Asymptotic Properties of the Discrete Stability Time Series with Missed Observations Between Two-Vec

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ASYMPTOTIC PROPERTIES OF THE DISCRETE STABILITY TIME SERIES WITH MISSED OBSERVATIONS BETWEEN TWO-VECTOR VALUED STOCHASTIC PROCESS M.A.Ghazal1, A.I.El-Deosokey2, M.A.Alargt3 1Department

of Mathematics, Faculty of Science, University of Damietta, Egypt. faculty of computer science and information system 6th of October University, Egypt.

3Department of Mathematics, Faculty of Science, University of Damietta, Egypt.

2Lecture

---------------------------------------------------------------------***--------------------------------------------------------------------t  0,1,2,..... with X (t ) - r vector-valued and

Abstract - In this paper, we defined the Expanded finite Fourier transform of the strictly stability ( r  s ) vector

Y (t ) s vector-valued.

valued time series where there are some randomly missed observations, asymptotic moments are derived and the application will be studied .

Key Words: Discrete time stability processes, Data tapers, Finite Fourier transform, Missing values, Complex Normal Distribution.

We assume the series (2.1) is (r  s ) stability vector-valued series with components X j (t ) Yi (t )T , j  1,2,..., r , i  1,2,....., s all of whose moments exist, we define the means as

EX (t )  C x , EY (t )  C y

(2.2)

1.INTRODUCTION

The covariances

Many authors, as e.g. Brillinger [1]; Dahlhaus[3]; Ghazal and Farag [4] studied ''The estimation of the spectral density, autocovariance function and spectral measure of continuous time stationary processes''; E.A,El-Desokey[9] studied ''Some properties of the discrete expanded finite Fourier transform with missed observations''; M.A.Ghazal, G.S. Mokaddis and A.El-Desokey[10],[11] are Studied ''The Spectral Analysis of strictly stationary continuous time series'' and ''Asymptotic Properties of spectral Estimates of Second-Order with Missed Observations''. The paper is organized as the following: Section1. Introduction, we

E [ X (t  u )  C x ][ X (t )  C x ]T  C xx (u ) ,













E [ X (t  u)  C x ][Y (t )  C y ]T  C xy (u) , E [Y (t  u)  C y ][Y (t )  C y ]T  C yy (u) , and the second-order spectral densities

f xx ( )  (2 ) 1 f xy ( )  (2 )

1

Z (t )  X (t ) Y (t ) ,

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xy

(u ) Exp(-iu )

yy

(u ) Exp(-iu )



∑C

f yy ( )  (2 ) 1



∑C

,

(2.4)

u  

,

for       . s -vector  , and an



(2.1)

Impact Factor value: 5.181

(u ) Exp(-iu ) ,

In this section we consider the problem of determining an

Consider an (r  s ) vector-valued stability series T

xx

u  

a(u ) In Section 2, the Asymptotic properties of Expanded

2. ASYMPTOTIC PROPERTIES OF ESTIMATES THE DESIRED  , a(u )



∑C u  

develop asymptotic properties of estimates the desired  , finite Fourier transform with missed observations was discussed in section 3, section 4 we will apply our theoretical study in two cases in climate and economy.

(2.3)

s r

filter {a (u )} , so that



 a(t  u ) X (u)

(2.5)

,

u  

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Which is close to Y (t ) . Suppose we measure closeness by the

s  s Hermitian matrix

    E [Y (t )     a(t  u ) X (u )][Y (t )     a(t  u ) X (u )]T  , (2.6) u   u    

      [Y (t )     a(t  u ) X (u )]T  E[Y (t )     a(t  u ) X (u )]  u   u    

 E[Y (t )   



  



 a(t  u) X (u)]  E[Y (t )     a(t  u) X (u)]

u  

Theorem 2.1

T

T

u  



Consider an (r  s ) vector-valued second-order of stability time series of the form (2.1) with mean (2.2) and autocovariance functions (2.3). Suppose c xx (u ) , c yy (u ) are absolutely summable and suppose f xx ( ) , f xy ( ) and f yx ( ) are given by (2.4) and Then the,

f xx ( ) is nonsingular,       .

 , and a(u ) that minimize (2.6) are given by 





  c y    a(u ) c x  c y  A(0)c x ,  u  



(2.7)

  [ f yy ( )  f yx ( ) f xx1 ( ) f xy ( )]d  



  [ A( ) f xx ( )  f yx ( )] f xx1 ( )  

 [ A( ) f xx ( )  f yx ( )]T d  [c y   

2

(2.8)

0

u  

A( )  f yx ( ) f xx ( )

1

(2.9)

,

the filter {a (u )} is absolutely summable. The minimum

0

]T

    (2.8) E [Y (t )     a(t  u ) X (u )][Y (t )     a(t  u ) X (u )]T   u   u    

  [ f yy ( )  f yx ( ) f xx1 ( ) f xy ( )]d 

let

achieved is 2

x



where



x

u  



 a(t  u )c

 [c y   

and

a(u)  (2 ) 1  A( ) Exp{iu} d ,



 a(t  u)c ] 

cy   

[ f yy ( )  f yx ( ) f xx ( ) 1 f xy ( )]d .

where A( ) is the transfer function of the

(2.10)

then

s  r filter

frequency

  cy 

x

0 ,



 a(t  u)c

u  

x

 c y  A(0)c x

,

and

A( ) f xx ( )  f yx ( )  0

.

Proof



Let A( ) , be the transfer function of a (u ) which defined as (2.8). We may write as,

  E [Y (t )     a(t  u ) X (u )][Y (t )     a(t  u ) X (u )]T  u   u     





u  

u  

 a(t  u) X (u )]  E[Y (t )     a(t  u) X (u)] 

 E[Y (t )   

2

[ f

yy

( )  f yx ( ) f xx1 ( ) f xy ( )]d .

