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EUR 5070 e

UNFOLDING OF COMPOSITE SPECTRA BY LINEAR REGRESSION by W. MATTHES

Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Nuclear Study

Luxembourg, January 1974 — 28 Pages — 15 Figures — B.Fr. 50.—

We consider the problem of resolving a measured pulse-height spectrum of a material mixture (e.g. Raman-spectrum, gamma-spectrum) into a weighted sum of the spectra of the individual components of the mixture. If the measured spectrum is contaminated with noise, the standard least-square method cannot be used to unfold the spectrum into its individual components for small signal-to-noise ratios. To improve the identification of the components in the mixture we constructed a "stepwise regression" method which gives very good results even for signal to noise ratios of the order of one. The new method is a combination of the least-square mechanism and repeated application of statistical tests.

EUR 5070e

UNFOLDING OF COMPOSITE SPECTRA BY LINEAR REGRESSION by W. MATTHES

Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Nuclear Study

Luxembourg, January 1974 — 28 Pages — 15 Figures — B.Fr. 50.—

We consider the problem of resolving a measured pulse-height spectrum of a material mixture (e.g. Raman-spectrum, gamma-spectrum) into a weighted sum of the spectra of the individual components of the mixture. If the measured spectrum is contaminated with noise, the standard least-square method cannot be used to unfold the spectrum into its individual components for small signal-to-noise ratios. To improve the identification of the components in the mixture we constructed a "stepwise regression" method which gives very good results even for signal to noise ratios of the order of one. The new method is a combination of the least-square mechanism and repeated application of statistical tests.

EUR 5070 e

UNFOLDING OF COMPOSITE SPECTRA BY LINEAR REGRESSION by W. MATTHES

Commission of the European Communities

Joint Nuclear Research Centre — Ispra Establishment (Italy) Nuclear Study

Luxembourg, January 1974 — 28 Pages — 15 Figures — B.Fr. 50.—

EUR 5070 e

COMMISSION OF THE EUROPEAN COMMUNITIES

UNFOLDING OF COMPOSITE SPECTRA

BY LINEAR REGRESSION

by

W. MATTHES

1974

Joint Nuclear Research Centre Ispra Establishment - Italy

ABSTRACT

We consider the problem of resolving a measured pulse-height spectrum of a material mixture (e.g. Raman-spectrum, gamma-spectrum) into a weighted sum of the spectra of the individual components of the mixture. If the measured spectrum is contaminated with noise, the standard least-square method cannot be used to unfold the spectrum into its individual components for small signal-to-noise ratios. To improve the identification of the components in the mixture we constructed a "stepwise regression" method which gives very good results even for signal to noise ratios of the order of one. The new method is a combination of the least-square mechanism and repeated application of statistical tests.

KEYWORDS

SPECTRA RESOLUTION PULSES

C O N T E N T S

page

1) Introduction 5 2) Stepwise linear regression 9

3) Application of the stepwise regression method 12 4) Conclusion 1 3

δ ­

Ι) Introduction

We consider the problem of resolving a measured pulse­height spectrum of a material mixture (e„g„ gamma­ray­spectrum, Raman­spectrum) into a

weighted sum of the spectra of the individual constituents of the mix­ ture l[l,2,3,4J . The analytical formulation of this problem will be based

on the following models

(1) the measured spectrum of the mixture is represented by the vector Ν = ) Ν., ,N„o. .N>T f , where Ν is the number of counts in channel >>

L 1' 2 Nj ' y r

(2) the vector S. = ÍS. ,S „. .S.„f represents the spectrum of con­ χ *· il i2 iNj

stituent i (i - 1,2...L) which might possibly be in che mixture (3) ß. is the magnitude of the contribution of constituent i to the measured spectrum Ν (ß. is proportional to the concentration of

1

constituent i in the mixture)„

The number of counts observed in channel V is then given by:

A,„- X / * * , v * 3 > .

