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A method for comparison of discrete spectra
L. Tsankov
To cite this version:
L. Tsankov. A method for comparison of discrete spectra. Journal de Physique III, EDP Sciences, 1993, 3 (7), pp.1525-1530. �10.1051/jp3:1993217�. �jpa-00249016�
Classification Physics Abstracts
02.70 06.50
A method for comparison of discrete spectra
L. T. Tsankov
Department of Physics, University of Sofia, BG-l126, Sofia, Bulgaria
(Received 28 July J992, revised 9 February J993, accepted 5 April J993)
Abstract. A numerical method for comparison of similar sets of experimental data (« spectra ») recorded at eventually different XY-scales is proposed. The method is invariant towards any
particular spectral line shape. It can be useful for a correct data acquisition in the long time experiments and also for comparison of recent and earlier measurements of similar objects.
Introduction.
The problem for comparison of recorded data sets (spectral distributions, time series, etc., referred below to as « spectra ») appears frequently in the conventional experimental work. At least two situations can be outlined when this problem is of considerable importance.
I) Comparison of successive runs during a long time experiment. The measured spectra are
recorded at (seeming) the same experimental conditions ; they are, however, really affected by
the long-term instabilities of the experimental setup. The progress in contemporary electronics reduces the problem to lower differences between the consecutive runs but does not remo~;e it.
ii) Comparison of results from a recent measurement with old (archival) data of similar measurements.
Any comparison procedure should give answer to the question are the compared spectra similar (I,e, statistically indistinguishable, apart from a trivial multiplication factor), or have
they essential differences ?
While the amplitude aspects of this problem do not bring any serious difficulties, this is not
the matter with its frequency aspects. We have in mind the case when an unknown
transformation of the abscissae (energy or frequency Scale) had meanwhile occurred between the compared records. Of course, the problem might be considerably Simplified (or even avoided) if it is possible to process each Spectrum separately (I.e, to extract the meaningful
information peaks positions, widths, etc, from the data) and then to compare those
parameters with each other. However, the spectra frequently contain more information than
one derivable by means of some restrictive (and sometimes inadequate) parametric models.
Moreover, the important information often exists in a latent form (due to poor statistics by low intensities, etc.) and our goal is then to recalibrate individual Spectra to a common X-Scale in order to add them properly, without smearing the peaks.
JOURNAL DE PHYSIQUE III -T 3 N'7 JULY I991 55
1526 JOURNAL DE PHYSIQUE III N° 7
The purpose of the present work is to suggest a general method for solving the discussed
problem. The method and its illustrations are developed in the context of nuclear spectra, but
the ideas used are rather general and may be applied straightforwardly to any type of
experimental data.
Method.
We suppose to have two data sets (spectra)
F(x~)= ~~~~dxf(x);
4l(f~)= ~~~~df~R(f),
I=0, n-I. (I)
,~ f~
We suppose that the spectra F, 4~ had been obtained by discretization of the continuously changing signals f(.<) and
~R (f integrated within the channel width the usual mode for the
Standard multichannel analyzers (ADC or MSC).
Further, it is assumed that f and
~R are of « Similar Shape » (I,e, they have many « common
features » and few « differences »), and that f
= T(x) is a continuous transformation of the argument scale (caused e-g- by long-term changes of the amplifier gain) which had occurred between the recording of F and 4l. Actually, the determination of f = T(.< solves the essential part of the problem.
Our idea is to use the presumed « proximity » of f with
~R to evaluate f = T(x) in a least- squares sense, minimizing an appropriate defined « distance », p ~f,
~R), between f and
~R as a
function of T(x). The least-squares functional Q should therefore have the general form : Q = P ~f,
~D ~j, (2>
Paying attention to the fact that the functions f and
~R are not explicitly measurable (instead,
we have only n values of observations of some of their integrals, (F and 4l), as already
defined by Eq. (I)), equation (2) is written in the form
n
Q
= z wi it iT(~j )i F (xi))~ (3)
, o
where w~ is a weight function accounting for the statistical errors of the measurements.
