Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise

Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise

Stochastic processes of counts have very broad applications 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 discuss recursive filters for partially observed discrete-valued time series...

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Опубліковано в: :Электронное моделирование
Дата:2011
Автор: Aggoun, L.
Формат: Стаття
Мова:Ukrainian
Опубліковано: Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України 2011
Теми:
Математические методы и модели
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/61760
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise / L. Aggoun // Электронное моделирование. — 2011 — Т. 33, № 3. — С. 13-21. — Бібліогр.: 10 назв. — англ.

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spelling Aggoun, L.
2014-05-11T10:08:21Z
2014-05-11T10:08:21Z
2011
Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise / L. Aggoun // Электронное моделирование. — 2011 — Т. 33, № 3. — С. 13-21. — Бібліогр.: 10 назв. — англ.
0204-3572
https://nasplib.isofts.kiev.ua/handle/123456789/61760
Stochastic processes of counts have very broad applications 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 discuss recursive filters for partially observed discrete-valued time series where the noise in the observations is a fractional Gaussian noise.
Стохастическая обработка отсчетов широко применяется в множестве задач, содержащих целочисленно-оцениваемые временные ряды, которыми нельзя удовлетворительно оперировать в рамках классических Гауссово-подобных рядов. Рассмотрены рекурсивные фильтры для частично наблюдаемых дискретно-оцениваемых рядов, в которых шумы наблюдений являются дробными Гауссовыми шумами.
Стохастична обробка відліків широко застосовується у багатьох задачах з цілочислооцінюваними часовими рядами, котрими не можна задовільно оперувати в рамках класичних Гаусово-подібних рядів. Розглянуто рекурсивні фільтри для частково спостережуваних дискретно-оцінюваних рядів, в котрих шуми спостережень є дробовими Гаусовими шумами.
uk
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
Электронное моделирование
Математические методы и модели
Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
spellingShingle Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
Aggoun, L.
Математические методы и модели
title_short Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
title_full Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
title_fullStr Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
title_full_unstemmed Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise
title_sort partially observed discrete-valued time series in fractional gaussian noise
author Aggoun, L.
author_facet Aggoun, L.
topic Математические методы и модели
topic_facet Математические методы и модели
publishDate 2011
language Ukrainian
container_title Электронное моделирование
publisher Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
format Article
description Stochastic processes of counts have very broad applications 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 discuss recursive filters for partially observed discrete-valued time series where the noise in the observations is a fractional Gaussian noise. Стохастическая обработка отсчетов широко применяется в множестве задач, содержащих целочисленно-оцениваемые временные ряды, которыми нельзя удовлетворительно оперировать в рамках классических Гауссово-подобных рядов. Рассмотрены рекурсивные фильтры для частично наблюдаемых дискретно-оцениваемых рядов, в которых шумы наблюдений являются дробными Гауссовыми шумами. Стохастична обробка відліків широко застосовується у багатьох задачах з цілочислооцінюваними часовими рядами, котрими не можна задовільно оперувати в рамках класичних Гаусово-подібних рядів. Розглянуто рекурсивні фільтри для частково спостережуваних дискретно-оцінюваних рядів, в котрих шуми спостережень є дробовими Гаусовими шумами.
issn 0204-3572
url https://nasplib.isofts.kiev.ua/handle/123456789/61760
citation_txt Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise / L. Aggoun // Электронное моделирование. — 2011 — Т. 33, № 3. — С. 13-21. — Бібліогр.: 10 назв. — англ.
