STATISTICAL METHODS APPLIED TO PSD DATA : CHARACTERIZATION ON THE DETECTOR RESPONSE

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STATISTICAL METHODS APPLIED TO PSD DATA : CHARACTERIZATION ON THE DETECTOR

RESPONSE

A. Antoniadis, J. Berruyer, A. Filhol

To cite this version:

A. Antoniadis, J. Berruyer, A. Filhol. STATISTICAL METHODS APPLIED TO PSD DATA :

CHARACTERIZATION ON THE DETECTOR RESPONSE. Journal de Physique Colloques, 1986,

47 (C5), pp.C5-17-C5-25. �10.1051/jphyscol:1986503�. �jpa-00225820�

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STATISTICAL METHODS APPLIED TO PSD DATA : CHARACTERIZATION ON THE DETECTOR RESPONSE

A .

ANTONIADIS*,

J.

BERRUYER"

and

A. FILHOL""

' ~ e p a r t m e n t of Mathematics, University of California, CA-92717 Irvine, U.S.A.

Departement de Mathematiques, Universite d e St-Etienne, F-42024 St-Etienne Cedex, France

* * *

Institut Laue-Langevin, 156X, F-38042 Grenoble Cedex, France

Rdsumd

-

On considPre plusieurs procedures pour tester numbriquement les performances de ddtecteurs

1

gaz .% localisation spatiale (DLS; en anglais : PSD) bidimensionnelles, La reponse d'un tel DLS

B

un flux homogsne de neutrons diffu- sds est moddlis6e par un processus de Poisson et des tests et techniques sta- tistiques spscifiques sont ddvelopp6s de fafon 1 dsterminer les caractbristiques du DLS. Deux approches sont utilisdes dans ce but : la premisre fait appel 2 une approche non parametrique pour reconnaitre l'inhomogenbitd des donnbes ex- pdrimantales ; la seconde est une extension de la technique classique d'analyse de la variance .% deux facteurs 2 un schema poissonien plus general en utilisant le princi~e du maximum de vraisemblance. La stabilite temporelle des donnses du DLS est aussi testde au moyen d'un nouveau test de comparaison de deux Lchan- tillons pour des donndes poissoniennes.

Des exemples num6riques sont prQsentds 1 ~artir de donnbes synthdtiques ou rdelles et les performances des algorithmes statistiques sont discutees.

Abstract

-

Several procedures are considered for testing numerically the performances of some two-dimensional position sensitive gas detectors (PSD)

.

The response of a PSD to a homogeneous flux of scattered neutrons is modelled by a two-dimensional Poisson process and specific statistical tests and techniques are developed in order to determine the PSD's characteristics. Two approaches are used: the first one employs a non-parametric probability scheme for recognizing the inhomogeneity in the experimental data. -The second one extends the classical two-way analysis of variance to a more general Poisson scheme using the maximum likelihood principle. The temporal stability of PSD data is also tested by means of a new two-sample comparison test on Poisson data.

Numerical examples from simulated data are presented and the performances of the statistical algorithms are discussed.

I -

INTRODUCTION

The present study is mainly devoted to the statistical analysis of the characteristics of two-dimensional position sensitive detectors's (2D-PSD), a first step towards data reduction of diffraction spectra.

Most of the algorithms given below are specifically designed for PSD's with coding discretization within the detecting medium and more precisely for gas neutron PSD's.

In the latter the signal is recorded by a grid of parallel anodes orthogonal to a grid of parallel cathodes. Pixels are thus physically located at the crossing of anode and cathode wires.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986503

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C5-18 JOURNAL DE PHYSIQUE

Under s t a n d a r d experimental c o n d i t i o n s t h e r e s u l t of t h e measurements i s a s e t of i n t e g e r s n

.

l < i < n where n i , j denotes t h e count recorded i n t h e i , j t h p i x e l of

i , j 7 I<icm

a n n x m d e t e c t o r . These accumulated counts can be reasonably considered a s r e a l i - z a t i o n s of P o i s s o n - d i s t r i b u t e d random v a r i a b l e s ( r . v . ) Nij w i t h parameter Xij. I f t h e counting r a t e i s n o t too h i g h , dead-time e f f e c t s a r e n e g l i g i b l e and one can assume t h a t t h e random v a r i a b l e s Nij ( i = l ,

...,

n; j = l ,

...,

m) a r e s t o c h a s t i c a l l y independent.

