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  1. F=? X=? B=? THE GENETIC ALGORITHM FOR A SIGNAL ENHANCEMENT L.Karimova,Y.Kuandykov, N.Makarenko Institute of Mathematics, Almaty, Kazakhstan, chaos@math.kz The measured signal might be corrupted by noise of different provenance and properties. Y - observed time series of paleodata X - clean signal (as if we have no noise) B - noise component Y=F(X,B)

  2. 2 • APPROACH • J. Levy Vehel, Signal enhancement based on Holder regularity analysis, • IMA Vol. In Math. And Its Applications, vol.132, pp. 197 -209 (2002) • Task • To find time series, which is less corrupted by noise and at the same time preserves relevant information about the structure • and method: • Time series enhancement based on the local Hölder regularity • Approach does not require any a priori assumption on noise structure and functional relation between original signal and noise • Signal may be nowhere differentiable with rapidly varying local regularity • Increment of the local Hölder exponent of the signal must be specified • New signal with prescribed regularity may be reconstructed using a few methods, particularly, the genetic algorithm.

  3. 3 exponent a Time series or signal is locally described by the polynomial and Geometrical interpretation of 0<a<1

  4. How to estimate a? S. Mallat, A Wavelet Tour of Signal Processing (1999) Jaffard S. //Pointwise smoothness, two-microlocalization and wavelet-coefficients, Publ. Mat. 35, No.1, p.155-168, 1991 4 • Wavelet transformation of : • has local exponent a in x0 if • has n vanishing moments: for

  5. 5 The scheme of the method K.Daoudi, J.LevyVehel, Y.Meyer, Construction of continuos function with prescribed local regularity, Constructive Approximation, 014(03), pp349-385 (1998) X Estimation of the local exponent Construction of a function with prescribed regularity Y + d INRIA software FracLab is available at http://www-rocq.inria.fr/fractales

  6. 6 • Y is close to in the norm • Local Hölderis prescribed, yj,k - wavelet coefficients of Y - wavelet coefficients of enhancedHaar wavelets How to construct a function with prescribed regularity? J. Levy Vehel, Signal enhancement based on Holder regularity analysis, IMA Vol. In Math. And Its Applications, vol.132, pp. 197-209 (2002) There are two conditions for the construction of a function with prescribed local regularity One can estimate and enhance the regularity structure by modification of wavelet decomposition coefficients, solving the next optimization problem

  7. 7 Steady State Genetic Algorithm for enhancement of time series It is imposed that where are real numbers 1.Initialization: random 2.Crossoverand mutation 3.The evolution function: is modifier 4. Replacement percentage is 60%

  8. Roulette wheel selection 8 Solutions = individuals of a population Function to be optimized is fitness =“adaptation to the environment” = f(x) Performance Convergence means a concentration of the population around the global optimum Initial random population Convergence of the population evolution Software C++GALib Wall M.//GALib homepage: http://lancet.mit.edu/ga

  9. 9 Enhancement of the cosmogenic isotopes time series by genetic algorithm and multifractal denoising. 14C annual data (1610-1760 AD) Enhanced data by genetic algorithm d=0.7 Multifractal denoising data d=0.7

  10. 10 Fourier spectra of original and enhanced 14C data ------ originaldata;------- multifractal denoising;------- genetic algorithm

  11. 11 Revealing deterministic dynamics from enhanced data Helama, S.et al., 2002: The supra-long Scots pine tree-ring record for Finnish Lapland: Part 2, The Holocene 12, 681-687. 3-D phase portraits of annual mean July temperature in northern Finnish Lapland, reconstructed from tree-ring widths of Scots pine. Correlation dimension of the time series. Enhanced data preserve their multifractal structure.

  12. 12 • CONCLUSION • 1. Enhancement based on local Holder regularity are useful when • signal is very irregular; • regularity may vary in time; • Hölder regularity bears essential information for further processing; • signal may be nonstationarity; • noise nature and its relation with “pure” signal are unknown; • 2. Advantages and drawbacks of Genetic Algorithm (GA) • GA is able to trace all (global and/or local) optima of functional of an arbitrary complexity • GA is well adapted to the task of signal enhancement • GA requires high computational capability

  13. GENES & DNA "Individuals" are characterized by there DNA (genome) which is composed of a string of genes. Numbers are represented in the computer by N bytes, which we call a genes. The DNA consists of a string of genes. Each individual carries one gene for each of the parameters in the parameter space P plus two extra ones, for the crossover rate Rc and for the mutations rate Rm. Also each individual has a performance measure M. The measure M is the enhancement times the efficiency

  14. Reproduction Each simulation year, depending on the population size, individuals reproduce by selecting a mate. Individuals with higher performance measure Mhave a higher probability of being selected as a mate. If the population is large, the rate of reproduction is smaller, and vice verse.

  15. Multifractal Denoising of 10Be time series (d=2). Wavelet transformation and Fourier spectra (1-real, 2-denoised) 11 1 2