Journal of Statistical and Econometric Methods, vol. 2, no.2, 2013, 21-27
ISSN: 2051-5057 (print version), 2051-5065(online)
Scienpress Ltd, 2013
Xuewen Xia
1
D-order processes is a sort of important stochastic processes, it is a class of useful stochastic processes in practices, its study is very value. In this paper, we study 1-order processes using haar wavelet and wavelet transform on R. We study its some properties and wavelet expansion.
Mathematics Subject Classification: 60H30, 60H35
Keywords : 1-order processes, wavelet, haar, expansion
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
1
Hunan Institute of Engineering,Xiangtan,411104,China
Hunan Normal University,Changsha,410082,China
Article Info: Received : February 3, 2013 . Revised : March 11, 2013.
Published online : June 1, 2013

22 Xuewen Xia 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]-[19].
In this paper, we study it using haar wavelet and wavelet transform on R. We study its some properties and wavelet expansion.
Definition 1: Let
E
 d
2
  
0 1 t
   d t d
(1) t R we call X(t) is D-order processes on R. we study 1-order processes. let d=1
We have:
  
1 t
( , )

( ) ( )


E (
   
0 1
)(
0 1 s )

0
2  
1
2 st
(2) st
( Let
 i
2 
1, i

0,1)
Definition 2: Let ｛ x(t) , t ∈ R ｝ is a stochastic processes on probability space （ Ω, g, P), we call:
W(s, x) ＝
1 s

R x ( t )
( x
 s t
) dt (3) is wavelet transform of x(t), where, ψ is mather wavelet([11]).
Then, we have:
 
)
 s
1 
R
 x s t
) dt (4)

The Study of 1-order Processes through Haar Wavelet 23
Definition 3: Let mother wavelet ψ(x) is function:
ψ(x)=



1 , 0
 x






0 ,
1 ,
1
2
 other x
1
2

1 (5)
ψ(x) is the Haar wavelet.
Then, we have:
 x t
( ) s
 
 

1, x s
2 t x x s t x s
2
(6)

( x
 s t
)


 
1, x x

 s
2 t x s t x

 s
(7)
2
We have:
R
 
[ ( , ) ( ,
 
)]

E [
1 s

R x t
 y t s
 dt

1
E [

R
2 x t x t
1
 s 2
1 
E x t x t s 2

1 s
2

Where,
R
2
(1
 st


 s s s
1 
R x t
1 s y y y
 y t
1 ) dt
1
] s t
1 ) dtdt
1
] s
 t
1 ) dtdt
1 s t
1 ) dtdt
1
= I s
I
 
1 
R
2
(1 s
2
Then, we have:
 st
 s y s t
1 ) dtdt
1
]
I


1
2 s
2
1
2 s
2
[
 y

R
2

( s y
/2
(1
 tt dt
1
) y
   y
 s /2 dt
1 s
 t
1 ) dtdt
1
]

 y
1
2 s
2

R
2 tt
1

(
/2
(1
 tt dt
1
)
 y y

 s s /2 dt
1 s y s t
1 ) dtdt
1
]

24 Xuewen Xia
  /2
(1
 tt dt
 y
/2
 s /2 dt
1
  /2
(1
 tt dt y
  y
 s s /2 dt
1
] the zero density degree of W(s , y) is:

R

(0)
2
R (0)
can be obtained.
The average density degree of w(s,y) is:
(8)
R
( 4 )
( 0 )

2 R ( 2 ) ( 0 )
Let real function

is standard orthogonal element of multiresolution analysis
{ V j
} j

( t )


Let

( t )
Z (see [4]), then exists h k
2

 k

( 2 t

2
 k
(

1 ) k k
) h
1
 k

( 2 t

 l
2
, have k )
Then wavelet express of ( ) in mean square is:
( )

2

J
2

K
C n
J

(2

J t n 2
 j d n j

(2
 j t
 n )
Where, C n j 
2

2 j

R d n j 
2

2 j

R
Then have:


 j t

)
 j t

) j k
E C C n m
2

2

R
2
    j t
 n
  k j k
E d d n m
2

2

R
2

( ) ( )
 
(2
 j t
 n
  k 
Where,

(2 j t
 m )


 j
(1/ 2

1, (1/ 2
 m )2
 j
 m )2
(1 j m )2
 j

(2 k s
 n )
 
 
 k
(1/ 2
1, (1/ 2
 n )2
 k
 n )2
(1
 k n )2
 k
(9)

The Study of 1-order Processes through Haar Wavelet 25
Use (8) and (10), we can obtain value of n j k
E d d m
.
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
(10)
 0, otherwise
Now we consider function

( t ) that exist compact support set on [
 k
1
, k
2
], k
1
, k
2

0 , and exist enough large M, have support set on [
 k
3
, k
4
] satisfy k
1

R
 t m
 k
2
Let
  jk
 ， ( , )

( t ) dt
 k
3

0 , 0
 m
 k
4
,
 jk k

3
, k
4


M
0 .

1 , then

exist compact
Let J is a constant, then,



2
J
2

( 2
J x
 k ), k

Z



 


2 2 j

( 2 j t
 k ), k

Z


 j

J
are a standard orthonormal basis of space L
2
( R ) , then have:
( )

2 2
J

( ,
 J t

K )
   j
2 2 b j K
 j t

K ) (11)
Therefore, the self-correlation function of
R b
( j , K ; m , n )

E
 b ( j , m ) b ( k , n )
 b ( j , m )

2

2

R
2

( ) ( )
 
(2 j t
 m

And have also the self-correlation function of a
( , ; , )
 
( , ) ( , )

( , )

2

2

R
2

( ) ( )
 
(2 j t
 m

K 
K
(12)
(13)
Then, we use (8) and (9) have: b
( , ; , )

2

2

R
2
 
2

2

R
2
(1
 ts
 j
 k
R a
( , ; , )

2

2

R
2



 
2

2

R
2
(1
 ts

We have: b
( , ; , )
 j
 k  j t
 m
 j t
 m
 k k

26 Xuewen Xia


2 2 [

 m 2
 m
(1/2
 m )2

2
 j
(1/2
 m )2
 j
 j j
(1

)
(1

)
 n
(1/2
 n )2

2
 k k
(1
 n )2


(1/2
 n )2
 k k ds
 ds
(1
 m )2


(1/2
 m )2
 j j
(1

)
 (1
 m )2


(1/2
 m )2
 j j
(1

)
(1
 n )2


(1/2
 n )2
 k
 n
(1/2
 n )2

2
 k k k ds ds ] (14)
We obtain:

(2 j t
 n )


1, 2
0, j   other
( n

1)2

(2 k j s m )

 0, m k   other
( m

1)2 k
Then, we have:
R ( , ; , ) = a

2

2

R
2
(1
 st
 j
 k 
ACKNOWLEDGEMENTS : The Project supported by Hunan Provinial Natural Science
Foundation of China (12JJ9004) and the Scientific Research Found of Hunan Provincial
Education Department (12A033).
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The Study of 1-order Processes through Haar Wavelet 27
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