For a w e l l designed gas PSD, an incoming neutron i s e q u a l l y l i k e l y t o be d e t e c t e d anywhere on t h e s e n s i t i v e a r e a ( i g n o r i n g edge e f f e c t s ) s i n c e t h e d e t e c t i n g medium (gas) i s homogeneous. I n o t h e r words, t h e d e t e c t o r counting e f f i c i e n c y i s homogeneous. However, t h e indivBdua1 p i x e l responses ( e f f e c t i v e c r o s s - s e c t i o n s ) a r e g e n e r a l l y d i f f e r e n t due t o imperfect wire s p a c i n g , unbalanced e l e c t r o n i c t u n i n g , e t c . .

.

Hence, f o r such PSD's, any l a c k i n homogeneity i n t h e o v e r a l l response appears b a s i - c a l l y rowwise o r columnwise.

I1

-

DATA CORRECTION FOR RESPONSE HETEROGENEITY

I n p r a c t i c e t h i s c o r r e c t i o n i s always n e c e s s a r y t o o b t a i n reasonably " f l a t " observed s p e c t r a from a homogeneous incoming background. The c o r ~ e c t i o n f a c t o r s a r e e s t i - mated from " c a l i b r a t i o n " s p e c t r a w i t h g l o b a l mean count N much l a r g e r than t h e maximum counts expected i n d i f f r a c t i o n s p e c t r a .

The c l a s s i c a l c o r r e c t i o n is t o m u l t i p l y t h e counting v a r i a b l e Nij of e a c h c e l l ( i , j ) by a c o r r e c t i o n f a c t o r a i j e s t i m a t e d from c a l r b r a t i o n measurements a s :

-

" i j

= i T ^,

N ! . = c l . 13 1 j N i j l j

This approach d e s t r o y s t h e Poisson n a t u r e of t h e recorded d a t a s i n c e t h e mean and t h e v a r i a n c e of t h e N!.'s a r e no Longer e q u a l , and t h i s may a f f e c t t h e behaviour of f u t u r e s t a t i s t i c a l t e B t s .

To p r e s e r v e a s much a s p o s s i b l e , t h e Poisson c h a r a c t e r i s t i c s of t h e d a t a we have developed an a l g o r i t h m which transforms t h e o r i g i n a l Poisson v a r i a t e s N i j i n t o a new s e t of random v a r i a b l e s N.'.=f3..Nij, u n c o r r e l a t e d , w i t h common v a r i a n c e and mean e q u a l t o t h e a r i t h m e t i c a v e r a i a o t J t h e X i i .

Such a t r a n s f o r m a t i o n r e l i e s on t h e following p r o p o s i t i o n , t h e proof of which can be found i n 131.

P r o p o s i t i o n : Let X I and X2 be two Poisson independently d i s t r i b u t e d random v a r i a b l e s w i t h parameters

X

and p r e s p e c t i v e l y . Then, t h e random v a r i a b l e s Y1 and Y 2 d e f i n e d by

a r e non-correlated and have t h e same mean and v a r i a n c e (X+u)/2.

With f o u r r . v . (Xl,X2,X3,X4) a two-step procedure i s necessary. The formula (11) a p p l i e d t o couples X , , X 3 and X2,X4 transforms them t o couples Y1,Y3 and Z2,Z4 r e s p e c t i v e l y and a g a i n transforms Y1,Z2 and Y3,Z4 t o f o u r non-correlated r . v . (U1, U2.U3,U4) with t h e same mean and v a r i a n c e . S i m i l a r l y a s e t of 2P r.v. (with p an i n t e g e r ) r e q u i r e s p i t e r a t i o n s .

