First order autoregressive representation of Markov bi-dimensional chains of 1-order

Sciences

&

Technologie A– N°24, Décembre (2006), pp. 36-40

© Université Mentouri Constantine, Algérie, 2006.

First order autoregressive representation of Markov bi-dimensional chains of 1-order

Reçu le 22/03/2005 – Accepté le 15/12/2006

Abstract

This paper suggests an extension of Lai’s results about the first order autoregressive representation of Markov bi-dimensional chain of 1-order. In the case of markov chain with independent components, we find of course the conditions validating these results for each component.

Keywords: Markov’chains , autoregressive process, spectral density, diagonal development of bivariate distribution

Résumé

Ce papier suggère une extension d'un résultat de Lai à propos de la représentation autorégressive des chaînes de Markov bi-dimenssionnelle d'ordre 1. Dans le cas où les composantes de la chaîne sont indépendantes, nous retrouvons naturellement les conditions validant ces résultats pour chaque composante.

Mots clés: Chaînes de Markov, processus autoregressif, densité spectrale, développement diagonal des distributions bivariées.

2000 Mathematics Subject Classification : 60K05

enerally, a markov chain is not an autoregressive process, except for first order homogeneous, aperiodic and irreducible markov chains at two states {0,1}. However, Lai (1977) has shown that under convenient hypothesis, a first order uni-dimensional markov chain, valued in a finite or non-finite countable set, belong to a first order autoregressives processes class. In this paper, we extend this result to the cases of first order bi-dimensional markov chains valued in

{ 0 , 1 ,..., N }

². Firstly, we study Lai’s analogue conditions and secondly the case of the bivariate distributions having a diagonal development.

G

M. BOUSSEBOUA F. L. RAHMANI

Département de Mathématique, Université Mentouri Constantine, Algérie

ﺞﺋﺎﺘﻧ ﻢﻴﻤﻌﺗ حﺮﺘﻘﻳ ﻞـــــﻤﻌﻟا اﺪـــــه LAI

ﺺﺨﺗ ﻲﺘﻟا

ﻞــــﻴﺜﻤﺘﺑ AR (1) ﻞـــﺳﻼﺴﻠﻟ Markov

ﺔﺟرﺪﻟا ﻦﻣ 1

.

ﺔــﻠﺴﻠﺳ ﺔﻟﺎﺣ ﻲﻓ Markov

ﺼﺤﺘﻧ ﺔﻠﻘﺘﺴﻣ تﺎﺒآﺮﻣ تاد ﻞ

طوﺮﺸﻟا ﻰﻠﻋ ﺎﻌﺒﻃ

LAI ﺔﺒآﺮﻣ ﻞﻜﻟ ﺔﺒﺴﻨﻟﺎﺑ ﺔﻘﻘﺤﺘﻣ

ﺔــــﻴﺣﺎﺘﻔﻤﻟا تﺎﻤﻠﻜﻟ

ا :

ﻞــــﻴﺜﻤﺗ AR (1) ﻞـــﺳﻼﺴﻠﻟ

ﺔــﻠﺴﻠﺳ Markov

ﺺﺨﻠﻣ

First order autoregressive representation of Markov bi-dimensional chains of 1-order

37 1. AR(1) representation of first order

bi-dimensional markov chains

Let

( Z

_t

= ( X

_t

, Y

_t

) : t ∈ Z )

be a first order bi- dimensional markov chain valued in

{ 0 , 1 ,..., N }

²_{. If the}

chain

( ) Z

t is stationary, the probabilities

( ) t P ( Z ( ) x y )

p

_x,y

=

_t

= ,

and the transition probabilities

( ) t P ( Z ( x y ) Z ( ) x y )

p

_x,y,x,'y'

=

_t+1

= ,' ' /

_t

= ,

are independent of time

t

. Let’s set down

P

as the transition probabilities matrix,

P

ⁿ _its

n

^th _power

( n ≥ 1 )

_and

( )

(

xy

≤ ^{x y} ≤ ^N )

= π

_,

: 0 ,

π

its stationary

distribution.