Considering F~ and 4~~ as statistical measurements having dispersions (variances) D(F~) and D(4~~), the weight function w~ has the form :
w~ =
,
i
=
0, n~- 1. (4)
D(F,)+D(4~~)
A concrete parameterization of the unknown transformation f
= T(x) is necessary now in order to find the minimum of the functional Q (Eq. (3 ii. AS an example, the Simplest way to
parameterize T(x) is to expand it in a power series :
n,
T(x)
= £ a~ x~ (5)
t =o
The method suggested here, however, does not depend on any particular form of
parameterization. So, in order to save the general form of the equations, we use the common notation f
= T( (a), x). The coefficients (a) are now regarded as fitting parameters and the
problem is to find those values of (a) corresponding to the minimum of the functional Q.
The gradient of Q must be 0 at the minimum of Q. So, we obtain the following system of equations for the unknown parameters (a)
fl=2~[~w~j4~iT(x,>i-F(x,>j.@) =o, k=o, m. (6>
Since (see Eq. (I))
T(.,, 4~iT(X,)] ~
=
df
~D (f),
= 0, n
,
(7)
T(1,
and 4l depends on (a) via the integration limits, so we have
t ~aiT(x~
+
>1. ~()~~ ~a iT(xj>1 ~()~~ ,
k
= o, m. (8>
In this way, the problem for comparison of the Spectra F and 4l can be completely reduced to
a standard nonlinear least-squares problem if we are able to calculate ~R(f) for each
f ~ [T(x~), T(x,
~
)], I
= 0, n I.
In principle, any interpolation operator can be used to approximate
~R (and, respectively, 4l) in the points other than (f~ Ii
= 0, n I) (note that f~ # T(x~) !). The formulation of the discussed problem and the structure of data suggest the use of a spline interpolation for
~R
(denoted here as #). Among the numerous splines we found as most convenient those described in ill- They are written in the form
2p
I (<)
= z A~~(< <~)~/k
,
~ i<~, <,
~
j, I
=
0, n
,
(9>
where the coefficients (A) must satisfy the following relations
j~'~'df # (f) = 4~ (f, ), ~" df j#iPi(f)j2 = min I = o, n i (io)
f, en iAi
Here ~p) stays for the p-th derivative of ji all other quantities have their usual meaning.
The interpretation of equation (lo) is straightforward: at each node f, the spline
Should equals the observation 4l~ (in the sense of the integrals defined by Eq. (I)), and, at the same time, we wish to use the rest arbitrariness in the determination of the spline
coefficients (A~~) to minimize the global curvature of the spline. As it is Seen from equation (9),
~Ris represented in each subinterval with a local polynomial of power 2 p. The coefficients of the polynomials, (A~~), are derived by Solving equation (lo), which result in a coupled
system of linear equations for (A) (note that the second equation in (10) defines actually a
simple linear least-squares problem). All other details regarding the construction of the splines
are discussed in the cited book ill-
Results and discussion.
For practical purposes, the functional Q (Eq. (3)) is rewritten in the form
ni ~j,,~~~ 2
Q= z ~(A.F(x~)+B-
d<~a(ii (ii)
,=i T(,,)
1528 JOURNAL DE PHYSIQUE III N° 7
here A and B are independent on x; they account for a possible (and trivial) linear transformation of the ordinates (due to different acquisition time for F and 4l, different
background level, etc. ). They are regarded as additional fitting parameters in the minimization
procedure.
It should be noted that although f = T( (a), xi may be a linear function of (a) (represented
in particular as a power series, cf. Eq. (5)), the least-squares problem I Ii is nonlinear. So, for evaluation of the minimum of Q we use the well known nonlinear Gauss-Newton method. In the most cases, convergence is achieved after 3-5 iteration steps. The processing time for a pair
of 512-channel spectra lies under I min (PC/AT computer with a 80 287 math co-processor
working at lo MHz).
