
ا43&'2 دV& W';2X
ع1P1& ق إ6;2ع و1P1!
4
;ب ات او3
ا
ء اول & ا76Y2
دةL 4P2 ر4
"& اولS آ7;
واARIMA ;[ام !ذجF. 7859 ا:(';
ا
SB6C ! آTime Series 4'&
;[ام ا
!;ت اF. 7859 ا:(';
4!3%&و
V!L 7N و7859 ا:(';
43;ر1O
ا42J;
ق ا6
"\ ا. ;ب إ3
ا, 7N
4859 ا4&%
;[ام اF. 4ت ا
راX%
وا4]&^ اP1;. S!B ;ب3
اء اLأ
*وN V!. ء59 ;") ) ا4+ رت1= W5 _&ا. 7 وهMinitab
.
!ن. ب6
ةN1;& 4&%
@ اAوه
ذن اF. ق6;2 ف1 "
ب ا
رات ا6
*L1&;ب و3
& ا7]
ا
ء ا
H21%;
ا4
و!ذج داIntervention Analysis H;
اH%C H]& VPا1!
ا
!;"دة4'&
و!ذج ا
!;ت اTransfer Function Models
*OL1!
ا4'&
و!ذج ا
!;ت اMultivariate Time Series Models
State !
و&^ آ4
%
`ء اN و!ذجVector Time Series Models
Threshold Time %
و!ذج اSpace Models and Kalman Filtering
:(';
ا7N OCJ(6C وGARCH و!ذجARCH و!ذجSeries Models
ت3(K
ق إ
ا6;' ! آFinantial Time Series Forecasting 7
!
ا
.7859 ا:(';
ا7N O& وإ;[اNeural Networks 4("
ا
4(;3!
اء اU9) و23
* اOL1
H!"
ا اA إز ه7N 7'JN12 ا & ا ان1Lا وارAه
.;ب3
ا اA هH]& ة إJb
ا4."
ا
2
S;9 ا43( Lا1;& ن13 1 ) وهW
= ي7& ;ب3
ا اAن ه13
http://www.abarry.ws/books/statisticalForecast.pdf VB1!
ا7N
.N1!
وا ا
e
:!
ا
ي. !5
( اL& ن.د
د1" f!
ا4"&L
هـ1422 "ةJ
ذو ا
م2002 2'2
3
ﺍﻟﻤﺤﺘﻮﻳﺎﺕ
&4&J
-1ا
Hbاول 4&J& :و"Cر9...................................................................e2
1-1
أ&] 4ا
!;ت ا
&'9 ....................................................... 4
2-1
ا
Yض & درا 4و H%Cا
!;ت ا
&'9 ................................. 4
3-1ا
[16ات ا
!;[Aة ('ء !1ذج 9 ................................................ :('C
"C 1-3-1ا
'!1ذج ...................................................................
9
(6C 2-3-1ا
'!1ذج 10 ..................................................................
o[KC 3-3-1وإ;(ر ا
'!1ذج 10 .....................................................
1C 4-3-1ا
;'(:ات 10 ....................................................................
5-3-1إ;[ام ا
;'(:ات وو VPا
Jارات 10 .........................................
"C 4-1ر e2و&(دئ أو
10 ................................................................... 4
7P& e2"C 1-4-1أو Cر r2ا
Iهة .............................................
10
e2"C 2-4-1ا
P%أو اsن .......................................................
10
e2"C 3-4-1أ6ء ا
;......................................................... (6
10
e2"C 4-4-1أ6ء ا
;'(............................................................ :
11
e2"C 5-4-1اJ;9ار 11 .................................................................
e2"C 6-4-1ا
` 4ا
(`ء 11 .........................................................
]& 7-4-1ل : 1ا
! 7Kا
"1Kا11 ................................................... 78
e2"C 8-4-1دا
4ا
; 2Yا
Aا.................................................... 7C
12
e2"C 9-4-1دا
4ا
;ا w.ا
Aا12 ................................................... 7C
]& 10-4-1ل : 2دا
4ا
;ا w.ا
Aا 4`
7Cا
(`ء 12 .........................
e2"C 11-4-1دا
4ا
;ا w.ا
Aا 7Cا
13 ...................................... 78
]& 12-4-1ل :3دا
4ا
;ا w.ا
Aا 7Cا
4`
78ا
(`ء 14 ..................
e2"C 13-4-1دا
4ا
;ا w.ا
Aا15 ......................................... 4'"
7C
e2"C 14-4-1دا
4ا
;ا w.ا
Aا 7Cا
16 .............................. 4'"
78
4
]& 15-4-1ل :4دا
4ا
;ا w.ا
Aا 7Cوا
;ا w.ا
Aا 7Cا
17 ......... 4'"
78
-2ا
Hbا
]! :7ذج ا%9ار ا
Aا-7Cا
!; w1ا
!;%ك ARMA
-3وإ;[ا& 7N OCا
;'(25 .........................................................................:
1-2
! e2"Cذج ا%9ار ا
Aا-7Cا
!; w1ا
!;%ك & ا
ر25 ....... (p,q) 4L
2-2
H& e2"Cا9زا 45ا
[25........................................................ 7b
3-2
H& e2"Cا9زا 45ا&&25 .................................................... 7
4-2
H& e2"Cا
;25 ............................................................... 2b
5-2
H& e2"Cا
;!25 ............................................................... V
6-2
أ&]26 ................................................................................... 4
7-2
! o8ذج ا%9ار ا
Aا-7Cا
!; w1ا
!;%ك 26 ...........................
1! 1-7-2ذج )26 ......................................................... ARMA(0,0
1! 2-7-2ذج )31 ................................................................... AR(1
1! 3-7-2ذج )31 ................................................................... AR(2
1! 4-7-2ذج )36 .................................................................... MA(1
1! 5-7-2ذج )39 ................................................................... MA(2
1! 6-7-2ذج )40 ........................................................... ARMA(1,1
1 7-7-2اص !ذج )47 ............................................... ARMA(p,q
-3ا
Hbا
]
! :Qذج ا
!;ت ا
&' z 4ا
!;Jة 58 ....................................
1-3
م اJ;9ار 7Nا
!;58 ............................................................ w1
2-3
م اJ;9ار 7Nا
;(59 ............................................................... 2
3-3
!ذج ا%9ار ا
Aا-7Cا
;-7&3ا
!; w1ا
!;%ك & ا
ر62 ..... (p,d,q) 4L
1! 1-3-3ذج )62 ...................................................... ARIMA(1,1,0
1! 2-3-3ذج )62 ....................................................... ARIMA(0,1,1
1! 3-3-3ذج ا
! 7Kا
"1KاF. 78اف 63 ................................................
4- 3
دا
4اوزان ) ψ (Bو! H]!Cذج )63 .............................. ARMA(p,q
5- 3
ا&]
4ا
4اوزان ("\ ا
'!ذج 64 .....................................................
1-5-3دا
4اوزان '!1ذج )64 .................................................... AR(1
2-5-3دا
4اوزان '!1ذج )65 .................................................... MA(1
5
3-5-3دا
4اوزان '!1ذج )65 .................................................... AR(2
4-5-3دا
4اوزان '!1ذج )66 ................................................... MA(2
5-5-3دا
4اوزان '!1ذج )66 ........................................... ARMA(1,1
6-5-3دا
4اوزان '!1ذج )67 ................................................... ARI(1
7-5-3دا
4اوزان '!1ذج ا
! 7Kا
"1Kا68 .................. ARIMA(1,0,1) 78
6- 3
1 \".اص دا
4اوزان ) 69 .................................................... ψ (B
-4ا
Hbا
ا :V.ا
;'(:ات ذات &; V.& w1ا
[ {6اد '!ذج )71 ....... ARMA(p,q
:2 42I 1-4أ6ء ا
;'(71 ......................................................................... :
41!& 2-4ا
!"&1ت 72 ................................................. Information Sets
:3 42I 3-4ا
!;'(| ذا &; V.& w1ا
[ {6اد 72 ............................................
B 4-4ة 72 ............................................................................................. 2
e2"C 5-4دا
4ا
;'(73 ................................................................................. :
6-4دوال ا
;'(!'
:ذج )73 ....................................................... ARIMA(p,d,q
1-6-4دا
4ا
;'(1!'
:ذج )73 ............................................................... AR(1
2-6-4ط ا!;9ار 73 ............................................................................
3-6-4دا
4ا
;'(1!'
:ذج )74 .............................................................. AR(2
4-6-4دا
4ا
;'(1!'
:ذج )74 ................................................. ARIMA(0,1,1
5-6-4دا
4ا
;'(1!'
:ذج )75 .............................................................. MA(1
6-6-4دا
4ا
;'(1!'
:ذج )75 .............................................................. MA(2
7-6-4دا
4ا
;'(1!'
:ذج )76 ..................................................... ARMA(1,1
5 7-4ود ا
;'(80 ......................................................................................... :
;N e2"C 1-7-4ة 4!J
:('Cا
!;80 ..................................................... 4(J
]& 2-7-4ل 81 ........................................................................................
-5ا
Hbا
[&~ )!C :و'.ء Iم :('Cإ82 ............................................ 785
"C 1-5أو 2%Cا
'!1ذج 82 .......................................................................
S(]C 1-1-5ا
;(83 .............................................................................. 2
2-1-5إ;ر در 4Lا
;83 ................................................................ d 2b
84 ............................................................................... p,q 2%C 3-1-5
4-1-5إ )"& 4NPإاف 84 ...................................................................
6
2JCا
'!1ذج 85 ..............................................................................
2-5
4J2= 1-2-5ا
"وم 85 ........................................................................
2JC 2-2-5ا
"وم ("\ ا
'!ذج 86 .......................................................
1!'
1-2-2-5ذج )86 .............................................................. AR(1
1!'
2-2-2-5ذج )86 ......................................................... MA(1
1!'
3-2-2-5ذج )87 .......................................................... AR(2
1!'
4-2-2-5ذج )87 ......................................................... MA(2
1!'
5-2-2-5ذج )87 ................................................ ARMA(1,1
4J2= 3-2-5ا
!".ت ا
ا
89 .............................................. 4=K
2JC 4-2-5ات ا
!".ت ا
ا
\"(
4=Kا
'!ذج 89 .........................
!'
1-4-2-5ذج )89 .........................................................AR(1
!'
2-4-2-5ذج )90 ....................................................... MA(1
o[KCوإ;(ر ا
'!1ذج94 .........................................................
3-5
o%N 1-3-5ا
(1ا94 .................................................................... 7B
1-1-3-5إ;(ر ا
!;1(
w1ا94 .................................................. 7B
2-1-3-5إ;(ر ا
"1Kا1(
48ا95 ................................................. 7B
3-1-3-5إ;(ر ا
;ا w.أو اJ;9ل (1ا95 ................................... 7B
4-1-3-5إ;(ر =(" 4ا
(1ا96 .................................................... 7B
\". 2-3-5ا
!" 2اXى ;9ر !1ذج &'96 ............................. W
1-2-3-5إ 485آ '
1و96 ........................................... ~3.
"& 2-2-3-5ر ا9م ا
Aا96 ......................................... AIC 7C
3-3-5أ&] 4وX5ت درا96 .......................................................... 4
-6ا
Hbا
دس! :ذج ا%9ار ا
Aا-7Cا
;-7&3ا
!; w1ا
!;%ك ا
!143 ...... 4!1
1-6
دوال ا
;ا w.ا
Aا 7Cوا
;ا w.ا
Aا 7Cا
\"(
78ا
'!ذج ا
!143 ..... 4!1
1!'
1-1-6ذج 144 ......................................... SARMA(0,1)(1,1)12
2- 6
دوال ا
;ا w.ا
Aا \"(
7Cا
'!ذج ا
!145 .................................. 4!1
145............................................. SARIMA(0,d,0)(0,D,1)s 1-2-6
145 ........................................... SARIMA(0,d,0)(1,D,1)s 2-2-6
145 ............................................ SARIMA(0,d,1)(0,D,1)s 3-2-6
7
146 ........................................... SARIMA(0,d,0)(1,D,1)s 4-2-6
146 ........................................... SARIMA(0,d,1)(1,D,0)s 5-2-6
146 ............................................ SARIMA(0,d,2)(0,D,1)s 6-2-6
3- 6
دا
4ا
;ا w.ا
Aا 7Cا
1!'
78ذج ا
! 7!1ا
;`147 .................... 7b
4- 6
أ&] 4ا
!;ت ا
&' 4ا
!147 ....................................... 4!1
5- 6
إ;Jق دوال ! \"(
:('Cذج ا
!;ت ا
! 4!1ا
;`148 .......... 4b
1-5-6دا
4ا
;'(1!'
:ذج 148 .................... SARIMA(0,0,0)(0,1,1)12
2-5-6دا
4ا
;'(1!'
:ذج 148 .................... SARIMA(0,1,1)(0,1,1)12
6- 6
أ&] 4وX5ت درا;!
4ت ا
&' 4ا
!149 .......................... 4!1
ا
ء ا
"!:7
-7ا
Hbا
:V.ورC 4Bر 7! W2ا
;'(1. :ا! 46ذج ا%9ار
ا
Aا-7Cا
!; w1ا
!;%ك 178 .......................................................................
-8ا
Hbا
]& ]& :ل H%Cا
(1ا 7Bو&" إ;ر !1ذج &'193 .................. W
-9ا
Hbا
; H%C :Vأو f3bCا
!; 4ا
&' 4إ
&آ(ت 202 .....................
-10ا
;! Oوا
;'(1. :ا 46ا
!; w1ا
!;%ك 225 ...........................................
1-10ا
w1ا
ري 225 .............................................................
-11ا
Hbا
%دي :Kا
;! Oوا
;'(1. :ا 46ا
;! Oا 7ا
(232 ............. w
-12ا
Hbا
] :K 7ا
;! Oوا
;'(1. :ا 46ا
;! Oا 7ا
!دوج 239 ...........
. 4J2= 1-12اون 239 ........................................................................
4J2= 2-12ه240 ........................................................................ S
1
3-12أ&]242 .................................................................................. 4
-13ا
Hbا
]
:K Qا
;! Oا 7ا
] 7Uوا
;'(1. :ا4J2= 46
و;ز !;ت ا
! 4!1ا
!'250 ..................................................... 4N
1-13ا
'!1ذج ا250 ............................................................... 7NP9
2-13ا
'!1ذج ا
;`242 .............................................................. 7b
]& 3-13ل ('ء !1ذج 259 ........................................................... :('C
]& 4-13ل '(
,ء !1ذج 267 ..................................................... :('C
& (1) %أ 4وإ.Lت ا(;9رات ا
275 ............................................... 4J.
ا
!ا358 ........................................................................................... VL
8
9
ﺍﻟﻔﺼﻞ ﺍﻷﻭﻝ
ور
:1
ا ا
Time Seriesه #
$ا" اهة !هة ا
%ا
) #او ' ا)ن (
ا
' -ا,ت ا
:
" -1اbBل f'. )Oا
2ض .&12
-2د ا
51ات ا
! 4.16ا( & 1ا;ج ".4'"& 4
)5 -3ا
!("ت .& 4" & 2O
)5 -4ا;9ج ا
wb'
7&1ا
[م .43!!
.
وا3ض #درا 0و ./ا,ت ا
ه:
)ON -1و!1K 4LAا 48ا
Iهة ا
!Kهة.
-2ا
;'( :ا
)Jا
!;I
4(Jهة ا
"1Kا.48
-3ا
;I
. )3%هة ا
"1Kا 48إذا ا& 3ذ
.f
ا
H3Kا
;
4;!
7ز&'K& 4هة وه( 7رة ا;9ج ا
J(
H=
. W%
7&1ة &
850
750
C1
650
550
70
60
50
40
10
30
20
10
Index
ا;:ات ا9:ة ء 7ذج :4
إن إ2د !1ذج &' 4;& 4 (6'C Wز&'K& 4هة & (;"2ا
!Oم ا
"( 4وا
;;%C 7ج
ا
ا
& ]3ا
( Q%وا
[(ة1 .ف ;"ض \".ا
[16ات ا
"'(
4`2ء !1ذج ر7P2
;'( 4;& :ز&':& 4
-1
#اذج أو /اذج :Model Identification
وهAا ). );2ا
!; 4ا
&'132 Q5 Time Plot !2 !N 4ن ا59ا 7Uا 7JNXه 1ا
&
وا
أ )5 7ا
Iهة ا
!Kهة و& )Uإ;ر !1ذج ر \". 7 2!;"& 7P2ا
!~2J
ا 4859ا
; 1! !Cذج ,و ا
[(ة ا
!;!ة & ا
رات وا%.Xث.
-2
;= اذج :Model Fitting
1! ^C ".ذج او اآ] آ'!1ذج &' e+1
Wا
!; 4ا
!Kهة 1Jم )
"& 2J;.هAا
ا
'!1ذج & ا
(ت ا
!Kهة [;F.ام =ق ا
; 2Jا 7859ا
[;!
. 4+ت ا
&'4
وهAا ا
'!1ذج ا
!^ A:2آ'!1ذج او
.J5X H2";
H.B 7
-3
>:وإ?ر اذج :Model Diagnostics
إLاء إ;(رات 4%bCأ6ء ا
;& 4N"!
Fitting Errors (6ى .6Cا
!Kهات
& Vا
)Jا
! & 4.1%ا
'!1ذج ا
!^ و&ى PN 4%+ت ا
'!1ذج 4
5 7N .إ;Lز ا
'!1ذج
ا
!^ @AOا(;9رات 1Jم !;F.دة ا* ا
'!1ذج ا
' 78Oو[;2م ;:('C 1ات )J
ا
!; 4(Jوإ1" Xد [16ة اXو
;" !1ذج .2L
-4
ا4ات :Forecast Generation
[;2م ا
'!1ذج ا
':('C 1;
78Oات
ا
)Jا
!; 4(Jو&
Forecast Errorsآ! ا;ت 2L )Bة &Kهة &
5 )Uب أ6ء ا
;'(:
ا
!; 4ا
&' 4و&ا 4(Bه@A
ا6ء 66[!. !2 & Nت ا
!ا Control Charts 4(Bوا
;1(J
VP1C 7ل {6 4('.
&" إذا Cوز 4Cأ6ء ا
;'("2 :د ا
' 7N Iا
'!1ذج و"Cد ا
ورة & 1! 2%;. 2Lذج
&^ .,
-5
إ:0ام ا4ات وو %Aاارات Implementation and Decision
JC :makingم ا
;'(:ات 7"
2JC Nا
Jار ' 7N Iإ;[ا&H3K
. O
ا
!'.W
11
:ر و
دئ او
&
. 4'&
ا4;!
ا1C 7;
ا48ا1K"
ا4!"
أو ا48ا1K"
هة اI
& ف1
{Z t }
4=(. { اوZ t , t ∈ {⋯, −1,0,1,2,⋯}} او ا;را
{⋯, Z −1 , Z 0 , Z1 , Z 2 ,⋯}
{z1 , z2 ,⋯, zn−1 , zn } &
. هةK!
ا4'&
ا4;!
و
:2
History ا!هةE' او رA$ ' z1 , z2 ,⋯ , zn −1 "ا
4LA!'
ا4! 7N اL )O& r2وا
;ر
: 3 نG او اA/ ' اzn ا
. هة اةK!
ا7وه
:4 ) ; ه ا" اzˆt JK et = zt − zˆt , t = 1, 2,..., n I,$ '; =;أ?;ء ا
Residuals N0 اواO اذج( و أ#
L .M/7 ا" ا
.ذج1!'
ا2JC ". ة5 وا4"N دO H% (6;
ء ا6 ان ا52و
مH3K. اوzn +1 , zn +2 , zn +3 ,... ز1&
. 4(J;!
هات اK!
& ف1 :!K,
مH3K. او
zn (1) , zn ( 2 ) , zn ( 3) ,...
&
. OCا:(';
&و
zn + ℓ , ℓ ≥ 0
zn ( ℓ ) , ℓ ≥ 0
12
:5 en ( ℓ ) = zn +ℓ − zn ( ℓ ) , ℓ ≥ 0
I,$ '; 4أ?;ء ا
4JJ%
) اJ
هت ا1م ا
& وJC !ى آX ا1C ة5ا1
';_ اC :(';
ء ا6وأ
:6 SK إذاStationary { ةz1 , z2 ,⋯ , zn −1 , zn } ل ان ا ا
اهة
:اوط ا
1) E ( zt ) = constant = µ , ∀t
constant = γ 0 , ∀t , ∀s, t = s
2) cov ( zt , zs ) =
f ( s − t ) , ∀t , ∀s, t ≠ s
Building Blocks ب ا
('ء1= ة او5 O13
اL 4!O& 4'& ز4;& ف "ف1 نsا
Oف ر1 7;
ا
'!ذج اV!
:7 White ءO اVO او اWhite Noise Series ءO اVO
ا
) ;$ ااW اهات اا#
$
# { ه رةat } Noise Process
زتLات اا ا )ن و3 ا#
$
L7ض اY7 7Kوا
#ي وY[ \0$ ( Independent, Identically Distributed (IID) $;
: أيσ 2 S$]
1) E ( at ) = 0, ∀t
σ 2 , ∀t , ∀s, t = s
2) cov ( at , a s ) =
0 , ∀t , ∀s , t ≠ s
at ~ WN ( 0,σ 2 ) $ L و
13
:1ل-
: Random Walk
ا اا
:7
;
{ آZ t } 48ا1K 4! 7'( ف1
Z1 = a1
Z 2 = a1 + a2
⋮
Z t = a1 + a2 + ⋯ + at
أو
Z t = Z t −1 + at
Z t نN j
&
' اe[
&م او اX ا7
اA:C 7;
ة ا16[
) ا5 1 هa j ا;( ان1
أي
t &
' ا78ا1K 7& VB1& 7ه
4!
"
اق ا
!ل ا1 اeC 7;
ا اL 4&O
& ا
'!ذج ا4;!
او ا4!"
@ اA ه:!K,
ة؟J;& 4!"
اH وهt , s )B V!
cov ( Z t , Z s ) وE ( Z t ) L او: 2!C
:8 : وف آAutocovariance Function ا9 ا3دا ا
γ t ,s = cov ( Z t , Z s ) , ∀t , ∀s
= E ( Z t − µ )( Z s − µ ) , ∀t , ∀s
نca Z t + k أوZ t −k #$ وZ t #$ .MY ة ا
اY اb7 اk : اa وإذا
:I,$ '; ا9 ا3دا ا
γ k = cov ( Z t , Z t −k ) , k = 0, ±1, ±2,⋯
= E ( Z t − µ )( Z t −k − µ ) , k = 0, ±1, ±2,⋯
!8 دا7]
اe2";
ف ;[م ا1 :!K,
14
:9 : وف آAutocorrelation Function (ACF) ا9\ ا$دا اا
ρk =
γk
, k = 0, ±1, ±2,⋯
γ0
:4
;
اص ا1[
اO
و
1. ρ 0 = 1
2. ρ − k = ρ k
3.
ρk ≤ 1
:2 ل-
ا
(`ء4`
ا4!"
7CاA
اw. ا
;ا4
ن داs; اK ف1
:7 ا
(`ء ه4`
ا4!"
7CاA
ا2Y;
ا4
دا
σ 2 , k = 0
γ k = cov ( at , at −k ) =
0, k ≠0
:7CاA
اw. ا
;ا4
داO'& و7 e2";
& اf
وذ
γ k 1 , k = 0
=
γ 0 0 , k ≠ 0
:7
;
اH3K
اO
و
Autocorrelation function of White Noise
1.0
Autocorr
ρk =
0.5
0.0
0
1
2
3
4
5
Lag
15
6
7
8
9
:10 Partial Autocorrelation Function (PACF) Vا ا9\ ا$دا اا
ات3 ا#
f\ ا$] ااg S إزا$ Z t −k وZ t #$ \$و; ار اا
قi K وأφkk $ k : اL وL$ I ااZ t −1 , Z t −2 ,..., Z t −k +1
:.- اa φkk Vار ا/7j ا.
بK مL$K
Z t = φk 1Z t −1 + φk 2 Z t −2 + ⋯ + φkk Z t −k + at
: φ11 ب5
Z t = φ11Z t −1 + at
VB1;
اA وأZ t −1 ـ. 4B"
ا7N= `ب.
E ( Z t −1 Z t ) = φ11 E ( Z t −1 Z t −1 ) + E ( Z t −1at )
أي
γ 1 = φ11γ 0
( J5X (' ! آE ( Z t −k at ) = 0, k = 1, 2,... مH3K. ) E ( Z t −1at ) = 0 Q5
γ 0 7 4!J
.و
φ11 = ρ1
16
:11 : آVا ا9\ ا$ م ف دا اا.)$
φkk =
k =0
k =1
1,
ρ1 ,
1
ρ1
ρ1
1
⋮
⋮
⋯ ρ k −2
⋯ ρ k −3
⋯
⋮
⋮
⋯ ρ1 ρ k
,
⋯ ρ k −2 ρ k −1
ρ k −1 ρ k −2
ρ1
1
ρ1
1 ⋯ ρ k −3
⋮
⋮
ρ k −1 ρ k −2
ρ1
ρ2
⋯
⋯
⋮
ρ1
k = 2,3,...
ρ k −2
⋮
1
4N1b& دة%& 7
& اC
Q5
4
ب دا%
, e2"C 76" ف1 اAO
(ة و3
اk )J
;[ام9 اW"+ .
اe2";
ا
:2ار3C 78
ا7CاA
اw.ا
;ا
: ب11 تI, ا#
)ارφkk N/
φ00 = 1, by definition
φ11 = ρ1
k −1
φkk =
ρ k − ∑φk −1, j ρ k − j
j =1
k −1
1 − ∑φk −1, j ρ j
, k = 2,3,...
j =1
JK
φkj = φk −1, j − φkkφk −1,k −1 ,
j = 1, 2,..., k − 1
17
: φ22 ب5
: ب11 e2"C &
φ22 =
ρ 2 − φ11 ρ1 ρ 2 − ρ12
=
1 − φ11 ρ1
1 − ρ12
. φ11 = ρ1 نf
وذ
:3 ل-
: ا
(`ء4`
ا4!"
78
ا7CاA
اw. ا
;ا4
ن داs; اK ف1
ب11 e2"C &
φ00 = 1, by definition
φ11 = ρ1 = 0
.
ا1 & &]لf
وذ
φkk
ب11 e2"C 7N \21";
.و
φ22 = φ33 = ⋯ = 0
:اA3وه
1, k = 0
0, k ≠ 0
φkk =
:7
;
اH3K
اO
و
18
Partial Autocorrelation function of White Noise
PACF
1.0
0.5
0.0
0
1
2
3
4
5
6
7
8
9
Lag
ا
(`ء4`
ا4!"
78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
& داH أن آ5X :!K,
او46. ا
!;اz 48ا1K"
ات اY;!
اV!L 4+ @A وه. اولe[;
& اb
وي اC
.f
A
7CاA
اw. ا
;ا4
;[م داC 78ا1K Y;!
هةK& )B . w.;(ر م ا
;ا9 .4J;!
ا
: 12 Sample Autocorrelation Function SACF ا9\ ا$دا اا
:I,$ '; وrk , k = 0,1, 2,... $ L وz1 , z2 ,⋯ , zn −1 , zn ز
هة
n −k
rk =
∑( z
t
t =1
− z )( zt +k − z )
n
∑( z
t =1
t
−z)
, k = 0,1,2,...
2
z=
1 n
∑ zt JK
n t =1
رJ&ُ O! ا. وρˆ k = rk , k = 0,1, 2,... أي7CاA
اw. ا
;ا4
اEstimator
رJ&ُ 7وه
:4
;
ا4'"
اص ا1[
اO
نFN اAO
ى وX 4' & 8ا1K Y;C إذا7ON
نFN ρ k = 0, k > q S إذا آ-1
q
1
V ( rk ) ≅ 1 + 2∑ ρ k2 , k > q
n
k =1
19
1
V ( rk ) ≅ , k > 0 نFN ρ k = 0, k > 0 &' 4+[
ا4
%
ا7Nو
n
مJ
اV6; 7
;
. و7"(= V2ز1C (2JC O
ن132 rk نFN ρ k = 0 (ة و3
اn )J
-2
:7
;
;(ر اX.
H 0 : ρk = 0
H1 : ρ k ≠ 0
:4859;[ام اF. f
وذ
rk
n
− 12
= n rk
n rk > 1.96 S إذا آH 0 \NC وα = 0.05 421'"& ى1;& ' f
وذ
corr ( rk , rk − s ) ≅ 0, s ≠ 0 نFN H 0 : ρ k = 0, ∀k 4Pb
اS%C -3
:7
;
آ4'"
7CاA
اw. ا
;ا4
'ت ا2(;
ر اJCُ -4
q
1
Vˆ ( rk ) ≅ 1 + 2∑ rk2 , k > q
n
k =1
:13 Sample Partial Autocorrelation Function Vا ا9\ ا$دا اا
$
L
و
هة
z1 , z2 ,⋯ , zn −1 , zn
ز
SPACF
:I,$ '; rkk , k = 0,1, 2,...
20
1,
r,
1
1
r1
1
r1
⋮
⋮
rkk = rk −1 rk −2
1
r1
1
r1
⋮
⋮
rk −1 rk −2
k =0
k =1
⋯ rk −2
⋯ rk −3
⋯ ⋮
⋯ r1
⋯ rk −2
⋯ rk −3
⋯
⋯
⋮
r1
r1
r2
⋮
rk
,
rk −1
rk −2
k = 2,3,...
⋮
1
:2ار3C 4'"
78
ا7CاA
اw. ا
;ا4
ب دا%
و
: ب13 تI, ا#
)ارrkk N/
r00 = 1, by definition
r11 = r1
k −1
rk − ∑ rk −1, j rk − j
rkk =
j =1
k −1
1 − ∑ rk −1, j rj
, k = 2,3,...
j =1
JK
rkj = rk −1, j − rkk rk −1,k −1 ,
j = 1, 2,..., k − 1
أي
7CاA
ا
4'"
78
ا
w.ا
;ا
4
ا
Estimator
رJ&
`2ا
7وه
نFN اAO
ى وX 4' & 8ا1K Y;C إذا7ON
رJ&ُ O! ا. وφˆkk = rkk , k = 0,1, 2,...
:4
;
ا4'"
اص ا1[
اO
21
1
V ( rkk ) ≅ , k > 0 -1
n
;(رX. مJ
اV6; 7
;
. و7"(= V2ز1C (2JC O
ن132 rkk نFN (ة3
اn )J
-2
:7
;
ا
H 0 : φkk = 0
H1 : φkk ≠ 0
:4859;[ام اF. f
وذ
rkk
n
− 12
= n rkk
n rkk > 1.96 S إذا آH 0 \NC وα = 0.05 421'"& ى1;& ' f
وذ
corr (φkk ,φk −s ,k − s ) ≅ 0, s ≠ 0 نFN H 0 : φkk = 0, ∀k 4Pb
اS%C -3
:7
;
آ4'"
7CاA
اw. ا
;ا4
'ت ا2(;
ّر اJCُ -4
1
Vˆ ( rkk ) ≅ , k > 0
n
:4 ل-
:&12 "& _;'& 7 W6
اH]!C 4
;
ا
(ت ا
158 222 248 216 226 239 206 178 169
:!O! وار4'"
78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;اW5أ
1 n
1
z = ∑ zt = (158 + 222 + ⋯ + 169 ) = 206.89 w1;!
اW% :Xاو
n t =1
9
4B"
& ا7CاA
اw. ا
;اW% :U
n −k
rk =
∑( z
t =1
t
− z )( zt +k − z )
n
∑( z
t =1
t
−z)
, k = 0,1,2,...
2
22
r1 =
(158 × 222 + 222 × 248 + ⋯ + 178 × 169 )
= 0.265116
(158 × 158 + 222 × 222 + ⋯ + 169 × 169 )
r2 =
(158 × 248 + 222 × 216 + ⋯ + 206 × 169 )
= -0.212
(158 × 158 + 222 × 222 + ⋯ + 169 × 169 )
r3 =
(158 × 216 + 222 × 226 + ⋯ + 239 × 169 )
= −0.076
(158 × 158 + 222 × 222 + ⋯ + 169 × 169 )
r8 = 0.230, r7 = 0.104, r6 = −0.242, r5 = −0.387, r4 = −0.183 اA3وه
& 'ت2(;
اW% :]
U
q
1
Vˆ ( rk ) ≅ 1 + 2∑ rk2 , k > q
n
k =1
1
Vˆ ( r1 ) ≅
9
(
)
1
1
2
Vˆ ( r2 ) ≅ (1 + 2r12 ) = 1 + 2 ( 0.265) = 0.1267
n
9
) (
(
(
1
1
2
2
Vˆ ( r3 ) ≅ 1 + 2 ( r12 + r22 ) = 1 + 2 ( 0.265) + ( −0.212 )
n
9
)) = 0.1367
Vˆ ( r4 ) ≅ 0.138 Vˆ ( r5 ) ≅ 0.1454 Vˆ ( r6 ) ≅ 0.1787
…r
ا
:4'"
78
ا7CاA
اw. ا
;اW% :".را
r00 = 1, by definition
r11 = r1 = 0.265
42ار3;
ت اB"
ت & ا6. ا
;ا7B. W% )U
k −1
rk − ∑ rk −1, j rk − j
rkk =
j =1
k −1
1 − ∑ rk −1, j rj
, k = 2,3,...
j =1
Q5
rkj = rk −1, j − rkk rk −1,k −1 ,
j = 1, 2,..., k − 1
23
1
r2 − ∑ r1, j r2− j
r22 =
j =1
1
1 − ∑ r1, j rj
=
r2 − r11r1 ( −0.212 ) − ( 0.265)( 0.265) −0.282225
=
=
1 − r11r1
1 − ( 0.265)( 0.265)
0.929775
j =1
= −0.30354
& W%C وr21 ;ج ا% r33 ب%
r21 = r11 − r22 r11 = 0.265 − ( −0.303)( 0.265) = 0.345295
2
r3 − ∑ r2, j r3− j
r33 =
j =1
k −1
1 − ∑ r2, j rj
=
r3 − ( r21r2 + r22 r1 )
1 − ( r21r1 + r22 r2 )
j =1
=
( −0.076 ) − ( ( 0.345)( −0.212 ) + ( −0.303)( 0.265) )
1 − ( ( 0.345)( 0.265) + ( −0.303)( −0.212 ) )
= 0.092
4'"
48
ت ا6. ا
;ا7B. W% اA3وه
r88 = 0.042, r77 = 0.013, r66 = −0.207, r55 = −0.294, r44 = −0.298
1 = 0.1111 (2JC ويC 'ت2(;
!" اL O
و
9
4'"
78
ا7CاA
اw. وا
;ا7CاA
اw. ر) دوال ا
;ا:&
Autocorrelation
Autocorrelation Function for Demand
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
0.27
-0.21
-0.08
-0.18
-0.39
-0.24
0.10
0.80
-0.59
-0.21
-0.49
-1.01
-0.57
0.24
0.87
1.50
1.60
2.26
5.96
7.89
8.43
8
0.23
0.52
13.66
24
7
8
Partial Autocorrelation
Partial Autocorrelation Function for Demand
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag PAC
1
2
3
4
5
6
7
0.27
-0.30
0.09
-0.30
-0.29
-0.21
0.01
4
5
6
T
Lag PAC
T
0.80
-0.91
0.27
-0.89
-0.88
-0.62
0.04
8 0.04
0.13
25
7
8
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻧﻲ
Autoregressive-Moving Average ك/\ ا0ا_ا9ار ا/7jذج ا7
:4 اa L
ا:0 وإModels
ك%;!
اw1;!
_ا7CاA
ار ا%9 !ذج اO 62 7;
ا
'!ذج ا
& آ(ة48 ه'ك
e;[& 7N ]ة3
ث ا%. اS;(U ا7;
واAutoregressive-Moving Average Models
.:(';
ا7N 42J;
ق ا6
ا7 H8O
اOB1bC 7 4J(6;
ا2ا
!د
:14 ARMA ( p, q ) b و
( p, q )
n ار#
ك/\ ا0ا_ا9ار ا/7jذج ا7
:.) اN) {z1 , z2 ,… , zn −1 , zn } ز
هة
zt = δ + φ1 zt −1 + φ2 zt −2 + ⋯ + φ p zt − p + at − θ1at −1 − θ 2 at −2 − ⋯ − θ q at −q
اي.- S$] "
−∞ < δ < ∞ ء وO$ VA at ~ WN ( 0,σ 2 ) JK
وAutoregressive Parameters ا9ار ا/7j ه " اφ1 ,φ2 ,… ,φ p و
Moving Average Operators ك/\ ا0 ه " اθ1 ,θ 2 ,… ,θ q
O"& H&";
اHO2 73
@ ا
'!ذجA هw(;
Operators Algebra ( ا
"!ل. "; ف1
اص: اb وB b وBackshift Operator Y: اKزاj ا.
:15 :ا
1 − Bzt = zt −1
2 − B m zt = B m −1 ( Bzt ) = B m−2 ( B ( Bzt ) ) = ⋯ = zt −m
3 − Bc = c, c is a constant
:7 هJ5X O
;ج ا% !ل ايL1C 7b[
ا45زا9 اH& 7
ا4NP9.
26
: ب15 : وف آF b وForewardshift Operator q اKزاj ا.
-1
F = B −1
: ∇ وف آb وDifference Operator =Y ا.
-2
∇ = (1 − B )
: وف آS b وSum Operator %V ا.
-4
S = ∇ −1 = (1 − B )
7 *(;3و
( p, q )
−1
4Lك & ا
ر%;!
اw1;!
_ا7CاA
ار ا%9ذج ا1! 7
د ا1" نsا
:H3K
ا
zt − φ1 zt −1 − φ2 zt −2 − ⋯ − φ p zt − p = δ + at − θ1at −1 − θ 2 at −2 − ⋯ − θ q at −q
zt − φ1 Bzt − φ2 B 2 zt − ⋯ − φ p B p zt = δ + at − θ1 Bat − θ 2 B 2 at − ⋯ − θ q B q at
(1 − φ B − φ B
1
2
2
− ⋯ − φ p B p ) zt = δ + (1 − θ1 B − θ 2 B 2 − ⋯ − θ q B q ) at
أو
φ p ( B ) z t = δ + θ q ( B ) at
Autoregressive 7CاA
ار ا%9 اH& 1 هφ p ( B ) = 1 − φ1 B − φ2 B 2 − ⋯ − φ p B p Q5
ك%;!
اw1;!
اH& 1 هθ q ( B ) = 1 − θ1 B − θ 2 B 2 − ⋯ − θ q B q وOperator
Moving Average Operator
:-
أ
W;32 وARMA ( 0,0 ) *
&2 وConstant Mean Model S.]
اw1;!
ذج ا1! -1
:H3K
ا7
φ 0 ( B ) z t = δ + θ 0 ( B ) at
او
(1) zt = δ + (1) at
zt = δ + at , at ~ WN ( 0,σ 2 )
27
:H3K
ا7 1 وهARMA (1,0 ) ≡ AR (1) 7
وX ا4L & ا
ر7CاA
ار ا%9ذج ا1! -2
φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
zt = δ + φ1 zt −1 + at , at ~ WN ( 0,σ 2 )
:H3K
ا7 1 وهARMA ( 0,1) ≡ MA (1) 7
وX ا4Lك & ا
ر%;!
اw1;!
ذج ا1! -3
φ0 ( B ) zt = δ + θ1 ( B ) at
zt = δ + (1 − θ1B ) at
zt = δ + at − θ1at −1 , at ~ WN ( 0,σ 2 )
:H3K
ا7 1 وهARMA ( 2,0 ) ≡ AR ( 2 ) 4]
ا4L & ا
ر7CاA
ار ا%9ذج ا1! -4
φ 2 ( B ) zt = δ + θ 0 ( B ) a t
(1 − φ B − φ B ) z
2
1
2
t
= δ + at
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at ~ WN ( 0,σ 2 )
7 *(;3 وARMA (1,1) (1و1) 4Lك & ا
ر%;!
اw1;!
_ا7CاA
ار ا%9ذج ا1! -5
:H3K
ا
φ1 ( B ) zt = δ + θ1 ( B ) at
(1 − φ1B ) zt = δ + (1 − θ1B ) at
zt = δ + φ1 zt −1 + at − θ1at −1 , at ~ WN ( 0,σ 2 )
28
:ك/\ ا0ا_ا9ار ا/7jذج ا7 >M?
ك%;!
اw1;!
_ا7CاA
ار ا%9! !ذج اC 7;
ا4859 اo8[
ف رس ا1
ذج1! 2%C ;" اوf
هة وذK& 4' & @ ا
'!ذجA ه5 ا7 ا
;"ف4b آ4N"&و
.هاتK!
اe2 W'&
:ARMA(0,0)S$-\ ا0ذج ا7 :rأو
H3K
ا7 W;32و
φ 0 ( B ) z t = δ + θ 0 ( B ) at
او
zt = δ + at , at ~ WN ( 0,σ 2 )
w. ا
;ا7;
( وداw1;!
)اVB1;
د ا2F. f
ذج وذ1!'
ا اAO
4859اص ا1[
; اK ف1
:7
;
آ78
ا7CاA
اw. وا
;ا7CاA
ا
E ( z t ) = δ + E ( at )
=δ
at ~ WN ( 0,σ 2 ) نf
وذ
δ = µ ن132 7
;
. وµ = E ( zt ) أيµ &
. E ( zt ) 4;!
اw1;!
& ف1
:ذج1!'
اW;32و
z t − µ = at
7N 4J.
ا4
ا
!"د7N= `ب78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
ق داJ;9
أيVB1;
اA{ وzt −k − µ
E ( zt −k − µ )( zt − µ ) = E ( zt −k − µ ) at
إذا8 e2"C & E ( zt −k − µ )( zt − µ ) = γ k 3
و
γ k = E ( zt −k − µ ) at , k = 0, ±1, ±2,⋯
:2ار3C 4B"
@ اA هH%و
k = 0 : γ 0 = E ( zt − µ ) at
29
أيVB1;
اA{ وat 7N zt − µ = at 7N= ! `ب2ف ا6
د ا29
E ( zt − µ ) at = E ( at at ) = σ 2
إذاat ~ WN ( 0,σ 2 ) نf
وذ
k = 0 : γ 0 = E ( zt − µ ) at = σ 2
k = 1: γ 1 = E ( zt −1 − µ ) at = 0
نf
وذ
zt −1 − µ = at −1
E ( zt −1 − µ ) at = E ( at −1at ) = 0
نFN 4JJ%
ا7N
zt −k − µ = at −k , k = 1,2,…
E ( zt −k − µ ) at = E ( at −k at ) = 0, k = 1, 2,…
أي
:1 ةI
σ 2 , k = 0
E ( zt −k − µ ) at = E ( at −k at ) =
0, k = 1,2,..
أي
γ0 =σ 2
γ k = 0, k = ±1, ±2,…
:7
داH3 7 VP1Cو
σ 2 , k = 0
γk =
0, k ≠ 0
γ 0 = σ 2 7 4!J
.و
30
ρk =
γ k 1, k = 0
=
γ 0 0, k ≠ 0
:7
;
اH3K
اO
و
Autocorrelation function of Constant Mean Model
Autocorr
1.0
0.5
0.0
0
1
2
3
4
5
6
7
8
9
Lag
11 e2";
& ا78
ا7CاA
اw. ا
;ا4
ن داs; اK
φ00 = 1, by definition
φ11 = ρ1 , by definition
φ11 = 0
1
ρ1 1 0
ρ2 0 0
=
=0
ρ1 1 0
ρ1
1
1
φ22 =
ρ1
0 1
31
φ33 =
1
ρ1
1
ρ1 1 0
ρ2 0 1
ρ3 0 0
=
ρ2 1 0
ρ1 0 1
ρ1
ρ2
1
ρ1
1
ρ1
ρ1
1
ρ1
ρ2
0
0
0
=0
0
0
0 0 1
⋮
1
ρ1
ρ1
1
⋮
⋮
ρ k −1 ρ k −2
ρ1
1
ρ1
1
φkk =
⋮
⋮
ρ k −1 ρ k −2
⋯ ρ1
1 0 ⋯ 0
ρ2
⋯
⋮
⋯
⋯
⋯
ρk
ρ k −1
ρ k −2
⋮
⋯
⋮
1
0
⋮
0
=
1
0
⋮
⋮
⋯
⋮
⋯
⋯
⋯
⋮
1
⋮
0
0
1
⋮
0
⋮
0 0
= = 0, k = 2,3,⋯
0 1
0
⋮
0 0 ⋯ 1
:7
داH3 7 VP1Cو
1, k = 0
0, k ≠ 0
φkk =
Partial Autocorrelation function of Constant Mean Model
:7
;
اH3K
اO
و
PACF
1.0
0.5
0.0
0
1
2
3
4
5
6
7
8
9
Lag
z w1;& *
ان7N X ا
(`ء ا4`
ذج ا1! ;قb2X S.]
اw1;!
ذج ا1! :!K,
يb+
32
ARMA(1,0) = AR(1) وr اn ار#
ا9ار ا/7jذج ا7 :7]
:H3K
ا7 1وه
φ1 ( B ) zt = δ + θ 0 ( B ) at
(1 − φ1B ) zt = δ + at
zt = δ + φ1 zt −1 + at , at ~ WN ( 0,σ 2 )
:78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
( وداw1;!
)اVB1;
اL1 ف1 .
ذج ا1!'
آ
(1 − φ1 B ) zt = δ + at
δ
zt =
(1 − φ1 )
E ( zt ) =
+ (1 − φ1 B ) at
−1
δ
−1
+ E (1 − φ1 B ) at
(1 − φ1 )
1! ه2ف ا6
ا7N 7]
ا%
ا
∞ j
−1
E (1 − φ1 B ) at = E ∑φ1 B j at
j =0
∞
∑φ
j =0
j
1
B j < ∞ 48O
ا4;!
ن ا13C انW2 78O
ع ا1!!
ا7 VB1;
دل ا9
Y;& دورW"2 نs اB H&"
إذا ا;( اf
وذφ1 < 1 S إذا آJ%;2 f
وذ4.رJ;&
ان.X 4JJ%
ا7N B = 1 سJ
و
* اB = a + ib H3K
* اComplex Variable W&آ
أيB > 1 ة أي51
ة ا8( رج دا1 − φ1 B ) = 0 رb+ور او اL ن13C انW6;
1 − φ1 B = 0
B=
1
φ1
B >1⇒
1
φ1
> 1 ⇒ φ1 < 1
4B"
ا7
د ا1" .ارJ;9 ط ا1ا هAوه
33
∞ j j
−1
E (1 − φ1 B ) at = E ∑φ1 B at
j =0
∞ j
= ∑φ1 B j E ( at )
j =0
=0, ∀t
ن132و
E ( zt ) =
δ
(1 − φ1 )
او
µ=
δ
(1 − φ1 )
∴δ = µ (1 − φ1 )
ذج1!'
ا4Y+ 7N δ
\21";
.و
zt = δ + φ1 zt −1 + at
= µ (1 − φ1 ) + φ1 zt −1 + at
= µ + φ1 ( zt −1 − µ ) + at
( zt − µ ) − φ1 ( zt −1 − µ ) = at
أيVB1;
اA{ وzt −k − µ 7N 4J.
ا4
ا
!"د7N= `ب
E ( zt −k − µ )( zt − µ ) − φ1 E ( zt −k − µ )( zt −1 − µ ) = E ( zt −k − µ ) at , k = 0, ±1, ±2,⋯
أي
γ k − φ1γ k −1 = E ( zt −k − µ ) at , k = 0, ±1, ±2,⋯
:72 ! آ2ار3C 4B"
@ اA هH%C و8 e2"C & f
وذ
k = 0 : γ 0 − φ1γ 1 = E ( zt − µ ) at
:7
;
. م1J !2ف ا6
د ا29
34
E at ( zt − µ ) − φ1 E at ( zt −1 − µ ) = E ( at at )
E at ( zt − µ ) − φ1 × ( 0 ) = σ 2
∴ E at ( zt − µ ) = σ 2
إذا
γ 0 − φ1γ 1 = σ 2
k = 1: γ 1 − φ1γ 0 = E ( zt −1 − µ ) at = 0
4JJ%
ا7N
γ k − φ1γ k −1 = 0, k = 1, 2,⋯
γ 0 7 اة4
ا
!"د4!J.
ρ k − φ1 ρ k −1 = 0, k = 1,2,⋯
أو
ρ k = φ1 ρ k −1 , k = 1, 2,⋯
:نFN ρ 0 = 1 ! ان.و
ρ1 = φ1 ρ 0 = φ1
ρ 2 = φ1 ρ1 = φ12
⋮
ρ k = φ1k
4
داH3K. أو
ρ k = φ1k , k = 0, ±1, ±2,⋯
أيρ k & WL1!
اK
ا+ن وs & اI' ف1 ρ − k = ρ k , ∀k نf
وذ
ρ k = φ1k , k = 0,1, 2,⋯
:7
;
اH3K
اO
4
@ ا
اAه
φ1 > 0 ن13C &' -1
35
Autocorrelation function of AR(1) Model
0.5
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
Lag
φ1 < 0 ن13C &' -2
Autocorrelation function of AR(1) Model
0.3
0.2
0.1
0.0
ACF
ACF
0.4
-0.1
-0.2
-0.3
-0.4
-0.5
0
1
2
3
4
5
6
7
8
9
10
Lag
78
ا7CاA
اw. ا
;ا4
ن داs; اK
11 e2"C &
36
φ00 = 1, by definition
φ11 = ρ1 = φ1 , by definition
1
φ22 =
ρ1
1
ρ1
1 φ1
ρ1
ρ 2 φ1 φ12
0
=
=
=0
1 φ1 1 − φ12
ρ1
1
φ1 1
⋮
φkk =
1
ρ1
ρ1
1
⋮
⋮
ρ k −1 ρ k −2
1
ρ1
1
ρ1
⋮
⋮
ρ k −1 ρ k −2
⋯ ρ1
⋯ ρ2
1
φ1
φ1
1
⋯ φ1
⋯ φ12
⋯ ⋮
⋮
⋮
⋯ ρk
φ1k −1 φ1k −2
=
⋯ ρ k −1
1
φ1
⋯ ρ k −2
1
φ1
⋯
⋮
⋮
⋮
⋯ 1
φ1k −1 φ1k −2
⋯ ⋮
⋯ φ1k
⋯ φ
⋯ φ
k
1
k −1
1
⋯
⋯
=
0
>0
⋮
1
W;3 وφ1 7N .د اول &`و1&"
وي ا2 د ا1&"
ا ن اb+ ويC w(
دة ا%&
:7
ا
اH3K
ا7 78
ا7CاA
اw. ا
;ا4
دا
1, k = 0
φkk = φ1 , k = 1
0, k ≥ 2
:7
;
اH3K
اO
و
φ1 > 0 ن13C &' -1
Partial Autocorrelation function of AR(1) Model
0.5
PACF
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
Lag
37
7
8
9
10
φ1 < 0 ن13C &' -2
Partial Autocorrelation function of AR(1) Model
0.0
PACF
-0.1
-0.2
-0.3
-0.4
-0.5
0
1
2
3
4
5
6
7
8
9
10
Lag
.4(
ل ا3 ا7N φ00 = 1 اوρ 0 = 1 & ) أيCX !8 دا:4I5&
: اذجI
t )B V!
S.U1 وهE ( zt ) = δ (1 − φ1 ) نFN (ارJ;9 )ط اφ1 < 1 ن13C &' -1
t &
ا7 !;"CX وwJN k e[;
4
دا7CاA
اw. ا
;ا4
دا-2
&[;C وφ1 > 0 ن13C &' ρ1 & ;اءا. إ5@ واC إ7N ;[& اC 7CاA
اw. ا
;ا4
دا-3
φ1 < 0 ن13C &' 4(
وا4(L1!
) اJ
ا. ا &;ددة
ن132 ( وφ00 7
اI'
م اV& ) 42b+ z ة5 وا4!B O
78
ا7CاA
اw. ا
;ا4
دا-4
φ1 وي2 ارهJ& وφ1 إرةW5 OهCإ
: ARMA(2,0) = AR(2) 7- اn ار#
ا9ار ا/7jذج ا7 :-]
:H3K
ا7 W;32و
φ 2 ( B ) z t = δ + θ 0 ( B ) at
(1 − φ B + φ B ) z
2
1
2
t
= δ + at
zt = δ + φ1 zt −1 + φ2 zt −2 + at , at ∼ WN ( 0, σ 2 )
:78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
وداw1;!
اL1 .
آ
38
(1 − φ B − φ B ) z
2
1
zt =
2
t
= δ + at
−1
δ
+ (1 − φ1 B − φ2 B 2 ) at
(1 − φ1 − φ2 )
E ( zt ) =
−1
δ
+ E (1 − φ1 B − φ2 B 2 ) at
(1 − φ1 − φ2 )
∞
VB1;
اH 73
وE ∑ψ j at − j H3K
ا78OX ع1!& !2ف ا6
ا7N 7]
ا%
ا
j =0
إذاJ%;2 اA وهV.!
اw1;!
ا7N 4.رJ;&
∞
∑ψ
j =0
a
j t− j
ن13C ان.X 78O
اV!;
اHدا
:4
;
وط اK
ا7CاA
ار ا%9 &"
) اSJJ5 إذاJ%;2 اAوه
∞
∑ψ
j =0
2
j
< ∞ إذا آنwJNو
φ2 − φ1 < 1
φ2 + φ1 < 1
−1 < φ2 < 1
رb+ور او أL ن1آ
& `2';_ اC وطK
@ اAار ) ه0jوط ا$ 7!C 7;
وا
نFN ارJ;9 وط اSJJ%C إذا. ( ة51
ة ا8( رج دا1 − φ1 B − φ2 B 2 ) = 0
−1
−1
E (1 − φ1 B − φ2 B 2 ) at = (1 − φ1 B − φ2 B 2 ) E ( at ) = 0, ∀t
ن132و
µ = E ( zt ) =
δ
(1 − φ1 − φ2 )
δ = (1 − φ1 − φ2 ) µ
ذج1!'
ا4Y+ 7N δ
\21";
. و
zt = (1 − φ1 − φ2 ) µ + φ1 zt −1 + φ2 zt −2 + at
= µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at
( zt − µ ) − φ1 ( zt −1 − µ ) − φ2 ( zt −2 − µ ) = at
: VB1;
اA{ وzt −k − µ 7N 4J.
ا4
`ب ا
!"د
39
E ( zt − µ )( zt −k − µ ) − φ1 ( zt −1 − µ )( zt −k − µ ) − φ2 ( zt −2 − µ )( zt −k − µ )
= E at ( zt −k − µ ) , k = 0, ±1, ±2,...
أي
E ( zt − µ )( zt −k − µ ) − φ1 E ( zt −1 − µ )( zt −k − µ ) − φ2 E ( zt −2 − µ )( zt −k − µ )
= E at ( zt −k − µ ) , k = 0, ±1, ±2,...
أو
γ k − φ1γ k −1 − φ2γ k −2 = E at ( zt −k − µ ) , k = 0, ±1, ±2,...
:72 ! آ2ار3C 4B"
@ اA هH% نs ا8 e2"C & f
وذ
k = 0 : γ 0 − φ1γ −1 − φ2γ −2 = E at ( zt − µ ) = σ 2 ⇒ γ 0 = φ1γ 1 − φ2γ 2 + σ 2
1 ةB & f
وذ
k = 1: γ 1 − φ1γ 0 − φ2γ 1 = 0 ⇒ γ 1 = φ1γ 0 − φ2γ 1
k = 2 : γ 2 − φ1γ 1 − φ2γ 0 = 0 ⇒ γ 2 = φ1γ 1 − φ2γ 0
مH3K.و
k ≥ 1: γ k = φ1γ k −1 + φ2γ k −2
γ 0 7 N6
ا4!J.
ρ k = φ1 ρ k −1 + φ2 ρ k −2 , k = 1, 2,...
ρ k − φ1 ρ k −1 − φ2 ρ k −2 = 0, k = 1,2,... H3K
ا7 4J.
ا4
ا
!"دVP1. :4I5& )
H5 ;[ام =قF. Y& H3K. O5 3!2 7;
وا4]
ا4L & ا
ر4BوN 4
&"دO ا
(7
%
ر اJ!
ق ا6 ا رجA ه3
و4Bوb
ت اXا
!"د
: ;
!; اوB 7
;ج ا%C 7;
وا42ار3;
ا4J26
. 4J.
ا4B"
اH% ف1
1 − ρ0 = 1
2 − ρ1 = φ1 ρ0 + φ2 ρ −1 ⇒ ρ1 =
φ1
1 − φ2
O'&و
40
φ12
ρ 2 = φ1 ρ1 + φ2 ρ0 ⇒ ρ 2 =
+ φ2
1 − φ2
…r
ا اA3وه
AR ( 2 ) 4!"
7CاA
اw. وال ا
;ا7 ه4
;
ل ا3ا
φ1 = 0.4, φ2 = 0.4 (1) H3K
ا-1
φ1 = 1.5, φ2 = −0.8 (2) H3K
ا-2
φ1 = 0.5, φ2 = −0.6 (3) H3K
ا-3
(1) H3
ACF
0.7
0.6
ACF
0.5
0.4
0.3
0.2
0.1
0.0
0
10
20
Lag
(2) H3
ACF
1.0
ACF
0.5
0.0
-0.5
0
10
Lag
41
20
(3) H3
ACF
ACF
0.5
0.0
-0.5
0
10
20
Lag
:7
;
آAR ( 2 ) 4!"
78
ا7CاA
اw. ا
;ا4
; داK نsا
φ00 = 1, by definition
φ11 = ρ1 , by definition
1
φ22 =
ρ1
1
ρ1
φ33 =
ρ1
ρ 2 ρ 2 − ρ12
=
≠0
ρ1
1 − ρ12
1
1
ρ1
ρ1
ρ2
1
1
ρ1
ρ2
1
ρ1
1
ρ2
ρ1
ρ3 ρ 2
=
ρ2
ρ1
ρ1
1
ρ1
ρ1
ρ1
1
ρ1
ρ1 = φ1 + φ2 ρ1
ρ 2 = φ1 ρ1 + φ2
ρ3 = φ1 ρ 2 + φ2 ρ1
>0
=0
f
A آ،7]
اول وا2د1!"
& ا76 WآC 1 هw(
دة ا%& 7N د ا1!"
ن اf
وذ
42
1
ρ1
ρ1
1
⋮
⋮
φkk =
⋯ ρ1
1
⋯ ρ2
ρ1
⋯ ⋮
⋮
⋯ ρk
ρ
= k −1
⋯ ρ k −1
ρ k −1 ρ k −2
1
ρ1
1 ⋯ ρ k −2
ρ1
⋮
⋮
ρ k −1 ρ k −2
⋯
⋯
ρ1
⋯
1
⋮
⋯
⋯
⋮
⋯ ρ k = φ1 ρ k −1 + φ2 ρ k −2
= 0, k = 3, 4,...
>0
ρ k −2
ρ1 = φ1 ρ 0 + φ2 ρ1
ρ 2 = φ1 ρ1 + φ2 ρ 0
⋮
1
إذا..
اW(
~ اb'
`2 اf
وذ
k =0
1,
ρ,
1
φkk = ρ 2 − ρ12
1− ρ2 ,
1
0,
k =1
k =2
k ≥3
AR ( 2 ) 4!"
78
ا7CاA
اw. وال ا
;ا7 ه4
;
ل ا3ا
φ1 = 0.4, φ2 = 0.4 (4) H3K
ا-4
φ1 = 1.5, φ2 = −0.8 (5) H3K
ا-5
φ1 = 0.5, φ2 = −0.6 (6) H3K
ا-6
(4) H3
PACF
0.7
0.6
PACF
0.5
0.4
0.3
0.2
0.1
0.0
0
10
Lag
43
20
(5) H3
PACF
PACF
1
0
-1
0
10
20
Lag
(6) H3
PACF
0.3
0.2
PACF
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
10
20
Lag
: ARMA(0,1) = MA(1) وq اn ار#
ك/\ ا0ذج ا7 :$را
:H3K
ا7 W;3Cو
φ 0 ( B ) z t = δ + θ 1 ( B ) at
zt = δ + (1 − θ1 B ) at
zt = δ + at − θ1at −1 , at ∼ WN ( 0, σ 2 )
:78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
وداw1;!
اL1 نsا
E ( zt ) = E (δ + at − θ1at −1 ) = δ
∴µ = δ
44
ذج1!'
اW;3و
zt − µ = at − θ1at −1
VB1;
اA وأzt −k − µ 7N 4
@ ا
!"دA`ب ه.
E ( zt − µ )( zt −k − µ ) = E ( zt −k − µ ) at − θ1 E ( zt −k − µ ) at −1 , k = 0, ±1, ±2,...
او
γ k = E ( zt −k − µ ) at − θ1 E ( zt −k − µ ) at −1 , k = 0, ±1, ±2,...
2ار3C O%.و
k = 0 : γ 0 = E ( zt − µ ) at − θ1 E ( zt − µ ) at −1
:7Cs آE ( zt − µ ) at −1 وE ( zt − µ ) at & H آL1
E ( zt − µ ) at = E ( at at ) − θ1 E ( at −1at ) = σ 2
E ( zt − µ ) at −1 = E ( at at −1 ) − θ1 E ( at −1at −1 ) = −θ1σ 2
∴γ 0 = σ 2 − θ1 ( −θ1σ 2 ) = σ 2 (1 + θ12 )
k = 1: γ 1 = E ( zt −1 − µ ) at − θ1 E ( zt −1 − µ ) at −1
∴ γ 1 = −θ1σ 2 ⇒ ρ1 =
−θ1
γ1
=
γ 0 1 + θ12
1 ةJ
;[ام اF. f
وذ
k = 2 : γ 2 = E ( zt −2 − µ ) at − θ1 E ( zt −2 − µ ) at −1
∴ γ 2 = 0 ⇒ ρ2 = 0
نFN مH3K. و1 ةB & `2أ
k ≥ 2 : γ k = 0 ⇒ ρk = 0
7 7 هMA (1) 7
او4Lك & ا
ر%;!
اw1;!
ذج ا1!'
7CاA
اw. ا
;ا4
ن داFN اA3وه
:H3K
ا
45
1,
k =0
−θ
ρk = 1 2 , k = 1
1 + θ1
0
k≥2
:7
;
اH3K
اO
و
θ1 = 0.8 &' -1
ACF
0.0
ACF
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
Lag
θ1 = −0.8 &' -2
ACF
0.5
ACF
0.4
0.3
0.2
0.1
0.0
0
10
Lag
46
20
MA (1) 7
او4Lك & ا
ر%;!
