F=?

B=?X=?

Y - observed time series of paleodata X - clean signal (as if we have no noise) B - noise component

Y=F(X,B)

The measured signal might be corrupted by noise of different provenance and properties.

THE GENETIC ALGORITHM FOR A SIGNAL ENHANCEMENT

L.Karimova,Y.Kuandykov, N.Makarenko

Institute of Mathematics, Almaty, Kazakhstan, chaos@math.kz

APPROACHJ. 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.

2

exponent Holder

Time series or signal is locally described by the polynomial and

expΗοlder onent

0

0 0

x

nf x P x x C x x

f x

α 1 is differentiable

α 1 is continuous, but it is not differentiable

= 0.5 white noise, 0.5 < α < 1 coloured (red) , α < blue nois nois 0.e e5

f x

f x

Geometrical interpretation of 0<

3

0x

How to estimate ?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

• Wavelet transformation of : f x 1,

x bWf b s dx f x

s s

• has local exponent in x0 ifHolder f x

1

2,Wf b s As

1

2, 2

j

j kx C

( )f x

( , )Wf b s

4

• has n vanishing moments: for 0 k n 0kx x dx ,

The scheme of the methodK.Daoudi, J.LevyVehel, Y.Meyer, Construction of continuos function with prescribed local regularity,

Constructive Approximation, 014(03), pp349-385 (1998)

0 100 200 300 400 500 600

0.5

1.0

1.5

2.0

2.5

3.0

X

X̂

Y

Estimation of thelocal exponentHolder

Construction of a functionwith prescribed regularity

+

5

INRIA software FracLab is available at http://www-rocq.inria.fr/fractales

yj,k - wavelet coefficients of Y - wavelet coefficients of enhanced Haar wavelets

,ˆ j kx X̂

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

• Y is close to in the normX̂ 2L• Local Hölder is prescribed, ˆα

X ˆα α δ YX

2, ,,

ˆ minj k j kj k

F y x

1, 1 21

ˆlog 1/ 2j n

n

j nj ij

s x M i

One can estimate and enhance the regularity structure by modification of wavelet decomposition coefficients, solving the next optimization problem

6

It is imposed that where are real numbers  1.  Initialization: random

2.  Crossover and mutation

 

3.  The evolution function:

is modifier

4.  Replacement percentage is 60%

, , ,ˆ j k j k j kx u y

, 0,1j ku

,j ku

0.8cP

0.001P

,j ku

Steady State Genetic Algorithm for enhancement of time series7

2 ˆ, ,,

ˆ1 ( ) ( )j k j k u Xj k i

F i t ne s s u x i i

Solutions = individuals of a population

Software C++GALibWall M.//GALib homepage: http://lancet.mit.edu/ga

Initial random population

Convergence of the population

Roulette wheel selection

Performance

Function to be optimized is

fitness =“adaptation to the environment” = f(x)

evolution

Convergence means a concentration of the population around the global optimum

8

2 ˆ, ,,

ˆ1 ( ) ( )j k j k u Xj k i

f x u x i i

Enhancement of the cosmogenic isotopes time seriesby genetic algorithm and multifractal denoising.

14C annual data (1610-1760 AD)

Enhanced data by genetic algorithm=0.7

9

Multifractal denoising data=0.7

Fourier spectra of original and enhanced 14C data

------ original data; ------- multifractal denoising; ------- genetic algorithm

5 10 15

0

10

20

Spectra

Period

10

Revealing deterministic dynamics from enhanced data

1.55cd

Helama, S.et al., 2002: The supra-long Scots pine tree-ring record for Finnish Lapland: Part 2, The Holocene 12, 681-687.

11

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.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

f()

original enhanced

Enhanced data preserve their multifractal structure.

CONCLUSION1. 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

12

"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 .

GENES & DNA

The measure M is the enhancement times the efficiency

Reproduction

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

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

1

2

11