0



 a(t  u) X (u)]

T

u  

     E [Y (t )     a(t  u ) X (u )]  E[Y (t )     a(t  u ) X (u )]   u   u     

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A( )  f yx ( ) f xx ( ) 1 ,

Using (2.7) and (2.8) the minimum achieved



 cov[Y (t )   

 a(t  u )c

u  

(2.10)

achieving the indicated minimum . we call A( ) , the complex regression coefficient of Y (t ) on X (t ) at



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E a (t )  0 ,

let ha ( ) be the discrete expanded finite Fourier (T )

(3.5)

transform which is defined as 2  T 1 ha(T ) ( )  2  d a(T ) (t )  t  0  





1 2 T 1

d t 0

(T ) a

(t )  a (t ) exp it,      

(3.1)

c xx (u ) Cov  a1 (t1 ), a2 (t 2 )  pa1a2  c yx (u ) (3.1)



c xy (u )  (3.6) T , A( )c xx (u ) A( ) 







Cov  a1 (t1 ), a2 (t 2 ) 

where

 a (t )  Ba (t )Z a (t ) , a  1,2,......, min( r , s) ,

(3.2)

, X a (t ) , Ya (t ) are the observations on the stability stochastic

    f a1a2 (v) exp{ivu}dv  p a1a2      A( )  f a1a2 (v) exp{ivu}dv  

   ,  T A( )  f a1a2 (v) exp{ivu}dvA( )    

f

a1a2

(v) exp{ivu}dvA( ) T

(3.7)

processes, Ba (t ) is Bernoulli sequence of random variable which is stochastically independent of X a (t ) , Ya (t ) which satisfies

1 Ba (t )   0

Since X (t ) is a strictly stability series and Ba (t ) is

, if X a (t ) , Ya (t ) are observed ;

(3.3)

, otherwise.

Let Ba (t ) be an independent and identically distributed

pa  qa  1 .

t The data window function d a(T ) (t )  d a(T ) ( ) , t  (0, T ) is T bounded has bounded variation and vanishes for all t outside the interval [0, T ] .

Assumption Let d a(T ) (t ), t  R , a  1, r has bounded variation and vanishes

 T  1 , t  0 then,

theorem will give the asymptotic properties of  a (t ) which is defined as (3.2).

Theorem 3.1

 a (t )  Ba (t )Z a (t ) , a  1,2,......, min( r, s) are stable

stochastic

missed

processes,

X a (t ), Ya (t ) , a  1,2,......, min( r, s) and Ba (t ) is Bernoulli sequence of random variables equations(3.1),(3.4) , Then, © 2017, IRJET

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 Ba (t1 ) X a1 (t1 ) Ba2 (t 2 ) X a2 (t 2 )  E 1  Ba1 (t1 )Ya1 (t1 ) Ba2 (t 2 ) X a2 (t 2 )

which

satisfies

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Ba1 (t1 ) X a1 (t1 ) Ba2 (t 2 )Ya2 (t 2 ) .

Ba1 (t1 )Ya1 (t1 ) Ba2 (t 2 )Ya2 (t 2 ) 



 

 EB



 



 E Ba1 (t1 ) Ba2 (t 2 ) Cov X a1 (t1 ), X a2 (t 2 )    E Ba (t1 ) Ba (t 2 ) Cov Ya (t1 ), X a (t 2 ) 1 2 1 2 





For       and a1 ,....,a k  1 ,2 ,....,r .The following

the

 Ba (t1 ) X a (t1 )  Ba (t 2 ) X a (t 2 ) T   1  1 2 2  Cov ,     Ba1 (t1 )Ya1 (t1 )   Ba2 (t 2 )Ya2 (t 2 )  



 k  G a1 ,..., ak (  )    d a(Tj ) (t )  exp  it , t  0  j 1 

on



p a1a2 Cov X a1 (t1 ), X a2 (t 2 )     p a a Cov   A( ) X a (t1 ), X a (t 2 ) 1 2  12

T 1

observations



Cov  a1 (t1 ), a2 (t 2 ) 

(3.4)

P[ Ba (t )  0]  q a ,

Let



Z a (t ) then (3.5) comes directly.

 Cov Ba1 (t ) Z a1 (t ), Ba2 (t ) Z a2 (t )

P[ Ba (t )  1]  p a ,

for t

independent of



random variables with

where

Proof

 



 



(t1 ) Ba2 (t 2 ) Cov X a1 (t1 ), Ya2 (t 2 )    E Ba1 (t1 ) Ba2 (t 2 ) Cov Ya1 (t1 ), Ya2 (t 2 )  a1





p a1a2 Cov X a1 (t1 ),   A( ) X a2 (t 2 )



 p a1a2 C X a1 X a2 (t1  t 2 )    p A( )C X a1 X a 2 (t1  t 2 )  a1a2



   p a1a2 Cov   A( ) X a1 (t1 ),   A( ) X a2 (t 2 )  





p a1a2 C X a1 X a2 (t1  t 2 ) A( ) T

   p a1a2 A( )C X a1 X a2 (t1  t 2 ) A( ) T  

 C a1a2 (t1  t 2 )  p a1a2   A( )C a1a2 (t1  t 2 )

C a1a2 (t1  t 2 ) A( ) T   A( )C a1a2 (t1  t 2 ) A( ) T 

from the stability and the independence then,  c a1a2 (u ) Cov  a1 (t1 ), a2 (t 2 )  p a1a2   A( )c a1a2 (u )



and





c a1a2 (u ) A( ) T   , A( )c a1a2 (u ) A( ) T 



Cov  a1 (t1 ), a2 (t 2 )  ISO 9001:2008 Certified Journal

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    f a1a 2 (v) exp{ivu}dv  pa1a 2       A( )  f a1a 2 (v) exp{ivu}dv  

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   .  T A( )  f a1a 2 (v) exp{ivu}dvA( )    

f

a1 a 2

(v) exp{ivu}dvA( )T

ha(T ) ( )    f a a (v) (aT1a)2 (1  v,  2  v)dv    12 N rc s  0, p a1a2  R (T )    A( ) f a1a2 (v) a1a2 (1  v,  2  v)dv R 

 f (v) A( )   A( ) f (v) T

a1a 2

(T ) a1a 2

a1a 2

(1  v,  2  v)dv   (1  v,  2  v)dv  

(T ) a1a 2

R

R

(3.8)

Definition: The complex normal distribution: Suppose X and Y are random vectors in

2k -dimensional

R k such that vec[X Y] is a

normal vector. Then we say that the

complex random vector Z  X  iY has the complex normal distribution. This distribution can be described with three parameters:

where



 (aT1a)2 (1  v, 2  v)  (2 ) 1 Ga(1Ta)2 (0)

complex vector, the covariance

matrix  must be Hermitian and non-negative definite; the relation matrix C should be symmetric. Moreover, matrices

 and C

T

1

are such that the matrix   C  C is also

non-negative definite. Matrices  and C are related to the covariance matrices of X and Y via expressions