To e x p l a i n t h e p r a c t i c a l use of t h i s procedure t o c o r r e c t a d i f f r a c t i o n spectrum f o r t h e d e t e c t o r response inhomogeneity l e t u s come back t o t h e c a s e of a two-cell de- t e c t o r . I f we assume t h a t t h e unequal e f f i c i e n c y of t h e c e l l s i s due t o unequal e f f e c t i v e c r o s s - s e c t i o n s , t h i s means t h a t one c e l l counts a p a r t of t h e i n c i d e n t p a r t i c l e s which, i n t h e c a s e of a homogeneous d e t e c t o r , would b e counted by t h e

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mi = Blml + B2m2 mi = B3ml + B4m2

The counts (n1,n2) of the calibration spectrum being large are good estimates of the parameters ( 1 , ~ ) of proportions

Bi

which, from (11) may be taken as:

Finally, one can remark that the a-cohhedon, when applied to the calibration spectra used for its determination leads to a non random spectrum while the B-cOJVLedOn as stated above preserves the randomness.

111 - TEMPORAL STABILITY OF THE DETECTOR RESPONSE.

Let us define the following experimental process:

(a) Calibration spectra recorded from a homogeneous background (b) aij or 6ij correction factors are estimated from (a)

(c) later (presumably after some diffraction experiments) new control spectra are recorded from homogeneous background.

In the case of

nehAive

tempokd n Z u b U y , the a or 6-correction applied to the control spectra would lead to hqmogeneous ones. Otherwise a row/column perturbation will be present. The aboolde n . t a b U q

06

kehponw for any future experiment made under identical conditions to the calibration step can be tested by another procedure. Such a procedure is based on an approximate two sample test for Poisson pro- cesses, developed in /3/ where the duration of the experiment needed to obtain a specified power is determined.

The subsequent tests are relevant to the relative stability case.

111.1

-

Tests without assumption of row/column heterogeneity

As defined in the introduction the observed countsnij(i=l,n; j=l,m) are realisations of a family

{ N ~ ~ )

of independent Poisson random variables. If the detector was homogeneous, these random variables would be identically distributed and the observed data would be a sample drawn from a Poisson distribution with parameter A. Now if X is a Poisson r.v.the mean of X, denoted by E(X), is equal to the variance of X, V(X) and the ratio

1

is equal to

1.

In our case the common mean and variance of

E -

the Nij's are unknown and have to be estimated by the sample mean and the sample variance of the observed counts,that is by:

which leads to the estimation statistic 7.

v

E

is her's

theorem /5/ states that if nm >

35

and

X

>

3

then the distribution of the random variable

A

E 2

can be adequately approximated by a chi-square

(X

) distribution with nm-1 degrees of freedom. These assumptions are satisfied in our case. Moreover, nm is usually big and the

X2

distribution with (nm-1) degrees is approximated by a normal distribution with mean

(m-1)

and variance 2(nm-1). Thus, we know that if the hypothesis of homogeneity Ho is true, the function T of the observations will be distributed

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PHYSIQUE

according t o a normal law whose parameters a r e known. L e t a be a l e v e l of s i g n i - f i c a n c e , t h a t i s t h e maximum p r o b a b i l i t y w i t h which we a r e w i l l i n g t o r e j e c t homo- g e n e i t y when a c t u a l l y t h e r e i s homogeneity;one can f i n d numbers c and d such t h a t t h e p r o b a b i l i t y t h a t T S c when Ho i s t r u e i s l e s s t h a n u/2 and t h e p r o b a b i l i t y t h a t T >, d i s l e s s t h a n 1312. Since a i s s m a l l , i t i s reasoned t h a t T h a s l i t t l e chance of t a k i n g v a l u e s o u t s i d e [ c , d ] when Ho i s t r u e . Thus i f one computes T, i f c 4 T S d t h e n one would n o t wish t o r e j e c t Ho. I n our c a s e t h e t e s t we use a t t h e a-nominal l e v e l i s ,

i f

1 v

> t 2 / 1 2 , r e j e c t homogeneity E

where t denotes t h e 1

- 5

p e r c e n t i l e of t h e s t a n d a r d normal d i s t r i b u t i o n .

