The Study of the Wiener Processes Base on Haar Wavelet
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This document reviews the application of Haar wavelets in studying Wiener processes, a type of stochastic process valuable in various fields such as astronomy, medical imaging, and data analysis. The paper explores wavelet transformations and their properties, focusing on how these mathematical techniques can efficiently represent and analyze signals. It concludes with definitions and mathematical formulations related to wavelet expansions and Wiener square processes.
![Scientific Review
ISSN(e): 2412-2599, ISSN(p): 2413-8835
Vol. 2, No. 8, pp: 90-99, 2016
URL: http://arpgweb.com/?ic=journal&journal=10&info=aims
90
Academic Research Publishing Group
The Study of the Wiener Processes Base on Haar Wavelet
Xuewen Xia College of Science, Hunan Institute of Engineering, Xiangtan,411104, Hunan Normal University, Changsha,
410082, China
1. Introduction
Wiener processes are a sort of importment stochastic processes. Wiener square processes is a class of useful
stochastic processes in practies, its study is very value.
We will take wavelet and use them in a series expansion of signal or function. Wavelet has its energy
concentrated in time to give a tool for the analysis of transient, nonstationary, or time-varying phenomena. It still has
the oscillating wavelike characteristic but also has the ability to allow simultaneous time and frequency analysis with
a flexible mathematical foundation. We take wavelet and use them in a series expansion of signals or functions much
the same way a Fourier series the wave or sinusoid to represent a signal or function. In order to use the idea of
multiresolution, we will start by defining the scaling function and then define the wavelet in terms of it.
With the rapid development of computerized scientific instruments comes a wide variety of interesting problems
for data analysis and signal processing. In fields ranging from Extragalactic Astronomy to Molecular Spectroscopy
to Medical Imaging to computer vision, One must recover a signal, curve, image, spectrum, or density from
incomplete, indirect, and noisy data .Wavelets have contributed to this already intensely developed and rapidly
advancing field .
Wavelet analysis consists of a versatile collection of tools for the analysis and manipulation of signals such as
sound and images as well as more general digital data sets ,such as speech, electrocardiograms ,images .Wavelet
analysis is a remarkable tool for analyzing function of one or several variables that appear in mathematics or in
signal and image processing .With hindsight the wavelet transform can be viewed as diverse as mathematics ,physics
and electrical engineering .The basic idea is always to use a family of building blocks to represent the object at hand
in an efficient and insightful way, the building blocks themselves come in different sizes, and are suitable for
describing features with a resolution commensurate with their size .
There are two important aspects to wavelets, which we shall call “mathematical” and “algorithmical”
.Numerical algorithms using wavelet bases are similar to other transform methods in that vectors and operators are
expanded into a basis and the computations take place in the new system of coordinates .As with all transform
methods such as approach hopes to achieve that the computation is faster in the new system of coordinates than in
the original domain, wavelet based algorithms exhibit a number of new and important properties .Recently some
persons have studied wavelet problems of stochastic process or stochastic system ([1-18]).
2. Basic Definitions
Definition 1
Let
2
( ) ( ), 0X t W t t (1)
W(t) is Wiener processes, we call X(t) is Wiener square processes.
We have
2 2 2
( ( ) ( ( )) ( ( ))E W t D W t E W t t
Let s<t, we have
Abstract: The stochastic system is very importment in many aspacts. Wiener processes is a sort of importment
stochastic processes. Wiener square processes is a class of useful stochastic processes in practies,its study is very
value.In this paper,we study Wiener square processes using haar wavelet and wavelet transform.we study its some
properties and wavelet expansion. Index Wiener Integral processes is a class of useful stochastic processes in
practies,its study is very value.In this paper,we study it using haar wavelet and wavelet transform on [0,t].we study
its some properties and wavelet expansion.
Keywords: Wiener square processes; Wavelet; Haar; Expansion; Index Wiener Integral processes.](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-1-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
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2 2
2 2
( , ) ( ) ( ) ( ( ) ( ))
( ( )( ( ) ( ) ( ))
R s t EX t X s E W s W t
E W s W t W s W s
2 2 2
( ( )( ( ) ( )) ( ( ))E W s W t W s E W s
2 2 4 2
4 4 2
( ) 3
( ) 3
s t s s
s t s s
（2）
Definition 2
Let ｛x(t) , t∈R｝is a stochastic processes on probability space （Ω, g, P）,we call
dt
s
tx
tx
s R
)()(
1
x)W(s,
(3)
is wavelet transform of x(t) . where,ψ is mather wavelet([11]).
Then, we have
1
( , ) ( ) ( )
R
x t
w s x x t dt
s s
(4)
Definition 3
Let mather waveletψ(x) is function:
other
x
x
,0
1
2
1
,1
2
1
0,1
(x) (5)
we call ψ(x) is the Haar wavelet.
