A possibilistic analogue to Bayes estimation with fuzzy data

Institut f. Statistik u. Wahrscheinlichkeitstheorie
1040 Wien, Wiedner Hauptstr. 8-10/107
AUSTRIA
http://www.statistik.tuwien.ac.at
A possibilistic analogue to Bayes estimation with
fuzzy data
M. Arefi, R. Viertl, and S. Taheri
Forschungsbericht SM-2010-3
Dezember 2010
Kontakt: [email protected]
! "! # !$ %# " %
%$ &!' ()(* +(( % +!
! ! ! " Æ # $ % $ % $ % $ % $ &! ! " " # $ $ ! ! $ ! % ! & # $
$ $! "' " (#)
#*+
& $ & $ $ ,, -! $ $ $! "' $ %. (/+ . (##+ ,,' $!
$ (#0
#+ 1 !
,, "1 ,, "'1 "
!"# $%&#' ( $% )*&+' ) * &''' ,(
(#2+ (#3+
4 (#/+
4 (+
56'
(*+ $! ,, $ 7 (2+
8 (3+
9 (#:+ ! ; < (0+ .! $ '1 %=4! > (3+ ? @ (#A+
! &
$ ,, ,, . > = (2+ (+ 1
> B . $ ,,
8 C (/+ 1 ! $ $ (*+
!, & #
2 $! ) & 3
!
! & /
! . $ $ $ ,, A $ :
! " " $ " 4$
$ " & $ $! ",, D " D (0 +
D"
0 D" E D" E D" " D !"
# " D $ " D D" E " E " % " E D" $ " & " " " E
" & D" " " " #
' "
( F # " $ % ) % ) * )
+ , " # ' - E
" " ( # " $ $)
" "
" "
E "
" E " " " G ( )
( .'* $ ,/// (
( ( 0/ ( ' ,0" 11+
& 2 & 2 & 2 & 2 & 2
#
2
)
3
/
A
:
:/#
2:)
)33
##A
)::
)#:
A2
3*:
*
0
#
2
)
3
/
)3
)0
2/A
##/
:A
0#3
#:)
2A
A
:
*
#0
#
##
#2
#)
3#
3/)
)##
)00
/3
)#A
)/*
22*
#3
#/
#A
#:
#*
20
2
2#
3)#
:2
3/
2::
32/
:0/
)3*
30/
22
2)
23
2/
2A
2:
2*
)0
3*0
/#)
2))
30#
30A
/)3
22:
3/A
# ( ( E "
0
0
( ( 0//" . 3 0/ 1/ / 4/ 5 - -- 6 ,
E "
2
20 )0 30 )0
30
/0
,! - $-! ./ 0
7 $)
6 " E "
" "
" E " G
H
" E G "
"
"
20 )0 30 E "" HEE 30G
G )0
30
/0
" E 30 E 0*:A2
,! 0 - $-! -$ $ ./ 0
& $ ! ,, & !
" " " ' " -- - 8 "
E "
E -- $
( -- 9: ; !" )
"
G E "
E
"
"
" "
"
"
" " E
G
G
"
"
"
"
G E " ( % ( " G E
"
< -- E "
( )
" G # " $
$)
" E "" G " " "
"
E "" G " )
" E < *
, % (
"
( ( (
# ( ( --
$ % -- E " E 003 E # )0 I " E
0
'
+
,
0
*
1
)
()*+ ,-.
(-), '/.
(,00 +-.
(++1 ''.
(,)) +,.
(,+) +'.
(1-' -1.
(0/) -2.
/
'2
''
'+
'',
'0
'*
(,0' +-.
(,'2 +'.
(-*1 ').
(++* ''.
()'1 ,'.
('2+0 0'.
(+), ',.
(-'1 '*.
'1
')
'/
+2
+'
++
++,
(0+' +*.
(0*, +).
(,++ +'.
(,22 +2.
(*'0 -'.
(,+1 +'.
(,*/ +-.
(--/ '1.
+0
+*
+1
+)
+/
-2
-'
-+
$ ( -- " G E
E
E
3
"
I "
I
"
(0,+ +1.
()'- ,'.
(0'* +*.
(-)) '/.
(0-* +1.
()2* ,2.
(,0/ +-.
(02* +0.
--,
-0
-*
-1
-)
-/
,2
(0/2 -2.
(*+, -'.
(-,, '1.
(02+ +0.
(021 +0.
(*,0 -+.
(--) '1.
(0*1 +).
7 *)
, $)
" E 6 1
I 20 )0
I )0 30
" E
I 30 /0
E " G " E " E 30G "30 E /0# 0 "
,! - $-! ./ 0
< ( 5 ( 1 # --
E 0 $ = " 0
# ( 0# % -- 6 0
#3
0
0#
" E
" 0# ,! - $-! ./ /
# - E 2 -- E 0 0: E 0) 0/ E 0# 0* 0 0 $ ( -- E " G E
"
E
E 000: "
"
"
"
I I 0 "0* "0# "0: "0 "0/ "0) "
I 0# " I 0)2: " I 0)2# = $)
6 (' . 4 (' . 4 (' . 4 2 2+
( . 3 (' . 4 (' . 4 (' . 4 (' . 2+ '
- E 0)2)0 E "" G " E " E 0)2)0G " E 0)2)0 E 00)*/
"
,! - $-! ./ < -- )
H
$ E " " " "H F
$ 1 . '
" (A+
// & 8 A
,, "
(:+ ' !" " " -- $ H " H E !" " " " A
7 J7 <$J ,, H E "
H E "
E " E "" G
" " "" G " 7 J7 J ,, H E "
""
G "
"
E
E
" ""
G "
7 J 7J ,, H E "
H E "
E " E ( "" G
" + " ( "" G " +
< *
$ )
> H E " E 30
" 3A/#*#0
H E "
E E
:)) E ):AA::
" ( . 3 ( . 4
H E ( . 3 '')',,
+ 3 0/21+
"
E )*0#/
H E "
E " E 30
" < *
1 $ )
> ?
