Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 39-50
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2012
The Marlet wavelet density degree of the two-order
polynomial stochastic processes
Xuewen Xia1
Abstract
In this paper, we use the wavelet transform to the two-order polynomial processes
by Marlet wavelet, we obtain some statistical properties about the transform and
density degree and wavelet express.
Mathematics Subject Classification: 60H15
Keywords: two-order polynomial processes, wavelet analysis, wavelet transform,
marlet wavelet, density degree.
1
Introduction
With the rapid development of computerized scientific instruments comes a
wide variety of interesting problems for data analysis and signal processing. In
1
Hunan Institute of Engineering, Xiangtan, 411104, China.
Hunan Normal University, Changsha, 410082, China, e-mail:xxw1234567@163.com
Article Info: Received : September 18, 2012. Revised : November 2, 2012
Published online : December 30, 2012
40
The Marlet wavelet density degree of the two-order polynomial...
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 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 to use a family of building blocks to
represent the object at hand in an efficient and insightful way, the building blocks
come in different sizes, and are suitable for describing features with a resolution
commensurate with their sizes. Some persons have studied wavelet problems of
stochastic process or stochastic system (see[1]-[12]). In this paper, we study a
class of random processes using wavelet analysis methods, and study its energy,
the energy express by density degree.
2
Basic Definition
Definition 1
If stochastic processes(see[14])
x(t)=At2+Bt+C
where, A and B and C
(1)
are independent random variable; E(A)=a, E(B)=b,
E(C)=c, D(A)=  12 , D(B)=  22 , D(C)=  32 ; We call x(t) as Two-order polynomial
processes.
Definition 2 If x(t) is a stochastic processes on probability space ( , p, ), and
E x(t )   ,
2
Xuewen Xia
41
W(s,x)=
1
xt
x(t ) (
)dt

s
s R
(2)
We call W(s,x) as wavelet alteration of x(t),Where  (t ) is mather
wavelet.(see[5])
Definition 3 Let  (t )  e  u , u  R, We call  (t ) as Marlet wavelet.
2
3
Some properties
We have relational function of x(t)
E[x(t)x(s)]=Rx ( s, t )  s 2t 2 (12  a 2 )  abs 2t  acs 2  abst 2 
( 22  a 2 ) st  bcs  cat 2  cbt   32  c 2
Let  (t )  e  u , u  R,
2
then we have the relational function of w(s,x)
xt
1
x    t1

1
R ( )  E[ w( s, x) w( s, x   )]  E   x(t ) (
)dt   x(t1 ) (
)dt1 
s
s R
s

s R
=
1
s2
1
= 2
s
=
1
s2

R2

R2
E[ x(t ) x (t1 )]  (
E[ x(t ) x1 (t )] e
 t
R2
(
xt
x    t1
) (
)dt1
s
s
x t 2
)
s
e
(
x  t1 2
)
s
dtdt1
t ( 12  a 2 )  abtt12  act12  abt1t 2  ( 22  b 2 )tt1
2 2
1

 bct1  act 2  cbt   32  c 2 e
=
1
s2
2 2
2
2
 2 t t1 (1  a )e
R
(
x t 2
)
s
e
(
(
x  t1 2
)
s
x t 2
)
s
e
(
x  t1 2
)
s
dtdt1 
dtdt1
42
The Marlet wavelet density degree of the two-order polynomial...
1
 2
s

R2
t t (  a )e
2 2
1
2
1
2
(
x   t1 2
x t 2
) (
)
s
s
e
dtdt1 
x  t
x   t
x t 2
x t 2
1 2
1 2
(
(
) (
)
) (
)
1
1
2
2
s
s
s
s
abtt
e
e
dtdt

act
e
e
dtdt1 
1
1
1
2 R 2
2 R 2
s
s
x   t1 2
x   t1 2
x t 2
x t 2
(
(
)
) (
)
) (
1
1
 2  2 abt1t 2 e s e s dtdt1  2  2 ( 22  b 2 )tt1e s e s dtdt1 
s R
s R
x   t1 2
x  t1 2
x t 2
x t 2
) (
)
) (
)
(
(
1
1
2
s
s
s
 2  2 bct1e
e
dtdt1  2  2 act e
e s dtdt1 
s R
s R
x   t1 2
x  t1 2
x t 2
x t 2
) (
)
) (
)
(
(
1
1
 2  2 cbte s e s dtdt1  2  2 ( 32  c 2 )e s e s dtdt1
s R
s R