اw1;!
ذج ا1!'
78
اw. ا
;ا4
; داK نsا
φ00 = 1, by definition
φ11 = ρ1 , by definition
1
φ22 =
ρ1
1
ρ1
φ33 =
ρ1
1 ρ1
−θ12 (1 − θ12 )
ρ2
ρ1 0
− ρ12
−θ12
=
=
=
=
ρ1
1 − ρ12
1 − ρ12 1 + θ12 + θ14
1 − θ16
1
1
ρ1
ρ1
ρ2
1
1
ρ1
ρ2
ρ1
ρ1
1
ρ1
1 ρ1 ρ1
ρ1
ρ2
ρ1 1 0
−θ13 (1 − θ12 )
0 ρ1 0
ρ3
ρ13
=
=
=
1 ρ1 0 1 − 2 ρ12
ρ2
1 − θ18
ρ1
ρ1 1 ρ1
1
0 ρ1 1
مH3K.و
1 − θ1 (
2 k +1)
, k >0
:7
;
اH3K
اO
و
θ1 = −0.8 &' &' -1
PACF
0.5
0.4
0.3
PACF
φkk =
−θ1k (1 − θ12 )
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
10
Lag
47
20
θ1 = 0.8 &' -2
PACF
0.0
PACF
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
Lag
: ARMA(0,2) = MA(2) 7- اn ار#
ك/\ ا0ذج ا7 :
?
:H3K
ا7 W;3Cو
φ 0 ( B ) zt = δ + θ 2 ( B ) a t
zt = δ + (1 − θ1 B − θ 2 B 2 ) at
zt = δ + at − θ1at −1 − θ 2 at −2 , at ∼ WN ( 0,σ 2 )
:78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
وداw1;!
اL1 نsا
E ( zt ) = E (δ + at − θ1at −1 − θ 2 at −2 ) = δ
∴µ = δ
ذج1!'
اW;3و
zt − µ = at − θ1at −1 − θ 2 at −2
VB1;
اA وأzt −k − µ 7N 4
@ ا
!"دA`ب ه.
E ( zt − µ )( zt −k − µ ) = E ( zt −k − µ ) at − θ1 E ( zt −k − µ ) at −1
− θ 2 E ( zt −k − µ ) at −2 , k = 0, ±1, ±2,...
او
48
γ k = E ( zt −k − µ ) at − θ1E ( zt −k − µ ) at −1 − θ 2 E ( zt −k − µ ) at −2 , k = 0, ±1, ±2,...
2ار3C O%.و
γ 0 = (1 + θ12 + θ 22 ) σ 2
γ 1 = ( −θ1 + θ1θ 2 ) σ 2
γ 2 = −θ 2σ 2
γ k = 0, k > 2
γ 0 7 4!J
.و
ρ1 =
−θ1 + θ1θ 2
1 + θ12 + θ 22
ρ2 =
−θ 2
1 + θ12 + θ 22
ρ k = 0, k > 2
4
داH3 7 W;3Cو
1,
k =0
−θ + θ θ
1 2 1 22 , k = 1
1 + θ1 + θ 2
ρk =
−θ 2
, k =2
1 + θ12 + θ 22
0,
k >2
MA ( 2 ) 4!"
7CاA
اw. وال ا
;ا7 ه4
;
ل ا3ا
θ1 = 0.4, θ 2 = 0.4 (7) H3K
ا-7
θ1 = 1.5, θ 2 = −0.8 (8) H3K
ا-8
θ1 = 0.5, θ 2 = −0.6 (9) H3K
ا-9
49
(7) H3
ACF
0.0
ACF
-0.1
-0.2
-0.3
0
10
20
Lag
(8) H3
ACF
0.2
0.1
0.0
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
0
10
20
Lag
(9) H3
ACF
0.4
0.3
0.2
0.1
ACF
ACF
-0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
Lag
50
20
& ك%;!
اw1;!
ذج ا1!'
78
ا7CاA
اw. ا
;ا4
اY& H3 د2ا إL W"
& ا
)J
2ار3C O! ورO.%
ب11 e2"C ف ;[م1 اAO
وMA ( 2 ) 4]
ا4Lا
ر
:4
;
ا
!"
) ا
θ1 = 0.4, θ 2 = 0.4 (10) H3K
ا
-10
θ1 = 1.5, θ 2 = −0.8 (11) H3K
ا
-11
θ1 = 0.5, θ 2 = −0.6 (12) H3K
ا
-12
(10) H3
PACF
0.0
PACF
-0.1
-0.2
-0.3
0
10
20
Lag
(11) H3
PACF
0.2
0.1
0.0
PACF
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
0
10
Lag
51
20
(12) H3
PACF
0.4
0.3
0.2
PACF
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
Lag
: ARMA(1,1) n ار#
ا9ار ا/7jا-ك/\ ا0ذج ا7 :0د0
:H3K
اW;32و
φ1 ( B ) zt = δ + θ1 ( B ) at
(1 − φ1B ) zt = δ + (1 − θ1B ) at
zt − φ1 zt −1 = δ + at − θ1at −1
zt = δ + φ1 zt −1 + at − θ1at −1 , at ∼ WN ( 0, σ 2 ) , φ1 ≠ θ1
&;خ9! ط ا2 , وه'ك طθ1 < 1 بJ9 وط اφ1 < 1 ارJ;9ط ا
ذج1! ذج إ1!'
`! م إ&;خ ا2 طK
ا اA وهφ1 ≠ θ1 1 وهDegeneracy Condition
4!J
.( و1 − φ1B ) zt = δ + (1 − θ1B ) at 4B"
! اN φ1 = θ1 ن1 آ4
5 7bN 4L درHBأ
ARMA ( 0, 0 ) 1 وهδ ′ =
δ
1 − φ1
Q5 zt = δ ′ + at ^(2 ذج1!'
( أن ا1 − φ1B )
:7
;
آw1;!
اL1
(1 − φ1B ) zt = δ + (1 − θ1B ) at
(1 − θ1B ) a
δ
zt =
+
t
1 − φ1 (1 − φ1 B )
δ
(1 − θ1B ) E a
E ( zt ) =
+
( t)
1 − φ1 (1 − φ1 B )
اA3 وهφ1 < 1 نf
وذ
E ( zt ) =
δ
1 − φ1
52
δ
\21";
. وδ = µ (1 − φ1 ) أوE ( zt ) = µ =
δ
1 − φ1
أي
zt = µ (1 − φ1 ) + φ1 zt −1 + at − θ1at −1
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1
N6
VB1;
اA ( وأzt −k − µ ) , k = 0, ±1, ±2,... %
. 4
ا
!"د7N= `ب.و
E ( zt −k − µ )( zt − µ ) − φ1 E ( zt −k − µ )( zt −1 − µ ) = E ( zt −k − µ ) at − θ1 E ( zt −k − µ ) at −1 ,
k = 0, ±1, ±2,...
O'&و
γ k − φ1γ k −1 = E ( zt −k − µ ) at − θ1E ( zt −k − µ ) at −1 , k = 0, ±1, ±2,...
2ار3C O%.و
k = 0 γ 0 − φ1γ 1 = E ( zt − µ ) at − θ1 E ( zt − µ ) at −1
4B"
`ب ا. E ( zt − µ ) at −1 وE ( zt − µ ) at & Hن آs اL1
( zt − µ ) − φ1 ( zt −1 − µ ) = at − θ1at −1
VB1;
اA وأat −1 وat & H آ7N
E ( zt − µ ) at − φ1 E ( zt −1 − µ ) at = E [ at at ] − θ1 E [ at −1at ]
1 ةJ
و& ا
E ( zt − µ ) at − φ1 ( 0 ) = σ 2 − θ1 ( 0 )
E ( zt − µ ) at = σ 2
و
E ( zt − µ ) at −1 − φ1E ( zt −1 − µ ) at −1 = E [at at −1 ] − θ1E [ at −1at −1 ]
E ( zt − µ ) at −1 − φ1σ 2 = 0 − θ1σ 2
∴ E ( zt − µ ) at −1 = σ 2 (φ1 − θ1 )
4J.
ا4Y
ا7N \21";
.و
k = 0 γ 0 − φ1γ 1 = σ 2 − θ1σ 2 (φ1 − θ1 )
∴γ 0 − φ1γ 1 = σ 2 1 − θ1 (φ1 − θ1 )
و
53
k = 1 γ 1 − φ1γ 0 = E ( zt −1 − µ ) at − θ1 E ( zt −1 − µ ) at −1
∴γ 1 − φ1γ 0 = −θ1σ 2
و
k = 2 γ 2 − φ1γ 1 = E ( zt −2 − µ ) at − θ1 E ( zt − 2 − µ ) at −1 = 0
∴ k ≥ 2 γ k − φ1γ k −1 = 0
تXو& ا
!"د
γ 0 − φ1γ 1 = σ 2 1 − θ1 (φ1 − θ1 )
و
γ 1 − φ1γ 0 = −θ1σ 2
γ0 =
γ1 =
1 + θ12 − 2φ1θ1 2
σ
1 − φ12
(1 − φ1θ1 )(φ1 − θ1 ) σ 2
1 − φ12
;J.
; اB"
و& ا
ρ1 =
γ 1 (1 − φ1θ1 )(φ1 − θ1 )
=
γ0
1 + θ12 − 2φ1θ1
4B"
و& ا
γ k − φ1γ k −1 = 0, k ≥ 2
γ 0 4!J
.و
ρ k − φ1 ρ k −1 = 0, k ≥ 2
وρ0 = 1 4
) اوJ
;[ام اF. k ≥ 2 )B V!
2ار3C 4
@ ا
!"دA هH5
]!N ρ1 =
ρ 2 = φ1 ρ1
ρ 2 = φ1
(1 − φ1θ1 )(φ1 − θ1 )
1 + θ12 − 2φ1θ1
ρ 3 = φ1 ρ 2
54
3!2و
(1 − φ1θ1 )(φ1 − θ1 )
1 + θ12 − 2φ1θ1
ρ 3 = φ12
(1 − φ1θ1 )(φ1 − θ1 )
1 + θ12 − 2φ1θ1
.اA3وه
H3K
اARMA (1,1) ذج1!'
7CاA
اw. ا
;ا4
داW;3
1,
k =0
(1 − φ1θ1 )(φ1 − θ1 )
, k =1
ρk =
2
1
+
−
2
θ
φ
θ
1
1
1
φ1 ρ k −1
k≥2
φ1 = 0.9,θ1 = −0.5 )J
7CاA
اw. ا
;ا4
دا76"2 13 H3
(13)H3
A C F o f A R M A (1 ,1 )
1 .0
0 .9
0 .8
0 .7
C1
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0
5
10
15
Lag
φ1 = −0.9,θ1 = −0.5 )J
7CاA
اw. ا
;ا4
دا76"2 14 H3
(14)H3
A C F o f A R M A ( 1 ,1 )
C1
0 .5
0 .0
-0 .5
0
5
10
15
Lag
55
. أو &;دد5@ واC إ7N ;[& اC ARMA (1,1) ذج1!'
7CاA
اw. ا
;ا4
ان دا5
&ى انAR (1) ذج1!'
7CاA
اw. ا
;ا4
!& داC *(KC اA ه7N 7 وه4(
وا4(L1!
) اJ
ا
( ρ k = φ1k −1 ρ1 , k ≥ 2 ه أن. ) ρ1 & (أ2 &[;
ا
:7
;
ب آ11 أو11 e2"C & W%C φkk 78
ا7CاA
اw. ا
;ا4
دا
2ار3C φkk L1 ب11 e2"C &
φ00 = 1, by definition
φ11 = ρ1 =
(1 − φ1θ1 )(φ1 − θ1 )
1 + θ12 − 2φ1θ1
ρ 2 − φ11 ρ1
1 − φ11 ρ1
ρ −φ ρ −φ ρ
φ33 = 3 21 2 22 1 , φ21 = φ11 − φ22φ11
1 − φ21 ρ1 − φ22 ρ 2
φ22 =
.2ار3C )J
ا4J. W%C اA3وه
φ1 = 0.9,θ1 = −0.5 )J
]!N
φ11 = 0.944186 φ22 = -0.384471 φ33 = 0.183710
φ44 = -0.908462 φ55 = 0.452979 φ66 = -0.226337
φ77 = 0.113154 φ88 = -0.565702 φ99 = 0.282834
15 H3 7N )J
@ اAو) ه
15 H3
P A C F o f A R M A (1 ,1 )
1 .0
C2
0 .5
0 .0
0
5
10
15
Lag
φ1 = −0.9,θ1 = −0.5 )J
78
ا7CاA
اw. ا
;ا4
( دا2 16 H3
56
16 H3
P A C F o f A R M A (1 ,1 )
0 .3
0 .2
0 .1
C2
0 .0
-0 .1
-0 .2
-0 .3
-0 .4
-0 .5
-0 .6
0
5
10
15
Lag
أو &;دد5@ واC إ7N ;[& اC ARMA (1,1) ذج1!'
78
ا7CاA
اw. ا
;ا4
ان دا5
MA (1) ذج1!'
78
ا7CاA
اw. ا
;ا4
!& داC *(KC اA ه7N 7 وه4(
وا4(L1!
) اJ
ا.
. φ11 = ρ1 4
او4!J
" ا. (أ2 &[;
&ى ان ا
: ARMA(p,q) ذج7 > ?اص:
AR(p) ذج7 :rأو
:7
;
. !;2و
.4(
وا
;[&ات ا4X & ا
;[&ات اw & ن13;C و8OX ;!C 7C ذاw.اC 4
دا-1
أيk > p تb[;
) اJ
رb+ن & أ13;C 78L 7C ذاw.اC 4
دا-2
φ11 = φ22 = φ33 = ⋯ = φ pp ≠ 0
φ p +1, p +1 = φ p + 2, p + 2 = ⋯ = 0
. k > p e[;
" ا. 78
ا7CاA
اw. ا
;ا4
دا7N "6B اA! ه2و
: MA(q) ذج7 :7]
:7
;
. !;2و
أيk > q تb[;
) اJ
رb+ن & أ13;C 7C ذاw.اC 4
دا-1
ρ1 = ρ 2 = ρ 3 = ⋯ = ρ q ≠ 0
ρ q+1,q+1 = ρ q+ 2,q +2 = ⋯ = 0
. k > q e[;
" ا. 7CاA
اw. ا
;ا4
دا7N "6B اA! ه2و
57
-2دا
C 4ا w.ذا 8OX ;!C 78L 7Cو13;Cن & & wا
;[&ات ا 4Xوا
;[&ات
ا
(.4
5Xا9زدوا1! . Duality 4Lذ AR 7Lو .MA
] :-اذج ا:ARMA(p,q) \:
و:7
;
. !;2
دوال ا
;ا w.ا
Aا 7Cوا
;ا w.ا
Aا 7Cا
1!'
78ذج ا
![; 8OX ;!C wو13;Cن & & w
ا
;[&ات ا 4Xوا
;[&ات ا
( 4ا
; 7O;'C 7إ
ا
bآ! زاد ا
;[13C &' . k eن
FN k > q − pن دا
4ا
;ا w.ا
Aا%;C 7Cد & Lء ا%9ار ا
Aا1!'
7Cذج و '& 13Cن
FN k > p − qن دا
4ا
;ا w.ا
Aا 7Cا
%;C 78د & Lء ا
!; w1ا
!;%ك '!1ذج.
58
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻟﺚ
:Nonstationar Time Series Models اةW ت ا,ذج ا7
:\0 اa ار0j م ا:rاو
ارJ;
اول
طK
ا
ان
ى
4'&ز
4;&
ارJ;9
6
e2"C
&
]!N ، &
ل ا1= S.U 4;!
اw1;& ن132 أنW6;2 E ( zt ) = µ = constant ∀t
76[
اف ا9ذج ا1!'
zt = b0 + b1t + at , at ∼ WN ( 0, σ 2 ) , b0 , b1 ∈ ( −∞, ∞ )
1 هw1;!
ان ا
E ( z )t = b0 + b1t
.4
%
@ اA ه7N J%;& z ار اولJ;9 اي ان ط ا، &
4('
. S.U z 1وه
ذج1!'
ا2b;
اH& (6;. f
∇ وذzt H21%;
ول ا%'
wt = ∇zt = zt − zt −1 = b0 + b1t + at − b0 − b1 ( t − 1) − at −1
=b1 + at − at −1 = b1 + ct
∴ wt = b1 + ct , ct ∼ WN ( 0,ν 2 )
( σ 2 وν 2 . 4B"
اL أو: 2!C )
wt ة2
ا4;!
اw1;& نsا
E ( wt ) = b1 = constant ∀t
اول2b;
اAة ) أي أJ;!
اz 4;!
( = ∇ ا1 − B ) H21%;
( ا6C أي ان
.ةJ;& 4;& إO
15 (4;!
7".;
اف ا9ذج ا1! , آ!]ل
zt = b0 + b1t + b2t 2 + at , at ∼ WN ( 0, σ 2 ) , b0 , b1 , b2 ∈ ( −∞, ∞ )
w1;!
د ا2F.
E ( zt ) = b0 + b1t + b2t 2
(7]
ا2b;
اA∇ ) أ2 zt H21%;
اA{. .J;& z ذج1!'
أي ان ا، &
";! ا2 1وه
59
∇ 2 zt = ∇2 ( b0 + b1t + b2t 2 + at )
(1 − 2 B + B ) z = (1 − 2 B + B )( b
2
2
0
t
+ b1t + b2t 2 + at )
wt = {b0 − 2b0 + b0 } + {b1t − 2b1 ( t − 1) + b1 ( t − 2 )} +
{b t
2
2
}
− 2b2 ( t − 1) + b2 ( t − 2 ) +
2
2
{at − 2at −1 + at −2 }
= 2b2 + {at − 2at −1 + at −2 }
=b′ + ht , ht ∼ WN ( 0,τ 2 )
اA3وه
wt = ∇ 2 zt = b′ + ht , ht ∼ WN ( 0,τ 2 )
E ( wt ) = b′ = constant
∀t
اO
15 ةJ;!
اz 4;!
( ا7]
ا2b;
اA∇ )أي ا2 H21%;
( ا6C أي ان
.ةJ;&
.( σ 2 وτ 2 . 4B"
اL أو: 2!C )
H3K
اJ;!
اz ذج1!'
م إذا آن اH3K.
zt = b0 + b1t + ⋯ + bd t d + at , at ∼ WN ( 0,σ 2 ) , b0 , b1 ,⋯ , bd ∈ ( −∞, ∞ )
.J;& ذج1! 1 هwt = ∇d zt أي ان،J;& ذج1!
* إ1%2 ∇d zt H21%;
ن اFN
:16 \0 اa اةW zt = b0 + b1t + ⋯ + bd t d + at , at ∼ WN ( 0,σ 2 ) , b0 , b1 ,⋯ , bd ∈ ( −∞, ∞ )
. إ' ةL/ d n= رY∇ وه اd zt ./ا
:# اa ار0j م ا:ً 7]
7]
ط اK
ا،4'& ز4;& ارJ;9 6 e2"C &
V ( zt ) = γ 0 = constant ∀t
. t )B V!
S.U 2(;
ن ا132 أنW6;2
78ا1K"
ا7K!
ذج ا1!'
]!N
60
zt = zt −1 + at , at ∼ WN ( 0,σ 2 )
ر3;!
\ ا21";
& ا
zt = a1 + a2 + ⋯ + at
2(;
واVB1;
اAF.و
E ( zt ) = 0 = constant ∀t
V ( zt ) = tσ 2
. t &
";! ا2 2(;
أن ا5و
اول2b;
اA{.
wt = ∇zt = zt − zt −1 = at
2(;
واVB1;
اAF.و
E ( wt ) = 0 = constant ∀t
V ( wt ) = σ 2 = constant ∀t
.ةJ;& 4;& إ2(;
ا7N ةJ;!
اz 4;!
ل ا15 اول2b;
إذًا ا
H3K
اY;& (w1;&) ى1;!
4
دا2(;
آن ا1
مH3K.
V ( zt ) = cf ( µt )
Y;2 w1;& ى أو1;& µt و4(
z 4!B 6"C 4N &"و4
داf (⋅) وS.U c > 0 Q5
4
د دا2 أي إT ( zt ) H21%C د2ول إ% '";! ا
& وه2 2(;
ن اFN 7
;
. ا
& وV&
. 2(;
ار اJ;9 T (⋅)
H21%;
ا
y t = T ( zt ) =
ztλ − 1
λ
76"2 7
;
ا
ول ا.H21%;
&") ا1 هλ ∈ ( −∞, ∞ ) Q5 2(;
ا7N ةJ;& 4;& 76"2
:O
4.J!
ت ا21%;
اV& λ )"!
&) اآ] إ;[اJ
ا
λ
-0.1
yt
1
zt
-0.5
0.0
0.5
1
zt
ln zt
zt
61
1.0
zt
:ل-
2(;
واw1;!
ا7N ةJ;& z 4;!
()اH3K
ا
zt
O r ig in a l S e r ie s
400
z(t)
300
200
100
In d e x
10
20
30
40
50
60
70
80
90
yt = ln zt H21%;
اء اLF. 2(;
اS(]C ". 4;!
)ب( اH3K
ا
T r a n s f o r m e d S e r ie s
6 .0
ln z(t)
5 .5
5 .0
In d e x
10
20
30
40
50
60
70
80
90
∇yt = yt − yt −1 اول2b;
اء اL" إ. yt 4
1%!
ا4;!
)ج( اH3K
ا
D if f e re n c e d a n d T ra n s f o rm e d S e rie s
0 .2
y(t)-y(t-1)
0 .1
0 .0
-0 .1
-0 .2
In d e x
10
20
30
40
50
60
70
80
90
. 2(;
واw1;!
& اH آ7N ةJ;& 4;!
اS%(+ اe آ5X
62
Autoregressive- (p,d,q) n ار#
ك/\ ا0ا-
)ا-ا9ار ا/7jذج ا7
Integrated-Moving Average Models ARIMA(p,d,q)
& ك%;& w1;&-7Cار ذا%ذج أ1! H3 wt = ∇d zt ةJ;!
ا4;!
ا4LA! 3!2
:7
;
( آp, q ) 4Lا
ر
φ p ( B ) wt = φ p ( B ) ∇d zt = δ + θ q ( B ) at , at ∼ WN ( 0, σ 2 )
أو
φ p ( B )(1 − B ) zt = δ + θ q ( B ) at , at ∼ WN ( 0, σ 2 )
d
( p, d , q ) 4Lك & ا
ر%;!
اw1;!
ا-7&3;
ا-7CاA
ار ا%9ذج ا1! !2 ذج1!'
ا اAوه
.اف9 &") اδ ∈ ( −∞, ∞ ) Q5
: ( p, d , q ) 4Lك & ا
ر%;!
اw1;!
ا-7&3;
ا-7CاA
ار ا%9 !ذج ا4]&أ
: ARIMA(1,1,0) =ARI(1,1) ( أو1,1) n ار#
)ا-ا9ارا/7jذج ا7 :rاو
H3K
اW;32و
φ1 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at ∼ WN ( 0,σ 2 )
(1 − φ1B )(1 − B ) zt = δ + at
{1 − (φ1 + 1) B + φ1B 2 } zt = δ + at
أي
zt = δ + (φ1 + 1) zt −1 − φ1 zt −2 + at , at ∼ WN ( 0, σ 2 ) ,
φ1 < 1
(1,1) n ار#
ك/\ ا0ا-
)ذج ا7 :7]
: ARIMA(0,1,1) = IMA(1,1) أو
H3K
اW;32و
φ0 ( B )(1 − B ) zt = δ + θ1 ( B ) at , at ∼ WN ( 0, σ 2 )
(1 − B ) zt = δ + (1 − θ1B ) at ,
at ∼ WN ( 0, σ 2 ) ,
zt − zt −1 = δ + at − θ1at , at ∼ WN ( 0, σ 2 ) ,
θ1 < 1
θ1 < 1
أي
zt = δ + zt −1 + at − θ1at , at ∼ WN ( 0,σ 2 ) ,
θ1 < 1
63
أوRandom Walk with Trend Model افV7c$ ذج ا اا7 :-]
: ARIMA(0,1,0)
H3K
اW;32و
φ0 ( B )(1 − B ) zt = δ + θ 0 ( B ) at , at ∼ WN ( 0, σ 2 )
(1 − B ) zt = δ + at ,
at ∼ WN ( 0,σ 2 )
أي
zt = δ + zt −1 + at , at ∼ WN ( 0,σ 2 )
:ARMA(p,q) ذج7 .- وψ ( B ) وزانqدا ا
H3K
اARMA ( p, q ) ( أن آ;(' !ذج
φ p ( B ) zt = δ + θ q ( B ) at , at ∼ WN ( 0, σ 2 )
w1;!
اف ا%9 اH3K. أو
φ p ( B )( zt − µ ) = θ q ( B ) at , at ∼ WN ( 0,σ 2 )
φ p ( B ) 7CاA
ار ا%9 اH& 4!J
. ;
%
آ; ا7N
zt =
δ
φ p (1)
zt − µ =
+
θq ( B )
at , at ∼ WN ( 0, σ 2 )
φp (B)
θq ( B )
at , at ∼ WN ( 0, σ 2 )
φp (B)
φ p ( B ) = 0 ورAL نf
وذ4.رJ;& 4& H3KC
7N w1;!
اف ا%9 اH3K. 7b;3 ف1 اAO
و
θq ( B )
4('
ة اJ;!
ان '!ذج ا5X
φp (B)
δ
φ p (1)
= µ `2ة ا51
ة ا8 رج داVJC
4
;
;' اKB'&
zt − µ =
θq ( B )
at , at ∼ WN ( 0,σ 2 )
φp ( B)
4.رJ;!
ا4!
ا
64
ψ ( B) =
θq ( B )
φp (B)
H3K
اW;3C 7;
وا
ψ ( B) =
θq ( B)
= ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + ⋯ , ψ 0 = 1
φp (B)
. اوزان4
! داC
:17 I,$ '; اARMA ( p, q ) وزان ذجqدا ا
ψ ( B) =
θq ( B)
= ψ 0 B 0 + ψ 1 B1 + ψ 2 B 2 + ψ 3 B 3 + ⋯ , ψ 0 = 1
φp (B)
ψ ( B) =
θq ( B) ∞
= ∑ψ j B j , ψ 0 = 1
φ p ( B ) j =0
ψ 0 = 1,ψ 1 ,ψ 2 ,ψ 3 ,⋯ وزان هq اJK
H]!C !2 zt − µ =
θq ( B )
at , at ∼ WN ( 0,σ 2 ) H3K
ي اA
ذج ا1!'
ا:!K,
φp ( B)
. ARMA ( p, q ) '!ذج78O
ك ا%;!
اw1;!
ا
: اذجv وزانq ا ا-
أ
: AR(1) وزان ذجqدا ا
H3K
اW;32 AR(1) ذج1!
φ1 ( B )( zt − µ ) = θ 0 ( B ) at , at ∼ WN ( 0,σ 2 )
(1 − φ1B )( zt − µ ) = at
zt − µ =
1
a
(1 − φ1B ) t
z t − µ = ψ ( B ) at
Q5
ψ (B) =
1
(1 − φ1B )
65
4
;
ا4J26
. ψ 1 ,ψ 2 ,ψ 3 ,⋯ اوزانL1 ف1
ψ ( B) =
1
(1 − φ1B )
ψ ( B )(1 − φ1B ) ≡ 1
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ⋯) (1 − φ1 B ) ≡ 1
.42 &;و4B"
ا7N= B j أي ان &"&ت:N3C 4B 7 اة ه4B"
أن ا5X
4B"
ا7N= B j !واة &"&ت.و
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ⋯) (1 − φ1 B ) ≡ 1,
φ1 < 1
B 0 : (1)(1) ≡ 1
B1 : ψ 1 − φ1 ≡ 0 ⇒ ψ 1 = φ1
B 2 : ψ 2 − ψ 1φ1 ≡ 0 ⇒ ψ 2 = ψ 1φ1 = φ12
B 3 : ψ 3 − ψ 2φ1 ≡ 0 ⇒ ψ 3 = ψ 2φ1 = φ13
⋮
B j : ψ j − ψ j −1φ1 ≡ 0 ⇒ ψ j = ψ j −1φ1 = φ1j
7 هAR(1) ذج1!'
أي ان اوزان
ψ j = φ1j , φ1 < 1
: MA(1) وزان ذجqدا ا
H3K
اW;32 MA(1) ذج1!
φ0 ( B )( zt − µ ) = θ1 ( B ) at , at ∼ WN ( 0,σ 2 )
( zt − µ ) = (1 − θ1B ) at
z t − µ = ψ ( B ) at
Q5
ψ ( B ) = (1 − θ1B )
4B"
ا7N= B j !واة &"&ت.
ψ 1 = −θ1 , ψ 2 = ψ 3 = ⋯ = 0
أي
1,
ψ j = −θ1 ,
0,
j=0
j =1
j≥2
66
: AR(2) وزان ذجqدا ا
H3K
اW;32 AR(2) ذج1!
φ2 ( B )( zt − µ ) = θ 0 ( B ) at , at ∼ WN ( 0, σ 2 )
(1 − φ B − φ B ) ( z − µ ) = a
2
1
zt − µ =
2
t
t
1
a
(1 − φ1B − φ2 B 2 ) t
z t − µ = ψ ( B ) at
Q5
ψ ( B) =
1
(1 − φ1B − φ2 B 2 )
4J.
ا4J26
. ψ 1 ,ψ 2 ,ψ 3 ,⋯ اوزانL1 و
ψ ( B ) (1 − φ1B − φ2 B 2 ) ≡ 1
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ⋯)(1 − φ1 B − φ2 B 2 ) ≡ 1
B1 : ψ 1 − φ1 = 0 ⇒ ψ 1 = φ1
2
B 2 : ψ 2 − φψ
1 1 − φ 2 = 0 ⇒ ψ 2 = φψ
1 1 + φ 2 = φ1 + φ2
B 3 : ψ 3 − φψ
1 2 − φ 2ψ 1 = 0 ⇒ ψ 3 = φψ
1 2 + φ2ψ 1
⋮
B j : ψ j − φψ
1 j −1 − φ2ψ j − 2 = 0 ⇒ ψ j = φψ
1 j −1 + φ 2ψ j − 2
7 هAR(2) ذج1!'
أي ان اوزان
1,
φ ,
1
ψj = 2
φ1 + φ2 ,
φψ
1 j −1 + φ2ψ j − 2 ,
j=0
j =1
j=2
j≥3
: MA(2) وزان ذجqدا ا
H3K
اW;32 MA(2) ذج1!
φ0 ( B )( zt − µ ) = θ 2 ( B ) at , at ∼ WN ( 0, σ 2 )
( zt − µ ) = (1 − θ1B − θ 2 B 2 ) at
z t − µ = ψ ( B ) at
Q5
ψ ( B ) = (1 − θ1B − θ 2 B 2 )
67
4B"
ا7N= B j !واة &"&ت.
ψ 1 = −θ1 , ψ 2 = −θ 2 , ψ 3 = ψ 4 = ψ 5 ⋯ = 0
أي
j=0
1,
−θ ,
ψj = 1
−θ 2 ,
0,
j =1
j=2
j≥2
: ARMA(1,1) وزان ذجqدا ا
H3K
اW;32 ARMA(1,1) ذج1!
φ1 ( B )( zt − µ ) = θ1 ( B ) at , at ∼ WN ( 0,σ 2 )
(1 − φ1B )( zt − µ ) = (1 − θ1B ) at
(1 − θ1B ) a
zt − µ =
(1 − φ1B ) t
z t − µ = ψ ( B ) at
Q5
ψ (B) =
(1 − θ1B )
(1 − φ1B )
4J.
ا4J26
. ψ 1 ,ψ 2 ,ψ 3 ,⋯ اوزانL1 و
ψ ( B )(1 − φ1B ) ≡ (1 − θ1B )
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ⋯) (1 − φ1B ) ≡ (1 − θ1 B )
B1 : ψ 1 − φ1 = −θ1 ⇒ ψ 1 = φ1 − θ1
B 2 : ψ 2 − φψ
1 1 = 0 ⇒ ψ 2 = φψ
1 1 = φ1 (φ1 − θ1 )
2
B 3 : ψ 3 − φψ
1 2 = 0 ⇒ ψ 3 = φψ
1 2 = φ1 (φ1 − θ1 )
⋮
j −1
B j : ψ j − φψ
(φ1 − θ1 )
1 j −1 = 0 ⇒ ψ j = φψ
1 j −1 = φ1
7 هARMA(1,1) ذج1!'
أي ان اوزان
j −1
ψ j = φψ
(φ1 − θ1 ) ,
1 j −1 = φ1
j ≥ 1,
φ1 < 1, φ1 ≠ θ1
68
: ARI(1) وزان ذجqدا ا
H3K
اW;32 ARI(1) ذج1!
φ1 ( B )(1 − B )( zt − µ ) = at , at ∼ WN ( 0, σ 2 )
zt − µ =
1
a
(1 − φ1B )(1 − B ) t
z t − µ = ψ ( B ) at
Q5
ψ ( B) =
1
(1 − φ1B )(1 − B )
4J.
ا4J26
. ψ 1 ,ψ 2 ,ψ 3 ,⋯ اوزانL1 و
ψ ( B )(1 − φ1B )(1 − B ) ≡ 1
(1 + ψ B + ψ
(1 + ψ B + ψ
1
2
1
2
B 2 + ψ 3B 3 + ⋯) (1 − φ1B )(1 − B ) ≡ 1
B 2 + ψ 3B 3 + ⋯) (1 − (φ1 + 1) B + φ1B 2 ) ≡ 1
B1 : ψ 1 − (φ1 + 1) = 0 ⇒ ψ 1 = φ1 + 1
B 2 : ψ 2 − (φ1 + 1)ψ 1 + φ1 = 0 ⇒ ψ 2 = (φ1 + 1)ψ 1 + φ1 = (φ1 + 1) + φ1
2
B 3 : ψ 3 − (φ1 + 1)ψ 2 + φψ
1 1 = 0 ⇒ ψ 3 = (φ1 + 1)ψ 2 − φψ
1 1
⋮
B j : ψ j − (φ1 + 1)ψ j −1 + φψ
1 j − 2 = 0 ⇒ ψ j = (φ1 + 1)ψ j −1 − φψ
1 j −2
7 هARI(1) ذج1!'
أي ان اوزان
1,
φ + 1,
1
ψj =
2
(φ1 + 1) + φ1 ,
(φ1 + 1)ψ j −1 − φψ
1 j −2 ,
j=0
j =1
j=2
j≥3
L1 وأا
ARIMA(1,0,1) أوRandom Walk Mdel وزان ذج ا ااqدا ا
H3K
اW;32و
zt = zt −1 + at , at ∼ WN ( 0,σ 2 )
أي
69
zt − zt −1 = at , at ∼ WN ( 0,σ 2 )
(1 − B ) zt = at
zt =
1
a
(1 − B ) t
Q5
ψ ( B) =
1
(1 − B )
4J.
ا4J26
. ψ 1 ,ψ 2 ,ψ 3 ,⋯ اوزانL1 و
ψ ( B )(1 − B ) ≡ 1
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ⋯) (1 − B ) ≡ 1
B1 : ψ 1 − 1 = 0 ⇒ ψ 1 = 1
B 2 :ψ 2 −ψ 1 = 0 ⇒ ψ 2 = ψ 1 = 1
B 3 :ψ 3 −ψ 2 = 0 ⇒ ψ 3 = ψ 2 = 1
⋮
B j : ψ j − ψ j −1 = 0 ⇒ ψ j = ψ j −1 = 1
7 هARIMA ( 0,1, 0 ) 78ا1K"
ا7K!
ذج ا1!'
أي ان اوزان
ψ j = 1,
j ≥1
: ψ ( B ) وزانq ?اص دا اv$
H3K
اARMA(p,q) 4Lك & ا
ر%;!
اw1;!
ا-7CاA
ار ا%9ذج ا1! '(;( أن آ
zt − µ = ψ ( B ) at , at ∼ WN ( 0,σ 2 )
H3K
* اB"
@ اA ه4.;3.و
zt − µ = at + ψ 1at −1 + ψ 2at −2 + ψ 3at −3 + ⋯
∞
= ∑ψ j at − j , ψ 0 = 1
j =0
:4
;
ا42I'
(ت اU إ3!2 *FN
∑
∞
ψ 2j < ∞ رب ايJ;C ' ان اوزانP;Nوإذا ا
j =0
70
:1 !7
N) ي9 ا واARMA(p,q) n ار#
ك/\ ا0ا-ا9ار ا/7jذج ا
.)' ا
∞
zt − µ = ∑ψ j at − j , at ∼ WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j < ∞
∞
j =0
\ ه0 ا-1
E ( zt ) = µ , ∀t
I,$ '; ا9\ ا$ دا اا-2
∞
ρk =
∑ψ ψ
j
j =0
j+k
∞
∑ψ 2j
, k = 0,1, 2,⋯
j =0
اوزان4
داJ. L وAR(1) 4L & ا
ر7CاA
ار ا%9ذج ا1!'
:ل-
ψ j = φ1j , φ1 < 1
7CاA
اw. ا
;ا4
دا
∞
ρk =
∑ψ ψ
j
j =0
∞
=
∞
∑ψ
j =0
∑φ φ
j +k
2
j
j =0
∞
j j+k
1 1
∑φ
2j
1
j =0
=
φ1k
1 − φ12
1
1 − φ12
= φ1k , k = 0,1, 2,⋯
4J.
ا4;'
~ اb 7وه
4
;
'!ذج ا7CاA
اw. دوال ا
;اL( أو2) 2 42I {;[ام. :#
AR(2), MA(1), MA(2), ARMA(1,1), ARMA(2,1), ARMA(1,2)
71
ﺍﻟﻔﺼﻞ ﺍﻟﺮﺍﺑﻊ
ا4ات ذات %$
\0ا g;:اqد '7ذج )ARMA(p,q
Minimum Mean Square Error Forecasts for ARMA(p,q) Models
ﻓﻲ ﺍﻝﻔﻘﺭﺓ ﺍﻝﺴﺎﺒﻘﺔ ﻜﺘﺒﻨﺎ ﻨﻤﻭﺫﺝ ﺍﻹﻨﺤﺩﺍﺭ ﺍﻝﺫﺍﺘﻲ-ﺍﻝﻤﺘﻭﺴﻁ ﺍﻝﻤﺘﺤﺭﻙ ﻤﻥ ﺍﻝﺩﺭﺠﺔ )ARMA(p,q
ﺍﻝﻤﺴﺘﻘﺭ ﻋﻠﻰ ﺍﻝﺸﻜل
∞
∞ < zt − µ = ∑ψ j at − j , at ∼ WN ( 0, σ 2 ) , ψ 0 = 1, ∑ j =0ψ 2j
∞
j =0
ﺃﻭ
⋯ zt − µ = at + ψ 1at −1 + ψ 2at −2 + ψ 3at −3 +
∞
= ∑ψ j at − j , ψ 0 = 1
j =0
ﻤﻼﺤﻅﺔ :ﻫﺫﺍ ﻴﻨﻁﺒﻕ ﺃﻴﻀﺎ ﻋﻠﻰ ﻨﻤﺎﺫﺝ ) ARIMA(p,d,qﺒﺸﻜل ﻋﺎﻡ.
ﻝﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﺓ } {z1 , z2 ,⋯ , zn−1 , zn
ﺍﻝﺘﻨﺒﺅﺍﺕ zn ( ℓ ) , ℓ ≥ 1
ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ
zn +ℓ , ℓ ≥ 1ﻴﻤﻜﻥ ﺍﻥ ﺘﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜل
zn ( ℓ ) = ξ0 an + ξ1an −1 + ξ 2 an − 2 + ⋯ , ℓ ≥ 1
ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ zn +ℓ , ℓ ≥ 1ﺘﻜﺘﺏ ﺒﺩﻻﻝﺔ ﺍﻝﻨﻤﻭﺫﺝ ﻜﺎﻝﺘﺎﻝﻲ
zn + ℓ − µ = an + ℓ + ψ 1an + ℓ−1 + ⋯ + ψ ℓ−1an +1 + ψ ℓ an + ψ ℓ+1an −1 + ⋯ , ℓ ≥ 1
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻝﺨﻁﺄ ﻴﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ )ﺃﻨﻅﺭ ﺘﻌﺭﻴﻑ ( 5
2
E zn + ℓ − zn ( ℓ ) = E an + ℓ + ψ 1an +ℓ −1 + ⋯ + ψ ℓ−1an +1 + (ψ ℓ − ξ0 ) an + (ψ ℓ+1 − ξ1 ) an −1 + ⋯
2
∞
= (1 + ψ 12 + ⋯ + ψ ℓ2−1 ) σ 2 + ∑ (ψ ℓ+ j − ξ j ) σ 2
2
j =0
ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻝﺨﻁﺄ ﺍﻷﺩﻨﻰ ﻴﻨﺘﺞ ﻤﻥ ﺘﺼﻐﻴﺭ ﺍﻝﻌﻼﻗﺔ ﺍﻝﺴﺎﺒﻘﺔ ﺒﺎﻝﻨﺴﺒﺔ ﻝﻸﻭﺯﺍﻥ ξ jﻝﺠﻤﻴﻊ ﻗﻴﻡ j
ﻭﻫﺫﺍ ﻴﻤﻜﻥ ﺇﺫﺍ ﻭﻓﻘﻁ ﺇﺫﺍ ﺤﻘﻘﺕ ﺍﻷﻭﺯﺍﻥ ξ jﺍﻝﻌﻼﻗﺔ ﺍﻝﺘﺎﻝﻴﺔ
j = 0,1, 2,⋯ , ℓ ≥ 1
ξ j = ψ ℓ+ j ,
ﻭﻋﻠﻴﻪ ﻓﺈﻥ ﺍﻝﺘﻨﺒﺅﺍﺕ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻝﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ
zn ( ℓ ) = ψ ℓan + ψ ℓ +1an −1 + ψ ℓ+ 2 an −2 + ⋯ , ℓ ≥ 1
72
ﻨﻅﺭﻴﺔ :2
ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ:
en ( ℓ ) = zn + ℓ − zn ( ℓ ) = an +ℓ + ψ 1an + ℓ−1 + ψ 2 an +ℓ −2 + ⋯ + ψ ℓ−1an +1 , ℓ ≥ 1
ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ:
V en ( ℓ ) = σ 2 (1 + ψ 12 + ψ 22 + ⋯ + ψ ℓ2−1 ) , ℓ ≥ 1
ﺍﻝﺼﻴﻐﺔ zn ( ℓ ) = ψ ℓan + ψ ℓ+1an −1 + ψ ℓ +2an −2 + ⋯, ℓ ≥ 1ﻏﻴﺭ ﻋﻤﻠﻴﺔ ﻹﻴﺠﺎﺩ ﺍﻝﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ
ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ zn +ℓ , ℓ ≥ 1ﻭﺫﻝﻙ ﻷﻨﻨﺎ ﻨﺤﺘﺎﺝ ﺇﻝﻰ ﻤﻌﺭﻓﺔ ﺍﻝﻘﻴﻡ } . {a1 , a2 ,⋯ , an −1 , an
ﺘﻌﺭﻴﻑ : 18
ﻤﺠﻤﻭﻋﺔ
ﺍﻝﻤﻌﻠﻭﻤﺎﺕ
)} I ({z1 , z2 ,⋯ , zn −1 , zn
ﺘﻜﺎﻓﺊ
ﻤﺠﻤﻭﻋﺔ
ﺍﻝﻤﻌﻠﻭﻤﺎﺕ
)} I ({a1 , a2 ,⋯, an−1 , anﻭﺫﻝﻙ ﺒﺎﻝﻤﻌﻨﻰ ﺃﻥ ﺍﻝﻤﺠﻤﻭﻋﺔ } {a1 , a2 ,⋯ , an −1 , anﺘﺤﺘﻭﻯ ﻋﻠﻰ ﻨﻔﺱ
ﺍﻝﻤﻌﻠﻭﻤﺎﺕ ﻋﻥ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ } . {z1 , z2 ,⋯ , zn−1 , zn
ﻤﻼﺤﻅﺔ :ﺍﻝﻤﺘﺴﻠﺴﺔ ﺍﻝﺯﻤﻨﻴﺔ } {z1 , z2 ,⋯, zn −1 , znﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﻭﻗﻴﺎﺴﻬﺎ ﻭﻝﻜﻥ ﺍﻝﻤﺘﻠﺴﻠﺔ
} {a1 , a2 ,⋯ , an −1 , anﻻﻴﻤﻜﻥ ﻤﺸﺎﻫﺩﺘﻬﺎ ﺃﻭ ﻗﻴﺎﺴﻬﺎ.
ﻨﻅﺭﻴﺔ : 3
ﺍﻝﻤﺘﻨﺒﺊ ﺫﺍ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻝﺨﻁﺄ ﺍﻷﺩﻨﻰ MMSE Forecastsﻴﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
ﺃﻱ ﻫﻭ ﺍﻝﺘﻭﻗﻊ ﺍﻝﺸﺭﻁﻲ ﻝﻠﻘﻴﻤﺔ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ zn +ℓ , ℓ ≥ 1ﻤﻌﻁﻰ } . {z1 , z2 ,⋯ , zn−1 , zn
ﺘﺴﺘﺨﺩﻡ ﻨﻅﺭﻴﺔ 2ﻋﻤﻠﻴﺎ ﻹﻴﺠﺎﺩ ﻗﻴﻡ ﺍﻝﺘﻨﺒﺅﺍﺕ ﺒﺩﻻ ﻤﻥ ﺍﻝﺼﻴﻐﺔ
zn ( ℓ ) = ψ ℓan + ψ ℓ +1an −1 + ψ ℓ+ 2 an −2 + ⋯ , ℓ ≥ 1
73
ﻭﺫﻝﻙ ﺘﺒﻌﺎ ﻝﻠﻤﻼﺤﻅﺔ ﺍﻝﺴﺎﺒﻘﺔ.
ﻗﺎﻋﺩﺓ :2
a , j ≤ 0
1 − E ( an + j zn , zn −1 ,⋯) = n + j
j>0
0,
j≤0
zn + j ,
2 − E ( zn + j zn , zn −1 ,⋯) =
zn ( j ) , j > 0
ﻨﻅﺭﻴﺔ 3ﻤﻊ ﺍﻝﻘﺎﻋﺩﺓ 2ﺘﻌﻁﻲ ﻁﺭﻴﻘﺔ ﻋﻤﻠﻴﺔ ﻭﺴﻬﻠﺔ ﻹﻴﺠﺎﺩ ﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ zn + ℓ , ℓ ≥ 1
ﺘﻌﺭﻴﻑ :19
ﺍﻝﺩﺍﻝﺔ zn ( ℓ ) , ℓ ≥ 1ﻜﺩﺍﻝﺔ ﻝﺯﻤﻥ ﺍﻝﺘﻘﺩﻡ ℓ ≥ 1ﻋﻨﺩ ﻨﻘﻁﺔ ﺍﻻﺼل ﻝﻠﺯﻤﻥ nﺘﺴﻤﻰ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ.
دوال ا 4ذج ): ARIMA(p,d,q
ﺍﻭﻻ :ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ ): AR(1
ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ } {z1 , z2 ,⋯ , zn−1 , znﺤﺘﻰ ﺍﻝﺯﻤﻥ nﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ
ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(1ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜل
) ∞ zt − µ = φ1 ( zt −1 − µ ) + at , at ∼ WN ( 0,σ 2 ) , φ1 < 1, µ ∈ ( −∞,
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ⋯ zn +1 , zn+2 , zn +3 ,ﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +ℓ , ℓ ≥ 1
ﻤﻥ ﻨﻅﺭﻴﺔ 3ﻨﺠﺩ
zn ( ℓ ) = E ( zn +ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=µ +E φ1 ( zn + ℓ−1 − µ ) + an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
=µ +E φ1 ( zn + ℓ−1 − µ ) zn , zn −1 ,⋯ + an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
= µ +φ1E ( zn + ℓ−1 zn , zn −1 ,⋯) − µ + E an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
ﺃﻱ
74
zn ( ℓ ) = µ +φ1E ( zn + ℓ−1 zn , zn −1 ,⋯) − µ + E an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ 2
ℓ = 1: zn (1) = µ +φ1E ( zn zn , zn −1 ,⋯) − µ + E an +1 zn , zn −1 ,⋯
) = µ +φ1 ( zn − µ
ℓ = 2 : zn ( 2 ) = µ +φ1E ( zn +1 zn , zn −1 ,⋯) − µ + E an + 2 zn , zn −1 ,⋯
= µ +φ1 zn (1) − µ
ℓ = 3 : zn ( 3) = µ +φ1E ( zn + 2 zn , zn −1 ,⋯) − µ + E an +3 zn , zn −1 ,⋯
= µ +φ1 zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( ℓ ) = µ +φ1 zn ( ℓ − 1) − µ , ℓ ≥ 1
ﻭﻫﻲ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ )AR(1
ﺘﻌﺭﻴﻑ :20
ﺸﺭﻁ ﺍﻹﺴﺘﻤﺭﺍﺭ Continuity Conditionﻴﺘﻁﻠﺏ ﺃﻨﻪ ﻋﻨﺩﻤﺎ ﺘﻜﻭﻥ ℓ = 1ﻓﺈﻥ
zn ( ℓ − 1) = zn ( 0 ) = zn
ﻤﻥ ﻨﻅﺭﻴﺔ 2ﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﺘﻌﻁﻰ ﻤﻥ ﺍﻝﻌﻼﻗﺔ
V en ( ℓ ) = σ 2 (1 + ψ 12 + ψ 22 + ⋯ + ψ ℓ2−1 ) , ℓ ≥ 1
ﺴﺒﻕ ﺃﻥ ﺍﺸﺘﻘﻘﻨﺎ ﺩﺍﻝﺔ ﺍﻷﻭﺯﺍﻥ ﻝﻨﻤﻭﺫﺝ ) AR(1ﻭﻫﻲ
ψ j = φ1j , φ1 < 1
ﻭﺒﺎﻝﺘﻌﻭﻴﺽ ﻓﻲ ﺼﻴﻐﺔ ﺍﻝﺘﺒﺎﻴﻥ ﻨﺠﺩ
), ℓ ≥1
(
( V en ( ℓ ) = σ 2 1 + φ12 + φ14 + ⋯ + φ1
)2 ℓ −1
1 − φ12 ℓ
, ℓ ≥1
1 − φ12
ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻝﻨﻤﻭﺫﺝ
75
=σ2
) zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at ∼ WN ( 0, 0.024
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻝﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﻝﻬﺎ.
ﺍﻝﺤل :ﻤﻥ ﺍﻝﺼﻴﻐﺔ zn ( ℓ ) = µ +φ1 zn ( ℓ − 1) − µ , ℓ ≥ 1ﻨﺠﺩ
) z156 (1) = 0.97+0.85 ( z156 − 0.97
= 0.97+0.85 ( 0.49 − 0.97 ) = 0.56
) z156 ( 2 ) = 0.97+0.85 ( z156 (1) − 0.97
= 0.97+0.85 ( 0.56 − 0.97 ) = 0.62
) z156 ( 3) = 0.97+0.85 ( z156 ( 2 ) − 0.97
= 0.97+0.85 ( 0.62 − 0.97 ) = 0.68
ﻭﺍﻝﺘﺒﺎﻴﻨﺎﺕ
1 − φ12 ℓ
, ℓ ≥1
1 − φ12
V en ( ℓ ) = σ 2
V e156 (1) = 0.024
4
= 0.041
= 0.054
)1 − ( 0.85
2
)1 − ( 0.85
6
)1 − ( 0.85
2
)1 − ( 0.85
V e156 ( 2 ) = 0.024
V e156 ( 2 ) = 0.024
ﺜﺎﻨﻴﺎ :ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ ): AR(2
ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ } {z1 , z2 ,⋯ , zn−1 , znﺤﺘﻰ ﺍﻝﺯﻤﻥ nﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ
ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) AR(2ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜل
zt = µ + φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at , at ∼ WN ( 0,σ 2 ) , µ ∈ ( −∞, ∞ ) ,
φ2 − φ1 < 1, φ2 + φ1 < 1, φ2 < 1
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ⋯ zn +1 , zn+2 , zn +3 ,ﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +ℓ , ℓ ≥ 1
ﻤﻥ ﻨﻅﺭﻴﺔ 3ﻨﺠﺩ
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=µ +E φ1 ( zn + ℓ−1 − µ ) + φ2 ( zn + ℓ− 2 − µ ) + an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
76
=µ +E φ1 ( zn + ℓ−1 − µ ) zn , zn −1 ,⋯ + φ2 ( zn + ℓ−2 − µ ) zn , zn −1 ,⋯ + an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
= µ +φ1E ( zn + ℓ−1 zn , zn −1 ,⋯) − µ + φ2 E ( zn +ℓ −2 zn , zn −1 ,⋯) − µ + E an + ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
ﺃﻱ
zn ( ℓ ) = µ +φ1E ( zn + ℓ−1 zn , zn −1 ,⋯) − µ + φ2 E ( zn + ℓ−2 zn , zn −1 ,⋯) − µ + E an +ℓ zn , zn −1 ,⋯ , ℓ ≥ 1
2 ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ
ℓ = 1: zn (1) = µ +φ1E ( zn zn , zn −1 ,⋯) − µ + φ2 E ( zn −1 zn , zn −1 ,⋯) − µ + E an +1 zn , zn −1 ,⋯
= µ +φ1 ( zn − µ ) + φ2 ( zn −1 − µ )
ℓ = 2 : zn ( 2 ) = µ +φ1E ( zn +1 zn , zn −1 ,⋯) − µ + φ2 E ( zn zn , zn −1 ,⋯) − µ + E an +2 zn , zn −1 ,⋯
= µ +φ1 zn (1) − µ + φ2 ( zn − µ )
ℓ = 3 : zn ( 3) = µ +φ1E ( zn + 2 zn , zn −1 ,⋯) − µ + φ2 E ( zn +1 zn , zn −1 ,⋯) − µ + E an +3 zn , zn −1 ,⋯
= µ +φ1 zn ( 2 ) − µ +φ2 zn (1) − µ
ℓ = 4 : zn ( 4 ) = µ +φ1E ( zn +3 zn , zn −1 ,⋯) − µ + φ2 E ( zn + 2 zn , zn −1 ,⋯) − µ + E an + 4 zn , zn −1 ,⋯
= µ +φ1 zn ( 3) − µ +φ2 zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( ℓ ) = µ +φ1 zn ( ℓ − 1) − µ + φ2 zn ( ℓ − 2 ) − µ , ℓ ≥ 1
AR(2) ﻭﻫﻲ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ
ﻭﺩﺍﻝﺔ ﺍﻷﻭﺯﺍﻥ2 ﻭﻴﻤﻜﻥ ﺤﺴﺎﺏ ﺘﺒﺎﻴﻨﺎﺕ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﻤﻥ ﻨﻅﺭﻴﺔ
1,
φ ,
1
ψj = 2
φ1 + φ2 ,
φψ
1 j −1 + φ2ψ j − 2 ,
j=0
j =1
j=2
j≥3
: ARIMA(0,1,1) ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ:ﺜﺎﻝﺜﺎ
ﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎn { ﺤﺘﻰ ﺍﻝﺯﻤﻥz1 , z2 ,⋯ , zn−1 , zn } ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ
ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜلARIMA(0,1,1) ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ
zt = zt −1 + at − θ1at −1 , at ∼ WN ( 0, σ 2 )
77
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ⋯ zn +1 , zn+2 , zn +3 ,ﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +ℓ , ℓ ≥ 1
ﻤﻥ ﻨﻅﺭﻴﺔ 3ﻨﺠﺩ
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=E ( zn + ℓ−1 zn , zn −1 ,⋯) + E ( an + ℓ zn , zn −1 ,⋯) − θ1E ( an + ℓ−1 zn , zn −1 ,⋯) , ℓ ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ 2
zn ( ℓ ) =E ( zn + ℓ−1 zn , zn −1 ,⋯) + E ( an +ℓ zn , zn −1 ,⋯) − θ1E ( an + ℓ−1 zn , zn −1 ,⋯) , ℓ ≥ 1
)⋯ℓ = 1: zn (1) =E ( zn zn , zn −1 ,⋯) + E ( an +1 zn , zn −1 ,⋯) − θ1 E ( an zn , zn −1 ,
= zn − θ1an
)⋯ℓ = 2 : zn ( 2 ) =E ( zn +1 zn , zn −1 ,⋯) + E ( an +2 zn , zn −1 ,⋯) − θ1 E ( an +1 zn , zn −1 ,
)= zn (1
)⋯ℓ = 3 : zn ( 3) =E ( zn +2 zn , zn −1 ,⋯) + E ( an +3 zn , zn −1 ,⋯) − θ1E ( an +1 zn , zn −1 ,
) = zn ( 2
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( ℓ ) = zn ( ℓ − 1) , ℓ ≥ 2
ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ ) ARIMA(0,1,1ﺘﻌﻁﻰ
ﺒﺎﻝﻌﻼﻗﺔ
zn − θ1an , ℓ = 1
zn ( ℓ ) =
zn ( ℓ − 1) , ℓ > 1
ﺭﺍﺒﻌﺎ :ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ ): MA(1
ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ } {z1 , z2 ,⋯ , zn−1 , znﺤﺘﻰ ﺍﻝﺯﻤﻥ nﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎ
ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ ) MA(1ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜل
) zt = µ + at − θ1at −1 , at ∼ WN ( 0, σ 2
ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ⋯ zn +1 , zn+2 , zn +3 ,ﺃﻭ ﺒﺸﻜل ﻋﺎﻡ . zn +ℓ , ℓ ≥ 1
ﻤﻥ ﻨﻅﺭﻴﺔ 3ﻨﺠﺩ
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=µ + E ( an + ℓ zn , zn −1 ,⋯) − θ1 E ( an + ℓ−1 zn , zn −1 ,⋯) , ℓ ≥ 1
ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ 2
78
zn ( ℓ ) =µ + E ( an + ℓ zn , zn −1 ,⋯) − θ1E ( an + ℓ−1 zn , zn −1 ,⋯) , ℓ ≥ 1
ℓ = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,⋯) − θ1 E ( an zn , zn −1 ,⋯)
= µ − θ1an
ℓ = 2 : zn ( 2 ) =µ + E ( an + 2 zn , zn −1 ,⋯) − θ1 E ( an +1 zn , zn −1 ,⋯)
=µ
ℓ = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,⋯) − θ1 E ( an + 2 zn , zn −1 ,⋯)
=µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( ℓ ) = µ , ℓ ≥ 2
ﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔMA(1) ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ
µ − θ1an , ℓ = 1
zn ( ℓ ) =
ℓ≥2
µ,
: MA(2) ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ: ﺨﺎﻤﺴﺎ
ﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎn { ﺤﺘﻰ ﺍﻝﺯﻤﻥz1 , z2 ,⋯ , zn−1 , zn } ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ
ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜلMA(2) ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ
zt = µ + at − θ1at −1 − θ 2 at −2 , at ∼ WN ( 0,σ 2 )
. zn +ℓ , ℓ ≥ 1 ﺃﻭ ﺒﺸﻜل ﻋﺎﻡzn +1 , zn+2 , zn +3 ,⋯ ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ
ﻨﺠﺩ3 ﻤﻥ ﻨﻅﺭﻴﺔ
zn ( ℓ ) = E ( zn +ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=µ + E ( an + ℓ zn , zn −1 ,⋯) − θ1 E ( an +ℓ −1 zn , zn −1 ,⋯) − θ 2 E ( an +ℓ −2 zn , zn −1 ,⋯) , ℓ ≥ 1
2 ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ
zn ( ℓ ) =µ + E ( an + ℓ zn , zn −1 ,⋯) − θ1 E ( an + ℓ−1 zn , zn −1 ,⋯) − θ 2 E ( an + ℓ−2 zn , zn −1 ,⋯) , ℓ ≥ 1
ℓ = 1: zn (1) =µ + E ( an +1 zn , zn −1 ,⋯) − θ1 E ( an zn , zn −1 ,⋯) − θ 2 E ( an −1 zn , zn −1 ,⋯)
= µ − θ1an − θ 2 an −1
ℓ = 2 : zn ( 2 ) =µ + E ( an +2 zn , zn −1 ,⋯) − θ1E ( an +1 zn , zn −1 ,⋯) − θ 2 E ( an zn , zn −1 ,⋯)
= µ − θ 2 an
ℓ = 3 : zn ( 3) =µ + E ( an +3 zn , zn −1 ,⋯) − θ1E ( an + 2 zn , zn −1 ,⋯) − θ 2 E ( an +1 zn , zn −1 ,⋯)
=µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
79
zn ( ℓ ) = µ , ℓ ≥ 3
ﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔMA(2) ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ
µ − θ1an − θ 2 an −1 , ℓ = 1
z n ( ℓ ) = µ − θ 2 an ,
ℓ=2
µ,
ℓ≥3
: ARMA(1,1) ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﻝﻨﻤﻭﺫﺝ: ﺴﺎﺩﺴﺎ
ﻭﺍﻝﺘﻲ ﻨﻌﺘﻘﺩ ﺍﻨﻬﺎn { ﺤﺘﻰ ﺍﻝﺯﻤﻥz1 , z2 ,⋯ , zn−1 , zn } ﻝﻨﻔﺘﺭﺽ ﺍﻨﻨﺎ ﺸﺎﻫﺩﻨﺎ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ
ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜلARMA(1,1) ﺘﺘﺒﻊ ﻨﻤﻭﺫﺝ
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at ∼ WN ( 0,σ 2 ) , φ1 ≠ θ1 , φ1 < 1
. zn +ℓ , ℓ ≥ 1 ﺃﻭ ﺒﺸﻜل ﻋﺎﻡzn +1 , zn+2 , zn +3 ,⋯ ﻨﺭﻴﺩ ﺃﻥ ﻨﺘﻨﺒﺄ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ
ﻨﺠﺩ3 ﻤﻥ ﻨﻅﺭﻴﺔ
zn ( ℓ ) = E ( zn +ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
=µ + φ1 E ( zn + ℓ−1 − µ ) zn , zn −1 ,⋯ + E ( an +ℓ zn , zn −1 ,⋯) − θ1E ( an + ℓ−1 zn , zn −1 ,⋯) , ℓ ≥ 1
2 ﻨﺤل ﻫﺫﻩ ﺍﻝﻌﻼﻗﺔ ﺘﻜﺭﺍﺭﻴﺎ ﻭﺒﺈﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺎﻋﺩﺓ
zn ( ℓ ) =µ + φ1 E ( zn + ℓ−1 − µ ) zn , zn −1 ,⋯ + E ( an + ℓ zn , zn −1 ,⋯) − θ1 E ( an +ℓ −1 zn , zn −1 ,⋯) , ℓ ≥ 1
ℓ = 1: zn (1) =µ + φ1 E ( zn − µ ) zn , zn −1 ,⋯ + E ( an +1 zn , zn −1 ,⋯) − θ1E ( an zn , zn −1 ,⋯)
= µ + φ1 ( zn − µ ) − θ1an
ℓ = 2 : zn ( 2 ) =µ + φ1E ( zn +1 − µ ) zn , zn −1 ,⋯ + E ( an + 2 zn , zn −1 ,⋯) − θ1 E ( an +1 zn , zn −1 ,⋯)
= µ + φ1 zn (1) − µ
ℓ = 3 : zn ( 3) =µ + φ1 E ( zn + 2 − µ ) zn , zn −1 ,⋯ + E ( an +3 zn , zn −1 ,⋯) − θ1E ( an + 2 zn , zn −1 ,⋯)
= µ + φ1 zn ( 2 ) − µ
ﻭﻫﻜﺫﺍ ﺒﺸﻜل ﻋﺎﻡ
zn ( ℓ ) = µ + φ1 zn ( ℓ − 1) − µ , ℓ ≥ 2
ﺘﻌﻁﻰARMA(1,1) ﻭﻫﻜﺫﺍ ﻓﺈﻥ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺫﺍﺕ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﻷﺨﻁﺎﺀ ﺍﻷﺩﻨﻰ ﻝﻨﻤﻭﺫﺝ
ﺒﺎﻝﻌﻼﻗﺔ
µ + φ1 ( zn − µ ) − θ1an ,
zn ( ℓ ) =
µ + φ1 zn ( ℓ − 1) − µ ,
ℓ =1
ℓ≥2
80
ﺘﻤﺭﻴﻥ:
ﻝﻨﻤﻭﺫﺝ ) ARMA(1,1ﻭﺍﻝﺫﻱ ﻴﻜﺘﺏ ﻋﻠﻰ ﺍﻝﺸﻜل
zt = µ + φ1 ( zt −1 − µ ) + at − θ1at −1 , at ∼ WN ( 0,σ 2 ) , φ1 ≠ θ1 , φ1 < 1
ﺒﺭﻫﻥ ﺍﻥ ﻋﻨﺩﻤﺎ ﺘﺅﻭل φ1 → 1ﻓﺈﻥ ) ARMA(1,1) → IMA(1,1ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ
ﻝﻨﻤﻭﺫﺝ ). IMA(1,1
ﺘﻤﺭﻴﻥ:
ﺃﻭﺠﺩ ﺩﻭﺍل ﺍﻝﺘﻨﺒﺅ ﻭﺘﺒﺎﻴﻥ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﻝﻜل ﻤﻥ ﺍﻝﻨﻤﺎﺫﺝ ﺍﻝﺘﺎﻝﻴﺔ:
ARIMA(1,2,0),
ARIMA(2,1,0),
ARIMA(0,1,2),
ARIMA(1,1,1),
ARIMA(0,2,1), ARIMA(0,2,0).
