Partially Observed Discrete-valued Time Series
The analysis of time series of counts is a rapidly developing area. It has very broad application in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical framework of Gaussian- like series. In this paper we derive recursive filters for partially...
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| Cite this: | Partially Observed Discrete-valued Time Series / Lakhlar Aggoun, Lakdare Benkherouf, Ali Benmerzouga // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 33-43. — Бібліогр.: 10 назв. — рос. |
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| citation_txt | Partially Observed Discrete-valued Time Series / Lakhlar Aggoun, Lakdare Benkherouf, Ali Benmerzouga // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 33-43. — Бібліогр.: 10 назв. — рос. |
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| description | The analysis of time series of counts is a rapidly developing area. It has very broad application in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical framework of Gaussian- like series. In this paper we derive recursive filters for partially observed discrete-valued time series. These processes are regulated by thinning binomial and multinomial operators (to be defined below).
Анализ временных последовательностей отсчетов — интенсивно развивающееся направление. Такой анализ широко используется для базовых целочисленных временных последовательностей, с которыми нельзя удовлетворительно работать в рамках классических последовательностей гауссовa типа. Получены рекурсивные фильтры для частично наблю даемых дискретизированных временных последовательностей. Показано, что эти процессы регулируются прореживающими биномиальными и полиномиальными операторами.
Аналіз часових послідовностей відліків — напрям, що інтенсивно розвивається. Такий аналіз широко використовується для базових цілочисельних часових послідовностей, з якими не можна задовільно працювати у рамках класичних послідовностей гаусовa типу. Отримано рекурсивні фільтри для частково спостерігаємих дискретизованих часових послідовностей. Показано, що ці процеси регулюються проріжуючими біноміальними та поліноміальними операторами.
|
| first_indexed | 2025-11-25T20:49:02Z |
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Lakhdar Aggoun *, Lakdere Benkherouf **, Ali Benmerzouga *
* Department of Mathematics and Statistics, Sultan Qaboos University
(P.O.Box 36, Al-Khod 123, Sultanate of Oman,
e-mail: laggoun@squ.edu.om)
** Department of Statistics and Operations Research College
of Science, Kuwait University
(P.O.Box 5969, Safat 13060, Kuwait,
e-mail: lakdereb@kuc01.kuniv.edu.kw)
Partially Observed Discrete-valued Time Series
(Recommended by Prof. E. Dshalalow)
The analysis of time series of counts is a rapidly developing area. It has very broad application in
view of the host of integer-valued time series which cannot be satisfactorily handled within the
classical framework of Gaussian- like series. In this paper we derive recursive filters for partially
observed discrete-valued time series. These processes are regulated by thinning binomial and
multinomial operators (to be defined below).
Àíàëèç âðåìåííûõ ïîñëåäîâàòåëüíîñòåé îòñ÷åòîâ — èíòåíñèâíî ðàçâèâàþùååñÿ íàïðàâ-
ëåíèå. Òàêîé àíàëèç øèðîêî èñïîëüçóåòñÿ äëÿ áàçîâûõ öåëî÷èñëåííûõ âðåìåííûõ ïîñëå-
äîâàòåëüíîñòåé, ñ êîòîðûìè íåëüçÿ óäîâëåòâîðèòåëüíî ðàáîòàòü â ðàìêàõ êëàññè÷åñêèõ
ïîñëåäîâàòåëüíîñòåé ãàóññîâa òèïà. Ïîëó÷åíû ðåêóðñèâíûå ôèëüòðû äëÿ ÷àñòè÷íî íàáëþ-
äàåìûõ äèñêðåòèçèðîâàííûõ âðåìåííûõ ïîñëåäîâàòåëüíîñòåé. Ïîêàçàíî, ÷òî ýòè ïðîöåññû
ðåãóëèðóþòñÿ ïðîðåæèâàþùèìè áèíîìèàëüíûìè è ïîëèíîìèàëüíûìè îïåðàòîðàìè.
K e y w o r d s: filtering, time series, change of measre, binomial thinning.