Then, we have
1,
2( )
1,
2
s
x t x
x t
ss
x s t x
(6)
1,
2( )
1,
2
s
x t x
x t
ss
x s t x
(7)
3. Some Results about Density Degree
We have
1
1 1
( ) [ ( , ) ( , )]
1 1
[ ( ) ( ) ][ ( ) ( ) ]
R R
R E w s y w s y
y ty t
E x t dt x t dt
s s s s
2
1
1 12
1
[ ( ) ( ) ( ) ( ) ]
R
y ty t
E x t x t dtdt
s s s
1
1 12
1
[ ( ) ( )] ( ) ( )
y ty t
E x t x t dtdt
s s s
2
4 4 2
1 1 12
1
1
1
[ ( ) 3 ] ( )
( )
R
y t
t t t t
s s
y t
dtdt
s
](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-2-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
92
2
2
4 1
1 1 12
4 2 1
1 1
1
[ ( ) ( ) ( )
3 ( ) ( ) ]
R
R
y ty t
t t t dtdt
s s s
y ty t
t dtdt
s s
2
2
4 1
1 12
4 2 1
1 1
1
[ ( ) ( )
2 ( ) ( ) ]
R
R
y ty t
t t dtdt
s s s
y ty t
t dtdt
s s
1 2I I
Where
2
2
4 1
1 1 12
4 2 1
2 1 12
1
( ) ( ) ,
1
2 ( ) ( )
R
R
y ty t
I t t dtdt
s s s
y ty t
I t dtdt
s s s
Then,we have
/2
4
1 1 1 1 12 /2 /2 /2
/2 /2 /2 /2
1 1 1 1/2
1
[
]
y y y y s
y s y s y s y s
y s y s y s y s
y s y s y s y s
I tdt t dt tdt t dt
s
tdt t dt tdt t dt
The same time, we have
2I 2
2 2 1
1 12
1
2 [ ( ) ( ) ]
R
y ty t
t dtdt
s s s
/2
2 2 2
1 1 1 12 /2 /2 /2
/2 /2 /2 /2
2 2
1 1 1 1/2
2
[
]
y y y y s
y s y s y s y s
y s y s y s y s
y s y s y s y s
dt t dt dt t dt
s
dt t dt dt t dt
We let 1 , through compute on above
Then, the zero density degree of W(s , y) is
2
(0)
(0)
R
R
can be obtained.
The average density degree of w(s, y) is
)0(
)0(
)2(2
)4(
R
R
can be obtain all.
4. Wavelet Expansion of System
In order to use the idea of multiresolution, we will start by defining the scaling function and then define the
wavelet in terms of it.
Let real function is standard orthogonal element of multiresolution analysis ZjVj }{ (see [7]), then exist
2
lhk , have
k
ktt )2(2)(
Let
k
k
k
ktht )2()1(2)( 1
Then wavelet express of ( )y t in mean square is
2
2
( ) 2 (2 )
2 (2 )
J
J J
n
K
j
j j
n
j J n Z
y t C t n
d t n
Where](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-3-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
93
2
2 ( ) (2 )
j
j j
n R
C x t t n dt
2
2 ( ) (2 )
j
j j
n R
d x t t n dt
Then have
2
2
2 ( ) ( )
(2 ) (2 )
j k
j k
n m R
j k
E C C E x t x s
t n s m dsdt
2
2
2 ( ) ( )
(2 ) (2 )
j k
j k
n m R
j k
E d d E x t x s
t n s m dsdt
Where
1, 2 (1/ 2 )2
(2 )
1,(1/ 2 )2 (1 )2
j j
j
j j
m t m
t m
m t m
(8)
1, 2 (1/ 2 )2
(2 )
1,(1/ 2 )2 (1 )2
k k
k
k k
n s n
s n
n s n
(9)
Use (8) and (10) ,we can obtain value of
j k
n mE d d .
If we let normalized scaling function to have compact support over [0,1],then a solution is a scaling function that is a
simple rectangle function
1,0 1
( )
0,
t
t
otherwise
(10)
Now we consider function )(t that exist compact support set on 0,],,[ 2121 kkkk , and exist enough large M,
have 10,0)( Mmdttt
R
m
, then exist compact support set on ],[ 43 kk satisfy
0,, 434321 kkkkkk .
Let ( , ) ( ), jkb j k y t
( , ) ( ), jka j k y t
Let J is a constant, then
Jj
j
j
J
J
ZkktZkkx
),2(2),2(2 22
are a standard orthonormal basis of space )(2
RL ,
then have
2
2
( ) 2 ( , ) (2 )
2 ( , ) (2 )
J
J
K Z
j
j
j J K Z
y t a J K t K
b j K t K
(11)
Therefore, the self-correlation function of ),( mjb
),(),(),;,( nkbmjbEnmKjRb
2
2
2 ( ) ( ) (2 ) (2 )
j K
j K
R
E x t x s t m s n dtds
(12)
And have also the self-correlation function of ( , )a j m
( , ; , ) ( , ) ( , )aR j K m n E a j m a k n
2
2
2 ( ) ( ) (2 ) (2 )
j K
j K
R
E x t x s t m s n dtds
(13)](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-4-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
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Then ,we use (8) and (9) have
2
2
( , ; ,
2 [ ( ) ( )] (2 ) (2 )
b
j k
j k
R
R j k m n
E x t x s t m s n dtds
）
2
2
2 (2 ) (2 )
j k
j k
R
ts s n s m dtds
2
22
2 2 (2 ) (2 )
j k
j k
R
t s n s m dtds
2
2
( , ; , )
2 [ ( ) ( )] (2 ) (2 )
a
j k
j k
R
R j k m n
E x t x s t m s n dtds
2
2
2 (2 ) (2 )
j k
j k
R
ts s n s m dtds
2
22
2 2 (2 ) (2 )
j k
j k
R
t s n s m dtds
We have
(1/2 )2 (1/2 )2
2
2 2
( , ; , ) 2 [
j k
j k
j k
m n
b m n
R j k m n tdt sds
(1/2 )2 (1 )2 (1 )2 (1/2 )2
2 (1/2 )2 (1/2 )2 2
j k j k
j k j k
m n m n
m n m n
tdt sds tdt sds
(1 )2 (1 )2
(1/2 )2 (1/2 )2
]
j k
j k
m n
m n
tdt sds
(1/2 )2 (1/2 )2
22
2 2
2 [ 2
j k
j k
j k
m n
m n
t dt ds
(1/2 )2 (1 )2
2
2 (1/2 )2
(1 )2 (1/2 )2
2
(1/2 )2 2
2
2
j k
j k
j k
j k
m n
m n
m n
m n
t dt ds
t dt ds
(1 )2 (1 )2
2
(1/2 )2 (1/2 )2
2 ]
j k
j k
m n
m n
t dt ds
We obtain
1, 2 ( 1)2
(2 )
0,
j j
j n t n
t n
other
1, 2 ( 1)2
(2 )
0,
k k
k m s m
s m
other
Then ,we have
( , ; , )aR j k m n 2
2
2 (2 ) (2 )
j k
j k
R
ts s n s m dtds
2
22
2 2 (2 ) (2 )
j k
j k
R
t s n s m dtds
5. Basic Definitions of Wiener Processes
Definition 1
Let
( )
0
( ) ( ), 0, 0
t
a t u
X t e dw u t a R
W(u) is Wiener processes, we call X(t) is Index
Wiener Integral processes.