V xx  E[( X   x )( X   x ) T ]  V xy  E[( X   x )(Y   y ) T ] 

T 1

t1 0

T 1

(T ) a1

(t1 )  d a(T2 ) (t 2 )  t 2 0

(3.9)

From equations (3.1)and (3.5) we have,

T

k -dimensional

1

Proof

where Z denotes matrix transpose, and Z denotes complex conjugate. Here the parameter  can be an arbitrary

 d

 exp i[(1  v)t1  i(2  v)t 2 ]

  E (Z ) ,   E[( Z   )( Z   ) T ] ,

C  E[( Z   )( Z   ) T ] ,

  ,  

1 Re[   C ], 2

Eha (t )  0 ,





Cov ha(1T ) (1 ), ha(T2 ) (2 )  1 2  T 1 T 1 2 (T )  Cov2  d a1 (t1 )   d a(1T ) (t1 )  a1 (t1 ) exp  i1t1,  t1  0  t1  0





 T 1 2 , 2  d a(T2 ) (t 2 )   t2 0 







1 Im[   C ], 2

(3.10)

 (2 ) 1 Ga(T1a)2 (0)



 d 1

1 2

T 1

t1  0

 (T )   d ( t )  ( t ) exp  i  t  a2 2 a2 2 2 2  t2 0  T 1

T 1

(T ) a1

(t1 ) exp i1t1 d a(T2 ) (t2 ) expi2t2  t2 0



 Cov  a1 (t1 ), a2 (t 2 )

1 V yx  E[(Y   y )( X   x ) ]  Im[   C ], 2 1 V yy  E[(Y   y )(Y   y ) T ]  Re[   C ], 2



T

 p a1a2 (2 ) 1 G a(1Ta)2 (0)

and conversely

  V xx  V yy  i (V yx  V xy ) ,   V xx  V yy  i(V yx  V xy ) .

 d 1

T 1

t1  0

(T ) a1

(t1 ) exp i1t1 

T 1 cxx (t1  t2 )   da(T2 ) (t2 ) expi2t2  t2 0 c yx (t1  t2 )

cxy (t1  t2 )  , A( )cxx (t1  t2 ) A( )T 

and

Theorem 3.2 Let a (t ) is missed observations on the stable stochastic process X a (t ) Ya (t )T ,

a  1,...., min( r , s) and Ba (t ) is

Bernoulli sequence of random variables which satisfies equations (3.3) and (3.4), Let ha ( ) be defined as (T )

(3.1),and d a ( ) satisfies assumption, then ha(T ) ( ) will be (T )

distributed approximately as,

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





Cov ha(1T ) (1 ), ha(T2 ) (2 )  (2 )1 Ga(1Ta)2 (0)

 d 1

T 1

t1  0

(T ) a1

(t1 ) exp i1t1

T 1

  d a(T2 ) (t 2 ) expi 2 t 2  t2 0

    f a1a2 (v) exp{iv (t1  t 2 )}dv  p a1a2      A( )  f a1a2 (v) exp{iv (t1  t 2 )}dv  

ISO 9001:2008 Certified Journal

     A( )  f a1a2 (v) exp{iv (t1  t 2 )}dvA( ) T    

f

a1a2

(v) exp{iv (t1  t 2 )}dvA( ) T



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p-ISSN: 2395-0072



 (2 ) 1 Ga(1Ta)2 (0)

 p a1a2

1

T 1

t1 0



 d 1

T 1

t1 0

satisfies the following property :

T 1

(T ) a1

(t1 )  d a(T2 ) (t 2 )  t 2 0

      A( )  f a1a2 (v) A( ) T dv    

f

a1a2

(v) A( ) T dv

(T ) a1

and

aa

(   ) A( ) T  (aaT ) ( )d

R



(T )  (aaT ) ( )  (2 ) 1 Gaa (0)

(t1 )  d a(T2 ) (t 2 ) 



1

Ga(T ) ( )

2

,

t 2 0

2   4 

where G a(T ) ( ) , a  1,2,..., min( r , s ) ,   R be defined in Assumption .

Proof From equation (3.11), we get  f aa (v) (aaT ) (  v)dv  R Dh ( )  p aa  (T )   A( ) f aa (v) aa (  v)dv R

(3.11)

(v) A( ) T  (aaT ) (  v)dv    , T (T ) R A( ) f aa (v)A( )  aa (  v)dv

f

(T ) a

where

aa

R

When

1  p a a 1

T 1 T 1  1 f a1a2 (v)(2 ) 1 Ga(1Ta)2 (0)  d a(1T ) (t1 )  d a(T2 ) (t 2 )  2  t1  0 t 2 0   





 exp i[(1  v)t1  i(2  v)t 2 ]dv  pa1a2  f a1a2 (v)

(T ) a1a2

1  2   ,   R and a1  a2  a , a  1,..., min( r , s) .

For

(1  v, 2  v)dv ,

a1a2



(v) A( ) T  (aT1a)2 (1  v, 2  v)dv ,

1.

R

 A( ) f

 3  pa a

1 2

a1a2

(v) (aT1a)2 (1  v, 2  v)dv ,

 A( ) f

(v) A( )  T

a1a2

(T ) a1a2

Now from equation (3.10) and (3.11) then equation (3.8) is obtained which complete the proof.

From equation (3.11) we can drive the following corollary by

1  2   , 1 , 2 ,   R.

h

 (aaT ) ( ) ,

(T ) aa

( )d  1, a  1,..., min( r , s ) ,   R

(3.13)



Lim  (aaT ) ( )d  Lim  (aaT ) ( )d  0 , .

T 



T 



(3.14)



3.

Lim   (aaT ) ( )d  1, T 



 a  1,..., min( r , s),   0 ,

  R . (3.15)

Theorem 3.4

X  R is bounded continuous at a point X   ,   R and

( ) , a  1,2,..., min( r , s) ,   R be defined as

(3.1), then the dispersion of

© 2017, IRJET



If the spectral density function f aa ( X ) , a  1,..., min( r , s) ,

Corollary 3.1 (T ) a

function

,   0 , a  1,..., min( r , s ) ,   R

(1  v, 2  v)dv

R

putting

the



2.

and

1 2

,



R

 4  pa a

R

properties:

f

1 2

any

a  1,..., min( r , s) is the kernel that satisfies the following

similarly

 2  pa a

  v   ,then formula (3.12) is obtained.