ﺤﺩﻭﺩ ﺍﻝﺘﻨﺒﺅ : Forecasting Limits
ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ zn ( ℓ ) , ℓ ≥ 1ﻋﻨﺩ ﻗﻴﻤﺔ ﻤﻌﻴﻨﺔ ﺘﻌﻁﻲ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻝﻨﻘﻁﺔ Point Forecastﻭﺍﻝﺫﻱ
ﻻﻴﻜﻔﻲ ﺍﻭ ﻴﻔﻴﺩ ﻓﻲ ﺇﺘﺨﺎﺫ ﻗﺭﺍﺭﺍﺕ ﺇﺤﺼﺎﺌﻴﺔ ﻋﻥ ﺍﻝﻅﺎﻫﺭﺓ ﺍﻝﻌﺸﻭﺍﺌﻴﺔ ﺍﻝﻤﺩﺭﻭﺴﺔ ﻷﻥ
P ( Z n + m = zn ( m ) ) = 0, for some m > 0
ﺃﻱ ﺃﻥ ﻤﻘﺩﺍﺭ ﺘﺄﻜﺩﻨﺎ ) ﺃﻭ ﺇﺤﺘﻤﺎل( ﻤﻥ ﺃﻥ ﺍﻝﻘﻴﻤﺔ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻝﻤﺭﺍﺩ ﺍﻝﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﺴﺎﻭﻱ ﺍﻝﻘﻴﻤﺔ
ﺍﻝﻤﻌﻁﺎﺓ ﻤﻥ ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺘﺴﺎﻭﻱ ﺍﻝﺼﻔﺭ ﺃﻱ ﺍﻨﻨﺎ ﻏﻴﺭ ﻤﺘﺄﻜﺩﻴﻥ ﺇﻁﻼﻗﺎ ﻭﺒﺎﻝﺘﺎﻝﻲ ﻻﻓﺎﺌﺩﺓ ﻤﻥ ﺍﻝﺘﻨﺒﺅ.
ﻝﻠﺘﻐﻠﺏ ﻋﻠﻰ ﺫﻝﻙ ﻭﺃﻹﺴﺘﻔﺎﺩﺓ ﻤﻥ ﺍﻝﺘﻨﺒﺅﺍﺕ ﻨﺴﺘﺨﺩﻡ ﻤﺎﻴﺴﻤﻰ ﺒﺘﻨﺒﺅ ﺍﻝﻔﺘﺭﺓ Interval Forecast
ﻭﻫﻲ ﻋﺒﺎﺭﺓ ﻋﻥ ﻓﺘﺭﺓ ﻤﺜل ] [a, bﻋﻠﻰ ﺨﻁ ﺍﻷﻋﺩﺍﺩ ﺍﻝﺤﻘﻴﻘﻴﺔ ﺒﺤﻴﺙ ﻴﻜﻭﻥ
) P ( a ≤ Z n + m ≤ b ) = (1 − α
ﻭﺒﻬﺫﺍ ﻨﺴﺘﻁﻴﻊ ﺃﻥ ﻨﺤﺩﺩ ﺩﺭﺠﺔ ﺘﺄﻜﺩﻨﺎ ﻤﻥ ﺃﻥ ﺍﻝﻘﻴﻤﺔ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ﺍﻝﻤﺭﺍﺩ ﺍﻝﺘﻨﺒﺅ ﻋﻨﻬﺎ ﺘﻘﻊ ﺒﻴﻥ ﺍﻝﻘﻴﻡ a
ﻭ bﺒﺩﺭﺠﺔ ﺘﺄﻜﺩ ﺃﻭ ﺇﺤﺘﻤﺎل ) 1 − αﺃﻭ ( 100 × (1 − α ) %ﻓﻤﺜﻼ ﻝﻭ ﻜﺎﻨﺕ α = 0.05ﻓﺈﻨﻨﺎ
ﻨﻜﻭﻥ ﻤﺘﺄﻜﺩﻴﻥ ﻭﺒﺈﺤﺘﻤﺎل 95%ﺍﻥ ﺍﻝﻘﻴﻤﺔ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ ﺘﻘﻊ ﺒﻴﻥ ﺍﻝﻘﻴﻡ aﻭ . b
ﺘﻌﺭﻴﻑ :21
ﻋﻠﻰ ﺇﻓﺘﺭﺍﺽ ﺃﻥ ) at ∼ N ( 0, σ 2ﻓﺈﻥ ﺤﺩﻭﺩ 100 × (1 − α ) %ﻓﺘﺭﺓ ﺘﻨﺒﺅ ﻝﻠﻘﻴﻤﺔ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ
zn +ℓ , ℓ ≥ 1ﺘﻌﻁﻰ ﺒﺎﻝﻌﻼﻗﺔ
81
}
12
{
zn ( ℓ ) ± uα 2 V en ( ℓ )
α
ﺤﻴﺙ uα 2ﺍﻝﻤﺌﻴﻥ 100 1 − ﻝﻠﺘﻭﺯﻴﻊ ). N ( 0,1
2
ﻓﻤﺜﻼ ﻋﻨﺩﻤﺎ α = 0.05ﻓﺈﻥ . u0.025 = 1.96
ﻤﻼﺤﻅﺔ :ﻓﻲ ﺍﻝﺘﻌﺭﻴﻑ ﺇﻓﺘﺭﻀﻨﺎ ﺃﻥ ﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﻀﺠﺔ ﺍﻝﺒﻴﻀﺎﺀ ) at ∼ N ( 0, σ 2ﻭﻫﺫﺍ ﻤﻤﻜﻥ
ﺇﻋﺘﻤﺎﺩﺍ ﻋﻠﻰ ﻨﻅﺭﻴﺔ ﻨﻬﺎﻴﺔ ﻤﺭﻜﺯﻴﺔ.
ﻤﺜﺎل :ﻤﺘﺴﻠﺴﻠﺔ ﺯﻤﻨﻴﺔ ﻤﺸﺎﻫﺩﻩ ﻭﺠﺩ ﺍﻨﻬﺎ ﺘﺘﺒﻊ ﺍﻝﻨﻤﻭﺫﺝ
) zt − 0.97 = 0.85 ( zt −1 − 0.97 ) + at , at ∼ N ( 0,0.024
ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻝﻤﺸﺎﻫﺩﺓ ﺍﻷﺨﻴﺭﺓ ﻫﻲ ، z156 = 0.49ﺃﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159
ﻭﺃﻭﺠﺩ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﻝﻬﺎ ﻭﻤﻥ ﺜﻡ ﺃﻭﺠﺩ ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ.
ﺍﻝﺤل :ﺴﺒﻕ ﺃﻥ ﺤﺴﺒﻨﺎ ﻓﻲ ﻤﺜﺎل ﺴﺎﺒﻕ ﺍﻝﺘﻨﺒﺅﺍﺕ ﻭ ﺃﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﻜﺎﻝﺘﺎﻝﻲ:
ﺍﻝﺘﻨﺒﺅﺍﺕ
z156 (1) = 0.56, z156 ( 2 ) = 0.62, z156 ( 3) = 0.68
ﻭﺍﻝﺘﺒﺎﻴﻨﺎﺕ
V e156 (1) = 0.024, V e156 ( 2 ) = 0.041, V e156 ( 2 ) = 0.054
ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ z157 , z158 , z159ﻨﻭﺠﺩﻫﺎ ﻤﻥ ﺼﻴﻐﺔ ﺘﻌﺭﻴﻑ 21
}
12
= 0.56 ± 1.96 0.024 = 0.56 ± 0.304
}
12
{
zn ( ℓ ) ± uα 2 V en ( ℓ )
{
{V e
{V e
1 − z156 (1) ± u0.025 V e156 (1)
= 0.62 ± 1.96 0.041 = 0.62 ± 0.397
}( 2 )
156
2 − z156 ( 2 ) ± u0.025
= 0.68 ± 1.96 0.054 = 0.68 ± 0.455
}( 3)
156
3 − z156 ( 3) ± u0.025
12
12
ﺃﻱ ﺃﻥ ) z157 ∈ ( 0.256,0.864ﺒﺈﺤﺘﻤﺎل 0.95ﻭ ﻭﻜﺫﻝﻙ ) z158 ∈ ( 0.223,1.017ﻭ ﻜﺫﻝﻙ ﺃﻴﻀﺎ
). z159 ∈ ( 0.225,1.135
82
ﺍﻟﻔﺼﻞ ﺍﻟﺨﺎﻣﺲ
"Mو$ء !7م 4إDesigning and Building Statistical MK
: Forecasting System
( أن ذآ ان ا
[16ة او
;!) Iم :('Cه'. 7ء !1ذج .إن !'. 4ء !1ذج
إ 785ه3C 4! 7ار13;C Iterative 42ن & 2%Cا
'!1ذج 2JC ،ا
'!1ذج )وO. J
)
"& 2JCا
'!1ذج( و إ;(ر ا
'!1ذج.
#أو /اذج ): Model Identification (Specification
2%C 45& 7Nا
'!1ذج ;[م ا
(ت أو ا
!Kهات ا
) 4J.ا
;ر (r2واي &"&1ت اى
ا
4b3ا
;
1C 7ت O.ا
!; 4وذ
;B9 fاح &! & 41ا
'!ذج ا
!'( .4و "C );2أو
2%Cا
'!1ذج W5ا
[16ات ا
" 4`2ا
;
:4
ا;:ة اrو' S- ./ :ا: Variance-stabilizing Transformation #
".ر) ا
!; w6[& 7N 4ز&' Time Plot 7وإLاء \".ا(;9رات ا4N"!
4859
!Nإذا آن ا
;( ،S.U 2و 4
5 7Nم (Uت ا
;( 2او إذا آن ا
;(1;& V& Y;2 2ى
ا
!; (6 'FN 4ا
; H21%ا
z1ر 7!Cا
!; 4وFN 2L & O%bذا S(]C )C
ا
;( 2وإ { Xإ
(6C 7أ 5ا
;21%ت ا
; 7ذآه L 7Nول .41 4%b+
ا;:ة ا :7-إ?ر در nا: d =Y
إذا آ Sا
!; 4أو J;& z O21%Cة 7Nا
!; 2%C ' WN w1در 4Lا
; d 2bا
;7
H"Cا
!!; 4أو J;& O21%Cة 7Nا
!; w1و1Jم A{.ا
; 2bاول o%b )Uا
;
:7
-1ا
![66ت ا
&' 4;!!
4أو .O21%C
66[& -2ت دا
; 7ا
;ا w.ا
Aا 7Cا
"' 7وا
;ا w.ا
Aا 7Cا
78ا
"' SACF 7و . SPACF
-3إLاء , 2bCإذا ا;5ج ا& وإدة ا
[16ات 1و 2ا
. ;J.
ا
![66ت ا
&';!
4ت zا
!;Jة 7N YC (Cا
!;1ى ودا
C 4ا w.ذا7' 7C
&;[&ة w(.ء آ! ان دا
4ا
;ا w.ا
Aا 7Cا
78ا
"' 4!B 76"C 7وا5ة & 4(2Bا
1ا5
ا
\Y.) ^%ا
' Iا9رة( و 4J.ا
L 4(2B )Jا & ا
.b
& :4I5در 4Lا
;13C & (
z d 2bن 0او 1او . 2
83
: q وp / :--;ة ا:ا
ار%9 ا4L در2%;. م1J w1;!
وا2(;
& اH آ7N ةJ;& 4;& H% " ان.
w. وا
;ا7'"
ا7CاA
اw. ا
;ا7;
أ!ط دا4رJ!. f
وذq ك%;!
اw1;!
ا4L ودرp 7CاA
ا
78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
ا42I'
ا!ط اV& 7'"
ا78
ا7CاA
ا
@A ه76"2 7
;
وا
ول ا38 4%b+ 7N رة1آA!
اARMA(p,q) اص !ذج1[. 2;&
:4"8K
اص ("\ ا
'!ذج ا1[
ا
PACF
ذج1!'
ا
ACF
φkk = 0, k > 1
&;دد7 أوا7[& اC
AR(1)( و1,d,0)
φkk = 0, k > 2
7(L &[C او7[& اC
AR(2)( و2,d,0)
φkk = 0, k > p
7(L &[C او/ و7[& اC
AR(p)( وp,d,0)
7[& اC O 62
ρ k = 0, k > 1
MA(1) ( و0,d,1)
7(L او7[& اC O 62
ρ k = 0, k > 2
MA(2) ( و0,d,2)
7(L او/ و7[& اC O 62
ρ k = 0, k > q
MA(q) ( و0,d,q)
7[& اC O 62 وoB';C
;[& اC وoB';C
1 e[;
& ا
ARMA(1,1) ( و1,d,1)
1 e[;
& ا
e[;
" ا. oB';C
&[;C وq - p e[;
" ا. oB';C
62 وp – q
q – p e[;
" ا. (L او/ ا و
ARMA(p,q) ( وp,d,q)
7[& اC O
". 7(L او/و
p – q e[;
ا
:افV7 " إaA إ:$;ة اا:ا
م1"& إاف4NP! إذا آن ' إN ' ا
;{آWN 2bC ;ج إ%C 4;!
اSإذا آ
{6[
اV& ةJ;!
ا4Bb!
ا4;!
w 4'"
اw1;& 4رJ!. );2 اAذج وه1!'
إ
اδ
w1;!
ا اAO
ا
!"ري
12
c
s.e ( w ) ≅ 0 (1 + 2r1 + 2 r2 + ⋯ + 2 rK )
n
84
;(ر9ن ا132 و. K 4L ر421'"!
ا4'"
ا4CاA
ت ا6. ا
;ا7 هr1 ,⋯ , rK وc0 = γˆ0 Q5
H0 : δ = 0
H1 : δ ± 0
.
w
> 1.96 S إذاآα = 0.05 ' H 0 \Nو
s.e ( w )
85
: Model Estimation اذج
f
وذσ 2 وθ1 ,… ,θ q وφ1 ,… ,φ p وδ ذج1!'
&"
) ا2JC & .X ذج1!'
اH3 2%C ".
.'2
ةN1;!
ا4[2;[ام ا
(ت ا
;رF.
;حJ!
ذج ا1!'
واz1 , z2 ,… , zn −1 , zn هةK!
ا4'&
ا4;!
' ا2
;ض انb'
φ p ( B ) wt = δ + θ q ( B ) at , at ∼ N ( 0, σ 2 )
أو
φ p ( B ) zt = δ + θ q ( B ) at , at ∼ N ( 0, σ 2 )
O"!L VJC φ p ( B ) = 0 4
ور ا
!"دAL و4;آK& ورAL O'. L12X θ q ( B ) وφ p ( B ) Q5
.(ارJ;9ة ) ط ا51
ة ا8رج دا
رJ!
ا اAق ه6 !P HC ;J2= wJN ' هO'& آA' )
"!
ا2J;
ه'ك =ق آ]ة
.4=K
"ت ا
ا.!
ا4J2= ا
"وم و4J2= !وه
: The Method of Moments اومi :rأو
"وم. rk 4'"
4CاA
ت ا6. وا
;اz 4'"
اw1;& H]& 4'"
";! &وات وم اCو
!"
) ا
!اد4('
. 4C'
ت اX ا
!"دH5 وρ k 7CاA
اw. ا
;ا4
وداµ w1;!
اH]& 42I'
ا
.ه2JC
:7
;
آAR(p) ذج1!'
4J26
ف ;"ض ا1
µˆ = z = ∑ i =1 zi n ايz رJ!
. µ w1;!
ر اJ2 -1
n
:4B"
;[م اφ1 ,…, φ p 2J;
-2
ρ k = φ1 ρ k −1 + φ2 ρ k −2 + ⋯ + φ p ρ k − p , k > 1
7N .VB1;
اA وأzt −k − µ %
. AR(p) ذج1!'
4N"!
ا4
ب ا
!"دP & _;'C 7;
وا
ل و12 تXت ا
!! &"دXم ا
!"دI H% k = 1,2,…, p VP1. 4J.
ا4
ا
!"د
:7
;
اYule-Walker ووآ
ρ1 = φ1 + φ2 ρ1 + ⋯ + φ p ρ p −1
ρ 2 = φ1 ρ1 + φ2 + ⋯ + φ p ρ p −2
⋮
ρ p = φ1 ρ p −1 + φ2 ρ p −2 + ⋯ + φ p
:7
;
آφˆ1 ,… ,φˆp )
"!
رات ا
"ومJ& H% rk رJ!
. ρ k \21";
. و
86
:7N1b!
اH3K
ل و ووآ ا12 تX &"دVP1.
r1
r1 1
r r
1
2= 1
⋮ ⋮
⋮
rp rp −1 rp −2
r2
⋯ rp −2
r1
⋯ rp −3
⋮
⋮
⋮
rp −3 ⋯
r1
rp −1 φˆ1
rp −2 φˆ2
⋮ ⋮
1 φˆ
p
)
"!
4
@ ا
!"دA هH%.و
φˆ1 1
r1
1
φˆ2 r1
⋮ = ⋮
⋮
φˆ rp −1 rp −2
p
r2
⋯ rp −2
r1
⋯ rp −3
⋮
⋮
⋮
rp −3 ⋯
r1
rp −1
rp −2
⋮
1
−1
r1
r
2
⋮
rp
7
;
آσ 2 رJC
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2 −⋯φˆp rp
)
Q5
γˆ0 =
1 n
2
( zt − z )
∑
n t =1
.4'"
ا2(C 1ه
: اذجv اوم
AR(1) ذج7 -1
zt − µ = φ1 ( zt −1 − µ ) + at , at ∼ N ( 0,σ 2 )
1 هφ1 )"!
ر ا
"ومJ&
φˆ1 = r1
1 هµ )"!
ر ا
"ومJ&
µˆ = z
1 هσ 2 )"!
ر ا
"ومJ&
(
σˆ 2 = γˆ0 1 − φˆ1r1
)
Q5
γˆ0 =
1 n
2
( zt − z )
∑
n t =1
87
MA(1) ذج7 -2
zt − µ = at − θ1at −1 , at ∼ N ( 0,σ 2 )
4B"
;[م اθ1 )"!
ر ا
"ومJ& د29
ρ1 =
−θ1
1 + θ12
OCراJ!. )
"!
\ ا21";.و
r1 =
−θˆ1
1 + θˆ12
θˆ1 رJ!
4
ا
!"دH%.و
−1 ± 1 − 4 r1
θˆ1 =
2 r1
نFN r1 = −0.4 S آ1
]!N . θˆ1 < 1 J%C 7;
ا4!J
اA{ θˆ1 رJ!
;!B 76"2 H%
ا اAه
. θˆ1 = −0.77 1 هθ1 )"!
ر ا
"ومJ& ن132 7
;
.( وθˆ1 ) = 3.27 ( وθˆ1 ) = −0.77
2
1
AR(2) ذج7 -3
zt − µ = φ1 ( zt −1 − µ ) + φ2 ( zt −2 − µ ) + at , at ∼ N ( 0,σ 2 )
7 هφ2 وφ1 )
"!
رات ا
"ومJ& ل ووآ12 تX;[ام &"دF.
φˆ1 1 r1 −1 r1
=
φˆ r1 1 r2
2
O'&و
φˆ1 =
2
r1 − r1r2
ˆ = r2 − r1
φ
,
2
1 − r12
1 − r12
1 هµ )"!
ر ا
"ومJ&
µˆ = z
1 هσ 2 )"!
ر ا
"ومJ&
(
σˆ 2 = γˆ0 1 − φˆ1r1 − φˆ2 r2
)
Q5
γˆ0 =
1 n
2
( zt − z )
∑
n t =1
88
MA(2) ذج7 -4
zt − µ = at − θ1at −1 − θ 2 at −2 , at ∼ N ( 0, σ 2 )
تB"
;[م اθ 2 وθ1 )
"!
رات ا
"ومJ& د29
ρ1 =
−θ1 (1 − θ 2 )
−θ 2
, ρ2 =
2
2
1 + θ1 + θ 2
1 + θ12 + θ 22
θ 2 وθ1 )
"!
رات ا
"ومJ& H% r2 وr1 راتJ!
\ ا21";.و
(
)
−θˆ1 1 − θˆ2
−θˆ2
r1 =
,
r
=
2
1 + θˆ12 + θˆ22
1 + θˆ12 + θˆ22
. θ 2 − θ1 < 1, θ 2 + θ1 < 1, θ 2 < 1 J%C 7;
ل ا1%
اA{ وθˆ2 وθˆ1 & H3
H%و
ARMA(1,1) ذج7 -5
zt − µ = φ1 ( zt −1 − µ ) + at − θ1at −1 , at ∼ N ( 0,σ 2 )
تB"
;[م اφ1 وθ1 )
"!
رات ا
"ومJ& د29
ρ1 =
(1 − φ1θ1 )(φ1 − θ1 ) , ρ = (1 − φ1θ1 )(φ1 − θ1 ) φ
2
1
2
2
1 + θ1 − 2φ1θ1
1 + θ1 − 2φ1θ1
φ1 وθ1 )
"!
رات ا
"ومJ& H% r2 وr1 راتJ!
\ ا21";.و
r1 =
(1 − φˆθˆ )(φˆ − θˆ ) ,
1 1
1
1
r2 =
1 + θˆ − 2φˆ1θˆ1
2
1
(1 − φˆθˆ )(φˆ − θˆ ) φˆ
1 1
1
1
1 + θˆ − 2φˆ1θˆ1
2
1
1
r1 رJ!
4N"!
ا4
ا
!"دr2 رJ!
4N"!
ا4
ا
!"د4!J.و
φˆ1 =
r2
r1
4N"!
ا4
ا
!"د7N φˆ1 ض1" θ1 )"!
ر ا
"ومJ& د29 . φ1 )"!
ر ا
"ومJ& 1وه
r1 رJ!
r2 ˆ r2 ˆ
1 − r θ1 r − θ1
1
1
r1 =
r
1 + θˆ12 − 2 2 θˆ1
r1
89
. θˆ1 < 1 J%C 7;
ا4!J
اA{ وθˆ1 رJ!
4C'
ا4".;
ا4
ا
!"دH%و
4
;
رات ا
"وم !"
) ا
'!ذج اJ& L أو:#ر
ARIMA(1,1,1),
ARIMA(2,1,0),
ARIMA(0,1,2),
ARIMA(1,2,0),
ARIMA(0,2,1), ARIMA(0,2,0).
.4Bرات أآ] دJ& د29 4
) أوJ;[م آC رات ا
"ومJ& :!K,
: Conditional Least Square Method i ا7ت ا$ اi :7]
H3K
اW;3C 7;
واARMA(p,q)
'!ذج
φ p ( B )( zt − µ ) = θ q ( B ) at , at ∼ N ( 0,σ 2 )
O"!L VJC θ q ( B ) = 0 4
ور ا
!"دAL و4;آK& ورAL O'. L12X θ q ( B ) وφ p ( B ) Q5
:7
;
آat ء6
.
ذج ا1!'
ا4.;دة آF. .(بJ9ة ) ط ا51
ة ا8رج دا
at =
φp (B)
( zt − µ )
θq ( B)
µ وθ = {θ1 ,θ 2 ,… ,θ q } وφ = {φ1 ,φ2 ,… ,φ p } )
"!
ا7N 4
إ;(رة آا3!2 !2ف ا6
ا
W;32 و
(1 − φ B − φ B
a ( φ, θ, µ ) =
(1 − θ B − θ B
1
2
1
2
t
2
2
−⋯ − φpB p )
−⋯ −θ pB p )
( zt − µ )
YC z = {z1 , z2 ,… , zn } ة6"& هاتK!
و4=K
"ت ا
ا.!
ا4J2= !;"C
4
ا
ا
min Sc ( φ, θ, µ ) =
φ ,θ , µ
n
∑ a ( φ, θ, µ z )
t = p +1
2
t
.راتJ!
4('
. 4
;
ا4C'
اNormal Equations 4"(6
ت اX ا
!"دH5و
90
∂
∂ n 2
Sc ( φ, θ, µ )
=
∑ at ( φ, θ, µ z ) φ=φˆ = 0
φ =φˆ
∂φ
∂φ t = p +1
ˆ
ˆ
θ=θ
µ = µˆ
θ= θ
µ = µˆ
∂
∂ n 2
Sc ( φ, θ, µ )
=
∑ at ( φ, θ, µ z ) φ=φˆ = 0
φ =φˆ
∂θ
∂θ t = p +1
θ =θˆ
θ =θˆ
µ = µˆ
µ = µˆ
∂
∂ n 2
Sc ( φ, θ, µ )
=
∑ at ( φ, θ, µ z ) φ=φˆ = 0
φ= φˆ
µ
∂µ
∂
t = p +1
ˆ
θ =θ
θ = θˆ
µ = µˆ
µ = µˆ
42 أي &وa p = a p −1 = ⋯ = a p +1−q = 0 )J
;ط ان اK ' ' ه4= !C راتJ!
@ اAه
.( t = p + 1 4!J
(أ & ا2 4J.
ت اX ا
!"د7N V!;
أن ا5X ) .O"B1;
& σ 2 2(;
ر اJ2
σˆ 2 =
(
Sc φˆ , θˆ , µ
)
n − ( p + q + 1)
: اذجv i ا7ت ا$ات ا
AR(1) ذج7 -1
zt − µ = φ1 ( zt −1 − µ ) + at , at ∼ N ( 0,σ 2 )
z رهJ!. µ ف ;(ل1 تBJ;9 اw(;
zt − z = φ1 ( zt −1 − z ) + at , at ∼ N ( 0,σ 2 )
ء6 اW;3 z = {z1 , z2 ,… , zn } ة6"& هاتK!
at (φ1 ) = ( zt − z ) − φ1 ( zt −1 − z ) , t = 2,3,⋯ , n
هاتK!
اH آV!
واN6
اV.Cو
at2 (φ1 ) = ( zt − z ) − φ1 ( zt −1 − z ) , t = 2,3,⋯ , n
2
n
n
t =2
t =2
Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ( zt − z ) − φ1 ( zt −1 − z )
2
b
42 &و4;'
ن ا13C وφ1 )"!
4('
. 4J.
ا4
; ا
!"دK ، wJN φ1 )"!
4
@ داAوه
أيφ1 = φˆ1 &'
91
n
n
Sc (φ1 ) = ∑ at2 (φ1 ) = ∑ ( zt − z ) − φ1 ( zt −1 − z )
t =2
2
t =2
2
∂
∂
Sc (φ1 ) =
( zt − z ) − φ1 ( zt −1 − z )
∑
∂φ1
∂φ1 t =2
n
n
= ∑ −2 ( zt −1 − z ) ( zt − z ) − φ1 ( zt −1 − z )
t =2
n
∂
Sc (φ1 ) ˆ = ∑ −2 ( zt −1 − z ) ( zt − z ) − φˆ1 ( zt −1 − z ) = 0
φ1 =φ1
∂φ1
t =2
n
∴ ∑ ( zt −1 − z ) ( zt − z ) − φˆ1 ( zt −1 − z ) = 0
t =2
n
∑(z
t =2
n
t −1
− z )( zt − z ) − φˆ1 ∑ ( zt −1 − z ) = 0
2
t =2
أي
n
φˆ1 =
∑(z
t =2
t −1
− z )( zt − z )
n
∑(z
t =2
t −1
−z)
2
. φ1 )"!
4=K
"ت ا
ا.!
ر اJ& 1وه
. φ1 )"!
ر ا
"ومJ&ر وJ!
ا اA ه. رنB : 2!C
MA(1) ذج7 -2
zt − µ = at − θ1at −1 , at ∼ N ( 0, σ 2 )
w1;!
4
"!
ا4;!
اH!"و
z رهJ!. µ ف ;(ل1 تBJ;9 اw(;
ذج1!'
(^ اN xt = zt − z
xt = at − θ1at −1 , at ∼ N ( 0, σ 2 )
H3K
اة ا4
ا
!"د4.;3.و
at = xt − θ1at −1
ء6 اW;3 = a0 = 0 VP1. وx1 , x2 ,… , xn ة6"& هاتK!
و
a1 = x1
a2 = x2 − θ1a1
a3 = x3 − θ1a2
⋮
an = xn − θ1an−1
92
7
;
.و
n
Sc (θ1 ) = ∑ at2
t =1
Q%(
ق ا6. Sc (θ1 ) YC 7;
واθ1 4!B د2 إ3!2 وθ1 )"!
ا7N 46 z 4J.
ا4
ا
ا
7N o[;C 7;
واC1-وسL 4J2= ( أو إ;[ام-1,1) ا
!ل7N 73(K
اQ%(
اH]& 42ا
"د
&] أيθ * 4
أو4!B ل15 θ1 )"!
46 4
ا. at = at (θ1 ) W2JC
at (θ1 ) ≈ at (θ
*
) + (θ
1
−θ
*
)
dat (θ * )
dθ1
4('
. at = xt − θ1at −1 4
ا
!"د7N= قJ;F. f
وذ2ار3C O.5 3!2
dat (θ * )
dθ1
4J;K!
ا
H%'
θ1 )"!
dat (θ1 ) θ1dat −1 (θ1 )
=
+ at −1 (θ1 )
d θ1
dθ1
4
ا
!"د.
at (θ1 ) ≈ at (θ * ) + (θ1 − θ * )
da0 (θ1 )
= 0 4
أو4!J. و
dθ1
dat (θ * )
dθ1
"ت.!
ع ا1!& YC ن3&X. 7
;
. وθ1 )"!
ا7N 46
n
Sc (θ1 ) = ∑ at2
t =1
رJ!
. θ * ;(الF. 4!"
@ اAر ه3 وθ1 )"!
H`N وأ2L رJ& 7 H%'
%C
"ت.!
ع ا1!& 7N oJ'
ا أو اL Y+ C 2رJ& . قb
(^ ا2 ;5 !; و2
ا
ربJC H% 73
θ * 4
أو4!J
د ا29 ا
"وم4J2= إ;[ام3!& .اL Y+
.f
A
W5 ;ج إ%C H. 2و2 );CX 4J.
ا4J26
=(" ا.V2
7;
ا46;[!
ك او ا
'!ذج ا%;!
اw1;!
!ذج ا4
5 7N ذج1!'
)
"!
ا2JC أن52
76 z H3K. ك%;!
اw1;!
ى &"
) ا1%C O اJ"C H3KC ك%;& w1;& ي1%C
."!L O6. أ1 وهMA(1) ذج1!'
ا4
5 7N آ! هO%
42;ج إ
=ق د%C اAO
و
93
4&[;!
ى اXق ا6
"\ ا. آA 3
و4J.
; اJ26
ا اAر هJ& 7N 7b;3 ف1
:H]& ذج1!'
&"
) ا2JC 7N
Maximum Likelihood Method !I"
ا4%L ار4J2= -1
Unconditional Least Squares Method 4=K
اz "ت ا.!
ا4J2= -2
Nonlinear Estimation Methods 46[
اz2J;
=ق ا-3
94
>:وإ?ر اذج : Model Checking and Diagnostics
".ا
;"ف !1ذج &( 78و )
"& 2JCهAا ا
'!1ذج ي \".ا
;[Kت ا
(1ا7B
أو أ6ء ا
;) (6ا'
(4 e2"C Iى &ى & 4J.6ا
'!1ذج !; 4ا
!Kهة ،و;b2ض
أن ا
(1ا 7BهJ& 7رات !; 4ا
` 4ا
(`ء atوا
;;b 7ض ا1& Oزw1;!. "(= 4
b+ي و . σ 2 2(Cا
(1ا4B"
. 6"C 7B
et = zt − zˆt = aˆt , t = 1, 2,..., n
أي ان ا
(1ا 7Bه 7ا
)Jا
!Kهة oBا
)Jا
!.4J(6
1J2م ا
; o[Kوا(;9رات o%Nا
(1ا 7Bوا
'& 7N Iى Pb
OJJ%Cت ا
'!1ذج
وا
; 7ه:7
b+ w1;& -1ي
-2ا
"1Kا48
-3م ا
;اw.
-4
&1ز4
1CزV2
=("7
)&;HJ
و&;.6
w1;!.
b+ي
و2(C
σ2
أي ) ( at ∼ IIDN ( 0,σ 2
AOا 'FNي o[KCوه & 41!& 1ا(;9رات 7ا
(1ا'
7Bي !Nإذا آJ%C S
ه @Aا
Kوط و 7Nه @Aا
(;" 4
%ا
'!1ذج ا
! X1(J& (6أ& إذا HKNا 5ه @Aا(;9رات
' WNإدة ا
' Iوإ;Bاح !1ذج ,
أو :rإ?ر ا:\0
H 0 : E ( at ) = 0
H 1 : E ( at ) ≠ 0
وه 1إ;(ر 2A.و;[م 4Nا4859
e
) se ( e
= uوا
;1C O
7ز'"N 7B 7"(= V2
&;1ى &"' (;" α = 0.05 421ان E ( at ) = 0إذا آ ) u < 1.96 SهAا 7إ;(ر ان
)5ا
"' 4اآ( & 30و5ة وهAا دا;!
J%;& !8ت ا
&' 4ا
; 7ر( O
95
: إ?ر اا:7]
b
ل ا15 وw1;!
ل ا15 Runs test إ;(ر ا
ي46ا1. 7Bا1(
ا48ا1K (;[
7N W
6
اOر2 48ا1K"
;(رات9 آ] & اL12 ) 4!"&
;(رات ا9 ا5 ا1وه
.(;(ر9ا اAO. ' ه7b;3 3
وQ%. 241 رJ!
ا
:ل,0j\ أو ا$ إ?ر اا:-]
Autocorrelation
7CاA
اw. إ;(ر ا
;ا46ا1. 7Bا1(
ل اJ; أو إw.اC (;[2
4
داV& O;رJ& و7Bا1(
SACF 4'"
ا4CاA
ت ا6.ب ور) ا
;ا%. f
وذtest
. ا
(`ء4`
ا4;!
7CاA
اw.ا
;ا
;(ر9ا
H 0 : ρ1 = 0
H1 : ρ1 ≠ 0
(;" α = 0.05 421'"& ى1;& '"N 7B 7"(= V2ز1C O
u =
r1
4859 اQ5
se ( r1 )
. u < 1.96 S إذا آρ1 = 0 ان
:I ااi إ?ر:$را
:H]& "ة =ق. f
=(" وذ4ز1& 7Bا1(
اS & إذا آ7N (;[
7!"&
;(ر ا9 و;[م اGoodness of Fit Test .6;
ا5 إ;(ر-1
. Kolmogorov-Smirnov Test ف1! -روف1L1!
1آ
. Normal Probability Plot 7"(6
;!ل ا59 اw6[& -2
. Q-Q Plot "ت.
ا-"ت.
اw6[& -3
:N0ذج ا7 ?رj ?ىr ا اv$
;[مC وLBQ ;[C وLjung-Box Q statistc ~آ1.-'
ـ1 آ485( إ1
:4Pb
;(ر ا9
H 0 : ρ1 = ρ 2 = ⋯ = ρ K = 0
:4B"
. 6"Cو
rk2
∼ χ 2 ( K − m)
k =1 n − k
K
Q = n (n + 2) ∑
.ذج1!'
ا7N رةJ!
د ا
!"
) اm Q5
96
6"C وAIC ;[C وAutomatic Information Criteria 7CاA
&ت ا1"!
( &"ر ا2
:4B"
.
AIC ( m ) = n ln σ a2 + 2m
min AIC ( m )
m
76"2 يA
ذج ا1!'
ذج و[;ر ا1!'
ا7N رةJ!
د ا
!"
) اm Q5
: Examples and Case Studies 0ت دراrK و-
أ
هةK& 4'& ز4;!
4
;
ا
(ت ا-1
z(t)
60.1815 59.5257 58.9275 56.4828 56.1346 57.2318 60.7196 59.9315
61.0640 61.4230 63.1547 63.9622 63.5049 64.6886 62.8556 61.0344
58.0059 58.7108 57.9813 59.1721 62.4654 60.5820 59.3191 60.6643
61.2223 61.4761 61.1856 60.9225 59.3054 58.3755 59.5353 60.5777
61.9753 62.1789 61.8108 58.1483 58.4174 60.1325 59.6004 59.9086
60.4833 61.7008 59.1609 59.4554 59.0903 58.0151 59.1455 62.2658
63.4411 60.5918 65.1325 61.7122 58.8802 59.5333 60.9492 61.9013
59.3478 59.4444 62.6899 61.6708 63.7261 55.7339 58.1690 54.5045
56.7241 57.3334 57.9363 58.5870 61.8370 58.9585 56.7437 55.8451
58.1281 62.1017 59.9443 60.2990 61.6337 61.1520 63.8189 59.3572
61.7840 57.3292 54.7163 58.2273 58.7564 59.0087 59.3402 61.8956
60.9021 63.1070 60.0538 63.6776 60.8942 60.5289 59.9246 59.7252
60.7001 58.1895 54.5550 54.6083 56.5413 59.1567 57.9624 58.4651
61.9462 61.9205 63.3933 62.3827 61.4310 60.3373 57.8803 61.2797
61.9448 56.2599 59.9569 57.8763 59.2086 55.4219 54.2185 58.0143
60.9805 62.1362 60.0855 60.3843 60.8605 62.3728 57.0642 56.6085
57.5151 58.4221 60.6919 63.5907 61.4451 60.1458 57.3940 56.8697
59.2145 60.8962 61.1852 58.1711 53.8560 57.5307 59.3236 57.2961
58.5278 60.3030 60.6201 59.9346 59.4119 61.5614 61.1107 59.6266
60.3550 60.7021 60.7227 58.0423 59.3488 60.0377 58.7336 58.1105
59.4242 58.5790 58.6501 55.4010 59.3839 60.8256 62.1957 61.9152
60.3319 57.1459 59.0970 59.0997 59.8597 59.0780 56.9972 59.0778
97
61.5555 60.9815 60.3563 59.5097 58.3583 63.1777 61.8685 58.2759
59.7755 60.2052 60.2513 59.2927 56.1494 56.0309 56.6666 59.5015
59.4755 60.9013 61.2179 61.1168 61.7218 59.2298 60.7356 63.4124
MINITAB 4859 ا4&%
;[ام اF. Time Plot 7'& زw6[& 7N 4;!
) اXاو
:7
;
آ
MTB > TSPlot 'z(t)';
SUBC>
Index;
SUBC>
SUBC>
SUBC>
TDisplay 11;
Symbol;
Connect;
SUBC>
Title "An obseved Time Series".
A n o b s e v e d T im e S e r ie s
z(t)
65
60
55
In d e x
50
100
150
200
&;[ام اF. 7'"
ا7CاA
اw. و) ا
;اW% U
MTB > %ACF 'z(t)';
SUBC>
MAXLAG 20;
SUBC>
TITLE"SACF of observed Time Series".
Executing from file: H:\MTBWIN\MACROS\ACF.MAC
98
Autocorrelation
S A C F o f o b s e rv e d T im e S e rie s
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
L a g C o rr
1
2
3
4
5
6
7
0 .5 1
0 .2 0
-0 .0 0
-0 .0 5
-0 .0 8
-0 .1 8
-0 .1 9
10
T
LBQ
7 .1 9
2 .3 2
-0 .0 1
-0 .5 9
-0 .9 5
-2 .0 5
-2 .0 9
5 2 .4 8
6 0 .7 8
6 0 .7 8
6 1 .3 4
6 2 .8 2
6 9 .9 2
7 7 .5 8
L a g C o rr
8
9
10
11
12
13
14
-0 .1 4
-0 .1 4
-0 .0 9
-0 .0 7
-0 .0 8
-0 .0 2
0 .0 3
15
T
LBQ
-1 .5 0
-1 .5 2
-0 .9 0
-0 .7 1
-0 .7 9
-0 .2 1
0 .3 2
8 1 .7 6
8 6 .1 4
8 7 .7 3
8 8 .7 3
8 9 .9 7
9 0 .0 5
9 0 .2 7
L a g C o rr
15
16
17
18
19
20
0 .0 7
0 .1 3
0 .1 7
0 .2 0
0 .1 2
0 .0 6
20
T
LBQ
0 .6 8
1 .3 3
1 .7 5
2 .0 6
1 .2 1
0 .6 1
9 1 .2 3
9 4 .8 6
1 0 1 .3 3
1 1 0 .6 3
1 1 3 .9 8
1 1 4 .8 6
&;[ام اF. 7'"
ا78
ا7CاA
اw. و) ا
;اW% ]
U
MTB > %PACF 'z(t)';
SUBC>
MAXLAG 20;
SUBC>
TITLE"SPACF of obseved Time Series".
Executing from file: H:\MTBWIN\MACROS\PACF.MAC
Partial Autocorrelation
S P A C F o f o b se ve d T im e S e rie s
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
L a g P AC
1
2
3
4
5
6
7
0 .5 1
-0 .0 8
-0 .1 0
0 .0 0
-0 .0 5
-0 .1 6
-0 .0 4
10
T
7 .1 9
-1 .0 7
-1 .3 8
0 .0 4
-0 .7 3
-2 .3 4
-0 .5 0
L a g P AC
8
9
10
11
12
13
14
-0 .0 1
-0 .1 2
0 .0 1
-0 .0 4
-0 .0 9
0 .0 3
0 .0 2
15
T
-0 .1 2
-1 .6 3
0 .1 6
-0 .6 0
-1 .3 4
0 .3 9
0 .3 2
L a g P AC
15
16
17
18
19
20
-0 .0 2
0 .0 9
0 .0 8
0 .0 6
-0 .0 3
0 .0 4
20
T
-0 .2 3
1 .2 8
1 .2 0
0 .8 6
-0 .4 5
0 .5 2
ذج1! V(;C 4;!
ان ا5 7'"
ا78
ا7CاA
اw. و ا
;ا7CاA
اw.& أ!ط ا
;ا
&;[ام اF. هاتK!
ا7 ;حJ!
ذج ا1!'
( ا6 اAO
وAR(1)
MTB > Name c7 = 'RESI1'
MTB > ARIMA 1 0 0 'z(t)' 'RESI1';
99
SUBC>
Constant;
SUBC>
Forecast 5 c4 c5 c6;
SUBC>
SUBC>
GACF;
GPACF;
SUBC>
GHistogram;
SUBC>
GNormalplot.
ARIMA Model
ARIMA model for z(t)
Estimates at each iteration
Iteration
SSE
Parameters
0
839.667
0.100
53.870
1
746.819
0.250
44.876
2
695.840
0.400
35.883
3
685.086
0.502
29.769
4
685.054
0.507
29.458
5
685.054
0.507
29.443
Relative change in each estimate less than
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.5073
0.0611
Constant
29.4429
0.1309
Mean
59.7571
0.2656
0.0010
T
8.30
224.98
Number of observations: 201
Residuals:
SS = 685.020 (backforecasts excluded)
MS =
3.442 DF = 199
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
10.8(DF=11)
27.6(DF=23)
35.9(DF=35)
45.0(DF=47)
100
Forecasts from period 201
95 Percent Limits
Period
Actual
Forecast
Lower
Upper
202
59.7079
56.0707
63.3451
203
59.7322
55.6537
63.8106
204
59.7445
55.5600
63.9290
205
59.7507
55.5394
63.9620
206
59.7539
55.5357
63.9721
:7
;
و;';_ ا
1;ح هJ!
ذج ا1!'
ا-1
zt = 59.76 + 0.51( zt −1 − 59.76) + at , at ∼ WN ( 0,3.44 )
:7 هO
t 4!B ا
!"ري وONا%رة وإJ!
ا
!"
) ا-2
( )
φˆ1 = 0.51, s.e. φˆ1 = 0.061, t = 8.3
µˆ = 59.76, s.e. ( µˆ ) = 0.66
( )
δˆ = 29.44, s.e. δˆ = 0.131, t = 224.98
σˆ 2 = 3.44, with d . f . = 199
:7Bا1(
اo%b ".را
7Bا1(
اw1;& إ;(ر-1
MTB > ZTest 0.0 1.855 'RESI1';
SUBC>
Alternative 0;
SUBC> GHistogram;
SUBC> GDotplot;
SUBC> GBoxplot.
Z-Test
Test of mu = 0.000 vs mu not = 0.000
The assumed sigma = 1.85
Variable
RESI1
N
Mean
StDev
SE Mean
Z
P
201
-0.002
1.851
0.131
-0.01
0.99
b
وي ا2 w1;!
{ن ا. 42b
ا4Pb
\ اNX
7Bا1(
ا48ا1K إ;(ر-2
101
MTB > Runs 0 'RESI1'.
Runs Test
RESI1
K =
0.0000
The observed number of runs =
94
The expected number of runs = 101.0796
107 Observations above K
94 below
The test is significant at 0.3149
Cannot reject at alpha = 0.05
48ا1K 7Bا1(
{ن ا. 42b
ا4Pb
\ اNX
7Bا1(
7CاA
اw. ا
;ا-3
ACF of Residuals for z(t)
(with 95% confidence limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
10
15
20
25
30
35
40
45
50
Lag
7Bا1(
78
ا7CاA
اw. ا
;ا-4
102
PACF of Residuals for z(t)
(with 95% confidence limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
10
15
20
25
30
35
40
45
50
Lag
ا
(`ء4`
ا4;& أ!طV(;C 78
ا7CاA
اw. وا
;ا7CاA
اw. ان أ!ط ا
;ا5
: 7Bا1(
ا4"(= إ;(ر-5
7Bا1(
اري3;
اV`!
) ا-ا
Histogram of the Residuals
(response is z(t))
Frequency
30
20
10
0
-5
0
5
Residual
:
اI' انW2 H. 7b32X اA وه.(2JC 7"(6
اV2ز1;
اH3 4
أ* &;' و5
Normal Probability Plot 7"(6
;!ل ا5X ر) ا-ب
103
Normal Probability Plot for RESI1
99
Mean:
-1.6E-03
StDev:
1.85070
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-5.0
-2.5
0.0
2.5
5.0
Data
:م1J و
;{آ4"(= 7Bا1(
^ & ا
) أن اPوا
7Bا1(
ا4"(6
K-S Test ;(رF. -ج
MTB > %NormPlot 'RESI1';
SUBC>
Kstest;
SUBC>
Title "Normal Test for Residuals".
Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
Normal Test for Residuals
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-5
0
5
RESI1
Average: -0.0016272
StDev: 1.85070
N: 201
Kolmogorov-Smirnov Normality Test
D+: 0.045 D-: 0.060 D : 0.060
Approximate P-Value: 0.074
:7
;
ا5و
1;(ر ه9ا
H 0 : Residuals ∼ N ( 0,3.44 )
H1 : Residuals§N ( 0,3.44 )
6ف ا1!-روف1L1!
1إ;(ر آ
D + = 0.045, D − = 0.06, D = 0.06
104
4Pb
\ اNX ' أي اα = 0.05
& ( أآ7 وه0.074 7 ;(ر هP-Value ا
ـ
.42b
ا
:ات4 4(J;& )B 5 :(';
ذج1!'
ا;[&' ا
Forecasts from period 201
95 Percent Limits
Period
Forecast
Lower
Upper
Actual
202
203
59.7079
59.7322
56.0707
55.6537
63.3451
63.8106
204
205
206
59.7445
59.7507
59.7539
55.5600
55.5394
55.5357
63.9290
63.9620
63.9721
7
;
& ا. O!و
Plot C4*C8 C5*C8 C6*C8;
SUBC>
Connect;
SUBC>
Type 1;
SUBC>
Color 1;
SUBC>
Size 1;
SUBC>
Title "Forecast
limits";
SUBC>
Overlay.
of
5
future
Forecast of 5 future value with 95% limits
64
63
62
C4
61
60
59
58
57
56
55
1
2
3
C8
105
4
5
value
with
95%
.:(';
;ات اNات و:(';
اV& 4;!
ا
ء ا & ا76"2 7
;
وا
) ا
Forecast of 5 future value with 95% limits
64
63
62
C9
61
60
59
58
57
56
55
180
190
200
C8
هةK& 4'& ز4;!
4
;
ا
(ت ا-2
z(t)
499.148 496.650 511.026 488.539 498.440 507.382 496.208 494.948
503.975 501.649 489.348 506.040 496.678 502.233 498.429 503.170
498.758 502.969 498.229 501.605 493.371 505.884 496.227 496.806
493.057 506.459 502.545 497.785 506.329 496.665 491.923 504.340
499.890 494.559 503.107 502.891 498.598 500.074 499.260 496.372
507.416 500.508 496.830 491.981 516.373 492.286 500.356 503.506
498.090 498.319 507.020 493.161 499.217 508.489 494.033 496.062
504.877 498.304 495.355 505.581 495.000 504.965 497.393 501.521
494.918 501.527 504.712 501.064 492.352 500.664 495.431 507.886
499.173 494.833 504.072 497.883 495.423 507.072 496.285 506.345
496.765 504.129 495.737 500.744 505.577 485.991 507.673 507.735
482.567 507.594 503.580 493.866 501.819 500.921 503.415 497.295
500.989 498.294 501.700 495.868 501.175 503.852 499.783 497.642
501.331 496.932 507.582 494.885 504.666 498.380 496.181 510.287
489.314 504.394 501.928 494.814 509.407 498.060 497.133 496.029
502.720 499.982 503.325 495.954 504.408 500.199 494.878 503.134
502.489 498.640 500.484 493.552 501.417 504.785 497.943 501.634
495.691 502.173 502.066 497.130 492.318 505.517 499.299 499.611
106
496.252 504.346 501.082 497.626 496.757 505.475 498.787 500.388
499.279 504.913 493.843 506.259 498.403 497.462 499.467 505.987
498.169 500.712 498.571 504.085 491.707 504.817 502.933 493.858
497.015 504.204 501.703 490.683 505.429 504.336 495.430 494.857
503.195 506.403 498.599 487.344 514.220 490.887 511.741 497.861
500.252 502.721 500.256 494.614 502.414 503.465 501.999 493.017
498.158 503.746 497.643 507.438 491.418 506.649 496.078 498.931
500.409 506.001 490.619 512.122 496.150 505.218 497.413 497.794
496.225 501.827 500.324 505.367 498.016 498.477 495.353 513.900
491.726 496.063 499.779 504.012 501.542 496.680 499.134 504.717
489.032 505.709 497.956 497.231 507.590 491.202 503.130 502.209
500.024 493.502 502.681 505.234 497.647 495.699 504.174 497.992
505.194 497.421 502.823 496.877 504.640 492.716 501.701 501.387
499.574 497.048
ﺍﻭﻻ :ﺴﻭﻑ ﻨﺭﺴﻡ ﻓﻘﻁ 50ﻤﺸﺎﻫﺩﺓ ﻤﻥ ﻫﺫﻩ ﺍﻝﻤﺘﺴﻠﺴﻠﺔ
510
)z(t
500
490
50
40
20
30
W% :Uو) ا
;ا w.ا
Aا 7Cا
"':7
107
10
In d e x
Autocorrelation
Autocorrelation Function for z(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag Corr
10
T
LBQ
1 -0.53 -8.38
2 -0.05 -0.64
3 0.12 1.52
4 0.04 0.52
5 -0.13 -1.61
6 0.11 1.39
7 -0.09 -1.13
71.03
71.68
75.39
75.83
80.08
83.33
85.52
Lag Corr
T
15
LBQ
8 0.10 1.19 87.98
9 -0.11 -1.31 91.01
10 0.15 1.77 96.64
11 -0.18 -2.16 105.32
12 0.11 1.25 108.35
13 0.05 0.56 108.97
14 -0.12 -1.36 112.61
Lag Corr
20
T
LBQ
15 0.05 0.60 113.33
16 -0.02 -0.26 113.47
17 0.07 0.82 114.85
18 -0.10 -1.16 117.65
19 0.07 0.75 118.81
20 -0.06 -0.73 119.93
:7'"
ا78
ا7CاA
اw. و) ا
;اW% :]
U
Partial Autocorrelation
Partial Autocorrelation Function for z(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
10
15
Lag PAC
T
Lag PAC
T
1 -0.53
2 -0.46
3 -0.29
4 -0.07
5 -0.11
6 0.02
7 -0.09
-8.38
-7.28
-4.52
-1.08
-1.71
0.30
-1.36
8 0.04
9 -0.07
10 0.12
11 -0.07
12 -0.04
13 0.08
14 -0.04
0.65
-1.09
1.94
-1.15
-0.71
1.19
-0.59
Lag PAC
20
T
15 0.03 0.49
16 -0.13 -2.09
17 0.06 1.01
18 -0.08 -1.32
19 0.03 0.44
20 -0.13 -2.07
ذج1!'
ا اA( ه6;. وMA(1) ذج1! V(;C B 4;!
ان ا5 هةK!
& ا!ط ا
MTB > Name c7 = 'RESI1'
MTB > ARIMA 0 0 1 'z(t)' 'RESI1';
SUBC>
Constant;
SUBC>
Forecast 5 c4 c5 c6;
SUBC> GACF;
SUBC> GPACF;
SUBC> GHistogram;
SUBC> GNormalplot.
ARIMA Model
108
ARIMA model for z(t)
Estimates at each iteration
Iteration
SSE
Parameters
0
6081.19
0.100
500.046
1
5265.34
0.250
500.004
2
4615.22
0.400
499.980
3
4109.70
0.550
499.967
4
5
3766.60
3727.32
0.700
0.841
499.960
499.959
6
3687.70
0.797 499.963
7
3687.08
0.790 499.962
8
3687.07
0.791 499.962
9
3687.07
0.790 499.962
Relative change in each estimate less than
Final Estimates of Parameters
Type
Coef
StDev
MA
1
0.7905
0.0386
Constant
499.962
0.051
Mean
499.962
0.051
0.0010
T
20.50
9708.40
Number of observations: 250
Residuals:
SS = 3684.13 (backforecasts excluded)
MS =
14.86 DF = 248
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
26.7(DF=11)
35.9(DF=23)
63.1(DF=35)
82.8(DF=47)
Forecasts from period 250
Period
251
252
253
254
Forecast
502.256
499.962
499.962
499.962
95 Percent Limits
Lower
Upper
494.700
509.812
490.330
509.593
490.330
509.593
490.330
509.593
109
Actual
255
499.962
490.330
509.593
:7
;
و;';_ ا
1;ح هJ!
ذج ا1!'
ا-1
zt = 499.962 + at − 0.7905at −1 , at ∼ WN ( 0,14.86 )
:7 هO
t 4!B ا
!"ري وONا%رة وإJ!
ا
!"
) ا-2
( )
θˆ1 = 0.7905, s.e. θˆ1 = 0.0386, t = 20.50
( )
µˆ = δˆ = 499.962, s.e. δˆ = 0.051, t = 9708.40
σˆ 2 = 14.86, with d . f . = 248
:7Bا1(
اo%b ".را
7Bا1(
اw1;& إ;(ر-1
MTB > ZTest 0.0 3.847 'RESI1';
SUBC>
Alternative 0.
Z-Test
Test of mu = 0.000 vs mu not = 0.000
The assumed sigma = 3.85
Variable
RESI1
N
Mean
StDev
SE Mean
Z
P
250
-0.007
3.847
0.243
-0.03
0.98
b
وي ا2 w1;!
{ن ا. 42b
ا4Pb
\ اNX
7Bا1(
ا48ا1K إ;(ر-2
MTB > Runs 0 'RESI1'.
Runs Test
RESI1
K =
0.0000
110
The observed number of runs = 134
The expected number of runs = 125.9920
126 Observations above K 124 below
The test is significant at
0.3103
Cannot reject at alpha = 0.05
48ا1K 7Bا1(
{ن ا. 42b
ا4Pb
\ اNX
7Bا1(
7CاA
اw. ا
;ا-3
A C F o f R e s id u a ls fo r z (t)
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s )
1 .0
0 .8
Autocorrelation
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
10
15
20
25
30
35
40
45
50
55
60
Lag
7Bا1(
78
ا7CاA
اw. ا
;ا-4
P A C F o f R e s id u a ls f o r z (t)
( w ith 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u to c o r r e l a t io n s )
1 .0
Partial Autocorrelation
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
10
15
20
25
30
35
40
45
50
55
60
Lag
ا
(`ء4`
ا4;& أ!طV(;C 78
ا7CاA
اw. وا
;ا7CاA
اw. ان أ!ط ا
;ا5
: 7Bا1(
ا4"(= إ;(ر-5
7Bا1(
اري3;
اV`!
) ا-ا
111
H istogram of the R esiduals
(resp on se is z(t))
Frequency
30
20
10
0
-10
0
10
R esidual
:
اI' انW2 H. 7b32X اA وه.(2JC 7"(6
اV2ز1;
اH3 4
أ* &;' و5
Normal Probability Plot 7"(6
;!ل ا5X ر) ا-ب
Normal Probability Plot for RESI1
99
Mean:
-6.9E-03
StDev:
3.84651
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-10
-5
0
5
10
Data
:م1J و
;{آ4"(= 7Bا1(
^ & ا
) أن اPوا
7Bا1(
ا4"(6
K-S Test ;(رF. -ج
MTB > %Qqplot 'RESI1';
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: H:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
112
Data
Mean:
: RESI1
-6.9E-03
StDev:
3.84651
MTB > %NormPlot 'RESI1';
SUBC>
Kstest.
Executing from file: H:\MTBWIN\MACROS\NormPlot.MAC
Normal Probability Plot
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-10
0
10
RESI1
Average: -0.0069004
StDev: 3.84651
N: 250
Kolmogorov-Smirnov Normality Test
D+: 0.034 D-: 0.051 D : 0.051
Approximate P-Value: 0.105
6ف ا1!-روف1L1!
1إ;(ر آ
D + = 0.034, D − = 0.051, D = 0.051
ان4Pb
\ اNX ' أي اα = 0.05
& ( أآ7 وه0.105 7 ;(ر هP-Value ا
ـ
."(= 4ز1& 7Bا1(
ا
:ات4 4(J;& )B 5 :(';
ذج1!'
ا;[&' ا
Forecasts from period 250
Period
251
252
253
254
255
Forecast
502.256
499.962
499.962
499.962
499.962
95 Percent Limits
Lower
Upper
494.700
509.812
490.330
509.593
490.330
509.593
490.330
509.593
490.330
509.593
Actual
:('C 95% ;اتN V& ات:(';
ا76"2 7
;
وا
) ا
113
Forecast of 5 future values with 95% limits
C4
510
500
490
1
2
3
4
5
C8
هةK& 4'& ز4;!
4
;
ا
(ت ا-3
z(t)
229.574 227.346 230.260 229.903 226.778 226.641 226.760 224.678
224.077 225.772 223.390 222.482 221.562 222.515 224.063 227.500
230.713 234.323 236.033 236.488 232.308 229.136 225.663 221.632
215.405 213.619 217.433 223.408 232.653 239.577 238.463 234.178
228.758 221.484 217.123 218.067 222.156 227.621 232.209 233.005
234.678 236.419 235.744 229.359 229.331 229.564 230.102 232.432
234.155 233.918 235.767 234.668 231.319 231.633 231.121 228.189
227.075 226.765 224.927 225.721 225.734 227.982 229.848 231.718
230.421 228.200 228.472 230.888 230.122 227.859 223.115 222.468
224.663 225.799 228.227 229.851 228.225 228.618 228.418 231.163
233.335 236.399 236.659 235.024 235.122 228.989 224.483 226.479
223.571 222.523 225.196 226.724 228.198 229.792 232.738 234.207
234.561 232.976 231.266 227.812 224.928 225.447 228.163 230.455
232.473 232.067 233.891 233.841 234.707 234.825 232.232 233.640
231.653 230.148 230.327 228.922 231.665 235.224 236.562 233.725
230.146 227.077 227.032 227.089 229.575 233.044 233.427 233.089
233.444 233.256 232.820 228.954 224.747 224.207 225.484 228.655
230.076 231.062 232.461 232.152 226.865 222.819 220.782 220.958
221.171 224.050 228.727 232.135 232.027 232.315 232.030 231.531
230.582 232.032 231.411 232.684 233.852 233.127 230.938 231.363
114
232.344 233.622 233.799 235.038 232.160 229.733 229.757 228.285
224.880 223.599 225.273 223.994 224.258 227.948 230.636 229.320
227.449 229.100 231.898 228.203 228.606 227.046 230.713 235.587
239.660 242.860 243.963 239.883 234.243 230.662 230.360 228.729
225.860 225.123 225.070 229.486 231.265 234.107 234.625 232.700
229.792 230.082 227.643 230.342 233.628 238.762 241.821 240.884
235.112 228.468 223.381 223.795 226.994 230.499 230.865 236.017
238.292 235.623 230.088 226.271 225.616 225.771 226.222 229.321
227.805 226.745 225.447 223.250 225.291 225.358 225.985 228.141
230.794 229.727 227.934 228.920 230.296 229.369 229.352 228.958
231.092 232.891 235.210 235.339 236.029 232.881 228.837 226.114
225.020 224.096
MINITAB 4859 ا4&%
;[ام اF. Time Plot 7'& زw6[& 7N 4;!