We have
( ) ( )
0 0
2
( ) ( )
2
( , ) ( ) ( )
( ) ( )
( )
2
1
s t
a s u a t u
a s t a t s
R s t EX t X s
E e dw u e dw u
e e
a
Let
](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-5-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
95
Definition 2
Let ｛x(t) , t∈R｝is a stochastic processes on probability space （Ω, g, P）,we call
W(s, x)＝ dt
s
tx
tx
s R
)()(
1
ψ
is wavelet transform of x(t) . where,ψ is mather wavelet([11]).
Then, we have
1
( , ) ( ) ( )
R
x t
w s x x t dt
s s
Definition 3
Let mather waveletψ(x) is function:
other
x
x
,0
1
2
1
,1
2
1
0,1
(x) (5)
we call ψ(x) is the Haar wavelet.
Then, we have
1,
2( )
1,
2
s
x t x
x t
ss
x s t x
1,
2( )
1,
2
s
x t x
x t
ss
x s t x
We have
1
1 1
( ) [ ( , ) ( , )]
1 1
[ ( ) ( ) ][ ( ) ( ) ]
R R
R E w s y w s y
y ty t
E x t dt x t dt
s s s s
2
1
1 12
1
[ ( ) ( ) ( ) ( ) ]
R
y ty t
E x t x t dtdt
s s s
1
1 12
1
[ ( ) ( )] ( ) ( )
y ty t
E x t x t dtdt
s s s
1 1
2
( ) ( )
2
1
1
1 1
( ) ( )
2
( )
a t t a t t
R
y t
e e
s a s
y t
dtdt
s
1
2
1
2
( ) 1
12
( ) 1
1
1 1
[ ( ) ( )
2
1
( ) ( ) ]
2
a t t
R
a t t
R
y ty t
e dtdt
s a s s
y ty t
e dtdt
a s s
1 2I I
Where
1
2
1
2
( ) 1
1 12
( ) 1
2 1
1 1
[ ( ) ( ) ,
2
1
( ) ( ) ]
2
a t t
R
a t t
R
y ty t
I e dtdt
s a s s
y ty t
I e dtdt
a s s
Then,we have](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-6-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
96
1
1
1
1
1 12 /2 /2
/2
1/2
/2 /2
1/2
/2 /2
1
1
[
2
]
y y
atat
y s y s
y y s
atat
y s y s
y s y s
atat
y s y s
y s y s
atat
y s y s
I e dt e dt
as
e dt e dt
e dt e dt
e dt e dt
The same time, we have
2I 1
2
1
12
1
( ) ( ) ]
2
at at
R
y ty t
e dtdt
as s s
1
1
1
1
12 /2 /2
/2
1/2
/2 /2
1/2
/2 /2
1
1
[
2
]
y y
atat
y s y s
y y s
atat
y s y s
y s y s
atat
y s y s
y s y s
atat
y s y s
e dt e dt
as
e dt e dt
e dt e dt
e dt e dt
Then, the zero density degree of W(s, y) is
2
(0)
(0)
R
R
can be obtained.
The average density degree of w(s, y) is
)0(
)0(
)2(2
)4(
R
R
can be obtain all.
We consider wavelet expansions of stochastics processes and show that for certain wavelets, the coefficients of
the expansion have negligible correlation for different scales. we can introduce a modification of the wavelets.
Certain nonstationary processes the wavelets may be chose to give uncorrelated coefficients.
In order to use the idea of multiresolution , we will start by defining the scaling function and then define the
wavelet in terms of it.
Let real function is standard orthogonal element of multiresolution analysis ZjVj }{ (see [7]), then exist
2
lhk , have
k
ktt )2(2)(
Let
k
k
k
ktht )2()1(2)( 1
Then wavelet express of ( )y t in mean square is
2
2
( ) 2 (2 )
2 (2 )
J
J J
n
K
j
j j
n
j J n Z
y t C t n
d t n
Where, 2
2 ( ) (2 )
j
j j
n R
C x t t n dt
2
2 ( ) (2 )
j
j j
n R
d x t t n dt
Then have](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-7-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
97
2
2
2 ( ) ( ) (2 ) (2 )
j k
n m
j k
j k
R
E C C
E x t x s t n s m dsdt
2
2
2 ( ) ( ) (2 ) (2 )
j k
n m
j k
j k
R
E d d
E x t x s t n s m dsdt
Where
1, 2 (1/ 2 )2
(2 )
1,(1/ 2 )2 (1 )2
j j
j
j j
m t m
t m
m t m
(8)
1, 2 (1/ 2 )2
(2 )
1,(1/ 2 )2 (1 )2
k k
k
k k
n s n
s n
n s n
(9)
Use (8) and (10) , we can obtain value of
j k
n mE d d .
If we let normalized scaling function to have compact support over [0,1],then a solution is a scaling function
that is a simple rectangle function
1,0 1
( )
0,
t
t
otherwise
(10) Now we consider function )(t that exist compact support set on
0,],,[ 2121 kkkk , and exist enough large M, have 10,0)( Mmdttt
R
m
, then exist compact
support set on ],[ 43 kk satisfy 0,, 434321 kkkkkk .