By putting

Theorem 3.3

R

let

  , T (T ) A (  ) f (    ) A (  )  (  ) d  aa aa R  

f

(3.12)

T 1

    f a1a2 (v) A( ) T dv    f a1a2 (v)dv    exp i[(1  v)t1  i(2  v)t 2 ]      T A (  ) f ( v ) dv A (  ) f ( v ) A (  ) dv    a1a2  a1a2    

  1  3

 f aa (   ) (aaT ) ( )d  R Dh ( )  p aa  (T )   A( ) f aa (   ) aa ( )d R (T ) a

    f a1a2 (v)dv exp  i1t1  i2 t 2  ivt1  ivt 2      A( )  f a1a2 (v)dv  

 (2 ) 1 Ga(1Ta)2 (0)

 p a1a2

 d

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h ( )

 (aaT ) ( X ), a  1,..., min( r , s) ,

X  R satisfies

the properties of theorem 3.3, then,

(T ) a

Impact Factor value: 5.181

the function

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Volume: 04 Issue: 04 | Apr -2017

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 f aa ( ) f aa ( ) A( )T  Lim Dh ( )  paa  , T T   A( ) f aa ( ) A( ) f aa ( ) A( )  (T ) a

, a  1,..., min( r , s ).

(3.16)

Proof

f aa ( ) A( ) T    0, A( ) f aa ( ) A( ) T 

Dh

 paa 



 

point



 f aa (   ) f aa (   ) A( )T  J 2  paa    A( ) f aa (   ) A( )T    A( ) f aa (   )

f aa ( ) A( )T  (T )   aa ( )d A( ) f aa ( ) A( )T 

  f aa (   )  f aa ( )  paa     A( ) f aa (   )  A( ) f aa ( )

 (aaT ) ( )d

 f aa (   ) A( )T  f aa ( ) A( )T  A( ) f aa (   ) A( )T  A( ) f aa ( ) A( )T 

    (aaT ) ( )d 



    (aaT ) ( )d Hence,

f aa ( ) A( ) T  (T )   aa ( )d  A( ) f aa ( ) A( ) T 

is

J 2   . Now J 2

very

small,

is very small according to any

consequently

J2  0

Suppose



that

f aa ( ) a  1,..., min( r, s),   R is bounded by a constant M, then 

 f aa (   ) f aa (   ) A( )T    T  A( ) f aa (   ) A( ) f aa (   ) A( ) 

 f aa ( )   A( ) f aa ( )

a



  f (   ) f aa (   ) A( )T   paa   aa  T    A( ) f aa (   ) A( ) f aa (   ) A( ) 



at



 f aa ( ) f aa ( ) A( ) T  ( )  p aa   T  A( ) f aa ( ) A( ) f aa ( ) A( ) 

 f aa ( )   A( ) f aa ( )

continuous

, a1 , a2  1,..., min( r, s) ,   R , then we get

Now , from corollary 3.1 we have, (T ) a

is

 f aa ( )   A( ) f aa ( )

To prove formula (3.16), we must prove that

 f aa ( ) Lim Dh a(T ) ( )  p aa  T   A( ) f aa ( )

f a1a2 ( )

Since

J 1  2 M   (aaT ) ( )d T  0 ,  

f aa ( ) A( ) T  (T )   aa ( )d  A( ) f aa ( ) A( ) T 

  f (   ) f aa (   ) A( )T   paa   aa  T   A( ) f aa (   ) A( ) f aa (   ) A( ) 

according to property (3.14). similarly J 3 T  0, therefore, 

 f aa ( ) Dh a(T ) ( )  p aa   A( ) f aa ( )

f aa ( ) A( ) T    0 .

A( ) f aa ( ) A( ) T  T 

which completes the proof of the theorem.

Lemma 3.1

 f aa ( )   A( ) f aa ( )

 (T ) f aa ( ) A( )  aa ( )d  T A( ) f aa ( ) A( )  T

(T )

If the data window function d a

 paa  

 f aa (   ) f aa (   ) A( )T    T  A( ) f aa (   ) A( ) f aa (   ) A( ) 

 f aa ( )   A( ) f aa ( )

T 1

d t 0

f aa ( ) A( )T  (T )   aa ( )d A( ) f aa ( ) A( )T 

 1] ; then 1

(T ) a

(t ) ~ T  d a(T ) (u ) du,

(3.17)

0

where,

1 T 1 (T ) d a (t ) T   d a(T ) (u ) du , a  1, r , T  1,2,... (3.18)   T t 0 0 1

 J1  J 2  J 3 . © 2017, IRJET

Lemma 3.2 |

Impact Factor value: 5.181

is

bounded and has bounded variations and equal zero outside the interval [0, T



(t ), t  R , a  1, r

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Volume: 04 Issue: 04 | Apr -2017

p-ISSN: 2395-0072

(t ), t  R, a  1, r

is bounded by a constant

L and satisfying the Lipschitz condition, T 1



4.APPLICATIONS

d a(T ) (t  u )  d a(T ) (t ) 

u 0

The proof comes directly from Lemma 3.3 and Lemma 3.1.

We will apply our theoretical study in two cases in climate and economy as in the following sections.

T 1

  d a(1T ) (t ) d a(T2 ) (t ) exp  it   u ,

(3.19)

4.1.Studying the temperature and solar radiation

t 0

The data manipulated in this research make up a monthly chronic series that represents the average of the monthly temperature and solar radiation in Tripoli in Libya. The data is extracted from the meteorological centre of Tripoli, for the period from January 2005 to December 2013.

then, T 1

d t 0

(T ) a1

(u  t ) d a(T2 ) (t ) exp it  L u ,

for all constant

(3.20)

 , u  [(T  1), (T  1)] and   [ ,  ] .

Lemma 3.3 For all

d

(T ) a

1 , 2  [ ,  ], (1  2 )  (mod 2 ) and

(t ), t  R , a  1,..., min( r , s) is bounded by a constant

4.1.1.Studying the temperature In this study we will comparison between our results, model of strictly stability time series (temperature) with some missing observations and the classical results, where all observations are available.