1. Introduction. The analysis of time series of counts is a rapidly developing
area [1–6] and the book by MacDonald [7]. It has very broad application in view
of the host of integer-valued time series which cannot be satisfactorily handled
within the classical framework of Gaussianlike series. Many of the statistical
which occur in practice are by their very nature discrete-valued (see [7] for more
details). These models are also adequate for the study of branching processes
with immigration [8].
In this paper we derive recursive filters for partially observed discretevalued
time series. The dynamics of these processes are regulated by thinning binomial
and multinomial operators.
The Binomial thining operator «�» [2, 5] is defined as follows. For any
nonnegative integer-valued random variable X and ��� {0, 1},
a X Y j
j
X
� �
�
�
1
,
(1)
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 33
where Y1, Y2, . . . is a sequence of of i.i.d. random variables independent of X,
such that P Y P Yj j( ) ( )� � � � �1 1 0 �.
2. Scalar dynamics. Consider a system whose state at time k is xk �
�
Z .
The time index k of the state evolution will be discrete and identified with � =
= {0, 1, 2, ..., }.
Let ( , , ) � P be a probability space upon which {vk}, {wk}, k�� are inde-
pendent and identically distributed (i.i.d.) sequences of random variables such
that, for all k, vk �
�
� has probability function
and wk is Gaussian random
variables, having zero means and variances 1 (N (0, 1)). Let {�k }, k�� be the
complete filtration (that is �
0
contains all the P-null events) generated by {x0,
x1, ..., xk}. The state of the system satisfies the dynamics
x X x vk k k k� �
� �
1 1
� ( )� . (2)
Here{ }X k k��
is a stochastic process with finite state space S X of size N which
we identify, without loss of generality, with the canonical basis {e1, ..., eN} of
�
N
. Since Xn takes only a finite number of values we may write
� � � � � ���( ) ( ( ),..., ( )) ( ,..., ))X e ek N N� � �
1 1
�
Therefore � ��( ) ,X Xk k� . Here . ,. denotes the inner product in �
N
.
Let’s assume the process X is a Markov chain with semimartingale representa-
tion [9, 10].
X AX Mk k k� �
�1
(3)
where{ }M k k��
is a sequence of martingale increments with respect to the com-
plete filtration generated by X and A denotes the probability transition matrix of
the Markov chain X.
A useful and simple model for a noisy observation of xk is to suppose it is given
as a linear function of xk plus a random «noise» term. That is, we suppose that for
some real numbers ck and positive real numbers dk our observations have the form
y c x d wk k k k k� � . (4)
We shall also write {�k }, k�� for the complete filtration generated by
{ , ,..., }y y yk0 1
.
Using measure change techniques we shall derive a recursive expression for
the conditional distribution of xk given �k .
Recursive estimation. Initially we suppose all processes are defined on an
«ideal» probability space ( , , ) � P ; then under a new probability measure P, to
be defined, the model dynamics (2) and (4) will hold.
Suppose that under P:
1) {xk}, k�� is an i.i.d. sequence with density function
( )x with support
in �
�
;
Lakhdar Aggoun, Lakdere Benkherouf, Ali Benmerzouga
34 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
2) {yk}, k�� is an i.i.d. N (0, 1) sequence with density function
�
�
( ) .
/y e y
�
�1
2
2
2
For l = 0, �
�
�
0
0
1
0 0 0
0 0
�
�
�
( ( ))
( )
d y c x
d y
and for l = 1, 2, ... define
�
�� ��
��
l
l l l l l l l
l l l
x X x d y c x
d x y
�
� �
� �
�
( , ( ( ))
( ( )
1 1
1
�
,
(5)
�k
l
k
l�
�
��
0
.
(6)
Let �k be the complete �-field generated by {x0, x1, ..., xk , �� , , ...X x
0 0
�
..., , ,�� X xk k� y0, y1, ..., yk} for k��.
Lemma 1. The process {�k }, k�� is a P-martingale with respect to the fil-
tration {�k }, k��.
P r o o f . Since �k is �k -measurable E Ek k k k k[ ] [ ]� � �
� �
�
1 1
� � . There-
fore we must show that E k k[ ]�
�
�
1
1� :
E E
x X x d y c x
k k
k k k k k k k
[ ]
( , ( ( )
�
�� ��
�
� �
�
� � �
�
� �
1
1 1
1
1 1 1
�
� )
( ( )d x yk k k
k
� � �
�
�
�
�
�
�
�
1 1 1
��
�
�
� �
�
�
�
�
� � �E
x X x
x
E
d y c xk k k
k
k k k k
�� �
�
�( ,
(
( ( ))1
1
1
1
1 1 1
�
d y
x
k k
k k k
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
1 1
1
� ( )
,� � .