Let ( , ) ( ), jkb j k y t ，
( , ) ( ), jka j k y t
Let J is a constant, then
Jj
j
j
J
J
ZkktZkkx
),2(2),2(2 22
are a standard orthonormal basis of space )(2
RL ,
then have
2
2
( ) 2 ( , ) (2 )
2 ( , ) (2 )
J
J
K Z
j
j
j J K Z
y t a J K t K
b j K t K
(11)
Therefore, the self-correlation function of ),( mjb
),(),(),;,( nkbmjbEnmKjRb
2
2
2 ( ) ( ) (2 ) (2 )
j K
j K
R
E x t x s t m s n dtds
(12)
And have also the self-correlation function of ( , )a j m
( , ; , ) ( , ) ( , )aR j K m n E a j m a k n 2
2
2 ( ) ( ) (2 ) (2 )
j K
j K
R
E x t x s t m s n dtds
(13)
Then , we use (8) and (9) have
2
2
( , ; , )
2 [ ( ) ( )] (2 ) (2 )
b
j k
j k
R
R j k m n
E x t x s t m s n dtds
2
( )2
2 (2 ) (2 )
j k
a t s j k
R
e s n s m dtds
2
( )2
2 (2 ) (2 )
j k
a t s j k
R
e s n s m dtds
](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-8-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
98
2
2
( , ; , )
2 [ ( ) ( )] (2 ) (2 )
a
j k
j k
R
R j k m n
E x t x s t m s n dtds
2
( )2
2 (2 ) (2 )
j k
a s t j k
R
e s n s m dtds
2
( )2
2 (2 ) (2 )
j k
a t s j k
R
e s n s m dtds
We have
(1/2 )2 (1/2 )2
2
2 2
( , ; , )
2 [
j k
j k
b
j k
m n
at as
m n
R j k m n
e dt e ds
(1/2 )2 (1 )2
2 (1/2 )2
(1 )2 (1/2 )2
(1/2 )2 2
j k
j k
j k
j k
m n
at as
m n
m n
at as
m n
e dt e ds
e dt e ds
(1 )2 (1 )2
(1/2 )2 (1/2 )2
]
j k
j k
m n
at as
m n
e dt e ds
(1/2 )2 (1/2 )2
2
2 2
2 [
j k
j k
j k
m n
at as
m n
e dt e ds
(1/2 )2 (1 )2
2 (1/2 )2
(1 )2 (1/2 )2
(1/2 )2 2
j k
j k
j k
j k
m n
at as
m n
m n
at as
m n
e dt e ds
e dt e ds
(1 )2 (1 )2
(1/2 )2 (1/2 )2
]
j k
j k
m n
at as
m n
e dt e ds
We obtain
1, 2 ( 1)2
(2 )
0,
j j
j n t n
t n
other
1, 2 ( 1)2
(2 )
0,
k k
k m s m
s m
other
Then, we have
( , ; , )aR j k m n 2
( )2
2 (2 ) (2 )
j k
a t s j k
R
e s n s m dtds
2
( )2
2 (2 ) (2 )
j k
a t s j k
R
e s n s m dtds
References
[1] Adam Bobrowski, 2007. Functional analysis for probability and stochastic processes. Cambridge
University Press.
[2] Cambancs, 1994. "Wavelet approximation of deterministic and random signals." IEEE Tran.on Information
Theory, vol. 40, pp. 1013-1029.
[3] Flandrin, 1992. "Wavelet analysis and synthesis of fractional brownian motion." IEEE Tran.on Information
Theory, vol. 38, pp. 910-916.
[4] Haobo Ren, 2002. "Wavelet estimation for jumps on a heterosedastic regression model." Acta Mathematica
Scientia, vol. 22, pp. 269-277.
[5] Hendi, A., 2009. "New exact travelling wave solutions for some nonlinear evolution equations."
International Journal of Nonlinear Science, vol. 7, pp. 259-267.
[6] Krim, 1995. "Multire solution analysis of a class nstationary processes." IEEE Tran.on Information Theory,
vol. 41, pp. 1010-1019.
[7] Meyer, Y., 1990. "In Ondelettes et operatears(M), Hermann."
[8] Pairick Flandrin, 1992. "Wavelet analysis and synthesis of fractional brownian motion." IEEE Tran.,On
Information Theory, vol. 38, pp. 910-916.
[9] Paul Malliavin, 2003. Stochastic analysis. Springer-Verlag.
[10] Xia Xuewen, 2012. "The study of Wiener Processes with linear-trend base on wavelet." Studies in
Mathematical Science, vol. 2012, p. 2.
[11] Xia Xuewen, 2012. "The energy of stochastic vibration system of a class of protein base on wavelet."
Advances in Nature Science, vol. 5, pp. 68-70.
[12] Xia Xuewen and Dai Ting, 2012. "The energy of convolution of 2-dimension exponential random variables
base on Haar wavelet." Progress in Applied Mathematics, vol. 3, pp. 35-38.
[13] Xuewen Xia, 2005. "Wavelet analysis of the stochastic system with coular stationary noise." Engineering
Science, vol. 3, pp. 43-46.
[14] Xuewen Xia, 2007. "Wavelet analysis of Browain motion." World Journal of Modelling and Simulation,
vol. 3, pp. 106-112.](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-9-320.jpg)
![Scientific Review, 2016, 2(8): 90-99
99
[15] Xuewen Xia, 2008. "Wavelet density degree of continuous parameter AR model." International Journal
Nonlinear Science, vol. 7, pp. 237-242.