We start writing the covariance function

γ

(.) of the chain

( ) Z

t under the form of a matrix product. After that we determine its spectral density which we shall compare to that of autoregressive process.

Proposition 1

Let

( ^Z

t

= ( ^X

t

, ^Y

t

) : ^t ∈ ^Z )

be a first order bi- dimensional markov chain valued in

{ 0 , 1 ,..., N }

²_{. If this}

chain is stationary, then its covariance matrix γ (.) has the form

( ) ^{n = A} [ ^P

ⁿ

^{− 1}

_(N+1)²

^{. Π} ] ^{B , ∀ n ≥ 0}

γ

( ) ^{1 . 1}

where

Π = ( π

.0

, π

.1

, . . . , π

.N

)

_with

(

j Nj

)

j

π π

π

_.

=

₀

, . . . ,

,

1

⁽ 1⁾2

( 1 , . . ., 1 )

t

N₊

=

a vector of (N+1)²

R

and A and B are conveniently chosen matrices.

Proof. Let us set down that the relation

( ) 1 . 1

determines entirely the covariance function

^γ () ^.

_{and a}

simple computation provides us the coefficients of the covariance matrix

^γ ( ) ⁿ

_:

( ) n =

i,j

σ

( ) ( ) ( )

^{( )}

( )

( )

⎪⎪

⎩

⎪⎪

⎨

⎧

∑ − =

∑ − >

− + +

−

−

− −

−

N y x y

x ij ij

j i j i N

y x y

x j j i i nxyxy ij ij

n if y

x y x

n if p

y x y x y x

' ,' ,

, , ,

2 4

' ,' ,

, 2 1 2 1 ,,,'' , ,

0 ,

.

0 .

, . ' '

β α π

β α π

with

( )

^{( )i j}

n

y x j

i

x x y

+

−

= ⎟⎟⎠

⎞

⎜⎜⎝

=⎛

∑

⁴

0 ,

,

. π ,

α

and

( , ) , , 1 , 2 .

2

0 ,

,

⎟⎟ ⎠ ∀ =

⎞

⎜⎜ ⎝

= ⎛

− +

∑

ⁿ=

^{y x y}

^{i j}

^{i j}

y

x j

i

π

β

Considering then the matrix

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

. .

1 . 0

. .

1 . 0

' . . . ' '

. . .

N N

A A

A

A A

A A

where

( ) ( ) ( )

( x x x N )

x

A

_x.

= . π , 0 π , 1 . . . π ,

,

( ) ( ) ( )

( x x N x N )

A '

_x.

= 0 , π , 1 , 2 π , 2 , . . . , π ,

and the matrix

( ) ( ) ( ) ( ) ^⎟⎟ _⎠

⎞

⎜⎜ ⎝

= ⎛

^{+ + + +}

N N

N N

B N

^N

t N

t N t N

t t

,..., 1 , 0 . . . ,..., 1 , 0 ,..., 1 , 0 ,..., 1 , 0

1 . . . . 1 . 2 1 . 1 1

.

0

_{1 1 1 1}

we verify easily that

( )

( ^P ) ^{B ( ) n}

A .

ⁿ

− 1

_N+12

. Π . = γ

This is being true for every

n

≥

0

, setting down of course ( 1)²

0

= I

_N+

P

the (N+1)²-order unitmatrix.