Now,
E
d y c x
d y
xk k k k
k k
k k
�
�
( ( ))
( )
,
�
�
� � �
� �
�
��
�
�
�
�
�
�
1
1
1 1 1
1 1
1
�
�
�
�
�
�
� �
�
�
�
�
�
( ( ))
( )
( )
d y c x
d y
y dyk k k
k
1
1
1 1
1
1
�
and
E
x X x
x
k k k
k
k
�� �
�
( ,
(
�
�
��
�
�
�
�
�
�
1
1
�
�
�
��
�
�
�
�
�
� �
� �
� �
� �E
x X x
x
x uk k
x
k
u
�� �
�
( ,
(
( ) ( )
�
� �
� 1.
Partially Observed Discrete-valued Time Series
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 35
Define P on { ,�} by setting the restriction of the Radon—Nykodim deriv-
ative
dP
dP
to �k equal to � k . Then:
Lemma 2. {vk}, k�� is an i.i.d. sequence with density function
�(x with
support in �
�
and {wk}, k�� are i.i.d. N(0, 1) sequences of random variables,
where
v x X xk k k k� �
� �
1 1
�
( ,�� �� ,
w d y c xk k k k k� �
��
( ( )
1
.
P r o o f. Suppose f g, : � �� are «test» functions (i.e. measurable func-
tions with compact support). Then with E (resp. E) denoting expectation under P
(resp. P) and using Bayes’ Theorem [9, 10]
E f v g w
E f v g w
E
k k k
k k k k k
k
[ ( ) ( ) ]
[ ( ) ( ) ]
[
� �
� � �
�
1 1
1 1 1
�
��
�
�
� k k�
�
1 � ]
�
� � �
E f v g wk k k k[ ( ) ( ) ]� 1 1 1
� ,
where the last equality follows from Lemma 1. Consequently
E f v g w E f v g wk k k k k k k[ ( ) ( ) ] [ ( ) ( ) ]
� � � � �
� �
1 1 1 1 1
� ��
�
� �
� �
�
� � �
�
E
x X x d y c x
d x
k k k k k k k
k
�� ��
�
( , ( ( ))
(
1 1
1
1 1 1
1
�
� ( )yk�
�
�
�
�
�
�
�
1
� � � �
� �
�
� � �
f x X x g d y c xk k k k k k k k( , ( ( )) ]
1 1
1
1 1 1
�� �� �
�
�
� �
�
�
�
�
�
�
E
x X x
x
f x X xk k k
k
k k k
�� �
�
�� �
( ,
(
( ,
1
1
1
�
�
�
�
�
�
� � �
� �
�
�
�
E
d y c x
d y
g d yk k k k
k k
k k
�
�
( ( ))
( )
( (
1
1
1 1 1
1 1
1
1
1
�
�
�
�
�
�
�
�
�
�� � �
c x xk k k k k1 1 1
)) ,� � .
Now
E
d y c x
d y
g d yk k k k
k k
k k
�
�
( ( ))
( )
( (
�
�
� � �
� �
�
�
�
�
�
1
1
1 1 1
1 1
1
1
1
c x xk k k k� � �
�
�
�
�
�
�
�
1 1 1
)) ,�
�
�
�
�
�
�
�
�
�
� �
�
�
�
( ( ))
( )
( ) ( (
d y cx
d y
y g d y c xk k
k
k k k
1
1
1
1
1
1
1 1
)) ( ) ( )dy u g u du
� �
� �
� �
Lakhdar Aggoun, Lakdere Benkherouf, Ali Benmerzouga
36 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
and
E
x X x
x
f x X xk k k
k
k k k k
�� �
�
�� �
( ,
(
( ,
�
�
�
�
�
�
�
�
�
�
�
�
1
1
1
�
� �
�
�
�
�
�
�
�
�
�
�
�
�
�E
x X x
x
x f x X x
x
k k
k k k
�
�� �
�
� �� �
( ,
(
( ( ,
�
� �
�( ( )z f z
x�
�
�
�
.