=I1+I2+…+I9
where
  u


2
2
1
I1= 2 bc  e ( s )du   ( x  t )e  u   sue  u du  ( s )
  



s
2
1
= 2
s
=



t (  a )e
2
2
1
2
(
x t 2
)
s

dt  t e

2
1
(
x  t1 2
)
s
dt1


2
2
1 2
( 1  a 2 )  ( x  su ) 2 e u ( s )du  ( x    su ) 2 e u ( s )du
2


s


= (12  a 2 )  ( su  x) 2 e u du  ( su  x   ) 2 e u du


2
2


= (12  a 2 )  ( s 2u 2  2sxu  x 2 )e u du  [ s 2u 2  2( x   ) su + ( x   ) 2 ]e  u du

2




2
2
2
= ( 12  a 2 )   s 2 u 2 e u du   2sxue u du   x 2 e u du  
 



  s 2u 2 e u 2 du  2( x   ) s ue u 2 du  ( x   ) 2 e u 2 du 


 



2
2
2
  1

= ( 12  a 2 )  s 2   ude u  2sx  ue u du  x 2  x 2 e u du  


  2

 s 2 u 2 e u 2 du  2s ( x   ) ue u 2 du  ( x   ) 2  e u 2 du 


 

2
Xuewen Xia
43

   2


= (12  a 2 )  s 2
s  0   ( x   )2 
 0   x2   
2

  2

1
1

= (12  a 2 ) ( s 2  x 2 )  s 2  ( x   ) 2 
2
2

x   t
x t 2
1 2
(
) (
)
1
2
s
s
abtt
e
e
dtdt1
1
2 R 2
s
x   t1 2
x t 2
(
(
)
)

1 
2
s
 2  abte
dt  t1 e s dt1

s 

2
2
1 
 2  ab( x  su )e  u ( s )du  ( x    su ) 2 e u ( s )du


s
I2 


 ab[  ( x  su )e  u du ][  ( x    su ) 2 e  u du ]
2

2


2
2
 ab   xe u du   sue  u du    ( x   ) 2 e  u du 
  

 


2
2
  2s ( x   )ue  u du   s 2u 2 e u du 





2

 2
s 
 ab[  x  0]   ( x   ) 2  0 
2


2

s 
 ab x ( x   ) 2  
2

I3 ＝
1
s2
1
＝ 2
s
＝
2
 2 act1 e
(
x t 2
)
s
R



ace
(
x t 2
)
s
e
(

x  t1 2
)
s
dt  t e

2
1
(
dtdt1
x  t1 2
)
s
dt1


2
2
1
ac  e u ( s )du  ( x    su ) 2 e u ( s )du
2


s



s2 
2
 ＝ ac ( x   )   
2
2 

＝  ac ( x   ) 2   s 2

I4 ＝
1
s2
2
 2 abt1t e
R
(
x t 2
)
s
e
(
x   t1 2
)
s
dtdt1
44
The Marlet wavelet density degree of the two-order polynomial...
1
＝ 2
s
＝


2
abt e

(
x t 2
)
s

dt  t1e
(
x  t1 2
)
s

dt1


2
2
1
ab  ( x  su ) 2 e u ( s )du  ( x   )  su  e  u ( s )du
2


s


＝ ab  ( x  su ) 2 e u du  ( x    su )e u du
2

2


 2
2
2
2
 

＝ ab   x 2 e u du  2 xs  ue u du  s 2  u e u du  





 ( x   ) 2 e u 2 du   sue u 2 du 

 


 2
s  [  ( x   )  0]
＝ ab   x  0 
2


＝ ab ( x 
1
I5 = 2
s

1 2
s )( x   )
2
(  b )tt1e
2
2
R2
2
  (
1
= 2 ( 22  b 2 )  te

s
(
x t 2
)
s
x t 2
)
s
(
x  t1 2
)
s

(
e
dt  t1e
dtdt1
x  t1 2
)
s

dt1


2
2
1 2
( 2  b 2 )  ( x  su )e u ( s )du  ( x    su )e u ( s )du
2


s




2
2
2
2
= ( 22  b 2 )   xe u du   sue u du    ( x   )e u du   sue u du 
 
  