) اXاو
(wJN هةK& 50) :7
;
آ
z(t)
2 4 2
2 3 2
2 2 2
In d e x
1 0
2 0
3 0
4 0
5 0
7'"
ا7CاA
اw. و) ا
;اW% U
115
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
Lag
1
2
3
4
5
6
7
C o rr
0
0
0
-0
-0
-0
-0
.8
.5
.1
.1
.3
.3
.2
4
1
5
6
3
3
0
T
13
5
1
-1
-2
-2
-1
.2
.2
.3
.4
.9
.8
.7
7
0
5
6
8
7
3
10
LBQ
1
2
2
2
2
3
3
7
4
4
5
8
1
2
8
4
9
6
3
1
1
.2
.2
.7
.2
.7
.2
.9
Lag
1
9
1
1
5
3
5
1
1
1
1
1
C o rr
T
8 -0 .0 2 -0
9 0 .1 4 1
0 0 .2 3 1
1 0 .2 3 1
2 0 .1 5 1
3 0 .0 2 0
4 -0 .1 2 -0
.1
.1
.9
.8
.2
.1
.9
7
8
5
8
2
5
6
15
LBQ
3
3
3
3
3
3
3
2
2
4
5
6
6
6
2
7
1
5
1
1
5
.0
.2
.4
.1
.1
.2
.0
5
4
8
6
7
5
4
Lag
1
1
1
1
1
2
5
6
7
8
9
0
C o rr
-0
-0
-0
-0
0
0
.2
.2
.1
.1
.0
.0
1
2
8
0
0
8
20
T
-1
-1
-1
-0
0
0
.6
.7
.4
.7
.0
.6
5
9
2
6
2
1
LBQ
3
3
3
4
4
4
7
8
9
0
0
0
6
9
8
1
1
3
.3
.9
.8
.3
.3
.0
2
3
1
7
7
5
7'"
ا78
ا7CاA
اw. و) ا
;اW% ]
U
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
Lag
1
2
3
4
5
6
7
PAC
10
T
0 .8 4 1 3 .2 7
-0 .6 6 - 1 0 .3 9
-0 .0 7
-1 .1 3
-0 .0 6
-1 .0 0
0 .1 1
1 .7 8
0 .1 4
2 .1 8
-0 .0 1
-0 .2 2
Lag
15
PAC
T
8 0 .0 7
9 -0 .0 7
1 0 0 .0 5
1 1 -0 .0 7
1 2 0 .0 6
1 3 -0 .1 5
1 4 0 .0 1
1 .0 6
-1 .1 2
0 .7 3
-1 .1 8
0 .9 0
-2 .4 1
0 .1 8
Lag
20
PAC
T
1 5 0 .0 4
1 6 -0 .0 6
1 7 0 .0 4
1 8 -0 .0 7
1 9 0 .0 9
2 0 -0 .0 9
0 .6 7
-0 .9 3
0 .6 0
-1 .1 4
1 .4 4
-1 .3 9
ذج1! V(;C 4;!
ان ا5 7'"
ا78
ا7CاA
اw. و ا
;ا7CاA
اw.& أ!ط ا
;ا
هاتK!
ا7 ;حJ!
ذج ا1!'
( ا6 اAO
وAR(2)
MTB > Name c7 = 'RESI1'
MTB > ARIMA 2 0 0 'z(t)' 'RESI1';
SUBC>
Constant;
SUBC>
Forecast 10 c4 c5 c6;
SUBC> GACF;
SUBC> GPACF;
SUBC> GHistogram;
SUBC> GNormalplot.
116
ARIMA Model
ARIMA model for z(t)
Estimates at each iteration
Iteration
0
SSE
4257.23
Parameters
0.100
0.100
183.784
1
3528.31
0.250
0.012
169.535
2
3
2889.23
2338.97
0.400
0.550
-0.076
-0.165
155.360
141.201
4
5
1877.39
1504.46
0.700
0.850
-0.253
-0.342
127.051
112.913
6
1220.13
1.000
7
1024.34
1.150
8
916.97
1.300
9
894.38
1.402
10
894.31
1.408
11
894.31
1.408
Relative change in each estimate
Final Estimates of Parameters
Type
Coef
StDev
AR
1
1.4079
0.0473
AR
2
-0.6720
0.0474
Constant
60.6458
0.1203
Mean
229.638
0.456
Number of observations:
Residuals:
-0.430
98.789
-0.519
84.690
-0.608
70.623
-0.668
61.154
-0.672
60.670
-0.672
60.646
less than 0.0010
T
29.78
-14.19
504.11
250
SS = 893.567
MS =
3.618
(backforecasts excluded)
DF = 247
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
Chi-Square 17.5(DF=10) 27.2(DF=22) 49.7(DF=34)
48
67.7(DF=46)
Forecasts from period 250
Period
251
252
Forecast
224.939
226.747
95 Percent Limits
Lower
Upper
221.211
228.668
220.308
233.186
117
Actual
253
228.725
220.642
236.808
254
230.296
221.546
239.045
255
256
231.177
231.363
222.311
222.494
240.044
240.233
257
231.033
222.070
239.996
258
230.442
221.327
239.558
259
229.833
220.600
239.067
260
229.372
220.090
238.655
:7
;
و;';_ ا
1;ح هJ!
ذج ا1!'
ا-1
zt = 60.6458 + 1.4079 zt −1 − 0.672 zt − 2 + at , at ∼ WN ( 0,3.618)
:7 هO
t 4!B ا
!"ري وONا%رة وإJ!
ا
!"
) ا-2
( )
s.e. (φˆ ) = 0.0474,
φˆ1 = 1.4079, s.e. φˆ1 = 0.0473, t = 29.78
φˆ2 = −0.672,
2
t = −14.19
µˆ = 229.638, s.e. ( µˆ ) = 0.456
( )
δˆ = 60.6458, s.e. δˆ = 0.1203, t = 504.11
σˆ 2 = 3.618, with d . f . = 247
:I> اا/Y7 $را
7Bا1(
اw1;& إ;(ر-1
MTB > ZTest 0.0 3.618 'RESI1';
SUBC>
Alternative 0.
Z-Test
Test of mu = 0.000 vs mu not = 0.000
The assumed sigma = 3.62
Variable
RESI1
N
250
Mean
-0.005
StDev
1.894
SE Mean
0.229
Z
-0.02
P
0.98
b
وي ا2 w1;!
{ن ا. 42b
ا4Pb
\ اNX
118
7Bا1(
ا48ا1K إ;(ر-2
MTB > Runs 0 'RESI1'.
Runs Test
RESI1
K =
0.0000
The observed number of runs = 125
The expected number of runs = 125.8720
129 Observations above K 121 below
The test is significant at 0.9119
Cannot reject at alpha = 0.05
48ا1K 7Bا1(
{ن ا. 42b
ا4Pb
\ اNX
7Bا1(
7CاA
اw. ا
;ا-3
A C F o f R e s id u a ls f o r z (t)
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s )
1 .0
0 .8
Autocorrelation
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
5
10
15
20
25
30
35
40
45
50
55
60
Lag
7Bا1(
78
ا7CاA
اw. ا
;ا-4
119
P A C F o f R e s id u a ls f o r z ( t)
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s )
1 .0
Partial Autocorrelation
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
5
10
15
20
25
30
35
40
45
50
55
60
Lag
ا
(`ء4`
ا4;& أ!طV(;C 78
ا7CاA
اw. وا
;ا7CاA
اw. ان أ!ط ا
;ا5
: 7Bا1(
ا4"(= إ;(ر-5
7Bا1(
اري3;
اV`!
) ا-ا
H istogram of th e R esiduals
(resp on s e is z(t))
Frequency
30
20
10
0
-5
0
5
R es idual
:
اI' انW2 H. 7b32X اA وه.(2JC 7"(6
اV2ز1;
اH3 4
أ* &;' و5
Normal Probability Plot 7"(6
;!ل ا5X ر) ا-ب
120
N orm al P rob ab ility P lot for R E S I1
99
M e a n:
-4 .6 E -03
S tD e v :
1 .8 94 3 6
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-6
-4
-2
0
2
4
D ata
:م1J و
;{آ4"(= 7Bا1(
^ & ا
) أن اPوا
7Bا1(
ا4"(6
K-S Test ;(رF. -ج
Normal Probability Plot
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-5
0
5
RESI1
Average: -0.0046305
StDev: 1.89436
N: 250
Kolmogorov-Smirnov Normality Test
D+: 0.020 D-: 0.029 D : 0.029
Approximate P-Value > 0.15
& ( أآ7 وه0.15 7 ;(ر هP-Value ا
ـ6ف ا1!-روف1L1!
1إ;(ر آ
.7Bا1(
ا4"(= 4PN \NX ' أي اα = 0.05
:ات4 4(J;& )B 10 :(';
ذج1!'
ا;[&' ا
Forecasts from period 250
Period
251
252
Forecast
224.939
226.747
95 Percent Limits
Lower
Upper
221.211
228.668
220.308
233.186
121
Actual
253
228.725
220.642
236.808
254
230.296
221.546
239.045
255
256
231.177
231.363
222.311
222.494
240.044
240.233
257
231.033
222.070
239.996
258
230.442
221.327
239.558
259
229.833
220.600
239.067
260
229.372
220.090
238.655
:('C ;اتN 95% ات و:(';
ا76"2 7
;
وا
) ا
Forecast of 10 future values with 95% limits
C4
240
230
220
0
1
2
3
4
5
6
7
8
9
10
C8
Forecast of 10 future values with 95% limits
245
C9
235
225
215
0
100
200
C8
122
Forecast of 10 future values with 95% limits
C9
240
230
220
0
10
20
30
40
50
60
C8
V& اة4!B ![
7]
اH3K
ات وا:(';
اV& O&3. 4'&
ا4;!
( ا2 اولH3K
ا
.:(';
ا4
داH3 ^P1;
ات:(';
ا
:0 دراK
ا6 أB & )إV'& إ;جw 7N 4("!
ت ا12b;
"د ا7&1
اw1;!
ا4
;
ا4;!
ا
(6.
Defects
1.20
1.50
1.54
2.70
1.95
2.40
3.44
2.83
1.76
2.00
2.09
1.89
1.80
1.25
1.58
2.25
2.50
2.05
1.46
1.54
1.42
1.57
1.40
1.51
1.08
1.27
1.18
1.39
1.42
2.08
1.85
1.82
2.07
2.32
1.23
2.91
1.77
1.61
1.25
1.15
1.37
1.79
1.68
1.78
1.84
4;!
7'&
اw6[!
ا
123
3 .5
Defects
3 .0
2 .5
2 .0
1 .5
1 .0
In d e x
10
20
30
40
78
ا7CاA
وا7CاA
اw.ا
;ا
Autocorrelation
Autocorrelation Function for Defects
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag
Corr
4
5
6
7
8
T
LBQ
Lag
Corr
T
LBQ
1 0.43 2.88
2 0.26 1.49
3 0.14 0.77
4 0.08 0.43
5 -0.09 -0.46
6 -0.07 -0.39
7 -0.21 -1.10
8.84
12.18
13.18
13.50
13.89
14.18
16.57
8
9
10
11
-0.11
-0.05
-0.01
-0.04
-0.57
-0.27
-0.04
-0.19
17.25
17.41
17.41
17.50
9
10
11
Partial Autocorrelation
Partial Autocorrelation Function for Defects
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag PAC
4
5
T
1 0.43 2.88
2 0.09 0.63
3 -0.00 -0.01
4 0.00 0.00
5 -0.16 -1.07
6 0.00 0.02
7 -0.18 -1.19
6
Lag PAC
7
8
T
8 0.07 0.44
9 0.05 0.35
10 0.01 0.09
11 -0.03 -0.23
124
9
10
11
:4B"
. 6"2 يA
واAIC 7CاA
&ت ا1"!
ف ;[م &"ر ا1 W'!
ذج ا1!'
;ر ا9
AIC ( m ) = n ln σ a2 + 2m
min AIC ( m )
m
76"2 يA
ذج ا1!'
ذج و[;ر ا1!'
ا7N رةJ!
د ا
!"
) اm Q5
:7
ا1;
ا7 ( ا
'!ذج6 ف1
MTB > ARIMA 1 0 0 'Defects' 'RESI1';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
Final Estimates of Parameters
Type
Coef
StDev
T
AR
1
0.4421
0.1365
3.24
Constant
0.99280
0.06999
14.19
Mean
1.7795
0.1254
Number of observations: 45
Residuals:
SS = 9.47811 (backforecasts excluded)
MS = 0.22042 DF = 43
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
48
Chi-Square
* (DF= *)
12
24
4.9(DF=11)
8.9(DF=23)
MTB > ARIMA 2 0 0 'Defects' 'RESI2';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
125
36
30.9(DF=35)
Final Estimates of Parameters
Type
AR
1
Coef
0.3999
StDev
0.1533
T
2.61
AR
0.0989
0.1531
0.65
0.89019
0.07047
12.63
1.7762
0.1406
2
Constant
Mean
Number of observations: 45
Residuals:
SS = 9.38567 (backforecasts excluded)
MS = 0.22347
DF = 42
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
4.0(DF=10)
8.1(DF=22)
28.8(DF=34)
* (DF= *)
MTB > ARIMA 1 0 1 'Defects' 'RESI3';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.5983
0.2691
MA
1
0.1926
0.3294
Constant
0.71334
0.05693
Mean
1.7759
0.1417
T
2.22
0.58
12.53
Number of observations: 45
Residuals:
SS = 9.39423 (backforecasts excluded)
MS = 0.22367 DF = 42
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
126
Lag
12
24
36
48
Chi-Square
* (DF= *)
4.1(DF=10)
8.3(DF=22)
29.1(DF=34)
MTB > ARIMA 0 0 1 'Defects' 'RESI4';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
Final Estimates of Parameters
Type
Coef
StDev
MA
1
-0.3409
0.1431
Constant
1.78480
0.09651
Mean
1.78480
0.09651
T
-2.38
18.49
Number of observations: 45
Residuals:
SS = 10.0362 (backforecasts excluded)
MS = 0.2334 DF = 43
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
8.0(DF=11)
13.2(DF=23)
35.7(DF=35)
* (DF= *)
MTB > ARIMA 0 0 2 'Defects' 'RESI5';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
127
Final Estimates of Parameters
Type
Coef
StDev
T
-0.3869
-0.1816
0.1516
0.1516
-2.55
-1.20
Constant
1.7839
0.1118
15.96
Mean
1.7839
0.1118
MA
MA
1
2
Number of observations:
Residuals:
45
SS = 9.61059
MS = 0.22882
(backforecasts excluded)
DF = 42
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
4.6(DF=10)
9.2(DF=22)
31.0(DF=34)
* (DF= *)
MTB > ARIMA 2 0 1 'Defects' 'RESI6';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.4134
1.5680
AR
2
0.0929
0.7113
MA
1
0.0136
1.5749
Constant
0.87675
0.07036
Mean
1.7761
0.1425
T
0.26
0.13
0.01
12.46
Number of observations: 45
Residuals:
SS = 9.38561 (backforecasts excluded)
MS = 0.22892 DF = 41
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
128
Lag
12
24
36
48
Chi-Square
* (DF= *)
4.0(DF= 9)
8.1(DF=21)
28.8(DF=33)
MTB > ARIMA 1 0 2 'Defects' 'RESI7';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
* ERROR * Model cannot be estimated with these data
MTB > ARIMA 2 0 2 'Defects' 'RESI8';
SUBC>
Constant;
ARIMA Model
ARIMA model for Defects
Final Estimates of Parameters
Type
Coef
StDev
AR
1
1.6720
0.1165
AR
2
-0.7263
0.1251
MA
1
1.3199
0.0184
MA
2
-0.3196
0.0731
Constant 0.096224
0.003323
Mean
1.77238
0.06121
T
14.35
-5.80
71.63
-4.37
28.95
Number of observations: 45
Residuals:
SS = 8.33225 (backforecasts excluded)
MS = 0.20831 DF = 40
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
48
129
36
Chi-Square
4.7(DF= 8)
8.9(DF=20)
29.7(DF=32)
* (DF= *)
:7
;
ول ا. f
ذo[و
Model
σˆ 2
m
AIC
__________
________
___
_________
AR (1)
0.22042
3
−62.0499
0.22347
4
−59.4315
0.23340
3
−59.4751
0.22882
4
−58.3669
0.22367
4
−59.3913
0.22892
5
−56.3472
−
−
−
0.20831
6
−58.5928
AR ( 2 )
MA (1)
MA ( 2 )
ARMA (1,1)
ARMA ( 2,1)
ARMA (1, 2 )
ARMA ( 2, 2 )
min AIC ( m ) = −62.0499
m
. AR(1) 1ذج ه1! H`Nأي ان أ
.ات:('C 1C و7Bا1(
اo%N 2!; آW
6
;ك2
130
Kدرا:0
ا
!; 4ا
;
4ه 7د Hا
!("ت ا
' 2!. 421ا
X2ت Kآ& 4
Sales
3.49 5.74 5.51 3.99 3.45 4.77 4.14 4.60 3.80 5.43
3.96 2.54 4.05 6.16 3.78 5.07 5.42 3.91 4.30 3.88
2.89 4.61 4.08 4.05 3.28 2.65 1.22 3.98 3.45 3.57
2.52 1.58 4.00 5.14 3.84 4.40 3.08 5.43 4.80 2.75
5.77 4.99 4.31 6.46 6.11 4.79 5.65 5.52 6.12 6.06
3.20 5.05 6.23 6.12 4.99 4.89 4.78 5.67 6.08 5.80
5.13 7.07 8.02 6.36 5.75 5.70 5.61 5.63 5.71 5.16
7.20 6.87 7.56 6.57 6.08 4.72 6.09 6.64 7.49 6.64
7.26 7.22 6.69 7.49 9.01 7.27 5.62 7.59 7.53 6.43
6.42 8.22 7.67 7.53 7.23 8.50 8.27 8.75 7.50 7.86
&[ w6ز&'4;!
7
9
8
7
6
Sales
5
4
3
2
1
100
90
80
70
60
50
دوال ا
;ا w.ا
Aا 7Cوا
Aا 7Cا
78ا
"'4
131
40
30
20
10
In d e x
Autocorrelation
Autocorrelation Function for Sales
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag Corr
1
2
3
4
5
6
7
0.71
0.60
0.65
0.64
0.59
0.59
0.51
T
LBQ
7.10 51.97
4.26 89.85
3.94 134.64
3.36 177.75
2.83 215.72
2.63 253.97
2.12 282.61
15
Lag Corr
8
9
10
11
12
13
14
0.56
0.49
0.49
0.51
0.42
0.38
0.45
T
LBQ
Lag Corr
2.22 317.32
1.87 344.49
1.79 371.67
1.82 401.71
1.46 422.60
1.29 439.91
1.50 464.06
15
16
17
18
19
20
21
0.41
0.35
0.31
0.30
0.36
0.31
0.26
T
25
LBQ
1.32 483.85
1.13 499.04
0.97 510.68
0.92 521.52
1.11 537.81
0.95 550.11
0.77 558.52
Lag Corr
22
23
24
25
0.22
0.17
0.21
0.25
T
LBQ
0.67 565.04
0.50 568.77
0.64 574.90
0.75 583.43
Partial Autocorrelation
Partial Autocorrelation Function for Sales
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag PAC
T
1 0.71 7.10
2 0.20 1.99
3 0.35 3.53
4 0.15 1.49
5 0.09 0.92
6 0.10 1.03
7 -0.13 -1.27
15
Lag PAC
25
T
Lag PAC
T
8 0.19 1.93
9 -0.17 -1.70
10 0.14 1.44
11 0.03 0.34
12 -0.15 -1.52
13 0.02 0.25
14 0.04 0.39
15 0.03
16 -0.10
17 -0.08
18 -0.01
19 0.14
20 -0.04
21 0.00
0.32
-0.96
-0.84
-0.10
1.43
-0.44
0.04
Lag PAC
T
22 -0.17 -1.68
23 -0.10 -1.00
24 0.15 1.55
25 0.03 0.34
.w1;!
ا7N ارJ;ل م إ2 B !& |6. &[C لC 7'"
ا7CاA
اw. ا
;ا4
دا
O! وwt = ∇zt 4;!
اول2b;
'ب ا
3
2
w(t)
1
0
-1
-2
-3
In d e x
10
20
30
40
50
132
60
70
80
90
100
ةJ;!
ا4;!
78
ا7CاA
وا7CاA
اw. دوال ا
;ا.نsة اJ;& 4;!
(و اC
Autocorrelation
Autocorrelation Function for w(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
Lag Corr
12
T
LBQ
1 -0.30 -3.00
2 -0.31 -2.86
3 0.11 0.90
4 0.04 0.31
5 -0.04 -0.33
6 0.14 1.15
7 -0.24 -1.98
9.26
19.32
20.49
20.63
20.80
22.80
29.02
Lag Corr
T
LBQ
8 0.19 1.53
9 -0.07 -0.53
10 -0.07 -0.52
11 0.16 1.27
12 -0.07 -0.53
13 -0.14 -1.04
14 0.15 1.10
33.07
33.58
34.08
37.12
37.67
39.85
42.37
Lag Corr
22
T
LBQ
15 0.00 0.00
16 0.02 0.13
17 -0.07 -0.52
18 -0.15 -1.09
19 0.20 1.48
20 0.01 0.05
21 -0.04 -0.25
42.37
42.40
42.99
45.65
50.73
50.74
50.90
Lag Corr
T
LBQ
22 0.05 0.35 51.22
23 -0.18 -1.28 55.44
24 0.03 0.20 55.55
Partial Autocorrelation
Partial Autocorrelation Function for w(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
12
22
Lag PAC
T
Lag PAC
T
Lag PAC
T
1 -0.30
2 -0.44
3 -0.22
4 -0.21
5 -0.17
6 0.06
7 -0.26
-3.00
-4.41
-2.23
-2.05
-1.72
0.59
-2.55
8 0.12
9 -0.16
10 -0.06
11 0.09
12 -0.06
13 -0.02
14 -0.07
1.17
-1.59
-0.60
0.93
-0.59
-0.17
-0.71
15 0.06
16 0.07
17 -0.03
18 -0.17
19 -0.03
20 -0.10
21 0.09
0.59
0.70
-0.32
-1.65
-0.31
-0.95
0.85
Lag PAC
T
22 0.08 0.80
23 -0.15 -1.53
24 -0.05 -0.55
:4B"
. 6"2 يA
واAIC 7CاA
&ت ا1"!
ف ;[م &"ر ا1 W'!
ذج ا1!'
;ر ا9
AIC ( m ) = n ln σ a2 + 2m
min AIC ( m )
m
76"2 يA
ذج ا1!'
ذج و[;ر ا1!'
ا7N رةJ!
د ا
!"
) اm Q5
:7
ا1;
ا7 ( ا
'!ذج6 ف1
MTB > ARIMA 1 1 0 'Sales';
SUBC>
NoConstant.
133
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
AR
1
-0.3114
0.0959
T
-3.25
Differencing: 1 regular difference
Number of observations:
differencing 99
Residuals:
SS = 133.134
MS =
1.359
Original
series
100,
after
(backforecasts excluded)
DF = 98
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
31.9(DF=11)
51.2(DF=23)
62.8(DF=35)
81.0(DF=47)
MTB > ARIMA 2 1 0 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
AR
1
-0.4532
0.0897
AR
2
-0.4656
0.0901
T
-5.05
-5.17
Differencing: 1 regular difference
Number of observations:
Original series 100, after
differencing 99
Residuals:
SS = 104.715 (backforecasts excluded)
MS =
1.080 DF = 97
134
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
48
12
Chi-Square
21.8(DF=10)
24
36
40.9(DF=22)
49.4(DF=34)
59.9(DF=46)
MTB > ARIMA 0 1 1 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
MA
1
0.7636
0.0648
T
11.78
Differencing: 1 regular difference
Number of observations:
Original series 100, after
differencing 99
Residuals:
SS = 101.411 (backforecasts excluded)
MS =
1.035 DF = 98
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
12.6(DF=11)
27.8(DF=23)
35.9(DF=35)
48.5(DF=47)
MTB > ARIMA 0 1 2 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
135
Final Estimates of Parameters
Type
MA
1
Coef
0.5756
StDev
0.0990
T
5.81
MA
0.2029
0.0998
2.03
2
Differencing: 1 regular difference
Number
of
observations:
differencing 99
Residuals:
SS = 99.2463
MS =
1.0232
Original
series
100,
after
(backforecasts excluded)
DF = 97
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
14.3(DF=10)
28.3(DF=22)
36.5(DF=34)
47.0(DF=46)
MTB > ARIMA 1 1 1 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.1283
0.1334
MA
1
0.8027
0.0799
T
0.96
10.04
Differencing: 1 regular difference
Number of observations:
Original series 100, after
differencing 99
Residuals:
SS = 100.421 (backforecasts excluded)
MS =
1.035 DF = 97
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
136
Lag
12
24
36
27.9(DF=22)
36.1(DF=34)
48
Chi-Square
48.2(DF=46)
13.3(DF=10)
MTB > ARIMA 2 1 1 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
* WARNING * Back forecasts not dying out rapidly
Final Estimates of Parameters
Type
Coef
StDev
AR
1
-1.1389
0.0987
AR
2
-0.1440
0.0983
MA
1
-0.9889
0.0002
T
-11.53
-1.47
-3987.49
Differencing: 1 regular difference
Number of observations:
Original series 100, after
differencing 99
Residuals:
SS = 134.250 (backforecasts excluded)
MS =
1.398 DF = 96
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
35.1(DF= 9)
53.5(DF=21)
66.6(DF=33)
83.2(DF=45)
MTB > ARIMA 1 1 2 'Sales';
SUBC>
NoConstant.
137
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
T
AR
1
-0.3476
0.4077
-0.85
MA
MA
1
2
0.2422
0.4506
0.3771
0.2656
0.64
1.70
Differencing: 1 regular difference
Number of observations:
differencing 99
Residuals:
SS = 97.2357
MS = 1.0129
Original
series
100,
after
(backforecasts excluded)
DF = 96
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
11.9(DF= 9)
25.0(DF=21)
32.1(DF=33)
41.8(DF=45)
MTB > ARIMA 2 1 2 'Sales';
SUBC>
NoConstant.
ARIMA Model
ARIMA model for Sales
Final Estimates of Parameters
Type
Coef
StDev
AR
1
-0.0691
0.3618
AR
2
-0.2941
0.1450
MA
1
0.5637
0.3737
MA
2
0.0840
0.3266
Differencing: 1 regular difference
138
T
-0.19
-2.03
1.51
0.26
Number
of
observations:
Original
series
100,
after
differencing 99
Residuals:
SS = 93.6368
MS = 0.9857
(backforecasts excluded)
DF = 95
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
36.5(DF=44)
11.0(DF= 8)
23.4(DF=20)
30.1(DF=32)
:7
;
ول ا. f
ذo[و
Model
σˆ 2
m
AIC
__________
________
___
_________
ARI (1,1)
1.359
2
34.368
1.080
3
13.619
1.035
2
7.4057
1.023
3
8.2706
1.035
3
9.4057
1.398
4
41.169
1.013
4
9.2689
0.986
5
8.5741
ARI ( 2,1)
IMA (1,1)
IMA (1, 2 )
ARIMA (1,1,1)
ARIMA ( 2,1,1)
ARIMA (1,1, 2 )
ARIMA ( 2,1, 2 )
min AIC ( m ) = 7.406
m
.ات:('C 1C و7Bا1(
اo%N 2!; آW
6
;ك2 . IMA(1,1) 1ذج ه1! H`Nأي ان أ
139
ﺍﻟﻔﺼﻞ ﺍﻟﺴﺎﺩﺱ
0ك ا/\ ا0ا ا)
ا9ار ا/7jذج ا7
Seasonal Autoregressive Integrated Moving Average Models
H]& "(
ا42 &;و4'&;ات زN ر3;C 4O.K;& ا!ط76"C 4!1!
ا4'&
ا
!;ت ا
.4' اوO ا4UU اوO Hم او آ2 ا4"( H او آ4 ونK و4". ارH آw!'
ر ا3;2 ان
@ ا
!;تA هH]& (C 4
;
ل ا3ا
S e a s o n a l T im e S e r ie s
z(t)
70
60
50
In d e x
50
100
150
S e a s o n a l T im e S e r ie s
1000
z(t)
900
800
700
600
In d e x
50
100
150
46ا1. O;LA! و=ق4!1!
ا4'&
اص ا
!;ت ا1 ف ;"ض1 Hb
ا اA ه7N
]!N SARIMA(p,d,q)(P,D,Q)s ك%;!
اw1;!
ا7&3;
ا7CاA
ار ا%9!ذج ا
H3K
اW;32 SARIMA(0,1,1)(1,1,0)12 ذج1!'
ا
140
(1 − Φ B ) (1 − B ) z = (1 − θ B ) a ,
s
t
1
t
1
at ∼ WN ( 0, σ 2 )
(p,d,q)(P,D,Q)s 4L
ر. ك%;!
اw1;!
ا7&3;
ا7CاA
ار ا%9ذج ا1! نFN مH3K.و
H3K
اW;32 SARIMA(p,d,q)(P,D,Q)s
φ p ( B ) Φ P ( B s ) (1 − B ) (1 − B s ) zt = δ + θ q ( B ) ΘQ ( B s ) at , at ∼ WN ( 0,σ 2 )
D
d
&ت7;
وا4!1!
اz ك%;!
اw1;!
وا7CاA
ار ا%9 !ل اθ q ( B ) وφ p ( B ) Q5
و7!1!
ا7CاA
ار ا%9 اH& Φ P ( B s ) = 1 + Φ1B s + Φ 2 B 2 s + ⋯ + Φ P B Ps وJ. '
اA! ه2 و7!1!
ك ا%;!
اw1;!
اH& ΘQ ( B s ) = 1 + Θ1B s + Θ2 B 2 s + ⋯ + ΘQ B Qs
.Multiplicative Seasonal Models 7b`;
ا7!1!
ذج ا1!'
.
at ∼ WN ( 0, σ 2 ) !' أنP &1Ob& ن13 4&دJ
ا
'!ذج اV!L 7N :4I5&
:0 اذج اv Vا ا9\ ا$ا واا9\ ا$دوال اا
wt = (1 − B ) (1 − B s ) zt ةJ;!
ا4!1!
ا4;!
ف ;[م ا1 4
;
ت اBJ;9 ا7N
d
D
SARMA(p,q)(P,Q)s ذج1!'
اV(;C 7;
وا
φ p ( B ) Φ P ( B s ) wt = δ + θ q ( B ) ΘQ ( B s ) at , at ∼ WN ( 0, σ 2 )
SARMA(0,1)(1,1)12 7b`;
ا7!1!
ذج ا1!'
7CاA
اw. ا
;ا4
; داK ف1
wt = Φ wt −12 + at − θ at −1 − Θat −12 + θ Θat −13 , at ∼ WN ( 0,σ 2 )
VB1;
اA وأwt 7N 4N"!
ا4
ا
!"د7N= `ب.
γ 0 = Φ γ 12 + σ 2 + θ 2σ 2 − Θ ( Φ − Θ ) σ 2 + θ Θ ( −Φ θ + θ Θ ) σ 2
=Φ γ 12 + σ 2 (1 + θ 2 ) + Θ ( Φ − Θ ) (1 + θ 2 )
=Φ γ 12 + σ 2 (1 + θ 2 ) 1 + Θ ( Φ − Θ )
VB1;
اA وأwt −12 7N 4N"!
ا4
ا
!"د7N= `ب.و
γ 12 = Φ γ 0 − Θσ 2 + θ Θ ( −θ ) σ 2
=Φ γ 0 − Θσ 2 (1 + θ 2 )
;J.
; اB"
اH%.و
1 + Θ2 − 2ΦΘ
1 − Φ2
2
Φ (Θ − Φ )
2
2
γ 12 = σ (1 + θ ) Φ − Θ +
1 − Φ 2
γ 0 = σ 2 (1 + θ 2 )
141
`2أ
γ 1 = E ( wt wt −1 )
=Φ γ 11 − θσ 2 − ΘE ( at −12 wt −1 ) + θ ΘE ( at −13wt −1 )
=Φ γ 11 − θσ 2 + θ Θ ( Φ − Θ ) σ 2
و
γ 11 = E ( wt wt −11 ) = Φ γ 1 + Θθσ 2
;J.
; اB"
اH%.و
( Θ − Φ )2
γ 1 = −θσ 1 +
1 − Φ 2
2
2
Φ (Θ − Φ )
γ 11 = θσ Θ − Φ −
1 − Φ 2
2
(ت أنU إ3!2 4J26
~ اb'.و
γ 2 = γ 3 = ⋯ = γ 10 = 0
γ 13 = γ 11
γ k = Φ γ k −12 , k > 13
7CاA
اw. ا
;ا4
داL1 4J.
ت اB"
اV!L &و
k =0
1,
θ
−
,
k =1
2
1+θ
0,
k = 2,...,10
γ
θ ( Θ − Φ )(1 − ΦΘ )
ρk = k =
, k = 11
γ 0 1 + θ 2 1 + Θ2 − 2ΦΘ
( Θ − Φ )(1 − ΦΘ )
,
k = 12
−
1 + Θ2 − 2ΦΘ
k = 13
ρ11 ,
Φρ ,
k > 13
k −12
142
:0 اذج اv ا9\ ا$دوال اا
wt = (1 − ΘB s ) at SARIMA(0,d,0)(0,D,1)s ذج1! -1
k =0
1,
Θ
ρk = −
,
k=s
2
1+ Θ
otherwise
0,
(1 − ΦB ) w
= at
s
t
SARIMA(0,d,0)(1,D,1)s ذج1! -2
k =0
1,
ks
ρ k = Φ , k = s, 2 s,...
0,
otherwise
wt = (1 − θ B ) (1 − ΘB s ) at
SARIMA(0,d,1)(0,D,1)s ذج1! -3
k =0
1,
θ
−
,
k =1
2
1+θ
θΘ
, k = s −1
2
2
ρ k = (1 + θ )(1 + Θ )
Θ
−
,
k=s
2
1+ Θ
ρ s −1 ,
k = s +1
otherwise
0,
(1 − ΦB ) w = (1 − ΘB ) a
s
SARIMA(0,d,0)(1,D,1)s ذج1! -4
s
t
t
k =0
1,
( Θ − Φ )(1 − ΦΘ ) k s −1
ρk = −
Φ , k = s, 2 s,...
2
1 + Θ − 2ΦΘ
otherwise
0,
(1 − ΦB ) w = (1 − θ B ) a
s
t
t
143
SARIMA(0,d,1)(1,D,0)s ذج1! -5
k =0
1,
θ
−
, k =1
2
1+θ
0,
k = 2,..., s − 2
ρk = θ Φ
−
, k = s −1
1+θ 2
k=s
Φ,
ρ s −1 ,
k = s +1
k > s +1
Φ ρ k − s ,
wt = (1 − θ1B − θ 2 B 2 )(1 − ΘB12 ) at
1,
− θ1 (1 − θ 2 ) ,
1 + θ12 + θ 22
θ2
−
,
1 + θ12 + θ 22
θ 2Θ
,
2
(1 + θ1 + θ 22 )(1 + Θ2 )
ρk =
θ1Θ (1 − θ 2 )
,
(1 + θ 2 + θ 2 )(1 + Θ2 )
1
2
Θ
,
−
2
1+ Θ
ρ s −1 ,
ρ ,
s −2
0,
SARIMA(0,d,2)(0,D,1)s ذج1! -6
k =0
k =1
k =2
k = s−2
k = s −1
k=s
k = s +1
k = s+2
otherwise
ء6 4!1!
ا4'&
!;ت ا7CاA
اw. ا
;ا4
&ت ا1
"\ ا. ف ;"ض1
.O
3ة أ3N
: SARIMA(0,d,1)(1,D,0)12 ذج1!'
4
;
ل ا3ا
144
(1) H3
Φ = 0.6, θ = 0.5
A C F
o f S A R I M
A ( 0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
C1
0 .5
0 .0
-0 .5
0
1 0
2 0
3 0
4 0
5 0
L a g
( 2) H3
Φ = 0.6, θ = −0.5
A C F
o f S A R I M
A (0 ,d ,1 ) ( 1 ,D ,0 ) 1 2
0 .7
0 .6
C1
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0
1 0
2 0
3 0
4 0
5 0
L a g
(3) H3
Φ = −0.6, θ = 0.5
A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2
C1
0 .5
0 .0
-0 .5
0
10
20
30
Lag
145
40
50
(4) H3
Φ = −0.6, θ = −0.5
A C F o f S A R IM A (0 ,d ,1 )(1 ,D ,0 )1 2
0 .5
C1
0 .0
-0 .5
5 0
4 0
20
3 0
1 0
0
L ag
دا اا \$ا9ا ا Vذج ا 0ا:YO
& ا
" 4.1إ;Jق و bCأ!ط دا
4ا
;ا w.ا
Aا 7Cا
!'
78ذج ا
! 4!1ا
;`4b
و
O'3و H3K.م FNن أLاء ا
'!1ذج ا
! 4!1و zا
! 4!1وا
;A!'C 7ج ا
!; w1ا
!;%ك
&[C 76"Cات ا 4و&[Cات ' 4(Lا
;[bت ا
! 4!1وzا
! 4!1و 7Nا
'!ذج ا
;7
1%Cي إ%ار ذاFN 7Cن ا
;ا6.ت ا
Aا 4Cا
. cut off "6B 76"C 48
ا3ل ا
;
69 4ء 3Nة \".دوال ا
;ا w.ا
Aا 7Cا
\"(
78ا
'!ذج :
H3 -1دا
4ا
;ا w.ا
Aا 7Cا
1!'
78ذج wt = (1 − ΘB12 ) at
ا( Θ = 0.6
A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2
0 .0
-0 .1
-0 .2
-0 .4
-0 .5
-0 .6
5 0
4 0
2 0
3 0
L a g
ب(
Θ = −0.6
146
1 0
0
C1
-0 .3
A C F o f S A R IM A (0 ,d ,0 )(0 ,D ,1 )1 2
0 .6
0 .5
0 .4
0 .3
C1
0 .2
0 .1
0 .0
-0 .1
-0 .2
4 0
5 0
2 0
3 0
0
1 0
L ag
H3 -2دا
4ا
;ا w.ا
Aا 7Cا
1!'
78ذج = at
(1 − ΦB ) w
12
t
ا( Φ = 0.6
A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2
0 .6
0 .5
0 .4
C1
0 .3
0 .2
0 .1
0 .0
4 0
5 0
2 0
3 0
0
1 0
L a g
ب( Φ = −0.6
A C F o f S A R IM A (0 ,d ,1 )(0 ,D ,0 )1 2
0 .0
-0 .1
-0 .2
C1
-0 .3
-0 .4
-0 .5
-0 .6
4 0
5 0
2 0
3 0
0
1 0
L a g
ﺃﻤﺜﻠﺔ :ﻝﻠﻤﺘﺴﻠﺴﻠﺔ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﻤﻭﺴﻤﻴﺔ )ﻓﻲ ﺠﻤﻴﻊ ﺍﻷﻤﺜﻠﺔ ﺍﻝﺘﺎﻝﻴﺔ ﺇﻗﺭﺃ ﺴﻁﺭﺍ ﺒﺴﻁﺭ(
)z(t
56.9
57.4
61.5
72.7
72.2
71.5
59.1
57.2
56.3
55.8
55.7
56.3
54.4
56.0
60.0
71.0
70.6
68.2
57.7
54.6
54.9
54.9
54.9
55.3
54.6
55.6
59.4
69.8
71.0
67.4
58.2
54.3
53.4
53.0
52.8
53.3
147
53.4
53.0
53.0
53.2
54.2
58.0
67.5
70.1
68.2
56.6
54.9
54.0
52.9
52.6
52.8
53.0
53.6
56.1
66.1
69.8
69.3
61.2
57.5
54.9
53.4
52.7
53.0
52.9
55.4
58.7
67.9
70.0
68.7
59.3
56.4
54.5
52.8
52.8
53.2
55.3
55.8
58.2
65.3
67.9
68.3
61.7
56.4
53.9
52.6
52.1
52.4
51.6
52.7
57.3
65.1
71.5
69.9
61.9
57.3
55.1
53.6
53.4
53.5
53.3
53.9
52.7
61.0
69.9
70.4
59.4
56.3
54.3
53.5
53.0
53.2
52.5
53.4
56.5
65.3
70.7
66.9
58.2
55.3
53.4
52.1
51.5
51.5
52.4
53.3
55.5
64.2
69.6
69.3
58.5
55.3
53.6
52.3
51.5
51.7
51.5
52.2
57.1
63.6
68.8
68.9
60.1
55.6
53.9
53.3
53.1
53.5
53.5
53.9
57.1
64.7
69.4
70.3
62.6
57.9
55.8
54.8
54.2
54.6
54.3
54.8
58.1
68.1
73.3
75.5
66.4
60.5
57.7
55.8 54.7 55.0 55.6 56.4 60.6 70.8 76.4 74.8 62.2
1 ه4;!
اH3
z(t)
7 0
6 0
5 0
In d e x
5 0
1 0 0
1 5 0
7CاA
اw. ا
;ا4
دا
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
2
Lag
1
2
3
4
5
6
7
8
9
10
11
12
C o rr
12
T
LBQ
Lag
C o rr
0 .7 7 1 0 .3 1
0 .3 3
2 .9 3
-0 .1 1 - 0 .9 9
-0 .3 8 - 3 .2 8
-0 .5 2 - 4 .2 2
-0 .5 6 - 4 .1 6
-0 .5 2 - 3 .5 0
-0 .3 8 - 2 .3 9
-0 .1 2 - 0 .7 1
0 .2 9
1 .7 6
0 .6 9
4 .1 5
0 .8 8
4 .8 6
1 0 8 .1 0
1 2 7 .3 4
1 2 9 .7 4
1 5 6 .8 3
2 0 7 .2 4
2 6 6 .5 5
3 1 6 .8 0
3 4 3 .7 1
3 4 6 .2 5
3 6 2 .0 8
4 5 3 .5 8
6 0 3 .8 7
13
14
15
16
17
18
19
20
21
22
23
24
0 .6 8
0 .2 8
-0 .1 2
-0 .3 6
-0 .4 9
-0 .5 3
-0 .4 9
-0 .3 7
-0 .1 4
0 .2 3
0 .6 1
0 .7 9
22
T
32
42
LBQ
Lag
C o rr
T
LBQ
Lag
C o rr
T
LBQ
3 .3 4 6 9 4 .2 8
1 .3 0 7 0 9 .8 2
-0 .5 3 7 1 2 .4 3
-1 .6 5 7 3 8 .3 0
-2 .2 1 7 8 6 .2 8
-2 .3 3 8 4 3 .0 4
-2 .1 0 8 9 2 .3 6
-1 .5 4 9 2 0 .3 9
-0 .5 6 9 2 4 .2 4
0 .9 5 9 3 5 .3 7
2 .4 6 1 0 1 1 .3 1
3 .1 0 1 1 4 0 .9 0
25
26
27
28
29
30
31
32
33
34
35
36
0 .6 1
0 .2 5
-0 .1 1
-0 .3 4
-0 .4 7
-0 .5 1
-0 .4 8
-0 .3 6
-0 .1 4
0 .1 9
0 .5 4
0 .7 2
2 .3 0
0 .9 1
-0 .4 2
-1 .2 4
-1 .6 7
-1 .8 0
-1 .6 5
-1 .2 3
-0 .4 9
0 .6 6
1 .8 2
2 .3 7
1 2 2 0 .0 7
1 2 3 3 .1 8
1 2 3 5 .9 8
1 2 6 1 .2 9
1 3 0 8 .3 2
1 3 6 4 .7 2
1 4 1 4 .0 8
1 4 4 2 .7 1
1 4 4 7 .3 4
1 4 5 5 .7 9
1 5 2 0 .8 9
1 6 3 6 .4 9
37
38
39
40
41
42
43
44
0 .5 7
0 .2 3
-0 .1 1
-0 .3 3
-0 .4 4
-0 .4 8
-0 .4 5
-0 .3 4
1 .8 2
0 .7 3
-0 .3 4
-1 .0 2
-1 .3 8
-1 .4 8
-1 .3 7
-1 .0 4
1 7 0 9 .3 6
1 7 2 1 .7 0
1 7 2 4 .4 3
1 7 4 9 .0 6
1 7 9 5 .0 0
1 8 4 9 .4 1
1 8 9 7 .0 5
1 9 2 5 .3 0
78
ا7CاA
اw.وا
;ا
148
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
2
12
22
32
42
Lag
PAC
T
Lag
P AC
T
Lag
P AC
T
Lag
P AC
T
1
2
3
4
5
6
7
8
9
10
11
12
0 .7 7
-0 .6 8
-0 .0 6
0 .0 1
-0 .4 2
-0 .1 8
-0 .1 1
-0 .2 2
0 .1 8
0 .5 7
0 .2 6
0 .1 6
1 0 .3 1
- 9 .0 1
- 0 .8 2
0 .0 7
- 5 .5 5
- 2 .4 5
- 1 .4 8
- 2 .9 6
2 .3 9
7 .6 5
3 .5 0
2 .1 6
13
14
15
16
17
18
19
20
21
22
23
24
- 0 .4 2
0 .2 9
0 .0 1
- 0 .0 7
0 .0 7
- 0 .0 1
- 0 .0 9
- 0 .0 5
- 0 .0 8
0 .1 0
- 0 .0 3
0 .0 3
- 5 .6 2
3 .8 9
0 .1 6
- 0 .9 2
0 .9 5
- 0 .1 1
- 1 .1 5
- 0 .7 2
- 1 .1 1
1 .3 8
- 0 .3 5
0 .3 7
25
26
27
28
29
30
31
32
33
34
35
36
- 0 .1 3
- 0 .0 0
0 .0 4
- 0 .1 0
- 0 .0 3
0 .0 1
0 .0 1
- 0 .0 4
- 0 .0 4
0 .0 1
- 0 .0 2
0 .0 7
- 1 .7 5
- 0 .0 5
0 .5 0
- 1 .3 0
- 0 .3 4
0 .1 9
0 .0 7
- 0 .5 5
- 0 .5 2
0 .1 5
- 0 .3 2
0 .9 0
37
38
39
40
41
42
43
44
- 0 .0 6
- 0 .0 5
0 .0 1
0 .0 0
- 0 .0 5
0 .0 5
- 0 .0 5
- 0 .0 3
- 0 .7 4
- 0 .6 2
0 .1 7
0 .0 0
- 0 .6 3
0 .6 6
- 0 .6 4
- 0 .4 2
.4J.
ل ا3 ا7N 4%P وا4!1!
ا!ط ا5
ﻤﺜﺎل ﺁﺨﺭ
z(t)
589
561
640
656
727
697
640
599
568
577
553
582
600
566
653
673
742
716
660
617
583
587
565
598
628
618
688
705
770
736
678
639
604
611
594
634
658
622
709
722
782
756
702
653
615
621
602
635
677
635
736
755
811
798
735
697
661
667
645
688
713
667
762
784
837
817
767
722
681
687
660
698
717
696
775
796
858
826
783
740
701
706
677
711
734
690
785
805
871
845
801
764
725
723
690
734
750
707
807
824
886
859
819
783
740
747
711
751
804
756
860
878
942
913
869
834
790
800
763
800
826
799
890
900
961
935
894
855
809
810
766
805
821
773
883
898
957
924
881
837
784
791
760
802
828
778
889
902
969
947
908
867
815
812
773
813
834
782
892
903
966
937
896
858
817
827
797
843
4;!
اH3
149
1000
900
z(t)
800
700
600
In d e x
50
100
150
7CاA
اw.ا
;ا
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1
0
0
0
0
0
-0
-0
-0
-0
-1
.0
.8
.6
.4
.2
.0
.2
.4
.6
.8
.0
2
L ag
1
2
3
4
5
6
7
8
9
10
11
12
C o rr
0
0
0
0
0
0
0
0
0
0
0
0
.8
.7
.6
.4
.4
.3
.4
.4
.5
.6
.7
.8
T
12
LB Q
9 1 1 .5 6 1 3 5 .9 4
8
6 .2 7 2 4 0 .1 3
2
4 .1 2 3 0 6 .7 2
9
2 .9 5 3 4 7 .9 7
3
2 .4 7 3 8 0 .0 9
8
2 .1 0 4 0 5 .0 2
1
2 .2 5 4 3 5 .5 4
5
2 .4 0 4 7 2 .3 7
6
2 .8 7 5 2 9 .0 7
9
3 .3 4 6 1 4 .2 7
7
3 .5 2 7 2 1 .7 2
4
3 .6 1 8 5 2 .4 1
Lag
1
1
1
1
1
1
1
2
2
2
2
2
3
4
5
6
7
8
9
0
1
2
3
4
22
C o rr
0
0
0
0
0
0
0
0
0
0
0
0
.7
.6
.4
.3
.3
.2
.2
.3
.4
.5
.6
.6
4
4
9
6
1
5
9
2
2
3
0
7
T
2
2
1
1
1
0
1
1
1
1
2
2
.9
.4
.7
.3
.0
.9
.0
.1
.4
.8
.0
.2
6
11
91
11
91
01
11
21
41
11
31
11
LB Q
9
0
0
0
1
1
1
1
1
2
3
4
5
3
7
9
1
2
4
6
9
5
2
1
4
0
4
9
7
9
5
5
9
3
5
5
.6
.0
.8
.6
.3
.7
.5
.4
.1
.8
.5
.2
8
9
5
8
9
6
9
2
3
1
1
9
Lag
2
2
2
2
2
3
3
3
3
3
3
3
5
6
7
8
9
0
1
2
3
4
5
6
32
C o rr
0
0
0
0
0
0
0
0
0
0
0
0
.5
.4
.3
.2
.1
.1
.1
.2
.2
.3
.4
.5
8
9
5
4
9
4
7
0
8
8
5
2
T
1
1
1
0
0
0
0
0
0
1
1
1
.8
.5
.0
.7
.5
.4
.5
.6
.8
.1
.3
.5
61
21
91
31
71
31
11
01
51
51
51
31
LB Q
4
5
5
5
5
5
5
5
6
6
6
7
8
3
5
6
7
7
8
9
1
4
8
4
3
0
6
8
5
9
5
3
0
1
5
2
.1
.9
.4
.2
.4
.5
.5
.6
.3
.5
.2
.9
1
9
5
0
4
5
2
2
5
7
7
5
Lag
3
3
3
4
4
4
7
8
9
0
1
2
42
C o rr
0
0
0
0
0
0
.4
.3
.2
.1
.0
.0
3
5
2
2
6
2
T
1
1
0
0
0
0
.2
.0
.6
.3
.1
.0
71
01
41
31
81
51
LB Q
7
8
8
8
8
8
8
1
2
2
2
2
4
0
1
4
5
5
.1
.6
.6
.6
.5
.6
7
0
8
7
6
2
78
ا7CاA
اw.وا
;ا
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
2
12
22
32
42
Lag
PAC
T
L ag
PAC
T
L ag
PAC
T
Lag
PAC
T
1
2
3
4
5
6
7
8
9
10
11
12
0 .8 9
-0 .0 8
-0 .2 8
0 .0 3
0 .3 5
-0 .0 8
0 .2 8
0 .0 9
0 .4 0
0 .3 0
0 .0 6
0 .2 2
1 1 .5 6
-1 .0 6
-3 .6 5
0 .4 2
4 .5 4
-1 .0 7
3 .6 7
1 .1 9
5 .1 7
3 .9 5
0 .8 1
2 .8 8
13
14
15
16
17
18
19
20
21
22
23
24
-0 .6 3
-0 .0 2
0 .0 7
-0 .0 4
-0 .0 9
-0 .0 4
-0 .0 5
0 .0 3
0 .0 4
0 .0 5
0 .0 5
0 .0 5
-8 .1 9
-0 .2 1
0 .9 5
-0 .5 2
-1 .1 2
-0 .4 9
-0 .6 0
0 .3 8
0 .4 6
0 .6 7
0 .6 0
0 .5 9
25
26
27
28
29
30
31
32
33
34
35
36
-0 .1 8
0 .0 8
0 .0 6
-0 .0 3
-0 .0 4
0 .0 0
-0 .0 6
-0 .0 1
-0 .0 1
0 .0 3
0 .0 0
0 .0 1
-2 .3 6
1 .0 6
0 .7 3
-0 .4 4
-0 .4 7
0 .0 2
-0 .7 2
-0 .1 1
-0 .1 8
0 .3 8
0 .0 3
0 .0 9
37
38
39
40
41
42
-0 .1 1
-0 .0 2
0 .0 4
-0 .0 3
-0 .0 8
0 .0 1
-1 .3 7
-0 .2 2
0 .5 1
-0 .4 2
-1 .0 6
0 .0 8
150
?z ل-
z(t)
302
262 218 175 100 077
242 181 107
056 049 047 047 071 151 244 280 230 185 148
098 061 046 045 055
049
042
043 047 049 069 152 205 246 294
046
074
048 115 185 276 220 181 151 083 055
103
200
237
247
215
040 044 063 085 185 247 231 167 117 079
182
080
046
065
045 040 038 041
069 152 232 282 255 161 107 053 040 039 034 035
056 097
210 260 257 210 125 080 042 035 031 032 050 092 189
256
250 198 136 073 039 032 030 031 045
H3K
اO
و
3 0 0
z(t)
2 0 0
1 0 0
0
In d e x
10
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
7CاA
اw. ا
;ا4
ودا
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1
0
0
0
0
0
-0
-0
-0
-0
-1
.0
.8
.6
.4
.2
.0
.2
.4
.6
.8
.0
5
L a g
1
2
3
4
5
6
7
C o rr
0
0
-0
-0
-0
-0
-0
.8
.4
.0
.4
.6
.7
.6
1
3
3
3
9
8
8
T
8
2
-0
-2
-4
-3
-3
.3
.8
.1
.7
.0
.9
.0
8
9
7
0
3
9
9
L B Q
1
1
2
2
7
9
9
1
6
3
8
2
2
2
3
6
5
9
.2
.3
.4
.1
.3
.1
.3
2
2
0
3
7
7
3
1 5
L a g
1
1
1
1
1
C o rr
T
8 -0 .4 3 -1 .7 8 3 1
9 -0 .0 4 -0 .1 8 3 1
0 0 .3 8 1 .5 4 3 2
1 0 .7 1 2 .8 2 3 8
2 0 .8 4 3 .1 0 4 7
3 0 .7 1 2 .4 2 5 3
4 0 .3 7 1 .2 0 5 5
L B Q
0
0
8
9
6
8
6
.6
.8
.2
.7
.2
.8
.1
6
8
2
2
2
4
0
L a g
1
1
1
1
1
2
2
5
6
7
8
9
0
1
2 5
C o rr
-0
-0
-0
-0
-0
-0
-0
.0
.3
.6
.6
.5
.3
.0
3
8
0
7
9
7
4
T
-0
-1
-1
-2
-1
-1
-0
.0
.1
.8
.0
.7
.0
.1
8
9
8
3
2
5
0
L B Q
5
5
6
6
7
7
7
5
7
2
7
2
4
4
6
4
0
8
4
2
2
.1
.0
.2
.3
.1
.2
.3
8
7
2
6
7
1
8
L a g
2
2
2
2
2
2
3
4
5
6
C o rr
0
0
0
0
0
.3
.6
.7
.5
.3
3
1
1
9
1
T
0
1
1
1
0
.9
.7
.9
.5
.8
2
1
3
5
0
L B Q
7
8
8
9
9
5
0
7
2
4
6
8
9
8
2
.9
.9
.4
.4
.4
8
7
2
1
6
78
ا7CاA
اw. ا
;ا4
ودا
151
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r z ( t )
1
0
0
0
0
0
-0
-0
-0
-0
-1
.0
.8
.6
.4
.2
.0
.2
.4
.6
.8
.0
5
L a g
1
2
3
4
5
6
7
P A C
0
-0
-0
-0
-0
-0
-0
.8
.7
.2
.2
.1
.3
.1
1
0
9
2
6
2
6
T
8
-7
-2
-2
-1
-3
-1
.3
.1
.9
.2
.6
.2
.6
1 5
L a g
8
7
8
8
1
9
0
1
1
1
1
1
P A C
8 -0 .0 3
9 0 .1 9
0 0 .2 7
1 0 .1 8
2 0 .0 7
3 -0 .1 0
4 -0 .1 5
-0
1
2
1
0
-0
-1
.3
.9
.7
.8
.7
.9
.5
T
L a g
1
1
6
2
4
9
2
1
1
1
1
1
2
2
2 5
P A C
5
0 .1
6
0 .0
7
0 .1
8
0 .0
9 -0 .0
0 -0 .0
1
0 .0
9
8
1
4
2
3
4
1
0
1
0
-0
-0
0
.9
.8
.1
.3
.2
.2
.4
T
L a g
3
4
0
9
2
9
6
2
2
2
2
2
P A C
2 0 .0
3 0 .0
4 -0 .0
5 -0 .0
6 0 .1
1
7
7
2
0
T
0
0
-0
-0
1
.1
.7
.7
.2
.0
5
0
2
3
1
.4%P وا4!1& (ي ا!طC @ ا
!;تA هHوآ
:YO ا0ت ا
ا,ذج ا7 v 4 إ{ق دوال
ن =قFN ARIMA & !ذج4+ 4
5 7 ه4!1!
ا4'&
! ان !ذج ا
!;ت ا.
ذج1!'
&"
) ا2JCذج و1!'
اH3 7 ا
;"فQ5 & 4J.
ق ا6
~ اb 7 هO"& H&";
ا
z '!ذجJ. در'ه7;
ت اXق وا
!"د6
اV!L .:(';
) اU & و4%b;
;(رات ا9وا
.wJN ^P1;
("\ ا
'!ذج:(';
; دوال اK ف1 .'( ه6'C 4!1!
ا
: SARIMA(0,0,0)(0,1,1)12 ذج1!'
:(';
ا4
دا-1
H3K
ذج ا1!'
اW;32و
(1 − B ) z = (1 − ΘB ) a
12
12
t
t
4Bوb
ا4
& ا
!"د
zn +ℓ = zn +ℓ−12 + an +ℓ − Θan + ℓ−12
7
;
ات آ:(';
ل ا1%
ا3!2
zn (1) = zn −11 − Θan −11
zn ( 2 ) = zn −10 − Θan −10
⋮
zn (12 ) = zn − Θan
zn ( ℓ ) = zn ( ℓ − 12 ) , ℓ ≥ 12
أو
zn +ℓ −12 − Θan + ℓ−12 ,
zn ( ℓ ) =
zn ( ℓ − 12 ) ,
ℓ = 1, 2,...,12
ℓ > 12
^ أنPوا
152
zn (1) = zn (13) = zn ( 25 ) = ⋯
zn ( 2 ) = zn (14 ) = zn ( 26 ) = ⋯
⋮
zn (12 ) = zn ( 24 ) = zn ( 36 ) = ⋯
:(';
ء ا6 أ2(C
V en ( ℓ ) = σ 2 (1 + ψ 12 + ⋯ + ψ ℓ2−1 )
(f
ه ذ.) 4B"
. 6"C اوزان4
ودا
1 − Θ,
0,
ψj =
j = 12, 24,...
otherwise
:(';
ء ا6 أ2(C 4Y+ 7N \ اوزان21";.و
ℓ − 1
2
V en ( ℓ ) = σ 2 1 +
(1 − Θ )
12
. x & ^%
ا
ء ا7'"C x Q5
: SARIMA(0,1,1)(0,1,1)12 ذج1!'
:(';
ا4
دا-2
H3K
ذج ا1!'
اW;32و
(1 − B ) (1 − B12 ) zt = (1 − θ B ) (1 − ΘB12 ) at
4Bوb
ا4
& ا
!"د
zn +ℓ = zn +ℓ −1 + zn +ℓ −12 − zn +ℓ −13 + an + ℓ − θ an +ℓ −1 − Θan + ℓ−12 + θ Θan + ℓ−13
7
;
ات آ:(';
ل ا1%
ا3!2
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12
zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11
⋮
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1
zn (13) = zn (12 ) + zn (1) − zn + θ Θan
zn ( ℓ ) = zn ( ℓ − 1) + zn ( ℓ − 12 ) − zn ( ℓ − 13)
4
) أوJ. اA3وه
153
zn (1) = zn + zn −11 − zn −13 − θ an − Θan −11 + θ Θan −12
zn ( 2 ) = zn (1) + zn −10 − zn −11 − Θan −10 + θ Θan −11
⋮
zn (12 ) = zn (11) + zn − zn −1 − Θan + θ Θan −1
zn (13) = zn (12 ) + zn (1) − zn + θ Θan
42ار3C 4Bو
zn ( ℓ ) = zn ( ℓ − 1) + zn ( ℓ − 12 ) − zn ( ℓ − 13) , ℓ > 13
.ات:(';
ب & ا16!
ا
"د ا1C 3!2
:0ت ا
ا, اv 0ت دراrK و-
أ
:4
;
هات اK!
( ا6'2 SARIMA 48 & ذج1! د2ول إ% ف1 : (1) ل-
z(t)
589
561
640
656
727
697
640
599
568
577
553
582
600
566
653
673
742
716
660
617
583
587
565
598
628
618
688
705
770
736 678
639
604
611
594
634
658
622
709
722
782
756
653
615
621
602
635
677
635
736
755
688
713
667
762
784
837
817
767
722
681
687
660
698
717
696
775
796
858
826
783
740
701
706
677
711
734
690
785
805
871
845
801
764
725
723
690
734
750
707
807
824
886
859
819
783
740
747
711
751
804
756
860
878
942
913
869
834
790
800
763
800 826
799
890
900
961
935
894
855
809
810
766
805
821
773
883
898
957
924
881
837
784
791
760
802
828
778
889
902
969
947
908
867
815
812
773
813
834
782
892
903
966
937
896
858
817
827
797
811
702
798
735
697
661
667
645
843
هاتK!