Therefore E f v g w z f z u g u duk k k
x
[ ( ) ( ) ] ( ( ) ( ) ( )
� �
�
�
�
� �1 1
�
� �
� �
and the lemma is
proved.
Using Bayes’ Theorem [10]
E I x x X
E I x x X
E
k k k
k k k k
k k
[ ( ) ]
[ ( ) ]
[ ]
� �
�
�
�
�
�
�
,
(7)
where E (resp. E) denotes expectations with respect to P (resp. P). Consider the
unnormalized, conditional expectation which is the numerator of (7) and write
E I x x X q x q x q xk k k k k k k
N
[ ( ) ] ( ) ( ( ), ..., ( ))� � � � ��
1
. (8)
If pk (.) denotes the normalized conditional density, such that E I xk[ ( �
� �x X p xk k k) ] ( )� , then from (7) we see that
p x q x q zk k k
z
( ) ( ) ( )�
�
�
�
�
�
��
�1
for x�
�
� , k�� .
Then we have the following result.
Theorem 1. The measure-valued process q satisfies the recursion
q x A z x q zk
z
k�
�
�
�
�1
( ) ( , ) ( )
�
B ,
where B ( , )z x is a diagonal matrix with entries
�
�
� �
( ( ))
( )
( ) (
d y cx
d y
x r
z
r
k
k r
z
i
r
i
�
�
� �
�
�
�
!
"
#
$ ��
1
1
1 0
1 )
z r�
.
P r o o f. In view of (3), (5) and (6)
E I x x Xk k k k k[ ( ) ]� �
� � � �
� �1 1 1 1
�
�
� �
�
�
� �
�
E
x X x d y c x
d x y
k
k k k k k
k k
�
�� ��
��
( , ( ( ))
( (
�
1
1
1 1
1 �
� �
�
�
�
�
�
�
�
�
1
1 1
)
( ]
�%&x X Mk k k�
Partially Observed Discrete-valued Time Series
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 37
�
�
�
�
�
� �
� � �
�
�
�
�
( ( ))
( )
(
d y c x
d y
E x xk k k
k k i
N
k i k
1
1
1 1
1 1 1
� � � X e Aek i k i, ]�
�
�
1
�
�
�
�
�
� �
� �
�
�
( ( ))
( )
d y c x
d y
k k k
k k
1
1
1 1
1 1
� �
�
!
"
#
$ �
� �
�
� �
i
N
k
r
x
k
i
r
i
x r
k i kE x r
x
r
X e
k
k
1 0
1�
� �( ) ( ) , �
�
�
�
�
�
�
�
�
1
] Aei
�
�
�
�
�
� �
� �
�
�
( ( ))
( )
d y c x
d y
k k k
k k
1
1
1 1
1 1
� �
�
!
"
#
$ � �
� � �
�
� � �
�
i
N
k
z r
z
i
r
i
z r
kE x r
z
r
I x
1 0
1�
�
� �( ) ( ) ( z X e Aek i k i) , ]�
�
�
�
�
�
�
.
The last equality follows from the fact that xk+1 has distribution
and is inde-
pendent of everything else under P. Also, note that given yk+1 we condition only
on �k to get an expression similar to notation (8), that is,
E I x x Xk k k k[ ( ) ]�
� � � �
� �
1 1 1 1
�
�
�
�
�
!
�
�
� � � �
� � �
�
�
�
( ( ))
( )
( )
d y cx
d y
x r
z
r
k
k i
N
z r
z1
1
1 1 0�
"
#
$ � �
�
� � �i
r
i
z r
k i iq z e Ae( ) (1
�
�
�
�A z x q z
z
k
�
B ( , ) ( ),
where B ( , )z x is a diagonal matrix with entries
�
�
� �
( ( ))
( )
( ) (
d y cx
d y
x r
z
r
k
k r
z
i
r
i
�
�
� �
�
�
�
!
"
#
$ ��
1
1
1 0
1 )
z r�
.
Which finishes the proof.