u 2

It i s n a t u r a l t o r e q u i r e such a t e s t t o g i v e t h e maximum p r o b a b i l i t y of r e j e c t i n g Ho when Ho i s n o t t r u e ; t h i s i s g e n e r a l l y n o t t h e c a s e w i t h t h e t e s t (111.11, t h e s e t of a l t e r n a t i v e s b e i n g c e r t a i n l y t o o l a r g e .

Another non-parametric t e s t which we use i s t h e u s u a l chi-square t e s t of "goodness of f i t " . It i s known t h a t t h e l i m i t d i s t r i b u t i o n of t h e X2 s t a t i s t i c depends gene- r a l l y on t h e bounds of t h e i n t e r v a l c l a s s e s i n t o which d a t a a r e grouped. I n prac- t i c a l a p p l i c a t i o n s t h e s e c l a s s e s a r e chosen according t o t h e d a t a and a r e t h e r e f o r e random. For t h e PSD measurements we use a r u l e of grouping and t h e r e l a t e d l i m i t d i s t r i b u t i o n given i n 141. A d i s c u s s i o n of t h e performances of t h e s e t e s t s i s given i n t h e l a s t s e c t i o n of t h i s p a p e r .

111.2

-

Homogeneity t e s t s f o r a row/column s t r u c t u r e

Suppose t h a t an e v e n t occurs randomly i n time and l e t X(1) b e t h e number of occur- r e n c e s of t h e event i n an i n t e r v a l I. A Poisson process w i t h mean measure

u

is c h a r a c t e r i z e d by t h e p r o p e r t y t h a t f o r every f i n i t e c o l l e c t i o n of non overlapping i n t e r v a l s 11, 12,.

. . ,

I k , X ( I I ) , X ( I Z ) ,

. . . ,

X(Ik) a r e independent Poisson d i s t r i b u t e d random v a r i a b l e s w i t h parameters ~ ( 1 1 ) ,

. . . ^,

~ ( 1 ~ ) . By convention a Poisson process on Lo, i s denoted by (Xt, t > o ) where Xt i s t h e number of occurrences i n t h e i n t e r v a l [ o , t [

.

A Poisson process i s homogeneous i f At = p ( [ o , t [ ) = v . t where v i s a p o s i t i v e c o n s t a n t .

The homogeneity t e s t we a r e going t o develop h e r e i s suggested by t h e following remark on homogeneous random Poisson p r o c e s s e s . Let T be a f i x e d time and l e t us assume t h a t NT = k , i . e . , t h a t t h e number of occurrences i n t h e time i n t e r v a l C0,T.C i s k . I f t 1 ,t 2 , .

. .

, t k a r e t h e ones a t which t h e e v e n t s have o c c u r r e d , ( t l , t q ,

. . .

^,tk!

can be considered a s a s e t of independent o b s e r v a t i o n s of a random v a r i a b l e UT uni- formly d i s t r i b u t e d on [O,T[. It t h e n f o l l o w s t h a t t h e random v a r i a b l e Nt/NT pro- v i d e s an unbiased e s t i m a t o r of t / T f o r any t i n [O,T[. A r e p h r a s i n g of t h e above remark provides u s w i t h a method f o r t e s t i n g t h e homogeneity of t h e c o n t r o l s p e c t r a c o r r e c t e d by t h e u ' s o r B ' s . For example i n o r d e r t o d e r i v e a t e s t f o r a row homo- g e n e i t y , we can c o n s i d e r t h e d i s c r e t i z e d Poisson process d e f i n e d f o r k i < n by

m Nf =

t.