=
= ( 22  b 2 )[  x  0][( x   )   0]
= ( 22  b 2 )x( x   )
I6 ＝
1
s2
＝
1
s2

R2



bct1e
bce
(
(
x t 2
x  t1 2
) (
)
s
s
x t 2
)
s
e

dt  t1e

(
dtdt1
x  t1 2
)
s
dt1
Xuewen Xia
45
  (
1
= 2 bc  e

s
x t
)
s

dt  ( x    su )e u ( s )du
2

  u


2
2
1
bc
e ( s )du   ( x  t )e u du   sue u du  ( s )
2

 



s
2
=
= bc  [( x  t )   0]  bc ( x   )
1
I7= 2
s

2
R2

act e
(
(
x t t1 2
x t 2
)
) (
s
s
e
x t 2
)
s
  (
dtdt1
x t t1 2
)
s
=
1
s2
=


2
2
1
ac  ( x  su ) 2 e u ( s )du  e u ( s )du
2


s


2
act e
dt  e
dt1



= ac  ( x  su ) 2 e u du  e u du
2

2





2
2
2
2
= ac   x 2e  u du  2sx  ue  u du  s 2  u 2e  u du    e u du
  


 

 2
1
= ac   x 2  o 
s    ac ( x 2  s 2 )
2
2


I8=
1
s2
 2 cbte
(
x t 2
x t t1 2
) (
)
s
s
R

(
x t 2
)
s
e
  (
dtdt1
x t t1 2
)
s
=
1
s2
=


2
2
1
cb  ( x  su )e u ( s )du  e u ( s )du
2


s


cbte
= cb 

= cb
 xo

I9=
1
s2

dt  e

xe  u du  s 
2


ue  u du  

2
  cbx
2
2
 2 ( 3  c )e
R

dt1
(
x t 2
x t t1 2
) (
)
s
s
e
dtdt1
46
The Marlet wavelet density degree of the two-order polynomial...
  (
1
= 2 ( 32  c 2 )  e

s
=
x t 2
)
s
  (
dt  e

x t t1 2
)
s
dt 1


2
2
1 2
( 3  c 2 )  e u ( s )du  e u ( s )du
2


s
= ( 32  c 2 )
1
1

Then, R ( ) = (12  a 2 ) ( s 2  x 2 )  s 2  ( x   ) 2  
2
2



s2 
s2 
abx ( x   ) 2    ac ( x   ) 2   
2
2


1 2
s )( x   )  ( 22  b 2 )x( x   )  bc ( x   )
2
1
+ ac ( x 2  s 2 )  cbx  ( 32  c 2 )
2
1
Then ， R ' ( )  (12  a 2 ) ( s 2  x 2 )2( x   )  2abx( x   )
2
1
+ 2ac( x   )  ab ( x  s 2 )  ( 22  b 2 )x  bc
2
1
R ( )  2( 12  a 2 ) ( s 2  x 2 )  2abx  2ac
2
ab ( x 
R( )  R ( 4 ) ( )  0
Then,
1
1 
1


R(0) = (12  a 2 ) ( s 2  x 2 )  s 2  x 2   abx  x 2  s 2 
2
2 
2


1
1 
1

+ ac  x 2  s 2   ab ( x  s 2 ) x  ( 22  b 2 )xx  bcx  ac ( x 2  s 2 )
2
2 
2

+ cbx  ( 32  c 2 )
1
1
 (12  a 2 )( s 2  x 2 ) 2   2ab x( x  s 2 )  2 acx 2
2
2
2
2
2
2
2
 ( 2  b ) x  2bc x  ( 3  c )
Xuewen Xia
47
1
R(0)  2 (12  a 2 )( s 2  x 2 )  2a (bx  c)
2
The use above, we can obtain the zero density of w(s,x):
R(0)
 2 R(0)
The same time, we have:
Theorem
The averge density of w(s, x) is zero.
Because(see[6])
R (4) (0)
0
 2 R (2) (0)
4
Wavelet representation
Let real function  is standard orthogonal element of multiresolution
analysis {V j } j  Z (see [13]), then exist hk  l 2 , have
 ( t )  2   ( 2t  k )
k
Let  (t )  2  ( 1) k h1k  ( 2t  k )
k
Then wavelet express of X (t ) in mean square is
X (t )  2

where, C  2
j
n
J
2

 C nJ  (2  J t  n)   2
j
2
K
j J

X (t ) ( 2 t  n )dt ,
R
j

j
2
 d  (2
nZ
j
n
d 2
j
n

j
2

Let
1, 0  t  1
0, other
 (t )  
R
j
t  n)
X (t ) ( 2  j t  n )dt
48
The Marlet wavelet density degree of the two-order polynomial...
Then have
E[Cnj ]  2
2

j
2

R

3j
2

j
2
E (d nj )  2

E[ x(t )] (2 j t  n)dt  2
a

R
2

j
2

( n 1)2 j
n 2 j
(a 2t 2  bt  c)dt
b
/ 3[(n  1)3 22 j  n3 22 j ]  2 j (2n  1)  c
2