Proposition 2

Let

( Z

_t

: t ∈ Z )

be a first order bi-dimensional markov chain with valued in

{ 0 , 1 ,..., N }

², irreducible, aperiodic and stationary and let be its stationary distribution. If the matrix

P

of transition probabilities of

( Z

_t

: t ∈ Z )

admits simple and non nul eigenvalues and if one at least of the following conditions

( ) C

is satisfied, then the spectral density of the chain

( ^Z

t

: ^t ∈ ^Z )

is given by :

( ) ( ) ( ) 0

cos . 2 1

1 2

1

2 2 2

2

2

γ

π λ

λ z z

f z

− +

= −

( ) 1 . 2

where

z

₂ is the secund large eigenvalue (in absolute value) of P

Conditions

( ) C

:

( )

^{( )}( )

( )

^{( )}( )

( )( )

( )( )

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

=

=

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

= +

= +

∑

∑

∑

∑

=

+ + +

=

+ + +

=

+ + +

=

+ + +

N

y x

y N x j N

y x

y N x j

N

y x

y N x j N

y x

y N x j

v y and v x ii u

y x y and

u y x x i

0 ,

1 1 0 ,

1 1

0 ,

1 1 0

,

1 1

0 .

0 .

) , 0 .

1 , .

0 .

1 , . )

π

π

( N ) and where u

^{( )j}

and v

^{( )j}

j

=

3 , . . .,

+

1

²

∀

are respectively the eigenvectors of matrices

P

and ^t

P

associated to the eigenvalue

z

_j. Proof.

Taking into account the factorization

( ) 1 . 1

of

γ ( ) n

and that

I − 1

_(N+1₎2

.

^t

Π = γ ( ) 0

is symetrical, we can write the series

∑ ( )

∈

− Z n

in

n

e

^λ

γ

under the form

( )

[

_{( )}

]

( )

[

( )

]

( )

[

^A ( ) ^B

]

B e e A

B e G A B e G A B I

A n e

N t i N

i

t i i N

Z n

in

. . 1 1 .

. 1 . 1 1 .

1

. . . . . . 1 .

2 2

2

1 1

1

− Π

−

− Π

−

+ +

Π

−

−

=

+

− +

−

∈ +

∑ −

λ λ

λ λ

λγ

( ) 1 . 3

where

G ( ) ( z = I − z . P )

⁻¹

: z ≤ 1

is the generating function of the transition probabilities matrix

P

. As

P

is supposed to be simple and its eigenvalues aren’t nul, then following result of Lancaster (1968) and taking into account

( ) C

conditions, we can write :

( )

[ ( ) ] ( ) ( ^{A u v B} )

B z v u z A

B z G A and

B v u z A

B v u z A B z G A

t t t t

t

t t

2 2 2 2 1

1

2 2 2 2 1

1

. .

. 1 . . 1

.

. .

. 1 . . 1

.

λ λ λ λ

− −

− −

=

− −

− −

=

where

u

₁

= 1

_(N+1)2

and v

₁

= Π

are the eigenvectors of

P

and ^t

P

associated to the eigenvalue

2

1

1 and λ

z

= indicate the inverse of the

eigenvalue

z

₂ (we shall notice that the vector is solution of the equation Π=Π

. P

because

π

is a stationary distribution). Reporting these expressions in

( ) 1 . 3

, this latter becomes

( ) [

_{( )}

] ^{( )}

(

^{. . .}

) ( )

^1.4

. . .

. . 1 .

. 1 . 1 .

. 1 . 1 .

2 2 2 2 2 2 2 2

1 1 1

1² 1

B v u e A

B v u e A

B v u e A B v u e A B I

A n e

t t i t i

t t i t

i t

N Z

n in

λ λ λ

λ γ

λ λ

λ λ

λ

− −

− −

− −

− − Π

−

−

=

−

−

+ −

∈

∑

−

The sum of the first three terms on the right hand side of

( ) 1 . 4

is reduced to

− AB

(because the matrices

AB

and

( )

B

A

^t

N

. .

1

.

2

1

Π

+ are symetrics), and taking into account

( ) C

conditions, we have (Lancaster (1968)) :

Au

2^t

v

2

B = AB − A . 1

_(N+1₎2

.