work_keys_str_mv AT aggounl partiallyobserveddiscretevaluedtimeseriesinfractionalgaussiannoise
first_indexed 2025-11-27T05:28:27Z
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fulltext Lakhdar Aggoun Department of Mathematics and Statistics, Sultan Qaboos University (Sultanate of Oman, E-mail: laggoun@squ.edu.om) Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise (Recommended by Prof. E. Dshalalow) Stochastic processes of counts have very broad applications in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical framework of Gaussi- an-like series. In this paper we discuss recursive filters for partially observed discrete-valued time series where the noise in the observations is a fractional Gaussian noise. Ñòîõàñòè÷åñêàÿ îáðàáîòêà îòñ÷åòîâ øèðîêî ïðèìåíÿåòñÿ â ìíîæåñòâå çàäà÷, ñîäåðæàùèõ öåëî÷èñëåííî-îöåíèâàåìûå âðåìåííûå ðÿäû, êîòîðûìè íåëüçÿ óäîâëåòâîðèòåëüíî îïå- ðèðîâàòü â ðàìêàõ êëàññè÷åñêèõ Ãàóññîâî-ïîäîáíûõ ðÿäîâ. Ðàññìîòðåíû ðåêóðñèâíûå ôèëüòðû äëÿ ÷àñòè÷íî íàáëþäàåìûõ äèñêðåòíî-îöåíèâàåìûõ ðÿäîâ, â êîòîðûõ øóìû íàáëþäåíèé ÿâëÿþòñÿ äðîáíûìè Ãàóññîâûìè øóìàìè. K e y w o r d s: change of measure, discrete-valued time series, fractional Gaussian noise. Introduction. The analysis of time series of counts is a rapidly developing area (e.g. [1—7]). It has very broad application in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical frame- work of Gaussian-like series. Many of the phenomena which occur in practice are by their very nature discrete-valued [4]. To start, we make use of The Binomial thinning operator � introduced in [2, 6], namely: for any nonnegative integer-valued random variable X and� �{ , }0 1 , � � X Y j j X � � � 1 , 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 �. With the operator � on hand and xk standing the realization of an integer- valued process in period k, let x x vk k k� � �� �1 1 1� � , where{ }vk is some integer- valued stochastic process. ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2011. Ò. 33. ¹ 3 13 One may think of xk as referring to the number of patients in a hospital in pe- riod k, then the number of patients in period k + 1 is made up of a portion of those patients who were present in period k (� � xk �1) and new arriving patients vk+1. Time series models incorporating � have been extensively examined in [1—3, 5—7]. Dynamics with fractional Gaussian noise. Let Z denote the set of integers, and Z + denote the set of non- negative integers. Following [8] we define a set of functions L on Z + with values in R. We suppose that if i < 0, then f (i) = 0. These functions could be considered as infinite sequences: f i f i( ) � , i = 0, 1, … . Then we define: if f 1, f 2 are in L the convolution product f 1 * f 2 is defined by ( * )( )f f n f f f fn i i i n i i i n 1 2 1 2 0 1 2 0 � �� � � � � � � . In this set of functions, consider the function u, which is defined as u = (u0, u1, ...) = (1, 1, ... ). The convolution powers of u are as follows: u0 = (1, 0, 0, ...), u2 = u * u = (1, 2, 3, ...), u3 = u2 * u = (1, 3, 6, ...) … u r r r r r rk � � � � � � �1 1 1 2 1 2 3 , ! , ( ) ! , ( ) ( ) ! ,... . Note that for any f in L, f u u f f* *0 0� � and for any s, r in R, u u ur s s r* � � . In particular u u ur r* � � 0. Let ( , , )� F P be a probability space upon which{ }wk , k �N are independent and identically distributed (i.i.d.) Gaussian random variables, having zero means and variances 1 (N (0, 1)). Then [8] the fractional Gaussian noise is defined as w u w n u wn r r i r n i i n � � � � �� ( * ) ( ) 0 . Then wr is a sequence of Gaussian random variables which have memory and are correlated. Also, E wn r[ ]�0, Var w un r i r i n ( ) ( )� � � 2 0 , Lakhdar Aggoun 14 ISSN 0204–3572. Electronic Modeling. 