H E " E 0)2)0
:
H E E " " " E " E 00020
0#A3 E 0)A#)
"
I " I " E " " H E H E "
E " E 0)/2)
E 00#A3
# E 002:
" E 00#32 E 030)
00)*/
" I
& K "
K ( & .
" " "
. > F #" F ! ! "
$ ( -- " " " #" E #" " " " $ #" -- " = E " " " " ! < *
$ )
?
= #" E "
( 6 4 7 = ( - E ):*233
*
,! # ! -$ ( . 3 ( . ./ = #" E
( 6 + 7 = (
- E )*/*)
,! * # ! -$ ( . 3 ./ < *
1 $ )
?
= #" E "
( - E 2 6 ! 7 =
( - E 0))*
,! # ! -$ ( . 3 ( . ./ *
0
= #" E ( - E 2 6 @ 7 = ( - E 0)23
,! 1 # ! -$ ( . 3 ./ *
$ ,, $
$ ' " -- E "
$ " $ %" "
&
E "
" % "& " H
&
"& H
E "& % = % "& E "&
" " E " "& $ 5 >
%"& %"& ?
,) $ %"& %"& -- ) $ %"& %"& 1) $ %"& A > )
< " $)
*
$ %" $)
" E 6 ,/
&
"&
% "& E "
" E
& E 0 " "& H
I 20 )0
E
I
)0 I
30
30 /0
$ B
/B $ $ ( ( ( / ,/// (
, ( 3
/ @/C ( 1/
2 '2
+2
-2
,2 02 *2
12
)2
/2
'22
$ 2 2,,0' 21+)) 2)/0' 2/11' ' 2/)+0 2/-), 2)1)' 2)2)) 21-0)
,! ( - $ $-! -$ $! ./ < *
1 $ $)
= $)
" E *
1
I "
I "
"
E
" " I " I "
7 E " "
% "& "& H
&
" E "
#
I I "
" 0
0# &
E 0 0#
E
(' . 4 (' . 4 (' . 4 ½ 4 ('
4 ('
4 ('
('
.
.
.
.
½
2 2+
2+ '
0/2#) 030) E
0
E
E E 0 = $)
" E " 0/2#)
030)
&
&
6 ,,
" G "
E
" "
0
00*:
E 00*: 0/)*#
0/)*# E 0
00*:
00*: 0/)*#
0/)*# ,! - $-! -$ ! ./ 1
7 6 , "
&
E "
E " "
% "& " " "& H
E
E
""
" 03):#
0A03)
&
&
" " &
E0
&
E
E 0 E 03):# 0A03) E
0
2
&
E 0 ,! 0 - $ $-! -$ ! ./ 1
& . Æ !
& $ '
",, " ! !
. ; !
! " ! " (#
)
3
:+ ? !"#
C . ! B 8 C -$ ! 4 ! $
(+ #00 C ,, %
D ; .
" (#+ 7 L4
@! @M
7 74 #000 N $
,, 6-- # #
3#'3#3
(2+ 7 LL **3 $)
.
# :33':/2
()+ 7$ N
4M *:/ 7! ! 6-- # #
'2
(3+ 8 8 #00/ C ! <
# O
2 )A'/*
(/+ 8 8
C 4 **A ! 6-- # #)
##2'#)0
(A+ 8 8
C 4 *:: ; $ ;
; & E(
)
!" 2 2 ; 7 # # ,@@1 F 3 %? 6--
' # ' G G . H I ; 2H
,/1),, @" 66)# # ,@@1 F -- = 6-- # #
?
0,)!
,/" J . < & J ; ,@! $ -- ? -- * D F G ?
4)10
,," 7 2 K G //0 6-- = %? # .
G
%
#
'-)2- . H #)K =H
1@ )1@!
, " I JD 6 $ ,@!! 6-- # L %
;)7 *)
<> &D
,1" ' # = = /// G ? = )
6-- # #
? ,,@),0/
,0" ' D6 //1 # . . '
2 # *
D 5 O # & D
," FM7 ' $ ,@+4 = N =
( ? /,) /1
,4" F( $ $( 7 ' ,@+! -- --
-- %
< ? ,1),0+
,+" G <; //, $ = < # * #)K 7
,!" G $D ,@@ 6-- ' * .J)7 & E(
,@" $ #. //1 $ -- D # ? 1@) +
/" $ #. = D //, = -- 6--
# #
? 1@)0!
," $ #. = D //4 F = -- -- % D ; .
? ,1@),0
" $( 7 F( $ I ,@+@ 6-- %? 6-- # $ J .. H &)
7 H 1/1)1 /
1" $ 7 = D //+ '( -- #
; ? /@) 0" $(P //! 6
* D
F G ? ,1!),4,
" K G //4 L -- <
# 2 ? ,11),0+
4" K G 7 2 //0 6-- % D 6--
#
? 01)4
+" 5 7< //1 $ -- -- -- * D F G ? ,/,),,0
3
!" 9 ' ,@4! ; -- D .
? 0 ,)0 +
@" 9
7D /// ) *
D F G ? ,@/),@!
/