Vector dynamics. Consider a system whose state at time k = 0, 1, 2, ..., is
X k
m
�
�
� and which can be observed only indirectly through another process
Yk
d
�� .
Let ( , , ) � P be a probability space upon which Vk and Wk are sequences of
random variables such that Wk is normally distributed with means 0 and
covariance identity matrices I d d� and Vk has probability distribution
with sup-
port in �
�
m
. Assume that Dk, k ' 0, are non singular matrices. Let {�k }, k��, be
the complete filtration generated by {X0, X1, ..., Xk}.
Lakhdar Aggoun, Lakdere Benkherouf, Ali Benmerzouga
38 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
Now we wish to generalize the operator � to vector-valued random variables
with non-negative integer-valued components.
For any vector X = (X 1
, ..., X m
)� in �
�
m
and any vector �� � �
i i
m
i
� �( ,..., )
1
such that � j
i
(0 and � j
i
i
� �1define
�� � �
i i i i
m
i i
j
i
j
Z
mj
i
j
X X X Y Y
i
) � ) ) � �
� �
�( , ..., ) , ...,
1 1
1
1
1
Z m
i
�
�
!
!
"
#
$
$
�
, (9)
where Zi
� , i, � �1, ..., m, are non-negative, integer-valued random variables such
that Z Xi
m
i
�
��
� �
1
. For each i, �, Y Yi
m
i
� �1
, ..., , are i.i.d. nonnegative, integer-valued
random variables with probability function * �
i
.
Let
A m
� ( , ..., ),� �
1 A X Xi
i
m
i
) � )
�
��
1
.
(10)
One possible interpretation of this model is that X X X m
� �( , ..., )
1
repre-
sents a population composed of m distinct groups of, say, cells. Some time later,
each cell in the population, regardless to which group it belongs, can mutate and
divide itself into a number of new cells of any of the m types. For instance, a cell
of type 1 may mutate with probability�
2
1
to produce through division a new gen-
eration of cells of type 2. Let �
2
1 1
2
1
1
2
1
) �
�
�X Y j
j
Z
is the (random) number of new
cells of type 2 with Z
2
1
parents of type1. In other words, for j = 1, ..., Z
2
1
, the j-th
parent cell of type 1 gave birth to Y j2
1
new cells of type 2. Here Y j2
1
is a random
variable with probability function *
2
1
with support in �
�
.
The state and observations of the system are given by the dynamics
X A X Vk k k k
m
� � �
� ) � �
1 1
� , (11)
Y C X D Wk k k k k
d
� � �� . (12)
Here Ck is a matrix of appropriate dimensions and A Xk k) is defined in (10).
We write again {�k }, k��, for the complete filtration generated by the
observed data {Y0, Y1, ..., Yk} up to time k. Using measure change techniques
we shall derive a recursive expression for the conditional distribution of Xk
given �k .
Partially Observed Discrete-valued Time Series
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 39
Recursive estimation. Initially we suppose all processes are defined on an
«ideal» probability space ( , , ) � P ; then under a new probability measure P, to
be defined, the model dynamics (11) and (12) will hold.
Suppose that under P:
1) {Xk}, k��, is an i.i.d. sequence with probability function
( )x defined
on �
�
m
;
2) {Yk}, k��, is an i.i.d. N (0, I d d� ) sequence with density function
�
�
( )
( )
/
/y e
d
y y
�
� �1
2
2
2
.
For any square matrix B write | B | for the absolute value of its determinant.
For l = 0, �
�
� %
0
0
1
0 0 0
0 0
�
�
�
( ( )
)
D Y C X
D Y
and for l = 1, 2, ... define
�
�
�� %
l
l l l l l l l
l l l
X A X D Y C X
D X Y
�
� ) �
� �
�
( ) ( ( )
( )
1 1
1
,
� �k l
l
k
�
�
�
0
.
Let {�k } be the complete �-field generated by {X0, X1, ..., Xk, Y0, Y1, ..., Yk}
for k��.
The process {�k }, k��, is an P-martingale with respect to the filtration {�k }.