Let  a (t )  Ba (t ) X a (t ) , a  1,2,......, r ,where

L and satisfying Lipschitz condition (3.19), then,



L



Cov ha(1T ) (1 ), ha(T2 ) ( 2 ) 

 d T 1

2

t1 ,t 2  0

for all

(1 )

 d 2

(T ) a2

( 2 )



2

 C a1a2 (u)   C a1a2 (u ) [ u  1], (3.21)    T 1)   T 1  T 1

T 1

For all

d

random variable of



 [ u  1] C   

a1a2

(u )  ,



X a (t ) which satisfies equations (3.3)

and (3.4), we suppose that the data

 1,  a (t )  X a (t ), which is the classical B  0, table 4.1.1 shows the comparison

of these results with and without missed observations.

Table-4.1.1: The comparison of the results with and without missed observations Series without missed observations

(3.22)

Series with missed observations

Time Series Plot of temperature



lim Cov ha(1T ) (1 ), ha(T2 ) (2 )  0,

(3.23)

Time Series Plot of Testi

30

30

25

25

20

10 1

a1 , a2  1,..., min( r, s).

20

15

15

10

for all

X a (t ) , t  (1,2,....., T ] is the

average of the monthly temperature, where all observations

random way, i.e.,

(t ), t  R , a  1,..., min( r , s) is bounded and

T 

Ba (t ) is Bernoulli sequence of independent

case suppose that there is some missing observations in a

1 , 2  [ ,  ], (1  2 )  (mod 2 ) and

then

time series and

are available, B

a1 , a2  1,..., min( r, s).

Theorem 3.5

(T ) a

X a (t ) , (t  0,1,.....) be a strictly stability r-vector valued

temperature

   Lc ( 1 12 ) / 2 

(T ) a1



Testi

(T )

Suppose d a

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11

22

33

44

55 Index

66

77

88

99

1

12

24

36

48

60 Index

Proof © 2017, IRJET

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72

84

96

108

The average monthly temperatures

The average monthly temperatures

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Volume: 04 Issue: 04 | Apr -2017

p-ISSN: 2395-0072

(with 5% significance limits for the autocorrelations) 1.0

0.8

0.8

0.6

0.6

0.4

0.4

Autocorrelation

Autocorrelation

(with 5% significance limits for the autocorrelations)

1.0

0.2 0.0 -0.2 -0.4

t  0,  1, .....), be a strictly stability s-vector valued time series and

0.0 -0.2 -0.4 -0.6

-0.8

-0.8 -1.0

1

5

10

15

20

25 Lag

30

35

40

45

1

50

5

10

15

20

25 Lag

30

35

40

45

50

Partial Autocorrelation Function for Def12Ln(Zt)

(with 5% significance limits for the partial autocorrelations)

0.8

0.8

0.6

0.6

Partial Autocorrelation

1.0

0.2 0.0 -0.2 -0.4 -0.6 -0.8

a (t )  Ya (t ), which is the

observations are available, B  1,

0.4

classical case, suppose that there is some missing

0.2 0.0 -0.2 -0.4

B  0, table 4.1.2 shows

observations in a random way, i.e.,

-0.6 -0.8

-1.0

stochastically independent of Ya (t ) which satisfies equations the average of the monthly temperature, where all

Partial Autocorrelation Function for def12Z**t

(with 5% significance limits for the partial autocorrelations)

1.0

Ba (t ) is Bernoulli sequence of random variable which is

(3.3) and (3.4), we suppose that the data Ya (t ), t  (1,2 ,.., T ] is

ACF of the seasonal difference

ACF of the seasonal difference

0.4

where Ya (t ),

0.2

-0.6 -1.0

 a (t )  Ba (t )Ya (t ) , a  1,2,......, s,

Let

Autocorrelation Function for def12Z**t

Autocorrelation Function for Def12Ln(Zt)

Partial Autocorrelation

www.irjet.net

-1.0 1

5

10

15

20

25 Lag

30

35

40

45

50

1

P

SE Coef

20

25 Lag

30

35

40

45

50

the comparison of these results with and without missed observations.

Table-4.1.2: The comparison of the results with and without missed observations of the solar radiation

ARIMA (2,0,0)  (0,1,2)12

Series without missed observations

Final Estimates of Parameters

Final Estimates of Parameters Coef

15

ARIMA Model: Temperature

ARIMA (2,0,0)  (0,1,1)12

Type

10

PACF of the seasonal difference

ACF of the seasonal difference ARIMA Model: Temperature

5

P

Type

Coef

0.001

AR 1

0.5700

T

SE Coef

T

Series with missed observations

Time Series Plot of solar(Xt)

P

Time Series Plot of Testi

600

30 500

0.1031

SMA 12 0.8445 0.0839

1.48

0.143

10.07 0.000

AR 2 -0.0155

Differencing: 0 regular, 1 seasonal of order 12 Number of observations: Original series 108, after differencing 96

0.1059

0.000

-0.15

0.884

25 400

Testi

5.39

300

SMA 12 1.5643

0.1035 15.12 0.1014

100

0.000

-6.41

0.000

1

Number of observations: Original series 108, after differencing 96

12

36

MS = 0.008526

77

88

10

99

1

Lag

12

24

36

Autocorrelation Function for Def12LnXt

0.6

0.6

0.4

0.4

0.2 0.0 -0.2 -0.4 -0.6

DF

8

P-Value

0.057

25.0

36.4

20

32

0.203 0.269

41.4 44 0.585

Chi-Square DF P-Value

10.7 7 0.153

20.8

44.1

-0.8

43

0.35

0.059

0.133

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Impact Factor value: 5.181

|

96

108

120

0.0 -0.4

-1.0 10

15

20

25

30

35

40

45

1

5

10

15

20

25

30

35

40

45

Lag

ACF of the seasonal difference

ACF of the seasonal difference

Partial Autocorrelation Function for def12X**t (with 5% significance limits for the partial autocorrelations)

1.0

1.0

0.8

0.8

0.6

0.6

0.4 0.2 0.0 -0.2 -0.4 -0.6

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

1

5

10

15

20

25

30

35

40

1

45

5

10

15

20

25

30

35

40

45

Lag

Lag

PACF of the seasonal difference

PACF of the seasonal difference

ARIMA Model: solar radiation without missed observations

ARIMA Model: solar radiation with missed observations

ARIMA (3,0,0)  (0,1,2)12 Final Estimates of Parameters

Final Estimates of Parameters

In this study we will comparison between our results, model of strictly stability time series (Solar Radiation) with some missing observations and the classical results, where all observations are available.

