TESTS FOR HARMONIC CO~Of\rF'~rTS
IN BIVARIATE TIME SERIES
by
Donald F. Morrison
University of North Carolina
Sponsored by the Office of Naval Research
under the contract for research in probability
and statistics at Chapel Hill. Reproduction
in whole or in part is permitted for any purpose
of the United States government.
Institute of Statis tics
nqimeogr cp h Series No. 165
dpri.l, 1957
ii
.
AC KN
a l(J
T., E D G F
~ ~
N T
I wish to express my deep appreciation to
Professor Harold Hotelling for introducing me to
the many unsolved problems of periodogram analysis,
and in narticular, for his suggestion of the need for
a bivariate test of
signi~icant
periods in parallel
time series.
I am especially indebted to
p1rs.
Hilda Kattsoff
for tyning these pages under the difficult conditions
imposed by my absence from Chapel Hill.
Finally, I wish to convey my thanks to the Office
of Naval Research for their generous support of my
work while at the University of North Carolina .
•
iii
TABLE OF CONTENTS
PAGE
i1
. ACKNOWLEDGEMENT
I.
INTRODUCTION
v
TESTS OF SIGNIFICANCE FOR PERIODICITIES
1
1.1
Introduction
1
1.?
The Least Squares Model .
1
1.3 The r,east Souares Estimates of the Parameters
. . . . • . • .
C, A 'R and e . . .
1.4
1.5
~ests
of Sipnificance of Trial Periods
?
4
Fisherls Distribution of the Ratio of the
Greatest of a Set of Mean Souares to their
Sum ••
7
v~.....
1.6 An Identity involving the Sum of the
Intensities
1.7
• • . . • •
The Expect at ion of the Periodogram Ordinate in
the \icinity of the True Period
1.8 The Power Function for Walker1s Test
II.
A TFST FOR THE
SIGNIFIC~WCE
IN BrVARIATw TIME SERIES •
-e
16
19
PFRTODIC COMPONENTS
•• • . .
2.1
Introduction . . •
2.2
The Bivariate Model
?3
Estimation of the Parameters C , C?' AI' B ,
l
l
?4
?5
•
O~
13
?5
A Bivariate Test for a Si~ple Sinusoid
31
Other Possible Statistics
37
iv
CHAPTER
III.
PAGE
CERTAIN EXPECTATIONS OF THE INTENSITY DETERMINANT..
Introduction . • • • • • • • • • • • . • • .
3.2
..
The Intensity Deterndnant under the Model of
Chapter II
••••.••••••••.•.
3.3 The FXoectatton of the Intensity Determinant for
Different Model Periods and Types of Estimates •
APPENDIX:
CERT UN
TRI('i)NO~ifF.TRIC
SUMS
BIBLIOGRAPHY . . . . . . . . . . . .
41
41
hI
45
,6
62
v
INTRODTJCTTONI
The purpose of this thesis is that of setting forth a test
..
for sinusoidal
co~ponents
A test based upon a
II
of enual period in two parallel time series .
generalized intensityll in the form of a determin-
ant is proposed and the exact distrihution of the greatest of these
intensities is easily obtained from existing distribution theory.
Chapter I is a review of the least squares analysis of periodicities in a s~le time series.
L-17_7
SChuster's
L-13_72
tests are discussed, and Fisher's 1929
L-5_7
and Walker's
distribution
of the ratio of the greatest int8nstty to the sum of the intensities
.-
is given in detail.
Some studies of the effect of the model period
being situated between two adjacent trial periods are presented.
The
power function for 1.Talker' s test is obtained, and is shown to be unbiassed.
~
bivariate test for simple
s~nusoids
current series is presented in Chapter It.
in each of two conA general theorem on
the estimation of parameters in multivariate regression is given,
and the estimates of the coefficients for the case of unenual model
periods are obtained to show their dependence upon certain nuisance
parameters.
An extremely simple exact test based on the "intensity
determinant" follows, with a short discussion of some of its pro-
1. Sponsored by the Office of ~aval Research under the contract
for research in probability and statistics at Chapel Hill. Reproduction
in whole or in part is nermitted for any purpose of the United States
povernment •
?
The numbers in souare brackets refer to the bibliography listed at the e nd •
perties.
Other possible test statistics are mentioned, none of which
appears to give rise to such a simple distribution as the intensity
determinant.
The expectation of the intensity determinant at a specified
point is obtained under a variety of conditions and model series in
Ch3pter
rlr.
The
e~pectations
of the twelve possible combinations of
equal, uneoual, and non-existent periods in the two series with equal
or
une~ual
trial periods and
~JO
tvoes of estimates of the parameters
based on eoual or unequal model periods, are computed at a point and
compared.
CHAPTER I
TESTS OF SIGNIFICANCE FOR PERIODICITIES
1.1.
Introduction
This chapter is an expository review of tests of significance
for a simple sinusoidal component in a single series of observations
ordered in time.
No attempt is made to summarize the entire field
of harmonic analysis as it
~as
set forth mainly by Fourier in l82?
Rather, this chapter will be restricted to the concepts arising from
I
Sir ~rthur Schuster's ~13_7 definition of a periodogram, and its
ramifications by later workers in modern statistical theory.
1.2.
-The Least Squares -Model
l~e
hypothesize that the observations arise from the structure
(1.?1)
For estimation purposes,
~t
is a random variable satisfying the
usual least squares conditions
(1.2.2)
E(e t )
=0
, where
2
=
5 (j
r s
rs
E(e e )
S
rs
=s
1,
r
0,
r :f s
=
To make tests of hypotheses concerning the parameters of (1.2.1),
it is further required that
variable.
(1.2.3)
8
is a normally distributed random
t
This condition will be denoted by
8 t: N( 0..
2
u ), t = 1, ••• , N •
2
The phase parameter
g
entering non-linearly into the model
o
may be changed into a more tractable form by writing the model equations as
(1.2.4)
or as,
,•
9
0
may be determined by the relation
go
=
tan- l
(B/A). While it
is sometimes desirable to put confidence bounds on the phase, we shall
be concerned only with the problem of estimat ing
e.
In all further
work, (1.2.$) will be taken as the univariate model for a simple
sinusoidal component.
1.3.
The Least Squares Estimates
2! ~
Parameters C, A, B, and 9
Unless otherwise specified, in all further work the restrictions will be made that N is odd, and
(1.3.1)
~~
N
n ~ -
Th;.s restriction on
' k
~
Q
1 , •.• , -,N-l
is of the form
~
n
9 is made to use the exact orthogonality of
certain trigonometric functions when summed over suitable intervals.
A variety of such sums needed in this study is given in the Appendix.
For a time series of
the parameters
C, ", B, and
N
observations, we wish to estimate
e so that the residual sum of squares
N
Z CUt - Acos tQ - Fsin t9 - C)2
t=l
3
is a mi..nimum.
Making the matrix definitions
1 cosQ
r
(1.3.3) x=
sinQ
.
I.
I 1 cosNQ
~I
'\
I
1
I'
==
b
I
SinNQJ
L~
the least squares estimates of C,
b
,
Jf.::O
fe'
f'J.,
=
l~
,
I
I
J
and B are given by
(x ,x )-1( x , y )
(1.3.1.~)
N
0
0
N
", -1
Z ut
t~1
~e
N
0
::0
'2
N
Z utcos tQ
0
I
I
0
0
N
"2
t
t=1
,
N
Z u sin tQ
J
t=l t
1
where the dia~ona1 matrix (x x) has resulted from the use of the
ADoendix sums (11.2.1) - (A. 2. S) •
~ence,
the least square estimates are simply
/'
1 l\T
C='N ZUt
t=l
A
2
A=-
N
Z u
N t=l t
1\
B ==
2
N
cos t9
N
Z u t sin tQ
t=l
To find the least souare estimate of Q, consider the error
4
sum of squares (1.3.?).
