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)
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
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<
a1
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

1760 AD)
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 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
10
Be time series (
d
=2).
Wavelet transformation and Fourier spectra (1

real,
2

denoised
)
1
2
11
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