N '

j = l i j '

Let i be a f i x e d v a l u e of i corresponding t o a homogeneous l i n e . For each i l > io the r g w s w i t h i n d i c e s l y i n g between io and i l w i l l be homogeneous i f t h e random va- r i a b l e s Ui i n t r o d u c e d above a r e unif ormly d i s t r i b u t e d on [io, i l

F.

Such a u n i f o r m i t y h y p o t h e s i s can be t e s t e d by u s i n g t h e Kolmogorov-Smyrnov s t a t i s t i c .

t h I f t h e t e s t r e j e c t s t h e homogeneity h y p o t h e s i s ( t h e d i s o r d e r d e t e c t e d a t t h e i l l o c a t i o n corresponds t o a non homogeneous row) i t i s processed no f u r t h e r . By a s i m i l a r technique we can c o n s t r u c t a t e s t f o r t h e column h e t e r o g e n e i t y .

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i n c e p t i o n a s a methodology. Much of i t s development i n p r a c t i c e i s due t o S c h e f f e

181.

As i n d i c a t e d e a r l i e r , t h e s t r u c t u r e of a PSD allows us t o c o n s i d e r t h e rows and t h e columns a s two f a c t o r s with n and m l e v e l s r e s p e c t i v e l y . Under a homogeneous f l u x t h e observed counts can b e a r r a n g e d i n a 2D a r r a y and a r e r e s p e c t i v e l y d i s t r i - buted a s Poisson random v a r i a b l e s w i t h mean A . . . The s t a t i s t i c a l q u e s t i o n i s "can t h e o b s e r v a t i o n s r e p r e s e n t e d by t h e nm countslde considered a s having been drawn from t h e same Poisson p o p u l a t i o n w i t h mean A?" The n u l l h y p o t h e s i s can be s t a t e d a s H o : i i . = A f o r any ( i , j ) . Hence we wish t o determine whether t h e d i f f e r e n c e s among t h e o d s e r v a t i o n s ( N i j ; l t i t n , 16jbm) a r e t o o g r e a t t o a t t r i b u t e t o t h e chance e p r o r s of drawing t h e samples f r o m t h e same population.

Before we d i s c u s s t h e procedures involved i n our p r e s e n t a p p l i c a t i o n we w i l l c o n s i d e r t h e g e n e r a l r a t i o n a l e u n d e r l y i n g t h e ANOVA p r i n c i p l e . I f t h e n u l l h y p o t h e s i s t h a t a l l t h e i i j a r e equal i s t r u e t h e n both t h e v a r i a t i o n s among t h e row-means and t h e v a r i a - t i o n s among t h e column-means r e f l e c t random e r r o r s of t h e sampling process and would be expected n o t t o d i f f e r s i g n i f i c a n t l y from each o t h e r . On t h e o t h e r hand, i f t h e n u l l h y p o t h e s i s i s f a l s e and f o r example t h e column means a r e indeed d i f f e r e n t t h e n t h e between-column v a r i a t i o n would n o t r e f l e c t sample e r r o r s only and w i l l be more important. Hence, a comparison of t h e between-column-variations and t h e between-row- v a r i a t i o n s y i e l d s i n f o r m a t i o n concerning d i f f e r e n c e s among t h e column and t h e row means. This i s t h e c e n t r a l i n s i g h t provided by t h e ANOVA technique whose main charac- t e r i s t i c i s t h a t t h e o b s e r v a t i o n s a r e assumed t o be r e a l i z a t i o n s of random v a r i a b l e s normally d i s t r i b u t e d w i t h t h e same v a r i a n c e 02.

I n t h e model considered h e r e such assumptions a r e n o t a p p l i c a b l e s i n c e t h e v a r i a n c e of a Poisson v a r i a b l e i s equal t o t h e mean. Hence e i t h e r t h e d a t a h a s t o be t r a n s - formed i n such a way t h a t t h e s e assumptions a r e s a t i s f i e d o r we have t o extend t h e s e i d e a s t o t h e Poisson c a s e . These two approaches a r e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s .