[16] Xuewen Xia, 2010. "The morlet wavelet density degree of the linear regress processes with random
coefficient." International Journal of Nonlinear Science, vol. 9, pp. 349-351.
[17] Xuewen Xia, 2011. "The study of a class of the fractional brownian motion based on wavelet." Inter.J.of
Nonlinear Science, vol. 11, pp. 290-294.
[18] Xuewen Xia and Ting Dai, 2009. "Wavelet density degree of a class of Wiener processes." International
Journal of Nonlinear Science, vol. 7, pp. 327-332.
A Project supported by Hunan Provinial Natural Science Foundation of China (12JJ9004);
A Project Supported by Scientific Research Found of Hunan Provincial Education Department (12A033).](https://image.slidesharecdn.com/sr2890-99-200902071511/85/The-Study-of-the-Wiener-Processes-Base-on-Haar-Wavelet-10-320.jpg)
The Study of the Wiener Processes Base on Haar Wavelet
- 1. Scientific Review ISSN(e): 2412-2599, ISSN(p): 2413-8835 Vol. 2, No. 8, pp: 90-99, 2016 URL: http://arpgweb.com/?ic=journal&journal=10&info=aims 90 Academic Research Publishing Group The Study of the Wiener Processes Base on Haar Wavelet Xuewen Xia College of Science, Hunan Institute of Engineering, Xiangtan,411104, Hunan Normal University, Changsha, 410082, China 1. Introduction Wiener processes are a sort of importment stochastic processes. Wiener square processes is a class of useful stochastic processes in practies, its study is very value. We will take wavelet and use them in a series expansion of signal or function. Wavelet has its energy concentrated in time to give a tool for the analysis of transient, nonstationary, or time-varying phenomena. It still has the oscillating wavelike characteristic but also has the ability to allow simultaneous time and frequency analysis with a flexible mathematical foundation. We take wavelet and use them in a series expansion of signals or functions much the same way a Fourier series the wave or sinusoid to represent a signal or function. In order to use the idea of multiresolution, we will start by defining the scaling function and then define the wavelet in terms of it. With the rapid development of computerized scientific instruments comes a wide variety of interesting problems for data analysis and signal processing. In fields ranging from Extragalactic Astronomy to Molecular Spectroscopy to Medical Imaging to computer vision, One must recover a signal, curve, image, spectrum, or density from incomplete, indirect, and noisy data .Wavelets have contributed to this already intensely developed and rapidly advancing field . Wavelet analysis consists of a versatile collection of tools for the analysis and manipulation of signals such as sound and images as well as more general digital data sets ,such as speech, electrocardiograms ,images .Wavelet analysis is a remarkable tool for analyzing function of one or several variables that appear in mathematics or in signal and image processing .With hindsight the wavelet transform can be viewed as diverse as mathematics ,physics and electrical engineering .The basic idea is always to use a family of building blocks to represent the object at hand in an efficient and insightful way, the building blocks themselves come in different sizes, and are suitable for describing features with a resolution commensurate with their size . There are two important aspects to wavelets, which we shall call “mathematical” and “algorithmical” .Numerical algorithms using wavelet bases are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in the new system of coordinates .As with all transform methods such as approach hopes to achieve that the computation is faster in the new system of coordinates than in the original domain, wavelet based algorithms exhibit a number of new and important properties .Recently some persons have studied wavelet problems of stochastic process or stochastic system ([1-18]). 2. Basic Definitions Definition 1 Let 2 ( ) ( ), 0X t W t t (1) W(t) is Wiener processes, we call X(t) is Wiener square processes. We have 2 2 2 ( ( ) ( ( )) ( ( ))E W t D W t E W t t Let s<t, we have Abstract: The stochastic system is very importment in many aspacts. Wiener processes is a sort of importment stochastic processes. Wiener square processes is a class of useful stochastic processes in practies,its study is very value.In this paper,we study Wiener square processes using haar wavelet and wavelet transform.we study its some properties and wavelet expansion. Index Wiener Integral processes is a class of useful stochastic processes in practies,its study is very value.In this paper,we study it using haar wavelet and wavelet transform on [0,t].we study its some properties and wavelet expansion. Keywords: Wiener square processes; Wavelet; Haar; Expansion; Index Wiener Integral processes.