Using the three estimates just bbtained; this
may be written as
(
SSE = N
~ u t=:l t
(1.3.6)
2""
- Acos t e -
BS1n tQ)2
1\..
N
=
2 t~lUt 2
N 2
~ ut-(--,r) - 2(A
N
t=l
+
2
B)
Thts is clearly at a minimum when the quantity
is at a maximum.
I
is called the 'intensity" associated with a
particular value of Q. Since we have restricted our attention to
2nk , k=l, 2, •..• n, our estimate of the true
e's of the form ~
period is N/k, where k is the integer from the set 1, ••. , n that
makes
I a maximum. 1."'e have thus connected Schuster's original
periodogram concept with the least squares criterion.
1.4.
Tesys of Significance of Trial Periods
- -7 is
Schuster /-13
apparently the first to have considered a
test for the significance of a single trial period.
the n
He proposed
th~
intensities be computed, and then be plotted on a chart
against k =: 1, ••• , n.
The period is taken to be N/k,
where
o
k0
is the argument corresponding to the greatest periodogram ordinate.
A1thou~h
not set forth in this more modern notation, Schuster went
further to propose a test of the hypothesis
(1.4.1)
5
against the alternative
Since A and B are real numbers, this test is equivalent to testing whether A and B are both 0, and hence whether a periodic component exists in the model.
We shall obtain the distribution of the sample intensity
12 + B2
under the null hypothesis (1.4.1) in modern terminology, then
discuss in what respect this differs from 9chuster's original result.
S~nce the £t are distributed with zero mean and variance
fol~oW8 from the properties of the trigonometric sums
of the
..,
e
~pDendtx
0
2
it
,
(A.?4)-(A.2.6)
that
N
2
== N ! cos tQ
t==l
1\.
(l.l.j..3 )
.A
and
A
B
:::t
2
N
N
t=l
sin tQ
~
each have a normal distribution with zero mean and variance
N
from the fact that
~
sin tQ cos te
Thus,
N(A
x
2
=
+
'B 2
2(/
)
has a X 2 distribution TN1th two degrees of freedom,
f ( x)dx
x
1 -'?
=:
? s . dx,
2
and
= 0, they are distrihuted inde-
t=l
nendently of each other.
20-
~,
x >
°
6
x =
The probability that
N(~2+B2)
2(/
,
will exceed the value x by
chance under the null hypothesis of a strictly random normal series
is riven hy the cumulative distribution
X
,
(1.4.6)
p(x > x ) = e
I
-2'
Th is exnress ion is generally referred to as rrSchuster I S
Probabili ty It
by contemporary writers on time series.
It will be interesting to compare this expression with Schuster's
-7 paper.
actual 1898 1-13
-
Schuster took
as
a measure of intensity the
sauare root of our expression (1.3.7), denoting it by
r
=
1
He then divided this by ao=
to give his measure
p
J / al'2
1
2
... b l
N
u ' for p :::: length of trial period,
p t=l t
-
rl
- --
- a •
o
~
Disre~arding
the fact that
a0
is a
random variable, Schuster states that the probahility element ef p
is
2
(1.4.8)
H( p )dp
From this he shows that
=
~ p exp(- .:u:..)d p
E( p)::::
4 .
fi,
IN
E( P 2)=
'?
exp (- ~). l'~ile these are valid for a
l±
,and
p( p > p )=
n
o
constant, it must not
have been apparent to Schuster that he was deal i.ng with the ratio of
two random variables, and not a single one.
7
Sir Gi.lbert 1nJalker ["17
7 notea
in 1914 that Schuster's test
failed to allow for the selection of the greatest intensity from a
set of
n, and gave the
should exceed x
,
proba~ility
that the greatest intensity x
as
x'
P(x > x') = 1 -
(1.4.9)
.I\?
1111'here x
1\
L-l -
e-':!
_7
n
,
2
= N( A +~:l provided all the n statistics x are inde20'
pendent.
The independence condition is satisfied if all these
n
trial periods are submultiples of the number N of observations.
Tables of this
-e
probability are availa"l)le in David,
L-3_7.
We shall
discuss the power function of Walker's test in Section 1.8.
1.,.
Fisher's Distribution of the Ratio of the Greatest of a Set of
~
Squares
.!2
their Sum
It remained for Sir Ronald Fisher to obtain the distribution for
the general case of 0'2 unknown.
- -7,
In mia 1929 paper /-,
Fisher
found the exact distribution of the ratio.
throuph the in<"1enious use of e-eometrical methods. 1nTe shall present
here a derivation in the same manner communicated to the author by
Professor Harold Hotellinr., and in much greater detail than the original
1929 paper.
An analytical derivation of the distribution of the r-th
-
greatest intensity ratio has been ~iven by Whittle /-18 7, p. 99,
-
through the use of the Levy inversion theorem fer the characteristic
function of the joint distribution of the intensi ties.
Cochran,[?_7
also obtained the distribution more straightforwardly by transforming
the
n
different grs to independent variates, then operating upon
their joint density.
The cumulative form of Fisher1s distribution for the ratio
g
given in (1.5.1) is
PF(g > gl)=
-e
m
Z (n )(_)j+1(1_jg)n-l
j=l j
7 i.e., the largest integer
where m= _/ - g -1 _'
~
g-1
,
This condition
is reauired in order that the non-zero terms (1 - jg)n-l
are always
posl ti ve.
We shall now obtain Fisher's distribution geometrically.
joint density of the
n
The
quantities
is
-n
f ( xl' •.• , x n ) :: 2
exp
t!
n
,
Z Xi J
i=l
It is apparent that the density is constant over hyperplanes of dimension
) and g may vary from n-1 ,for all intensities equal,
(n-l,
to 1, for all but one equal to
O.
Os
~
varies over this range,
9
the Dortions of the hyperplane in which a qiven Xi
is greatest
m~
overlap, so that the distribution has discontinuities for certain
definite values of
the
foll~wing
g.
The boundaries of these portions intersect in
sets of points:
;" x =n -1
n
.• g
• ; x =(n_l)-l = g
n
:x =(n_2)'..1
n
-e
=>
g
Using this knowledge of the discontinuities at these vertices,
we may write the cumulative distribution in the form
P
F
1'lhere k:::
P (n
-1
L-g-1 _7,
) = 1.
k
=
~
x=l
a. ( 1-xg )
x
n-l
,
i.e. , the greatest integer
~
g-1 • Since g -> 1-1 ,
Thus,
n-1
1=
x)n-l
r,a;1-(
x""l x
n
Similarly, the first (n-l) derivatives of the cumulative distribution
function must vanish at
g
=0
n -1
This requires that
n-l
O = ~~ ~ xrCl - !)n-r-l r:= 1
x=l x
n
u.
"
••• ,
n- 2
10
"Ie are thus- p.rovided with a system of (n-I) ecu8'tions in the (n-l)
unkn::>wns
a 1 , .•. " an-I'
The matrix
1 n-l
r
I (1-'ll)
I
1 n-?
(1- -)
n
(1.5.8)
of this system is
2 n-l
(1- -)
n
?(1... ~)
n
n... ?
.
(1- ~)
n
n-l
1 n-2
(n-I)(l- ~)
n
I
I
1
I (1- ri)
We wish to show that the rank of this
that there exists a unioue
equations.
00 lution
~atrix
is
n-l, hence
(aI' ... , an_I) t9 th~ above
After some minor changes, the matrix becomes
1
-n:I
n
(1.5.9)
n-2
t
...
iT
~e
(n-I)
1
may f ae t or n -n+i from the ith row f or all
• i , an d th en mu It ip1y
the jth column by (n_j)l-n for all j.
This leaves
11
r~
(1.5.10)
1
1
2
(n-l)
n-2
•
I
eonst.