For d a t a of t h e type considered h e r e we assume t h e mathematical model (111.3)

X..

_1.1= p + ui + v _j i = l

,.. .

^{, n}^;^{j = l}

,.. .

^,m

where !J i s t h e g e n e r a l mean, u; i s t h e e f f e c t of t h e i t h row and v j i s t h e e f f e c t of t h e j t h column. The hypotheses of i n t e r e s t are:

Hr : row homogeneity u i = u f o r a l l i i n i l , . . , n ) H, : column homogeneity

v j = . v f o r a l l j i n ( I , . . , m l H : H, and Hc t o t a l homogeneity

111.3.1

-

S q w r e r o o t transform of a Poisson v a r i a t e

By t h e c e n t r a l l i m i t theorem, i f X i s a Poisson v a r i a t e w i t h mean U, then, a s p +

*,

t h e random v a r i a b l e

(x-u)/&

i s d i s t r i b u t e d a s a s t a n d a r d normal d i s t r i b u t i o n N ( 0 , l ) . I n / 7 / it i s shown t h a t under t h e same c o n d i t i o n s ,

&

i s d i s t r i b u t e d a s a normal random v a r i a b l e w i t h mean

4

and v a r i a n c e

1.

It was found by ANSCOMBE / I / t h a t t h e

4

t r a n s f o r m a t i o n

&,

where c i s a s u i t a b l y determined c o n s t a n t

,

has some t h e o r e t i - c a l advantages. This t r a n s f o r m a t i o n has been r e v i s i t e d by KIHLBERG e t a l . /6/ _who s u g g e s t c=31/80 a s t h e b e s t c o n s t a n t f o r n o r m a l i t y and s t a b i l i s a t i o n of t h e v a r i a n c e . This i s t h e t r a n s f o r m a t i o n we used f o r t h e f i r s t ANOVA approach of our problem.

111.3.2

-

T e s t s f o r transformed d a t a

I f t h e counts f o r each c e l l a r e l a r g e (25) t h e s t a t i s t i c a l model o b t a i n e d by applying t h e square r o o t t r a n s f o r m a t i o n i s normal with t h e same v a r i a n c e b u t t h e a d d i t i v i t y s t a t e d i n ( I I I . 3 ) no longer h o l d s . More p r e c i s e l y , t h e transformed v a r i a b l e s a r e now Z i j = /~;j+0.386 and i f mij denotes t h e mean of Z i j we have, w i t h no l o s s of

g e n e r a l i t y :

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C5-22 JOURNAL DE PHYSIQUE

where, m12 represents an interaction between the ith and jth level introduced by the square root transformation. However by inspecting analytically the relations among Xij and mij we have, for the hypotheses Hr and Hc defined earlier:

Hr : for all i and all j ml(i) = 0 and m12(i,j) = 0 Hc : for all i and all j m2(j) = 0 and mlz(i,j) = 0

Hence, if an additivity model is assumed for the initial data, the tests of Hr, Hc and H can be processed by the usual ANOVA techniques (see for example

181).

This approach while well suited to the relative stability test of the detector is not applicable if interactions (e.g. peaks in Bragg diffraction spectra) are present in the original data.

111.4.

-

ANOVA for Poisson data

Instead of transforming the original data, we now consider a statistical model that is a natural generalization of classical ANOVA. This generalization largely removes the normality assumptions and the additivity of systematic effects and therefore may also be extended to estimate the Bragg intensity in diffraction experiments as discussed later.

Let us assume that the detector has a stable response and that

Y

is the random inte- ger valued vector whose p = nm components are obtained from the random variables Nij by:

Hence, the components Yi of

Y

are independent Poisson variates with corresponding means pi. Let

y

be the p-dimensional vector with components

u;

and let

y

be an observed value of

Y.