- 2. Scientific Review, 2016, 2(8): 90-99 91 2 2 2 2 ( , ) ( ) ( ) ( ( ) ( )) ( ( )( ( ) ( ) ( )) R s t EX t X s E W s W t E W s W t W s W s 2 2 2 ( ( )( ( ) ( )) ( ( ))E W s W t W s E W s 2 2 4 2 4 4 2 ( ) 3 ( ) 3 s t s s s t s s （2） Definition 2 Let ｛x(t) , t∈R｝is a stochastic processes on probability space （Ω, g, P）,we call dt s tx tx s R )()( 1 x)W(s, (3) is wavelet transform of x(t) . where,ψ is mather wavelet([11]). Then, we have 1 ( , ) ( ) ( ) R x t w s x x t dt s s (4) Definition 3 Let mather waveletψ(x) is function: other x x ,0 1 2 1 ,1 2 1 0,1 (x) (5) we call ψ(x) is the Haar wavelet. Then, we have 1, 2( ) 1, 2 s x t x x t ss x s t x (6) 1, 2( ) 1, 2 s x t x x t ss x s t x (7) 3. Some Results about Density Degree We have 1 1 1 ( ) [ ( , ) ( , )] 1 1 [ ( ) ( ) ][ ( ) ( ) ] R R R E w s y w s y y ty t E x t dt x t dt s s s s 2 1 1 12 1 [ ( ) ( ) ( ) ( ) ] R y ty t E x t x t dtdt s s s 1 1 12 1 [ ( ) ( )] ( ) ( ) y ty t E x t x t dtdt s s s 2 4 4 2 1 1 12 1 1 1 [ ( ) 3 ] ( ) ( ) R y t t t t t s s y t dtdt s
- 3. Scientific Review, 2016, 2(8): 90-99 92 2 2 4 1 1 1 12 4 2 1 1 1 1 [ ( ) ( ) ( ) 3 ( ) ( ) ] R R y ty t t t t dtdt s s s y ty t t dtdt s s 2 2 4 1 1 12 4 2 1 1 1 1 [ ( ) ( ) 2 ( ) ( ) ] R R y ty t t t dtdt s s s y ty t t dtdt s s 1 2I I Where 2 2 4 1 1 1 12 4 2 1 2 1 12 1 ( ) ( ) , 1 2 ( ) ( ) R R y ty t I t t dtdt s s s y ty t I t dtdt s s s Then,we have /2 4 1 1 1 1 12 /2 /2 /2 /2 /2 /2 /2 1 1 1 1/2 1 [ ] y y y y s y s y s y s y s y s y s y s y s y s y s y s y s I tdt t dt tdt t dt s tdt t dt tdt t dt The same time, we have 2I 2 2 2 1 1 12 1 2 [ ( ) ( ) ] R y ty t t dtdt s s s /2 2 2 2 1 1 1 12 /2 /2 /2 /2 /2 /2 /2 2 2 1 1 1 1/2 2 [ ] y y y y s y s y s y s y s y s y s y s y s y s y s y s y s dt t dt dt t dt s dt t dt dt t dt We let 1 , through compute on above Then, the zero density degree of W(s , y) is 2 (0) (0) R R can be obtained. The average density degree of w(s, y) is )0( )0( )2(2 )4( R R can be obtain all. 4. Wavelet Expansion of System In order to use the idea of multiresolution, we will start by defining the scaling function and then define the wavelet in terms of it. Let real function is standard orthogonal element of multiresolution analysis ZjVj }{ (see [7]), then exist 2 lhk , have k ktt )2(2)( Let k k k ktht )2()1(2)( 1 Then wavelet express of ( )y t in mean square is 2 2 ( ) 2 (2 ) 2 (2 ) J J J n K j j j n j J n Z y t C t n d t n Where
- 4. Scientific Review, 2016, 2(8): 90-99 93 2 2 ( ) (2 ) j j j n R C x t t n dt 2 2 ( ) (2 ) j j j n R d x t t n dt Then have 2 2 2 ( ) ( ) (2 ) (2 ) j k j k n m R j k E C C E x t x s t n s m dsdt 2 2 2 ( ) ( ) (2 ) (2 ) j k j k n m R j k E d d E x t x s t n s m dsdt Where 1, 2 (1/ 2 )2 (2 ) 1,(1/ 2 )2 (1 )2 j j j j j m t m t m m t m (8) 1, 2 (1/ 2 )2 (2 ) 1,(1/ 2 )2 (1 )2 k k k k k n s n s n n s n (9) Use (8) and (10) ,we can obtain value of j k n mE d d . If we let normalized scaling function to have compact support over [0,1],then a solution is a scaling function that is a simple rectangle function 1,0 1 ( ) 0, t t otherwise (10) Now we consider function )(t that exist compact support set on 0,],,[ 2121 kkkk , and exist enough large M, have 10,0)( Mmdttt R m , then exist compact support set on ],[ 43 kk satisfy 0,, 434321 kkkkkk . Let ( , ) ( ), jkb j k y t ( , ) ( ), jka j k y t Let J is a constant, then Jj j j J J ZkktZkkx ),2(2),2(2 22 are a standard orthonormal basis of space )(2 RL , then have 2 2 ( ) 2 ( , ) (2 ) 2 ( , ) (2 ) J J K Z j j j J K Z y t a J K t K b j K t K (11) Therefore, the self-correlation function of ),( mjb ),(),(),;,( nkbmjbEnmKjRb 2 2 2 ( ) ( ) (2 ) (2 ) j K j K R E x t x s t m s n dtds (12) And have also the self-correlation function of ( , )a j m ( , ; , ) ( , ) ( , )aR j K m n E a j m a k n 2 2 2 ( ) ( ) (2 ) (2 ) j K j K R E x t x s t m s n dtds (13)