7'&
اw6[!
وا
154
1 00 0
z(t)
90 0
80 0
70 0
60 0
In d e x
5 0
1 00
15 0
7!Uرz1
H21%;. X أو2(;
اS(] f
A
w1;!
وا2(;
ا7N ةJ;& z 4;!
ان ا52
O
7'&
اw6[!
و) اyt = ln ( zt ) أي
6 .9
6 .8
y(t)
6 .7
6 .6
6 .5
6 .4
6 .3
In d e x
5 0
1 0 0
1 5 0
قb
اA{ f
A
w1;!
ا7N ةJ;& z الCX 3
و2(;
ا7N تJ; ا4;!
ان ا5
7
;
اH3K
اO
وxt = (1 − B ) yt = (1 − B ) ln ( zt ) اول
0 .1 5
y(t)-y(t-1)
0 .1 0
0 .0 5
0 .0 0
-0 .0 5
-0 .1 0
In d e x
5 0
1 0 0
1 5 0
w. وا
;ا7CاA
اw. إ
دوال ا
;اI''
.w1;!
وا2(;
& اH آ7N ةJ;& نs ا4;!
ا
O
78
ا7CاA
ا
155
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
10
20
30
L ag
C o rr
T
LB Q
Lag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1
2
3
4
5
6
7
8
9
10
11
12
0 .0 1
0 .2 5
- 0 .0 8
- 0 .3 7
- 0 .0 6
- 0 .5 0
- 0 .0 4
- 0 .3 5
- 0 .0 5
0 .2 3
0 .0 1
0 .9 0
0 .1 2
3 .2 3
-1 .0 1
-4 .4 8
-0 .6 8
-5 .4 1
-0 .3 3
-3 .3 0
-0 .4 3
2 .0 1
0 .1 1
7 .7 4
0 .0 2
1 0 .7 0
1 1 .8 8
3 5 .6 0
3 6 .2 8
7 9 .9 7
8 0 .1 9
1 0 2 .3 7
1 0 2 .8 0
1 1 2 .2 9
1 1 2 .3 2
2 6 1 .4 1
13
14
15
16
17
18
19
20
21
22
23
24
0 .0 2
0 .2 3
- 0 .0 7
- 0 .3 4
- 0 .0 6
- 0 .4 6
- 0 .0 3
- 0 .3 2
- 0 .0 5
0 .2 1
0 .0 1
0 .8 2
0 .1 2
1 .5 1
- 0 .4 7
- 2 .1 8
- 0 .3 9
- 2 .8 8
- 0 .2 0
- 1 .9 3
- 0 .2 6
1 .2 0
0 .0 8
4 .7 6
2 6 1 .4 7
2 7 1 .2 9
2 7 2 .3 0
2 9 3 .7 3
2 9 4 .4 5
3 3 4 .6 7
3 3 4 .8 8
3 5 4 .9 6
3 5 5 .3 6
3 6 3 .5 5
3 6 3 .5 8
4 9 7 .3 0
25
26
27
28
29
30
31
32
33
34
35
36
0 .0 2
0 .2 1
-0 .0 6
-0 .3 1
-0 .0 6
-0 .4 2
-0 .0 2
-0 .3 0
-0 .0 5
0 .1 8
0 .0 0
0 .7 6
0 .0 8
1 .1 0
- 0 .2 9
- 1 .5 7
- 0 .2 9
- 2 .0 9
- 0 .1 2
- 1 .4 7
- 0 .2 4
0 .8 9
0 .0 2
3 .6 3
4 9 7 .3 5
5 0 6 .5 0
5 0 7 .1 6
5 2 6 .4 5
5 2 7 .1 4
5 6 3 .0 5
5 6 3 .1 6
5 8 2 .0 5
5 8 2 .5 8
5 8 9 .8 4
5 8 9 .8 5
7 1 3 .1 4
40
L ag
C o rr
T
LBQ
3 7 0 .0 1 0 .0 5
3 8 0 .2 0 0 .9 0
3 9 - 0 .0 4 -0 .1 9
4 0 - 0 .2 7 -1 .2 1
4 1 - 0 .0 5 -0 .2 4
7 1 3 .1 6
7 2 2 .0 6
7 2 2 .4 5
7 3 8 .8 6
7 3 9 .5 1
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
10
20
30
40
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1
2
3
4
5
6
7
8
9
10
11
12
0 .0 1
0 .2 5
-0 .0 9
-0 .4 6
-0 .0 2
-0 .3 5
-0 .1 4
-0 .4 8
-0 .4 1
-0 .2 3
-0 .5 8
0 .6 3
0 .1 2
3 .2 3
- 1 .2 0
- 5 .9 7
- 0 .2 2
- 4 .5 2
- 1 .8 2
- 6 .2 6
- 5 .3 6
- 2 .9 9
- 7 .5 5
8 .0 8
13
14
15
16
17
18
19
20
21
22
23
24
- 0 .0 4
- 0 .3 3
0 .0 0
0 .1 8
0 .0 1
0 .0 8
- 0 .0 8
0 .0 8
0 .0 2
- 0 .0 4
0 .0 2
0 .0 8
- 0 .4 8
- 4 .2 6
0 .0 6
2 .2 9
0 .0 7
1 .0 6
- 1 .0 4
0 .9 8
0 .2 9
- 0 .5 3
0 .2 9
1 .0 4
25
26
27
28
29
30
31
32
33
34
35
36
-0 .1 3
-0 .0 5
0 .0 6
0 .0 0
-0 .0 7
0 .0 3
0 .0 7
-0 .0 1
-0 .1 0
0 .0 6
0 .0 3
0 .0 1
- 1 .6 7
- 0 .6 1
0 .8 4
0 .0 5
- 0 .9 4
0 .3 4
0 .9 3
- 0 .0 9
- 1 .3 1
0 .8 0
0 .3 9
0 .1 3
37
38
39
40
41
-0 .0 4
0 .0 2
0 .0 4
0 .0 6
-0 .0 2
-0 .5 1
0 .3 0
0 .4 8
0 .8 1
-0 .2 0
و12 تb[;
' اO!B ! ن1& ةJ;& z 4;!
ان ا7CاA
اw. ا
;ا4
& دا52
wt = (1 − B12 ) (1 − B ) ln ( zt ) اول7!1!
ق اb
اA{ f
A
ءw(. &[;C 36 و24
2b;
ا اA" ه. O!و
y(t)-y(t-1)12
0 .0 5
0 .0 0
-0 .0 5
In d e x
50
100
150
O
78
ا7CاA
اw. وا
;ا7CاA
اw. دوال ا
;اL1
156
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
15
25
Lag
C o rr
T
LBQ
Lag
C o rr
T
LBQ
Lag
C o rr
T
LBQ
1
2
3
4
5
6
7
8
9
10
11
12
- 0 .2 1
- 0 .0 1
0 .1 0
- 0 .1 3
- 0 .1 0
- 0 .0 2
0 .1 2
0 .0 5
- 0 .0 5
0 .1 3
- 0 .0 1
- 0 .4 4
-2 .6 5
-0 .1 3
1 .1 4
-1 .4 9
-1 .1 1
-0 .2 8
1 .3 3
0 .5 4
-0 .5 8
1 .4 5
-0 .0 9
-4 .9 0
7 .1 5
7 .1 7
8 .6 3
1 1 .2 0
1 2 .6 7
1 2 .7 7
1 4 .9 5
1 5 .3 1
1 5 .7 5
1 8 .4 9
1 8 .5 0
5 0 .8 8
13
14
15
16
17
18
19
20
21
22
23
24
0 .1 8
-0 .0 7
-0 .0 5
0 .0 3
0 .1 2
-0 .0 0
-0 .1 1
0 .0 3
-0 .0 2
-0 .0 9
0 .1 1
-0 .0 4
1 .8 0
-0 .7 0
-0 .4 8
0 .2 6
1 .1 3
-0 .0 2
-1 .0 8
0 .2 5
-0 .2 0
-0 .8 0
0 .9 9
-0 .3 8
5 6 .6 4
5 7 .5 4
5 7 .9 7
5 8 .1 0
6 0 .5 8
6 0 .5 8
6 2 .9 1
6 3 .0 4
6 3 .1 2
6 4 .4 5
6 6 .5 2
6 6 .8 2
25
26
27
28
29
30
31
32
33
34
35
36
0 .0 7
- 0 .0 0
- 0 .0 6
0 .0 3
- 0 .1 0
0 .0 1
0 .0 4
0 .0 0
0 .0 2
0 .0 0
0 .0 9
- 0 .0 6
0 .6 2
- 0 .0 2
- 0 .5 7
0 .3 0
- 0 .9 0
0 .0 8
0 .3 8
0 .0 4
0 .1 9
0 .0 3
0 .8 0
- 0 .5 1
6 7 .6 7
6 7 .6 7
6 8 .3 8
6 8 .5 8
7 0 .4 0
7 0 .4 1
7 0 .7 4
7 0 .7 4
7 0 .8 3
7 0 .8 3
7 2 .3 6
7 2 .9 9
35
L ag
C o rr
T
LBQ
3 7 - 0 .0 6 - 0 .5 4
3 8 0 .0 3 0 .2 6
7 3 .7 1
7 3 .8 8
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r y ( t ) - y ( t
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
5
15
25
35
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1
2
3
4
5
6
7
8
9
10
11
12
-0 .2 1
-0 .0 6
0 .0 8
-0 .0 9
-0 .1 5
-0 .1 0
0 .1 1
0 .1 1
-0 .0 4
0 .0 6
0 .0 5
-0 .4 2
- 2 .6 5
- 0 .7 3
1 .0 6
- 1 .1 6
- 1 .8 2
- 1 .1 9
1 .4 1
1 .4 2
- 0 .5 1
0 .8 0
0 .5 9
- 5 .2 3
13
14
15
16
17
18
19
20
21
22
23
24
- 0 .0 0
- 0 .0 2
0 .0 0
- 0 .1 1
0 .0 4
0 .0 1
- 0 .0 4
- 0 .0 3
- 0 .0 3
0 .0 3
0 .0 7
- 0 .2 7
- 0 .0 3
- 0 .2 5
0 .0 0
- 1 .4 0
0 .4 7
0 .1 2
- 0 .5 2
- 0 .3 5
- 0 .3 1
0 .3 2
0 .9 1
- 3 .4 1
25
26
27
28
29
30
31
32
33
34
35
36
0 .1 2
-0 .0 1
-0 .1 2
-0 .0 7
0 .0 4
-0 .0 4
-0 .0 5
0 .0 2
-0 .0 1
-0 .0 2
0 .1 8
-0 .2 1
1 .5 1
- 0 .1 5
- 1 .4 4
- 0 .9 1
0 .4 4
- 0 .5 3
- 0 .6 0
0 .1 9
- 0 .1 2
- 0 .1 9
2 .2 1
- 2 .5 7
37
38
0 .1 0
-0 .0 6
1 .2 3
-0 .7 9
wt = (1 − B12 ) (1 − B ) ln ( zt ) 4;!
78
ا7CاA
اw. وا
;ا7CاA
اw.& ا!ط دوال ا
;ا
ذج1!'
( ا6 اي0 و0 7 ه4'3!!
اq وp )B ان
(1 − B ) (1 − B ) ln ( z ) = (1 − ΘB ) a
12
12
t
t
ذج1!'
ا اA( ه62 MINITAB 7N 7
;
ا& اSARIMA(0,1,0)(0,1,1)12 1ه
ARIMA 0 1 0 0 1 1 12 'y(t)' ;
NoConstant.
zt = e yt H21%;
ي ا48O'
_ ا8;'
ل ا1%
وyt = ln ( zt ) '&[; ا' ا5X
:fا
MTB > Name c14 = 'RESI3' c15 = 'FITS3'
MTB > ARIMA 0 1 0 0 1 1 12 'y(t)' 'RESI3' 'FITS3';
SUBC>
NoConstant;
SUBC>
Forecast 24 c7 c8 c9;
157
SUBC>
GACF;
SUBC>
GPACF;
SUBC>
SUBC>
GHistogram;
GNormalplot.
ARIMA Model
ARIMA model for y(t)
Estimates at each iteration
Iteration
SSE
Parameters
0
0.0228597
0.100
1
0.0204943
0.250
2
0.0187066
0.400
3
0.0174234
0.550
4
0.0169841
0.684
5
0.0169841
0.683
6
0.0169841
0.683
Relative change in each estimate less than
Final Estimates of Parameters
Type
Coef
StDev
SMA 12
0.6831
0.0610
0.0010
T
11.20
Differencing: 1 regular, 1 seasonal of order 12
Number of observations:
Original series 168, after
differencing 155
Residuals:
SS = 0.0165799 (backforecasts excluded)
MS = 0.0001077 DF = 154
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
9.0(DF=11)
29.9(DF=23)
44.5(DF=35)
59.4(DF=47)
Forecasts from period 168
95 Percent Limits
158
Period
Forecast
Lower
Upper
Actual
169
170
6.76750
6.70901
6.74716
6.68024
6.78784
6.73778
171
6.83815
6.80292
6.87338
172
6.85381
6.81313
6.89450
173
6.92288
6.87739
6.96836
174
6.89349
6.84366
6.94331
175
176
6.84654
6.80008
6.79272
6.74255
6.90035
6.85761
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
6.74395
6.75028
6.70664
6.75999
6.79052
6.73203
6.86117
6.87684
6.94590
6.91651
6.86956
6.82310
6.76697
6.77330
6.72966
6.78301
6.68293
6.68596
6.63918
6.68952
6.71514
6.65203
6.77680
6.78832
6.85342
6.82023
6.76962
6.71963
6.66009
6.66312
6.61627
6.66649
6.80497
6.81461
6.77410
6.83045
6.86590
6.81203
6.94554
6.96535
7.03838
7.01279
6.96950
6.92657
6.87385
6.88349
6.84305
6.89952
1 ه4;!
@ اAO
;حJ!
ذج ا1!'
أي أن ا
(1 − B ) (1 − B ) ln ( z ) = (1 − 0.683B ) a ,
12
12
t
t
at ∼ N ( 0, 0.0001077 )
ان5X
Θ = 0.683, s.e. ( Θ ) = 0.061, t = 11.2
.421'"!
ا7
)"!
أي ان ا
:I> اا/a
\0إ?ر ا
MTB > ZTest 0.0 0.0103778 'RESI3';
159
SUBC>
Alternative 0.
Z-Test
Test of mu = 0.000000 vs mu not = 0.000000
The assumed sigma = 0.0104
Variable
RESI3
StDev
SE Mean
Z
P
155 -0.000111 0.010375
N
Mean
0.000834
-0.13
0.89
اb+ 7Bا1(
اw1;& \ أنNX اي0.05 & ( اآ7 وهP-value=0.89 ان ا
ـ5X
Iإ?ر ا اا
MTB > Runs 0 'RESI3'.
Runs Test
RESI3
K =
0.0000
The observed number of runs = 70
The expected number of runs = 78.1097
72 Observations above K
83 below
The test is significant at 0.1893
Cannot reject at alpha = 0.05
7Bا1(
ا48ا1K 4PN \NX ' اي ا0.1893 ' ي1'"& ;(ر9ا
:Iل اا,0إ?ر إ
78
ا7CاA
اw. وا
;ا7CاA
اw.دوال ا
;ا
160
A C F o f R e s id u a ls f o r y ( t )
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s )
1 .0
0 .8
Autocorrelation
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
P A C F o f R e s id u a ls f o r y ( t )
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s )
1 .0
Partial Autocorrelation
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
.4J;& 7ON 4"(= S وإذا آ46. &;اz O ا
(`ء أي ا4`
ا!ط ا76"C O ا5
:I ااi إ?ر
161
Histogram of the Residuals
(response is y(t))
Frequency
30
20
10
0
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
Residual
Normal Probability Plot of the Residuals
(response is y(t))
0.04
0.03
Residual
0.02
0.01
0.00
-0.01
-0.02
-0.03
-3
-2
-1
0
1
2
3
Normal Score
K-S test for Residuals
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
RESI3
Average: -0.0001115
StDev: 0.0103754
N: 155
Kolmogorov-Smirnov Normality Test
D+: 0.074 D-: 0.045 D : 0.074
Approximate P-Value: 0.041
' ي1'"& ;(ر9 اذا ا0.05 & HB ا7 وه0.041 76"2 K-S ;(ر9 P-value ان ا
ـ5X
.7Bا1(
ا4"(= 4PN \NX ايα = 0.05
162
:ام اذج:0c$ 4ا
)
. O! و:('C ;اتN 95% V& 4(J;& 4!B 24 :(';
. '!B 4J.
ت اL[!
ا7N
:7
;
ا
1150
Forecast
1050
950
850
750
0
5
10
15
20
25
T im e
:(';
ود ا5ات و:(';
اV& 4;!
7
;
وا
) ا
1150
Forecast
1050
950
850
750
650
550
0
100
200
Time
: 0 دراK
:4
;
هات اK!
( ا6'2 SARIMA 48 & ذج1! د2ول إ% ف1
z(t)
56.3
55.7
55.8
56.3
57.2
59.1
71.5
72.2
72.7
61.5
57.4
56.9
55.3
54.9
54.9
54.9
54.6
57.7
68.2
70.6
71.0
60.0
56.0
54.4
163
53.3
52.8
53.0
53.4
54.3
58.2
67.4
71.0
69.8
59.4
55.6
54.6
53.4
53.0
53.0
53.2
54.2
58.0
67.5
70.1
68.2
56.6
54.9
54.0
52.9
52.6
52.8
53.0
53.6
56.1
66.1
69.8
69.3
61.2
57.5
54.9
53.4
52.7
53.0
52.9
55.4
58.7
67.9
70.0
68.7
59.3
56.4
54.5
52.8
52.8
53.2
55.3
55.8
58.2
65.3
67.9
68.3
61.7
56.4
53.9
52.6
52.1
52.4
51.6
52.7
57.3
65.1
71.5
69.9
61.9
57.3
55.1
53.6
53.4
53.5
53.3
53.9
52.7
61.0
69.9
70.4
59.4
56.3
54.3
53.5
53.0
53.2
52.5
53.4
56.5
65.3
70.7
66.9
58.2
55.3
53.4
52.1
51.5
51.5
52.4
53.3
55.5
64.2
69.6
69.3
58.5
55.3
53.6
52.3
51.5
51.7
51.5
52.2
57.1
63.6
68.8
68.9
60.1
55.6
53.9
53.3
53.1
53.5
53.5
53.9
57.1
64.7
69.4
70.3
62.6
57.9
55.8
54.8
54.2
54.6
54.3
54.8
58.1
68.1
73.3
75.5
66.4
60.5
57.7
55.8
54.7
55.0
55.6
56.4
60.6
70.8
76.4
62.2
7'&
اw6[!
ا
7 0
z(t)
74.8
6 0
5 0
In d e x
5 0
1 0 0
1 5 0
wt = (1 − B ) zt w1;!
ار اJ;9 ق اولb
اA{.
164
y(t)
1 0
0
-1 0
In d e x
5 0
1 0 0
1 5 0
78L 7C ذاw.اC و7C ذاw.اC دوالO
و
Autocorrelation
Autocorrelation Function for y(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
8
9
10
11
12
0.47
-0.02
-0.37
-0.28
-0.21
-0.19
-0.21
-0.27
-0.32
-0.00
0.46
0.86
6.32
-0.22
-4.10
-2.82
-2.00
-1.77
-1.92
-2.46
-2.85
-0.02
3.91
6.71
40.60
40.67
65.83
80.10
87.92
94.41
102.29
115.89
135.47
135.48
176.37
318.37
13
14
15
16
17
18
19
20
21
22
23
24
0.43
-0.01
-0.33
-0.25
-0.19
-0.17
-0.19
-0.25
-0.31
-0.01
0.42
0.79
2.76
-0.03
-2.00
-1.49
-1.12
-0.97
-1.09
-1.42
-1.76
-0.06
2.35
4.28
354.69
354.69
375.77
388.04
395.15
400.68
407.71
419.89
439.21
439.24
475.10
602.75
25
26
27
28
29
30
31
32
33
34
35
36
0.41
-0.00
-0.29
-0.23
-0.17
-0.16
-0.17
-0.23
-0.28
-0.01
0.37
0.72
2.06
-0.01
-1.43
-1.08
-0.82
-0.78
-0.83
-1.06
-1.31
-0.06
1.70
3.27
638.60
638.60
656.89
667.77
674.14
679.93
686.56
697.63
714.89
714.93
745.11
860.42
37
38
39
40
41
42
43
44
0.40
0.02
-0.27
-0.21
-0.17
-0.15
-0.16
-0.20
1.72
0.07
-1.14
-0.89
-0.71
-0.61
-0.65
-0.83
896.48
896.54
913.29
923.75
930.47
935.55
941.37
950.82
2
12
22
32
42
Partial Autocorrelation
Partial Autocorrelation Function for y(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
Lag PAC
1
2
3
4
5
6
7
8
9
10
11
12
0.47
-0.32
-0.30
0.09
-0.23
-0.25
-0.16
-0.43
-0.64
-0.31
-0.16
0.41
12
T
6.32
-4.21
-4.00
1.22
-3.05
-3.33
-2.09
-5.67
-8.56
-4.11
-2.18
5.42
Lag PAC
13
14
15
16
17
18
19
20
21
22
23
24
-0.27
-0.01
0.01
-0.07
-0.01
0.02
0.03
0.07
-0.12
-0.02
-0.06
0.13
22
T
-3.62
-0.17
0.13
-0.95
-0.16
0.24
0.44
0.89
-1.62
-0.26
-0.86
1.74
32
Lag PAC
25
26
27
28
29
30
31
32
33
34
35
36
-0.02
-0.06
0.07
-0.00
-0.02
-0.04
0.01
0.03
-0.02
0.00
-0.09
0.05
T
-0.33
-0.74
0.99
-0.05
-0.26
-0.60
0.20
0.37
-0.25
0.06
-1.17
0.61
42
Lag PAC
37
38
39
40
41
42
43
44
0.03
-0.03
-0.01
0.02
-0.04
0.03
-0.00
0.03
T
0.38
-0.35
-0.17
0.23
-0.56
0.35
-0.00
0.43
4;!
';_ ا2 وwt = (1 − B12 ) (1 − B ) zt أي12 4(C
& ا7!1& 2bC ;ج إ%C Oى ا
4
;
ا
165
5
w(t)
0
-5
In d e x
50
100
150
78L 7C ذاw.اC و7C ذاw.اC دوالO
و
Autocorrelation
A u t o c o r r e la t io n F u n c t io n f o r w ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
- 1 .0
10
20
30
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
L ag
C o rr
T
LBQ
1
2
3
4
5
6
7
8
9
10
11
12
-0 .0 1
-0 .1 9
-0 .2 4
-0 .0 0
0 .0 0
-0 .0 4
-0 .0 4
-0 .0 3
0 .2 2
0 .1 0
0 .0 8
-0 .4 2
-0 .1 9
-2 .4 8
-2 .9 7
-0 .0 3
0 .0 3
-0 .5 0
-0 .4 9
-0 .3 1
2 .5 6
1 .1 6
0 .9 3
-4 .6 3
0 .0 4
6 .3 1
1 6 .1 0
1 6 .1 0
1 6 .1 0
1 6 .4 1
1 6 .7 1
1 6 .8 3
2 5 .2 6
2 7 .1 3
2 8 .3 7
5 9 .4 0
13
14
15
16
17
18
19
20
21
22
23
24
-0 . 0 8
0 .1 0
0 .1 2
-0 . 0 2
-0 . 0 3
0 .1 0
0 .0 7
-0 . 0 1
-0 . 2 0
-0 . 0 2
0 .1 0
0 .1 4
-0 .8 4
0 .9 8
1 .1 5
-0 .1 7
-0 .2 8
1 .0 1
0 .7 0
-0 .0 5
-1 .9 4
-0 .2 3
0 .9 5
1 .3 0
6 0 .6 9
6 2 .5 0
6 5 .0 1
6 5 .0 7
6 5 .2 2
6 7 .2 4
6 8 .2 3
6 8 .2 4
7 5 .9 4
7 6 .0 5
7 8 .0 2
8 1 .7 8
25
26
27
28
29
30
31
32
33
34
35
36
-0 .0 8
-0 .1 3
-0 .0 1
0 .0 6
0 .1 2
-0 .1 5
-0 .0 8
-0 .0 3
0 .1 9
0 .1 7
-0 .1 0
-0 .2 2
-0 .7 6
-1 .1 8
-0 .1 3
0 .5 6
1 .0 9
-1 .3 6
-0 .7 2
-0 .2 8
1 .6 6
1 .4 9
-0 .8 6
-1 .9 2
8 3 .0 9
8 6 .2 8
8 6 .3 3
8 7 .0 7
8 9 .9 4
9 4 .4 7
9 5 .8 0
9 6 .0 1
1 0 3 .2 0
1 0 9 .1 9
1 1 1 .2 4
1 2 1 .7 7
40
Lag
C o rr
T
LBQ
3 7 0 .0 5 0 .4 6
3 8 0 .1 0 0 .8 4
3 9 0 .0 2 0 .1 5
4 0 -0 .0 2 -0 .1 6
4 1 -0 .1 2 -1 .0 1
1 2 2 .4 0
1 2 4 .5 2
1 2 4 .5 9
1 2 4 .6 7
1 2 7 .8 4
Partial Autocorrelation
P a r t ia l A u t o c o r r e la t io n F u n c t io n f o r w ( t )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
10
20
30
L ag
PAC
T
L ag
PAC
T
L ag
PAC
T
1
2
3
4
5
6
7
8
9
10
11
12
-0 .0 1
-0 .1 9
-0 .2 6
-0 .0 7
-0 .1 1
-0 .1 4
-0 .1 1
-0 .1 3
0 .1 4
0 .0 7
0 .1 7
-0 .3 2
-0 .1 9
-2 .4 8
-3 .2 8
-0 .8 5
-1 .4 3
-1 .8 4
-1 .4 6
-1 .6 6
1 .7 6
0 .9 5
2 .1 9
-4 .0 8
13
14
15
16
17
18
19
20
21
22
23
24
-0 .0 5
0 .0 1
-0 .0 4
-0 .0 3
-0 .0 2
0 .0 6
0 .0 5
-0 .0 3
-0 .0 5
0 .0 5
0 .1 9
0 .0 1
-0 .6 9
0 .1 5
-0 .5 0
-0 .3 3
-0 .2 3
0 .7 8
0 .6 2
-0 .4 2
-0 .6 8
0 .6 3
2 .5 0
0 .0 9
25
26
27
28
29
30
31
32
33
34
35
36
-0 .1 2
-0 .0 8
-0 .0 3
-0 .0 3
0 .0 7
-0 .1 1
-0 .0 2
-0 .1 0
-0 .0 1
0 .1 8
0 .0 7
-0 .0 9
-1 .4 9
-0 .9 7
-0 .3 2
-0 .3 6
0 .8 7
-1 .3 9
-0 .2 9
-1 .2 9
-0 .0 9
2 .3 1
0 .9 5
-1 .1 8
Lag
40
PAC
T
3 7 0 .0 2
3 8 -0 .0 8
3 9 0 .0 3
4 0 0 .0 5
4 1 -0 .0 3
0 .2 4
- 1 .0 6
0 .4 4
0 .6 3
- 0 .3 6
1 هW'!
ذج ا1!'
ن ا132 B 78
ا7CاA
اw. وا
;ا7CاA
اw.هة وال ا
;اK!
& ا!ط ا
أيSARIMA(1,1,1)(0,1,1)12
166
(1 − φ B ) (1 − B12 ) (1 − B ) zt = (1 − θ B ) (1 − ΘB12 ) at
:7
;
آ4;!
ذج ا1!'
ا اA( ه6
MTB > ARIMA 1 1 1 0 1 1 12 'z(t)' 'RESI2';
SUBC>
NoConstant;
SUBC>
GACF;
SUBC>
SUBC>
GPACF;
GHistogram;
SUBC>
GNormalplot.
ARIMA Model
ARIMA model for z(t)
Estimates at each iteration
Iteration
SSE
Parameters
0
307.653
0.100
0.100
1
281.217
0.100
0.100
2
262.275
0.226
0.231
3
262.027
0.376
0.381
4
261.770
0.526
0.531
5
261.426
0.675
0.681
6
260.905
0.824
0.831
7
260.036
0.970
0.981
8
227.926
0.835
0.980
9
221.838
0.748
0.980
10
221.665
0.738
0.980
11
221.637
0.738
0.980
12
221.610
0.737
0.980
13
221.585
0.737
0.980
Relative change in each estimate less than
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.7374
0.0620
MA
1
0.9796
0.0017
SMA 12
0.5898
0.0736
167
T
11.89
582.86
8.01
0.100
0.250
0.400
0.401
0.401
0.402
0.403
0.405
0.536
0.576
0.586
0.589
0.589
0.590
0.0010
Differencing: 1 regular, 1 seasonal of order 12
Number
of
observations:
Original
series
178,
after
differencing 165
Residuals:
SS = 214.393
MS =
1.323
(backforecasts excluded)
DF = 162
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
67.1(DF=45)
15.7(DF= 9)
30.9(DF=21)
61.6(DF=33)
1 ه4;!
@ اAO
;حJ!
ذج ا1!'
أي أن ا
(1 − 0.74 B ) (1 − B12 ) (1 − B ) zt = (1 − 0.98B ) (1 − 0.59 B12 ) at ,
at ∼ N ( 0,1.323)
ان5X
φ = 0.74, s.e. (φ ) = 0.062, t = 11.89
θ = 0.96, s.e. (θ ) = 0.0017, t = 582.86
Θ = 0.59, s.e. ( Θ ) = 0.074, t = 8.01
.421'"!
ا4
)
"!
أي ان ا
:I> اا/a
\0إ?ر ا
MTB > ZTest 0.0 1.15 'RESI1';
SUBC>
Alternative 0.
Z-Test
Test of mu = 0.0000 vs mu not = 0.0000
The assumed sigma = 1.15
Variable
RESI1
N
165
Mean
-0.0144
StDev
1.1433
SE Mean
0.0895
Z
-0.16
P
0.87
اb+ 7Bا1(
اw1;& \ أنNX اي0.05 & ( اآ7 وهP-value=0.87 ان ا
ـ5X
Iإ?ر ا اا
168
MTB > Runs 0 'RESI1'.
Runs Test
RESI1
K =
0.0000
The observed number of runs =
67
The expected number of runs =
82.4061
73 Observations above K
92 below
The test is significant at
0.0149
اءL;ج إ
إ%2 اA وه7Bا1(
ا48ا1K 4PN \N ' اي ا0.05 ' ي1'"&z ;(ر9ا
:7
;
اw1
اSign Test رة9 إ;(ر اH]& ة & إ;(ر ا
ي1B ] أآ, إ;(ر
MTB > STest 0.0 'RESI1';
SUBC>
Alternative 0.
Sign Test for Median
Sign test of median = 0.00000 versus
RESI1
N
165
N*
13
Below
92
Equal
0
not =
Above
73
0.00000
P
0.1611
Median
-0.08139
w1
اWC
رات ا9 ن1آ13
و
;{آ ي إ;(ر و0.1611 ' ي1'"& ;(ر9وا
7
;
ا
MTB > WTest 0.0 'RESI1';
SUBC>
Alternative 0.
Wilcoxon Signed Rank Test
Test of median = 0.000000 versus median not = 0.000000
RESI1
N
165
Number
N for
Wilcoxon
Missing
13
Test
165
Statistic
6321.0
Estimated
P
0.392
Median
-0.05940
0.392 ' ي1'"& `2;(ر ا9وا
:Iل اا,0إ?ر إ
169
78
ا7CاA
اw. وا
;ا7CاA
اw.دوال ا
;ا
A C F o f R e s id u a ls f o r z ( t)
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e a u t o c o r r e l a t io n s )
1 .0
0 .8
Autocorrelation
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
P A C F o f R e s id u a ls f o r z ( t)
( w it h 9 5 % c o n f id e n c e l im it s f o r t h e p a r t ia l a u t o c o r r e l a t io n s )
1 .0
Partial Autocorrelation
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
3
6
9
12
15
18
21
24
27
30
33
36
39
Lag
.4J;& 7ON 4"(= S وإذا آ46. &;اz O ا
(`ء أي ا4`
ا!ط ا76"C O ا5
:I ااi إ?ر
Histogram of the Residuals
(response is z(t))
50
Frequency
40
30
20
10
0
-6
-5
-4
-3
-2
-1
Residual
170
0
1
2
3
4
Normal Probability Plot of the Residuals
(response is z(t))
4
3
2
Residual
1
0
-1
-2
-3
-4
-5
-3
-2
-1
0
1
2
3
Normal Score
K-S Test for Residuals
.999
Probability
.99
.95
.80
.50
.20
.05
.01
.001
-5
-4
-3
-2
-1
0
1
2
3
RESI1
Average: -0.0144171
StDev: 1.14327
N: 165
Kolmogorov-Smirnov Normality Test
D+: 0.117 D-: 0.140 D : 0.140
Approximate P-Value < 0.01
ايα = 0.05 ' ي1'"& ;(ر9 اذا ا0.01 & HB أK-S ;(ر9 P-value ان ا
ـ5X
.7Bا1(
ا4"(= 4PN \NX
:ام اذج:0c$ 4ا
:('C ;اتN 95% V& 4(J;& 4!B 36 :(';
. م1J'
Forecasts from period 178
Period
179
180
181
182
Forecast
57.7885
55.8516
54.7429
54.1820
95 Percent Limits
Lower
Upper
55.5332
60.0437
53.0220
58.6812
51.6264
57.8594
50.9063
57.4578
171
Actual
57.9994
51.2601
54.6298
183
58.3430
51.4875
54.9152
184
59.1294
62.8688
52.1986
55.8869
55.6640
59.3778
185
186
72.2015
65.1833
68.6924
187
77.5925
70.5472
74.0698
188
77.4983
70.4317
73.9650
189
67.1286
60.0447
63.5866
190
62.6347
60.6128
55.1843
52.9407
58.9095
56.7768
191
192
59.4305
58.8105
59.2131
59.4653
60.1905
63.8885
73.1929
78.5647
78.4575
68.0793
63.5664
61.5364
60.3514
59.7315
60.1357
60.3903
61.1184
64.8194
74.1270
79.5021
79.3983
69.0235
51.6169
50.9022
51.2380
51.4409
52.1278
55.7947
65.0729
70.4217
70.2940
59.8968
55.0418
52.7962
51.4676
50.7472
51.0774
51.2749
51.9568
55.6189
64.8927
70.2374
70.1057
59.7046
55.5237
54.8564
55.2256
55.4531
56.1592
59.8416
69.1329
74.4932
74.3758
63.9881
59.3041
57.1663
55.9095
55.2394
55.6066
55.8326
56.5376
60.2191
69.5099
74.8697
74.7520
64.3640
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
و! )
. Oا
;
:7
172
Forecast
8 0
7 0
6 0
5 0
0
1 0
2 0
3 0
4 0
T im e
:(';
;ات اNات و:(';
اV& O&3. 4;!
7
;
ا
) ا
Forecast
80
70
60
50
0
100
T im e
173
200
ﺍﻟﻔﺼﻞ ﺍﻟﺴﺎﺑﻊ
ا9ار ا/7jا-ك/\ ا0ذج ا7 ;0ا$ 4 اN رIور
Forecasting By ARMA Models
4'& ز4;& H3 7 4& 48ا1K هةI
4
;
هات اK!
ا
12.0
20.5
21.0
15.5
15.3
23.5
24.5
21.3
23.5
28.0
24.0
15.5
17.3
25.3
25.0
36.5
36.5
29.6
30.5
28.0
26.0
21.5
19.7
19.0
16.0
20.7
26.5
30.6
32.3
29.5
28.3
31.3
32.2
26.4
23.4
16.4
:7
;
اH3K
اO
و
MTB > TSPlot C1;
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
C
1
30
20
10
Index
10
20
30
78
ا7CاA
اw. وا
;ا7CاA
اw. دوال ا
;اL1 Xاو
174
MTB > %ACF C1.
Autocorrelation
Autocorrelation Function for C1
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
0.63
0.30
0.14
-0.05
-0.24
-0.30
-0.24
3.79
1.35
0.59
-0.20
-1.01
-1.22
-0.94
15.62
19.27
20.07
20.17
22.71
26.71
29.40
5
Lag
6
Corr
7
T
LBQ
8 -0.20 -0.78
9 -0.12 -0.44
31.42
32.10
8
9
MTB > %PACF C1.
Executing from file: E:\MTBWIN\MACROS\PACF.MAC
Partial Autocorrelation
Partial Autocorrelation Function for C1
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag PAC
1
2
3
4
5
6
7
0.63
-0.16
0.04
-0.20
-0.18
-0.04
0.02
4
5
6
T
Lag PAC
T
3.79
-0.98
0.22
-1.20
-1.09
-0.23
0.12
8 -0.07
9 0.05
-0.39
0.29
7
8
9
ARMA (1,1) ذج1! & ن13C B هاتK!
& ا!ط ا
ا
; ان ا5
;حJ!
ذج ا1!'
( ا6
MTB > Name c17 = 'RESI1'
MTB > ARIMA 1 0 1 C1 'RESI1';
SUBC>
Constant;
SUBC>
Forecast 5 c14 c15 c16;
SUBC> GACF;
SUBC> GPACF;
SUBC> GNormalplot.
ARIMA Model
175
ARIMA model for C1
Estimates at each iteration
Iteration
SSE
Parameters
0
1337.71
0.100
0.100
21.918
1
936.95
0.250
-0.049
18.193
2
849.78
0.211
-0.199
19.106
3
751.53
0.215
-0.349
18.941
4
5
658.66
592.30
0.266
0.372
-0.499
-0.649
17.594
14.890
6
580.80
0.433
7
579.30
0.455
8
579.11
0.464
9
579.08
0.467
10
579.08
0.468
11
579.08
0.468
Relative change in each estimate
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.4684
0.1755
MA
1
-0.7221
0.1380
Constant
12.345
1.154
Mean
23.221
2.170
-0.699
13.314
-0.714
12.698
-0.719
12.470
-0.721
12.386
-0.722
12.356
-0.722
12.345
less than 0.0010
T
2.67
-5.23
10.70
Number of observations: 36
Residuals:
SS = 523.365 (backforecasts excluded)
MS = 15.860 DF = 33
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square 7.2(DF=10)
15.9(DF=22) * (DF= *)
* (DF=
*)
Forecasts from period 36
Period
Actual
37
Forecast
14.7649
95 Percent Limits
Lower
6.9578
176
22.5720
Upper
38
19.2606
7.1228
31.3985
39
21.3663
8.4715
34.2610
40
41
22.3524
22.8143
9.2975
9.7245
35.4074
35.9041
:1;ح هJ!
ذج ا1!'
ا
zt = 12.345 + 0.4684 zt −1 + at − 0.7221at −1 , at ∼ WN ( 0,15.86 ) ∀t
Q5
( )
θˆ = −0.7221 se (θˆ ) = 0.1380 t = −5.23
δˆ = 12.345 se (δˆ ) = 1.154 t = 10.70
φˆ1 = 0.4684 se φˆ1 = 0.1755 t = 2.67
1
1
σˆ 2 = 15.86 df = 33
4Pb
!] اN α = 0.05 ' 421'"& راتJ!
اV!L ان5و
H 0 : φ1 = 0
H1 : φ1 ≠ 0
أيα = 0.05 ' 421'"& 7 وهt =
φˆ1
0.4684
=
= 2.6689 4859. [;(ه
0.1755
se φˆ1
( )
ىXرات اJ
اV!
H]!
. وφ1 = 0 \ انN 'ا
I> اا/a 7]
:7Bا1(
اw1;& إ;(ر
1;(ر ه9ا
H 0 : µ a = 0, H1 : µa ≠ 0
MTB > TTest 0.0 'RESI1';
SUBC>
Alternative 0.
T-Test of the Mean
Test of mu = 0.000 vs mu not = 0.000
Variable
N
Mean
StDev
177
SE Mean
T
P
RESI1
36
0.344
3.851
0.642
0.54
أي انα = 0.05 & ( اآ7 وه0.6 7 هO
P-Value
0.60
وا
ـt = 0.54 ان5X
b
وي ا2 7Bا1(
اw1;& إ;(ر3!2 ي أي1'"& z ;(ر9ا
:Iإ?را اا
Runs Test إ;(ر ا
يf
A
و;[م
MTB > Runs 'RESI1'.
Runs Test
RESI1
K =
0.3443
The observed number of runs = 21
The expected number of runs = 19.0000
18 Observations above K
18 below
The test is significant at 0.4989
Cannot reject at alpha = 0.05
α = 0.05 ' 7Bا1(
ا48ا1K \N'' ر3!2X
:I\ اا$إ?ر ا
7CاA
اw. إ;(ر ا
;اf
A
و;[م
ACF of Residuals for C1
(with 95% confidence limits for the autocorrelations)
1.0
0.8
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
8
9
Lag
PACF of Residuals for C1
(with 95% confidence limits for the partial autocorrelations)
1.0
0.8
Partial Autocorrelation
Autocorrelation
0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
Lag
178
6
7
8
9
7K!;C ا!طOIC O أي ا7Bا1(
4b;[!
) اJ
ا. 4L & أي درw.اC أيL12X * ا5
`ء. 4P 4;& O1 آV&
Normal Probability Plot
ـ. 7Bا1(
ا4"(= (;[ واا
Normal Probability Plot of the Residuals
(response is C1)
Residual
10
0
-10
-2
-1
0
1
2
Normal Score
( 1 ) ل1(J& 1وه
('& ;حJ!
ذج ا1!'
إ;(ر ا3!2 إذا
O
:('C 95% ;اتN و4(J;& )B 4![
ات:(';
7
;
ا
) ا
MTB > TSPlot C14 C15 C16;
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect;
SUBC> overlay.
Time Series Plot for C1
(with forecasts and their 95% confidence limits)
36
C1
26
16
6
5
10
15
20
Time
179
25
30
35
:?z ل-
4'& ز4;& H3 7 4& 48ا1K هةI
4
;
هات اK!
ا
10.38
11.86
10.97
10.80
9.79
10.39
10.42
10.82
11.40
11.32
11.44
11.68
11.17
10.53
10.01
9.91
9.14
9.16
9.55
9.67
8.44
8.24
9.10
9.09
9.35
8.82
9.32
9.01
9.00
9.80
9.83
9.72
9.89
10.01
9.37
8.69
8.19
8.67
9.55
8.92
8.09
9.37
10.13
10.14
9.51
9.24
8.66
8.86
8.05
7.79
6.75
6.75
7.82
8.64
10.58
9.48
7.38
6.90
6.94
6.24
6.84
6.85
6.90
7.79
8.18
7.51
7.23
8.42
9.61
9.05
9.26
9.22
9.38
9.10
7.95
8.12
9.75
10.85
10.41
9.96
9.61
8.76
8.18
7.21
7.13
9.10
8.25
7.91
6.89
5.96
6.80
7.68
8.38
8.52
9.74
9.31
9.89
9.96
MTB > TSPlot C10;
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
12
11
C
1
0
10
9
8
7
6
Index
10
20
30
40
50
60
70
80
90
78
ا7CاA
اw. وا
;ا7CاA
اw. دوال ا
;اo%b
180
Autocorrelation
Autocorrelation Function for C10
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
Lag Corr
1
2
3
4
5
6
7
0.83
0.61
0.46
0.37
0.33
0.28
0.26
T
12
LBQ
Lag Corr
8.24 69.92
3.91 107.90
2.56 129.56
1.95 143.87
1.65 155.04
1.40 163.68
1.28 171.23
8
9
10
11
12
13
14
0.26
0.26
0.18
0.09
0.04
0.03
0.04
T
LBQ
1.26 178.83
1.21 186.14
0.84 189.86
0.43 190.87
0.20 191.09
0.13 191.19
0.19 191.39
22
Lag Corr
T
LBQ
Lag Corr
15 0.05 0.21 191.63
16 0.04 0.16 191.78
17 0.00 0.02 191.78
18 -0.03 -0.15 191.91
19 -0.05 -0.24 192.26
20 -0.05 -0.24 192.60
21 0.01 0.07 192.63
T
LBQ
22 0.10 0.47 194.02
23 0.18 0.81 198.12
24 0.20 0.89 203.24
Partial Autocorrelation
Partial Autocorrelation Function for C10
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
12
Lag PAC
T
Lag PAC
T
1 0.83
2 -0.27
3 0.13
4 0.03
5 0.06
6 -0.02
7 0.09
8.24
-2.64
1.29
0.34
0.61
-0.21
0.91
8 0.05
9 0.00
10 -0.20
11 0.02
12 0.01
13 0.01
14 0.03
0.45
0.03
-1.98
0.19
0.09
0.12
0.34
22
Lag PAC
15
16
17
18
19
20
21
-0.01
-0.03
-0.07
-0.03
0.06
0.02
0.21
T
Lag PAC
T
-0.15
-0.25
-0.73
-0.26
0.60
0.20
2.03
22 0.05
23 0.06
24 -0.07
0.51
0.59
-0.65
AR ( 2 ) 4]
ا4L & ا
ر7Cار ذا%ذج إ1! ;حJC ان ا!ط5
;حJ!
ذج ا1!'
( ا6
MTB > Name c17 = 'RESI1'
MTB > ARIMA 2 0 0 C10 'RESI1';
SUBC>
Constant;
SUBC>
Forecast 5 c14 c15 c16;
SUBC> GSeries;
SUBC> GACF;
SUBC> GPACF;
SUBC> GNormalplot.
ARIMA Model
181
ARIMA model for C10
Estimates at each iteration
Iteration
SSE
Parameters
0
126.398
0.100
0.100
7.283
1
2
103.515
84.535
0.250
0.400
0.043
-0.014
6.434
5.586
3
4
69.407
58.132
0.550
0.700
-0.071
-0.128
4.738
3.887
-0.184
-0.239
-0.256
-0.255
-0.255
-0.255
less than
3.030
2.163
1.838
1.816
1.814
1.814
0.0010
5
50.724
0.850
6
47.212
1.000
7
46.918
1.053
8
46.916
1.054
9
46.916
1.054
10
46.916
1.054
Relative change in each estimate
Final Estimates of Parameters
Type
Coef
StDev
AR
1
1.0542
0.0992
AR
2
-0.2547
0.0993
Constant
1.81360
0.07092
Mean
9.0480
0.3538
Number of observations:
Residuals:
T
10.63
-2.56
25.57
98
SS = 46.7518
MS = 0.4921
(backforecasts excluded)
DF = 95
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
48
Chi-Square
7.2(DF=10)
13.7(DF=22)
21.3(DF=34)
28.8(DF=46)
Forecasts from period 98
Period
Actual
Forecast
95 Percent Limits
Lower
182
Upper
99
9.7950
8.4198
11.1703
100
9.6033
7.6050
11.6016
101
102
9.4432
9.3232
7.1234
6.8446
11.7629
11.8018
103
9.2375
6.6825
11.7925
1;ح هJ!
ذج ا1!'
ا
zt = 1.8136 + 1.0542 zt −1 − 0.2547 zt −2 + at , at ∼ WN ( 0,0.4921) ∀t
7 هO
t )B و42 ا
!"رOCNا%رات ا
!"
) وإJ&
( )
φˆ = −0.2547 se (φˆ ) = 0.0993 t = −2.56
δˆ = 1.8136 se (δˆ ) = 0.07092 t = 25.57
φˆ1 = 1.0542 se φˆ1 = 0.0992 t = 10.63
2
2
σˆ 2 = 0.4921 df = 95
α = 0.05 ' 421'"& راتJ!
اV!L ان5
7Bا1(
ا7 ي إ;(رات:(';
W'& * ا7 ذج1!'
ا اAO. H(J 73
MTB > TTest 0.0 'RESI1';
SUBC>
Alternative 0.
T-Test of the Mean
Test of mu = 0.0000 vs mu not = 0.0000
Variable
RESI1
N
98
Mean
StDev
-0.0082 0.6942
SE Mean
0.0701
T
-0.12
P
0.91
42b
ا4Pb
;!ل ان ا5 إ7 هP-Value ا
ـ4I5& ) ي1'"& z ا ان ا;(رL ^Pوا
( 4%%+
7Bا1(
ا48ا1K (;[
MTB > Runs 'RESI1'.
Runs Test
RESI1
K =
-0.0082
The observed number of runs = 47
The expected number of runs = 49.9184
183
47 Observations above K
51 below
The test is significant at
0.5529
Cannot reject at alpha = 0.05
7Bا1(
ا48ا1K 4PN \N'' ر3!2X أي
78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;اo%b
ACF of Residuals for C10
(with 95% confidence limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
4
6
8
10
12
14
16
18
20
22
24
Lag
PACF of Residuals for C10
(with 95% confidence limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
4
6
8
10
12
14
Lag
16
18
20
ا
(`ء4`
ا ا!ط اL ^Pوا
22
24
7Bا1(
ا4"(= o%N 7J(2
184
Normal Probability Plot of the Residuals
(response is C10)
2
Residual
1
0
-1
-2
-3
-2
-1
0
1
2
3
Normal Score
( at ∼ IIDN ( 0,0.4921) ) أي7"(= V2ز1C O
7Bا1(
ل ان ا1J انV6;و
O
:('C 95% ;اتN و4(J;& )B 4![
ات:(';
7
;
ا
) ا
MTB > TSPlot C14 C15 C16;
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect;
SUBC> overlay.
Time Series Plot for C10
(with forecasts and their 95% confidence limits)
12
11
C10
10
9
8
7
6
10
20
30
40
50
60
70
80
90
Time
_8;'
ا. رنB وAR (1) ذج1! 4J.
هات اK!
ا7 (= : 2!C
185
186
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻣﻦ
N0 و
إ?ر اذج اI اا./ ' ل-
: Example on Residual Analysis and Model Selection Criteria
ة6"& هاتK& !N 4J(6!
) اJ
اoB هةK!
) اJ
اO ا7 7Bا1(
اJ. 'N J
:7Bا1(
اW;3C وzˆ1 , zˆ2 ,..., zˆn 4J(6& )B '2
_;'2 (6& ذج1! وz1 , z2 ,..., zn
ei = zi − zˆi , i = 1,2,..., n
J%C انW2 اAO
وaˆi = ei , i = 1,2,..., n ذج أي1!'
ا7N ء6رات اJ& 7 ه7Bا1(
وا
:O'& 7;
ذج وا1!'
ا اA ه7N ء6 ا7 4Pوb!
وط اK
ا
b
وي ا2 ء6 اw1;& -1
ء6;ض ان اb آ] & ا
'!ذج7N ) و4J;& أو46. &;اz و48ا1K ء6 ا-2
( at ∼ IIDN ( 0,σ 2 ) أيσ 2 2(Cي وb+ w1;!. .6;& وHJ;& 7"(= V2ز1C O
@A هJ%C S! إذا آN 'ي7Bا1(
ا7 ;(رات9 & ا41!& 1 وه%C ' يFN اAO
WN ;(رات9@ اA ه5 اHKN أ& إذاX1(J& (6!
ذج ا1!'
";( ا4
%
@ اA ه7Nوط وK
ا
, ذج1! ;احB وإI'
' إدة ا
w1;!
إ;(ر ا:Xأو
H 0 : E ( at ) = 0
H 1 : E ( at ) ≠ 0
'"N 7B 7"(= V2ز1C O
7;
واu =
e
se ( e )
4859 ا4N و;[م2A. إ;(ر1وه
إ;(ر ان7 اA ) هu < 1.96 S إذا آE ( at ) = 0 ";( انα = 0.05 421'"& ى1;&
( O ر7;
ا4'&
!;ت اJ%;& !8ا داAة وه5 و30 & ( اآ4'"
) ا5
:ل-
Metals Y;!
ا7 4]
]
ا4Lك & ا
ر%;& w1;& (6C &]ل7
د ا1" ف1
MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'.
Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW
Worksheet was saved on
6/ 5/1996
MTB > TSPlot 'Metals';
SUBC>
Index;
187
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
M
e
ta
ls
50
45
40
Index
10
20
30
40
50
60
MTB > Name c4 = 'AVER1' c5 = 'FITS1' c6 = 'RESI1'
MTB > %MA 'Metals' 3;
SUBC>
Averages 'AVER1';
SUBC>
Fits 'FITS1';
SUBC>
Residuals 'RESI1'.
Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average
Data
Metals
Length
60.0000
NMissing
0
Moving Average
Length: 3
Accuracy Measures
MAPE: 1.55036
MAD:
0.70292
MSD:
0.76433
188
Moving Average
Actual
Predicted
M
etals
50
Actual
Predicted
45
Moving Average
Length:
40
0
10
20
30
40
50
3
MAPE:
1.55036
MAD:
0.70292
MSD:
0.76433
60
Time
~&[
د ا1!"
ا7N 4J(6!
) اJ
د ا
دس وا1!"
ا7N 7Bا1(
ا' ا5X
MTB > print c3 c6 c5
Data Display
Row
Metals
RESI1
FITS1
1
44.2
*
*
2
44.3
*
*
3
44.4
*
*
4
43.4
-0.90000
44.3000
5
42.8
-1.23333
44.0333
6
44.3
0.76667
43.5333
7
44.4
0.90000
43.5000
8
44.8
0.96667
43.8333
9
44.4
-0.10000
44.5000
10
43.1
-1.43333
44.5333
11
42.6
-1.50000
44.1000
12
42.4
-0.96667
43.3667
13
42.2
-0.50000
42.7000
14
41.8
-0.60000
42.4000
15
40.1
-2.03333
42.1333
189
16
42.0
0.63333
41.3667
17
42.4
1.10000
41.3000
18
43.1
1.60000
41.5000
19
42.4
-0.10000
42.5000
20
43.1
0.46667
42.6333
21
43.2
0.33333
42.8667
22
42.8
-0.10000
42.9000
23
43.0
-0.03333
43.0333
24
42.8
-0.20000
43.0000
25
42.5
-0.36667
42.8667
26
42.6
-0.16667
42.7667
27
42.3
-0.33333
42.6333
28
42.9
0.43333
42.4667
29
43.6
1.00000
42.6000
30
44.7
1.76667
42.9333
31
44.5
0.76667
43.7333
32
45.0
0.73333
44.2667
33
44.8
0.06667
44.7333
34
44.9
0.13333
44.7667
35
45.2
0.30000
44.9000
36
45.2
0.23333
44.9667
37
45.0
-0.10000
45.1000
38
45.5
0.36667
45.1333
39
46.2
0.96667
45.2333
40
46.8
1.23333
45.5667
41
47.5
1.33333
46.1667
42
48.3
1.46667
46.8333
43
48.3
0.76667
47.5333
44
49.1
1.06667
48.0333
45
48.9
0.33333
48.5667
46
49.4
0.63333
48.7667
47
50.0
0.86667
49.1333
48
50.0
0.56667
49.4333
49
49.6
-0.20000
49.8000
190
50
49.9
0.03333
49.8667
51
49.6
-0.23333
49.8333
52
50.7
1.00000
49.7000
53
50.7
0.63333
50.0667
54
50.9
0.56667
50.3333
55
50.5
-0.26667
50.7667
56
51.2
0.50000
50.7000
57
50.7
-0.16667
50.8667
58
50.3
-0.50000
50.8000
59
49.2
-1.53333
50.7333
60
48.1
-1.96667
50.0667
7Bا1(
اw1;& (;[ نsا
MTB > TTest 0.0 'RESI1';
SUBC>
Alternative 0.
T-Test of the Mean
Test of mu = 0.000 vs mu not = 0.000
Variable
N
RESI1
57
Mean
StDev
SE Mean
T
0.158
0.868
0.115
1.37
P
0.17
z ( 2(;
اف ا
!"ري ) او ا%9ن ا132 &' &_ م. ;[م2 Minitab 7N :4I5&
أي1.96
& HB ا7 وهT=1.37 7 ه4859 ا4!B ان5X . Ttest * 62&"وف و
42b
ا4Pb
\ اNX
b
ل ا15 وw1;!
ل ا15 Runs test إ;(ر ا
ي46ا1. 7Bا1(
ا48ا1K (;[ :U
: ا
!]لV.C
MTB > Runs 'RESI1'.
Runs Test
191
RESI1
K =
0.1579
The observed number of runs =
17
The expected number of runs =
29.4211
30 Observations above K
27 below
The test is significant at
0.0009
MTB > Runs 0 'RESI1'.
Runs Test
RESI1
K =
0.0000
The observed number of runs =
17
The expected number of runs =
28.7895
33 Observations above K
24 below
The test is significant at
0.0013
7Bا1(
ا48ا1K \NX ;
%
آ; ا7N * ا5
Autocorrelation test 7CاA
اw. إ;(ر ا
;ا46ا1. 7Bا1(
ل اJ; أو إw.اC (;[ :]
U
: ا
!]لV.C
MTB > %ACF 'RESI1'.
Executing from file: E:\MTBWIN\MACROS\ACF.MAC
192
Autocorrelation Function for RESI1
Autocorrelation
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
4
9
14
Lag
Corr
T
LBQ
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
0.56
0.24
-0.01
0.04
0.02
-0.07
-0.14
4.24
1.39
-0.06
0.21
0.11
-0.38
-0.81
18.93
22.31
22.32
22.40
22.42
22.72
24.07
8
9
10
11
12
13
14
-0.04
0.12
0.29
0.24
0.20
0.05
0.03
-0.22
0.68
1.64
1.30
1.05
0.25
0.17
24.17
25.21
31.36
35.67
38.71
38.90
38.99
أيρ1 = 0 \ انN ' أي ا4.24 ويC اولe[;
' ا7CاA
اw. ;اT ان ا
ـ5X
;(ر9 ا7N وX ا4L & ا
ر7Bا1(
ا. w.اC L12
H 0 : ρ1 = 0
H1 : ρ1 ± 0
r1
= 4.24 7 ه4859 اQ5
se ( r1 )
"(= 4ز1& 7Bا1(
اS & إذا آ7N (;[ :".را
: ا
!]لV.C
MTB > %NormPlot 'RESI1';
SUBC>
Kstest.
Executing from file: E:\MTBWIN\MACROS\NormPlot.MAC
Normal Probability Plot
.999
P
ro
b
a
b
ility
.99
.95
.80
.50
.20
.05
.01
.001
-2
-1
0
1
RESI1
Average: 0.157895
StDev: 0.867525
N: 57
Kolmogorov-Smirnov Normality Test
D+: 0.054 D-: 0.084 D : 0.084
Approximate P-Value > 0.15
193
V2ز1;
ا4PN \NX أي0.05 & ( اآ7 وه0.15 ويC 4C'
اP-Value ا
ـ5X
( &ي2 يA
واQ-Q Plot ا
ـ1 ه4"(6
, ` إ;(ر2 ه'ك اα = 0.05 ' 7"(6
ا
"& V2ز1C V& & هاتK& .6C
MTB > %Qqplot 'RESI1';
SUBC>
Table;
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: E:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
Data
: RESI1
Mean:
0.157895
StDev:
0.867525
Normal Probability Plot for RESI1
99
Mean:
0.157895
StDev:
0.867525
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-2
-1
0
1
2
Data
7"(= V2ز1C O
7Bا1(
\ ان اNX 'FN ;
%
آ; ا7N ان5X
194
& 4I5اة(2 :و ان ا
(1ا )I"& J%C 7Bا
Kوط !Nي ا
;ا w.ا
Aي . L12ا
)J
ا
!;;
4وهAا 1L \N '"2دة ا
; 4J26
(6ا
!; w1ا
!;%ك & ا
ر 4Lا
]
]Q5 4
ادي ا
1. 7ا;& 7Bا.46.
195
ﺍﻟﻔﺼﻞ ﺍﻟﺘﺎﺳﻊ
:Decomposition Method )| ا ا آتY او./
213;
O`". V& ة%;& اءL & ة &آ(ت أو ا413& O ا4'&
ا4;!
إ
اI'2
،4;!
@ اAه
O;LA! 3!2 * أ4.;
. L وJ
. z1 , z2 ,..., zn @هK!
ا4'&
ا4;!
' ا2
;ض انb'
H3K
ا
zt = Tt + St + Ct + Et , t = 1, 2,..., n
يA
@ ا
"م اC9ج اA!'C 7;
ا7اف وه9 ا4( &آTt هة وK!
ا4'&
ا4;!
اZt Q5
1( وهL )إذا و7!1!
اU{;
ج اA!'C و4!1& 4( &آSt و4;!
'ف ا
* اC أو7%'C
4( &آCt و421'
وا42OK
اH]& 4!1!
ات اU{;
ا4; 4;!
ث%2 يA
اY;
ا
Et و4!1& z 421= 4'&;ات زN ". ر3;2 ةC أو إ%'& جA!'C ت( وL )إذا و42دور
O;LA! 3!2X 7;
وا4;!
اU:C 7;
ى اX اH&ا1"
اV!L H!KC{ و6[
ا4(&آ
Additive 7NP9ذج ا1!'
. 7!2 .
ذج ا1!'
ا.OB اوOCهK& 3!2X 7;
!' أو اP
H]& ل اى3 ه'ك أ.ذج1!'
ا7N 7NP إH3K. HC ا
!آ(تH ن آf
وذModel
zt = Tt St + Ct + Et , t = 1, 2,..., n
zt = Tt St Ct + Et , t = 1, 2,..., n
. Multiplicative Models 4b`;
'!ذج ا. !C 7;
وا
ن13C& درا42 ا
ور4( ن ا
!آf
وذCt 42 ا
ور4( ا
!آH!O ف1 ى1;!
ا اA ه7N
اL 421= هاتK& ;ج ا%C O ( 4216
ة أو اJ
ا
!;ت ا7N دة1L1&
.د1J"
&ي د آ( & ا
H3K
ا7
'!ذج. 7b;3و
zt = Tt + St + Et , t = 1, 2,..., n
zt = Tt St + Et , t = 1, 2,..., n
b
:!