2011. V. 33. ¹ 3 Cov w w u un r n r n i r n i r i n ( , )� � � � � � � ��1 1 0 1 1. Now 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 N �{ , , , ...}0 1 2 . The state of the system satisfies the dynamics x x vk k k k� �� �1 1� � . (1) Here { }vk , k �N are independent and identically distributed random variables such that, for all k, vk � �Z has probability distribution �. 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 r� � . Following [8] let z u y kk r� �( * ) ( ). Therefore z c u x k d w c h x x d wk k r k k k k k k� � � ��( * ) ( ) ( , ,...) � 0 1 , (2) where w is a sequence of i.i.d. N (0, 1). Write{ }Z k , k �N for the complete filtra- tion generated by{ , ,..., }z z zk0 1 . Using measure change techniques we shall derive a recursive expression for the conditional distribution of xk given Z k . Filtering. Initially we suppose all processes are defined on an «ideal» prob- ability space ( , , )� F P ; then under a new probability measure P, to be defined, the model dynamics (1) and (2) will hold. Suppose that under P: 1) { }xk , k �N is an i.i.d. sequence with probability distribution � ( )x with support in Z� ; 2){ }zk , k �N is an i.i.d. N (0, 1) sequence with density function � � ( ) /z e z� �1 2 2 2. Let � � � 0 0 1 0 0 0 0 0 0 � ��( ( ( ))) ( ) d z c h x d z and for l = 1, 2, ... . � � � � � l l l l l l l l l l x x d z c h x x d x � � �� � �( ) ( ( ( ,..., ))) ( 1 1 1 0� l lz�� ( ) . Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2011. Ò. 33. ¹ 3 15 Set �k l l k � � � � 0 . (3) Let G k be the complete �-field generated by { , , ..., , , ...x x x xk0 1 0 0� � ..., , , , ..., }� k k kx z z z� 0 1 for k �N. Lemma 1. The process{ }�k , is a P-martingale with respect to the filtration {G k }, with k �N. P r o o f. Since �k isG k -measurable E Ek k k k k[ ] [ ]� �� ��1 1|G |G� . There- fore we must show that E k k[ ]� � �1 1|G : E k k[ ]� � �1|G � � �� � � � � � �E x x d y c h x xk k k k k k k k� � �( ) ( ( ( , ..., )1 1 1 1 1 1 0 1� )) ( ( )d x zk k k k � � � � � � � � � � 1 1 1� �� |G � �� � � �� � � � � � �E x x x E d y c h xk k k k k k k k� � � �( ) ( ) ( ( (1 1 1 1 1 1 1� 0 1 1 1 , ..., ))) ( ) x d z k k k � � � � � � � � � � ���|G |Gk k kx, | ]1 . Now, E d z c h x x d z k k k k k k k � � ( ( ( , ..., ))) ( ) � � � � � � � � �1 1 1 1 1 0 1 1 1 |G k kx, � � � � � � � �1 � � � � � � � �R � � � ( ( ( , ..., ))) ( ) ( ) d z c h x x d z z dk k k k k 1 1 1 1 0 1 1 z �1; E x x x E x x x k k k k k x k k� � � � � � ( ) ( ) ( ) ( � � � �� � � � � � � � � �1 1 � � |G Z ) ( ) ( )� �x uk u |G � � � � � � � � � � � Z 1. Define P on ( , )� F by setting the restriction of the Radon-Nykodim derivative dP dP to G k equal to �k . A key result which relates expectation under P and P is given by a Bayes’s like formulae ([9, 10]) E I x x E I x x E k k k k k k k [ ( ] [ ( ] [ ] � � � |G |G |G � � , where E (resp. E) denotes expectations with respect to P (resp. P). Lakhdar Aggoun 16 ISSN 0204–3572. Electronic Modeling. 