Define P on { ,�} by setting the restriction of the Radon—Nykodim de-
rivative
dP
dP
to �k equal to �k . It can be shown that on { ,�} and under P, Wk is
normally distributed with means 0 and covariance identity matrix I d d� , and Vk
has probability function
defined on �
�
m
where
V X A Xk k k k� �
� � )
1 1
�
, W D Y C Xk k k k k� �
�� 1
( ) ,
write
E I x x X q xk k k k n[ ( ) ] ( )� � �� .
Then we have the following result.
Theorem 2. For k ' 0
q x
D Y C x
D Y
k
k k k
k k
�
�
�
� �
� �
�
�
�
1
1
1
1 1
1 1
( )
( ( )
)
�
� %
�
�
!
!
"
#� � � � � �
�
� � � �
u i
m
z z u i
m
k
i
i
m
i
m i
m
i i
x
z z
� 1 1 1
1
...
...
$
$
�%� %�
1
1
i z
m
i zi
m
i
) ... )
Lakhdar Aggoun, Lakdere Benkherouf, Ali Benmerzouga
40 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
� �
�
!
!
"
#
$
$
�
�
!
!
! � � �
� � �
x y y
i
m
j
i
j
z
mj
i
j
zi
m
i
1
1
1 1
1
,...,
"
#
$
$
$ � �
��
i
m
i
j
i
k
j
z
y q u
i
,
( ) ( )
�
� �
�
1 1
* .
P r o o f. The proof is similar to the scalar case and is skipped.
A sampling observation model. The state of the system is again given by
the dynamics in (11). Write N Xk k
i
i
m
�
�
�
1
and Ï (N k ) for the set of all partitions of
N k into m summands; that is, x N k�+ ( ) if x x x x m
� ( , , ..., )
1 2
where each x i
is a
non-negative integer and x x x Nm
k
1 2
� � � �... . In this section we assume that
the total number of individual N k is approximately known but it is practically
very difficult to measure directly their distribution between the m types. There-
fore the population is sampled by withdrawing, (with replacement), at each time
k, n individuals and observing to which type they belong. That is, at each time k a
sample
Y Y Y Y nk k k k
m
� �( , , ..., ) ( )
1 2
+
is obtained, where Ï (n) is the set of partitions of n.
We assume that
P Y y X x
n
y y y
x
N
x
N
k k m
k
y
k
( )
...
� � �
�
!
"
#
$
�
!
"
#
$
�
!
"
#
$
1 2
1 2
1 y m
k
y
x
N
m2
...
�
!
"
#
$ . (13)
Clearly this sequence of samples, Y (0), Y (1), Y (2), ... enables us to revise
our estimates of the state Xk.
Recursive estimates. Initially we suppose all processes are defined on an
«ideal» probability space ( , , ) � P ; then under a new probability measure P, to
be defined, the model dynamics (11) and (13) will hold.
Suppose that under P:
1) {Xk}, k��, is an i.i.d. sequence with probability function � ( )x defined
on �
�
m
;
2) {Yk}, k��, is an i.i.d. sequence such that for y � Ï (n) ,
P Y y
n
y y y m
k k m
n
( )
...
� �
�
!
"
#
$
�
!
"
#
$�
1 2
1
.
For l = 0, �
0
1� and for l = 1, 2, ... define
�
�
l
l l l
l
n k
k
Y
k
k
X A X
X
m
X
N
X
N
k
�
� ) �
!
!
"
#
$
$
�
!
!
� �
�
�
( )
(
1 1
1 2
1
"
#
$
$
�
!
!
"
#
$
$
Y
k
m
k
Yk k
m
X
N
2
... , � �k l
l
k
�
�
�
0
.
Partially Observed Discrete-valued Time Series
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 41
Let {�k } be the complete �-field generated by {X0, X1, ..., Xk, Y0, Y1, ..., Yk}
for k��. The process {�k }, k��, is an P-martingale with respect to the filtra-
tion {�k }.
Define P on { ,�} by setting the restriction of the Radon—Nykodim de-
rivative
dP
dP
to�k equal to �k . It can be shown that on { ,�} and under P, Vk has
probability function �( )x defined on �
�
m
whereV X A Xk k k k� �
� � )
1 1
�
and (13) is
true. For r �Ï(Nk+1) write q r E I X rk k k k� � � �
� �
1 1 1 1
( ) [ ( ) ]� � .