•
I
l
1 n-2
. (n:!)
n-
But
th~s
2 n-2
(-:._)
n-2
is a Vandermonde matrix with rank
basic elements
i(n_i)-l
are equal.
• • • (n_1)n-2
J
n-l, since none of the
Hence, the system has a unique
solutton.
He shall state the solution as
--
,
(1.5.11)
and then show by finite difference
unique one.
ope~ations
that this must be the
An attempt was made to find an inverse to (1.5.8), but
because of the asymmetry, no simply patterened inverse appears to be
readily available.
From the equations
where 00r is the Kronecker O.
_ r(l
ux - x
(1.5.6) and (1.5.7) ,
Let
x)n-r-l
-n
Then, the n-th difference, 6nUx ' of this (n-l)-degree polynomial
is zero. l.rtle further employ the "shift operator I' E such that
12
E f(x)
=
f(x + 1) = (1 + ~) f(x)
x=o,
we have
(1., .1,)
Since un
=0
by (1.,.13), the last term in this expansion can be
omitted.
Cancelling the (_)n as well, and rearranging terms,
For r= 0,
U
x
= ( 1-
(_)x-l(n) = a
x
x
nx)n-l ,so that
o =1.
U
For r > 0,
U
o =- O.
Hence
is the uni~ue solution of the system of equations.
Fisher's cumulative distribution becomes
,
(1.,.17)
as was to be shown.
Fisher
-;-7- 7
considered the enuivalence of this distribution
with the probability obtained by W. L. Stevens L-l,_7 for a certain
geometrical problem.
Fisher also noted that Stevens' mora general
13
result could be used to obtain the distribution of the second greatest
ratio of intensities in the set of n.
Fisher has made reference to
- -7 on
this distribution in his 1939 paper ;-6
the distributions of
statistics arising from non-linear equat ions, and cons! dered therein
the mean of the distribution for large n.
not be
re~uired
Since these results will
in the later chapters, we shall not consider them
here.
~
1.6.
Identity Involving
~ ~
of the Intensities
We shall now establish in two ways a special case of Bessel's
Inequality:
--
This enables us to find the sum of all n intensities without the
laborious computation associated with each.
This identity can be shown most elegantly as follows:
The
residual sum of squares for the complete Fourier sequence is identically zero, or,
- ...
(L6.?)
:;;; 0
If we expan d the sunnnation and note from the Appendix that
N
Z
t=l
2
N
sin t9 = -
N
2N
Z cos to
2' tl=l
N
= '2 ' Z sin
t=l
to cos tQ =- 0, this bec",mes
14
Thus,
,
as was to be shown.
A more pedestrian derivation is given by using (A.l.4) of
the Appendix to sum
~ u
cos (r-s)2nk
s=l r s
N
(1.6.3)
Thi-s
becomes, for r
u
r s,
N-l
2n
n+l 2 n )
)2n
N cos(---)(r-s)
-- - cos (---)--(r-s -l+cos(r-s -?
N
2 N
N
r . Z Z
S
S(1_cos(r-s)2nk)
N
N
7r
,
where all trigonometric terms cancel, leaving the contribution for
r ~ s as
2
N
N
- ~ Z E u u.
N r s r s
Summing over k for r = s, each cosine term
in (1.6.3) is 1, so that the sum of the n intensities for ?n+l odd is
n (.. . 2
Z
k=l
A 2)
k+B k ..,
A
4 -I-(N-l)
-""2
-.,...
N
?
=-
N
2
1 NN
Z u - -2 Z Z u u
t=l t
r s r s
_7
rl s
N
Z (u
N t=l t
-ii)
2
,
as we wished to show.
For N even, so that it is of the form N = 2(n+l), Fisher
L-5_7
suggests that only the first n complete harmonics be consider-
15
ed, and states
2
(1.6.4)
that the sum of these
~n ~ffect
-N
N
_ 2
1: (ut-u)
tal
2L-1:~(_)r+l
- •
2
J
n intensities is
2
N
(1:u (_)r+l
z(u -u)2 _
Actually, Fisher I s result is
r
t
-
7'?
N
but the presence of our factor ~ is due to the fact that his esimates of A
a1
/'i,
d B involve
i.Te shall now find the
sum
and not
2
N'
as
chosen for this study.
of the n complete intensities.
Starting with (1+.2.4), this sum is, (for 9
-e
(1.6.5)
cos
(
N-:?
N9
e( ~)-C08
2' -
=:
1+008
e
2n(~-s»,
N 2
)+ n ~ U
tal t
?(l-cos e)
_7 ,
where the first expression is eaua1 to 0 for r-s odd, and equal to
-~N ~ ur vs for r-s even. (1.6.5) can thus be written as
r,s
r's
(1.6.6)
Since
the above is equal to
-e
+
N N
r s
Z 6 u u (-) - ) 7.
..L
r s
r,B
16
0tnce N = 2(n+l), this becomes
To show the similarity to the case
N = 2n+l, this may be written as
,
.
as originally stated in (1.6.4) •
1.7.
~
Fxpectation of
~
Periodogram Ordinate in the Vicinity
of the True Period
Let the model be of the usual standard univariate form
(1.7.1)
U
t '"
+ A cos tA + B sin tA + e
wlth the assumptions (1.2.2) on the residuals
us comnute the expectation of the intensity
period corresponding to
9 "" 2~k
e
t
t
,
satisfied.
A2 + B2 for
Let
the trial
17
(1.7.2)
N
+
~;- Z ut(A +A cos tA + B sin tA + 8 t )sin tQ_~
Nt: - t=l
0
N
Z cos tA sin te)2
t==l
4B 2
N
+ "?"'"( Z sin tA cos tQ)? +
Nt=l
N
+
2
N
2n(k+
A ==
c
N
N
tA sin te)(z sin tA sin te)
t=l
~)
N -, the model period is farthest from any of
the possible trial periods of the usual form 2~k.
nometric suma
2
~(
Z cos tA cos t9)(t~lsin tA cos te)
NG: tzl
-
N
+ BAB( Z cos
N? t=l
For
4
N~ (t~lsin tA sin t9)
Using the trigo-
(A,5.1) - (A.5.8) of the Apnendix, we may evaluate
the expectation (1.7.2) when the true period is half-way between two
consecutive sub-multiples of N as
.18
+
4B? /-
if -
Nealecting terms in N- 2
and
N 4k+2 7? BAB L- N (4k+2) 7
it 4k+I - +"iT - n (4k+1) - .
N-1 , we are left with
It follows from these computations that the expectation of
the neriodogram ordinate when the true period is one-half a unit
between it and the next ordinate is about
tion intensity
2 2
A +B , or
0.406 that of the popula-
0.406 times the expectation of the ordinate
resulting from trial and model periods being exactly equal.
In similar fashion, the expectation of the ordinate
for
A = n(~~+l) , may be found from the 4ppendix sums (A. 6. 1) - (A.6.6).
Omitting the details of computation, save to note that the terms in
AB are conveniently of the order N- 2, and that all the coefficients
of A and B are asymptotically eaua1, the expectation for this value
of I is
(1.7.5)
19
It is evident that a model period as close as one-auarter of a unit
to any trial period would, for large N, be reflected in that trial
period's intensity.
These results have been known for some time, the first one
having been discussed under slightly different circumstances by
Whittle ~18_7, p. 107.
They have been presented here not only to
illustrate the effect of non-integral sub-multiple periods upon the
periodogram, but also because it is felt that more explicit derivations of the summations involved would be useful.
1.8.
The Power Function for Walker's Test
-----The proba'l:>ility of not rejecting the null hypothesis of any
-e
simple sinusoidal component when
simoly.