Convergence problems may appear in the least-squares procedure if the initial values of the fitted parameters are too away from their « best » values. The convergency radius may be, however, considerably enlarged if a proper smoothing is applied to the data at the first stage of the iterative procedure. Such a smoothing can be included in a natural way in the algorithm by replacing the interpolation spline (Eqs. (9, lo)) with a smoothing one, I-e- replacing equation (10) with the more general requirement
n-> f,~, 2 f,,
£ df ~2~(f) 4~(f~) + £Y df[~2j~~(f)j~
# mIn, (12)
1=0 f> f0 (Al
where the parameter a regulates the smoothness (and the closeness) of the approximation. This method was realized in the program and combined with a procedure for optimal choice of a based on the residuals criterion (see [I]). In order to save completely the individuality of the
original spectra, equation (12) may be replaced by equation (lo) at the last stage of the minimization procedure (when desired coefficients (a) are already close to their final values), thus restoring the original scheme.
Some illustrations of the method are presented in the figures.
Figure la shows two M6ssbauer transmission spectra of a 25 ~Lm thick Fe foil measured at rather different (roughly twice) velocity scales. The resultant difference between the spectra
after application of the above recalibration procedure is displayed in figure16. The
parameterization of T(x) given by equation (5) was used (restricted to a linear function, which is relevant to the physical situation). The highest power ~p) of the spline approximation (Eq. (9)) was also restricted to I, corresponding to quadratic splines. Under these conditions the best values obtained for the coefficients are ao = 98.31± 0.02, aj
=
0.5128 ± 0.0001 (the quoted uncertainties correspond to one standard deviation) the normalized x~ is 2.03.
It is clearly seen from the figure that even in this difficult case the method gives satisfactory
results. The noise level of residuals in figure 16 exceeds the statistical fluctuations only around the major peaks. The last fact is explained by the well known geometric (cosine) factor in
Mossbauer experiments, resulting in some deformation of the peaks depending on the chosen
velocity scale.
Another way to evaluate the applicability of the method is to compare the least-squares
estimates of the peaks positions and widths for the first spectrum and those obtained for the
sum of the spectra (the second spectrum being added after correction). The mean difference in the peaks positions estimates is 0.031 channels, which is comparable with their estimated
standard deviations (vr
= 0.0212). The peaks of the summary spectrum are broader (with respect to those of the first spectrum in Fig. la) by about (mean value, averaged over all peaks)
2.5 % (0.I 13 ± 0.066 channels). The latter number may be regarded as a realistic evaluation of the spectrum deterioration due to the influence of the recalibration procedure.
The method works equally well also in the case when the spectrum does not contain any
sharp peaks. Figure 2 illustrates a beta-spectrum of a very weak source detected with a plastic
x10'
70
G60w
I
~
50
a)
25 100 200 300
Channels b)
Fig. I. a) M6ssbauer transmission spectra of a 25 ~m thick metallic Fe foil measured at different velocity calibration of the transducer. b) The net difference between the spectra of figure la after
application of the comparison procedure.
3,00
~
§ 2,00
o O cn
i,oo
0,00
0 100 200 300 600
Channels
Fig. 2. Beta-spectrum of a weak 9°Sr + 9°Y
source measured by a plastic scintillation detector [2].
1530 JOURNAL DE PHYSIQUE III N° 7
scintillator [2]. The signal consists of a continuous curve with a very low amplitude imposed
on an exponentially decreasing background. In order to prove the discussed recalibration method we compared each with other two successive records of the same signal, and then the second spectrum was shifted by one channel with respect to the first one. The resultant spectrum shift was evaluated by the program as ao = 1,125 ± 0.830 channels.
Conclusion.
In our opinion, the method for comparison of spectra suggested in this work appears to be
simple and fast enough to be applicable if either long-time measurements are performed, or spectral data have to be compared with results of other similar measurements. Since the formulation of the method is not restricted to any particular spectral line shape, it might have a
quite large application area.
A two-dimensional generalization of that method may be also developed.
References
[1] VASILENKO V. A., Splajn-Funkcii: Teorija, Algoritmi, Programi, Nauka, Novosibirsk (1983)
(Russian).
[2] VAPIREV E. i., private communication.