E[ x(t )] (2 j t  n)dt

E d nj d mk  2

j k
2
 EX (t ) X (s) (2
R2
j
t  n) (2  k s  m)dsdt
We can obtain the value on above. We may also can analyse C nj as above.
Now we consider function  (t ) that exist compact support set on
[ k1 , k 2 ], k1 , k 2  0 , and exist enough large M, have
t

then
exist
 (t )dt  0,0  m  M  1 ,
m
R
compact
support
set
on
[ k 3 , k 4 ]
k1  k 2  k 3  k 4 , k 3 , k 4  0 .
Let
b( j , k )  X (t ), jk  , a ( j, k )  X (t ),  jk 
Let J is a constant, then
 J2
  2j

J
j
2
(
2

),


x
k
k
Z

  2  ( 2 t  k ), k  Z 

 
 j J
are a standard orthonormal basis of space L2 ( R ) , then have
J
X (t )  2 2
j
 a( J , K ) (2 J t  K )    2 2 b( j, K ) (2 j t  K )
KZ
j  J KZ
Therefore, the self-correlation function of b( j , m)
satisfy
Xuewen Xia
49
Rb ( j , K ; m, n)  E b( j , m)b(k , n)
2

jK
2
 EX (t ) X (s) (2
R2
j
t  m) (2 K s  n)dtds
Let
F (2 j  K , t )    (2 j  K s  t ) ( s )ds .
R
If  (t ) have ( M  1) -order waning moments, then F (2 j K , t ) have
(2M  1) order waning moments. In actual,

R2
t m (2 j  K s  t ) ( s )dsdt    2 (2 j  K s  t ) m (t ) ( s )dsdt
R
  
R2
C
n
m
(2 j  K s ) m  n (t ) n (t ) ( s )dsdt  0, m  2 M
n
Therefore we have:
Theorem Let X (t ) is solution process of system (1),  (t ) have compact
supported set on [ K 1 , K 2 ] , K 1 , K 2 >0, and  (t ) have ( M  1) -order waning
moments, and  (t ) is standard orthonormal wavelet function. Then stochastic
process b( J , m) are stationary process.
Acknowledgements. 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).
References
[1] Cambancs, Wavelet Approximation of Deterministic and Random Signals,
IEEE Tran. on Information Theory, 40(4), (1994), 1013-1029.
50
The Marlet wavelet density degree of the two-order polynomial...
[2] Flandrin, Wavelet Analysis and Synthesis of Fractional Brownian Motion,
IEEE Tran.on Information Theory, 38(2), (1992), 910-916.
[3] Krim, Multire solution Analysis of a class of Nonstationary Processes, IEEE
Tran.on Information Theory, 41(4), (1995), 1010-1019.
[4] Haobo Ren, Wavelet estimation for jumps on a heterosedastic regression
model, Acta Mathematica Scientia, (2002), 22(2), 269-277.
[5] Xuewen Xia, Wavelet analysis of the stochastic system with coular stationary
noise, Engineering Science, 3, (2005), 43-46.
[6] Xuewen Xia, Wavelet density degree of continuous parameter AR model,
International Journal Nonlinear Science, 7(4), (2008).
[7] Xuewen Xia and Kai Liu, Wavelet analysis of Browain motion, World
Journal of Modelling and Simulation, 3, (2007).
[8] I. Daubechies, Different perspective on wavelets, Proceedings of Symposia in
Applied Mathematics, 46, (1993).
[9] M. Rosenblatt, Random Processes, Springer, New York,1974.
[10] S. Mallat and W. Hwang, Singularity detection and processing with wavelets,
IEEE Trans. On Information Theory, 38, (1992), 617-643.
[11] J. Zhang and G.. Walter, A wavelet based K-L-like expansion for wide-sense
stationary process, IEEE Trans. Signal Processing, (1994).
[12] Xuewen Xia and Ting Dai, Wavelet density degree of a class of Wiener
processes, International Journal Nonlinear Science, 8(3), (2009).
[13] Y. Meyer, In Ondelettes et operatears (M), Hermann, 1990.
[14] Adam Bobrowski, Functional analysis for probability and stochastic
processes, Cambridge University Press, 2007.