2
0
is known can be found quite
Consider v.Talker's cumulative distribution (1.1-1-.9):
p(x >
I
X ) ""
1- ["1
where in the usual notation,
"" x ~
corresoonding to a Type I error of
~rescribed
size a, we wish to
know the probability
,
(1.8.1)
that is, the probability of accepting Ho : A2 + B2 = 0, given that
2
Ha : A2 + B "" K is true. If this alternative is correct, then one
of the
n Xi has a non-central
X?
distribution with 2 degrees of
20
freedom and non-centrality parameter
A=
!IT 2
2(A
maining (n-l) of the xi have central X-(2}
Fisher
centr al
-/-4- 7,
2
X, , i. e •,
==
2
B ), while the re-
distributions.
p. 670, gives the probability element for a nonX '2 , vari ate with
non-centrality parameter A
For n
+
n
degrees of freedom and
as
2, this reduces to
'2
_
(1.8.3) ~( X , A, 2)oX
I?
X'2+A
12
(X: )jA j d X'2
e
2
ro
= --- ~
(jl)24j
2
j=O
The probability of accepting H ' given H true, is then
o
a
~
-x o
=1
- { 1 -
x0
.fl-.~7n-l J
¢(x'2,).,2)dx' 2
o
-x
(1.8.4)
==
["l-e . zC: 7n-1 (
-
The power function becomes
12+A
xg
I
e
2
co
2
j=O
~
J
21
'2
X.
(1.8.5)
-~oo
e
/
o
2::
j=O
No simpler expression apnears to be availalbu.e at this time. We may
put a very loose upper bound on (1.8.5) by interchanfing the integral
and summation
signs, so that the power function may be written as
Xo
( 1.8. 6) 1-_/ - l-e-""2 _ 7n - 1
o
This is justified by the uniform convergence of the series
00
2::
'A.j
j=l (jl)
That
I5U
'A. j
ch convergence hold':: is obvious if we note that
'A. j co 'A. j
2::
<
, 2:: j=1 (j ! ) 2 - j! j=l j t
co
=e
A
'
sn that Weierstrass's lVI-test is satisfied.
x
Since each of the integrals
j'
u
o
e
- ~
j
u du is positive, it is evi-
o
dent that the power function is bounded above by
-e
2
1
22
For j a positive integer, the integral may be integrated by
parts repeatedly, so that
(1.8.8)
~mp10ying
this in (1.8.6), the power function becomes
X
-
( 1.R.9) 1- ( l-e
~~
A
o
~
1
~)n-
e
•
-.".
00
t:.
~
'\
Xo .
- -2 J
r 1-e
J
~
1\
j=o jt2 J -
1
X o j-i 7
Z (j-i)r(
~)
-'
i=o
shall now demonstrate that Walker's test is unbiassed, i.e.,
that the power is greater than the probability of the first kind of
error, a.
We thus wish to show that
(1.8.10)
A
Xo
-2
l-(l-e
n-1
)
- e
-? 00
») /_
~
_:-:I-a
j=o j!2 J -
This simplifies to
X
e
X+x
o
-2
<e
-r-o
00
j
~
Y,
j=o i=o
. '2 j (·J-~.
')'
J.
or
()
~
e
••• +
,
23
Since x o ' X ~ 0, it is readily seen that the inequality holds.
1~alker's test is unbiassed.
Hence,
CHAPTER II
A
•
TEST FOR SIGNIFICANCE OF PERIODIC COMPONENTS
IN BIVARIATE TIME SERIES
2.1.
Introduction
We shall now consider the case of two parallel series of ob-
servations ordered in time, and each with a single sinusoidal component.
In the case of the same model period in each series, an
exact test for the significance of a trial period is obtained.
This
test is based upon the determinant of the matrix of intensities and
cross-intensities of the two series.
The motivation for such a model and test might for example
be provided from astronomy by a series of simultaneous observations
on the brightness and radial velocity of a variable star.
2.2
The
~ivariate
The
~eneral
Model
bivariate model used for the results of this
chanter is
(2.2.1)
t runs through the integers 1,2, •.. , N, N being restricted to be
odd to take advantar'e of certain properties of sums of intensities
of a complete
~ourier se~uencee
k and m are integers from the set
1, ?, .ee, n, where, as in the univariate model, n will be used
N-l
henceforth to denote
~.
The test of significance developed in this
chapter will be based upon k
=m to
avoid the introduction of nuisance
parameters, although the effects of different model periods, as well
as ones not of the integral sub-multiple form will be considered in
~ 2.3 and Chapter III.
The following assumntions on the errors will be made:
2
E(er,e s ) = 0,2_ 51'S ,E(6,0)
r s = cr 20rs
,
,
where
°rs is the Kronecker 0,
s, and the value 0
for
r
assuming the value 1 for r eaual to
and s
A more general ~heme
different.
of correlation will be assumed in Chapter III to determine its effect
upon certain expected values.
We assume that
p
2
< 1 D
this is
necessary and sufficient fur the covariance matrix to be positive
definite, and therefore for the consistency of the assumptions (2.2.2).
For the test of significance obtained in this chapter, it is
further necessary to assume that the (e , at) have a bivariate normal
t
distribution with means and covariance matrix determined by (2.2.2).
2.3.
Estimation of the Parameters
We shall begin this section with a theorem on multivariate
least squares.
For the case of eaual periods in each of the series of
(?2.1), this theorem states that the estimates of C , A , B , and
1 1 1
e
26
C2 , A2, B are the same as those given by (1.3.5) for a single series.
2
First define the following matrices for a general multivariate
regression model:
Matrix of Predictors:
r~ll
xlp
I
(2.3.1)
X(N x p)
-
I ··
·
I
P~edictand
(2.3.2)
yeN
I
~l
=
.
,
I
~N
Matrix:
Y1l
-e
,
I
xJ
I
lXNl
~ultiple
1
x q)
.··
,
Yl q
.xl
=
=
YNl
.··
YNq
,
Y
-N J
Regression Coefficients:
r~11
(2.).3)
13(0 x q)
~lq
=:
-
=
~pl
.··
B )
(~l' ••• , ....q
~pq
Frrors:
(2.).4)
b(N x q)
=:
~1
==
e
-e
I
e1q
ell
N1
.•.
I
eNq
e
. -N
27
The elements
of
e
ij
~
are random variables with
E(e ij ) = 0, (i = 1, ... , N; j = 1, ... , q)
E( M ') = /\ (q x q) ,non-sinpular •
For each value of i, (i = 1, "', N), the elements
I
vector
e.
-J.
e ij of the row
have the multivariate normal distribution
N L-2 I (q x 1),.1\ (qx q)
_7.
These N distributions are independent,
so that E( eI'm ean ) = 6rs p. mn (1m(Jn ,where 6 rs
is the Kroneoker
symbol described in Section 2.2.
We shall now prove the following theorem:
Theorem:
-
~
For the multiple predictand regression model
yeN x q) = X(N x p) B(p x q)
(2.3.6)
the pq estimates
+ ~(N
of the elements of B
time by the least
squa~es
x q) ,
are given p
at a
solutions for the q predictands taken
separately.
Proof:
From the conditions (2.3.,), the likelihood of the matrix
may be written as
L( tl)
= const.
1
Nfl
exp /- - - ~!t L\. - !t 7
2 t=l
.-
~
28
= const.
exp
N
~ E
q"\ kf
l:
etketd
r- - -12 t=l
k=l f<tl
/(
~
~
Taking logarithms and substituting far the
e
t
t
_7 •
from the multiple
predict and model (2.3.6), t(~) is a monotonic decreasing fUnction
of the expo:1ent:
(2.3.8)
Differentiation with respect to the
-e
~jk'
with the resulting
derivatives set equal to 0, yields
Next interchange the summations over t and
f:
If we adopt the notation of single-predict and least squares theory,
where
,
a == X (p x N) X ( N x p)
=.L
N
l:
t=l
xti x t ·
J
_7,
a
r/
°
29
x
=-
f
(p x N) leN x 1)
we finally obtain
In this set of q equations
since the covariance
matrix~.