Neglecting a constant that does not involve

g,

the logarithm of the likelihood function of our model, given y, is:

-

P

If the row/column structure heterogeneity is additive one can show easily that the mean E(Nij) of Nij may be written as:

n m

(III.4.b) E(N. .) = rn

1 J +

ai +

Bj with

1

ai =

1 Bj

= 0

;=1 j = 1

such a decomposition being unique. Hence, if

2

denotes the (n+m-1)-dimensional vector defined by:

the mathematical expectation &of

Y

can be expressed as:

1

= AV where A is a known matrix of rank n+m-I. To elctend the analysis of variance, "go3h" estimates of

2

are needed. To do this, we will apply the maximum likelihood principle 171.

The maximum-likelihood algorithm we present is derived as a modification of the Newton-Raphson algorithm and may be interpreted as an iterative weighted least- squares method. More precisely, the maximum-likelihood equations are obtained by differentiating (III.4.a) with respect to each of the components of

2,

i.e. : (111.4.~) S(v) =

Since the maximum likelihood equations are not linear with respect to

2,

the method of scoring (171 p.

305)

has been used to develop an algorithm to find a root of

(111.4 .c)

.

We obtain the following adjustment at the tth iteration:

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A

and I ( l ( v ( t ) ) ) = i s t h e information m a t r i x a t t h e tth i t e r a t i o n .

To s t a r t t h e i t e r a t i o n s we t a k e r ( s ( ~ ) ) = y and ~ ( O ) = ~ ( O ) . The procedure i s r e p e a t e d u n t i l a s t a b l e s o l u t i o n i s reached. For more d e t a i l s t h e r e a d e r i s r e f e r r e d t o /3/

where we have shown t h a t t h e e s t i m a t o r s o b t a i n e d by t h i s method have good asymptotic p r o p e r t i e s . Once t h e maximum-likelihood e s t i m a t o r s of v and p a r e obtained we can develop l i k e l i h o o d r a t i o t e s t s / 7 / f o r homogeneity. Denoting t h e deviance of our model a t

g

by D(g) = 2 ( L ( y , ~ )

- L(y,g))

t h e v a r i o u s t e s t s a r e based on t h e s t a t i s t i c s :

,. ^A

)

-

D(ll>

-

¹ ^A

where u o = ( y , 1,

...,x)

w i t h

y

=

- ¹

^{yr and}

1

i s t h e maximuni-likelihood e s t i m a t o r P r = l

of g under t h e a p p r o p r i a t e homogeneity h y p o t h e s i s . By a s u i t a b l e large-sample ar- gument we have t h a t , a s y m p t o t i c a l l y under t h e v a r i o u s t e s t i n g hypotheses, t h e a p p r o p r i a t e s t a t i s t i c s a r e X2 d i s t r i b u t e d . Moreover a s l i g h t e x t e n s i o n of t h e S- method of H. Scheffe /8/ f o r c o n t r a s t s among rows o r columns provides us w i t h a way t o f i n d which rows o r columns a r e heterogeneous when t h e homogeneity h y p o t h e s i s i s r e j e c t e d .

I V . Numerical r e s u l t s and conclusions

For any s t a t i s t i c a l d e c i s i o n r u l e it i s n a t u r a l t o ask f o r t h e r e j e c t i o n of t h e homo- g e n e i t y hypothesis a s f r e q u e n t l y a s p o s s i b l e when t h e r e i s a l a c k of homogeneity.

This maximum p r o b a b i l i t y of t h e r i g h t d e c i s i o n when t h e r e i s h e t e r o g e n e i t y i s c a l l e d t h e

power

of t h e d e c i s i o n r u l e . However t h e power of a t e s t depends on i t s

&-

f i c a n c e l e v e l and (e .g. f o r t h e ANOVA t e s t ) i s not always o b t a i n a b l e a n a l y t i c a l l y . A p p l i c a t i o n computer programs w r i t t e n i n PASCAL have been developed f o r t h e algo- r i t h m s p r e s e n t e d h e r e and t h e power of t h e t e s t s have been e s t i m a t e d o r checked by means of s i m u l a t e d experiments.

