ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2012, Vol. 76, No. 3, pp. 303–304. © Allerton Press, Inc., 2012. Original Russian Text © E.R. Shaimukhametova, D.Z. Galimullin, M.E. Sibgatullin, D.I. Kamalova, M.Kh. Salakhov, 2012, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2012, Vol. 76, No. 3, pp. 347–349.

Influence of the Character of Noise on the Decomposition of Complex Spectral Contours E. R. Shaimukhametova, D. Z. Galimullin, M. E. Sibgatullin, D. I. Kamalova, and M. Kh. Salakhov Kazan Federal University, Kazan, 420008 Russia email: [email protected] Abstract—The effectiveness of decomposing complex spectral contours into components using the least squares method is compared to that of a genetic algorithm with distortion of the input signal by lowfrequency fractal noise. DOI: 10.3103/S1062873812030276

INTRODUCTION In experimental studies, spectral contours with unresolved internal structures produced by the super position of adjacent components of bands are fre quently observed, and are characteristic, e.g., of the vibrational spectra of matter in the condensed state. Such spectra require further processing, since useful information can be obtained from them only by deter mining the parameters of the individual bands in the complex spectrum. The problem of finding the num ber of band components and their shape, width, amplitude, and position on the frequency scale thus arises. The parameters of absorption bands are initially selected and, using the nonlinear leastsquares method, the best match between the model solution and the experimental data is achieved by minimizing the residual function. This task belongs to the class of optimization, since the objective function depends on the set of parameters. The genetic algorithms (GAs) proposed by John Holland for solving these problems are based on Charles Darwin’s principles of natural selection. GAs are stochastic heuristic search methods and are suc cessfully applied in various fields of endeavor (eco nomics, physics, engineering, etc.) to solve optimiza tion problems [1]. As distinct from most optimization methods, GAs are relatively resistant to falling into the local optima [2], and they can be used in problems with a changing environment [3]. In this work, we compare the efficiency of decom posing complex spectral contours using the least squares method and that of a genetic algorithm in the case of signal distortion by simulated lowfrequency noise. METHODS USED FOR DECOMPOSING THE SPECTRA The Least Squares Method As one method for approximating functions, this finds wide practical application in various fields of

knowledge when dealing with experimental data. In decomposing complex absorption spectra into com ponents, the parameters of the absorption bands are initially chosen and, using nonlinear LSM, the best match between a model solution and the experimental data is achieved by minimizing the residual function. Despite the apparent simplicity of this algorithm, dif ficulties arise when it is implemented: the iterative procedure is often divergent; i.e., the subsequent step of calculations does not lead to a reduction in residual values, especially when there is a high number of vari ables. The algorithm used in this work is a version of the subspace trust method; it is based on the interior reflective method of Newton described in [4]. Each iteration involves the approximate solution of a large linear system using the method of preliminary conju gate gradients. Genetic Algorithm (GA) This is an adaptive search algorithm used for solv ing optimization and modeling problems by sequential selection, combination, and variation of unknown parameters using mechanisms that resemble biological evolution [5]. The GA is based on the genetic pro cesses of biological organisms: mechanisms that simu late the processes of genetic inheritance and natural selection are used in implementing the algorithm to find a solution. Operators of selection, crossover, mutation, etc., are introduced as analogues of these mechanisms. A hybrid algorithm that we proposed based on con tinuous wavelet analysis and a GA for decomposing complex contours into components is described in [5]. The use of continuous wavelet analysis allows us to determine the number of absorption bands and their position, and this is used as the input parameters for applying the GA. It was shown in [5] that the method is relatively resistant to flat noise.

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Fig. 1. Recovery of the model contour using GA and LSM with a noise level of 10% and Hurst index H = 0.9 (А is amplitude; ν is frequency).

using GA. We compared the effectiveness of GA and LSM in the case of signal distortion by lowfrequency fractal noise. To investigate the possibility of recover ing the complex noise contours using these tech niques, we used a model spectrum consisting of six components and having a Gaussian shape. Lowfre quency noise with a value in the range of 0 to 10% was added to the model signal. Fractal noise with different Hurst indices—0.5 (white), 0.7, and 0.9 (lowfre quency fractal noise)—was used as the model noise. The resulting noise signal was split into elementary components by means of LSM and GA, and the rela tive error of recovery was calculated. CONCLUSIONS

Fig. 2. The relative error of recovery according to the noise level when using GA and LSM and applying noise with dif ferent Hurst indices (H).

Figure 1 shows the results from the recovery of a model contour using GA and LSM with a noise level of 10% and Hurst index H = 0.9, while Fig. 2 shows the dependence of the relative recovery error from noise levels with different Hurst indices. From the figures, it is clear that in the case of flat noise (Н = 0.5) and a low relative noise level, using LSM to recover the complex contour yields better results than using GA. However, when the noise has a lowfrequency structure or noise with a relatively high level is superimposed on the model spectral contour, the solution obtained using LSM is incorrect. At the same time, it is clear from Fig. 2 that the error of recovery for the complex contour using GA does not exceed 1% and depends weakly on the Hurst index in the investigated range of relative noise levels up to 10%. The application of GA thus ensures consis tency in searching for a solution regardless of the noise structure and its relative level.

OUR MODELS OF NOISE

REFERENCES

In our mathematical experiments, we used a ran dom fractal noise as our model of noise. Random frac tal processes occupy an intermediate position between deterministic and accidental [7, 8]. This model can be useful in describing real physical processes, and par ticularly in analyzing experimental noise in optical spectroscopy. Note especially that random fractal noise is used only as a convenient model for the occurrence of a random process. It allows us first of all to simulate low frequency noise, the calculating of which is most important in processing spectroscopic signals. Sec ond, it is possible to create noise with a guaranteed known (within the calculation error) value of the Hurst index specified in the simulation.

1. Holland, J.H., Scientific American, 1992, nos. 9–10, p. 32. 2. Gladkov, L.A., Kureichik, V.V., and Kureichik, V.M., Geneticheskie algoritmy (Genetic Algorithms), Mos cow: Fizmatlit, 2006. 3. Wise, B.M., Holt, B.R., Gallagher, N.B., and Lee, S., Chemometrics Intell. Lab. Syst., 1995, vol. 30, p. 81. 4. Coleman, T.F. and Li, Y., SIAM J. Opt., 1996, vol. 6, p. 418. 5. Shaimukhametova, E.R., Galimullin, D.Z., Sibgatul lin, M.E., et al., Bull. Russ. Acad. Sci. Phys., 2010, vol. 74, no. 7, p. 959. 6. Shaimukhametova, E.R., Galimullin, D.Z., Sibgatul lin, M.E., et al., Uch. Zap. Kaz. Univ., Ser. Fiz.Mat. Nauki, 2010, vol. 152, book 3, p. 185. 7. Mandelbrot, B.B., Fractal Geometry of Nature, San Francisco: Freeman, 1982. 8. Pallikari, F., Chaos. Solitons Fractals, 2001, vol. 12, p. 1499.

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RESULTS AND DISCUSSION In [6], we investigated the effect of flat noise on the reliability of the decomposition of complex contours

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2012