= 0, (f
==
=>
~f
- ~K
• Then,
Yf
whose only solution is given
1, ••. , q), or,
-e
1\
~{( == a
Thus, the
Yf
is non-singular, we have a set of
homogeneous linear equations in the
by Yf
let
esti~ates
-1
E.f!
of the elements of B are given by the q matrix
eouations, each containing p estimates, as was to be shown.
C , ~,
l
B , C , A , B , for equal model -periods are the same as those given
From this theorem it follows that the estimates of
I
2
2
2
by Chapter I, (1.3.,), or in the notation of the bivariate model,
/\
1
Cl = 'FJ'
N
1: u
t=l t
,
,
A
2 N
? k
"
2
+
A=>- 1: v cos ~ t ,
2
N t=-l t
n
-e
B2 = 'IV
N
1: vt sin
t=l
2 k
t
30
For the case of different model periods, the X matrix (?3.1)
is not the same for each series, so that the theorem does not apply.
The resulting estimates are rather interesting, and will now be obtained.
We may write the likelihood (2.3.7) as
L
= const. exp _ - -21
N I
Z e
Z-1~t _ 7
t:l -t
"l
~t
r
,
where
u t - °1 -
~
cos
- Bl sin 1 1 t
j-
,
vt - C2 - A2 cos A2t - B2 sin A t
2
-e
Taking logarithms and expanding the summation over r and s, the likelihood (2.3.21) is proportional to
(::'.3.16)
N
-2 ~lcr2 t~iUt-CI-AlcosAlt-B1SinAlt)(Vt-C2-A2cOsA2t-B2SinA2t).
Differentiating with respect to
C , C , AI' B , A , and B ,
1
2
l
2
2
using the sums (A.2.1) - (A.2.6) of the Appendix, and setting the
-e
31
resulting expressions equal to 0, the estimates are:
A
1
N
C1 =-.Eu
N t=l t
•
•
P
-e
0'2
0'1
?
-N
N
~
U cosA t
2
t:1 t
It is especially noteworthy that the assumption of unequal
periods has introduced nui.sance parameters
the estimates.
0'2
p-
0'1
and
0'1
p-
0'2
into
For this reason, the test proposed in this chapter
will be concerned only with equal model periods.
Some pronerties of
the more general model will be treated in Chapter III.
2.4.
1.
Bivari~
rest
.f2E ~
Simple Sinusoid
From the complete sample Fourier sequence of estimates given
by (2.3.18)-(2.3.2 0 ), form th8 2n intensities
-e
32
h2
1'2
Ilk = Alk + Blk
12k
,
1'2
= -"2
A2k + B2k
and the n "cross intensities II
1\
I l2k
1\
1\
"
= ~kA2k+BlkB2k
These form a matrix whose determinant is
It has long been known ~3,p.71_r, from Parseval's Theorem
A
A
A
"
that terms of the form AlkA2k+BlkB2k measure the covariance between
the two series of the model (2.2.1).
In fact, under this model, and
the usual error scheme (2.2.4),
where
k
6kmis the usual Kronecker symbol, assuming the value 1 for
= m,
that is, for trial and model periods equal and corresponding
2nk
to arguments of the form Nt, and 0 for other trial periods in the
orthogonal set.
Thus, since the matrix of Sk is a covariance
matrix associated with the contribution of the kth trial period, we
-e
shall take
~le
determinant Sk
univariate
peri~dogram
as the bivariate analogue of the
intensity.
In accordance with the univariate
procedure, we now wish the distribution of the greatest of the
l!
33,
intensity determinants, or better, the distribution of some "s tudentized" function of them.
To obtain this distribution, consider Wilks t
tion of the determinant
~
L-18_7
distribu-
of the matrix of product-sums of bi-
variate normal variates:
,
F( ~)d~ ::
~
2: 0 ,
where
1
A=
-e
Wilks' original notation in the 1932 paper involved the sample size
N, which has been replaced here by the number of degrees of freedom,
n
=:
N-l. Wilks t A also contains the factor
of product-sums.
determinant of mean squares,
£4
W,
since his
~
is a
while our intensity determinant is one
For the intensity determinant, A thus becomes
1
Var(A 1 )Var( ~)(l- p 2 LT.
Henceforth,
population covariance determinant
Z will denote the
22
o (1-p )
40 12
2
2
N
• For n
=
2 degrees
of freadom, the distributiop of a sample intensity determinant for no
periodic component and errors satisfying the conditions (2.?2) is
(2.4.,)
G(S
)dS ...
34
If we let
S
= 2Z l /?Sl/2, the distribution of s is
S
f(s)ds
or merely a
X
2,i
= 2~ e -?E ds
,
distribution with 2 degrees of freedom.
Hence
Fisher's g becomes
,
and has the cumulative distribution (1.5.2) of Chapter I.
Thus,
using the intensity determinant as a test statistic, we are provided with an exact significanoe test for a periodio component in two
parallel time series.
The sum
~
k=l
A
n
(2.4.8)
~
k=l
of the denominc.;t or of g can be written as
N
N
Z u.... sin
r s
s= 1
r-s)
?n:k_(N
_I
Because the positive souare root is taken in each case, no exact expression is available at this time for the sum, nor has any close
praotioal upper bound been obtained for it.
Examination for small
values of N has indicated that its value may involve irregular
sums of cross products of the u,
r vs , whose terms are multiplied
by the often irrational value of the sine factor involved.
Even for
positive observations, it does not appear that relations similar to
-e
35
those for the sums of univariate intensities exist.
~ I S~/2 I in
We may obtain a lower bound on the sum
the
k=l
following manner.
Disregard the absolute value signs, and write the
sum as
4
~
N
n
E
k=l
N
~
r=l
N
~
u v
5=1 r s
sin
2n(r-s)k
N
Using the fundamental sum (A.lo5) of the Appendix, with the appropriate values of a, b, 9, and p inserted, this becomes
NN
4
Z Z u v
~
N r s
-e
r
sin(N:l)?n~r-s) _ sin(N~l)?n(r-s)
r
s -
2N
2N_
(
2(1-cos 2n r-s
rls
)
N
)
sin 2n(r-s)
N
+-------2(1-c05 2n(r-s) )
N
With the aid of some trigonQmetric identities, this simplifies
further to
4
1\T
N
r s
. nCN-s) Lr cos n(r-s)
N
-
N
Sln
-2~I,UV
r s
r/=s
2 N N
=-;:;Z~uv
{'
r s N r s
r
-e
ria
cos
nC r-s)
N
sin
-
()r-s
nCr-s)
N
-
.
7
36
Since it is well known that
we have the lower bound
N N
2
6
r
Y
N
r
()r-8
cos •n(r-s) - .N n(r-s)
sm
N
r
6 u v
s rs-
Is
n
_7
<
r,
- k:l
1
I~l
Although Chapter III is devoted completely to the expectation
of the intensity determinant under a variety of assumptions, we shaQl
give some basic results here as a justification of its use as the
test statistic.
Under the basic bivariate model (2.2.1), with equal
neriods, the expectation for model period exactly equal to the trial
period can be found from the general expression (3.2.3) of Chapter
ITI to be
~ A
A
A
2
E(~~?-A2Bl) :
(
2 2
A1B2-A 2B1 ) + N
2 ? 2 222
Lql(A
1+B 1 )+(j2(A 2+B 2 )_7
~
2 2
4 r'\(jl(j2
N(A1A2+B1B2 ) +
80"1°2
N2
2
(1- p )
tTnder the null hypothesis of no periodic component, that is, far
1 , Bl ,
simply
4
A?,
B2 .all enual to
zer~
the determinant's expectation is
,
37
as may
he
checked by direct integration for the first moment o:f the
dtstribution of
Sk
in
pothesis t8 best. found
(2.4.5). The variance under the null hy-
ovreovurs.e
to
the 01 st.r1bllt.ioll flln~ti(lll
(2.4.,)
as well, and is
From the first expectation above, one serious objection to the
use of the intensity determinant as a test statistic is immediately
evident.