(1)

¿«­/

where D . is SJI error­term for channel y? „ For these error­terms we as­ sume that they are statistically independent and have expectation vuiue E(D ) = 0. Each Ν . nas a Poisson distribution with mean­value ana vari­ ance equal to

Ù

x. - Χ Λ'

5/

-The probability for the measured spectrum Ν becomes

<P(N)~

¡I

e

(3)

Now we apply the principle of Maximum Likelihood ¿J5 I to obtain estimated the unknown ß.. Due to this principle

operations :

values for the unknown ß.. Due to this principle we have to perform the

2ßt (i = 1,2...L) (4)

on (3) and obtains

Nu

*=4

X

/Vu \

z_±y

v

9/i,·

For practical applications we may assume that Ν will deviate not too much from \ . so that we can put

—

Ä

Ί+ "Ζ

ì

——' *=■

Λ—

r~^

(6)

and therefore

s­~

* 'fa­

­r)

cr)

Using (7) and (2) in (5) we finally arrive at the system of linear equat­

ions for the ß, s

k

2Λ/Ζ^7= Xiv

^-4

— 7

distribution properties is important however for further statements about the accuracy of the solutions of (8), for the construction of confidence intervals and for testing assumptions about the ß*s. To ob­ tain this information about the statistical behaviour of the B's we refer to the assumption made above, that Nv will deviate not too much

from X, ­. Under this assumption N . (greater than about 20 counts) will behave approximately as a random variable with a normal distribut­ ion of mean value E(N . ) =%» and variance &~, = %­ ,.

o

The variance (?L of N in our case is a function of the β*s which are

~ ν

not known. On the other hand we have up to terms of higher order:

>%_-_Vr: ^

A^, - V r

(9)

il7T¿

We assume therefore further QTV to be given and replace it for numerical calculations by /N^/ .

We introduce the quantities:

J

tv

m

ZP;Xi·' ι ï ~llob-1 J

(10)

which give equation (1) the form:

y - Σ Α · ^ / -

+ t~

V / s / f

or (11)

— 8 —

where Z.^, is normally distributed with mean value 0 and variance 1.

Our problem can then be stated in the following way:

Given a vector y in an N dimensional space V with components y

(j> = 1,2...N). Determine that vector £ = 2 Z b· \ · i n t n e subspace

V , spanned by the L linear independent vectors χ. , which makes

(12)

?-

/y-

¿7 = Ζ Λ"

¿*~

-

i *"

- ~ i W

to a minimum.

The corresponding b. are found out of the system of equations (equivalent to (8)):

Σ ¿Ζ *a )&*

=

Cx

' ^

This means that the error vector

­9

r=y~

J t

-*¿

(14)

is orthogonal to V and

lì

V -

S

(15)

2

/\, distribution of (N­L) degrees of freedom.

The initial problem (1) to (8) is thus reduced to a multiple linear re­ gression problem as formulated in (11) to (15). It is well known /6,7,8J that the solutions b,j& of the system (13) are normally distributed with mean value E(b* ) = ß, and variance (>, = a where a is a dia­

*~ +-

b

* ­ .*

2) Stepwise linear regression

A straightforward solution of the system (13) for the b*s could only be feasible if L is not too large and if the constituents­spectra x. were

ι accurately known. These are just two points which are not fulfilled in our case. We shall have to deal with the problem that the number L of constituents which could be in the mixture under investigation is very large (L ^ 50 or more) but that the number of constituents which actually are in the mixture is very small (^10). Further, the library spectra x. are in general found experimentally and as such are contaminated with a noise component. Thus taking into account all L components when solving (13) for the b. the allowance for components which are not in the mixture (β a« 0) will strongly influence the accuracy of those b. for constituents which actually are in the mixture. To solve this difficulty we shall con­ struct a stepwise regression procedure which selects a most probable com­ bination of constituents from the set of library spectra by means of re­

peated application of statistical tests and the least square mechanism. The final result will give both the constituents contained in the mixture and their respective (relative) strength (contribution to the measured spectrum y ) . To arrive at this stepwise regression procedure we employ the statistical test for the hypothesis that some of the ß's are iden­ tically zero. This test­situation can be described in the following terms

fs,9] :

Let us assume that only the Q spectra out of the set Τ =j χ ,x .. X­0>

contribute to the measured spectrum y and no other spectrum out of the remaining set Tn = J χ ,χ ,.,.χ I will give any "significant con­

*■ \ J1 3¿ JLi­W J

tribution".