2011. V. 33. ¹ 3 The next result shows that the original model can be recovered by some simple transformations. Lemma 2. The stochastic process{ }vk , k �N is an i.i.d sequence with den- sity function � ( )x with support in Z� and{ }wk , k �N are i.i.d. N (0, 1) sequences of random variables, where v x xk k k k� �� �1 1 � ( )� � , w d z c h x xk k k k k k� ��� 1 0( ( ,..., )) . P r o o f. Suppose f, g :R R! are «test» functions (i.e. measurable functions with compact support). Then with E (resp. E) denoting expectation under P (resp. P) and using Bayes’ Theorem 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|G |G� � � [ ]�k k� � 1|G � E f v g wk k k k[ ( ) ( ) ]� � � �1 1 1 |G , where the last equality follows from Lemma 1. Consequently E f v g wk k k[ ( ) ( ) ]� � �1 1 |G � � �� � � � � � �E x x d z c h x xk k k k k k k k� � �( ) ( ( ( ,..., )1 1 1 1 1 1 0 1� )) ( ( )d x zk k k� � � � � � � 1 1 1� �� � � �� � � � � � �f x x g d z c h x xk k k k k k k k( ) ( ( ( ,..., ))1 1 1 1 1 1 0 1� � ) ]|G k � � �� � � � �� � �E x x x f x xk k k k k k k � � � � ( ) ( ) ( )1 1 1 � � � �� � � � � � � � E d y c h x x d z k k k k k k k � � ( ( ( ,..., ))) ( ) 1 1 1 1 1 0 1 1 1 � � � � � �� � � � � � �g d z c h x x xk k k k k k k k( ( ( ,..., ))) , ]1 1 1 1 1 0 1 1|G |G ] . Now E d y c h x x d z k k k k k k k � � ( ( ( ,..., ))) ( ) � � � � � � � � �� 1 1 1 1 1 0 1 1 1� � � � � �� � � � � � �g d z c h x x xk k k k k k k( ( ( ,..., ))) , ]1 1 1 1 1 0 1 1|G Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2011. Ò. 33. ¹ 3 17 � � � � � � � � �R � � � ( ( ( ,..., ))) ( ) ( ) d z c h x x d z zk k k k k 1 1 1 1 0 1 1 � � �� � � � � g d z c h x x dz u g u duk k k k( ( ( ,..., ))) ( ) ( )1 1 1 1 0 1 R � and E x x x f x xk k k k k k k k � � � � ( ) ( ) ( )� � � � � � � � � � � �1 1 1 � � |G � � � � � � � � � � � �� � � �E x x x x f x x x k k k k k tZ Z � � � � � ( ) ( ) ( ) ( ) � � |G � ( ) ( )t f t . Therefore E f v g w t f t u g u duk k k t [ ( ) ( ) ] ( ) ( ) ( ) ( )� � � � � � 1 1 |G Z R � � . The lemma is proved. Consider the un-normalized, conditional expectation which is the numerator of (3) and write E I x x q xk k k k[ ( ) ] ( )� � �|Z . If pk ( )" denotes the normalized conditional density, such that E I x x p xk k k[ ( ) ] ( )� �|Z , and from (4) we see that p x q x q tk k t k( ) ( ) ( )� � � � � � � � � � � Z 1 , k �N . Then we have the following result. Theorem 1. q x d z c h x d d z z k k k � � � � � � 1 0 1 0 0 0 0 0 1 0 ( ) ( ( ( ))) ... ( ) ... ( � � � 1 ) � � � � � � � � � � � x xk d z c h x x 1 1 1 1 1 1 0 1 Z Z ... ( ( ( , )))� � � �� � � � �� ( ( ( , ..., , )))d z c h x x xk k k k k1 1 1 1 1 0 � � � � � � � �� � � �( ) ( )x m x m k m x k k k m k x m k k k k k 0 1 � � �( ) ( )x m x mm x m x m 1 0 0 0 0 0 0 0 0 0 0 01� � � � � � �� . P r o o f. In view of (3) E I x xk k k k[ ( ) ]� � � � �� �1 1 1|Z Lakhdar Aggoun 18 ISSN 0204–3572. Electronic Modeling. 2011. V. 33. ¹ 3 � � �� � � � �E x x d z c h x x x k k k k k k k k� � � �( ) ( ( ( , ..., , ))� 1 1 1 1 1 0 ) ( ( ) ( ) d x z x k k k � � � � � � � � � � 1 1 1 � �� � |Z � � � � � � �E d z c h x x d x z k x k k k k k k k kk � 1 1 0 Z � � �� ( ( ( ,..., ))) ( ( ) ( ) ( )� � �x x xk k k k� � � � �� �1 1� � � � � � � � � �� � � �( ) ( )x m x m k m x k k k m k x m k k k k k 0 1 � �� � � � � � � � � ( ( ( ,..., , ))) ( ) d z c h x x x d z k k k k k k k 1 1 1 1 1 0 1 1 |Z k � � � �1 ... ... ( ( ( ))) ... ( ) ... ( ) � � � d z c h x d d z zk k 0 1 0 0 0 0 0 1 0 1 � � � � � � � � � � � � � � x xk d z c h x x 1 1 1 1 1 1 0 1 Z Z ... ( ( ( , ))) ...� ... ( ( ( , ..., , )))� d z c h x x xk k k k k� � � � �� �1 1 1 1 1 0 � � � � � � � �� � � �( ) ( )x m x m k m x k k k m k x m k k k k k 0 1 ... � � �( ) ( )x m x mm x m x m 1 0 0 0 0 0 0 0 0 0 0 01� � � � � � �� . Approximate recursion. In this section, we give recursive approximate es- timates of the hidden states. Assume that x0 is known and let x x0 0�� ~ , ~ ( ) ( ( (~ )) ( ) ( )q x d z c h x d z xx0 0 1 0 0 0 0 0 0 0 � ��� � # ; ~ ~ ( )x t p tk k t � � � � Z , where ~ ( ) ~ ( ) ~ ( )p x q x q tk k k t � � � � � � � � � � � � Z 1 . Theorem 2. The un-normalized density qk+1(x) is approximately computed