Unlike the univariate intens ity, the part of the expectation
free of terms in N-
l
can vanish if
(2.4.16)
Further, the complete absence of a periodic component in one series
could obscure the presence of a real one in the second series.
?,.
Other Possible Statistics
The deficiency mentioned in the last paragraph of the previous
Section might possibly be alleviated by considering some functions
of the roots of the determinantal eouation
Isk
where
-
2~
T
sk* is an indeoendent matrix of the error sums of squcres and
cross products obtained aft·er the contribution due to a given kth
-e
sinusoid has been removed.
It is
~nown
that
Sk
and
*
sk
are inde-
38
pendently distributed.
miE'ht be used.
Or, the roots of the equation
there is a correspondence between these
\~tually,
N( 5 +8 + •.• 8
) so that the roots
sk* = ~
n-Sk'
1 2
Ask of the first may be expressed as a function of those (If the
two enuations, since
second by means of the
One
~ossible
ra~atiDn
statistic might be the
~
of the roots of (2.5.1).
The distribution of this sum was obtained by Hotelling
£9_7
as an
incomplete Beta integral, and the sum was shown to be free from the
effect of collinearity that causes the intensity determinant to be
zero when a real period may be present.
However, the distribution
has not been obtained, and since Sk* changes
each intensity determinant
for
Sk' the problem is further complicated.
Because the determinant of a sum of matrices is not necessarily equal
to the sum of the determinants, the use of Cochran's transformations
L-?_7
to derive Fisher's distribution is ~rohibited.
The Eroduct
of the roots of the equation (2.5.1) might con-
ceivably form a test criterion.
determinant of 8
Since this product is equal to the
divided by that of
s*, it is quite similar to the
mere i ltensity determinant, and has been shown by Wilks /,-18
have the same regrettable
~roperty
_7
to
of vanishing whenever all of the
39
observations lie on a straight line.
Again, the same problems men-
tioned in the previous par3graph would arise in attempting to find
the distribution of the maximum product of roots, although unlike
the sum, the distribution of a single product is an elementary closed
function readily transformable into the variance-ratio distribution.
The
t~
for general use.
of the intensity matrix
~ven
Sk seems tQ be ruled out
if both series were measured in the same units,
it seems likely that the variances and covariance would enter as
nuisance parameters in such a way as to resist "studentisation" by
dividing by, say, the sum of the traces for the complete Fourier sequence.
Many other possible statistics might be investigated, notably
the greatest or least roots of the second equation,
Sk)
1=
O.
t S-Q(Sl+
This eauation has a slightly greater intuitive appeal
over (2.5.1), since it is so formally analogous to Fisher's
~s
•••
g ratio.
for the sum of the roots, the distribution of the greatest root
involves incomplete integrals.
In closing this discussion of other statistics than the intensity determinant, it would be interesting to note a property of that
quantity that makes the need of alternative statistics imperat ive if
one wishes to extend these methods beyond the bivariate case.
general p-series problem, the intensity matrix becone s
For the
•
2
• A +B
p p
J
With a little insight. this may be £actorized into the product of a (p x 2) matrix and its transpose:
Hence, the rank of any intensity matrix is at most 2, so that the
use of its determinant as a test statistic is meaningless.
CHAPT'I'1l III
CERTAIN EXPECT lI.TIONS OF THE. TJ\TTENSITY
DETERMINANT
3.1.
Introduction
It is the purpose of this chapter to obtain the expectation
of the intensity determinant under the alternative hypothesis of
Chapter II, (2.2.1), and f~r different estimates of the AI' A2 , Bl ,
B • It has been necessary to restrict nearly all of the results to
2
the point at which the trial and model periods are equal, and then to
r~8trict
...,
the angular argument to be of the form
(N-l)/?
3.2.
2nk!N, k
= 1,
2,
Otherwise, the trigonometric sums would be retractable •
The Intensity Determinant under the Model of Chapter II
We assume that the parallel series may be represented by the
basic model of Chapter II, that is,
(J.2.1)
t , and at' except
are assumed to have been made
At this point no assumptions are made on AI' A?,
that the estimates of
8
AI' A , Bl , B
2
2
for Al = A2, with the oeriod a sub-multiple of the number of observations in the series. Under this representation of u ' v ' the int
t
tensity determinant may be expanded as
~ f(Z(AOl+AlCOs"ltcoset+"f3i5inAl.t
cos Qt
oJ-
et cos Qt»
x
X(Z( 802+A2cos''2t sin et +B sin A t sin Ot + 0t sin et)
2
2
2
The summations here, and elsewhere in this chapter, are allover the
values 1,
o •• ,
N
of t.
For the trial argument 9 of the form ?nk/N,
the terms involving
Aol ' Ao2 ' vanish, and for other values of 8, they
are of order small enough to be neglected. If we make that elimination,
distribute the summation operators, and carry out multiplication between
the quantities in parentheses. this may be written as
~ L:f~A2(zcosetcosAlt)(ZcosA2tsinQt)+~B2(ZcosAltcOSQt)(ZSin"2tsinet)
+BIA2(zsin"ltcosQt)(ZcosA2tsinet)+BlB2(zsinAltcoset)(zsin"2tsinGt)
-A?Al(ZCOSQtcos"2t)(l:Cos"-ItsinQt)-A2Bl(ZCOsA2tcOsQt) (l:sinA1tsinQt)
-B2Al(ZsinA2tcosgt)(r.~cos~tsinOt)-BlB2(ZS1n"2tcosOt)(Zsin~tsingt~
+
{(~l:CosgtCQaAlt+BlzSin"ltcOSQt)ZOtSinQt
+(A2l:cos"2tsinet+B2l:Sin).2tsinOt)Z~tCosQt
,-e
43
-(A2ZCosetcosA2t+B2i.sinA2tcos9t)ZetSinQ~
_(AlZcosAltsin9t+Bli.SinAlts1net)Z5tCOS9t~
2
We shall denote the three braced
terms
as I, II, and III.
The expectation of the intensity determinant under the assumptions (2.?2) of uncorrelated residuals of zero expectation may now
be obtained.
Expanding the trinomial in I, II, III, one obtains the
individual expectations:
,
(3 . 2 . 4)
= 0,
E(I.II)
since E~~t
= E~u =
t
° .'
E(II.III) = 0, since all expectations are either f.Llll,1J. 21 or
N
E(I.III)
= 0; since Z sin et cos gt = 0
t=l
ElII
2
=
(1'120'2(1- ()2)N2
2
2
2
EII
2
Na2
=>
2
L { ~ZcosetcosAlt+Blzsin~tcosgt
44
If we now let
help of
A = A ~
2
l
e
=
2~~, for m .,. 1.. •.... n with the
.\pPBndix Sums (A.2.1)-(A.2,6), EI 2 and EII 2 become.
and
respectively. Hence the expectation of the intensity determinant as
given previously in Chapter II is
In the manner of Chapter I, Section 1.7, it will be instructive
to consider the effect of the model period exactly half-way between
two trial periods of the integral sub-multiple form N/k.
~!e
shall
45
omit all terms in N- 1 and N-~· considering only the terms I of (3.2.3)•
. Ustn,:r the lIppendix surns (A.$.1)-(A.5.8). I becomes
•
This is a oonsiderable proportionate reductton in the expectation.
It
is the square of the proportionate reduction in the univariate oase ef
Section 1.7, and may indicate that the ability of the intensity determinant to deteot a non-integral Bub-multiple period may be quite low.
1h!
3.3,
Expeotation~ ~
MOdel Periods
~
Intensity Determinant
~
Ditterent
Types of Estimates
It is the purpose of this seotion to oompare the expeotations .
ot the intensity determinant under twelve combinations ot hypotheses,
trial periods, and estimates ot the parameters
A, B , B ,
2 l
2
twelve situations oan be outlined in the follOWing manner.