Two types of experimental d e s i g n have been used f o r t h e simulations:

A: T o t a l homogeneity, i.e.,X..=X ff.

1~ l , j '

B: Column homogeneity, but row h e t e r o g e n e i t y due t o a p e r t u r b a t i o n a t an a r b i t r a r y f i x e d row io, i.e.:

ffj A . =),

+

a Y i , j

i 2 i 0

Xij=X l o j

The power of t h e Poisson ANOVA t e s t has been parametrized by t h e s i g n a l t o n o i s e r a t i o SNR given by:

SNR = a

Jm J;;;

where n r i s t h e t o t a l number of simulated s p e c t r a f o r a nm PSD submitted t o a uniform background whose mean r a t e i s X and w i t h one row perturbed by an average amounta.

A hundred s i m u l a t i o n s have been performed f o r v a r i o u s v a l u e s of SNR, u s i n g a random g e n e r a t i o n procedure f o r Poisson d i s t r i b u t i o n s developed i n

1 2 1 .

For each e x p e r i - mental d e s i g n t h e r e l a t i v e frequency of r e j e c t i o n s i n 100 t r i a l s provides an e s t i - mate of t h e power of t h e t e s t . The nominal s i g n i f i c a n c e l e v e l of our t e s t was f i x e d a t 10%.

The r e s u l t s ( f i g u r e 1) show t h a t t h e estimated s i g n i f i c a n c e l e v e l of t h e ANOVA t e s t a g r e e s w i t h t h e nominal one ( s e t t o 10%). The t e s t h a s a good power w i t h a d e t e c t i o n l i m i t of SNRa0.75 ( t h e v a l u e f o r which t h e d e c i s i o n r u l e i s b e t t e r t h a n a fa-

-

d e c i s i o n ) .

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JOURNAL DE PHYSIQUE

Similar simulations on the other tests in this paper, have shown that the ANOVA test is equal or superior in power especially when compared to the aggregation test and the chi-square goodness of fit test. The power of the disorder test is of the same magnitude but its significance level can only be estimated by simulations since it is a multiple decision rule.

Finally the

ANOVA

test has the following advantages over the others: it is more general (no Gaussian approximation), its significance level is known theoretically, its algorithm may be extended to the peak detection in our "inclined" background in the following way:

We model the Xij by:

where the non linear tecms

y . .

are due to the peak and detect by ANOVA techniques the cells at which

y i j

#

0 .

"'This will be discussed in a subsequent paper.

Preliminary tests of the ANOVA algorithm on real calculations and control spectra from the neutron diffractometer D19A of the Institute Laue-Langevin have given promising results.

0.5 1 .O 1.5

SNR

Fig. 1 - Power of the ANOVA test as a function of SNR. The nominal significance level is set to

10%.

The dashed lines bound the region in which fall the results obtained for a large spread of background (b) and repetition (nr) parameter values .

REFERENCES

/I/ Anscombe, F.J., Biometrika 35 (1948) 246-254.

/2/ Antoniadis, A., Berruyer, ~ T a n d Filhol, A., "MultidBtecteurs bidimensionnels:

(I) Simulation d'un spectre avec un ou plusieurs pics de Bragg", Internal report 85AN19T, Institut Laue-Langevin, Grenoble, France (1985).

/3/ Antoniadis, A., Berruyer, J. and Filhol,

A.,

"MultidBtecteurs bidimensionnels:

If) Homog6nLitd de rgponse et stabilitB1', Internal report 85AN26T, Institut

Laue-Langevin, Grenoble , France

( 1985)

.

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/ 5 / F i s h e r , R.A., B i o m e t r i c s

5

(1950) 17-24.

161 K i h l b e r g , J.K., Herson, J.H. and S c h o t z , W.E., J.R.S.S., s Q r i e C, Vol. 1 , (1971) 76-81.

171 Rao, C.R., "Linear s t a t i s t i c a l i n f e r e n c e and i t s a p p l i c a t i o n s " , Second e d i t i o n , J. Wiley & Sons Ed. (1973).

181