^t

Π . B

Substiting these results in relation

( ) 2 . 4

, we obtain :

( ) Au v B

z z

n

^t

Z n

n

2 2 2

2

2

1

. λ

γ λ

−

= −

∑

∈

Now, we deduce the autoregressive representation of the chain

( ^Z

t

: ^t ∈ ^Z )

.

Corollary 1

Let

( Z

_t

: t ∈ Z )

be a first order bi-dimensional markov chain with states in

{ 0 , 1 ,..., N }

².

If this chain satisfies the conditions in the proposition 2, then this chain is a first order autoregressive and admits as representation

( ) 1 . 5

.

₁

1 t t

t

z Z

Z =

₋

+ ε

where

( ε

t

; ^t ∈ ^Z )

is a sequence of an uncorrelated random variable, having as covariance matrix

Γ

1

= ( ¹ − z

1²

) ^. γ ^{( ) 0 .}

The representation

( ) 1 . 5

allows us to see that the components

X

_t

and Y

_t of the chain

( ) Z

t satisfy a first order autoregressive equation without being markov chains. This result is then legitimely proved by the fact that if the components

( ) X

_t

and ( ) Y

_t are stochastically independent, so each component defines a first order markov chain valued in

{ 0 , 1 ,..., N }

², which are homogeneous, aperiodic and irreducible, and if

2

1

and P

P

are the transition probabilities matrices of

t

t

and Y

X

respectively, then we have

P

=

P

₁ ⊗

P

₂ (tensorial product of

P

₁

and P

₂) and it is easy to see that the covariance matrix

R

is in this case ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

2 1

0 0 γ γ

where

γ

₁

and γ

₂ are the covariance matrices of

( ) X

t

and ( ) Y

t respectively. On the other hand, if

λ

is an eigenvalue of

P

₁

and

^t

( x

₁

, . .. .., x

_N+1

)

is an eigenvector associated to

λ

, then the latter is also eigenvalue of

P

associated to the eigenvector

(

₁

, . .. ..,

₊₁

)

=

^t

u u

_N

u

with

u

_j =^t

( x

₁

^{, . .. ..,} x

_N+₁

)

:

N j = 1 ,...,

∀

, and if

λ

is an eigenvalue of

P

₂ and

(

₁

, . .. ..,

_N+1

)

t

y y

is an eigenvector associated to

λ

, then so

λ

is also an eigenvalue of P associated to the eigenvector

ω =

^t

( ω

₁

, . .. .., ω

_N+1

)

with

ω

_j

= y

_j

. 1

_N+1 :

1 ., . . . ,

1 +

=

∀ j N

. From this, we show that the

conditions

( ) C

are reduced to Lai’s conditions for each chain

( ) X

_t

and ( ) Y

_t . This confirms the individual autoregressive representation of

( ) X

_t

and ( ) Y

_t

First order autoregressive representation of Markov bi-dimensional chains of 1-order

39 02. Diagonal Development

According once more to Lai, let’s suppose that the transition probabilities of a bi-dimensional markov chain

( ) Z

t satisfy the relation

( ) ∑ ( ) ( ) ( )

=

= ^N

m n

m n m n m n y

x y

x p x y x y x y

p

0 ,

, ,

, '

,' ,

, ,' ' ρ .θ , .θ ,' ' 2.1

where

. ( ) θ

n_,m

: ⁿ , ^m ∈ { 0 ,...., ^N }

is a sequence of orthogonal functions relatively to the law

( ) ( ) { }

{ p x , y = P X

_t

= x , Y

_t

= y : x , y ∈ 0 ,...., N }

( ) ( ) ( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

∑

=

=

symbol s

kronec is

y x y

x y

x p e

i

_{nn mm i j}

N

m n

m n m

n

, . , . , ker'

. , .

.