Note that
I X rk
r N k
( )
( )
�
�
� �� 1
1
+
so that
q r Ek k k
r N k
� � �
�
�� 1 1 1
( ) [ ]
( )
�
+
� .
We then have the following recursion.
Theorem 3. If Y Y Y Y y y y Nk k k k
m m
k� � �( , , ..., ) ( , , ..., ) ( )
1 2 1 2
+ ,
q r m
r
N
r
N
r
N
k
n
k
y
k
y m
k
ym
( ) ...�
�
!
"
#
$
�
!
"
#
$
�
!
"
#
$ �
1 2
1 2
�
�
!
� � � � � �
�
� � � �
s N i
m
z z s i
m i
i
m
i
k
i
m
i i
s
z z
+( ) ...
...
1 1
1 1 1
!
"
#
$
$
�%� %�
1
1
i z
m
i zi
m
i
) ... )
� �
�
!
!
"
#
$
$
�
�
!
!
! � � �
� � �� r y y
i
m
j
i
j
z
mj
i
j
zi
m
i
1
1
1 1
1
, ...,
"
#
$
$
$ �
�
�
��
i
m
i
j
i
k
j
z
y q s
i
,
( ) ( )
�
� �
�
1
1
1
* .
(Note we take 0
0
= 1.)
P r o o f .
q r E I X rk k k k( ) [ ( ) ]� � �� �
� � � �
�
E I X r Y y y yk k k k
m
[ ( ) , ( , , ..., )]� �
1
1 2
� � � �
� �
E I X r Y y y yk k k k k
m
[ ( ) , ( , , ..., )]� 1 1
1 2
� �
�
�
!
"
#
$
�
!
"
#
$
�
!
"
#
$
�
m
r
N
r
N
r
N
E In
k
y
k
y m
k
y
k
m
1 2
1
1 2
... [ (� X rk � )
�
�
( )
(
]
r A X
r
k k
k
� )
�
�
�
1
1
�
�
Lakhdar Aggoun, Lakdere Benkherouf, Ali Benmerzouga
42 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
�
�
!
"
#
$
�
!
"
#
$
�
!
"
#
$
�
�
m
r
N
r
N
r
N
En
k
y
k
y m
k
y
k
s
m
1 2
1
1 2
... [�
+( )
( ) ( ) ]
N
k k k
k
r A s I X s
�
� � ) �
� �
1
1 1
� � ,
using the definition of the operator ) in (9) and (10) yields the result.
Remark.
P X r E I X r
q r
q s
k k k k
k
s N
k
k
( ) ) [ ( ) ]
( )
( )
( )
� � � �
�
�
� �
+
.
To obtain the expected value of Xk given the observations �k we consider
the vector of values r = r 1
, r 2
, ..., r m
) for any r � Ï (Nk). Then
E X
q r r
q s
k k
r N
k
s N
k
k
k
[ ]
( )
( )
( )
( )
� �
�
�
�
�
+
+
.
Àíàë³ç ÷àñîâèõ ïîñë³äîâíîñòåé â³äë³ê³â — íàïðÿì, ùî ³íòåíñèâíî ðîçâèâàºòüñÿ. Òàêèé
àíàë³ç øèðîêî âèêîðèñòîâóºòüñÿ äëÿ áàçîâèõ ö³ëî÷èñåëüíèõ ÷àñîâèõ ïîñë³äîâíîñòåé, ç
ÿêèìè íå ìîæíà çàäîâ³ëüíî ïðàöþâàòè ó ðàìêàõ êëàñè÷íèõ ïîñë³äîâíîñòåé ãàóñîâa òèïó.
Îòðèìàíî ðåêóðñèâí³ ô³ëüòðè äëÿ ÷àñòêîâî ñïîñòåð³ãàºìèõ äèñêðåòèçîâàíèõ ÷àñîâèõ ïî-
ñë³äîâíîñòåé. Ïîêàçàíî, ùî ö³ ïðîöåñè ðåãóëþþòüñÿ ïðîð³æóþ÷èìè á³íîì³àëüíèìè òà
ïîë³íîì³àëüíèìè îïåðàòîðàìè.
1. Aly A. A, Bouzar N. On some integer-valued autoregressive moving average models// J. of
Multivariate Analysis. — 1994. — 50. — P. 132—151.
