# THE GENETIC ALGORITHM FOR A SIGNAL ENHANCEMENT

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23 Οκτ 2013 (πριν από 3 χρόνια και 13 μέρες)

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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

APPROACH

J. Levy Vehel,
Signal enhancement based on Holder regularity analysis
,

IMA Vol. In Math. And Its Applications, vol.132, pp. 197
-
209 (2002)

To

find

time

series,

which

is

less

corrupted

by

noise

and

at

the

same

time

preserves

relevant

information

the

structure

and

method
:

Time

series

enhancement

ba se d

on

the

local

Hölder

regularity

Approach

does

not

require

any

a

priori

assumption

on

noi se

st r uct ure

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
a

Time series or signal is locally described by the polynomial and

Geometrical
interpretation

of 0<
a1

3

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

Wavelet transformation of :

has local exponent
a

in
x
0

if

4

has
n

vanishing moments: for

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

Y

Estimation of the

local exponent

Construction of a function

with prescribed regularity

+
d

5

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

y
j,k
-

wavelet coefficients of
Y

-

wavelet coefficients of enhanced

Ha a r wa v e l e t s

How to construct a f uncti on wi th prescri bed regul ari ty
?

J. Le vy Ve he l,
Si g na l e nha nc e me nt ba s e d o n Ho l de r r e g ul a r i t y a na l y s i s
,

I MA Vo l. I n Ma t h. And I t s Appl i c a t i o ns, vo l.1 3 2, pp. 1 9 7
-
2 0 9 ( 2 0 0 2 )

Th e r e a r e t wo c o n d i t i o ns f o r t h e c o n s t r u c t i o n o f a f u n c t i o n wi t h p r e s c r i b e d l o c a l r e gu l a r i t y

Y

i s cl os e t o i n t he nor m

L
ocal H
öl de r

i s prescri bed,

On e c a n e s t i ma t e a n d e n h a n c e t h e r e gu l a r i t y s t r u c t u r e b y mo d i f i c a t i o n o f wa ve l e t
d e c o mp o s i t i o n c o e f f i c i e n t s, s o l vi n g t h e n e xt o p t i mi z a t i o n p r o b l e m

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
%

Steady State Genetic Algorithm for enhancement of time series

7

Solutions = individuals of a population

Software

C++GALib

Wall

M
.
//GALib

homepage
:

http
:
//lancet
.
mit
.
edu/ga

Initial
random

population

Convergence

of the population

Roulette wheel s
election

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

Enhancement of the cosmogenic isotopes time series

by
genetic algorithm

and
multifractal denoising.

14
C annual data (1610
-

Enhanced data

by genetic algorithm

d
=0.7

9

Multifractal denoising data

d
=0.7

Fourier spectra of original and enhanced
14
C data

------

original

data;

-------

multifractal denoising
;

-------

genetic algorithm

10

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.

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.

Enhanced data
preserve their
multifractal
structure.

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 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
10
Be time series (
d
=2).

Wavelet transformation and Fourier spectra (1
-
real,
2
-
denoised
)

1

2

11