I,
~.
These
A Real Sinu80idal Component with the Same Period tor Both
Series.
A. Conventional Estimates (2.).14)
46
Equal Trial Periods
2.
B.
II.
Di~ferent
Trial
Estimates (?3.17)
Period~
B~sed
on Different Model Periods
1.
Equal Trial Periods
2.
Different 'fl.·tal Periods
A Real Sinusoidal Component with a Different Period in Each
Series
A.
B.
III.
Conventional Estimates (2.3.14 )
1.
Equal Trial Periods
2.
Different Trial Periods
Estimates (2.3.17) Based on Different Model Periods
1.
Equal Trial Periods
2.
Different Trial Periods
No Periodic Component in Either Series
A.
B.
Conventional Estimates (2.3.14)
1.
Equal Trial Periods
2.
Different Trial Periods
Estimates (2.3.17) Based on Different Model Periods
1.
Eoual Trial Periods
2.
Different Trial Periods
Throughout all expectations involving the estimates (2.3.17) it has
been necessary to assume the nuisance parameters P~2
as
0'1
known.
We shall now obtain in detail ten of these expectations.
The
47
cases of (I-A-l) and (III-A-l) of the outline have already been given
in (3.?S) and (2.4.14).
the point at which the
All ~f these expectations are evaluated at
tl~al
period coincides with one or both of the
model periods.
(a)
Equal model periods, conventional estimates, different
periods:
Using (A.2.S) in the basic expression (3.2.2),
(3.3. 1 )
The effect of correlation has been lost, as well as all terms free of
the number of observations, N.
(b)
Equal model periods,eoual trial periods, estimates (2.3.17):
Using the sums (A.2.l.)-(A.2.6), the estimates (2.].17) become
A1=?N
-1 AIN
-1 PO'l A2N
(~+ ZetcosQt)-2N
0'2 (~ + ~5tcos9t)
PO'l
-1
P 0'1
~(Al- -0 A2 )Y 2N z(e t - 0'2 0t)cos Qt ,
2
-1 BIN
-1 PO'I B 2N
Bl =2N (~+ ~etsinQt)-2N
0'2 (~ +~otsin 8t)
pO'l.
=(B1-
1
0'2~)+2N- Z(e t -
p 0'1
0'2 8t )sin Gt
,
48
-1 A.?N
1 P0"2 AlN
A2=2N (-,- + EO t COB9t)-?N- ---(--0"1 2
:ll-(A'C...
B =?N
2
+
Zetcoset)
P0"2
1
P0"2
0"1 ~)+2N- Z(Ot-St O'l)cOS 9t ,
-1 B2N
-1 P0"2 BIN
( - + l:0ts1n et )-2N
- " ( _ . + 1:6 Sinet)
t
2
0'1 2
=(R 2 -
P0"2
1
P0'2
0'1B1 )+2N- Z(5 t - O'let)sin at
fhus,
-1
PO"l
P ()2
+?N (A - ---ft )l:(Ot- -St)sin
1
0'2 2.
0"1
et
(c)
BQual model periods, e"ua1 trial periods, Estimates (2.3.17):
In the same manner as the previous expectation,
er
P '1
A=A--A
A
1
1
E1
= (B 1 -
/\
0'2
-1
+2N-
2
pO'l
0'2
--- B ) + 2N2
,
1
,
Z( e t
,
E
j-'AiJ2-A2Bl_72
= Ej
_
-1
2H
l.
-"l·N
-2
P,(11
(A1-
1:( e t
0'2 A2 ) Z( 0t- et
P(12
(Jl)Sin e2t
P (J2
(11 et ) cos
2
e2t 7
50
(d)
Diff~rent
rondel
Deriods~
Anual trial periods, Estimates
(::>.3.14):
Taking the expectation ~ the point
G1
=
~1 8 2
'
(3.3.6)
(e)
Di.fferent model neriods, different trial periods, Estimates
(2.3.14):
Evaluating the expectation at the point
EL-A 1B2 -'A;B 1 _7?
G1
= ~1' Q?
:=
1J'2'
51
The effect of correlation has been lost in this choice of trial and
model periods.
(f)
--
Different model periods, equal trial periods, Estimates
(2.3.17):
Expecting at the point
j"
~
=
Q '~l
~l + 2N-
1
1 ~2
with estimates given by
pal
Z(e t - ~ 8t ) cos Gt
?N- l Z(6 t
pa 2
,
.
---a., et)s~n 9t
,
.1.
2
0"2
. -1 2 2 2 -
p ~ + 2J\T
a
l
a (A +R
2 l l
)L
4
2
(l-p )+2(1~p)_7
52
( g)
Diff~rent
model periods, different trial periods, Estimates
(?3.17):
Let the
the point 91
est imat es
:=.
mod~l
1J.
Peri-ods be 1J.1 1J.2'
1 , U2-:::t IJ.?
lie wiEh the expectation at
Using the ne-aessar.v Appendix sums, the
AI' A2 , B1 :; ~2 become
-e
Inserting these in the intensity determinant and expecting, after
some simplifioation we have
53
(h)
Null hypothesis of no periodio component in either series,
different trial periods, Estimates (~.3.l4);
The estimates become simply
A..A -- 2N-1 Ll8tcos
...
C! t
1
l
(3.3.12)
0_
'
Hence,
(1.3.13)
The use of different trial periods and conventional estimates under
the null hypothesis has removed the effect of inter-series correlation
from this expectation.
(i)
Null hypothesis of no periodic component in either series,
equal trial periods, Estimat as (2.3.17):
For
Q
the common trial oeriod, the expected value of the
tensity determinant under the above conditions is
in~
54
The usual expectation under the null hypothesis is decreased by the
factor (1- p'2)2.
(j)
Null hypothesis of no periodic component in either series,
different trial periods, ~stimates (?).17):
Write the estimates (2.3.25)-(2.3.28) as
lations:
Ex x='O
23
,
,
55
hold frcm the c:I,-thog~nal1.ty of the trigonometric
HWTlS
involved we
have: .
,
Thus, ur..d er the condit.ions imposed above,
The usual null hypothesis expectation has been decreased by the factor
(1-
-e
r}) .
56
APPENDIX
CTiRT AIN TRI GONOME'ffiIG SUMS
1.1.
Introduction
Throughout this thesis it has been necessary to evaluate many
tr1p.onometric sums of the type
N
t
r
E 0 cos
t==l
where
r
or
s
21t1lI
N
t
.s 2nn t
sJ.n N
is usually small, if not zero, and
p
is usually 1.
These results have long been known to workers in applied mathematics,
but a detailed table for certain values of r,
s,
m, and
n, is
aiven in this Apoendix because of their freauent use in Chapters II
and III.
The sum
b
E e
(A.l.l)
1St
pt
t=a
is simoly a geometric orogression, and may be written as
Pb+ae is(b+a) - p ae iSa
(A.l.2)
pe
ie
-1
Multiplying the numerator and denominator by the complex conjup-ate
of the latter, this becomes
p
b+l+a
e
ie(b+a-l)
-
iQ(a~l) a iea
eb+aeie(b+a) - pa+l
. e
.
+p e·
2
p - pe
ie
- pe
-i9
+ 1
57
Taki.tlg re.al, and
(
im8~i.nary
parts separately, we have
~.1.4)
b
b+l+a .
."
Z p t~~'t=- .E.
cos(b+a-l)gt=a
Q
b+a
(
a+1'
a
cos b+a)Q- p
cos( B-])g+ P cosQa •
2
p -2 p cos 9 + 1
(A.L5)
b
t
p b+l+8 sin (b+a_l)Q_ p b+a sin ( b+a)- p 8+lsin ( a-1)0+ p BeinQa
t~a p sinQt=
2
p
-? p cosQ + 1
These general expressions form the basis for the following table of
summations.