_{, ' , ' ,}

0

,

θ

,

θ

,' '

δ δ δ

and

( ) ρ

n,m is a sequence of parameters caracterizing the bivariate distribution of

( ) Z

t and

( ) Z

t₋₁ which we find directly from

( ) 1 . 4

. We find the following result :

Lemma 1

If the markov chain

( ) Z

t is irreducible and if we suppose

( ) , 1 , ,

1

₀₀

01 10

00

= ρ = ρ = θ x y =

ρ

₁₀

( ) x , y = α

₁₀

x + β

_1,0

and θ

₀₁

( ) x , y = α

₀₁

x + β

₀₁

, ∀ x , y ∈ { 0 , 1 ,...., N } and θ

where

α

₁₀

and α

₀₁

, β

₁₀

and β

₀₁ are real constants with

α

₁₀

≠ 0 and β

₁₀

≠ 0 ,

then we have

( Z

_t

.

_n,m

( ) Z

_t

) = 0 ;

E θ

( , ) { ∈ 0 , 1 ,...., } ( ) ( ) ( )

²

− { 0 , 0 , 1 , 0 , 0 , 1 } .

∀ n m N

If the components of the chain are independent, with the same initial law and for each transition

probability admitting a diagonal development, then the chain

( ) Z

_t admits an ease to write diagonal development. In reverse, if the transition probabilities of the

bivariate chain admit a diagonal

development and is at independent components and if

( ) x y

_n

( ) ( ) x

_m

y

m

n

θ θ

θ

_,

, = .

and

( )

x y _n

( ) ( )

x _m y

m

n ρ ρ

ρ _, , = . , then the transition probabilities of each chain

( ) X

t

and ( ) Y

t admit a diagonal development.

A markov bi-dimensional chain of which the transition probabilities are of the form

( ) 2 . 1

is necessarily stationary.

Lemma 2

The eigenvalues of

P

are

( ρ

n_,m

: ⁿ , ^m = 0 ,..., ^N )

with for eigenvectors on the right

( ) ( )

(

nm nm

^N ) ^where

nm

( ) ^j (

nm

( ) ^j

nm

( ^{N j} ) )

m

n

~ ~ 0 , ,...., ,

,...,

~ 0

, ,

, ,

,

,

θ θ θ θ θ

θ = =

and as eigenvectors on the left

( ) ( )

( ^{Q Q N} )

Q

_{nm nm nm}

t

, ,

,

,..., ~

~ 0

= where

( ) ^j ( ^Q ( ) ^{j Q} ( ^{N j} ) ) ^with

Q ~

_{nm nm}

0 , ,....,

_nm

,

, ,

,

=

( ) ( )

_,

( ) ( ) { }

²

,

x , y p x , y . x , y ; x , y 0 , 1 ,...., N

Q

_nm

= θ

_nm

∀ ∈

. Moreover

( )

.

.

, 2

0 ,

, nm n m

t N

m n

m

n

Q I

₊

=

∑ ^θ

= Proof.

Let’s set down

( ^p

x_,y_,x_,'y_'

: 0 ≤ ^x , ^x ,' ^y , ^y ' ≤ ^N )

the transition probabilities matrix and note

Π

_n,m the

( n, m )

^th

row :

(

nm nmN nm nmN nm N nmNN

)

m

n_, =

p

_,,0,0

, .. ., p

_{,, ,0}

, p

_,,0,1

, .. ., p

_{,, ,1}

, . .. .. ., p

_,,0,

, .. ., p

_{,, ,} Π

By using diagonal development

( ) 3 . 1

of the transition probabilities

p

_n,m,x,y and orthogonality of the sequence

( θ

n_,m

( ) .,. : 0 ≤ ⁿ , ^m ≤ ^N )

, we easily set up without any difficulty that

ρ

_n,m is an eigenvalue of

P

associated to the eigenvector

θ

_n,m. Similarly, we check that

Q

_n,m is an eigenvector of the matrix ^t

P

associated to the eigenvalue

n,m

ρ

. Let’s show now that

θ

_n,m satisfies the dual orthogonality relationship. Let’s note Θ the eigenvectors matrix,

θ

_n,m, and let

Θ

⁻¹ be its inverse. Let’s set down

( ) ( ) ( ) ( ) ( )

( ) ( )

^⎟⎟_⎠^⎞

⎜⎜

⎝

=⎛

y x y

x

y x y

x y x y

x y

x

N N N

N N

, ., ..