84

-0.8

5

ARIMA (3,0,0)  (0,1,2)12

4.1.2.Studying the solar radiation

72

0.2

Lag

53.4

31

60 Index

-0.2

(with 5% significance limits for the partial autocorrelations)

19

48

-0.6

-1.0

15.1

36

Autocorrelation Function for def12X**t

0.8

Partial Autocorrelation Function for Def12LnXt

48

24

(with 5% significance limits for the autocorrelations) 1.0

-0.8

Chi-Square

12

The average monthly solar radiation

0.8

DF = 91

Modified Box-Pierce (Ljung-Box) ChiSquare statistic 48

66

-1.0

DF = 92

24

55 Index

Partial Autocorrelation

Lag

44

1.0

Residuals: SS = 0.775840 (back forecasts excluded)

Modified Box-Pierce (Ljung-Box) ChiSquare statistic

33

(with 5% significance limits for the autocorrelations)

Partial Autocorrelation

MS = 0.01122

22

The average monthly solar radiation

1

Residuals: SS = 1.03233 (back forecasts excluded)

11

Constant 0.0011289 0.000935 1.21 0.230 Differencing: 0 regular, 1 seasonal of order 12

20

15

200

SMA 24 -0.6502 Constant 0.002439 0.002641 0.92 0.358

0.1058

Autocorrelation

AR 2 0.1522

3.33

solar(Xt)

0.1036

Autocorrelation

AR 1 0.3446

Type

Coef

SE Coef

AR 1

0.6443

0.1035

AR 2

0.1997

0.1218

Type

Coef

6.22 0.000

AR 1

0.5507

1.64 0.105

AR 2

0.4379 0.1044 4.19

T

P

ISO 9001:2008 Certified Journal

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SE Coef 0.0974

T

P

5.65 0.000

Page 2059

0.000

International Research Journal of Engineering and Technology (IRJET)

e-ISSN: 2395 -0056

Volume: 04 Issue: 04 | Apr -2017

p-ISSN: 2395-0072

AR 3 -0.3299

0.1004

www.irjet.net

-3.29 0.001

AR 3

8.09

0.000

SMA 12 1.0427 0.1089

SMA 24 -0.0570 0.1637 -0.35

0.729

SMA 24 -0.2015 0.1577 -1.28 0.205

Constant -0.1524 0.8091 -0.19 0.851

Constant -0.0518 0.7478 -0.07 0.945

Differencing: 0 regular, 1 seasonal of order 12

Differencing: 0 regular, 1 seasonal of order 12

Number of observations: Original series 108, after differencing 96

Number of observations: Original series 108, after differencing 96

Residuals: SS = 113197 (back forecasts excluded)

Residuals: SS = 122927 (back forecasts excluded)

SMA 12 0.9057 0.1119

MS = 1258

-0.4197 0.0970 -4.33

DF = 90

MS = 1366

0.000

temperature 12.685 1.055

9.58 0.000

12.02 0.000

S = 58.5263 R-Sq = 57.0% R-Sq(adj) = 56.6%

Analysis of Variance

Analysis of Variance

Source

DF

SS

MS

F

Chi-Square 10.0

16.0

35.1

44.7

Chi-Square 8.5

24

36

48

21.3

41.1

51.2

6

18

P-Value 0.123

30

0.589

0.240

42

DF

6

0.358

P-Value

18

30

Total

107

3626

908341

Total

0.206 0.266

P

107

843621

Probability Plot of RESI2 Normal

Mean StDev N KS P-Value

99 95 90

99.9

-8.86863E-14 59.93 108 0.077 0.116

80 70 60 50 40 30 20

Mean StDev N KS P-Value

99 95 90

-3.16586E-13 58.25 108 0.072 >0.150

80 70 60 50 40 30 20

10

10

5

5

1

42

F

Residual Error 106 363085 3425

0.1

DF

MS

Residual Error 106 384352

Percent

12

SS

Regression 1 480536 480536 140.3 0.00

Percent

Lag

DF

Normal

Lag

48

Source

Probability Plot of RESI1

Modified Box-Pierce (Ljung-Box) ChiSquare statistic

36

P

Regression 1 523989 523989 144.5 0.00

DF = 90

Modified Box-Pierce (Ljung-Box) ChiSquare statistic 24

12.726 1.074 11.84 0.000

S = 60.2159 R-Sq = 57.7% R-Sq(adj) = 57.3%

99.9

12

Test.

1

-200

-100

0 RESI1

100

200

Normal-plot of standardized Residuals

0.1

-200

-100

0 RESI2

100

200

Normal-plot of standardized Residuals

0.085 0.15

4.1.4.Conclusion 4.1.3. Studying The Regression Between Solar Radiation And Temperature In this section we adjust the regression model which represents the relationship between Monthly rate of solar radiation in watt /m^2 rate and the average monthly temperature in the period from 2005 to 2013.

In this study we will comparison between our results with some missing observations and the classical results where all observations are available.

Let Z (t )  X (t ) Y (t )T where

X (t ) is the series of average of

temperature and Y (t ) is the series of the average of solar radiation, first we consider that the observations are available P  1,  (t )  B(t ) Z (t )  pZ (t )  Z (t ), then consider that there are some missing of observations randomly, P  0. We used SPSS,MINITAB to investigate our results which is shown in table 4.1.3 Table - 4.1.3: The comparison of the results with and without

missed observations of the regression analysis Without missed observations

The regression equation is

solar radiation = - 10.4 + 12.7 temperature

Solar = - 9.9 + 12.7 Temperature

Predictor

Coef

SE Coef

Predictor

Coef

Constant

-10.42

22.36

Constant

-9.94 22.71

© 2017, IRJET

T

P

-0.47 0.642

|

SE Coef

T

P

-0.44 0.663

Impact Factor value: 5.181

4.2. Studying the Export and the Gross domestic product The data manipulated in this research make up chronic series that represents the Export and the Gross domestic product. The data is extracted from the Central Bank of Libya for the period from 1970 to 2012.

4.2.1. Studying the Export In this study we will comparison between our results, model of strictly stability time series (Export) with some missing observations and the classical results, where all observations are available.