FORECASTING: METHODS AND APPLICATIONS آ;بIأ
141-131 صMAKRIDAKIS/ WHEELWRIGHT/ McGEE
أي42 ا
ور4(ون ا
!آ. 4b`;
وا4NP9 اH%;
=ق ا7
;
ا
!]ل ا7N ف ;"ض1
ا
'!ذج
196
zt = Tt + St + Et , t = 1, 2,..., n
zt = Tt St + Et , t = 1, 2,..., n
1960 4' & 'ة3. 12 او;ر4'2& 7N ا
;ات2!. 2'(
ا7 W6
ا7 ه4
;
ا
(ت ا
1975 4' ;5و
GasDemand
MONTHLY GASOLINE DEMAND ONTARIO GALLON MILLIONS 1960-1975
87695
86890
107677 108087
140735
96442
98133 113615 123924 128924 134775 117357 114626
92188
88591
98683
99207 125485 124677 132543
124008 121194 111634 111565 101007
140318
94228 104255 106922
130621
125251
146174
122318
128770
117518
115492
108497
100482
106140
118581
132371
132042
151938
150997
130931
137018
121271
123548
109894
106061
112539
125745
136251
140892
158390
148314
144148
140138
124075
136485
109895
109044
122499
124264
142296
150693
163331
165837
151731
142491
140229
140463
116963
118049
137869
127392
154166
160227
165869
173522
155828
153771
143963
143898
124046
121260
138870
129782
162312
167211
189689
166496
160754
155582
145936
139625
137361
138963
172897
155301
172026 165004 185861 190270 163903 174270 160272 165614 146182 137728
148932 156751 177998 174559 198079 189073 175702 180097 155202 174508
154277 144998 159644 168646 166273 190176 205541 193657 182617 189614
174176 184416 158167 156261 176353 175720 193939 201269 218960 209861
198688 190474 194502 190755 166286 170699 181468 174241 210802 212262
218099 229001 203200 212557 197095 193693 188992 175347 196265 203526
227443 233038 234119 255133 216478 232868 221616 209893 194784 189756
193522 212870 248565 221532 252642 255007 206826 233231 212678 217173
199024 191813 195997 208684 244113 243108 255918 244642 237579 237579
217775 227621
7'& زw6[& 7N 4;!
) اXأو
MTB > TSPlot 'GasDemand';
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
197
G
asD
em
and
250
200
150
100
Index
50
100
150
X ا
ا4N'& و4!1& 4'&
ا4;!
ان ا5
zt = Tt + St + Et , t = 1, 2,..., n
:7NP9ذج ا1!'
( ا6C :Xاو
MTB > %Decomp 'GasDemand' 12;
SUBC> Additive ;
SUBC>
Forecasts 24;
SUBC>
Title "Forecast of Gasoline Demand";
SUBC> Start 1.
Time Series Decomposition
Data
Length
NMissing
GasDeman
192.000
0
Trend Line Equation
Yt = 96.4074 + 0.680579*t
Seasonal Indices
Period
1
2
3
4
Index
-20.5625
-26.8125
-14.8958
-11.0625
198
5
9.89583
6
11.8958
7
8
22.7708
25.1875
9
5.64583
10
7.27083
11
-4.81250
12
-4.52083
Accuracy of Model
MAPE:
3.6952
MAD:
5.6622
MSD:
52.7851
Forecasts
Row Period Forecast
1
193
207.197
2
194
201.627
3
195
214.225
4
196
218.738
5
197
240.377
6
198
243.058
7
199
254.614
8
200
257.711
9
201
238.850
10
202
241.155
11
203
229.753
12
204
230.725
13
205
215.364
14
206
209.794
15
207
222.391
16
208
226.905
17
209
248.544
18
210
251.225
19
211
262.780
20
212
265.878
21
213
247.017
22
214
249.322
23
215
237.919
24
216
238.892
199
(1) H3
Forecast of Gasoline Demand
Seasonal Indices
Original Data, by Seasonal Period
30
250
20
10
200
0
150
-10
-20
100
-30
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period
Residuals, by Seasonal Period
30
20
10
10
0
5
-10
-20
0
-30
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
(2) H3
Forecast of Gasoline Demand
Original Data
Detrended Data
50
40
30
20
10
0
-10
-20
-30
-40
250
200
150
100
0
100
200
0
Seasonally Adjusted Data
100
200
Seasonally Adj. and Detrended Data
250
30
20
200
10
150
-10
0
-20
100
-30
0
100
200
0
100
200
(3) H3
200
Forecast of Gasoline Demand
Actual
Predicted
250
GasDeman
Forecast
Actual
Predicted
Forecast
200
150
100
MAPE:
MAD:
MSD:
0
100
3.6952
5.6622
52.7851
200
Time
:_8;'
ا4KB'&
(2 ا & ا
رH3K
N ، Seasonal Indices 4!1!
ات ا:!
^ اP12 (1) H3
oJ ث%2 4 و3 و2 و1 و12 و11 O ا7bN 4'
& ا4b;[!
اO ا7N W6
اU{C
^(2 ;5 2;ا2 )U 2 OK
ا7N *
&"لHB إ
أH2 ;5 2رC oB';2 إذW6
ا7N
( آH3K. oJ'2 )U 8 OK
ا7N 4(L1& 4!B B اH2 ;5 2;ا2 و5 OK
ا7N (L1&
4ز1& 4+هات اK!
Box Plot ر) ا
'وق76"2 !
ا & اH3K
ا.A8".
. Out Liers 4L) ا
[رJ
واO Hهات آK!
ر اK; وإV2ز1C ^P12 1 وهO ا
H3K
ا.(O )ا4!1!
;ات اb
ي ا1!
ا7('
اY;
ا76"2 & ا
رHb اH3K
ا
.O ا4ز1& ء6 أو ا7Bا1(
ر) ا
'وق76"2 !2 اHbا
76"2 ا & ا
رH3K
ا،4+هات اK!
ا76"2 !
ا & اH3K
( ا2) H3
اف أي9 ا4( &آ45" إزا. هاتK!
ا
wt = zt − Tt , t = 1, 2,..., n
=St + Et , t = 1, 2,..., n
أي4!1!
ا4( ا
!آ45" إزا. 4+هات اK!
ا76"2 & ا
رHb اH3K
ا
yt = zt − St , t = 1, 2,..., n
=Tt + Et , t = 1, 2,..., n
اف9 ا7;( &آ45" إزا. 7Bا1(
أو ا
Et {6[
ا4( &آ76"2
!2 اHb اH3K
ا
أي4+هات اK!
& ا4!1!
وا
et = zt − Tt − St , t = 1, 2,..., n
=Et , t = 1, 2,..., n
.(6;
ا4B~ د2J& V& 4(J;!
ا24 )J
ات:(';
ا76"2 (3) H3
201
zt = Tt St + Et , t = 1, 2,..., n
:7b`;
ذج ا1!'
( ا6C :U
MTB > %Decomp 'GasDemand' 12;
SUBC>
Forecasts 24;
SUBC>
Title "Forecast of Gasoline Demand";
SUBC> Start 1.
Executing from file: D:\MTBWIN\MACROS\Decomp.MAC
Macro is running ... please wait
Time Series Decomposition
Data
Length
NMissing
GasDeman
192.000
0
Trend Line Equation
Yt = 96.4074 + 0.680579*t
Seasonal Indices
Period
Index
1
2
3
4
5
6
7
8
9
10
11
12
0.860355
0.828555
0.892431
0.936273
1.06124
1.07274
1.15775
1.17075
1.03409
1.05059
0.966300
0.968923
202
Accuracy of Model
MAPE:
MAD:
3.6338
5.7720
MSD:
56.8996
Forecasts
Row
1
Period
193
Forecast
195.954
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
189.275
204.474
215.156
244.596
247.977
268.415
272.227
241.154
245.718
226.660
227.935
202.980
196.042
211.762
222.803
253.263
256.738
277.870
281.789
249.599
254.298
234.552
235.848
(4) H3
203
Forecast of Gasoline Demand
Seasonal Indices
Original Data, by Seasonal Period
1.2
250
1.1
200
1.0
150
0.9
100
0.8
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Percent Variation, by Seasonal Period
Residuals, by Seasonal Period
14
12
10
8
6
4
2
0
20
10
0
-10
(5) H3
-20
-30
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Forecast of Gasoline Demand
Original Data
Detrended Data
250
1.3
1.2
200
1.1
1.0
150
0.9
100
0.8
0
100
200
0
Seasonally Adjusted Data
100
200
Seasonally Adj. and Detrended Data
240
220
200
180
160
140
120
100
20
10
0
-10
(6) H3
-20
-30
0
100
200
0
100
200
Forecast of Gasoline Demand
Actual
280
Predicted
GasDeman
Forecast
Actual
Predicted
Forecast
180
MAPE:
MAD:
MSD:
80
0
100
3.6338
5.7720
56.8996
200
Time
:_8;'
ا4KB'&
.(3) ( و2) ( و1) ل3 ا7N ! آb;
~ اb O
(6) ( و5) ( و4) ل3ا
204
دور7C{2 ' وه،ذج1! H`N أن [;ر اWN هةK!
ا4;!
اLذ1! 'J(= '! ا.
:4B~ د2J& 4UU '2
،_&(
';_ & اC 7;
( وا6;
ا4B~ د2J&
MAPE أوMean Absolute Percentage Error 6!
ا7('
{ ا6[
اw1;& -1
4B"
. 6"2و
zt − zˆt
zt
× 100, zt ≠ 0
n
n
∑
t =1
MAPE =
4B"
. 6"2 وMAD أوMean Absolute Deviation 6!
اف ا%9 اw1;& -2
n
MAD =
∑z
t =1
− zˆt
t
n
4B"
. 6"2 ( وMSE )أوMSD (V.!
{ ا6[
اw1;& )أوV.!
اف ا%9 أw1;& -3
n
MSD =
∑( z
t =1
t
− zˆt )
2
n
]س اآJ!
ا،سJ!
ا اAO
4!B HB أ76"2 يA
ذج ا1!'
~ [;ر ا2J!
ة اA ه5;ر أF.
.'ف ه1 يA
ا1 وهMSE أوMSD 1 ه1إ;[ا& و
:7 ه4B
~ ا2J& 7NP9ذج ا1!'
MAPE:
3.6952
MAD:
5.6622
MSD:
52.7851
:7b`;
ذج ا1!'
و
MAPE:
3.6338
MAD:
5.7720
MSD:
56.8996
ذج1!'
ا اAر إ;[ام هJ f
A
وMSD سJ!
4!B HB ا6 ا7NP9ذج ا1!'
أن ا5
.W6
ا4;!
4(J;!
) اJ
ا:(';
205
:Decomposition Method )| ا ا آتY او./ i }A
إ;ج4
;
ا
(ت ا. إ
&آ(ت4;!
اf3bC أوH%C 4J2= 7
;
!]ل ا. ^P1 ف1
امL 13
. ا
!ارع5 أ7N W%
&12 168
MTB > Read "E:\Mtbwin\milk.dat" c1.
Entering data from file: E:\Mtbwin\milk.dat
168 rows read.
MTB > name c1='MilkProd'
MTB > print c1
Data Display
MilkProd
589
561
553
582
583
587
678
639
782
756
736
755
713
667
660
698
701
706
801
764
886
859
860
878
826
799
766
805
784
791
908
867
966
937
640
600
565
604
702
811
762
717
677
725
819
942
890
821
760
815
896
656
566
598
611
653
798
784
696
711
723
783
913
900
773
802
812
858
727
653
628
594
615
735
837
775
734
690
740
869
961
883
828
773
817
697
673
618
634
621
697
817
796
690
734
747
834
935
898
778
813
827
640
742
688
658
602
661
767
858
785
750
711
790
894
957
889
834
797
599
716
705
622
635
667
722
826
805
707
751
800
855
924
902
782
843
568
660
770
709
677
645
681
783
871
807
804
763
809
881
969
892
577
617
736
722
635
688
687
740
845
824
756
800
810
837
947
903
:&وا. 4J.
و) ا
(ت ا
MTB > TSPlot 'MilkProd';
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
206
1000
MilkProd
900
800
700
600
Index
50
100
150
:7NP9ذج ا1!'
( ا6C :Xاو
73
zt = Tt + St + Et , t = 1, 2,..., n H3K
ذج ا1!'
اW;3 z1 , z2 ,..., zn هاتK!
:4'3!& ;J2= ف ;"ض1 4J.
ا
!آ(ت ا7
ا4'&
ا4;!
@ اA هf3b;. م1J
:7
وX ا4J26
ا
: أيTt افX ا4( &آ2J;
&
ا7 هاتK!
w. 76 ار%( إ6 -1
Tˆt ≡ zˆt = a + bt , t = 1, 2,...,168
:أي
MTB > set c2
DATA> 1:168
DATA> end
MTB
>
name
c1='MilkProd'
c2='Time'
c3='Trend'
c5='Detrend' c6='Index' c8='Fitted' c9='Resid'
MTB > regr c1 1 c2;
SUBC> fits c3.
Regression Analysis
The regression equation is
MilkProd = 612 + 1.69 Time
Predictor
Constant
Time
Coef
611.682
1.69262
StDev
9.414
0.09663
207
T
64.97
17.52
P
0.000
0.000
S = 60.74
R-Sq = 64.9%
R-Sq(adj) = 64.7%
Analysis of Variance
Source
DF
SS
MS
F
1
1132003
1132003
306.83
P
Regression
0.000
Error
166
612439
Total
167
1744443
3689
1اف ه9 اH3و
900
Trend
800
700
600
Index
50
100
150
4
&ا4;!
. 7!2& 7 H%'N 4+Xهات اK!
اف & ا9 ا4(ح &آ6 -2
zt − zˆt = zt − Tˆt , t = 1,2,...,168 أيDetrended Series اف9ا
MTB > let c5=c1-c3
:7
;
اH3K
اO
و
Detrend
100
0
-100
Index
50
100
208
150
zt − Tˆt = St + Et , t = 1, 2,...,168 نwJN 4!1& S%(+ اO ا5
:7
;
آSeasonal Indices 4!1!
ات ا:!
اL1 4!1!
ا4( ا
!آ2J;
-3
OK
7!1!
ا:!
اI1 Q5 I s , s = 1,2,...,12 &
. 4!1!
ات ا:!
&'
d t = zt − Tˆt , t = 1, 2,...,168 ـ. &'
ا وA3 وه7]
اOK
7!1!
ا:!
اI 2 اول و
:7
;
ات آ:!
@ اAر هJC
1
( d1 + d13 + d 25 + ⋯ + d157 )
14
1
I 2 = ( d 2 + d14 + d 26 + ⋯ + d158 )
14
⋮
I1 =
I12 =
1
( d12 + d 24 + d 36 + ⋯ + d168 )
14
:4
;
;[ام اوا& اF. f
;) ذ2و
MTB > set c4
DATA> 14(1:12)
DATA> end
MTB > stat c5;
SUBC> by c4;
SUBC> mean c6.
MTB > Stack 'Index' 'Index' 'Index' 'Index' 'Index'
'Index' 'Index' &
CONT>
'Index' 'Index' 'Index' 'Index' 'Index' 'Index'
'Index' c7.
MTB > let c8=c3+c7
MTB > let c9=c1-c8
MTB > set c10
DATA> 1:12
DATA> end
MTB > print c10 c6
Data Display
209
Index
Season
Row
-18.328
-57.806
1
2
1
2
34.716
3
3
49.595
110.616
4
5
4
5
82.281
32.517
6
7
6
7
-9.747
-52.297
-48.775
-79.754
-43.018
8
9
10
11
12
8
9
10
11
12
-4ا
;'(:ات 1Cآ
;
:7
z168 ( ℓ ) = 612 + 1.69 ( ℓ + 168 ) + I ( ℓ mod 12 ) , ℓ = 1, 2,...
]!Nا
;'( ' :ا
1م 169ه1
z168 (1) = 612 + 1.69 (169 ) + I1
=897.61 + ( −18.328 ) = 879.282
ا
4J26ا
]:4
وه 7ا
;: Minitab _&. O&[;2 7
-1آ
4J26اXو
(6إ%ار K!
w. 76هات 7ا
& ;& 2Jآ( 4اXاف
~b H%'N Ttا
'; 4آ! 7Nا
4J26اXو
)(1
-2ا `2ه' 6ح &آ( 4ا9اف & ا
!Kهات ا 7 H%'N 4+Xا
!;& 4ا
4
ا9اف Detrended Series
(6 -3اsن &;%;& w1ك & در 4Lا
! )1و *61اذا ا;5ج ا&X
6 -4ح ا
!;61ت ا
!;%آI & 4ا 7N OCا
!;& 4ا
4ا9اف 7 H%'N
&;1%C 4ي ا
!آ(ت ا
!4!1
JC -5ر ا
!آ(ت ا
! 4!1آ
;
:7
210
I1 = Median ( d1 , d13 , d 25 ,⋯, d157 )
I 2 = Median ( d 2 , d14 , d 26 ,⋯, d158 )
⋮
I12 = Median ( d12 , d 24 , d 36 ,⋯ , d168 )
.
ات آ:(';
ا1C -6
:7
;
ا آAف ;"ض ه1و
MTB > Read "E:\Mtbwin\milk.dat" c1.
Entering data from file: E:\Mtbwin\milk.dat
168 rows read.
MTB > name c1='MilkProd'
MTB > set c2
DATA> 1:168
DATA> end
MTB > name c2='Time'
MTB > regr c1 1 c2;
SUBC> fits c3.
Regression Analysis
The regression equation is
MilkProd = 612 + 1.69 Time
Predictor
Constant
Time
Coef
611.682
1.69262
S = 60.74
StDev
9.414
0.09663
R-Sq = 64.9%
T
64.97
17.52
P
0.000
0.000
R-Sq(adj) = 64.7%
Analysis of Variance
Source
P
Regression
0.000
Error
Total
DF
SS
MS
F
1
1132003
1132003
306.83
166
167
612439
1744443
3689
211
Unusual Observations
Obs
Time
St Resid
113
MilkProd
Fit
StDev Fit
Residual
113
942.00
802.95
5.44
139.05
125
961.00
823.26
6.11
137.74
2.30R
125
2.28R
R
denotes
an
observation
with
a
large
standardized
residual
MTB
MTB
MTB
MTB
>
>
>
>
name c3='Trend'
let c4=c1-c3
name c4='Detrend'
Name c5 = 'AVER1'
:*61 و12 4Lك & ا
ر%;& w1;& (6 4
;
ة ا16[
ا7N
MTB > %MA 'Detrend' 12;
SUBC> Center;
SUBC>
Averages 'AVER1'.
Executing from file: E:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
Length
NMissing
Detrend
168.000
0
Moving Average
Length: 12
Accuracy Measures
MAPE: 111.68
MAD:
52.36
MSD: 3564.77
ON ا
!ال إا4;!
& ا461!
ا4آ%;!
ت ا61;!
ح ا6
212
MTB > let c6=c4-c5
MTB > name c6='DeSeason'
MTB > set c2
DATA> 14(1:12)
DATA> end
MTB > stat c6;
SUBC> by c2;
SUBC> median c7.
MTB > name c7='SeasInx'
Data Display
Row
Season
SeasInx
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
-20.750
-58.958
35.625
50.083
109.542
81.292
33.917
-10.000
-52.792
-50.250
-79.958
-44.375
:7+ ا
(&_ اV& 'ه2L ا7;
ت ا.%
رن اJ
MTB > %Decomp 'MilkProd' 12;
SUBC> Additive ;
SUBC> Start 1.
Executing from file: E:\MTBWIN\MACROS\Decomp.MAC
Macro is running ... please wait
Time Series Decomposition
213
Data
Length
MilkProd
168.000
NMissing
0
Trend Line Equation
Yt = 611.682 + 1.69262*t
Seasonal Indices
Period
1
2
3
4
5
6
7
8
9
10
11
12
Index
-20.1979
-58.4062
36.1771
50.6354
110.094
81.8437
34.4687
-9.44792
-52.2396
-49.6979
-79.4063
-43.8229
Accuracy of Model
MAPE:
MAD:
MSD:
1.583
12.088
244.406
.;ن2( &;و2JC !O ا
';; ا4رJ!.و
:7
;
آ%Decomp _&(
;[ام ا. ات:('C 1 ف1
MTB > %Decomp 'MilkProd' 12;
SUBC> Additive ;
214
SUBC>
SUBC>
Forecasts 12;
Start 1.
:ات:(';
ا76"C 7;
وا
Forecasts
Row
Period
Forecast
1
169
877.54
2
170
841.02
3
4
5
6
7
8
9
10
11
12
171
172
173
174
175
176
177
178
179
180
937.30
953.45
1014.60
988.04
942.36
900.13
859.04
863.27
835.25
872.53
) اةJ
. OرBة و6"!
; اJ26
;[ام اF. 7&1
اW%
;ج ا9 ات:('C و: 2!C
_&(
& ا4C'
ا
215
ﺍﻟﻔﺼﻞ ﺍﻟﻌﺎﺷﺮ
Using Moving Average Smoothing ك/\ ا0; ا0ا$ 4 و اLا
for Forecasting
'2
آن1
]!N ء6 ا2(C HJ;. f
هات وذK!
اO!;
ك%;!
اw1;!
;[م ا2
m 4Lك & ا
ر%;!
اw1;!
N z1 , z2 , z3 ,… , zn −2 , zn −1 , zn 4'& ز4;& & هاتK&
4B"
& اW%2 هاتK!
zˆt =
1
( zt + zt −1 + zt −2 + ⋯ + zt −m+1 ) , t = m, m + 1,..., n
m
أو
zˆt = zˆt −1 +
1
( zt − zt −m ) , t = m, m + 1,..., n
m
. n − m + 1 O!;
" ا. ^(+هات اK!
ان د ا5X
1 ه4]
]
ا4Lك & ا
ر%;!
اw1;!
ن اFN m=3 S آ1
]!N
1
( z3 + z2 + z1 )
3
1
zˆ4 = ( z4 + z3 + z2 ) or
3
⋮
zˆ3 =
zˆn =
zˆ4 = zˆ3 +
1
( zn + zn−1 + zn −2 ) or
3
1
( z4 − z1 )
3
zˆn = zˆn −1 +
1
( z n − zn − 3 )
3
ذج1!'
اV(;C هاتK!
;ض ان اb'
ء6 ا2(C HJ;
O!;
اH!"2 e ى آ73
و
zt = µ + at , at ∼ WN ( 0,σ 2 ) , t = 1, 2,..., n
ن13N
V ( zt ) = σ 2 , ∀t
7
;
.و
V ( zˆt ) =
σ2
m
, t = m, m + 1,..., n
216
O!;
ا اA وه4+هات اK!
& اe"P m ـ. Y+ أO'2(C ^(+ة اO!!
هات اK!
أي ان ا
.ء6 اU{C & 6Y& او1N& آن4;!
ا7N w! أيOI2 ء6
.ةO!!
) اJ
اw1C W';'
f
وذ42دN !8 داm A:C :4I5&
:ك/\ ا0ام ا:0c$ 4ا
:ك%;!
اw1;!
ا4(J;!
) اJ
|(';! آA:2
zn ( ℓ ) = zˆn −1 , ℓ > 0
:ل-
EMPLOY.MTW H!"
ا4B ا
(ت & ورH!% MINITAB 485 ا4&%
;[ام اF.
MTB > Retrieve
'E:\Mtbwin\DATA\EMPLOY.MTW'.
اتY;& & ي1%C &ذاI'
MTB > info
Information on the Worksheet
Column
C1
C2
C3
Count
60
60
60
Name
Trade
Food
Metals
Metals Y;!
هاتK!
ف ;[م ا1
Metals
44.2
44.8
40.1
42.8
43.6
45.2
48.3
49.9
50.7
44.3
44.4
42.0
43.0
44.7
45.0
49.1
49.6
50.3
44.4
43.1
42.4
42.8
44.5
45.5
48.9
50.7
49.2
43.4
42.6
43.1
42.5
45.0
46.2
49.4
50.7
48.1
42.8
42.4
42.4
42.6
44.8
46.8
50.0
50.9
44.3
42.2
43.1
42.3
44.9
47.5
50.0
50.5
44.4
41.8
43.2
42.9
45.2
48.3
49.6
51.2
:هاتK!
@ اA) ه
217
MTB > TSPlot 'Metals';
SUBC>
Index;
SUBC>
SUBC>
TDisplay 11;
Symbol;
SUBC>
Connect.
Metals
50
45
40
Index
10
20
30
40
50
60
ات:('C L1 وm=3 4Lك & ا
ر%;!
اw1;!
;[ام اF. هاتK!
@ اAO
اO!C نs( ا6
:*(J;& )B 6
ـ
MTB > %MA 'Metals' 3;
SUBC>
Forecasts 6;
SUBC>
Title "Smoothing and Forecasting Metals".
Executing from file: E:\MTBWIN\MACROS\MA.MAC
Moving average
Data
Metals
Length
60.0000
NMissing
0
Moving Average
Length: 3
Accuracy Measures
MAPE: 1.55036
218
MAD:
0.70292
MSD:
0.76433
Row
Period
Forecast
Lower
Upper
1
61
49.2
47.4865
50.9135
2
3
62
63
49.2
49.2
47.4865
47.4865
50.9135
50.9135
4
5
64
65
49.2
49.2
47.4865
47.4865
50.9135
50.9135
6
66
49.2
47.4865
50.9135
Smoothing and Forecasting Metals
Actual
Predicted
Metals
50
Forecast
Actual
Predicted
Forecast
45
Moving Average
Length:
3
MAPE: 1.55036
40
0
10
20
30
40
50
MAD:
0.70292
MSD:
0.76433
_8;'
ا4KB'& :U
60
Time
zˆ59 =
7
%
ا
!]ل ا7N
50.3 + 49.2 + 48.1 147.6
=
= 49.2
3
3
ا ا
!]لA ه7N أوzn +1 , zn + 2 ,..., zn +6 )J
أي4(J;!
ا6 ) ا
ـJ
ات:(';
اA:C
:7
;
آz61 , z62 ,..., z66
z60 (1) = z60 ( 2 ) = ⋯ = z60 ( 6 ) = 49.2
أي
zn ( ℓ ) ± 1.96σˆ , ℓ > 0 !ت3
ا
W%
95% :('C
;اتN
ب%
ر ـJ! آMSD = 0.76433 4!J
اA{ ، 4(J;!
ات ا:(';
) اB V!
[49.2 ± 1.96σˆ ]
219
)J
اV!
95% :('C ;ةN ن13C * وσˆ = 0.8743 ن13N σˆ 2 = 0.76433 أيσ 2
:7 ه4(J;!
ا
49.2 ± 1.96 ( 0.8743) = [ 49.2 ± 1.7135] = [47.4865,50.9135]
:أي
z60+ℓ ∈ [47.4865,50.9135] , ℓ > 0 with probability 0.95
7Cs آMSD W%C :4I5&
n −1
MSD = σˆ =
2
∑( z
− zˆi )
i
i =2
n−2
: 2!C
:(';
H`N اO2ر اB و4J.
هات اK!
ا7 7 و5 تL & ا
ر4آ%;& ت61;& (=
.؟4(J;!
) اJ
ا
Running Median ريV\ ا0ا
4N6;!
ا4!J
N 4N6;!
او اOutliers 4Lهات ا
[رK!
. آ]اU{;2 ك%;!
اw1;!
ا
هاتK!
ا2
Sآ1
]!N 4
;;!
ا4آ%;!
ت ا61;!
& اm 7 U:C ة5ا1
ا
z(t)
5
7
3
13
18
8
20
9
6
10
12
1500
11
15
H3K
اO
و
1500
z(t)
1000
500
0
In d e x
5
10
220
3 4Lك & ا
ر%;& w1;& A{.
M o v in g A v e r a g e
A c tu a l
1500
P re d ic te d
A c tu a l
P re d ic te d
z(t)
1000
M o v in g A v e ra g e
500
L e n gth:
M APE:
0
0
5
10
3
1081
M AD :
273
MSD:
268811
15
T im e
.4N6;!
ا4!J
. )B 4UU 4N تU{C _C'
ك ا%;!
اw1;!
ان ا5X
76 z O!!دي آb
ل ا16
ا
ري ذا اw1
;[م ا2 ت.1"
@ اA هH]& WY;
.4N6;!
) اB. U{;2X يA
وا
W%2 z1 , z2 , z3 ,… , zn −2 , zn −1 , zn هاتK!
j = 2i + 1 ديb
ل ا16
ا
ري ذا اw1
ا
4B"
& ا
zɶt = med ( zt −i ,..., zt ,..., zt +i ) , j = 2i + 1
3 ل16
ري ذا اL w وA{. وzɶt = med ( zt −1 , zt , zt +1 ) 4B"
(^ اC j = 3 4!J
]!N
4J.
هات اK!
smoothz(t)
15
10
5
In d e x
2
4
6
8
10
12
7
;
اH3K
اO
4JJ%
هات اK!
ن اFN 1500 ~
و15 7 هz9 ـ4JJ%
ا4!J
اSواذا آ
221
20
z(t)
15
10
5
Index
5
10
. ;;'
ا. رنB
222
ﺍﻟﻔﺼﻞ ﺍﻟﺤﺎﺩﻱ ﻋﺸﺮ
Using Single Exponential \ ا0r اL; ا0ا$ 4 و اLا
: Smoothing for Forecasting
4!2J
) اJ
ن اFN 7
;
. و4!~ اهb ا
(تV!L 76"2 ك%;!
اw1;!
ا46ا1. O!;
ا
7X اO!;
ا،%%+ 4!"
ا45'
ن & ا132X B اA وه4]2%
) اJ
آU{;
~ اb U:C 1
]) اآJ
ا76"2 ~3"
ا
'2
آن1
]!N .O&B V& اoB';C 4! اه6"C يX) اJ
أآ( وا4! أه4Uا5
m 4Lك & ا
ر%;!
اw1;!
N z1 , z2 , z3 ,… , zn −2 , zn −1 , zn 4'& ز4;& & هاتK&
4B"
& اW%2 هاتK!
zˆt =
1
( zt + zt −1 + zt −2 + ⋯ + zt −m+1 ) , t = m, m + 1,..., n
m
O;.; آ3!2 7;
وا
zˆt =
1
1
1
1
zt + zt −1 + zt −2 + ⋯ + zt −m+1 , t = m, m + 1,..., n
m
m
m
m
zˆt = β zt + β zt −1 + β zt −2 + ⋯ + β zt −m+1 , t = m, m + 1,..., n, β =
1
m
β زن1
~ اb ا
(تV!L 76"2 ك%;!
اw1;!
أي ان ا
7
;
آzn ةP%
ا4!J
هات اK!
ُ" ا. V& اoB';C ' ا
(ت اوزان6 أ1
نsا
st = α zt + α (1 − α ) zt −1 + α (1 − α ) zt −2 + ⋯ , t = 1, 2,..., n, 0 < α < 1
2
O!;
. !2& اA وه4J.
) اJ
اV!
اoB';C {وزان. زون1& w1;& 7 هst 4!J
ا
اري3C H3K. W;32 وw(
ا7Xا
st = α zt + (1 − α ) st −1 , t = 1, 2,..., n, s0 = z
ات:(';
اA:Cو
zn ( ℓ ) = sn , ℓ ≥ 1
223
:ل-
EMPLOY.MTW H!"
ا4B ا
(ت & ورH!%C
MTB > Retrieve
'E:\Mtbwin\DATA\EMPLOY.MTW'.
Metals Y;
ا7N هاتK!
ف ;[م ا1
Metals
44.2
44.3
44.4
43.4
42.8
44.3
44.4
44.8
44.4
43.1
42.6
42.4
42.2
41.8
40.1
42.0
42.4
43.1
42.4
43.1
43.2
42.8
43.0
42.8
42.5
42.6
42.3
42.9
43.6
44.7
44.5
45.0
44.8
44.9
45.2
45.2
45.0
45.5
46.2
46.8
47.5
48.3
48.3
49.1
48.9
49.4
50.0
50.0
49.6
49.9
49.6
50.7
50.7
50.9
50.5
51.2
50.7
50.3
49.2
48.1
:هاتK!
@ اA) ه
MTB > TSPlot 'Metals';
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
M
etals
50
45
40
Index
10
20
30
224
40
50
60
α = 0.2 O!C S.U A{ وw(
ا7X اO!;
;[ام اF. هاتK!
@ اAO
اO!C نs( ا6
:*(J;& )B 6 ات ـ:('C L1و
MTB > %SES 'Metals';
SUBC>
Weight 0.2;
SUBC>
Forecasts 6;
SUBC>
Title "Smoothing and Forecasting Metals";
SUBC>
Initial 6.
Single Exponential Smoothing
Data
Metals
Length
60.0000
NMissing
0
Smoothing Constant
Alpha: 0.2
Accuracy Measures
MAPE: 2.17304
MAD:
1.00189
MSD:
1.45392
Row
Period
Forecast
Lower
Upper
1
61
49.7216
47.2670
52.1763
2
62
49.7216
47.2670
52.1763
3
63
49.7216
47.2670
52.1763
4
64
49.7216
47.2670
52.1763
5
65
49.7216
47.2670
52.1763
6
66
49.7216
47.2670
52.1763
225
Smoothing and Forecasting Metals
Actual
Predicted
Forecast
Metals
50
Actual
Predicted
Forecast
45
Smoothing Constant
Alpha:
40
0
10
20
30
40
50
0.200
MAPE:
2.17304
MAD:
1.00189
MSD:
1.45392
60
Time
_8;'
ا4KB'& :U
α = 0.2 O!C S.]. z1 , z2 ,… , zn −1 , zn −2 هاتK!
w(
ا7X اO!;
اW%2 -1
:42ار3;
ا4B"
& ا
si = α zi + (1 − α ) si −1 , i = 1,2,..., n
W%C 7;
واs0 4
وX ا4!J
ا7
;ج ا% ة اO!!
) اJ
ب ا%
42ار3;
ا4B"
(أ ا73
w1;!
42 &وs0 VP و7 هO&[;' 7;
ق وا6
@ اA ه5 أ،"ة =ق.
m
'
]& 7bN s0 =
s0 =
∑z
i
i =1
m
, m = 6 ( or n, if n<6 )
44.2 + 44.3 + 44.4 + 43.4 + 42.8 + 44.3
= 43.9
6
ن132 7
;
.و
s1 = α z1 + (1 − α ) s0 = 0.2 ( 44.2 ) + 0.8 ( 43.9 ) = 8.84 + 35.12 = 43.96
s2 = α z2 + (1 − α ) s1 = 0.2 ( 44.3) + 0.8 ( 43.96 ) = 8.86 + 35.168 = 44.028
:7
;
';_ اN هةK& , 7;5 !; اA3وه
Time
Metals
SMOO1
FITS1
RESI1
1
44.2
43.9600
43.9000
0.30000
2
44.3
44.0280
43.9600
0.34000
3
44.4
44.1024
44.0280
0.37200
4
43.4
43.9619
44.1024
-0.70240
226
5
42.8
43.7295
43.9619
-1.16192
6
44.3
43.8436
43.7295
0.57046
7
44.4
43.9549
43.8436
0.55637
8
44.8
44.1239
43.9549
0.84510
9
44.4
44.1791
44.1239
0.27608
10
43.1
43.9633
44.1791
-1.07914
11
42.6
43.6906
43.9633
-1.36331
12
42.4
43.4325
43.6906
-1.29065
13
42.2
43.1860
43.4325
-1.23252
14
41.8
42.9088
43.1860
-1.38601
15
40.1
42.3470
42.9088
-2.80881
16
42.0
42.2776
42.3470
-0.34705
17
42.4
42.3021
42.2776
0.12236
18
43.1
42.4617
42.3021
0.79789
19
42.4
42.4494
42.4617
-0.06169
20
43.1
42.5795
42.4494
0.65065
21
43.2
42.7036
42.5795
0.62052
22
42.8
42.7229
42.7036
0.09642
23
43.0
42.7783
42.7229
0.27713
24
42.8
42.7826
42.7783
0.02171
25
42.5
42.7261
42.7826
-0.28264
26
42.6
42.7009
42.7261
-0.12611
27
42.3
42.6207
42.7009
-0.40089
28
42.9
42.6766
42.6207
0.27929
29
43.6
42.8613
42.6766
0.92343
30
44.7
43.2290
42.8613
1.83875
31
44.5
43.4832
43.2290
1.27100
32
45.0
43.7866
43.4832
1.51680
33
44.8
43.9892
43.7866
1.01344
34
44.9
44.1714
43.9892
0.91075
35
45.2
44.3771
44.1714
1.02860
36
45.2
44.5417
44.3771
0.82288
37
45.0
44.6334
44.5417
0.45830
38
45.5
44.8067
44.6334
0.86664
227
39
46.2
45.0853
44.8067
1.39331
40
46.8
45.4283
45.0853
1.71465
41
47.5
45.8426
45.4283
2.07172
42
48.3
46.3341
45.8426
2.45738
43
48.3
46.7273
46.3341
1.96590
44
49.1
47.2018
46.7273
2.37272
45
48.9
47.5415
47.2018
1.69818
46
49.4
47.9132
47.5415
1.85854
47
50.0
48.3305
47.9132
2.08683
48
50.0
48.6644
48.3305
1.66947
49
49.6
48.8515
48.6644
0.93557
50
49.9
49.0612
48.8515
1.04846
51
49.6
49.1690
49.0612
0.53877
52
50.7
49.4752
49.1690
1.53101
53
50.7
49.7202
49.4752
1.22481
54
50.9
49.9561
49.7202
1.17985
55
50.5
50.0649
49.9561
0.54388
56
51.2
50.2919
50.0649
1.13510
57
50.7
50.3735
50.2919
0.40808
58
50.3
50.3588
50.3735
-0.07353
59
49.2
50.1271
50.3588
-1.15883
60
48.1
49.7216
50.1271
-2.02706
FITS1 ~&[
د ا1!"
اsi , i = 1,2,...,60 ة أيO!!
) اJ
ى ا1%2 SMOO1 V.د ا
ا1!"
ا
ء6ي ا1%2 RESI1 ~&[
د ا1!"
اzˆi = si −1 , i = 1, 2,...,60 أي4J(6!
) اJ
ى ا1%2
ei = zi − zˆi , i = 1, 2,...,60 ( أيResiduals 7Bا1(
)ا
:ة أيO!& 4!B , 4(J;!
) اJ
|(';! آA:2 -2
zn ( ℓ ) = sn , ℓ > 0
7
%
ا
!]ل ا7bN
z60 ( ℓ ) = 49.7216, ℓ > 0
228
ا ا
!]لA ه7N أوzn +1 , zn +2 ,..., zn +6 )J
أي4(J;!
ا6 ) ا
ـJ
ات:(';
اA:C
:7
;
آz61 , z62 ,..., z66
z60 (1) = z60 ( 2 ) = ⋯ = z60 ( 6 ) = 49.7216
أي
zn ( ℓ ) ± 1.96σˆ , ℓ > 0 !ت3
ا
W%
95% :('C
;اتN
ب%
-3
MSD = 1.45392 4!J
اA{ ، 4(J;!
ات ا:(';
) اB V!
[49.7216 ± 1.96σˆ ]
95% :('C ;ةN ن13C * وσˆ = 1.205786 ن13N σˆ 2 = 1.45392 أيσ 2 ر ـJ!آ
:7 ه4(J;!
) اJ
اV!
49.7216 ± 1.96 (1.205786 ) = [49.7216 ± 2.3633] = [47.35826,52.08494]
:أي
z60+ℓ ∈ [47.3582,50.0849] , ℓ > 0 with probability 0.95
7Cs آMSD W%C :4I5&
n
MSD = σˆ =
2
∑( z
i
i =1
− zˆi )
n −1
: 2!C
H`N اO2ر اB وα = 0.3,0.4,0.5 &[;& 4J.
هات اK!
ا7 w. 7 اO!C (=
.؟4(J;!
) اJ
ا:(';
229
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻧﻲ ﻋﺸﺮ
Using Double Exponential ادوج0r اL; ا0ا$ 4 و اLا
: Smoothing for Forecasting
:Brown’s Method اون. 4J2= :Xأو
:7
;
اL1 0 < α < 1 O!C S.]
وz1 , z2 ,… , zn −1 , zn −2 هاتK!
st( ) = α zt + (1 − α ) st(−1) , t = 1,2,..., n
1
1
اA ه4L در7
& اC (1) ( وw(
ا7X اO!;
اI )اst( ) = st w. 7 اO!C st( ) Q5
1
1
O!;
ا
st( ) = α st( ) + (1 − α ) st(−1) , t = 1, 2,..., n
2
1
2
O!;
ا اA ه4L در7
& اC ( 2 ) &دوج و7 اO!C st( ) Q5
2
at = 2 st( ) − st( ) , t = 1, 2,..., n
1
bt =
2
α
1
2
st( ) − st( ) , t = 1, 2,..., n
1−α
(
)
4
& ا
!"د4J(6!
) اJ
اW%C
zˆt = at + bt t ,
t = 1, 2,..., n
& zn +ℓ , ℓ > 0 4(J;!
) اJ
ات:(';
اW%Cو
zn ( ℓ ) = an + bn ℓ, ℓ > 0
4J.
ت اB"
& ا: s0( ) وs0( ) 4
وX) اJ
ب ا5
2
s0( ) = a0 −
1
1−α
s0( ) = a0 − 2
2
α
b0
1−α
α
1
b0
ن132 وzt = α + β t + et ,
t = 1,2,..., n
&
ا7 هاتK!
ار ا%F. b0 وa0 L1
b0 = βˆ وa0 = αˆ
230
:Holt’s Method S
1 ه4J2= :U
:7
;
اL1 0 < γ < 1 و0 < α < 1 O!C 7;.]
وz1 , z2 ,… , zn −1 , zn −2 هاتK!
st = α zt + (1 − α )( st −1 + bt −1 ) ,
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 ,
t = 1, 2,..., n
t = 1, 2,..., n
& 4J(6!
) اJ
اW%
zˆt = st + bt t ,
t = 1, 2,..., n
& 4(J;!
) اJ
ات:(';
وا
zn ( ℓ ) = sn + bn ℓ, ℓ > 0
& b0 وs0 4
وX) اJ
اW%
s0 = z1
b0 = z2 − z1
b0 =
or
( z2 − z1 ) + ( z3 − z2 ) ( z3 − z1 )
=
or
2
2
( z − z ) + ( z3 − z2 ) + ( z4 − z3 ) ( z4 − z1 )
b0 = 2 1
=
3
3
:ل-
EMPLOY.MTW H!"
ا4B ا
(ت & ورH!%C
MTB > Retrieve 'E:\Mtbwin\DATA\EMPLOY.MTW'.
Metals Y;
ا7N هاتK!
ف ;[م ا1
Metals
44.2
44.3
44.4
43.4
42.8
44.3
44.4
44.8
44.4
43.1
42.6
42.4
42.2
41.8
40.1
42.0
42.4
43.1
42.4
43.1
43.2
42.8
43.0
42.8
42.5
42.6
42.3
42.9
43.6
44.7
44.5
45.0
44.8
44.9
45.2
45.2
45.0
45.5
46.2
46.8
47.5
48.3
48.3
49.1
48.9
49.4
50.0
50.0
49.6
49.9
49.6
50.7
50.7
50.9
50.5
51.2
50.7
50.3
49.2
48.1
231
:هاتK!
@ اA) ه
MTB > TSPlot 'Metals';
SUBC>
Index;
SUBC>
TDisplay 11;
SUBC>
Symbol;
SUBC>
Connect.
Metals
50
45
40
Index
10
20
30
40
50
60
نs ;[م ا،اون. 4J26. ا
!دوج7X اO!;
;[ام اF. هاتK!
@ اAO
اO!C نs( ا6
هA{ ف1 42{وزان &;و. WEIGHT 7b
& اX اV& %DES (Macro) _&(
ا
0.2
MTB > %DES 'Metals';
SUBC>
Weight 0.2 0.2;
SUBC>
Forecasts 6;
SUBC>
Title "Brown's Double Exponential Smoothing";
SUBC>
Table.
Double Exponential Smoothing
Data
Metals
Length
60.0000
232
NMissing
0
Smoothing Constants
Alpha (level): 0.2
Gamma (trend): 0.2
Accuracy Measures
MAPE: 2.16187
MAD:
0.97032
MSD:
1.62936
Time Metals
Smooth
Predict
Error
1
44.2
41.7739
41.1674
3.03257
2
44.3
42.4976
42.0470
2.25303
3
44.4
43.1686
42.8607
1.53927
4
43.4
43.5546
43.5933
-0.19330
5
42.8
43.7373
43.9716
-1.17163
6
44.3
44.1459
44.1074
0.19257
7
44.4
44.4990
44.5238
-0.12377
8
44.8
44.8575
44.8719
-0.07189
9
44.4
45.0620
45.2275
-0.82751
10
43.1
44.9391
45.3989
-2.29891
11
42.6
44.6673
45.1841
-2.58407
12
42.4
44.3271
44.8088
-2.40884
13
42.2
43.9378
44.3723
-2.17229
14
41.8
43.4769
43.8962
-2.09617
15
40.1
42.7011
43.3514
-3.25142
16
42.0
42.3565
42.4456
-0.44557
17
42.4
42.1465
42.0831
0.31694
18
43.1
42.1286
41.8857
1.21426
19
42.4
42.0132
41.9164
0.48355
233
20
43.1
42.0763
41.8204
1.27964
21
43.2
42.1877
41.9347
1.26533
22
42.8
42.2374
42.0967
0.70327
23
43.0
42.3396
42.1745
0.82549
24
42.8
42.4078
42.3098
0.49024
25
42.5
42.4181
42.3976
0.10244
26
42.6
42.4495
42.4119
0.18809
27
42.3
42.4207
42.4509
-0.15090
28
42.9
42.5129
42.4161
0.48394
29
43.6
42.7420
42.5276
1.07245
30
44.7
43.1797
42.7996
1.90036
31
44.5
43.5507
43.3133
1.18668
32
45.0
43.9854
43.7317
1.26826
33
44.8
44.3338
44.2172
0.58280
34
44.9
44.6511
44.5889
0.31112
35
45.2
44.9749
44.9187
0.28133
36
45.2
45.2430
45.2538
-0.05376
37
45.0
45.4157
45.5197
-0.51967
38
45.5
45.6373
45.6716
-0.17162
39
46.2
45.9491
45.8863
0.31368
40
46.8
46.3285
46.2106
0.58938
41
47.5
46.7909
46.6136
0.88637
42
48.3
47.3492
47.1115
1.18850
43
48.3
47.8339
47.7173
0.58266
44
49.1
48.4002
48.2253
0.87469
45
48.9
48.8413
48.8267
0.07332
46
49.4
49.2966
49.2707
0.12930
47
50.0
49.7849
49.7311
0.26890
48
50.0
50.1841
50.2302
-0.23017
49
49.6
50.4162
50.6202
-1.02022
50
49.9
50.6292
50.8114
-0.91145
51
49.6
50.7104
50.9880
-1.38797
52
50.7
50.9509
51.0137
-0.31368
53
50.7
51.1334
51.2417
-0.54169
234
54
50.9
51.3019
51.4024
-0.50244
55
50.5
51.3407
51.5509
-1.05093
56
51.2
51.4782
51.5477
-0.34770
57
50.7
51.4770
51.6712
-0.97120
58
50.3
51.3649
51.6311
-1.33115
59
49.2
51.0127
51.4659
-2.26587
60
48.1
50.4384
51.0230
-2.92300
Row
Period
Forecast
Lower
Upper
1
61
50.3318
47.9545
52.7091
2
62
50.2252
47.7984
52.6520
3
63
50.1186
47.6384
52.5987
4
64
50.0120
47.4749
52.5490
5
65
49.9054
47.3080
52.5027
6
66
49.7988
47.1381
52.4594
Brown's Double Exponential Smoothing
Actual
Predicted
Forecast
Metals
50
Actual
Predicted
Forecast
45
Smoothing Constants
Alpha (level): 0.200
Gamma (trend):0.200
MAPE:
MAD:
MSD:
40
0
10
20
30
Time
235
40
50
60
2.16187
0.97032
1.62936
: b0 وa0 د2إ
MTB > set c4
DATA> 1:60
DATA> end
MTB > regr c3 1 c4
Regression Analysis
The regression equation is
Metals = 41.0 + 0.152 C4
W% O'& وb0 = 0.152 وa0 = 41.0 إذا
1
s0(
2)
1−α
0.8
( 0.152 ) = 41.608
α
0.2
1−α
0.8
= a0 − 2
b0 = 41.0 − 2
( 0.152 ) = 42.216
α
0.2
s0( ) = a0 −
b0 = 41.0 −
s1( ) = ( 0.2 )( 44.2 ) + ( 0.8 )( 41.608 ) = 42.1264
1
s1( ) = ( 0.2 )( 42.1264 ) + ( 0.8 )( 42.216 ) = 42.19808
2
a1 = ( 2 )( 42.1264 ) − 42.19808 = 42.05472
( 0.2 )
( 42.19808 − 42.05472 ) = 0.03584
( 0.8 )
zˆ1 = 42.05472 + ( 0.03584 )(1) = 42.09056
b1 =
… r
ا اA3وه
.4J.
ا4]& ا7N ! آMSD ;[امF. :(';
;ات اN W%C
:ل-
WEIGHT 7b
& اX اV& %DES (Macro) _&(
ن اs ;[م اS
1 ه4J2= (6;
γ = 0.3 وα = 0.2 A{ ف1 4b;[& {وزان.
MTB > RETR 'E:\Mtbwin\DATA\EMPLOY.MTW'.
Retrieving worksheet from file: E:\Mtbwin\DATA\EMPLOY.MTW
Worksheet was saved on
6/ 5/1996
236
MTB > %DES 'Metals';
SUBC>
Weight 0.2 0.3;
SUBC>
Forecasts 6;
SUBC>
Title "Holt's Double Exponential Smoothing";
SUBC>
Table.
Double Exponential Smoothing
Data
Metals
Length
60.0000
NMissing
0
Smoothing Constants
Alpha (level): 0.2
Gamma (trend): 0.3
Accuracy Measures
MAPE: 2.15656
MAD:
0.96328
MSD:
1.56274
Time
Metals
Smooth
Predict
Error
1
44.2
41.7739
41.1674
3.03257
2
44.3
42.5461
42.1076
2.19238
3
44.4
43.2891
43.0113
1.38868
4
43.4
43.7501
43.8376
-0.43760
5
42.8
43.9779
44.2724
-1.47237
6
44.3
44.3895
44.4118
-0.11184
7
44.4
44.7334
44.8167
-0.41671
8
44.8
45.0685
45.1356
-0.33560
9
44.4
45.2405
45.4506
-1.05057
237
10
43.1
45.0676
45.5595
-2.45952
11
42.6
44.7113
45.2391
-2.63911
12
42.4
44.2595
44.7244
-2.32443
13
42.2
43.7466
44.1332
-1.93322
14
41.8
43.1634
43.5043
-1.70426
15
40.1
42.2751
42.8188
-2.71884
16
42.0
41.8139
41.7674
0.23263
17
42.4
41.5361
41.3202
1.07985
18
43.1
41.5057
41.1072
1.99283
19
42.4
41.4371
41.1964
1.20365
20
43.1
41.5799
41.1999
1.90008
21
43.2
41.8054
41.4568
1.74322
22
42.8
41.9895
41.7869
1.01314
23
43.0
42.2254
42.0317
0.96829
24
42.8
42.4205
42.3257
0.47431
25
42.5
42.5395
42.5493
-0.04933
26
42.6
42.6522
42.6653
-0.06528
27
42.3
42.6793
42.7741
-0.47413
28
42.9
42.7982
42.7728
0.12724
29
43.6
43.0394
42.8993
0.70070
30
44.7
43.4861
43.1826
1.51743
31
44.5
43.8762
43.7202
0.77977
32
45.0
44.3257
44.1571
0.84285
33
44.8
44.6858
44.6573
0.14275
34
44.9
45.0007
45.0259
-0.12590
35
45.2
45.3066
45.3333
-0.13327
36
45.2
45.5449
45.6312
-0.43116
37
45.0
45.6749
45.8436
-0.84361
38
45.5
45.8384
45.9230
-0.42295
39
46.2
46.0888
46.0610
0.13895
40
46.8
46.4159
46.3199
0.48014
41
47.5
46.8406
46.6757
0.82428
42
48.3
47.3799
47.1499
1.15014
43
48.3
47.8665
47.7582
0.54181
238
44
49.1
48.4419
48.2774
0.82265
45
48.9
48.9016
48.9020
-0.00205
46
49.4
49.3693
49.3617
0.03832
47
50.0
49.8653
49.8317
0.16832
48
50.0
50.2702
50.3378
-0.33779
49
49.6
50.4979
50.7224
-1.12240
50
49.9
50.6862
50.8827
-0.98275
51
49.6
50.7296
51.0121
-1.41206
52
50.7
50.9166
50.9708
-0.27079
53
50.7
51.0532
51.1415
-0.44152
54
50.9
51.1813
51.2516
-0.35162
55
50.5
51.1869
51.3586
-0.85860
56
51.2
51.2901
51.3127
-0.11267
57
50.7
51.2673
51.4092
-0.70916
58
50.3
51.1350
51.3438
-1.04381
59
49.2
50.7591
51.1489
-1.94889
60
48.1
50.1448
50.6560
-2.55603
Row
Period
Forecast
Lower
Upper
1
61
49.8884
47.5283
52.2484
2
62
49.6319
47.1597
52.1041
3
63
49.3755
46.7803
51.9707
4
64
49.1190
46.3915
51.8466
5
65
48.8626
45.9946
51.7306
6
66
48.6061
45.5908
51.6215
239
Holt's Double Exponential Smoothing
Actual
Predicted
Forecast
Metals
50
Actual
Predicted
Forecast
45
Smoothing Constants
Alpha (level): 0.200
Gamma (trend):0.300
MAPE:
MAD:
MSD:
40
0
10
20
30
40
50
2.15656
0.96328
1.56274
60
Time
.6C مVB1;2 :4I5& ) 2و2 )J
"\ ا. V(;;. 4J.
ت ا.%
ا4%+ & J%C : 2!C
7N f
A وآ4(%
ا4
s واW%
ا. اادH]!C 2= ;ف9 f
!& وذC ت.%
ا
(!O'& H ذاآت آ7N مB ار2[C
240
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻟﺚ ﻋﺸﺮ
aV ا0ت ا, ز7 وi ;0ا$ 4] و ا,- ا0r اLا
Triple Exponential Smoothing: Winters' Three-Parameter Trend
and Seasonality Smoothing Method
&ى4!1!
اه ا1I
اH%;
Vb'C X Hb
ا اA ه7N در'ه7;
ا4J.
ق ا6
اV!L
Winters' trend and
و;ز4J2= وDecomposition Method
f3b;
ا4J2=
' هOP"; ف1 7;
اseasonal smoothing
4N'!
ا4!1!
و;ز !;ت ا4J2=
4B"
. ا آO!C هاتK!
اO!C :Xأو
st = α
zt
+ (1 − α )( st −1 + bt −1 ) , t = 1,2,..., n
St − s
اف9 اO!C :U
bt = γ ( st − st −1 ) + (1 − γ ) bt −1 , t = 1,2,..., n
4!1!
اO!C :]
U
St = β
zt
+ (1 − β ) St −s , t = 1, 2,..., n
st
4!1!
دورة ا7 هs وi &
' ا4!1!
ا4( ا
!آ7 هSi Q5
4B"
. 76"C 4J(6!
) اJ
ا
zˆt = ( st + bt t ) St −s , t = 1, 2,..., n
4B"
ات & ا:(';
وا
zn ( ℓ ) = ( sn + bn ℓ ) Sn −s +ℓ , ℓ > 0
ارز&ت1[. W%C 4
وX) اJ
ان اQ5 42ت ا
و.%
. و;ز4J2= V(;C اL W"
& ا
.W%
& ا4L[!
_ ا8;'
. 7b;3 ه' وOP"; اAO
وW%
;[ام اF. 46 z
:ل-
EMPLOY.MTW H!"
ا4B ا
(ت & ورH!%C
241
MTB > Retrieve
'E:\Mtbwin\DATA\EMPLOY.MTW'.
Food Y;!
ا7N هاتK!
ف ;[م ا1
Food
53.5
53.0
53.2
52.5
53.4
56.5
65.3
70.7
66.9
58.2
55.3
53.4
52.1
51.5
51.5
52.4
53.3
55.5
64.2
69.6
69.3
58.5
55.3
53.6
52.3
51.5
51.7
51.5
52.2
57.1
63.6
68.8
68.9
60.1
55.6
53.9
53.3
53.1
53.5
53.5
53.9
57.1
64.7
69.4
70.3
62.6
57.9
55.8
54.8 54.2 54.6 54.3 54.8 58.1 68.1 73.3 75.5 66.4 60.5 57.7
هاتK!
و) ا
75
Food
70
65
60
55
50
Index
10
20
30
40
50
60
:7
;
و;ز آ4J2= نs( ا6 12 ورة. 4!1& هةI
ان ا5
Additive Model 7NP9ذج ا1!'
:Xأو
zt = bt + St + et , t = 1, 2,..., n
MTB > %Wintadd 'Food' 12;
SUBC>
Weight 0.2 0.2 0.2;
SUBC>
Forecasts 12;
SUBC>
Title "Wintrs' Trend and Seasonal Smoothing";
242
SUBC>
Table.
Winters' additive model
Data
Food
Length
60.0000
NMissing
0
Smoothing Constants
Alpha (level):
0.2
Gamma (trend):
0.2
Delta (seasonal): 0.2
Accuracy Measures
MAPE: 1.94769
MAD:
1.15100
MSD:
2.66711
Time
Food
Smooth
Predict
Error
1
53.5
48.7755
49.4303
4.06965
2
53.0
49.6020
50.4197
2.58027
3
53.2
51.0736
51.9944
1.20556
4
52.5
52.0733
53.0424
-0.54244
5
53.4
53.5117
54.4591
-1.05914
6
56.5
57.4851
58.3901
-1.89013
7
65.3
66.2299
67.0593
-1.75932
8
70.7
71.7852
72.5443
-1.84430
9
66.9
71.8932
72.5785
-5.67851
10
58.2
62.3206
62.7787
-4.57874
11
55.3
57.5208
57.7958
-2.49577
12
53.4
55.1544
55.3296
-1.92957
13
52.1
55.0393
55.1373
-3.03734
14
51.5
53.6493
53.6258
-2.12584
243
15
51.5
53.1185
53.0100
-1.50996
16
52.4
52.2661
52.0971
0.30287
17
53.3
52.6528
52.4960
0.80401
18
55.5
55.7616
55.6369
-0.13695
19
64.2
63.8483
63.7181
0.48187
20
69.6
68.8787
68.7678
0.83218
21
69.3
68.0386
67.9610
1.33902
22
58.5
59.2825
59.2585
-0.75851
23
55.3
55.0979
55.0435
0.25650
24
53.6
53.0432
52.9991
0.60092
25
52.3
53.0377
53.0177
-0.71765
26
51.5
52.1394
52.0907
-0.59067
27
51.7
51.9889
51.9165
-0.21651
28
51.5
51.7214
51.6403
-0.14031
29
52.2
52.1875
52.1009
0.09913
30
57.1
55.0750
54.9923
2.10774
31
63.6
63.7515
63.7531
-0.15314
32
68.8
68.8427
68.8382
-0.03822
33
68.9
68.0160
68.0099
0.89007
34
60.1
58.9061
58.9356
1.16436
35
55.6
55.3220
55.3981
0.20190
36
53.9
53.4419
53.5262
0.37384
37
53.3
53.3084
53.4076
-0.10757
38
53.1
52.6717
52.7666
0.33345
39
53.5
52.9095
53.0177
0.48233
40
53.5
52.9745
53.1020
0.39803
41
53.9
53.7952
53.9386
-0.03858
42
57.1
57.2065
57.3484
-0.24838
43
64.7
65.2747
65.4066
-0.70661
44
69.4
70.4039
70.5076
-1.10758
45
70.3
69.6200
69.6794
0.62065
46
62.6
60.5655
60.6497
1.95031
47
57.9
57.0392
57.2014
0.69858
48
55.8
55.3721
55.5623
0.23773
244
49
54.8
55.2403
55.4399
-0.63993
50
54.2
54.6681
54.8422
-0.64220
51
54.6
54.8138
54.9622
-0.36218
52
54.3
54.7366
54.8705
-0.57048
53
54.8
55.3001
55.4112
-0.61119
54
58.1
58.5310
58.6176
-0.51765
55
68.1
66.4168
66.4827
1.61731
56
73.3
71.8806
72.0112
1.28878
57
75.5
71.8794
72.0616
3.43843
58
66.4
63.7240
64.0437
2.35629
59
60.5
60.3141
60.7281
-0.22810
60
57.7
58.6397
59.0446
-1.34455
Row
Period
Forecast
Lower
Upper
1
61
58.6167
55.7968
61.4366
2
62
58.3236
55.4449
61.2023
3
63
58.8195
55.8775
61.7614
4
64
58.9840
55.9746
61.9935
5
65
59.8723
56.7913
62.9532
6
66
63.4804
60.3243
66.6365
7
67
72.0757
68.8410
75.3104
8
68
77.4486
74.1321
80.7651
9
69
77.7540
74.3528
81.1552
10
70
68.9067
65.4180
72.3954
11
71
64.6434
61.0647
68.2221
12
72
62.7731
59.1020
66.4441
245
Wintrs' Trend and Seasonal Smoothing
Actual
80
Predicted
Forecast
Actual
Predicted
Forecast
Food
70
60
Smoothing Constants
Alpha (level): 0.200
Gamma (trend):0.200
Delta (season):0.200
50
MAPE:
MAD:
MSD:
0
10
20
30
40
50
60
1.94769
1.15100
2.66711
70
Time
Multiplicative Model 7b`;
ذج ا1!'
:U
zt = bt St + et , t = 1, 2,..., n
MTB > %Wintmult 'Food' 12;
SUBC>
Weight 0.2 0.2 0.2;
SUBC>
Forecasts 12;
SUBC>
Title "Winters' Trend and Seasonal Smoothing";
SUBC>
Table.