­J»

In other words, if we would add an arbitrary spectrum χ of Τ to Τ and try to fit y within the set Τ ^ = ^ χ ,x. ... x 0?x i / ( Wj » t n e assumption

says that no χ. is necessary to fit "y* or that the corresponding b . are indentically zero for all *A* . If our hypothesis is true, then (for each

A*») the quantity

— 10 —

has a F, N_Q_­.·) distribution with 1 and (H.­Q­1) degrees of freedom, where

P%

Due to this fact we can find a number F,, „ Λ „. such that

(1,N­Q­1)

(17)

(18)

choosing for Ρ values of the order of 1%, 5% or 10% we can be (almost) sure

</*f) P%

that all f* will be smaller than F,, „ Λ ,. if our hypothesis is true

_^ ^ (1,M­Q­1) **

(that no χ. is necessary to fit y ) .

The test of this hypothesis will therefore be performed in four steps:

1) Calculate f ^ for all x. out of Τ .

Of

1

2) If f ^ ^ F^° N_Q_1 ) for all¿»*. go to step 4.

3) If for some ^c we have f ^ ^ F ° ., we choose the m largest of

(**) tijN­y— 1)

these f ^ (usually m = 1) and add the corresponding spectra to Τ , re­ move then from Τ and go back to step (1) to repeat this "forward se­

lection procedure".

4) The hypothesis is accepted that the constituents of Τ only make up the mixture and no component of Τ will make any further "significant contribution" to y.

It might still be possible that during this "forward selection procedure" we added to many components to Τ and some of the x.^iof the final Τ ) have a negligible influence on y,

To test for this possibility we make the hypothesis that e.g. x... has no significant influence on y and to fit y it is enough to use the spectra of set Τ „ only, where T~ contains all spectra of Τ except x.

this hypothesis is true, then the quantity

— 11 —

0

(f)

.

Se&)-SCrz) ,(„-tj\

<1β)

has a F., „ '■ « distribution. (1,N­Q)

Now we apply the above "forward"­procedure in the reverse sense and perform the following four steps:

1) Calculate g^* for all x. ' out of Τ .

Vf

°

2) If g^\p> F , ' N_ Q ) f o r a 1 1 (*· SO t o s t e p 4 . <<*.) p %

3) If for some (*~ we have ζ S. F, „,­>·>> we choose the η smallest of (>».) ^ ιΙ,ΝΗί;

these g (usually η = 1), remove the corresponding spectra from Τ o and go back to step (1) to repeat this "backward elimination proce­ dure" .

4) The hypothesis is accepted that no spectrum of Τ may be neglected. This means that the experimental data (y) are consistent with the as­ sumption that all components of Τ (and not more) are necessary to fit y. This stepwise regression method consists therefore of the two essential steps:

a) the "forward selection procedure" selects by repeated application of the statistical tests out of the set of all L possible library spectra a subset Τ of vectors which should all be included in the regression. _o All remaining vectors in Τ are rejected as the final test supports the

hypothesis that all ß*s of the vectors in Τ could be assumed to vanish. b) The "backward"elimination procedure" eliminates from Τ found in step

(a) further variables for which a test supports the hypothesis that their corresponding ß's can be assumed to vanish.

— 12

3) Application of the stepwise regression method

In order to test the efficiency of the stepwise regression method (SRM) we transformed the procedures described in chapter 2) in a computer pro­ gramme. This programme was then used (to simulate a practical application) to unfold test­spectra which were obtained by mixing some spectra (out of a library of arbitrarily chosen spectra) and adding a noise component.