by the recursion ~ ( ) ( ( (~ , ..., ~ , ))) q x d z c h x x x d k k k k k k k � � � � � �� � 1 1 1 1 1 1 0� � � � 1 1� ( )zk Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2011. Ò. 33. ¹ 3 19 � � � � � � � � � � � � t r t k r k t r kx r t r q t Z � � �( ) ( ) ~ ( ) 0 1 . (4) P r o o f . E I x xk k k k[ ( ) ]� � � � �� �1 1 1|Z � � �� � � � �E x x d z c h x x x k k k k k k k k� � � �( ) ( ( ( ,..., , ))� 1 1 1 1 1 0 ) ( )d zk k k � � � � � � � � � 1 1 1 � |Z . Now we replace x xk0 ,..., with ~ ,..., ~x xk0 which, of course, are Z k �1 measurable: E I x xk k k k[ ( ) ]� � � � �� �1 1 1|Z � � �� � � � �E x x d z c h x x x k k k k k k k k� � � �( ) ( ( (~ ,..., ~ ,� 1 1 1 1 1 0 ))) ( )d zk k k � � � � � � � � � � 1 1 1 � |Z � �� � � � � � � � � ( ( (~ , ..., ~ , ))) ( d z c h x x x d z k k k k k k k 1 1 1 1 1 0 1 1 ) [ ( ) ]E x xk k k k� � �� ��� |Z 1 � �� � � � � � � � � ( ( (~ , ..., ~ , ))) ( d z c h x x x d z k k k k k k k 1 1 1 1 1 0 1 1 ) ; E x r z r I x tk t r t k r k t r k k� � � � � � � � � � � � � � Z � � �( ) ( ) ( ) 0 1 |Z � � � � � � � �� � � � � � � � � ( ( (~ , ..., ~ , ))) ( d z c h x x x d z k k k k k k k 1 1 1 1 1 0 1 1 ) � � � � � � � � � � � � � � t r t k r k t r k kx r z r E I x t Z � � �( ) ( ) [ ( ) 0 1 � |Z k ]� � �� � � � � � � � � ( ( (~ , ..., ~ , ))) ( d z c h x x x d z k k k k k k k 1 1 1 1 1 0 1 1 ) � � � � � � � � � � � � � t r t k r k t r kx r z r q t Z � � �( ) ( ) ( ) 0 1 . Taking ~ ( )q tk as an approximation of q tk ( ) the result follows. In this paper an integer-valued process state space model is proposed. The state space model is not directly observed. The observed process is corrupted with a fractional Gaussian noise. Filters for the partially observed dynamics are derived. Lakhdar Aggoun 20 ISSN 0204–3572. Electronic Modeling. 2011. V. 33. ¹ 3 Ñòîõàñòè÷íà îáðîáêà â³äë³ê³â øèðîêî çàñòîñîâóºòüñÿ ó áàãàòüîõ çàäà÷àõ ç ö³ëî÷èñëî- îö³íþâàíèìè ÷àñîâèìè ðÿäàìè, êîòðèìè íå ìîæíà çàäîâ³ëüíî îïåðóâàòè â ðàìêàõ êëà- ñè÷íèõ Ãàóñîâî-ïîä³áíèõ ðÿä³â. Ðîçãëÿíóòî ðåêóðñèâí³ ô³ëüòðè äëÿ ÷àñòêîâî ñïîñòåðå- æóâàíèõ äèñêðåòíî-îö³íþâàíèõ ðÿä³â, â êîòðèõ øóìè ñïîñòåðåæåíü º äðîáîâèìè Ãàó- ñîâèìè øóìàìè. 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 Anal .— 1987. — 8.— P. 261—275. 3. Freeland R. K., McCabe B. P. M. Analysis of low count time series data by poisson autoregression// J. Time Series Anal. — 2004. — 25, ¹ 5. — P. 701—722. 4. MacDonald I. L. Hidden Markov and Other Models for Discrete-Valued Time Series. — Chapman & Hall, 1997. 5. Jung R. C., Tremayne A. R. Testing for serial dependence in time series models of counts// J. Time Series Anal. — 2003. — 24, Iss. 1. — Ð. 65—84. 6. McKenzie E. Some simple models for discrete variate time series// Water Res Bull.— 1985. — 21. — Ð. 645—650. 7. McKenzie E. Some ARMA models for dependent sequences of Poisson counts// Advances in Applied Probability. — 1988. — 20. — Ð. 822—835. 8. Elliott R. J., Deng J. A Filter for a state space model with fractional Gaussian noise// Automatica. — 2010. — 46. — P. 1689—1695. 9. Aggoun L., Elliott R. J. Measure Theory and Filtering: Introduction with Applications// Cambridge Series in Statistical and Probabilistic Mathematics, 2004. 10. Elliot R. J., Aggoun L., Moore J. B. Hidden Markov Models: Estimation and Control. — New-York: Springer-Verlag, 1995. — Applications of Mathematics N 29. Submitted 30.12.10 Partially Observed Discrete-Valued Time Series in Fractional Gaussian Noise ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2011. Ò. 33. ¹ 3 21 22 ISSN 0204–3572. Electronic Modeling. 2011. 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