, , ., . .

. . . , , ., ..

, , , , ., ..

, , ,

, ,

0

1 , 1

, 0 0

, 0

, 0

θ θ

θ θ

θ θ θ

the

( ) x, y

^th row vector of Θ and

( ) ( ) ( ) ( )

( ) ( ) ( )

^⎟⎟_⎠^⎞

⎜⎜

⎝

= ⎛

y x y

x y

x

y x y x y

y x x

N N N

N t N

, .,

..

, , ., . . . . . , , .,

..

, , , , ., ..

, , ,

, ,

0 1

,

1 , 0 0

, 0

, 0

α α

α

α α

α α

the

( ) x, y

^th column vector of

Θ

⁻¹

.

We have

( ) ( ) ∑ ( ) ( )

=

=

=

^N

m n

y y x x m

n m

n

x y x y

y x y x

0 ,

' , ' , ,

,

, . ,' ' .

' ,' .

, α θ α δ δ

θ

Since the sequence

( θ

_n,m

( ) .,. )

is

p ( ) .,.

-orthogonal, then the coefficients

α

_n,m

( x ,' y ' )

are provided by

α

_n,m

( x ,' y ' ) = p ( x ,' y ' ) . θ

_n,m

( x ,' y ' )

for every

x ,' y '

, and so,

( ) ( ) ( )

( ) ( )

( ) ( )

∑ ∑

= =

=

=

N

m n

N

m n

y y x x m

n m

n

m n m

n m

n

y x p y

x p

y x y

x

y x y

x y

x

0

, , 0

' , ' , ,

,

, ,

,

' ,' . '

,'

' ,' .

,

' ,' .

' ,' .

,

δ α δ

θ

α θ

θ

Now, the

( ) x, ^y

^th coefficient of

Π

_n,m is

( ) , .

_,

( ,' ' ) ( . ,' ' )

,_m

x y Q

_nm

x y p x y

θ

n , then

( )

.

.

2

0 ,

, ,

0 ,

, n m

N

m n

m n m

n t N

m n

m

n

Q I

₊

=

=

= Π

= ∑

∑ ^θ

Theorem

Let

( ) Z

_t be an irreducible markov chain for which the transition probabilities

p

_x,y,x,'y' satisfy the diagonal development

( ) 2 . 1

. If the eigenvalues

ρ

_n,m are simple

and real and

ρ

_0,0

= 1

and if

( )

⎩ ( )

⎨ ⎧

+

=

+

=

1 , 0 1 , 0 1

, 0

0 , 1 0 , 1 0

, 1

. ,

. ,

β α

θ

β α

θ

x y

x

x y

x

, then the process

( ) Z

_t is a first order autoregressive process.

Proof.

The required conditions in the lemma 1 and 2 being satisfied, then the

( ) ^C

conditions of the proposition 2 are also satisfied, and consequently the chain

( ) Z

t is indeniably a first order autoregressive process.

References

1. Brockwell, P. J. and Davis R. A., (1987). Time series : Theory and methods (Springer Verlag).

2. Feller, W., (1968). An introduction to probability theory and its application Vol 1 (John Wiley & sons)

3. Fuller, W.A. (1996). Introduction to Statistical Times series (Wiley Series in Probability and statistics).

4. Lai, C.D.,(1977). First order autoregressive markov processes. Stochastic processes and their applications, 3 1-4.

5. Lancaster, P., (1968.). Theory of matrices (Academic Press).