The L(=50) library spectra were constructed by a linear superposition of I (= 10) Gaussian curves with random aplitudes, random widths and random

M

positions with an average distance of — between the peak positions, where M (= 10 cm) is the range over which the spectra were chosen to extend (the range M is chosen to correspond to N (=100) channels). Figs. la,b,c show some typical library spectra. Some of these library spectra were now line­ arly superposed with different weighting factors to obtain a pure signal spectrum and to obtain measured spectra each signal spectrum was con­ taminated with (Gaussian random­) noise of mean zero and variance 1 and a constant background (=5.0) was added (see Fig. 2 ) .

The contamination of the signal spectra with noise was done for different signal­to­noise­ratios (SNR) to simulate measured spectra for different experimental conditions. The signal­to­noise­ratio is defined by

SMR

-

- ^ —

7 }

:

(20)

X

'-y

where: S(i) = Signal value in channel i

N(i) = Noise value in channel i Ν

1 * ■­

~ ^ ~ S (i); mean value of signal i=l

A typical sample of "measured" spectra for different SNR's is shown in Figs. 3a,b,c,d„

— 13 —

Many calculations were made with varying numbers K of components in the signal spectrum (1 to 20 components out of 50) and different signal-to-noise ratios (o.5,l,....30).

The results of these calculations were all similar to the special case mentioned above and shown in Figs. 3 and 4.

4) Conclusion

The main conclusions out of these calculations can be summarized as fol-lows:

The least-square method cannot be used to resolve a mixture of K(^ 10) out of L ( ^ 20) spectra in the range of SNR<^5.0.

— 14 —

Figure Captions

Fig. la,b,c: Some of the arbitrarily chosen library spectra.

Fig. 2 Superposition of a constant background and a random Gaussian noise component of mean zero and variance 1. The background is also considered as a library spectrum (no. 1) and can be chosen arbitrarily.

Fig. 3: a) Signal spectrum obtained from a linear superposition of the library spectra no. / 3,6,9,13,19J with the weight fac-tors ^0.45, 0.15, 0.28, 0.37, 1.12 J

b) Measured spectrum obtained from a linear superposition of the signal spectrum of Fig. 3a with the noise and back-ground of Fig. 2 and arranged for a SNR of 0.5

c) same as Fig. 3b for a SNR =1.0

d) same as Fig. 3b for a SNR =5.0

Fig. 4: a) Component spectrum of the signal spectrum of Fig. 3a

b) Least-square solution for the component spectrum of Fig. 3c (SNR = 1.0)

c) Least square solution for the component spectrum of Fig. 3d (SNR = 5.0)

d) Stepwise solution for the component spectrum of Fig. 3c. Components 6 and 13 of Fig. 4a are not identified.

e) Stepwise solution for the component spectrum of Fig. 3d. Only component 6 of Fig. 4a is not identified.

Fig. 5: a) Component spectrum for a mixture of 10 components, each with the same weight-factor of 0.5.

b) Stepwise solution for the component spectrum of Fig. 5a for a SNR = 5.0.

Only component 39 has been wrongly identified.

— 15

References

(1) D.F. COVELL et al., Nucí. ïnstr. and Meth. 80 (1970) 55.

(2) M.A. HOGAN et al., Nucl. Instr. and Meth. 80 (1970) 61

(3) M.H. YOUNG, Nucl. Inst, and Meth. 45 (1966) 287

(4) YUNG SUN SU et al., Nucl. Instr. and Meth. 41 (1966) 129

(5) L. JÂNOSSY, Theory and Practica of the evaluation of measurements", The international Series of monographs in physics, Oxford 1965

(6) H. SCHEFFE', The analysis of variance, Wiley publication in mathe-matical statistics, John Wiley & Sons, 1959

(7) N.R. DRAPER and H. SMITH, Applied Regression Analysis, John Wiley & Sons, 1967

(8) J. HEINHOLD and K.W. GAEDE, Ingenieur-Statistik, R. Oldenburg, München Wien 1968

(9) S. WILKS, Mathematical Statistics, Wiley 1962

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OFFICE FOR OFFICIAL PUBLICATIONS OF THE EUROPEAN COMMUNITIES Boite postale 1003 — Luxembourg 6326

Boite postale 1003 — Luxembourg 6326