Winters' multiplicative model
Data
Food
Length
60.0000
NMissing
0
Smoothing Constants
Alpha (level):
0.2
Gamma (trend):
0.2
Delta (seasonal): 0.2
Accuracy Measures
MAPE: 1.88377
MAD:
1.12068
MSD:
2.86696
246
Time
Food
Smooth
Predict
Error
1
53.5
48.7870
49.3853
4.11470
2
53.0
49.6755
50.4303
2.56966
3
53.2
51.1521
52.0132
1.18677
4
52.5
52.1675
53.0746
-0.57458
5
53.4
53.6181
54.5132
-1.11323
6
56.5
57.6509
58.5541
-2.05414
7
65.3
66.6199
67.5607
-2.26072
8
70.7
72.4105
73.3280
-2.62800
9
66.9
72.5679
73.3777
-6.47768
10
58.2
62.7837
63.2634
-5.06337
11
55.3
57.9154
58.1732
-2.87320
12
53.4
55.5108
55.6485
-2.24849
13
52.1
54.4920
54.5392
-2.43920
14
51.5
53.2117
53.1621
-1.66212
15
51.5
52.8118
52.6957
-1.19573
16
52.4
52.0929
51.9302
0.46985
17
53.3
52.5894
52.4439
0.85611
18
55.5
55.7388
55.6209
-0.12087
19
64.2
63.7189
63.5782
0.62178
20
69.6
68.7087
68.5838
1.01617
21
69.3
67.9722
67.8890
1.41104
22
58.5
59.4594
59.4361
-0.93606
23
55.3
55.4037
55.3468
-0.04680
24
53.6
53.4103
53.3536
0.24639
25
52.3
52.6818
52.6356
-0.33562
26
51.5
51.8659
51.8071
-0.30705
27
51.7
51.8002
51.7290
-0.02902
28
51.5
51.6271
51.5549
-0.05492
29
52.2
52.1643
52.0890
0.11103
247
30
57.1
55.0424
54.9676
2.13244
31
63.6
63.6079
63.6199
-0.01988
32
68.8
68.6702
68.6823
0.11774
33
68.9
67.9561
67.9727
0.92727
34
60.1
59.1021
59.1487
0.95133
35
55.6
55.6210
55.7003
-0.10032
36
53.9
53.7881
53.8609
0.03912
37
53.3
53.0479
53.1211
0.17892
38
53.1
52.4502
52.5294
0.57055
39
53.5
52.7444
52.8467
0.65329
40
53.5
52.8747
53.0029
0.49714
41
53.9
53.7689
53.9188
-0.01879
42
57.1
57.2790
57.4374
-0.33743
43
64.7
65.4702
65.6357
-0.93567
44
69.4
70.6713
70.8095
-1.40954
45
70.3
69.8908
69.9719
0.32815
46
62.6
60.7552
60.8370
1.76302
47
57.9
57.1925
57.3348
0.56523
48
55.8
55.5181
55.6775
0.12253
49
54.8
54.8764
55.0383
-0.23826
50
54.2
54.3244
54.4749
-0.27486
51
54.6
54.5372
54.6769
-0.07694
52
54.3
54.5298
54.6661
-0.36612
53
54.8
55.1925
55.3155
-0.51551
54
58.1
58.6054
58.7141
-0.61410
55
68.1
66.7739
66.8698
1.23016
56
73.3
72.4056
72.5622
0.73784
57
75.5
72.3385
72.5236
2.97638
58
66.4
63.6729
63.9378
2.46217
59
60.5
60.0395
60.3781
0.12191
60
57.7
58.3023
58.6338
-0.93381
Row
Period
Forecast
Lower
248
Upper
1
61
57.8102
55.0645
60.5558
2
62
57.3892
54.5864
60.1921
3
63
57.8332
54.9687
60.6977
4
64
57.9307
55.0005
60.8609
5
65
58.8311
55.8313
61.8309
6
66
62.7415
59.6686
65.8145
7
67
72.1849
69.0354
75.3344
8
68
78.1507
74.9215
81.3798
9
69
78.5092
75.1976
81.8208
10
70
68.6689
65.2721
72.0657
11
71
63.9258
60.4414
67.4103
12
72
61.8189
58.2446
65.3933
Winters' Trend and Seasonal Smoothing
Actual
80
Predicted
Forecast
Actual
Predicted
Forecast
Food
70
60
Smoothing Constants
Alpha (level): 0.200
Gamma (trend):0.200
Delta (season):0.200
50
MAPE:
MAD:
MSD:
0
10
20
30
40
50
60
1.88377
1.12068
2.86696
70
Time
:تI5&
S.U γ و73
اO!;
اS.U α 7 &"
) ه4UU )B إ;ر7
;ج ا% و;ز4J2= (6;
4UU 7N Optimization 4`N أ4! @A وه4!1!
اO!C S.U β اف و9 اO!C
;[امF. 8JC O.5 ا
!;[م7859`ء ا
!"
) ( إ& أن ;ك (&_ اN ) "د.ا
.)J
اf;. _&(
&اد اF. % م1J ا
(&_ أوH دا4'(& 46 z ارز&ت1
Q
]
اz &ة &"! وH آ7N S(U . α = γ = β = 0.2 A ا4J.
ا4]& ا7N : 2!C
MSD HB أ7 H%C 7;5
249
:4 ذج7 ل ء-
MINITAB _&(
ا
(ت41!& & CPI.MTW H!"
ا4Bف ;[م ور1
MTB > Retrieve
'C:\MTBWIN\STUDENT9\CPI.MTW'.
CPIChange Y;!
اA{ ف1
CPIChnge
1.7
1.0
1.0
1.3
1.3
1.6
2.9
3.1
4.2
5.5
5.7
4.4
3.2
6.2
11.0
9.1
5.8
6.5
7.6
11.3
13.5
10.3
6.2
3.2
4.3
3.6
1.9
3.6
4.1
4.8
5.4
4.2
3.0
4'"
ا48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.;
اL1 و4;!
) ا
MTB > TSPlot
14
12
CPIChnge
10
8
6
4
2
0
Index
5
10
15
20
25
30
MTB > %acf c2
Autocorrelation
Autocorrelation Function for CPIChnge
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
Lag
3
Corr
4
T
LBQ
1 0.79 4.53
2 0.46 1.77
3 0.29 1.02
4 0.26 0.88
5 0.26 0.86
6 0.16 0.52
7 -0.02 -0.06
22.42
30.32
33.59
36.24
38.97
40.05
40.07
5
Lag
Corr
6
T
LBQ
8 -0.16 -0.51
41.23
250
7
8
MTB > %pacf c2
Partial Autocorrelation
Partial Autocorrelation Function for CPIChnge
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
Lag PAC
1
2
3
4
5
6
7
0.79
-0.42
0.35
-0.04
0.08
-0.28
-0.05
5
6
T
Lag PAC
T
4.53
-2.44
2.01
-0.24
0.43
-1.62
-0.29
8 -0.09
-0.52
7
8
ARMA(1,1) ذج1!'
ن ا132 B 4'"
ا48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.;
& ا!ط ا
4(J;!
) اJ
ات:('C 5 1C;ح وJ!
ذج ا1!'
( ا6C 4
;
اوا& ا،4;!
( ا6'2
MTB >
SUBC>
SUBC>
SUBC>
SUBC>
SUBC>
SUBC>
arima 1 0 1 c2;
fore 5 c3 c4 c5;
gser;
gacf;
gpacf;
ghist;
gnormal.
ARIMA Model
ARIMA model for CPIChnge
Estimates at each iteration
Iteration
SSE
Parameters
0
323.251
0.100
0.100
1
200.616
0.250
-0.050
2
182.146
0.184
-0.200
3
163.067
0.135
-0.350
251
4.522
3.745
4.067
4.308
4
142.864
0.107
-0.500
4.434
5
121.402
0.111
-0.650
4.407
6
7
99.668
77.036
0.150
0.268
-0.800
-0.950
4.197
3.590
8
67.550
0.418
-0.956
2.828
9
62.802
0.568
-0.964
2.062
10
62.108
0.637
-0.973
1.687
11
62.030
0.644
-0.979
1.619
12
13
62.003
61.996
0.647
0.651
-0.982
-0.985
1.584
1.549
14
61.996
0.651
-0.986
1.539
Unable to reduce sum of squares any further
Final Estimates of Parameters
Type
Coef
StDev
AR
1
0.6513
0.1434
MA
1
-0.9857
0.0516
Constant
1.5385
0.4894
Mean
4.412
1.403
T
4.54
-19.11
3.14
Number of observations: 33
Residuals:
SS = 61.8375 (backforecasts excluded)
MS = 2.0613 DF = 30
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
Chi-Square 9.6(DF=10)
17.0(DF=22)
* (DF= *)
(DF= *)
48
*
Forecasts from period 33
Period
34
35
36
37
38
Forecast
3.1362
3.5810
3.8708
4.0594
4.1823
95 Percent Limits
Lower
Upper
0.3216
5.9507
-1.8180
8.9801
-2.3061
10.0476
-2.4192
10.5380
-2.4201
10.7848
Actual
1;ح هJ!
ذج ا1!'
أي ان ا
252
zt = 1.54 + 0.65zt −1 + at − 0.99at −1 , at ∼ N ( 0, 2.06 )
7 هt إ;(ر4!B و42 ا
!"رOCNا%رات ا
!"
) وإJ&
( )
θˆ = −0.9857, s.e. (θˆ ) = 0.0516, t = −19.11
δˆ = 1.5385, s.e. (δˆ ) = 0.4894, t = 3.14
φˆ1 = 0.6513, s.e. φˆ1 = 0.1434, t = 4.54
1
1
σˆ 2 = 2.0613, with d . f . = 30
.421'"& )
"!
اV!L ان5
:7Bا1(
اo%b نsا
ACF of Residuals for CPIChnge
(with 95% confidence limits for the autocorrelations)
1.0
0.8
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
8
Lag
PACF of Residuals for CPIChnge
(with 95% confidence limits for the partial autocorrelations)
1.0
0.8
Partial Autocorrelation
Autocorrelation
0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
Lag
253
6
7
8
4P V2ز1C V(;C 7Bا1(
ل أن اC 7Bا1(
78
ا7CاA
اw. وا
;ا7CاA
اw.أ!ط ا
;ا
:7Bا1(
ا4"(= o%b'
،46. &;اz `ء أي.
اري3;
ر) ا
!رج ا
Histogram of the Residuals
(response is CPIChnge)
8
7
Frequency
6
5
4
3
2
1
0
-3
-2
-1
0
1
2
3
4
Residual
.|K
"\ ا. ';& (و2
:7Bا1(
7"(6
;!ل ا59 اw6[& إI''
Normal Probability Plot of the Residuals
(response is CPIChnge)
4
3
Residual
2
1
0
-1
-2
-3
-2
-1
0
Normal Score
254
1
2
.(2JC 4"(= 7Bا1(
ل ان ا1J أنV6;
.4(J;!
) اJ
ات:('C 5 V& 4;!
7
;
ا
) ا
Time Series Plot for CPIChnge
(with forecasts and their 95% confidence limits)
CPIChnge
10
5
0
5
10
15
20
25
30
Time
:7
;
آ4;!
اAR(2) ذج1! (6C ول% 'د
MTB > arima 2 0 0 c2
Type
AR
1
AR
2
Constant
Mean
Coef
1.1872
-0.4657
1.3270
4.765
StDev
0.1625
0.1624
0.2996
1.076
T
7.31
-2.87
4.43
Number of observations: 33
Residuals:
SS = 88.6206 (backforecasts excluded)
MS = 2.9540 DF = 30
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
36
Chi-Square 19.8(DF=10)
25.4(DF=22)
* (DF= *)
(DF= *)
255
48
*
1;ح هJ!
ذج ا1!'
أي ان ا
zt = 1.33 + 1.187 zt −1 − 0.4657 zt −1 + at , at ∼ N ( 0, 2.95)
7 هt إ;(ر4!B و42 ا
!"رOCNا%رات ا
!"
) وإJ&
( )
φˆ = −0.4657, s.e. (θˆ ) = 0.1624, t = −2.87
δˆ = 1.327, s.e. (δˆ ) = 0.2996, t = 4.43
φˆ1 = 1.1872, s.e. φˆ1 = 0.1625, t = 7.31
2
1
σˆ 2 = 2.954, with d . f . = 30
;(ر9 إ
اI''
H 0 : φ2 = 0
H1 : φ2 ± 0
4859ا
t0 =
φˆ2
−0.4657
=
= −2.8676
ˆ
0.1624
s.e. φ2
( )
&. O
P-value ا
ـL1
MTB > cdf -2.8676;
SUBC> t 30.
Cumulative Distribution Function
Student's t distribution with 30 DF
x
-2.8676
P( X <= x)
0.0037
φ2 = 0 \ انNX أي0.05
& HB أ7 وه0.0037 ويC O
P-value أي ا
ـ
. AR(2) ذج1!'
\ اN 7
;
.و
:#
يL أ،ذج1! H`N وإ;ر أ4J.
ا4;!
ا4('& ( !ذج اى6C ول5
. AIC ` ا
!"ر2 وا;[م ا4('!
;(رات ا9ا
256
-ل ?zء 7ذج :4
1ف %ول '.ء !1ذج !;4
)z(t
-9.1
-2.5
-103.2
-76.7
-52.8
-33.1
-391.4
-339.9
-291.3
-246.0
-204.0
-165.9
-132.4
-836.1
-766.7
-698.2
-631.6
-566.7
-504.8
-446.4
-1307.3
-1242.9
-1177.4
-1111.3
-1044.2
-975.1
-905.3
-1736.9
-1679.0
-1620.8
-1561.5
-1500.0
-1436.5
-1371.8
-2097.5
-2055.4
-2010.0
-1960.6
-1906.9
-1851.1
-1794.5
-2363.4
-2328.5
-2290.6
-2250.5
-2210.8
-2173.3
-2136.4
-2721.4
-2651.9
-2589.5
-2533.0
-2482.0
-2437.6
-2398.7
-3353.3
-3253.4
-3156.1
-3060.8
-2968.6
-2880.4
-2797.6
-4153.0
-4028.2
-3906.6
-3788.7
-3675.0
-3564.3
-3456.9
-5068.7
-4937.2
-4805.0
-4673.3
-4542.3
-4412.0
-4281.7
-5848.8
-5760.6
-5663.8
-5558.0
-5444.4
-5323.6
-5197.8
-6387.3
-6317.8
-6244.3
-6167.9
-6090.4
-6011.8
-5931.6
-6834.5
-6773.3
-6711.8
-6649.0
-6584.6
-6518.9
-6453.5
-7259.9
-7193.3
-7131.7
-7073.1
-7015.6
-6957.1
-6896.0
-7891.4
-7784.7
-7683.2
-7586.4
-7495.3
-7411.2
-7332.9
-8791.2
-8649.8
-8512.3
-8379.1
-8249.2
-8124.0
-8004.6
-9878.4
-9713.3
-9552.0
-9394.6
-9239.6
-9086.3
-8936.4
-11071.9
-10903.2
-10734.0
-10564.0
-10392.1
-10219.0
-10047.2
-12157.5
-12011.7
-11863.9
-11713.8
-11560.2
-11402.0
-11238.9
-13190.3
-13039.8
-12889.7
-12740.8
-12593.4
-12447.5
-12302.5
-14196.3
-14051.7
-13910.1
-13769.9
-13629.3
-13486.3
-13339.6
-15335.9
-15158.3
-14986.4
-14819.8
-14657.5
-14499.7
-14345.9
-16697.1
-16492.7
-16290.9
-16091.5
-15895.8
-15705.0
-15518.4
-18155.6
-17951.5
-17745.2
-17535.7
-17324.4
-17113.5
-16904.2
-19547.0
-19356.3
-19161.7
-18965.1
-18766.2
-18564.2
-18360.1
-20827.7
-20655.7
-20478.0
-20294.5
-20107.3
-19919.5
-19733.5
-21908.3
-21762.1
-21614.0
-21463.7
-21310.9
-21154.4
-20993.7
-22474.4
-22335.1
-22195.6
-22053.2
و
Oا
) ا
&' 7ا
;
:7
257
-19.2
O r ig in a l T im e S e r ie s
-1 0 0 0 0
-2 0 0 0 0
In d e x
50
100
150
200
:7 ه48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.ا
;ا
Autocorrelation
Autocorrelation Function for z(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
10
Lag Corr
1
2
3
4
5
6
7
8
9
10
11
12
T
LBQ
0.9813.93 196.91
0.97 8.00 388.73
0.95 6.15 575.44
0.94 5.15 757.06
0.92 4.50 933.59
0.91 4.041105.06
0.89 3.681271.47
0.88 3.391432.85
0.86 3.151589.23
0.84 2.951740.66
0.83 2.781887.17
0.81 2.622028.81
Lag Corr
13
14
15
16
17
18
19
20
21
22
23
24
0.80
0.78
0.76
0.75
0.73
0.72
0.70
0.68
0.67
0.65
0.64
0.62
20
T
LBQ
2.492165.64
2.372297.73
2.252425.14
2.152547.94
2.062666.22
1.972780.05
1.892889.51
1.822994.69
1.753095.67
1.683192.54
1.623285.39
1.563374.32
30
Lag Corr
25
26
27
28
29
30
31
32
33
34
35
36
0.61
0.59
0.58
0.56
0.55
0.53
0.52
0.50
0.49
0.47
0.46
0.45
T
LBQ
1.503459.42
1.453540.78
1.403618.50
1.353692.67
1.303763.38
1.263830.74
1.213894.84
1.173955.76
1.134013.61
1.094068.47
1.054120.44
1.014169.59
40
Lag Corr
37
38
39
40
41
42
43
44
45
46
47
48
0.43
0.42
0.41
0.39
0.38
0.36
0.35
0.34
0.32
0.31
0.30
0.28
T
LBQ
0.984216.02
0.944259.81
0.914301.04
0.874339.79
0.844376.14
0.814410.16
0.774441.93
0.744471.52
0.714498.99
0.684524.43
0.654547.90
0.624569.48
50
Lag Corr
T
LBQ
49 0.27 0.594589.24
50 0.26 0.564607.24
P artial A utocorrelation Function for z(t)
Partial Autocorrelation
z(t)
0
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
10
20
30
Lag
PAC
T
Lag
PAC
T
Lag
PAC
T
1
2
3
4
5
6
7
8
9
10
11
12
0.98
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
13.93
-0.17
-0.17
-0.16
-0.16
-0.16
-0.15
-0.15
-0.15
-0.14
-0.14
-0.13
13
14
15
16
17
18
19
20
21
22
23
24
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.13
-0.12
-0.12
-0.11
-0.11
-0.11
-0.11
-0.10
-0.10
-0.09
-0.09
-0.09
25
26
27
28
29
30
31
32
33
34
35
36
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.09
-0.09
-0.09
-0.08
-0.08
-0.08
-0.08
-0.08
-0.08
-0.08
-0.09
-0.09
40
Lag PAC
37
38
39
40
41
42
43
44
45
46
47
48
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
-0.01
T
-0.09
-0.09
-0.10
-0.10
-0.11
-0.11
-0.12
-0.12
-0.12
-0.13
-0.13
-0.13
50
Lag
PAC
T
49 -0.01
50 -0.01
-0.14
-0.14
.w1;!
ا7N ةJ;& z zt 4;!
ا ان اL ^Pوا
O! وwt = zt − zt −1 وXوق اb
اA{
258
F irs t D if f e re n c e s w (t)= z (t)-z (t-1 )
w(t)
0
-1 0 0
-2 0 0
In d e x
50
100
150
200
:7 ه4Bb!
ا4;!
48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.ا
;ا
Autocorrelation
Autocorrelation Function for w(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag Corr
1
2
3
4
5
6
7
8
9
10
11
12
T
LBQ
0.9913.93 197.10
0.97 7.98 389.03
0.95 6.12 574.74
0.93 5.10 753.55
0.91 4.43 924.81
0.89 3.941087.93
0.86 3.561242.44
0.83 3.251388.10
0.81 2.981524.83
0.78 2.761652.58
0.75 2.561771.42
0.72 2.381881.49
15
Lag Corr
13
14
15
16
17
18
19
20
21
22
23
24
0.69
0.66
0.63
0.60
0.57
0.54
0.52
0.49
0.47
0.45
0.43
0.41
T
LBQ
2.211983.03
2.072076.38
1.932161.98
1.812240.33
1.702311.99
1.602377.42
1.502437.13
1.412491.57
1.332541.23
1.262586.61
1.192628.22
1.132666.48
25
Lag Corr
25
26
27
28
29
30
31
32
33
34
35
36
0.39
0.38
0.37
0.36
0.35
0.34
0.34
0.34
0.34
0.34
0.34
0.34
T
LBQ
1.082701.91
1.032734.99
1.002766.16
0.962795.84
0.942824.46
0.922852.35
0.912879.79
0.902907.03
0.892934.32
0.892961.85
0.892989.82
0.893018.34
35
Lag Corr
37
38
39
40
41
42
43
44
45
46
47
48
0.34
0.35
0.35
0.35
0.35
0.35
0.35
0.34
0.34
0.33
0.33
0.32
T
45
LBQ
0.903047.50
0.903077.32
0.903107.68
0.903138.42
0.903169.33
0.893200.21
0.883230.81
0.873260.99
0.853290.64
0.843319.59
0.823347.55
0.793374.17
Lag Corr
T
LBQ
49 0.31 0.763399.16
Partial Autocorrelation
P a rtia l A u to c o rre la tio n F un c tio n fo r w (t)
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-0 .2
-0 .4
-0 .6
-0 .8
-1 .0
5
Lag
P AC
T
1
2
3
4
5
6
7
8
9
10
11
12
0 .9 9
-0 .1 4
-0 .1 1
-0 .0 7
-0 .0 6
-0 .0 7
-0 .0 6
-0 .0 3
-0 .0 3
-0 .0 3
-0 .0 2
-0 .0 2
1 3 .9 3
-1 .9 7
-1 .5 1
-0 .9 6
-0 .9 0
-0 .9 7
-0 .7 8
-0 .4 8
-0 .3 7
-0 .3 7
-0 .3 4
-0 .2 2
15
Lag
P AC
T
1 3 -0 .0 1
1 4 -0 .0 0
1 5 0 .0 1
1 6 0 .0 2
1 7 0 .0 2
1 8 0 .0 1
1 9 0 .0 1
2 0 0 .0 1
2 1 0 .0 1
2 2 0 .0 3
2 3 0 .0 2
2 4 0 .0 2
-0 .1 6
-0 .0 2
0 .1 0
0 .2 3
0 .2 5
0 .1 0
0 .1 2
0 .1 1
0 .1 6
0 .3 5
0 .3 1
0 .2 7
25
Lag
35
45
P AC
T
Lag
P AC
T
2 5 0 .0 4
2 6 0 .0 4
2 7 0 .0 3
2 8 0 .0 4
2 9 0 .0 4
3 0 0 .0 2
3 1 0 .0 2
3 2 0 .0 1
3 3 0 .0 1
3 4 0 .0 1
3 5 0 .0 0
3 6 -0 .0 0
0 .5 2
0 .6 2
0 .4 7
0 .5 5
0 .5 6
0 .3 4
0 .2 2
0 .1 6
0 .1 8
0 .1 5
0 .0 4
-0 .0 7
37
38
39
40
41
42
43
44
45
46
47
48
-0 .0 1
-0 .0 2
-0 .0 4
-0 .0 4
-0 .0 4
-0 .0 4
-0 .0 3
-0 .0 0
0 .0 1
-0 .0 1
-0 .0 4
-0 .0 5
-0 .0 7
-0 .2 9
-0 .6 3
-0 .6 0
-0 .5 4
-0 .5 5
-0 .4 2
-0 .0 7
0 .0 8
-0 .1 8
-0 .5 6
-0 .7 6
Lag
P AC
T
4 9 -0 .0 3
-0 .4 8
.w1;!
ا7N ةJ;& z الCX wt 4;!
ا ان اL ^Pوا
O!( و4+ ا4;!
7]
ق اb
ا اA ان ه5X) yt = wt − wt −1 وXوق اb
اA{
259
F ir s t D if f e r e n c e s
y (t)= w (t)-w (t-1 )
1 0
y(t)
5
0
-5
In d e x
5 0
1 0 0
1 5 0
2 0 0
:7 ه4Bb!
ا4;!
@ اAO
48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.ا
;ا
Autocorrelation
A utocorrelation Function for y(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag C orr
1
2
3
4
5
6
7
8
9
10
11
12
T
LB Q
0.9313.08 173.62
0.80 6.83 303.36
0.70 4.89 401.81
0.62 3.89 479.62
0.56 3.26 542.93
0.50 2.77 593.77
0.43 2.32 632.35
0.36 1.90 659.84
0.29 1.50 677.66
0.22 1.11 687.71
0.15 0.75 692.41
0.08 0.41 693.82
15
Lag C orr
T
LBQ
13 0.01 0.06 693.85
14 -0.05 -0.23 694.31
15 -0.09 -0.46 696.12
16 -0.14 -0.70 700.34
17 -0.19 -0.97 708.51
18 -0.25 -1.24 722.16
19 -0.31 -1.53 743.30
20 -0.37 -1.83 774.40
21 -0.44 -2.13 818.29
22 -0.51 -2.40 876.73
23 -0.56 -2.57 948.00
24 -0.60 -2.671030.79
25
Lag Corr
T
LBQ
25 -0.64 -2.731124.21
26 -0.66 -2.721224.05
27 -0.66 -2.621324.28
28 -0.63 -2.441418.06
29 -0.59 -2.221500.63
30 -0.55 -2.011572.45
31 -0.50 -1.801632.42
32 -0.43 -1.531677.44
33 -0.36 -1.251708.46
34 -0.29 -1.001728.63
35 -0.22 -0.761740.51
36 -0.16 -0.531746.42
35
Lag C orr
T
45
LB Q
37 -0.09 -0.301748.28
38 -0.01 -0.031748.29
39 0.08 0.261749.70
40 0.15 0.501755.14
41 0.20 0.681765.20
42 0.23 0.781778.55
43 0.24 0.821793.44
44 0.25 0.861810.08
45 0.28 0.961831.09
46 0.34 1.131860.92
47 0.40 1.341903.36
48 0.45 1.491956.81
Lag Corr
T
LBQ
49 0.48 1.562017.77
Partial Autocorrelation
P artial A utocorrelation Function for y(t)
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
Lag PAC
1
2
3
4
5
6
7
8
9
10
11
12
0.93
-0.46
0.33
-0.14
0.12
-0.13
0.00
-0.06
-0.08
-0.04
-0.04
-0.09
T
13.08
-6.44
4.71
-1.97
1.72
-1.78
0.01
-0.81
-1.11
-0.55
-0.58
-1.21
15
25
Lag PAC
T
Lag
PAC
T
-0.05
0.05
-0.07
-0.09
-0.04
-0.08
-0.12
-0.11
-0.15
-0.07
-0.06
-0.15
-0.75
0.67
-0.95
-1.30
-0.55
-1.11
-1.71
-1.50
-2.12
-0.93
-0.87
-2.12
25
26
27
28
29
30
31
32
33
34
35
36
-0.03
-0.03
0.00
0.08
-0.01
-0.04
0.15
0.06
-0.00
0.04
0.03
0.04
-0.46
-0.44
0.04
1.13
-0.14
-0.56
2.17
0.88
-0.05
0.49
0.46
0.50
13
14
15
16
17
18
19
20
21
22
23
24
35
45
Lag PAC
T
0.08
0.11
0.04
0.04
-0.05
-0.05
-0.14
-0.01
0.07
0.06
-0.01
-0.09
1.16
1.56
0.51
0.52
-0.74
-0.66
-1.95
-0.10
0.92
0.79
-0.17
-1.24
37
38
39
40
41
42
43
44
45
46
47
48
Lag PAC
T
0.07
1.00
49
ةJ;& S%(+ اO ا48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6. و ا
;ا4;!
اH3 & 5
. d=2 اي انw1;!
ا7N
260
!& اولe[;
;[& & اC O ى ا48
ا4CاA
ت ا6. وا
;ا4CاA
ت ا6.& ا!ط ا
;ا
&. ذج1!'
ا اA( ه6 ف1 وzt 4+ ا4;!
ARIMA(1,2,1) ذج1! ^2
MTB > ARIMA 1 2 1 'z(t)' 'RESI2' 'FITS2';
SUBC>
NoConstant;
SUBC>
Forecast 10 c4 c5 c6;
SUBC>
GACF;
SUBC>
GPACF;
SUBC>
GHistogram;
SUBC>
GNormalplot.
ARIMA Model
ARIMA model for z(t)
Estimates at each iteration
Iteration
SSE
Parameters
0
2462.77
0.100
0.100
1
1345.58
0.250
-0.050
2
1170.63
0.203
-0.200
3
984.83
0.182
-0.350
4
782.47
0.200
-0.500
5
560.15
0.278
-0.650
6
363.93
0.428
-0.765
7
259.20
0.578
-0.814
8
202.76
0.728
-0.842
9
185.51
0.861
-0.859
10
185.36
0.873
-0.860
11
185.36
0.875
-0.860
12
185.36
0.875
-0.860
Relative change in each estimate less than
Final Estimates of Parameters
Type
Coef
StDev
261
T
0.0010
AR
1
0.8749
0.0353
24.75
MA
1
-0.8599
0.0357
-24.12
Differencing: 2 regular differences
Number
of
observations:
Original
series
200,
after
differencing 198
Residuals:
SS = 183.717
(backforecasts excluded)
MS =
DF = 196
0.937
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
12
24
Chi-Square
3.9(DF=10)
36
48
13.0(DF=22)
33.1(DF=34)
46.0(DF=46)
Forecasts from period 200
95 Percent Limits
Period
Forecast
Lower
Upper
Actual
201
-22615.2
-22617.1
-22613.3
202
-22757.2
-22764.6
-22749.9
203
-22900.4
-22917.2
-22883.5
204
-23044.5
-23075.3
-23013.7
205
-23189.5
-23238.8
-23140.1
206
-23335.2
-23407.8
-23262.6
207
-23481.5
-23582.1
-23380.9
208
-23628.4
-23761.8
-23495.0
209
-23775.8
-23946.7
-23605.0
210
-23923.7
-24136.6
-23710.7
1;ح هJ!
ذج ا1!'
ا
zt = 0.875t −1 z + at − 0.859at −1 , at ∼ N ( 0,0.937 )
7 هt إ;(ر4!B و42 ا
!"رOCNا%رات ا
!"
) وإJ&و
( )
s.e. (θˆ ) = 0.0357,
φˆ1 = 0.8749, s.e. φˆ1 = 0.0353, t = 24.75
θˆ1 = −0.8599,
1
t = −24.12
σˆ 2 = 0.937, with d . f . = 196
262
.421'"& )
"!
ان ا5
:7Bا1(
اo%b نsا
ACF of Residuals for z(t)
(with 95% confidence limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
10
15
20
25
30
35
40
45
Lag
PACF of Residuals for z(t)
(with 95% confidence limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
5
10
15
20
25
30
35
40
45
Lag
4P V2ز1C V(;C 7Bا1(
ل أن اC 7Bا1(
78
ا7CاA
اw. وا
;ا7CاA
اw.أ!ط ا
;ا
:7Bا1(
ا4"(= o%b'
،46. &;اz `ء أي.
اري3;
ر) ا
!رج ا
263
Histogram of the Residuals
(response is z(t))
Frequency
20
10
0
-3
-2
-1
0
1
2
Residual
.|K
"\ ا. ';& (و2
:7Bا1(
7"(6
;!ل ا59 اw6[& إI''
Normal Probability Plot of the Residuals
(response is z(t))
2
Residual
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
Normal Score
.(2JC 4"(= 7Bا1(
ل ان ا1J أنV6;
.:('C ;اتN 95% V& 4(J;!
) اJ
ات:('C 10 ـ7
;
ا
) ا
Forecast of 20 Future values with 95% limits
-22500
z(t)
-23000
-23500
-24000
0
1
2
3
4
5
Time
264
6
7
8
9
10
ﻤﻠﺤﻕ )(1
أ v$ ~0ا?jرات ا $و v$ا$njت اL /
265
ﺒﺴﻡ ﺍﷲ ﺍﻝﺭﺤﻤﻥ ﺍﻝﺭﺤﻴﻡ
ﻗﺴﻡ ﺍﻻﺤﺼﺎﺀ ﻭﺒﺤﻭﺙ ﺍﻝﻌﻤﻠﻴﺎﺕ
ﻜﻠﻴﺔ ﺍﻝﻌﻠﻭﻡ ﺠﺎﻤﻌﺔ ﺍﻝﻤﻠﻙ ﺴﻌﻭﺩ
ﺍﻻﺨﺘﺒﺎﺭ ﺍﻝﻨﻬﺎﺌﻰ ﻝﻠﻔﺼل ﺍﻻﻭل 1420/1419ﻫـ
ﺍﻝﻤﺎﺩﺓ 221ﺒﺤﺙ ﻁﺭﻕ ﺍﻝﺘﻨﺒﺅ ﺍﻻﺤﺼﺎﺌﻰ
ﺍﺠﺏ ﻋﻠﻰ ﺠﻤﻴﻊ ﺍﻻﺴﺌﻠﺔ ﺍﻝﺘﺎﻝﻴﺔ:
ﺍﻝﺴﺅﺍل ﺍﻻﻭل:
ﺍﻝﺒﻴﺎﻨﺎﺕ ﺍﻝﺘﺎﻝﻴﺔ ﺘﻤﺜل ﻋﺩﺩ ﺍﻝﺴﻴﺎﺭﺍﺕ ﺍﻝﻤﺒﺎﻋﺔ ﺍﺴﺒﻭﻋﻴﺎ ﻝﺩﻱ ﻤﻭﺯﻉ ﻤﺎ
ﺍﻻﺴﺒﻭﻉ
1
2
3
4
5
6
7
8
9
10
ﺍﻝﻌﺩﺩ
75
75
79
83
69
78
71
80
77
85
ﺍﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻝﻌﺩﺩ ﺍﻝﺴﻴﺎﺭﺍﺕ ﺍﻝﺘﻰ ﺴﺘﺒﺎﻉ ﻓﻰ ﺍﻻﺴﺒﻭﻋﻴﻥ ﺍﻝﺘﺎﻝﻴﻴﻥ ﻭﺍﻭﺠﺩ ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻬﺫﻩ
ﺍﻝﺘﻨﺒﺅﺍﺕ ﻜﻠﻤﺎ ﺍﻤﻜﻥ ﺫﻝﻙ ﺒﺎﺴﺘﺨﺩﺍﻡ:
ﺍ( ﻨﻤﻭﺫﺝ ﺍﻨﺤﺩﺍﺭ ﺨﻁﻰ ﻝﻠﻌﺩﺩ ﺍﻝﻤﺒﺎﻉ ﻤﻊ ﺍﻝﺯﻤﻥ ﺒﺎﻻﺴﺎﺒﻴﻊ.
ﺏ( ﺍﻝﺘﻤﻬﻴﺩ ﺒﻭﺍﺴﻁﺔ ﻤﺘﻭﺴﻁ ﻤﺘﺤﺭﻙ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻝﺜﺎﻝﺜﺔ.
ﺝ( ﺍﻝﺘﻤﻬﻴﺩ ﺍﻻﺴﻰ ﺍﻝﺒﺴﻴﻁ ﻤﺴﺘﺨﺩﻤﺎ . α = 0.3
ﺍﻝﺴﺅﺍل ﺍﻝﺜﺎﻨﻲ:
ﻝﻠﻨﻤﻭﺫﺝ (1 − φ1B − φ 2 B2 ) zt = δ + (1 − θB) atﺤﻴﺙ ) at ~ WN (0,σ 2ﻭ φ1 , φ2 ,δ ,θﻫﻰ ﻤﻌﺎﻝﻡ
ﺍﻝﻨﻤﻭﺫﺝ ﻭ Bﻫﻭ ﻋﺎﻤل ﺍﻻﺯﺍﺤﺔ ﺍﻝﺨﻠﻔﻰ ﺍﻭﺠﺩ:
ﺍ( E ( zt ) ∀t
ﺏ( ﺩﺍﻝﺔ ﺍﻝﺘﺭﺍﺒﻁ ﺍﻝﺫﺍﺘﻰ ρk ∀k ≥ 0
ﺝ( ﺩﺍﻝﺔ ﺍﻝﺘﺭﺍﺒﻁ ﺍﻝﺫﺍﺘﻰ ﺍﻝﺠﺯﺌﻰ φ kk ∀k ≥ 0
ﺩ( ﺩﺍﻝﺔ ﺍﻻﻭﺯﺍﻥ ψ j ∀j ≥ 0
ﺍﻝﺴﺅﺍل ﺍﻝﺜﺎﻝﺙ:
266
zt = 38.5 + 12ﺤﻴﺙ ) at ~ WN (0,4ﻭﺍﺫﺍ ﻜﺎﻨﺕ t = 10ﻭ
ﻝﻠﻨﻤﻭﺫﺝ . zt −1 − 0.7 zt − 2 + at − 0.4at −1
z9 = 77ﻭ z10 = 85ﻭ a10 = −1.6ﺍﻭﺠﺩ:
ﺍ( ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺍﻝﺘﻰ ﻝﻬﺎ ﺍﺩﻨﻰ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﺨﻁﺎﺀ
ﺏ( ﺘﺒﺎﻴﻥ ﺩﺍﻝﺔ ﺍﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﺤﺘﻰ ﺯﻤﻥ ﺍﻝﺘﻘﺩﻡ ℓ = 3
ﺝ( ﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ z11ﻭ z12
ﺩ( ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻠﺘﻨﺒﺅﺍﺕ ﺍﻝﺴﺎﺒﻘﺔ
ﻫـ( ﺍﺫﺍ ﻋﻠﻤﺕ ﺍﻥ z11 = 81ﻓﺠﺩﺩ ﺍﻝﺘﻨﺒﺅ ﻝﻠﻘﻴﻤﻪ z12ﻭﻝﻔﺘﺭﺓ ﺘﻨﺒﺅﻫﺎ.
267
ﺒﺴﻡ ﺍﷲ ﺍﻝﺭﺤﻤﻥ ﺍﻝﺭﺤﻴﻡ
ﻗﺴﻡ ﺍﻻﺤﺼﺎﺀ ﻭﺒﺤﻭﺙ ﺍﻝﻌﻤﻠﻴﺎﺕ
ﻜﻠﻴﺔ ﺍﻝﻌﻠﻭﻡ ﺠﺎﻤﻌﺔ ﺍﻝﻤﻠﻙ ﺴﻌﻭﺩ
ﺍﻻﺨﺘﺒﺎﺭ ﺍﻝﻨﻬﺎﺌﻰ ﻝﻠﻔﺼل ﺍﻻﻭل 1420/1419ﻫـ
ﺍﻝﻤﺎﺩﺓ 221ﺒﺤﺙ ﻁﺭﻕ ﺍﻝﺘﻨﺒﺅ ﺍﻻﺤﺼﺎﺌﻰ
ﺍﺠﺏ ﻋﻠﻰ ﺠﻤﻴﻊ ﺍﻻﺴﺌﻠﺔ ﺍﻝﺘﺎﻝﻴﺔ:
ﺍﻝﺴﺅﺍل ﺍﻻﻭل:
ﺍﻝﺒﻴﺎﻨﺎﺕ ﺍﻝﺘﺎﻝﻴﺔ ﺘﻤﺜل ﻋﺩﺩ ﺍﻝﺴﻴﺎﺭﺍﺕ ﺍﻝﻤﺒﺎﻋﺔ ﺍﺴﺒﻭﻋﻴﺎ ﻝﺩﻱ ﻤﻭﺯﻉ ﻤﺎ
Week
1 2 3 4 5 6 7 8 9 10
No. of cars 75 75 79 83 69 78 71 80 77 85
ﺍﻭﺠﺩ ﺘﻨﺒﺅﺍﺕ ﻝﻌﺩﺩ ﺍﻝﺴﻴﺎﺭﺍﺕ ﺍﻝﺘﻰ ﺴﺘﺒﺎﻉ ﻓﻰ ﺍﻻﺴﺒﻭﻋﻴﻥ ﺍﻝﺘﺎﻝﻴﻴﻥ ﻭﺍﻭﺠﺩ ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻬﺫﻩ
ﺍﻝﺘﻨﺒﺅﺍﺕ ﻜﻠﻤﺎ ﺍﻤﻜﻥ ﺫﻝﻙ ﺒﺎﺴﺘﺨﺩﺍﻡ:
ﺍ( ﻨﻤﻭﺫﺝ ﺍﻨﺤﺩﺍﺭ ﺨﻁﻰ ﻝﻠﻌﺩﺩ ﺍﻝﻤﺒﺎﻉ ﻤﻊ ﺍﻝﺯﻤﻥ ﺒﺎﻻﺴﺎﺒﻴﻊ.
ﺏ( ﺍﻝﺘﻤﻬﻴﺩ ﺒﻭﺍﺴﻁﺔ ﻤﺘﻭﺴﻁ ﻤﺘﺤﺭﻙ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻝﺜﺎﻝﺜﺔ.
ﺝ( ﺍﻝﺘﻤﻬﻴﺩ ﺍﻻﺴﻰ ﺍﻝﺒﺴﻴﻁ ﻤﺴﺘﺨﺩﻤﺎ . α = 0.3
ﺍﻝﺴﺅﺍل ﺍﻝﺜﺎﻨﻲ:
ﻝﻠﻨﻤﻭﺫﺝ (1 − φ1B − φ 2 B2 ) zt = δ + (1 − θB) atﺤﻴﺙ ) at ~ WN (0,σ 2ﻭ φ1 , φ2 ,δ ,θﻫﻰ ﻤﻌﺎﻝﻡ
ﺍﻝﻨﻤﻭﺫﺝ ﻭ Bﻫﻭ ﻋﺎﻤل ﺍﻻﺯﺍﺤﺔ ﺍﻝﺨﻠﻔﻰ ﺍﻭﺠﺩ:
ﺍ( E ( zt ) ∀t
ﺏ( ﺩﺍﻝﺔ ﺍﻝﺘﺭﺍﺒﻁ ﺍﻝﺫﺍﺘﻰ ρk ∀k ≥ 0
ﺝ( ﺩﺍﻝﺔ ﺍﻝﺘﺭﺍﺒﻁ ﺍﻝﺫﺍﺘﻰ ﺍﻝﺠﺯﺌﻰ φ kk ∀k ≥ 0
ﺩ( ﺩﺍﻝﺔ ﺍﻻﻭﺯﺍﻥ ψ j ∀j ≥ 0
ﺍﻝﺴﺅﺍل ﺍﻝﺜﺎﻝﺙ:
268
zt = 38.5 + 12ﺤﻴﺙ ) at ~ WN (0,4ﻭﺍﺫﺍ ﻜﺎﻨﺕ t = 10ﻭ
ﻝﻠﻨﻤﻭﺫﺝ . zt −1 − 0.7 zt − 2 + at − 0.4at −1
z9 = 77ﻭ z10 = 85ﻭ a10 = −1.6ﺍﻭﺠﺩ:
ﺍ( ﺩﺍﻝﺔ ﺍﻝﺘﻨﺒﺅ ﺍﻝﺘﻰ ﻝﻬﺎ ﺍﺩﻨﻰ ﻤﺘﻭﺴﻁ ﻤﺭﺒﻊ ﺍﺨﻁﺎﺀ
ﺏ( ﺘﺒﺎﻴﻥ ﺩﺍﻝﺔ ﺍﺨﻁﺎﺀ ﺍﻝﺘﻨﺒﺅ ﺤﺘﻰ ﺯﻤﻥ ﺍﻝﺘﻘﺩﻡ ℓ = 3
ﺝ( ﺘﻨﺒﺅﺍﺕ ﻝﻠﻘﻴﻡ ﺍﻝﻤﺴﺘﻘﺒﻠﻴﺔ z11ﻭ z12
ﺩ( ﻓﺘﺭﺍﺕ ﺘﻨﺒﺅ 95%ﻝﻠﺘﻨﺒﺅﺍﺕ ﺍﻝﺴﺎﺒﻘﺔ
ﻫـ( ﺍﺫﺍ ﻋﻠﻤﺕ ﺍﻥ z11 = 81ﻓﺠﺩﺩ ﺍﻝﺘﻨﺒﺅ ﻝﻠﻘﻴﻤﻪ z12ﻭﻝﻔﺘﺭﺓ ﺍﻝﺘﻨﺒﺅ
269
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
ﻗﺴﻡ ﺍﻹﺤﺼﺎﺀ ﻭﺒﺤﻭﺙ ﺍﻝﻌﻤﻠﻴﺎﺕ
ا
!دة = :ق ا
;'( :اQ%. 221 785X
ﺍﻻﺨﺘﺒﺎﺭ ﺍﻻﻭل ﻝﻸﻋﻤﺎل ﺍﻝﻔﺼﻠﻴﺔ
ا
Hbاول 1421-1420هـ
ا
& ; :
أ V!L 7 WLا 4ا
;
:4
ا
:ال اول 4;!
:ز&'{zt } 4
-1أذآ وط اJ;9ار
-2ف دا
4ا
;ا w.ا
Aا7C
ا
:ال ا
] 4;!
:7ا
;
4
9 10 11 12 13 14 15
8
7
6
5
4
3
2
1
53 43 66 48 52 42 44 56 44 58 41 54 51 56 38
w (= -1إ%ار ( t , zt ) .و& )Uأوz16 , z17 L
%;& w1;& (= -2ك & ا
ر 4Lا
]
] 4و& )Uأوz16 , z17 L
O!C (= -3ا α = 0.5 w. 7و& )Uأوz16 , z17 L
-4إذا آ z16 = 56 Sو {N z17 = 49ي & ا
6ق ا
4J.أآ] د 4Bوذ
[;F. fام &"ر
W'& {6
270
t
zt
:Q
]
ال ا:
ا
zt = 20 − 0.9 zt −1 + at
, at ~ WN ( 0, 4 ) ذج1!'
؟J;& ذج1!'
اH ه-1
µ L أو-2
k = 0,1,2,...,5 )J
φkk وρ k L أو-3
271
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
ﻗﺴﻡ ﺍﻹﺤﺼﺎﺀ ﻭﺒﺤﻭﺙ ﺍﻝﻌﻤﻠﻴﺎﺕ
ﺍﻝﻤﺎﺩﺓ :ﻁﺭﻕ ﺍﻝﺘﻨﺒﺅ ﺍﻻﺤﺼﺎﺌﻲ 221ﺒﺤﺙ
ﺍﻻﺨﺘﺒﺎﺭ ﺍﻝﻨﻬﺎﺌﻲ ﻝﻠﻔﺼل ﺍﻝﺜﺎﻨﻲ 1421-1420ﻫـ
ﺍﻝﺯﻤﻥ :ﺃﺭﺒﻊ ﺴﺎﻋﺎﺕ
أ V!L 7 WLا 4ا
;
:4
ا
:ال اول( 4& 20 ) :
أآ! Hا
bاzت ا
;
:4
(1
(
!; 4ز&' {Z } 4دا
4ا
; 2Yا
Aا) 4B"
. 6"C 7C
γ k = Cov Z t , Zودا
4
t
ρ k = γ kو
Oا
[1اص
ا
;ا w.ا
Aا4B"
. 7C
= ρ0و 1
ρkو
& 5 ) ρ k = ρت (
4;!
(2ز&'K& 4هة z1 , z2 ,⋯ , znدا
4ا
;ا w.ا
AاJC 4'"
7Cر &
⋯− z ) , k = 0,1,
n
t
∑( z
i =1
ا
"4B
) rk = ∑ ( zt − z )( zt +k − zو ا%9اف ا
!"ري
i =1
s.e ( rk ) ≃ 1آ! ان دا
4ا
;ا w.ا
Aا7C
ا
4ا
;ا w.ا
AاJC 4'"
7Cر & ا
"4B
k
rk +1 − ∑ rkj r
ا
JC 4'"
78ر &
j =1
k
ا
"4B
rj
, j = 1,...,
1− ∑r
= rk +1,k +1و ا
" 4Bا
!ة
j =1
rk +1, j = rkj − rk +1,k +1rk ,و
5 Oود 4L5
272
n
& 8 ) ±ت (
4B"
.
6"2
(
AR
H3K
ا7 W;32 يA
و ا1 − φ1
− φ2
2
( )
ذج1! (3
) Z = a , a ∼ WN (
t
t
t
( &ت7 ) Z t = δ + φ1
+ φ2
,
)
+ at
( 4& 40 ) :7]
ال ا:
ا
( !
أ & ا
رB )إ:تX2
ف اX. & 4آK
4&1
) ا
!("ت ا5 7 ه4
;
ا
(ت ا
29.3
20.0
25.8
29.0
31.0
27.5
32.7
26.8
33.6
30.6
28.9
28.5
28.2
26.1
27.8
28.2
27.6
26.7
29.9
30.0
30.8
30.5
36.6
31.4
30.8
27.1
33.2
33.7
30.2
36.6
29.0
28.1
30.3
29.4
33.6
17.5
30.3
23.7
20.1
24.2
32.4
32.4
29.4
23.5
23.6
30.6
28.1
32.3
29.9
31.6
28.0
24.1
29.2
34.3
26.4
21.7
28.8
21.5
21.3
24.7
33.6
36.5
35.7
33.7
29.3
30.6
25.1
29.1
27.2
28.5
32.0
31.9
31.7
29.0
31.9
26.6
24.3
28.9
22.7
28.3
28.2
28.6
30.7
30.6
20.8
31.8
16.6
32.5
25.2
30.3
26.1
19.0
24.3
31.5
32.0
31.7
29.1
23.2
48 & ذج1!'
ا7 "فC Minitab 7N %pacf و%acf
2&;[ام اF. (1
( &ت7 ) ;("* ا
(تC يA
واARIMA ( p, d , q ) ا
'!ذج
273
"ت ا.!
ا4J2= ا
"وم و4J26. * ذج ا
!;"ف1!'
!"
) ا4
رات أوJ& L( أو2
( &ت9 ) 4=K
ا
7 ذج1!'
اW; أو ا
"وم أآ4=K
"ت ا
ا.!
ا4J26. )
"!
رات اJ& ;[امF. (3
Q5
Z t = δ + φ1Z t −1 + ⋯ + φ p Z t − p + at − θ1at −1 − ⋯ − θ q at −q
H3K
ا
( &ت4 ) at ∼ WN ( 0, σ 2 )
أنS! وإذاZ100 وZ 99 4(J;!
) اJ
ات:('C L) أوU &ذج و1!'
:(';
ا4
داL( أو4
( &ت10) Z100 وZ 99 4(J;!
) اJ
95% :('C ;اتN LوN σ 2 = 10.83
HON z100 = 32.4 S وإذا آZ100 4(J;!
ا4!J
:(';
د اN z99 = 26.7 أنS! ( إذا5
( &ت5 ) ؟2;
" ا. امH(B H`N أ:(';
ا
وforecast 5 C2 C3 C4 4b
واوا& اarima p d q C1 &;[ام اF. (6
f8; V& رنB وJ%C gfit وgnormal وghist وgpacf وgacf وgseries
( &ت5 ) .4J.
ات اJb
ا7N
274
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
)Bا59ء و1%.ث ا
"!ت
إ;(ر ا
Hbاول 1421هـ1422/هـ
!دة = ) Q%. 221ق ا
;'( :ا( 7859
ﺍﻝﺯﻤﻥ 3ﺴﺎﻋﺎﺕ
أ V!L WLا 4ا
;
:4
ا
:ال اول:
-1ف ا
;
;F. 7ر :ا
` 4ا
(`ء ،ا
;ا w.ا
Aا ،7Cا
;! ،OاJ;9ار
-2اآ; Wا
!"دXت ا
!"!'
4Nذج ا
;
AR(2), MA(1), ARMA(1,2) :4
ج( '!1ذج ) AR(2أي & ا
!"
) ا
;
J%C 4اJ;9ار:
1) φ1 = 1.2, φ2 = −0.8
2) φ1 = −1.2, φ2 = −0.8
3) φ1 = 0.8, φ2 = −0.8
4) φ1 = −0.8, φ2 = 1.2
ا
:ال ا
]:7
ﺍﻝﺒﻴﺎﻨﺎﺕ ﺍﻝﺘﺎﻝﻴﺔ ﺘﻤﺜل ﻤﺒﻴﻌﺎﺕ ﺃﺠﻬﺯﺓ ﺍﻝﺤﺎﺴﺏ ﻓﻲ ﺃﺤﺩ ﺍﻝﺸﺭﻜﺎﺕ ﺸﻬﺭﻴﺎ ) ﺇﻗﺭﺃ ﻤﻥ ﺍﻝﻴﺴﺎﺭ ﻝﻠﻴﻤﻴﻥ
ﺴﻁﺭﺍ ﺒﺴﻁﺭ(
19
27
28
16
21
26
25
23
20
21
16
26
25
23
26
21
21
25
25
17
-1أد Hا
(ت 7Nور Minitab _&(
H! 4Bو أر! Oآ!; 4ز&'.4
(= -2ا
(ت ا
'!ذج ا
;
:4
i) Linear Trend Model
ii) Simple Moving Average Model of order 3
iii) Simple Exponential Smoothing Model with α =0.3
275
ج( =( ا
(ت !1ذج & ARMA( p, q) 48وذ
";
. fف p, qا
!'(4
و& B )Uر ا
!"
) '!1ذج ا
!;Jح.
د( أي !1ذج & ا
'!ذج ا
7N ) 4J.ا
CJbب و ج ( e2ا
!Kهات H3K.أH`N؟
هـ( [;F.ام ا
'!1ذج ا H`Nو
:('Cات 2OKا
;
;b.ات 95% :('C
276
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
ﺍﻟﻤﺎﺩﺓ :ﻃﺮﻕ ﺍﻟﺘﻨﺒﺆ ﺍﻻﺣﺼﺎﺋﻲ 221ﺑﺤﺚ
ﺍﻻﺧﺘﺒﺎﺭ ﺍﻻﻭﻝ ﻟﻸﻋﻤﺎﻝ ﺍﻟﻔﺼﻠﻴﺔ
ﺍﻟﻔﺼﻞ ﺍﻟﺜﺎﻧﻲ 1422/1421ﻫـ
ا
& ; :
أ V!L 7 WLا 4ا
;
:4
ا
:ال اول1 :هت ا
!; 4ا
&' 4ا
;
) :4إBا & ا
ر ! 6ا (6.
38
55
71
64
23
64
47
35
40
71
57
48
59
41
37
51
55
74
80
44
58
57
45
45
50
60
50
57
56
50
71
74
50
59
25
54
55
36
48
54
45
58
44
43
62
64
50
57
45
41
49
55
53
59
38
52
68
50
45
38
54
35
34
277
60
39
59
40
54
57
23
:7
;
اWL وأMinitab _&. 7N Worksheet H! 4b%+ 7N هاتK!
اHأد
(ة5 وا4&) ة؟ !ذا؟J;& 4;!
(و اC H هTsplot 46ا1. هاتK!
اo%N)أ( إ
4&) ؟ و
!ذا؟4;!
@ اAO
Trend Analysis اف9 اH%C 4J2= ;[مH)ب( ه
(ة5وا
4&) ؟ و
!ذا؟4;!
@ اAO
Decomposition Method f3b;
ا4J2= ;[مH)ج( ه
(ة5وا
@AO
Moving Average Smoothing ك%;!
اw1;!
. O!;
ا4J2= ;[مH)د( ه
(ة5 وا4&) ؟ و
!ذا؟4;!
ا
@AO
Simple Exponential Smoothing w(
ا7X اO!;
ا4J2= ;[مH)هـ( ه
(ة5 وا4&) ؟ و
!ذا؟4;!
ا
@AO
Double Exponential Smoothing 78']
ا7X اO!;
ا4J2= ;[مH)و( ه
(ة5 وا4&) ؟ و
!ذا؟4;!
ا
(ة5 وا4&) ؟ و
!ذا؟4;!
@ اAO
Winters’ Method و;ز4J2= ;[مH)ز( ه
4'&
ا4;!
اH%;
W ا7 هwJN ة5 وا4J2= L1C .
ال ا:
& ا:7]
ال ا:
ا
. 95% :('C ;اتN V& 4(J;& )B 4![
ات:('C L أو4('!
ا4J26
;[ام اF. !هةK!
ا
( &ت8)
278
إ 4.Lا
:ال اول:
)أ (
80
70
60
Sales
50
40
30
20
70
60
50
40
30
20
10
Index
ا
!;(C 4و &;Jة Q5ا15 Y;C Oل &;1ى S.U
)ب( 3!2إ;[ام = H%C 4J2ا9ف و
4('& z O'3ه' "م و1Lد إ*;C "& @C
* ا
!;4
)ج( 3!2إ;[ام = 4J2ا
; f3bو
W'C O'3أآ] ا
!;ت ا
;& ON 7آ(ت إاف و
&4!1
)د( W'2ا
;! w1;!
. Oا
!;%ك هAا ا
'1ع & ا
!;ت ا
&' 4أآ] & & @z
ا
6ق * آ! ى ه @Aا
!;;%C 4ج إ
w. O!Cإذ ~ O.أي إاف أو &4!1
)هـ( ا
;! Oا 7Xا
( H]!
Vb'2 wه @Aا
!;ت أ `2و 4+إذا آ1%C Sي إاN
w.
)و( ا
;! Oا 7Xا
]' b2 78أآ] 4
5 7Nا
!;ت ا
;1%C 7ي إا 76 z Nوا
Aي
(2Xو & Cف ا
!; 4ا
!Kهة
)ز( = 4J2و;ز ;!
Vb'Cت ا
! 4!1ا
;1%C 7ي &آ( 4إاف 76 z
إ 4.Lا
:ال ا
]:7
(2و أن = 4J2ا
;! w1;!
. Oا
!;%ك ه 7اآ] &'( H%;
4ا
!; 4ا
!Kهة'
.ب
ة &;61ت &;%آ O!;
4ا
!; 4وإ2د :('Cات .O
)'
(1ب &;%;& w1ك & ا
ر 4Lا
]!&) 4آ(:
;MTB > %MA 'Sales' 2
279
SUBC>
Center;
SUBC>
Forecasts 5.
Executing from file: G:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
Sales
Length
70.0000
NMissing
0
Moving Average
Length: 2
Accuracy Measures
MAPE: 21.843
MAD:
MSD:
10.045
154.429
Row
Period
Forecast
Lower
Upper
1
71
47
22.6432
71.3568
2
3
72
73
47
47
22.6432
22.6432
71.3568
71.3568
4
5
74
75
47
47
22.6432
22.6432
71.3568
71.3568
280
Moving Average
Actual
80
Predicted
Forecast
70
Actual
Predicted
Forecast
Sales
60
50
Moving Average
40
Length:
2
30
MAPE:
21.843
20
0
10
20
30
40
50
60
MAD:
10.045
MSD:
154.429
70
Time
:4]
]
ا4Lك & ا
ر%;& w1;& ( 'ب2)
MTB > %MA 'Sales' 3;
SUBC>
Forecasts 5.
Executing from file: G:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
Sales
Length
70.0000
NMissing
0
Moving Average
Length: 3
Accuracy Measures
MAPE: 23.909
MAD:
MSD:
281
11.184
183.098
Row
Period
Forecast
Lower
Upper
1
71
44.6667
18.1452
71.1881
2
3
72
73
44.6667
44.6667
18.1452
18.1452
71.1881
71.1881
4
5
74
75
44.6667
44.6667
18.1452
18.1452
71.1881
71.1881
Moving Average
Actual
80
Sales
Predicted
70
Forecast
60
Actual
Predicted
Forecast
50
Moving Average
40
Length:
3
MAPE:
23.909
30
20
0
10
20
30
40
50
60
MAD:
11.184
MSD:
183.098
70
Time
:( )&!آ4". ا
ا4Lك & ا
ر%;& w1;& ( 'ب3)
MTB > %MA 'Sales' 4;
SUBC> Center;
SUBC>
Forecasts 5.
Executing from file: G:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
282
Sales
Length
70.0000
NMissing
0
Moving Average
Length: 4
Accuracy Measures
MAPE:
MAD:
MSD:
21.161
9.733
146.200
Row
Period
Forecast
Lower
Upper
1
2
71
72
48
48
24.3010
24.3010
71.6990
71.6990
3
4
73
74
48
48
24.3010
24.3010
71.6990
71.6990
5
75
48
24.3010
71.6990
Moving Average
82
Actual
Sales
Predicted
72
Forecast
62
Actual
Predicted
Forecast
52
Moving Average
42
32
22
0
10
20
30
40
50
60
Length:
4
MAPE:
21.161
MAD:
9.733
MSD:
146.200
70
Time
:4&[
ا4Lك & ا
ر%;& w1;& ( أا ب4)
MTB > %MA 'Sales' 5;
SUBC>
Forecasts 5.
283
Executing from file: G:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
Sales
Length
70.0000
NMissing
0
Moving Average
Length: 5
Accuracy Measures
MAPE: 21.724
MAD:
MSD:
10.040
156.195
Row
Period
Forecast
Lower
Upper
1
2
71
72
46.6
46.6
22.1043
22.1043
71.0957
71.0957
3
4
73
74
46.6
46.6
22.1043
22.1043
71.0957
71.0957
5
75
46.6
22.1043
71.0957
284
Moving Average
Actual
80
Predicted
Forecast
70
Actual
Predicted
Forecast
Sales
60
50
Moving Average
40
Length:
5
30
MAPE:
21.724
20
0
10
20
30
40
50
60
MAD:
10.040
MSD:
156.195
70
Time
ـ4!B HB أ76"2 ا
!!آ4". ا
ا4Lط & ا
ر%;!
اw1;!
أن ا4J.
_ ا8;'
& ا
146.2 ويC (MSD (Mean Square Deviation
7 ه95% :('C ;اتN V& 4(J;!
) اB 4![
ات:(';
ا
Row
Period
Forecast
Lower
Upper
1
71
48
24.3010
71.6990
2
3
72
73
48
48
24.3010
24.3010
71.6990
71.6990
4
5
74
75
48
48
24.3010
24.3010
71.6990
71.6990
)5
! ا5
) ا ا.
ث ا
"!ت1%.ء و59) اB
م1"
ا4آ
د1" f!
ا4"&L
285
ا(;9ر ا
]! 7ل ا
Hbا
] 1422/1421 7هـ
!دة = ) Q%. 221ق ا
;'( :ا( 7859
ا
& 3 :ت
أ V!L WLا 4ا
;
:4
ا
:ال اول:
'!1ذج
)− 65) = (1 − 0.4 B ) at , at ~ WN ( 0,1
(1 − 1.2 B + 0.6B ) ( z
2
t
)أ( & J%Cان ا
'!1ذج &; JوJ
H.Bب.
)ب( أو Lآ ρ k & Hو . k = 1, 2,...,5 )J
φkk
)ج( أو Lدا
4اوزان . j = 1, 2,...,5 )J
ψ j
ا
:ال ا
]:7
'!1ذج ا
.إذا ! Sأن . z76 = 60.4, z77 = 58.9, z78 = 64.7, z79 = 70.4, z80 = 62.6
)أ( أو:('C Lات )Jا
!;. z81 , z82 , z83 , z84 4(J
)ب( أو;N Lات :(';
95% :('Cات 7Nا
Jbة ا
.4J.
ا
:ال ا
]
:Q
'!1ذج
)(1 − 0.43B )(1 − B ) zt = at , at ~ WN ( 0,1
)أ( ه Hا
'!1ذج &;J؟ و
!ذا؟
)ب( إذا آ W5{N z49 = 33.4, z50 = 33.9 Sا
;'(:ات ) . ℓ = 1, 2,...,5 )J
z50 ( ℓ
)ج( أو;N Lات :(';
95% :('Cات 7Nا
Jbة ا
.4J.