With missed observations

The regression equation is

1. Tables 4.1.1 and 4.1.2 shows the study of time series with missed observations and the original time series and we investigated that they have the same results.

2. Table 4.1.3 shows the study of regression model between Monthly average of solar radiation and average monthly temperature with some missed observations which had the same results of the study of the classical regression model.

|

Let

 a (t )  Ba (t ) X a (t ) , a  1,2,......, r , where X a (t ) , (t  0,1,.....) be

a strictly stability r-vector valued time series and ISO 9001:2008 Certified Journal

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Ba (t ) is

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Bernoulli sequence of independent random variable of

X a (t ) which satisfies equations (3.3) and (3.4), we suppose that the data X a (t ) , (t  (1,2,....., T ] is the Export, where all observations are available, B  1,  a (t )  X a (t ), which is the classical case suppose that there is some missing observations in a random way, i.e.,

B  0, table 4.2.1 shows

the comparison of these results with and without missed observations.

Residuals: SS = 2806388520 (back forecasts excluded) MS = 73852329

Residuals: SS = 3338821601 (back forecasts excluded)

DF = 38

MS = 87863726

DF = 38

Modified Box-Pierce (Ljung-Box) ChiSquare statistic

Modified Box-Pierce (Ljung-Box) ChiSquare statistic

Lag

Lag

12

24

36

48

12

24

36

48

Chi-Square

12.8

13.3

14.7

Chi-Square 7.6

7.8

8.1

DF

9

21

33

DF

21

33

P-Value

0.173

0.899

0.998

P-Value

0.996

1.000

9 0.570

Table -4.2.1: The comparison of the results with and without missed observations Series without missed observations

Series with missed observations

Time Series Plot of Export X(t) 80000

80000

70000

70000

60000

60000

50000

50000 X^(t)

Export X(t)

4.2.2. Studying the Gross domestic product

Time Series Plot of X^(t)

40000

In this study we will comparison between our results, model of strictly stability time series (Gross domestic product) with some missing observations and the classical results, where all observations are available.

40000

30000

30000

20000

20000 10000

10000

0

0 4

8

12

16

20 24 Index

28

32

36

4

40

8

12

16

20 24 Index

28

32

36

40

Let

The Export

The Export

a (t )  Ba (t )Ya (t ) , a  1,2,......, s, where Ya (t), (t  0,1,.....), be a

strictly stability s-vector valued time series and Autocorrelation Function for X(t-2)

Autocorrelation Function for X^(t-2)

(with 5% significance limits for the autocorrelations)

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0.0 -0.2 -0.4

0.0

-0.6 -0.8

-1.0

-1.0 2

3

4

5

6

7

8

9

10

1

Lag

3

4

5

Partial Autocorrelation Function for X(t-2)

7

8

9

10

0.8

0.6

0.6

Partial Autocorrelation

0.8 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8

3

4

5

6

7

8

9

0.2 0.0 -0.4 -0.6

10

1

2

3

4

5

Lag

7

8

9

10

Table-4.2.2 The comparison of the results with and without missed observations of the Gross domestic product Series without missed observations

PACF of the second difference ARIMA Model: Export

Final Estimates of Parameters

Final Estimates of Parameters SE Coef

T

P

Type

Coef

SE Coef

T

P

AR 1

-1.1606

0.1112

-10.44 0.000

MA 1

0.0554

0.2152

0.26

MA 1 -0.0757

Constant -197

1271

0.798

-0.15 0.878

-1.1559

120000

100000

100000

80000

80000

60000

Constant 776

1596

-0.34

0.738

0.49

0.630

Differencing: 2 regular differences

Differencing: 2 regular differences

Number of observations: Original series 43, after differencing41

Number of observations: Original series 43, after differencing 41

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Impact Factor value: 5.181

60000

40000

40000

20000

20000

0

0.1081 -10.69 0.000 0.2246

Time Series Plot of Y^(t)

120000

0 4

AR 1

Series with missed observations

Time Series Plot of Gross domestic product Y(t)

ARIMA (1,2,1)

ARIMA (1,2,1)

Coef

6 Lag

PACF of the second difference ARIMA Model: Export

is

is the Gross domestic product, where all observations are available, B  1, a (t )  Ya (t ), which is the classical case,

Gross domestic product Y(t)

2

which

results with and without missed observations.

0.4

-1.0 1

variable

-0.2

-0.8

-1.0

random

suppose that there is some missing observations in a random way, i.e, B  0, table 4.2.2 shows the comparison of these

Partial Autocorrelation Function for X^(t-2)

(with 5% significance limits for the partial autocorrelations) 1.0

Type

6

ACF of the second difference

(with 5% significance limits for the partial autocorrelations)

Partial Autocorrelation

2

Lag

ACF of the second difference

1.0

of

(3.3) and (3.4), we suppose that the data Ya (t ) , t  (1,2,....., T ]

-0.4

-0.8

sequence

Ba (t ) is

stochastically independent of Ya (t ) which satisfies equations

0.2 -0.2

-0.6

1

Bernoulli

Y^(t)

Autocorrelation

Autocorrelation

(with 5% significance limits for the autocorrelations)

|

8

12

16

20 24 Index

28

32

36

4

40

The Gross domestic product

ISO 9001:2008 Certified Journal

8

12

16

20 24 Index

28

32

36

The Gross domestic product

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International Research Journal of Engineering and Technology (IRJET)

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p-ISSN: 2395-0072

available P  1,  (t )  B(t ) Z (t )  pZ (t )  Z (t ), then consider that

Partial Autocorrelation Function for Y^(t-2)

Partial Autocorrelation Function for Y(t-2)

(with 5% significance limits for the partial autocorrelations)

(with 5% significance limits for the partial autocorrelations) 1.0

1.0

0.8

0.8

0.6

0.6

Partial Autocorrelation

Partial Autocorrelation

www.irjet.net

0.4 0.2 0.0 -0.2 -0.4 -0.6

there are some missing of observations randomly, P  0. We used SPSS,MINITAB to investigate our results which is shown in table 4.2.3

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8

-0.8

-1.0

-1.0 1

2

3

4

5

6

7

8

9

1

10

2

3

4

5

6

7

8

9

10

Lag

Lag

Table-4.2.3: The comparison of the results with and without missed observations of the regression analysis