2. Al-Osh M. N., Alzaid A. A. First order integer-valued autoregressive (INAR(1)) process// J.
Time Series Analysis. — 1987. — 8. — P. 261—275.
3. Freeland R. K., McCabe B. P. M. Analysis of low count time series data by poisson autoreg-
ression// Ibid. — 2004. — 25. — No 5. — P. 701—722.
4. Jung Robert C., Tremayne A. R. Testing for serial dependence in time series models of
counts// Ibid. — 2003. — Vol. 24. — P. 65.
5. McKenzie E. Some simple models for discrete variate time series// Water Res Bull.— 1985. —
21. — P. 645—650.
6. McKenzie E. Some ARMA models for dependent sequences of Poisson counts// Advances
in Applied Probability. — 1988. — 20. — No 44. — P. 822—835.
7. Iain L. MacDonald Hidden Markov and other models for Discrete-Valued Time Series. —
Chapman & Hall, 1997.
8. Dion J. -P. , Gauthier G., Latour A. Branching Processes with Immigration and Integer-Val-
ued Time Series// Serdica Mathematical J. — 1995. — Vol. 21, No 2.
9. Aggoun L., Elliott R.J. Measure Theory and Filtering: Introduction with Applications// Cam-
bridge Series In Statistical and Probabilistic Mathematics.—2004.
10. Elliot R. J., Aggoun L., Moore J. B. Hidden Markov Models: Estimation and Control// Ap-
plications of Mathematics. — 1995. —No. 29.
Ïîñòóïèëà 21.12.06
Partially Observed Discrete-valued Time Series
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 43
|
| id | nasplib_isofts_kiev_ua-123456789-101814 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-11-25T20:49:02Z |
| publishDate | 2007 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Lakhdar Aggoun Lakdere Benkherouf Ali Benmerzouga 2016-06-07T14:31:11Z 2016-06-07T14:31:11Z 2007 Partially Observed Discrete-valued Time Series / Lakhlar Aggoun, Lakdare Benkherouf, Ali Benmerzouga // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 33-43. — Бібліогр.: 10 назв. — рос. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101814 The analysis of time series of counts is a rapidly developing area. It has very broad application in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical framework of Gaussian- like series. In this paper we derive recursive filters for partially observed discrete-valued time series. These processes are regulated by thinning binomial and multinomial operators (to be defined below). Анализ временных последовательностей отсчетов — интенсивно развивающееся направление. Такой анализ широко используется для базовых целочисленных временных последовательностей, с которыми нельзя удовлетворительно работать в рамках классических последовательностей гауссовa типа. Получены рекурсивные фильтры для частично наблю даемых дискретизированных временных последовательностей. Показано, что эти процессы регулируются прореживающими биномиальными и полиномиальными операторами. Аналіз часових послідовностей відліків — напрям, що інтенсивно розвивається. Такий аналіз широко використовується для базових цілочисельних часових послідовностей, з якими не можна задовільно працювати у рамках класичних послідовностей гаусовa типу. Отримано рекурсивні фільтри для частково спостерігаємих дискретизованих часових послідовностей. Показано, що ці процеси регулюються проріжуючими біноміальними та поліноміальними операторами. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Математические методы и модели Partially Observed Discrete-valued Time Series Article published earlier |
| spellingShingle | Partially Observed Discrete-valued Time Series Lakhdar Aggoun Lakdere Benkherouf Ali Benmerzouga Математические методы и модели |
| title | Partially Observed Discrete-valued Time Series |
| title_full | Partially Observed Discrete-valued Time Series |
| title_fullStr | Partially Observed Discrete-valued Time Series |
| title_full_unstemmed | Partially Observed Discrete-valued Time Series |
| title_short | Partially Observed Discrete-valued Time Series |
| title_sort | partially observed discrete-valued time series |
| topic | Математические методы и модели |
| topic_facet | Математические методы и модели |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101814 |
| work_keys_str_mv | AT lakhdaraggoun partiallyobserveddiscretevaluedtimeseries AT lakderebenkherouf partiallyobserveddiscretevaluedtimeseries AT alibenmerzouga partiallyobserveddiscretevaluedtimeseries |