286
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
)Bا59ء و1%.ث ا
"!ت
آ 4ا
"1م
4"&Lا
!1" fد
ا(;9ر ا
' Hb
78Oا
] 1422/1421 7هـ
!دة = ) Q%. 221ق ا
;'( :ا(7859
ﺍﻝﺯﻤﻥ 3ﺴﺎﻋﺎﺕ
أ V!L WLا 4ا
;
:4
ا
:ال اول:
=("(& 4;& 'Jت !1ذج ) ARIMA (1,1,0ا
;
:7
)at ∼ WN ( 0, 25
(1 − 0.7 B )(1 − B ) zt = at ,
ا
!Kه Cاة هz143 = 770, z144 = 800 7
(أ( ه Hا
'!1ذج &; Jأم Xو
!ذا؟
)ب( أو Lدا
4اوزان . j = 1, 2,...,5 )J
ψ j
)ج( ا W5ا
;'(:ات ]ث ) (3ا
)Jا
!; 4(Jا
;
.4
)د( او;N Lات :(';
95% :('Cات ا
.4J.
ا
:ال ا
]:7
!; 4ا
&' 4ا
;
) : 4إBأ & ا
ر ! 6ا (6.
4.14
5.43
4.77
3.80
3.45
4.60
3.99
5.51
5.74
3.49
5.42
3.88
5.07
4.30
3.78
3.91
6.16
4.05
2.54
3.96
1.22
3.57
2.65
3.45
3.28
3.98
4.05
4.08
4.61
2.89
287
2.52
5.77
3.20
5.13
7.20
7.26
6.42
1.58
4.99
5.05
7.07
6.87
7.22
8.22
4.00
4.31
6.23
8.02
7.56
6.69
7.67
5.14
3.84
4.40
3.08
6.46
5.43
6.11
4.80
4.79
2.75
5.65
6.12
5.52
4.99
6.12
4.89
6.06
4.78
6.36
5.67
5.75
6.08
5.70
5.80
5.61
6.57
5.63
6.08
5.71
4.72
5.16
6.09
7.49
6.64
9.01
7.49
7.27
6.64
5.62
7.53
7.59
7.23
7.53
8.50
6.43
8.27
8.75
7.50
7.86
:7
;
اL أوMINITAB ;[امF.
Time Series Plot . ;[امF. 4'&
ا4;!
)أ( أر) ا
.)
اo%N ". 4;!
اO!;
4J2= ;حB)ب( أ
S! إذا آN 4Pb
وأ;( اResiduals 7Bا1(
اo%b. )B 45;J& 4J2= H3
()ج
Kolmogorov-Smirnov
;[ام إ;(رF. f
وذ7"(= V2ز1C 4ز1& 7Bا1(
ا
Test .
Mean Absolute Percentage Error
Mean
وMean
Absolute
ء6
4B
ا2"& رنB ()د
Deviation
(MAD) ( وMAPE)
] اآ4J26
) أ;[م اU & و45;J& 4J2= H3
Squared Deviation (MSD)
95% . :('C ;اتN د2 إV& 4(J;!
) اJ
ا4![
ات:('C 1;
4Bد
:Q
]
ال ا:
ا
(6. ا6 !
أ & ا
رB )إ:4
;
ا4'&
ا4;!
1.20
1.76
1.50
2.00
1.54
2.09
288
2.70
1.95
2.40
1.89
3.44
1.80
2.83
1.25
1.58
2.25
2.50
1.08
2.07
2.05
1.27
2.32
1.37
1.46
1.18
1.23
1.79
1.54
1.42
1.57
1.39
1.40
1.42
1.51
2.08
2.91
1.85
1.77
1.82
1.61
1.68
1.25
1.78
1.15
1.84
:7
;
اL أوMINITAB ;[امF.
Time Series Plot . ;[امF. 4'&
ا4;!
)أ( أر) ا
ARIMA ( p, d , q ) 48 & W'& ذج1! ;حB أO'& وSPACF وSACF & H آL)ب( أو
. q وd وp & H;" آ. f
وذ
2ة ا
!"
) &وA هH أو آ5! إذا آن أN تPb
;ح وأ;( اJ!
ذج ا1!'
)
"!
ر اB ()ج
.b
V2ز1C 4ز1& 7Bا1(
اS! إذا آN 4Pb
وأ;( اResiduals 7Bا1(
اo%b. )B ()د
Kolmogorov-Smirnov Test . ;[ام إ;(رF. f
وذ7"(=
95% . :('C ;اتN د2 إV& 4
;
ا4(J;!
) اJ
ا4![
ات:('C W5)هـ( أ
289
)5
! ا5
) ا ا.
هـ1422/1421 7]
اHb
78O'
;(ر ا4!;%& ت.Lإ
Q%. 221
!دة
:ال اول:
ا4.Lإ
(1 − B ) 2b;
اH& ى1%2 * J;& z ذج1!'
)أ( ا
()ب
(1 − 0.7 B )(1 − B ) zt = at
(1 − 1.7 B + 0.7 B ) z
2
t
= at
1
at
1 − 1.7 B + 0.7 B 2
= ψ ( B ) at
∴ zt =
1
1 − 1.7 B + 0.7 B 2
∴ (1 + ψ 1B + ψ 2 B 2 + ψ 3 B 3 + ⋯)(1 − 1.7 B + 0.7 B 2 ) ≡ 0
∴ψ ( B ) =
B : ψ 1 − 1.7 = 0 ⇒ ψ 1 = 1.7
B 2 : ψ 2 − 1.7ψ 1 + 0.7 = 0 ⇒ ψ 2 = 1.7ψ 1 − 0.7 = 2.19
B 3 : ψ 3 − 1.7ψ 2 + 0.7ψ 1 = 0 ⇒ ψ 3 = 1.7ψ 2 − 0.7ψ 1 = 2.53
⋮
B j : ψ j − 1.7ψ j −1 + 0.7ψ j −2 = 0 ⇒ ψ j = 1.7ψ j −1 + 0.7ψ j −2 , j = 2, 3,...
∴ψ 4 = 1.7ψ 3 − 0.7ψ 2 = 2.768
ψ 5 = 1.7ψ 4 − 0.7ψ 3 = 2.9346
ψ 1 = 1.7,ψ 2 = 2.19,ψ 3 = 2.53,ψ 4 = 2.77,ψ 5 = 2.93 :7 ه4.16!
إذا اوزان ا
:7
;
ات آ:(';
اW% ()ج
290
∵ (1 − 1.7 B + 0.7 B 2 ) zt = at
∴ zt = 1.7 zt −1 − 0.7 zt −2 + at
∴ zt ( ℓ ) = E zt + ℓ zt , zt −1 ,⋯ , ℓ ≥ 0
= E 1.7 zt + ℓ−1 − 0.7 zt + ℓ−2 + at + ℓ zt , zt −1 ,⋯ , ℓ ≥ 0
= 1.7 E zt + ℓ−1 zt , zt −1 ,⋯ − 0.7 E zt +ℓ −2 zt , zt −1 ,⋯ + E at +ℓ zt , zt −1 ,⋯ , ℓ ≥ 0
∴ ℓ = 1: zt (1) = 1.7 E zt zt , zt −1 ,⋯ − 0.7 E zt −1 zt , zt −1 ,⋯ + E at +1 zt , zt −1 ,⋯
= 1.7 zt − 0.7 zt −1
ℓ = 2 : zt ( 2 ) = 1.7 E zt +1 zt , zt −1 ,⋯ − 0.7 E zt zt , zt −1 ,⋯ + E at + 2 zt , zt −1 ,⋯
= 1.7 zt (1) − 0.7 zt
ℓ = 3 : zt ( 3) = 1.7 E zt + 2 zt , zt −1 ,⋯ − 0.7 E zt +1 zt , zt −1 ,⋯ + E at +3 zt , zt −1 ,⋯
= 1.7 zt ( 2 ) − 0.7 zt (1)
∴ ℓ ≥ 3 : zt ( ℓ ) = 1.7 zt ( ℓ − 1) − 0.7 zt ( ℓ − 2 )
∵ t = 144
∴ z144 (1) = 1.7 z144 − 0.7 z143 = 1.7 ( 800 ) − 0.7 ( 770 ) = 821
z144 ( 2 ) = 1.7 z144 (1) − 0.7 z144 = 1.7 ( 821) − 0.7 ( 800 ) = 835.7
z144 ( 3) = 1.7 z144 ( 2 ) − 0.7 z144 (1) = 1.7 ( 835.7 ) − 0.7 ( 821) = 845.99
:7 ه4(J;!
) اB ات ]ث:(';
إذا ا
z144 (1) = 821, z144 ( 2 ) = 835.7, z144 ( 3) = 845.99
4B"
. 6"C :('C (1 − α )100% ;اتN ()د
z (ℓ) ± u
α 2 V
et ( ℓ ) , ℓ ≥ 0
t
أي أنuα 2 = 1.96 نFN α = 0.05 انQ5 و7J
ا7"(6
اV2ز1;
95 !
ا1 هuα 2 Q5
zt + ℓ ∈ zt ( ℓ ) ± 1.96 V et ( ℓ ) w.p. 0.95, ℓ ≥ 0
V et ( ℓ ) = σ 2 (1 + ψ 12 + ψ 22 + ⋯ + ψ ℓ2−1 ) , ℓ ≥ 0 :4B"
& ا:(';
ء ا6'ت أ2(C W% Xأو
V et (1) = σ 2 = 25
(
)
V et ( 2 ) = σ 2 (1 + ψ 12 ) = 25 1 + (1.7 ) = 97.25
(
2
)
V et ( 3) = σ 2 (1 + ψ 12 + ψ 22 ) = 25 1 + (1.7 ) + ( 2.19 ) = 217.1525
2
2
:4.16!
ا:(';
;ات اN O'&و
291
z145 ∈ 821 ± 1.96 25 = [811.2,830.8] , w. p. 0.95
z146 ∈ 835.7 ± 1.96 97.25 = [816.37,855.03] , w. p. 0.95
z145 ∈ 845.99 ± 1.96 217.1525 = [817.11,874.87] , w. p. 0.95
:7]
ال ا:
4.Lإ
( )أ
7
6
Sales
5
4
3
2
1
Index
10
20
30
40
50
ق6
ا5ه أO!C 7N ;[مf
A
و4!1& z 4;!
أن ا.
)ب( & ا
) ا
:4
;
ا
Moving Average Smoothing ك%;!
اw1;!
ا-1
Single Exponential Smoothing w(
ا7 اO!;
ا-2
Double Exponential Smoothing 78']
ا7 اO!;
ا-3
ك%;!
اw1;!
اXأو
292
Smoothing Sales Series by Moving Avg. of Order 3
Actual
8
Sales
Predicted
7
Forecast
6
Actual
Predicted
Forecast
5
4
Moving Average
3
Length:
2
MAPE: 26.5366
1
0
10
20
30
40
3
MAD:
0.9077
MSD:
1.2322
50
Time
MTB > %MA 'Sales' 3;
SUBC>
Forecasts 5;
SUBC>
Title "Smoothing Sales Series by Moving
Avg. of Order 3";
SUBC>
Residuals 'RESI1'.
Executing from file: G:\MTBWIN\MACROS\MA.MAC
Macro is running ... please wait
Moving average
Data
Length
Sales
50.0000
NMissing
0
Moving Average
Length: 3
Accuracy Measures
MAPE: 26.5366
MAD:
0.9077
293
MSD:
1.2322
Row
Period
Forecast
Lower
Upper
1
51
5.9
3.72428
8.07572
2
3
52
53
5.9
5.9
3.72428
3.72428
8.07572
8.07572
4
5
54
55
5.9
5.9
3.72428
3.72428
8.07572
8.07572
w(
ا7 اO!;
اU
Smoothing Sales Series by Single Exponential Smoothing
8
Actual
Predicted
7
Forecast
Actual
Predicted
Forecast
Sales
6
5
4
Smoothing Constant
3
Alpha:
2
MAPE: 25.5159
1
0
10
20
30
40
0.258
MAD:
0.9002
MSD:
1.1638
50
Time
MTB > %SES 'Sales';
SUBC>
Forecasts 5;
SUBC>
Title "Smoothing Sales Series by Single
Exponential Smoothing";
SUBC>
Residuals 'RESI2'.
Executing from file: G:\MTBWIN\MACROS\SES.MAC
Macro is running ... please wait
294
Single Exponential Smoothing
Data
Length
Sales
50.0000
NMissing
0
Smoothing Constant
Alpha: 0.257773
Accuracy Measures
MAPE: 25.5159
MAD:
0.9002
MSD:
1.1638
Row
Period
Forecast
Lower
Upper
1
2
51
52
5.67586
5.67586
3.47035
3.47035
7.88137
7.88137
3
4
53
54
5.67586
5.67586
3.47035
3.47035
7.88137
7.88137
5
55
5.67586
3.47035
7.88137
78']
ا7 اO!;
] اU
295
Smoothing Sales Series by Double Exponential Smoothing
Actual
11
Predicted
Sales
Forecast
Actual
Predicted
Forecast
6
Smoothing Constants
Alpha (level): 0.681
Gamma (trend):0.019
MAPE:
MAD:
MSD:
1
0
10
20
30
40
27.1322
0.9880
1.5458
50
Time
MTB > %DES 'Sales';
SUBC>
Forecasts 5;
SUBC>
Title "Smoothing Sales Series by Double
Exponential Smoothing";
SUBC>
Residuals 'RESI3'.
Executing from file: G:\MTBWIN\MACROS\DES.MAC
Macro is running ... please wait
Double Exponential Smoothing
Data
Length
Sales
50.0000
NMissing
0
Smoothing Constants
Alpha (level): 0.680728
Gamma (trend): 0.019421
Accuracy Measures
MAPE: 27.1322
MAD:
296
0.9880
MSD:
1.5458
Row
Period
Forecast
Lower
Upper
1
51
6.04349
3.62298
8.4640
2
3
52
53
6.06254
6.08160
3.04085
2.40519
9.0842
9.7580
4
5
54
55
6.10065
6.11971
1.74005
1.05736
10.4613
11.1821
:7Bا1(
اo%N ()ج
ك%;!
اw1;!
Autocorrelation
Autocorrelation Function for RESI1
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
0.01
-0.33
-0.11
-0.01
0.10
0.20
-0.07
0.05
-2.26
-0.69
-0.06
0.64
1.22
-0.39
0.00
5.55
6.19
6.19
6.79
9.03
9.28
297
6
Lag
7
Corr
8
T
LBQ
8 0.08 0.47
9 -0.11 -0.66
10 0.00 0.02
11 0.04 0.25
9.66
10.43
10.43
10.55
9
10
11
Partial Autocorrelation
Partial Autocorrelation Function for RESI1
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
Lag PAC
T
1 0.01
2 -0.33
3 -0.12
4 -0.14
5 0.03
6 0.17
7 -0.02
0.05
-2.26
-0.81
-0.93
0.19
1.14
-0.16
6
7
Lag PAC
8
9
10
11
T
8 0.25 1.74
9 -0.11 -0.76
10 0.15 1.01
11 -0.06 -0.39
Frequency
10
5
0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
RESI1
Normal Probability Plot for RESI1
99
Mean:
3.15E-02
StDev:
1.12161
95
90
80
Percent
70
60
50
40
30
20
10
5
1
-3
-2
-1
0
1
2
3
Data
MTB > %Qqplot 'RESI1';
SUBC>
298
Table;
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: G:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
Data
Mean:
StDev:
: RESI1
3.15E-02
1.12161
Percentile Estimates
95% CI
95% CI
Approximate
Upper Limit
P
Percentile
Approximate
Lower Limit
0.01
-2.57777
-3.19506
-1.96047
0.02
0.03
-2.27202
-2.07803
-2.83741
-2.61158
-1.70663
-1.54447
0.04
0.05
-1.93210
-1.81339
-2.44238
-2.30524
-1.42181
-1.32155
0.06
0.07
-1.71236
-1.62377
-2.18891
-2.08723
-1.23581
-1.16032
0.08
0.09
-1.54445
-1.47231
-1.99647
-1.91417
-1.09244
-1.03046
0.10
0.20
-1.40591
-0.91248
-1.83864
-1.28563
-0.97318
-0.53934
299
0.30
-0.55668
-0.89868
-0.21469
0.40
0.50
-0.25267
0.03149
-0.57843
-0.28917
0.07309
0.35215
0.60
0.70
0.31565
0.61966
-0.01012
0.27767
0.64141
0.96165
0.80
0.90
0.97546
1.46889
0.60232
1.03616
1.34860
1.90162
0.91
0.92
1.53529
1.60743
1.09344
1.15542
1.97715
2.05945
0.93
0.94
1.68675
1.77534
1.22330
1.29879
2.15021
2.25189
0.95
0.96
1.87637
1.99508
1.38453
1.48479
2.36822
2.50536
0.97
0.98
2.14101
2.33499
1.60745
1.76961
2.67456
2.90038
0.99
2.64074
2.02345
3.25804
Kolmogorov-Smirnov Test for Residuals of MA
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-2
-1
0
1
2
RESI1
Average: 0.0314894
StDev: 1.12161
N: 47
Kolmogorov-Smirnov Normality Test
D+: 0.069 D-: 0.067 D : 0.069
Approximate P-Value > 0.15
w(
ا7 اO!;
300
Autocorrelation
Autocorrelation Function for RESI2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
7
Lag
T
LBQ
1 -0.02 -0.16
2 -0.21 -1.51
3 0.16 1.07
4 0.01 0.07
5 0.10 0.68
6 0.21 1.40
7 -0.08 -0.53
Corr
0.03
2.48
3.87
3.87
4.47
7.18
7.61
Lag
Corr
12
T
LBQ
8 0.17 1.05
9 -0.13 -0.79
10 -0.08 -0.47
11 0.13 0.81
12 -0.06 -0.38
9.33
10.37
10.76
11.95
12.24
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
7
Lag PAC
T
1 -0.02
2 -0.21
3 0.15
4 -0.03
5 0.18
6 0.20
7 -0.02
-0.16
-1.51
1.09
-0.24
1.28
1.42
-0.14
Lag PAC
12
T
8 0.25 1.76
9 -0.27 -1.89
10 0.04 0.26
11 -0.11 -0.75
12 -0.09 -0.62
9
8
7
Frequency
Partial Autocorrelation
Partial Autocorrelation Function for RESI2
6
5
4
3
2
1
0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
RESI2
301
Normal Probability Plot for RESI2
99
Mean:
9.40E-02
StDev:
1.08561
95
90
80
Percent
70
60
50
40
30
20
10
5
1
-3
-2
-1
0
1
2
3
Data
MTB > %Qqplot 'RESI2';
SUBC>
Table;
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: G:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
Data
Mean:
StDev:
: RESI2
9.40E-02
1.08561
Percentile Estimates
95% CI
302
95% CI
Approximate
Approximate
P
Percentile
Lower Limit
Upper Limit
0.01
0.02
-2.43146
-2.13552
-3.01074
-2.66609
-1.85218
-1.60495
0.03
0.04
-1.94776
-1.80651
-2.44846
-2.28537
-1.44706
-1.32766
0.05
0.06
-1.69162
-1.59383
-2.15318
-2.04103
-1.23006
-1.14663
0.07
0.08
-1.50809
-1.43131
-1.94300
-1.85549
-1.07317
-1.00713
0.09
0.10
-1.36149
-1.29722
-1.77614
-1.70330
-0.94684
-0.89113
0.20
0.30
-0.81962
-0.47525
-1.16979
-0.79618
-0.46946
-0.15431
0.40
0.50
-0.18099
0.09405
-0.48669
-0.20686
0.12471
0.39496
0.60
0.70
0.36908
0.66334
0.06338
0.34241
0.67479
0.98427
0.80
0.90
1.00772
1.48531
0.65756
1.07923
1.35789
1.89140
0.91
0.92
1.54959
1.61941
1.13494
1.19523
1.96423
2.04359
0.93
0.94
1.69618
1.78193
1.26127
1.33473
2.13110
2.22913
0.95
0.96
1.87972
1.99461
1.41816
1.51575
2.34128
2.47347
0.97
0.98
2.13586
2.32362
1.63516
1.79305
2.63655
2.85419
0.99
2.61955
2.04028
3.19883
303
Kolmogorov-Smirnov Test for Residuals of SES
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-2
-1
0
1
2
RESI2
Average: 0.0940483
StDev: 1.08561
N: 50
Kolmogorov-Smirnov Normality Test
D+: 0.077 D-: 0.068 D : 0.077
Approximate P-Value > 0.15
78']
ا7 اO!;
ا
Autocorrelation
Autocorrelation Function for RESI3
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
7
Lag
Corr
T
LBQ
1
2
3
4
5
6
7
-0.22
-0.29
0.22
-0.07
0.00
0.17
-0.16
-1.53
-1.98
1.39
-0.43
0.03
1.03
-0.95
2.50
7.12
9.81
10.10
10.10
11.82
13.38
Lag
Corr
12
T
LBQ
8 0.16 0.96
9 -0.16 -0.89
10 0.01 0.04
11 0.19 1.06
12 -0.15 -0.82
15.06
16.62
16.62
18.99
20.53
Partial Autocorrelation
Partial Autocorrelation Function for RESI3
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
7
Lag PAC
T
1 -0.22
2 -0.36
3 0.07
4 -0.11
5 0.07
6 0.14
7 -0.05
-1.53
-2.52
0.49
-0.81
0.47
1.02
-0.34
304
Lag PAC
12
T
8 0.24 1.69
9 -0.23 -1.61
10 0.13 0.92
11 0.01 0.04
12 -0.03 -0.25
9
8
Frequency
7
6
5
4
3
2
1
0
-3
-2
-1
0
1
2
3
RESI3
Normal Probability Plot for RESI3
99
Mean:
-3.0E-03
StDev:
1.25593
95
90
80
Percent
70
60
50
40
30
20
10
5
1
-3
-2
-1
0
1
2
3
Data
MTB > %Qqplot 'RESI3';
SUBC>
Table;
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: G:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
305
Data
: RESI3
Mean:
-3.0E-03
StDev:
1.25593
Percentile Estimates
95% CI
Approximate
95% CI
Approximate
P
Percentile
Lower Limit
Upper Limit
0.01
0.02
-2.92473
-2.58237
-3.59489
-3.19618
-2.25457
-1.96855
0.03
0.04
-2.36515
-2.20174
-2.94440
-2.75573
-1.78589
-1.64775
0.05
0.06
-2.06882
-1.95569
-2.60279
-2.47305
-1.53485
-1.43832
0.07
0.08
-1.85649
-1.76767
-2.35964
-2.25840
-1.35334
-1.27694
0.09
0.10
-1.68689
-1.61254
-2.16659
-2.08233
-1.20719
-1.14274
0.20
0.30
-1.06001
-0.66161
-1.46512
-1.03289
-0.65491
-0.29032
0.40
0.50
-0.32118
-0.00300
-0.67484
-0.35112
0.03248
0.34512
0.60
0.70
0.31519
0.65562
-0.03847
0.28433
0.66885
1.02690
0.80
0.90
1.05402
1.60655
0.64892
1.13675
1.45913
2.07634
0.91
0.92
1.68090
1.76168
1.20120
1.27095
2.16060
2.25241
306
0.93
1.85050
1.34735
2.35365
0.94
0.95
1.94969
2.06283
1.43233
1.52886
2.46706
2.59680
0.96
0.97
2.19575
2.35915
1.64176
1.77990
2.74973
2.93840
0.98
0.99
2.57637
2.91874
1.96256
2.24858
3.19019
3.58890
Kolmogorov-Smirnov Test for Residuals of DES
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-3
-2
-1
0
1
2
RESI3
Average: -0.0029958
StDev: 1.25593
N: 50
Kolmogorov-Smirnov Normality Test
D+: 0.077 D-: 0.070 D : 0.077
Approximate P-Value > 0.15
تPb
اJ%C 7Bا1(
ان ا4J.
ت اX%
اH آ7N
46. &;اz -1
ة5 و2(Cي وb+ w1;!. 7"(= V2ز1C (2JC O
-2
ء6
4B
~ ا2J& o[2 7
;
)د( ا
ول ا
MAPE
MAD
MSD
MA
26.5366
0.9077
1.2322
SES
25.5159
0.9002
1.1638
307
DES
27.1322
0.9880
1.5458
.4; H`N أ76"2 w(
ا7 اO!;
أن ا52و
:(';
95% ;اتN وw(
ا7 اO!;
;[ام اF. ات:(';
ا
Period
of
Forecast
Forecast
Lower
Upper
51
52
5.67586
5.67586
3.47035
3.47035
7.88137
7.88137
53
54
5.67586
5.67586
3.47035
3.47035
7.88137
7.88137
55
5.67586
3.47035
7.88137
:Q
]
ال ا:
4.Lإ
( )أ
3.5
Defects
3.0
2.5
2.0
1.5
1.0
Index
10
308
20
30
40
Autocorrelation
Autocorrelation Function for Defects
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag
Corr
4
5
6
7
8
9
T
LBQ
Lag
Corr
T
LBQ
1 0.43 2.88
2 0.26 1.49
3 0.14 0.77
4 0.08 0.43
5 -0.09 -0.46
6 -0.07 -0.39
7 -0.21 -1.10
8.84
12.18
13.18
13.50
13.89
14.18
16.57
8
9
10
11
-0.11
-0.05
-0.01
-0.04
-0.57
-0.27
-0.04
-0.19
17.25
17.41
17.41
17.50
10
11
Partial Autocorrelation
Partial Autocorrelation Function for Defects
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
Lag PAC
4
5
T
1 0.43 2.88
2 0.09 0.63
3 -0.00 -0.01
4 0.00 0.00
5 -0.16 -1.07
6 0.00 0.02
7 -0.18 -1.19
6
7
Lag PAC
8
9
10
11
T
8 0.07 0.44
9 0.05 0.35
10 0.01 0.09
11 -0.03 -0.23
ذج1! ;حJ2 اA وهr1,1 ". V6B O
SPACF ;[& أ وC SACF أنH3K
^ & اPوا
q = 0 وd = 0 وp = 1 أي أنARIMA (1, 0, 0 )
:)
"!
ا2JC ()ج
MTB > ARIMA 1 0 0 'Defects' 'RESI1' 'FITS1';
SUBC>
Constant;
SUBC>
Forecast 5 c3 c4 c5;
SUBC> GACF;
SUBC> GPACF;
SUBC> GHistogram;
SUBC>
GNormalplot;
SUBC> GFits;
SUBC>
309
GOrder.
ARIMA Model
ARIMA model for Defects
Estimates at each iteration
Iteration
SSE
Parameters
0
11.2419
0.100
1.700
Relative
1
2
10.0858
9.5649
0.250
0.400
1.393
1.086
3
4
9.5316
9.5309
0.436
0.441
1.006
0.995
5
6
9.5309
9.5309
0.442
0.442
0.993
0.993
change
in
each
estimate
less
than
0.0010
Final Estimates of Parameters
Type
AR
1
Constant
Coef
0.4421
StDev
0.1365
T
3.24
0.99280
Mean
0.06999
1.7795
14.19
0.1254
Number of observations:
Residuals:
SS
=
9.47811
(backforecasts
excluded)
MS = 0.22042
Modified
Box-Pierce
(Ljung-Box)
Lag
12
36
310
45
DF = 43
Chi-Square
statistic
24
48
Chi-Square
4.9(DF=11)
8.9(DF=23)
30.9(DF=35)
* (DF= *)
Forecasts from period 45
95 Percent Limits
Period
Forecast
Lower
Upper
Actual
46
47
1.80627
1.79135
0.88588
0.78503
2.72665
2.79767
48
49
1.78476
1.78184
0.76248
0.75648
2.80703
2.80721
50
1.78055
0.75459
2.80652
1;ح هJ!
ذج ا1!'
أن ا4J.
ت اL[!
& ا
zt = 0.9928 − 0.4421zt −1 + at , at ∼ WN ( 0, 0.22042 )
or
( zt − 1.7795) = −0.4421( zt −1 − 1.7795) + at
Q5
( )
φˆ1 = 0.4421, s.e φˆ1 = 0.1365, with t-value = 3.24
ي1'"& 7CاA
ار ا%9 اH&"& أي أن
( )
δˆ = 0.9928, s.e δˆ = 0.06999, with t-value = 14.19
ي1'"& δ ى1;!
` ا2وأ
:7Bا1(
اo%N ()د
w. م ا
;ا-1
311
ACF of Residuals for Defects
(with 95% confidence limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
8
9
10
11
10
11
Lag
PACF of Residuals for Defects
(with 95% confidence limits for the partial autocorrelations)
1.0
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
2
3
4
5
6
7
8
9
Lag
:7"(6
اV2ز1;
ا
Histogram of the Residuals
(response is Defects)
10
Frequency
Partial Autocorrelation
0.8
5
0
-1.0
-0.5
0.0
0.5
Residual
312
1.0
1.5
Normal Probability Plot of the Residuals
(response is Defects)
Residual
1
0
-1
-2
-1
0
1
2
Normal Score
Normal Probability Plot for RESI1
99
95
Mean:
8.21E-03
StDev:
0.464050
90
80
Percent
70
60
50
40
30
20
10
5
1
-1.0
-0.5
0.0
0.5
1.0
1.5
Data
MTB > %Qqplot 'RESI1';
SUBC>
Table;
SUBC>
Conf 95;
SUBC>
Ci.
Executing from file: G:\MTBWIN\MACROS\Qqplot.MAC
Distribution Function Analysis
Normal Dist. Parameter Estimates
Data
313
: RESI1
Mean:
StDev:
8.21E-03
0.464050
Percentile Estimates
95% CI
95% CI
Approximate
Upper Limit
P
Percentile
Approximate
Lower Limit
0.01
-1.07134
-1.33235
-0.81033
0.02
0.03
-0.94484
-0.86458
-1.18390
-1.09018
-0.70577
-0.63897
0.04
0.05
-0.80420
-0.75509
-1.01996
-0.96306
-0.58844
-0.54712
0.06
0.07
-0.71329
-0.67663
-0.91478
-0.87260
-0.51179
-0.48067
0.08
0.09
-0.64382
-0.61397
-0.83494
-0.80080
-0.45269
-0.42714
0.10
0.20
-0.58650
-0.38235
-0.76947
-0.54012
-0.40353
-0.22457
0.30
0.40
-0.23514
-0.10936
-0.37975
-0.24710
-0.09054
0.02838
0.50
0.60
0.00821
0.12577
-0.12738
-0.01197
0.14379
0.26351
0.70
0.80
0.25155
0.39876
0.10695
0.24098
0.39616
0.55654
0.90
0.91
0.60291
0.63038
0.41994
0.44355
0.78588
0.81721
0.92
0.93
0.66023
0.69305
0.46911
0.49708
0.85136
0.88901
314
0.94
0.72970
0.52820
0.93120
0.95
0.96
0.77150
0.82061
0.56353
0.60485
0.97947
1.03638
0.97
0.98
0.88099
0.96125
0.65539
0.72219
1.10659
1.20031
0.99
1.08775
0.82674
1.34876
Normal Probability Plot
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-0.5
0.0
0.5
1.0
RESI1
Average: 0.0082067
StDev: 0.464050
N: 45
Kolmogorov-Smirnov Normality Test
D+: 0.166 D-: 0.080 D : 0.166
Approximate P-Value < 0.01
4J(6!
) اJ
اV& 7Bا1(
أ!ط اo%b
Residuals Versus the Fitted Values
(response is Defects)
Residual
1
0
-1
1.5
2.0
2.5
Fitted Value
ا
(تWCC V& 7Bا1(
وا
315
Residuals Versus the Order of the Data
(response is Defects)
Residual
1
0
-1
5
10
15
20
25
30
35
40
45
Observation Order
ة6"!
ا
(ت ا7 (6;
W'& ;حJ!
ذج ا1!'
;';_ أن ا4J.
ت ا+1%b
& ا
()هـ
95%
Forecasts from period 45 :('C ;اتN و4(J;!
) اJ
ات [!~ ا:(';
ا
95 Percent Limits
Period
Forecast
Lower
Upper
46
1.80627
0.88588
2.72665
47
1.79135
0.78503
2.79767
48
1.78476
0.76248
2.80703
49
1.78184
0.75648
2.80721
50
1.78055
0.75459
2.80652
316
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
ﻗﺴﻡ ﺍﻹﺤﺼﺎﺀ ﻭﺒﺤﻭﺙ ﺍﻝﻌﻤﻠﻴﺎﺕ
ا
!دة = :ق ا
;'( :اQ%. 221 785X
ﺍﻻﺨﺘﺒﺎﺭ ﺍﻻﻭل ﻝﻸﻋﻤﺎل ﺍﻝﻔﺼﻠﻴﺔ
ا
Hbاول 1421-1420هـ
ا
& ; :
أ V!L 7 WLا 4ا
;
:4
ا
:ال اول:
ا-
VP
ا
'!1ذج
ا
;
7
7
z n + j = β 0 + β1 j + ε n + j , j = 0,±1,...
z n + j = f ′( j )β + ε n + j , j = 0,±1,...وذ
2%;. fآ& H
βو ) f ′( jأوL
ا
H3K
ان
Lو.ه
) f ′( j + 1) = Lf ′( j
]). X ′ = [f (− n + 1), f (− n + 2 ),..., f (1), f (0ه
ب y ′ = (z1 ,..., z n ) VP1. -و
n −1
n −1
j =0
j =0
ان
) X ′X = ∑ f (− j )f ′(− jوان X ′y = ∑ f (− j )z n − jو& )UاوJ& Lر !") β
ا
:ال ا
]:7
!; 4ا
&' 4ا
;
:4
12
11
10
9
8
7
6
5
4
3
2
1
t
28
26
24
25
22
19
15
17
13
12
8
7
zt
ا (= -و%;& wك & ا
ر 4;!
3 4Lا
!" @6و& )Uاو(';& Lت )Jا
!;4
z13 , z14
ب O!C (= α = 0.5 A{. -ا 4;!
w. 7ا
!" @6و& )Uاو(';& Lت )Jا
!;4
z13 , z14
317
:Q
]
ال ا:
ا
و
φ1 < 1
وS.U ارJ& 0 < δ < ∞
Q5
z t = δ + φ1 z t −1 + at
7
;
ذج ا1!'
at ~ WN (0, σ 2 )
k = 0,1,...,5 )J
O! وارρ k , ∀k 7CاA
اw. ا
;ا4
داL او-ا
k = 0,1,...,5 )J
O! و وارφk ,k , ∀k 78
ا7CاA
اw. ا
;ا4
داL او-ب
318
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
4"&Lا
!1" fد – آ 4ا
"1م
)Bا59ء و1%.ث ا
"!ت
إ;(ر أ!ل HNأول 1423/1422هـ
!دة =) Q%. 221ق ا
;'( :ا7859
ﺍﻝﺯﻤﻥ 2 :ﺴﺎﻋﺔ
أ V!L WLا 4ا
;
:4
ا
:ال اول:
'!1ذج
z = β0 + β1t + x t
) x t = φ1x t −1 + at , at ∼ WN (0, σ 2
إ;Nاض أن ا
!"
) .4&1"& β0 , β1, φ1, σ 2
.ه أن ا
!;'(| ا
[ 76ذا أد &;16[
{6 V.& w1ة ℓإ
ا&م 4B"
. 6"2
ℓ≥0
zt (ℓ ) = β0 + β1 (t + ℓ ) + φ1ℓ (zt − β0 − β1t ),
ا
:ال ا
]:7
إ;!دا &;K& n = 200 O
1= 4هة =( !1ذج ) AR (2و '5ا
;ا6.ت
ا
Aا1(
4Cا 7Bا
;
r1 = 0.13, r2 = 0.13, r3 = 0.12 4إذا آφˆ1 = 1.1, φˆ2 = −0.8 S
HONا
;ا6.ت ا
Aا1(
4Cا )C 7Bأن ا
'!1ذج ه AR (2) 1آ5 Hة أو &;!"4؟
ا
:ال ا
]
:Q
!; 4ا
!;Jة ا
;
B 6, 5, 4, 6, 4 4ر µ, γ 0, ρ1
319
:V.ال ا
ا:
ا
4
;
!ت ا3
اS(5 هةK& n = 100 O
1= 4'& ز4;!
r1 = 0.8, r2 = 0.5, r3 = 0.4, z = 2, s 2 = 5
H3K
اAR (2) ذج1! & هاتK!
;ض ان اNإذا ا
z = δ + φ1z t −1 + φ2z t −2 + at , at ∼ WN (0, σ 2 )
φ1, φ2, δ, σ 2 )
"!
رات ا
"ومJ& Lأو
320
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
4"&Lا
!1" fد – آ 4ا
"1م
)Bا59ء و1%.ث ا
"!ت
ا(;9ر ا
' Hb
78Oاول 1423/1422هـ
!دة =) Q%. 221ق ا
;'( :ا(7859
ﺍﻝﺯﻤﻥ 3 :ﺴﺎﻋﺎﺕ
أ V!L WLا 4ا
;
:4
ا
:ال اول:
'!1ذج
z = β0 + β1t + x t
) x t = φ1x t −1 + at , at ∼ WN (0, σ 2
إ;Nاض أن ا
!"
) .4&1"& β0 , β1, φ1, σ 2
.ه أن ا
!;'(| ا
[ 76ذا أد &;16[
{6 V.& w1ة ℓإ
ا&م 4B"
. 6"2
ℓ≥0
z t (ℓ ) = β0 + β1 (t + ℓ ) + φ1ℓ (z t − β0 − β1t ),
ا
:ال ا
]:7
إ;!دا &;K& n = 200 O
1= 4هة =( !1ذج ) AR (2و '5ا
;ا6.ت
ا
Aا1(
4Cا 7Bا
;
r1 = 0.13, r2 = 0.13, r3 = 0.12 4إذا آφˆ1 = 1.1, φˆ2 = −0.8 S
HONا
;ا6.ت ا
Aا1(
4Cا )C 7Bأن ا
'!1ذج ه AR (2) 1آ5 Hة أو &;!"4؟
ا
:ال ا
]
:Q
!; 4ا
!;Jة ا
;
B 6, 5, 4, 6, 4 4ر µ, γ 0, ρ1
ا
:ال ا
ا:V.
!; 4ز&'K& n = 100 O
1= 4هة S(5ا
!3ت ا
;
4
321
r1 = 0.8, r2 = 0.5, r3 = 0.4, z = 2, s 2 = 5
إذا ا;Nض ان ا
!Kهات & !1ذج ) AR (2ا
H3K
) z = δ + φ1z t −1 + φ2z t −2 + at , at ∼ WN (0, σ 2
أوJ& Lرات ا
"وم !"
) φ1, φ2, δ, σ 2
ا
:ال ا
[&~:
و Lأن ا
!("ت ا
' 2!. 421ا
X2ت Kآ V(;C & 4ا
'!1ذج
) zt = 5 + 1.1zt −1 − 0.5zt −2 + at , at ∼ WN ( 0, 2
إذا آ Sا
!("ت '1ات 1419و 1420و 1421هـ ه 7ا
;1ا
10 7و 11و 2& 9
ر2ل
-1أو:('C Lات !("ت 1422و 1423و 1424هـ
-2أ W5اوزان
ψj , j = 1,2, 3, 4.
ج( أ;N W5ات "(!
95% :('Cت '1ات 1422و 1423و 1424هـ.
322
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
4"&Lا
!1" fد – آ 4ا
"1م
)Bا59ء و1%.ث ا
"!ت
ا(;9ر ا
' Hb
78Oاول 1423/1422هـ
!دة =) Q%. 221ق ا
;'( :ا(7859
ﺍﻝﺯﻤﻥ 3 :ﺴﺎﻋﺎﺕ
أ V!L WLا 4ا
;
:4
ا
:ال اول:
'!1ذج
z t = β0 + β1t + x t
) x t = φ1x t −1 + at , at ∼ WN (0, σ 2
إ;Nاض أن ا
!"
) .4&1"& β0 , β1, φ1, σ 2
.ه أن ا
!;'(| ا
[ 76ذا أد &;16[
{6 V.& w1ة ℓإ
ا&م 4B"
. 6"2
ℓ≥0
z t (ℓ ) = β0 + β1 (t + ℓ ) + φ1ℓ (z t − β0 − β1t ),
ا
:ال ا
]:7
إ;!دا &;K& n = 200 O
1= 4هة =( !1ذج ) AR (2و '5ا
;ا6.ت
ا
Aا1(
4Cا 7Bا
;
r1 = 0.13, r2 = 0.13, r3 = 0.12 4إذا آφˆ1 = 1.1, φˆ2 = −0.8 S
HONا
;ا6.ت ا
Aا1(
4Cا )C 7Bأن ا
'!1ذج ه AR (2) 1آ5 Hة أو &;!"4؟
ا
:ال ا
]
:Q
!; 4ز&'K& n = 100 O
1= 4هة S(5ا
!3ت ا
;
4
r1 = 0.8, r2 = 0.5, r3 = 0.4, z = 2, s 2 = 5
إذا ا;Nض ان ا
!Kهات & !1ذج ) AR (2ا
H3K
323
) z = δ + φ1z t −1 + φ2z t −2 + at , at ∼ WN (0, σ 2
أوJ& Lرات ا
"وم !"
) φ1, φ2 , δ, σ 2
ا
:ال ا
ا:V.
و Lأن ا
!("ت ا
' 2!. 421ا
X2ت Kآ V(;C & 4ا
'!1ذج
) zt = 5 + 1.1zt −1 − 0.5zt −2 + at , at ∼ WN ( 0, 2
إذا آ Sا
!("ت '1ات 1419و 1420و 1421هـ ه 7ا
;1ا
10 7و 11و 2& 9
ر2ل
-3أو:('C Lات !("ت 1422و 1423و 1424هـ
-4أ W5اوزان
ψj , j = 1,2, 3, 4.
ج( أ;N W5ات "(!
95% :('Cت '1ات 1422و 1423و 1424هـ.
324
)5
! ا5
) ا ا.
هـ1423/1422 اولHb
78O'
;(ر ا4Lذ1! ت.Lإ
( 7859 ا:(';
) =ق اQ%. 221
!دة
:ال اول:
4.Lإ
H3K
ذج ا1!'
4N"!
ت اX ا
!"دV`
z t = β0 + β1t + φ1x t −1 + at , at ~ WN (0, σ 2 )
4B"
& ا6"2 إ
ا&مℓ ة16[
{6 V.& w1;& ذا أد76[
ا
!;'(| ا
zt (ℓ ) = E (z t +ℓ | zt , z t −1,...), ℓ ≥ 0
∵ x t = z t − β0 − β1t
∴ z t − β0 − β1t = φ1 [z t −1 − β0 − β1 (t − 1)] + at , at ~ WN (0, σ 2 )
zt (ℓ ) − β0 − β1 (t + ℓ ) = E (φ1 [zt +ℓ−1 − β0 − β1 (t + ℓ − 1)] + at +ℓ ) | z t , zt −1,... , ℓ ≥ 0
= E (φ1 [zt +ℓ−1 − β0 − β1 (t + ℓ − 1)] | zt , zt −1,...) + E (at +ℓ | z t , z t −1,...), ℓ ≥ 0
ℓ = 1 : z t (1) − β0 − β1 (t + 1) = E (φ1 (z t − β0 − β1t ) | zt , zt −1,...) + 0
= φ1 (z t − β0 − β1t )
ℓ = 2 : zt (2) − β0 − β1 (t + 2) = φ1 [zt (1) − β0 − β1 (t + 1)] = φ12 (z t − β0 − β1t )
ℓ = 3 : zt (3) − β0 − β1 (t + 3) = φ1 [z t (2) − β0 − β1 (t + 2)] = φ13 (z t − β0 − β1t )
ℓ = 4 : z t (4) − β0 − β1 (t + 4) = φ1 [zt (3) − β0 − β1 (t + 3)] = φ14 (zt − β0 − β1t )
مH3K. اA3وه
z t (ℓ ) − β0 − β1 (t + ℓ ) = φ1 [z t ( ℓ − 1) − β0 − β1 (t + ℓ − 1)] = φ1ℓ (z t − β0 − β1t )
ود%
اWCC ". أو
z t (ℓ ) = β0 + β1 (t + ℓ ) + φ1ℓ (z t − β0 − β1t ), ℓ ≥ 0
.ب16!
ا1وه
325
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
&دة =) Q%. 221ق ا
;'( :ا( 7859
ا(;9ر ا
' Hb
78Oا
] 1423/1422 7هـ
ا
& 3ت
أ V!L WLا 4ا
;
:4
ا
:ال اول:
'!1ذج
)zt = 200 + 1.2 zt −1 − 0.7 zt − 2 + at + 0.5at −1 , at ∼ N ( 0,1
-1أو Lدا
4ا
;ا w.ا
Aا k = 1, 2,...,5 )J
ρ k 7Cوار!.O
-2أو Lدا
4ا
;ا w.ا
Aا 7Cا
k = 1, 2,...,5 )J
φkk 78وار!.O
-3أو Lدا
4اوزان j = 1, 2,...,5 )J
ψ jوار!.O
-4أو Lدا
4ا
;'(. zn ( ℓ ) , ℓ ≥ 0 :
ا
:ال ا
]:7
ا
!Kهات ا
;
4;!
4ز&' ) :4إBأ & ا
ر 6ا ( 6.
197
197
198
199
201
200
200
202
198
326
203
202
201
196
193
195
197
199
201
201
201
203
200
197
204
202
200
200
198
198
199
204
206
203
200
200
198
199
201
201
201
204
205
205
201
198
197
203
194
195
197
201
204
201
198
198
196
193
194
206
202
204
206
205
202
:7
;
اWL وأMINITAB _&. 7N هاتK!
اHاد
;[امF. f
هات وذK!
( ا62 ARIMA 48 & W'& ذج1! "فC -1
AIC ( m ) = n ln σ a2 + 2m :4B"
. 6"2 يA
واAIC 7CاA
&ت ا1"!
&"ر ا
.4('!
;(رات ا9 اV!L O 2& 7Bا1(
اo%b. )B ;حJ!
ذج ا1!'
-2
. 95% :('C ;اتN V& مJC 4'& أز8 ;5 4(J;!
) اJ
ات:('C ;ح وJ!
ذج ا1!'
& ا-3
327
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ
هـ1423/1422 7]
اHb
78O'
;(ر ا4!;%& ت.Lإ
( 7859 ا:(';
)=ق اQ%. 221 &دة
:ال اول:
4.Lإ
تNا% إH3 zt = 200 + 1.2 zt −1 − 0.7 zt −2 + at + 0.5at −1 , at ∼ N ( 0,1) ذج1!'
اV`
ايµ =
δ
200
=
= 400 w1;!
ا
(1 − φ1 − φ2 ) 1 + 0.7 − 1.2
zt − 400 = 1.2 ( zt −1 − 400 ) − 0.7 ( zt − 2 − 400 ) + at + 0.5at −1 , at ∼ N ( 0,1)
:7
;
آρk L1 -1
E {( zt − 400 )( zt − k − 400 )} − 1.2 E {( zt −1 − 400 )( zt − k − 400 )} + 0.7 E {( zt − 2 − 400 )( zt − k − 400 )} =
E {at ( zt − k − 400 )} + 0.5 E {at −1 ( zt − k − 400 )}
γ k − 1.2γ k −1 + 0.7γ k −2 = E {at ( zt −k − 400 )} + 0.5E {at −1 ( zt −k − 400 )}
k = 0 : γ 0 − 1.2γ 1 + 0.7γ 2 = E {at ( zt − 400 )} + 0.5 E {at −1 ( zt − 400 )}
= σ 2 + 0.5 ( 0.7σ 2 ) = 1.35σ 2
k = 1: γ 1 − 1.2γ 0 + 0.7γ 1 = E {at ( zt −1 − 400 )} + 0.5 E {at −1 ( zt −1 − 400 )}
= 0.5σ 2
k = 2 : γ 2 − 1.2γ 1 + 0.7γ 0 = E {at ( zt − 2 − 400 )} + 0.5 E {at −1 ( zt − 2 − 400 )}
=0
k ≥ 2 : γ k − 1.2γ k −1 + 0.7γ k − 2 = 0
) σ 2 = 1 VP1.) ان4J.
ت اB"
& ا
ρ1 = 0.74436
γ 0 4!J
. اة و4B"
و& ا
γ k = 1.2γ k −1 − 0.7γ k −2 , k = 2,3,...
ρ k = 1.2 ρ k −1 − 0.7 ρ k −2 , k = 2,3,...
∴ ρ 2 = 1.2 ρ1 − 0.7 ρ0 = 1.2 ( 0.74436 ) − 0.7 (1) = 0.193232
328
ρ3 = 1.2 ρ 2 − 0.7 ρ1 = -0.289173
ρ 4 = -0.482271
ρ5 = -0.376304
ρ6 = -0.113975
ρ7 = 0.126642
ρ8 = 0.231754
ρ9 = 0.189455
ρ10 = 0.651180
:7
;
) آCو
ACF of the Model
ACF
0.5
0.0
-0.5
0
1
2
3
4
5
6
7
8
9
10
Lag
φkk L1 -2
329
φ00 = 1, by definition
φ11 = ρ1 = 0.744361, by definition
k −1
φkk =
ρ k − ∑ φk −1, j ρ k − j
j =1
k −1
1 − ∑ φk −1, j ρ j
, k = 2,3,...
j =1
φkj = φk −1, j − φkkφk −1,k −1 ,
φ22 =
j = 1, 2,..., k − 1
ρ 2 − φ11 ρ1 0.193233 − ( 0.744361)( 0.744361) −0.3608402
= −0.8091915
=
=
1 − φ11 ρ1
1 − ( 0.744361)( 0.744361)
0.4459268
φ33 = 0.343852
φ44 = -0.165706
φ55 = 0.821095
φ66 = -0.409620
φ77 = 0.204689
φ88 = -0.102327
φ99 = 0.511617
φ10,10 = -0.255822
:7
;
آO!و
PACF of the Model
0.8
0.6
PACF
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0
1
2
3
4
5
6
7
8
9
10
Lag
ψ j اوزان4
دا-3
330
:H3K
ذج ا1!'
اV`
zt = 200 + 1.2 zt −1 − 0.7 zt − 2 + at + 0.5at −1
zt − 1.2 zt −1 + 0.7 zt − 2 = 200 + at + 0.5at −1
(1 − 1.2B + 0.7 B ) z
2
zt =
t
= 200 + (1 + 0.5 B ) at
(1 + 0.5B ) a
200
+
t
1 − 1.2 + 0.7 (1 − 1.2 B + 0.7 B 2 )
= 400 + ψ ( B ) at
:7 اوزان ه4
دا
ψ (B) =
(1 − 0.5B )
(1 − 1.2B + 0.7 B 2 )
= 1 + ψ 1B + ψ 2 B 2 + ψ 3 B 3 + ...
:7
;
اوزان آL1
(1 + ψ B + ψ
1
2
B 2 + ψ 3 B 3 + ...)(1 − 1.2 B + 0.7 B 2 ) ≡ (1 + 0.5B )
B :ψ 1 − 1.2 = 0.5 ⇒ ψ 1 = 1.7
B 2 :ψ 2 − 1.2ψ 1 + 0.7 = 0 ⇒ ψ 2 = 1.2ψ 1 − 0.7 = 1.34
B 3 :ψ 3 − 1.2ψ 2 + 0.7ψ 1 = 0 ⇒ ψ 3 = 1.2ψ 2 − 0.7ψ 1 = 0.418
⋮
B j :ψ j = 1.2ψ j −1 − 0.7ψ j − 2
ψ 4 = 1.2ψ 3 − 0.7ψ 2 = -0.4364
ψ 5 = -0.81628
ψ 6 = -0.674056
ψ 7 = -0.237471
ψ 8 = 0.186874
ψ 9 = 0.390478
ψ 10 = 0.337762
:7
;
اH3K
اO
و
331
Psi Weights of the Model
2
Psi
1
0
-1
0
5
10
j
:4B"
. 6"C zn ( ℓ ) , ℓ ≥ 0 :(';
ا4
دا-4
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
V&
a , j ≤ 0
E ( an + j z n , z n −1 ,⋯) = n + j
j>0
0,
j≤0
zn+ j ,
E ( zn + j zn , z n −1 ,⋯) =
zn ( j ) , j > 0
:إذا
zn ( ℓ ) = E ( zn + ℓ zn , zn −1 ,⋯) , ℓ ≥ 1
= E ( 200 + 1.2 zn + ℓ −1 − 0.7 zn + ℓ − 2 + an + ℓ + 0.5an + ℓ −1 | zn , zn −1 ,⋯) , ℓ ≥ 1
= 200 + 1.2 E ( zn + ℓ −1 | zn , zn −1 ,⋯) − 0.7 E ( zn + ℓ − 2 | zn , zn −1 ,⋯)
+ E ( an + ℓ | zn , zn −1 ,⋯) + 0.5 E ( an + ℓ −1 | zn , zn −1 ,⋯) , ℓ ≥ 1
ℓ = 1: zn (1) = 200 + 1.2 zn − 0.7 zn −1 + 0.5an
ℓ = 2 : zn ( 2 ) = 200 + 1.2 zn (1) − 0.7 zn
ℓ ≥ 3 : zn ( ℓ ) = 200 + 1.2 zn ( ℓ − 1) − 0.7 zn ( ℓ − 2 )
:4
داH3K.و
332
200 + 1.2 zn − 0.7 zn −1 + 0.5an ,
ℓ =1
zn ( ℓ ) = 200 + 1.2 zn (1) − 0.7 zn ,
ℓ=2
200 + 1.2 z ( ℓ − 1) − 0.7 z ( ℓ − 2 ) , ℓ ≥ 3
n
n
:7]
ال ا:
4.Lإ
4;!
) ا
Observed Series
205
200
195
Index
10
20
30
40
50
60
2(;
واVB1;
ا7N ةJ;& O(و ا2
78
ا7CاA
اw. وا
;ا7CاA
اw. ا
;ا7;
) آ & دا
333
Autocorrelation
Autocorrelation Function for Observed
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
6
Partial Autocorrelation
Lag Corr
T
LBQ
1 0.77 6.18
2 0.31 1.67
3 -0.09 -0.48
4 -0.26 -1.33
5 -0.21 -1.06
6 -0.06 -0.32
7 0.06 0.29
39.96
46.47
47.06
51.72
54.89
55.19
55.45
11
Lag Corr
T
LBQ
8 0.11 0.55
9 0.08 0.38
10 -0.00 -0.01
11 -0.07 -0.33
12 -0.06 -0.27
13 0.02 0.08
14 0.11 0.52
56.38
56.84
56.84
57.21
57.47
57.49
58.44
Lag Corr
16
T
LBQ
15 0.16 0.75 60.52
16 0.13 0.63 62.00
Partial Autocorrelation Function for Observed
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
6
Lag PAC
T
1 0.77 6.18
2 -0.71 -5.68
3 0.24 1.91
4 0.07 0.56
5 -0.08 -0.67
6 0.08 0.62
7 -0.04 -0.34
11
Lag PAC
T
8 0.05 0.44
9 -0.11 -0.87
10 0.01 0.06
11 0.05 0.42
12 0.10 0.77
13 -0.04 -0.35
14 0.09 0.74
Lag PAC
16
T
15 0.01 0.04
16 -0.07 -0.56
& 41!& (6 ARMA ( p, q ) H3K
ا7ة وهJ;& 4;!
( ان اC ;
آ & ا
ا
)J
ا
'!ذج
:7
;
آAIC ( m ) = n ln σ a2 + 2m !"ر4!B HB اA{ وp = 0,1, 2 q = 0,1, 2
334
ARMA (1, 0 ) 1-
Type
AR
1
Constant
Coef
StDev
T
0.7841
43.1776
0.0803
0.2587
9.77
166.90
Mean
Residuals:
SS
199.954
1.198
Number of observations: 64
= 264.844
(backforecasts
MS =
excluded)
4.272 DF = 62
= 64 ln(4.272)+2(3) = 64(1.452)+6 = 98.928 AIC ( m ) = n ln σ a2 + 2m
ARMA ( 2, 0 ) 2-
Type
AR
1
AR
2
Constant
Coef
1.3568
StDev
0.0870
T
15.60
-0.7422
77.0740
0.0872
0.1768
-8.51
435.96
Mean
Residuals:
SS
200.013
0.459
Number of observations: 64
= 121.868
(backforecasts
MS =
excluded)
1.998 DF = 61
= 64 ln(1.998)+2(4) = 64(0.692)+8 = 52.288 AIC ( m ) = n ln σ a2 + 2m
ARMA ( 0,1) 3-
Type
MA
1
Constant
Coef
-0.8772
StDev
0.0775
T
-11.32
200.032
Mean
0.464
200.032
430.96
0.464
Number of observations:
335
64
Residuals:
SS
=
252.640
MS =
(backforecasts
excluded)
4.075 DF = 62
= 64 ln(4.075) + 2(3) = 64(1.40487) + 6 = 95.912 AIC ( m ) = n ln σ a2 + 2m
ARMA ( 0, 2 ) 4-
Type
MA
1
Coef
-1.3321
StDev
0.1042
T
-12.78
MA
2
Constant
-0.6491
200.038
0.1032
0.580
-6.29
344.69
Mean
Residuals:
SS
200.038
0.580
Number of observations: 64
= 149.291
(backforecasts
MS =
excluded)
2.447 DF = 61
= 64 ln(2.447) + 2(4) = 64(0.89486) + 8 = 65.271 AIC ( m ) = n ln σ a2 + 2m
ARMA (1,1) 5-
Type
AR
1
MA
1
Constant
Coef
0.6823
StDev
0.1023
T
6.67
-0.6832
63.5287
0.1054
0.3342
-6.48
190.09
Mean
Residuals:
SS
199.954
1.052
Number of observations: 64
= 153.610
(backforecasts
MS =
excluded)
2.518 DF = 61
= 64 ln(2.518) + 2(4) = 64(0.92346) + 8 = 67.101 AIC ( m ) = n ln σ a2 + 2m
ARMA ( 2,1) 6-
Model cannot be estimated with these data
336
ARMA (1, 2 ) 7-
Type
AR
MA
1
1
MA
2
Constant
Coef
StDev
T
0.5112
-1.0134
0.1492
0.1518
3.43
-6.68
-0.4821
97.7668
0.1482
0.4536
-3.25
215.55
Mean
Residuals:
SS
200.014
0.928
Number of observations: 64
= 126.688
(backforecasts
MS =
excluded)
2.111 DF = 60
= 64 ln(2.111) + 2(5) = 64(0.74716) + 10 = AIC ( m ) = n ln σ a2 + 2m
57.818
ARMA ( 2, 2 ) 8-
Type
AR
1
AR
MA
2
1
MA
2
Constant
Coef
1.1047
StDev
0.2011
T
5.49
-0.5789
-0.4394
0.1608
0.2275
-3.60
-1.93
-0.1918
94.8688
0.1973
0.2824
-0.97
335.90
Mean
Residuals:
SS
200.041
0.596
Number of observations: 64
= 113.051
(backforecasts
MS =
excluded)
1.916 DF = 59
= 64 ln(1.916) + 2(6) = 64(0.65) + 12 = 53.6 AIC ( m ) = n ln σ a2 + 2m
:7
;
ا
ول ا7N O[و
AIC ( m )
ARMA ( p, q )
337
min AIC ( m )
ARMA ( 0, 0 )
NA
ARMA (1, 0 )
98.928
ARMA ( 2, 0 )
52.288
ARMA ( 0,1)
95.912
ARMA ( 0, 2 )
65.271
ARMA (1,1)
67.101
ARMA ( 2,1)
NA
ARMA (1, 2 )
57.818
ARMA ( 2, 2 )
53.6
*
ARMA ( 2, 0 ) ذج1!'
AIC 7CاA
&ت ا1"!
!"ر ا4!B HB ان أ5
:7Bا1(
وإ;(ر اo%N -2
ACF of Residuals for Observed
(with 95% confidence limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
4
6
8
Lag
338
10
12
14
16
PACF of Residuals for Observed
(with 95% confidence limits for the partial autocorrelations)
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
2
4
6
8
10
12
14
16
Lag
Normal Probability Plot of the Residuals
(response is Observed)
4
3
2
Residual
Partial Autocorrelation
1.0
1
0
-1
-2
-3
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
Normal Score
339
1.0
1.5
2.0
2.5
Histogram of the Residuals
(response is Observed)
Frequency
10
5
0
-3
-2
-1
0
1
2
3
4
Residual
Norm al Probability Plot
.999
Probability
.99
.95
.80
.50
.20
.05
.01
.001
-3
-2
-1
0
1
2
3
RESI1
Average: 0.0042206
StDev: 1.39083
N: 64
Ko lmo go rov-Smirno v No rmality Test
D+: 0.067 D-: 0.068 D : 0.068
Appro ximate P-Value > 0.15
.W'& ;حJ!
ذج ا1!'
ان ا7'"2 اA;(رات وه9 اH;ز آC 7Bا1(
أن ا5
95% :('C ;اتN V& مJC 4'& أز8 ;5 4(J;!
) اJ
ات:('C -3
340
Forecasts from period 64
95 Percent Limits
Period
Forecast
Lower
Upper
Actual
65
66
197.419
198.729
194.648
194.059
200.190
203.400
67
68
200.196
201.215
194.621
195.480
205.772
206.949
69
70
201.507
201.148
195.756
195.181
207.258
207.116
71
72
200.445
199.756
194.202
193.378
206.687
206.134
207
C6
202
197
192
In d e x
10
20
30
40
341
50
60
70
:%nاا
1- Abraham, B. and Ledoter, J. (1983). Statistical Methods for
Forecasting, John Wiley, New York.
2- Anderson, T. W. (1971). The Statistical Analysis of Time Series,
John Wiley, New York.
3- Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis
Forecasting and Control, 2nd ed., Holden-Day, San Francisco.
4- Montgomery, D. C., Johnson, L. A. and Gardiner, J. S. (1990).
Forecasting and Time Series Analysis, 2nd ed., McGraw-Hill
International Edition.
5- Makridakis, S., Wheelwright, S. C. and McGee, V. E. (1983).
Forecasting Methods and Applications, 2nd ed., John Wiley, New
York.
6- Wei, W. W. S. (1990). Time Series Analysis Univariate and
Multivariate Methods, Addison Wesley.
7- Minitab Reference Manual, Release 11 for Windows. (1998).
ة6Y!
ى ا
!دة ا1;%& & ءL أوH آ76YC 4. VL &ا7! W5 L1CX 2K
اe
;ب أن3
او اVL!
ا اA هH]!. )"2 أو &رسQ5. أوW
= ا & أي1L;ب وأر3
ا اA ه7N
:7;و3
9 ا2(
ا4I5& 7
H2
[email protected] أو[email protected]
342
© Copyright 2026 Paperzz