- 5. Scientific Review, 2016, 2(8): 90-99 94 Then ,we use (8) and (9) have 2 2 ( , ; , 2 [ ( ) ( )] (2 ) (2 ) b j k j k R R j k m n E x t x s t m s n dtds ） 2 2 2 (2 ) (2 ) j k j k R ts s n s m dtds 2 22 2 2 (2 ) (2 ) j k j k R t s n s m dtds 2 2 ( , ; , ) 2 [ ( ) ( )] (2 ) (2 ) a j k j k R R j k m n E x t x s t m s n dtds 2 2 2 (2 ) (2 ) j k j k R ts s n s m dtds 2 22 2 2 (2 ) (2 ) j k j k R t s n s m dtds We have (1/2 )2 (1/2 )2 2 2 2 ( , ; , ) 2 [ j k j k j k m n b m n R j k m n tdt sds (1/2 )2 (1 )2 (1 )2 (1/2 )2 2 (1/2 )2 (1/2 )2 2 j k j k j k j k m n m n m n m n tdt sds tdt sds (1 )2 (1 )2 (1/2 )2 (1/2 )2 ] j k j k m n m n tdt sds (1/2 )2 (1/2 )2 22 2 2 2 [ 2 j k j k j k m n m n t dt ds (1/2 )2 (1 )2 2 2 (1/2 )2 (1 )2 (1/2 )2 2 (1/2 )2 2 2 2 j k j k j k j k m n m n m n m n t dt ds t dt ds (1 )2 (1 )2 2 (1/2 )2 (1/2 )2 2 ] j k j k m n m n t dt ds We obtain 1, 2 ( 1)2 (2 ) 0, j j j n t n t n other 1, 2 ( 1)2 (2 ) 0, k k k m s m s m other Then ,we have ( , ; , )aR j k m n 2 2 2 (2 ) (2 ) j k j k R ts s n s m dtds 2 22 2 2 (2 ) (2 ) j k j k R t s n s m dtds 5. Basic Definitions of Wiener Processes Definition 1 Let ( ) 0 ( ) ( ), 0, 0 t a t u X t e dw u t a R W(u) is Wiener processes, we call X(t) is Index Wiener Integral processes. We have ( ) ( ) 0 0 2 ( ) ( ) 2 ( , ) ( ) ( ) ( ) ( ) ( ) 2 1 s t a s u a t u a s t a t s R s t EX t X s E e dw u e dw u e e a Let
- 6. Scientific Review, 2016, 2(8): 90-99 95 Definition 2 Let ｛x(t) , t∈R｝is a stochastic processes on probability space （Ω, g, P）,we call W(s, x)＝ dt s tx tx s R )()( 1 ψ is wavelet transform of x(t) . where,ψ is mather wavelet([11]). Then, we have 1 ( , ) ( ) ( ) R x t w s x x t dt s s Definition 3 Let mather waveletψ(x) is function: other x x ,0 1 2 1 ,1 2 1 0,1 (x) (5) we call ψ(x) is the Haar wavelet. Then, we have 1, 2( ) 1, 2 s x t x x t ss x s t x 1, 2( ) 1, 2 s x t x x t ss x s t x We have 1 1 1 ( ) [ ( , ) ( , )] 1 1 [ ( ) ( ) ][ ( ) ( ) ] R R R E w s y w s y y ty t E x t dt x t dt s s s s 2 1 1 12 1 [ ( ) ( ) ( ) ( ) ] R y ty t E x t x t dtdt s s s 1 1 12 1 [ ( ) ( )] ( ) ( ) y ty t E x t x t dtdt s s s 1 1 2 ( ) ( ) 2 1 1 1 1 ( ) ( ) 2 ( ) a t t a t t R y t e e s a s y t dtdt s 1 2 1 2 ( ) 1 12 ( ) 1 1 1 1 [ ( ) ( ) 2 1 ( ) ( ) ] 2 a t t R a t t R y ty t e dtdt s a s s y ty t e dtdt a s s 1 2I I Where 1 2 1 2 ( ) 1 1 12 ( ) 1 2 1 1 1 [ ( ) ( ) , 2 1 ( ) ( ) ] 2 a t t R a t t R y ty t I e dtdt s a s s y ty t I e dtdt a s s Then,we have
- 7. Scientific Review, 2016, 2(8): 90-99 96 1 1 1 1 1 12 /2 /2 /2 1/2 /2 /2 1/2 /2 /2 1 1 [ 2 ] y y atat y s y s y y s atat y s y s y s y s atat y s y s y s y s atat y s y s I e dt e dt as e dt e dt e dt e dt e dt e dt The same time, we have 2I 1 2 1 12 1 ( ) ( ) ] 2 at at R y ty t e dtdt as s s 1 1 1 1 12 /2 /2 /2 1/2 /2 /2 1/2 /2 /2 1 1 [ 2 ] y y atat y s y s y y s atat y s y s y s y s atat y s y s y s y s atat y s y s e dt e dt as e dt e dt e dt e dt e dt e dt Then, the zero density degree of W(s, y) is 2 (0) (0) R R can be obtained. The average density degree of w(s, y) is )0( )0( )2(2 )4( R R can be obtain all. We consider wavelet expansions of stochastics processes and show that for certain wavelets, the coefficients of the expansion have negligible correlation for different scales. we can introduce a modification of the wavelets. Certain nonstationary processes the wavelets may be chose to give uncorrelated coefficients. In order to use the idea of multiresolution , we will start by defining the scaling function and then define the wavelet in terms of it. Let real function is standard orthogonal element of multiresolution analysis ZjVj }{ (see [7]), then exist 2 lhk , have k ktt )2(2)( Let k k k ktht )2()1(2)( 1 Then wavelet express of ( )y t in mean square is 2 2 ( ) 2 (2 ) 2 (2 ) J J J n K j j j n j J n Z y t C t n d t n Where, 2 2 ( ) (2 ) j j j n R C x t t n dt 2 2 ( ) (2 ) j j j n R d x t t n dt Then have
- 8. Scientific Review, 2016, 2(8): 90-99 97 2 2 2 ( ) ( ) (2 ) (2 ) j k n m j k j k R E C C E x t x s t n s m dsdt 2 2 2 ( ) ( ) (2 ) (2 ) j k n m j k j k R E d d E x t x s t n s m dsdt Where 1, 2 (1/ 2 )2 (2 ) 1,(1/ 2 )2 (1 )2 j j j j j m t m t m m t m (8) 1, 2 (1/ 2 )2 (2 ) 1,(1/ 2 )2 (1 )2 k k k k k n s n s n n s n (9) Use (8) and (10) , we can obtain value of j k n mE d d . If we let normalized scaling function to have compact support over [0,1],then a solution is a scaling function that is a simple rectangle function 1,0 1 ( ) 0, t t otherwise (10) Now we consider function )(t that exist compact support set on 0,],,[ 2121 kkkk , and exist enough large M, have 10,0)( Mmdttt R m , then exist compact support set on ],[ 43 kk satisfy 0,, 434321 kkkkkk . Let ( , ) ( ), jkb j k y t ， ( , ) ( ), jka j k y t Let J is a constant, then Jj j j J J ZkktZkkx ),2(2),2(2 22 are a standard orthonormal basis of space )(2 RL , then have 2 2 ( ) 2 ( , ) (2 ) 2 ( , ) (2 ) J J K Z j j j J K Z y t a J K t K b j K t K (11) Therefore, the self-correlation function of ),( mjb ),(),(),;,( nkbmjbEnmKjRb 2 2 2 ( ) ( ) (2 ) (2 ) j K j K R E x t x s t m s n dtds (12) And have also the self-correlation function of ( , )a j m ( , ; , ) ( , ) ( , )aR j K m n E a j m a k n 2 2 2 ( ) ( ) (2 ) (2 ) j K j K R E x t x s t m s n dtds (13) Then , we use (8) and (9) have 2 2 ( , ; , ) 2 [ ( ) ( )] (2 ) (2 ) b j k j k R R j k m n E x t x s t m s n dtds 2 ( )2 2 (2 ) (2 ) j k a t s j k R e s n s m dtds 2 ( )2 2 (2 ) (2 ) j k a t s j k R e s n s m dtds