--
A.2.
A Table of Trigonometric Sums
~_2fik
e---r'
( a)
N
(P.2.l)
a""
1
,
b
""
N
,
k = 1,2, •.. N-l;
cos Elt = 0
~
t=l
N
sin Elt
Z;
=.
0
t=l
N
Z sin tg cos tEl ""
t=l
N
~
2
lIT
sin te = -
2
t=l
(A.2.5)
·e
(~.2.6)
N
L: cos
t=l
N
°
2
to
1\1
=> . -
?
2nm
2nn
Z sin ~t sin ---N--- t
t=l
= 0, mIn
58
(b)
(A.3.1)
9
= 2~k
N
~ sin(t-l)9
t=2
, a = 2. b = N. p= 1; k=l.?, ... , N-lg
sin
tQ
N
=-
N
(A.3.2)
cos 9
2
E sin(t-l)e cos tQ
:::0
t=?
-~ sin e
N
2
E cos (t-l)Q
t=2
(A.3.4)
N
E cos(t-l)e cos tQ
t=?
N
2
Z sin (t-l)Q
t=2
(A.3.6)
N
~ cos(t-l)Q
t=2
liT
~ cos( t-l)e
t=?
(A.3.8)
N
E
t=>?
(c)
( .A .4 .1)
Q
N
~
t=?
sin(t-l)e
:::0
2nk
N'
=
(N-?) cos 9
2
flY
= -2
sin(t-l)Q = 0
sin te
=
(N 2)
-isin Q
=0
b = N, p a real number, k
:::0
1,2, •. "., N..l:
t
pN+a+1 cos( a-1)Q _
p N+acos ae
peas tg = .;..,----r§2r--.:.....;.;..~--;,."...--.,;;~.....::.;;;.
p
... 2 p cos Q + 1
59
-
For a
(
a+l
,
a
cose a-1)0+ p cos a9
p -2 pcos 9 + 1
p
= 1,
N t
~.4.?)
~
p
cos tQ ::
p
N+?
2
p
t=l
N
t
Z p sin t9
t=?
co
- p
P N+l+a
2
p
2
- p
N+l
cos Q + P cos Q
,
- 2 p cos 9 + 1
. (1)~
a- ~ - p.N+2 sin aO
-2p cos Q + 1
s~n
a+1
sine a-1)e + p a sin aO
- p
p
:.'
-? p cos
e
+ 1
For a ;: 1,
N t
Z p sin
te = -
t=l
(d
)
N+l
p
nk
e -_ N
N
Z sin
t=l
p
,A. ::
te ::
?
?n( 1
N
k+,?),
sin e
I-cos 9
sin e + p sin e
- 2 p cos Q + 1
a =1, b =N, p =<1, k :: 1, 2, ... , N-1:
, k odd
:: 0,
k even
= -1,
k odd
o.
k even
N
(1.5.2)
Z cos t9
t=l
=:
N
(A.5.)
Z cos t
t=l
=-1
60
(A.'.4)
N
~
t=l
cos t(k+2Q)
~
-1
N
r. cos tl cos 2tQ
(A.,.,)
t=l
•
=~
N
E
t=?
sin tA sin 2tQ
== 0
N
2: sin tA cos 2tO ::
t=l
1:
•
sin n(4k+1)
2-
n
·i
j-
+ sin
1-cos n(4k+1)
N
!
N "";;}
1-cos N
n
r 4k+i4k+2 7
n-
~J
"
E sin 2t9 cos tA =
t=l
1
i n(4k+l)
s n
N
'2 £ ----.;.-1
-cos
n(4k+l)
N
sin n
N
I-cos
n
N
-N4k
nfik+!""
( ~)
..,
Q
~
= N'
?nk '=
n(4k+1)
I\.
2N
, a=1 , ~LJ""On,
N
(A.6.1)
Z sin tA
t=1
N
(A.6.?)
E cos tA
t=1
= 1:
2
+
= -1-2 +
sin A
?(I-cosA)
1 k
p "",
~
sin A
2(I-c08 A)
~
= 0 , 1 ,"', N-1 :
1+ 2N
2 (4k+l)
1:
2
+
?N
(4k+1)
n
in n(8k+1)
N I s i n 11N
s
( A.6 .3) Z sin tA cos tQ = - r l +
+ _ _--.,;?..;.N:..-,__
t=l
22(l-cos-~) ?(l-cos
7
nCB:;l»-
61
.! + ! /?
•
(A.6.4)
11 -
8k+2
-gm
J
~~ ~
11
N
l cos t\ cos te ~
t=l
r
1
N
-w+~
11 -
.......
.... ...
~
(A.6.5)
N I l sin
sin t'A. sin tQ ". "2
+
r--
Z
t=l
-
.....
•
1t
1
ft(8~;1»
+
"2 -
sin
~
J
2(1-0082N)
l:!
11
n( 8k+1)
1 si n
2N
1
Z cos t\ sin tQ = _ / - v ~ j--;;2 - 2 2(1 -cos n(tiK+l)
c
t=:l
2N
J\T
(A.6.6)
7~-N
1t(8~Nl)
2(1-cos
2
8k
nN 8k+l
.'
8k+2"
~
OK+,L -
1
_ 1
- - (8k+l)
N
Bk
:.:.
11
N
11
It is interesting to note that the last four sums are all
a'Poroximately equal to Nn-1 in absolute value.
k
h~s
'!he range of values of
been"chossn to only aupply the needed sums for Section 1.7.
6?
BIBLIOGRAPHY
...
Bartlett, M. S., An Introduction to Stochastic Processes,
Cambridge, Cambridge University Press, 1955 .
•
Cochran 9 liITilliam G., "The Distributi.on of the Lar~est of a
Set of Estimated Variances as a Fraction of Their Total, I(
A~nals of Eup.enics, Vol XI(1941), 47-52.
Davis, H. T., The Analysis of Bconomic
ton, Principia Press, 1941.
~
Series, Blooming-
Fisher, Ronald A., "The General Sampling Distribution of the
Multiple Correlation Coefficient. II Proceedings of the
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Proceedings of the Royal Society of London, Vol. 125 (1929),
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--
Fisher, R. A., liThe Sampling Distribution of Some Statistics
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-Fisher, R. A., "On the Similarity of the Distributtons found
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stevens r Problem in Geometrical Probability," Annals of
Eugenics, Vol. X (1940), 14-17.
Hartley, H. 0., IITests of Significance in Harmonic Analysis,"
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Hotelling, Harold, IIA Generalized T test and J1easure of Multivariate Dispersion," Proceedings of the Second Berkeley
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-
Nanda, D. i\T., "Distribution of a Root of a Determinantal Equation," Annals of Mathematical Statistics, Vol. XIX (1948),
340-350.
/-117
-
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--
Pillai, K.C.S., On Some Distribution Problems in Multivariate
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L-l?~ Rao, C.
R., Advanced Statistical Methods in Biometric
New York, Wiley, 1952.
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63
Schuster, Sir .Arthur, nOn the Investigation of Hidden Periodicities with ~pDlication to a Supposed 26 Day Period of
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•
starkey, Daisy M., "The Distribution of the Multiple Correlation Coefficient in Periodogram Analysis," Annals ££. Mathe~tical Statistics, Vol. X (1939), 321.
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Stumpff, K., "Ermittlung und Realitat von Periodizitaten,"
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,,,Talker, Gilbert, "On the Criterion for the Reality of Relationshios or Pe1 iodicities," Indian Meterological lI1emoirs,
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4
Whittle,
Pet~r,
Hypothesis Testing in Time Series Analysis.
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1ATilks, S. S., "Certain Generalizations in the Analysis of Variance, It Biomet:dka, XXIV (1932), 471-494.
.
1J1)"old, Herman, A Study in the Anal ysis of Stationary Time
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