PACF of the second difference

PACF of the second difference

ARIMA Model: Gross domestic product without missed observations

ARIMA Model: Gross domestic product without missed observations

Final Estimates of Parameters

Final Estimates of Parameters Coef

SE Coef

T

P

Type

Coef

SE Coef

T

P

0.1694

-6.65

0.000

0.2419

0.57

0.574

1725

0.28

0.780

AR 1

-1.1536 0.1637

-7.05

0.000

AR 1

-1.1266

MA 1

0.2757 0.1961

1.41

0.168

MA 1

0.1373

0.999

Constant 486

Constant 1

1429

With missed observations

Regression Analysis: Y(t) versus X(t)

Regression Analysis: Y^(t) versus X^(t)

The regression equation is

The regression equation is

Y(t) = 4029 + 1.52 Export X(t)

Y^(t) = 4013 + 1.55 X^(t)

ARIMA (1,2,1)

ARIMA (1,2,1)

Type

Without missed observations

0.00

Differencing: 2 regular differences

Differencing: 2 regular differences

Number of observations: Original series 43, after differencing 41

Number of observations: Original series 43, after differencing 41

Residuals: SS = 5983868951 (back forecasts excluded)

Residuals: SS = 6027685315 (back forecasts excluded) MS = 158623298

MS = 157470236 DF = 38

SE Coef

Constant 4029.2

724.2

Modified Box-Pierce (Ljung-Box) ChiSquare statistic

Lag

12

24

36

Lag

Chi-Square

7.3

7.6

DF

9

24

36

8.6

Chi-Square 8.3

8.5

8.9

33

DF

21

33

P

5.56 0.000

SE Coef

Constant 4012.5

807.0

Analysis of Variance

Analysis of Variance

DF

SS

MS

F

Source

2819.14

DF

F

0.00

P

2254.98

Total

Total

42 45865442435

42 45458180623 Probability Plot of RESI3 Normal

99

Mean StDev N KS P-Value

P-Value

0.509 0.993

1.000

70

70

5

-10000

In this study we will comparison between our results with some missing observations and the classical results where all observations are available.

Let Z (t )  X (t ) Y (t )T where X (t ) is the series of the Export average and Y (t ) is the series of the Gross domestic product, first we consider that the observations are © 2017, IRJET

|

Impact Factor value: 5.181

-5000

0 RESI2

5000

10000

Normal-plot of standardized Residuals

In this section we adjust the regression model which represents the relationship between the Gross domestic product and Export in the period from 1970 to 2012 million Libyan dinars.

|

60 50 40 30

10

5

1

-2.96115E-12 3962 43 0.133 0.056

20

10

product and Export

90 80

60 50 40 30

Mean StDev N KS P-Value

95

80

20

4.2.3. Studying the regression between Gross domestic

99

-1.80207E-11 3957 43 0.133 0.053

Percent

1.000

0.00

Residual Error 41 811760984 19799048

Percent

0.996

MS

Residual Error 41 657479494 16036085

90

0.608

SS

Regression 1 44646419639 44646419639

P

95

P-Value

4.97 0.000

S = 4449.61 R-Sq = 98.2% R-Sq(adj) = 98.2%

Normal

9

P

S = 4004.51 R-Sq = 98.6% R-Sq(adj) = 98.5%

Probability Plot of RESI2

21

T

Export X^(t) 1.5544 0.03273 47.49 0.000

Source

48

Predictor Coef

Export X(t) 1.52488 0.02872 53.10 0.000

DF = 38

12

T

Regression 1 45207962941 45207962941

Modified Box-Pierce (Ljung-Box) ChiSquare statistic 48

Predictor Coef

1

-10000

-5000

0 RESI3

5000

10000

Normal-plot of standardized Residuals

4.2.4. Conclusion 1. Tables 4.2.1 and 4.2.2 shows the study of time series with missed observations and the original time series and we investigated that they have the same results.

2. Table 4.2.3 shows the study of regression models between Gross domestic product and Export with some missed observations which had the same results of the study of the classical regression models.

ISO 9001:2008 Certified Journal

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International Research Journal of Engineering and Technology (IRJET)

e-ISSN: 2395 -0056

Volume: 04 Issue: 04 | Apr -2017

p-ISSN: 2395-0072

www.irjet.net

REFERENCES

[1]

[2]

[3]

[4]

[5]

[6] [7]

D.R. Brillinger and M. Rosenblatt, ''Asymptotic theory of estimates of k-th order spectra'', in: B.Harris (Ed.), ''Advanced Seminar on Spectral Analysis of Time Series'', Wiley, New York, 1967,pp. 153–188.

D.R. Brillinger, ''Asymptotic properties of spectral estimates of second order'', Biometrika 56 (2),(1969) 375–390.

R. Dahlhaus, ''On a spectral density estimate obtained by averaging periodograms'', J. Appl. Probab. 22 (1985) 592–610.

M.A. Ghazal, E.A. Farag,'' Estimating the spectral density, autocovariance function and spectral measure of continuous time stationary processes'', in: Annual Conference, ISSR, Cairo, vol. 33, no. part I, 1998, pp. 1– 20.

M.A. Ghazal, ''On a spectral density estimate on noncrossed intervals observation'', Int. J. Appl. Math. 1 (8)(1999) 875–882.

D.R. Brillinger, Time series data analysis and theory; SIAM,2001.

A.A.M. Teamah, H.S. Bakouch,''Multitaper Multivariate spectral estimators of time series with distorted observations'', International Journal of Pure and Applied Mathematics, Vol.14 , No.1 (2004),45-57.

Chris Chatfield, The Analysis of Time Series, An Introduction, Sixth Edition, 2005.

[9] M.A. Ghazal, Farag, E.A,El-Desokey, ''Some properties of the discrete expanded finite Fourier transform with missed Observations'', Ain Shams University, Faculty of Engineering, No.3,2005,pp.887-902.

[10] M.A.Ghazal ,G.S. Mokaddis and A.El-Desokey, ''Spectral Analysis of strictly stationary continuous time series''; Journal of Mathematical Sciences; Vol.3,No.1(2009) 115.

[11] M.A.Ghazal ,G.S. Mokaddis and A.El-Desokey, ''Asymptotic Properties of spectral Estimates of SecondOrder with Missed Observations''; Journal of Mathematics and statistics 6(1):10-16,2010.

[8]

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