- 9. Scientific Review, 2016, 2(8): 90-99 98 2 2 ( , ; , ) 2 [ ( ) ( )] (2 ) (2 ) a j k j k R R j k m n E x t x s t m s n dtds 2 ( )2 2 (2 ) (2 ) j k a s t j k R e s n s m dtds 2 ( )2 2 (2 ) (2 ) j k a t s j k R e s n s m dtds We have (1/2 )2 (1/2 )2 2 2 2 ( , ; , ) 2 [ j k j k b j k m n at as m n R j k m n e dt e ds (1/2 )2 (1 )2 2 (1/2 )2 (1 )2 (1/2 )2 (1/2 )2 2 j k j k j k j k m n at as m n m n at as m n e dt e ds e dt e ds (1 )2 (1 )2 (1/2 )2 (1/2 )2 ] j k j k m n at as m n e dt e ds (1/2 )2 (1/2 )2 2 2 2 2 [ j k j k j k m n at as m n e dt e ds (1/2 )2 (1 )2 2 (1/2 )2 (1 )2 (1/2 )2 (1/2 )2 2 j k j k j k j k m n at as m n m n at as m n e dt e ds e dt e ds (1 )2 (1 )2 (1/2 )2 (1/2 )2 ] j k j k m n at as m n e dt e ds We obtain 1, 2 ( 1)2 (2 ) 0, j j j n t n t n other 1, 2 ( 1)2 (2 ) 0, k k k m s m s m other Then, we have ( , ; , )aR j k m n 2 ( )2 2 (2 ) (2 ) j k a t s j k R e s n s m dtds 2 ( )2 2 (2 ) (2 ) j k a t s j k R e s n s m dtds References [1] Adam Bobrowski, 2007. Functional analysis for probability and stochastic processes. Cambridge University Press. [2] Cambancs, 1994. "Wavelet approximation of deterministic and random signals." IEEE Tran.on Information Theory, vol. 40, pp. 1013-1029. [3] Flandrin, 1992. "Wavelet analysis and synthesis of fractional brownian motion." IEEE Tran.on Information Theory, vol. 38, pp. 910-916. [4] Haobo Ren, 2002. "Wavelet estimation for jumps on a heterosedastic regression model." Acta Mathematica Scientia, vol. 22, pp. 269-277. [5] Hendi, A., 2009. "New exact travelling wave solutions for some nonlinear evolution equations." International Journal of Nonlinear Science, vol. 7, pp. 259-267. [6] Krim, 1995. "Multire solution analysis of a class nstationary processes." IEEE Tran.on Information Theory, vol. 41, pp. 1010-1019. [7] Meyer, Y., 1990. "In Ondelettes et operatears(M), Hermann." [8] Pairick Flandrin, 1992. "Wavelet analysis and synthesis of fractional brownian motion." IEEE Tran.,On Information Theory, vol. 38, pp. 910-916. [9] Paul Malliavin, 2003. Stochastic analysis. Springer-Verlag. [10] Xia Xuewen, 2012. "The study of Wiener Processes with linear-trend base on wavelet." Studies in Mathematical Science, vol. 2012, p. 2. [11] Xia Xuewen, 2012. "The energy of stochastic vibration system of a class of protein base on wavelet." Advances in Nature Science, vol. 5, pp. 68-70. [12] Xia Xuewen and Dai Ting, 2012. "The energy of convolution of 2-dimension exponential random variables base on Haar wavelet." Progress in Applied Mathematics, vol. 3, pp. 35-38. [13] Xuewen Xia, 2005. "Wavelet analysis of the stochastic system with coular stationary noise." Engineering Science, vol. 3, pp. 43-46. [14] Xuewen Xia, 2007. "Wavelet analysis of Browain motion." World Journal of Modelling and Simulation, vol. 3, pp. 106-112.
- 10. Scientific Review, 2016, 2(8): 90-99 99 [15] Xuewen Xia, 2008. "Wavelet density degree of continuous parameter AR model." International Journal Nonlinear Science, vol. 7, pp. 237-242. [16] Xuewen Xia, 2010. "The morlet wavelet density degree of the linear regress processes with random coefficient." International Journal of Nonlinear Science, vol. 9, pp. 349-351. [17] Xuewen Xia, 2011. "The study of a class of the fractional brownian motion based on wavelet." Inter.J.of Nonlinear Science, vol. 11, pp. 290-294. [18] Xuewen Xia and Ting Dai, 2009. "Wavelet density degree of a class of Wiener processes." International Journal of Nonlinear Science, vol. 7, pp. 327-332. A Project supported by Hunan Provinial Natural Science Foundation of China (12JJ9004); A Project Supported by Scientific Research Found of Hunan Provincial Education Department (12A033).