Factorial Fractional Hidden Markov Models

Conventional hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive Gaussian noise. A lesser known extension of this class of models, is the so called Factorial Hidden Model (FHMM). FHMM’s also have numerous applications, notably in machine learn...

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Дата:	2012
Автор:	Aggoun, L.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України 2012
Назва видання:	Электронное моделирование
Теми:	Информационные технологии
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/61826
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Factorial Fractional Hidden Markov Models / L. Aggoun // Электронное моделирование. — 2012 — Т. 34, № 3. — С. 59-67. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-618262014-05-13T03:01:18Z Factorial Fractional Hidden Markov Models Aggoun, L. Информационные технологии Conventional hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive Gaussian noise. A lesser known extension of this class of models, is the so called Factorial Hidden Model (FHMM). FHMM’s also have numerous applications, notably in machine learning and speech recognition. In this article we consider FHMM’s with additive fractional Gaussian noise in the observed process. Общепринятые марковские модели скрытия информации представляют собой марковскую цепь, полученную с помощью линейного преобразования, искаженного аддитивным гауссовым шумом. Менее известным расширением этого класса моделей является так называемая факториальная модель скрытия информации (FHMM), которая также имеет множество приложений, в частности при обучении машин и распознaвании речи. Рассмотрены FHMM с аддитивным дробным гауссовым шумом в наблюдаемом процессе. Загальновідомі марковські моделі приховування інформації являють собою ланцюг Маркова, отриманий за допомогою лінійного перетворення, викривленого адитивним гауссовим шумом. Меньше відомим розширенням цього класу моделей є так звана факторіальна модель приховування інформації (FHMM), яка також широко застосовується, наприклад, при навчанні машин и розпізнаванні мови. Розглянуто FHMM з адитивним дробовим гауссовим шумом у процесі, що спостерігається. 2012 Article Factorial Fractional Hidden Markov Models / L. Aggoun // Электронное моделирование. — 2012 — Т. 34, № 3. — С. 59-67. — Бібліогр.: 11 назв. — англ. 0204-3572 http://dspace.nbuv.gov.ua/handle/123456789/61826 en Электронное моделирование Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Информационные технологии Информационные технологии
spellingShingle	Информационные технологии Информационные технологии Aggoun, L. Factorial Fractional Hidden Markov Models Электронное моделирование
description	Conventional hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive Gaussian noise. A lesser known extension of this class of models, is the so called Factorial Hidden Model (FHMM). FHMM’s also have numerous applications, notably in machine learning and speech recognition. In this article we consider FHMM’s with additive fractional Gaussian noise in the observed process.
format	Article
author	Aggoun, L.
author_facet	Aggoun, L.
author_sort	Aggoun, L.
title	Factorial Fractional Hidden Markov Models
title_short	Factorial Fractional Hidden Markov Models
title_full	Factorial Fractional Hidden Markov Models
title_fullStr	Factorial Fractional Hidden Markov Models
title_full_unstemmed	Factorial Fractional Hidden Markov Models
title_sort	factorial fractional hidden markov models
publisher	Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
publishDate	2012
topic_facet	Информационные технологии
url	http://dspace.nbuv.gov.ua/handle/123456789/61826
citation_txt	Factorial Fractional Hidden Markov Models / L. Aggoun // Электронное моделирование. — 2012 — Т. 34, № 3. — С. 59-67. — Бібліогр.: 11 назв. — англ.
series	Электронное моделирование
work_keys_str_mv	AT aggounl factorialfractionalhiddenmarkovmodels
first_indexed	2025-07-05T12:46:21Z
last_indexed	2025-07-05T12:46:21Z
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fulltext	Lakhdar Aggoun Department of Mathematics and Statistics, College of Science, Sultan Qaboos University (P.O. Box 36, Al-Khodh 123, Oman; e-mail: laggoun@squ.edu.om) Factorial Fractional Hidden Markov Models (Recommended by Prof. E. Dshalalow) Conventional hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive Gaussian noise. A lesser known extension of this class of models, is the so called Factorial Hidden Model (FHMM). FHMM’s also have numerous applications, nota- bly in machine learning and speech recognition. In this article we consider FHMM’s with addi- tive fractional Gaussian noise in the observed process. Îáùåïðèíÿòûå ìàðêîâñêèå ìîäåëè ñêðûòèÿ èíôîðìàöèè ïðåäñòàâëÿþò ñîáîé ìàðêîâñ- êóþ öåïü, ïîëó÷åííóþ ñ ïîìîùüþ ëèíåéíîãî ïðåîáðàçîâàíèÿ, èñêàæåííîãî àääèòèâíûì ãàóññîâûì øóìîì. Ìåíåå èçâåñòíûì ðàñøèðåíèåì ýòîãî êëàññà ìîäåëåé ÿâëÿåòñÿ òàê íàçûâàåìàÿ ôàêòîðèàëüíàÿ ìîäåëü ñêðûòèÿ èíôîðìàöèè (FHMM), êîòîðàÿ òàêæå èìååò ìíîæåñòâî ïðèëîæåíèé, â ÷àñòíîñòè ïðè îáó÷åíèè ìàøèí è ðàñïîçíaâàíèè ðå÷è. Ðàñ- ñìîòðåíû FHMM ñ àääèòèâíûì äðîáíûì ãàóññîâûì øóìîì â íàáëþäàåìîì ïðîöåññå. Key words: factorial hidden Markov chains, change of measure, fractional Gaussian noise. 1. Introduction. Hidden Markov models (HMM) have been heavily researched and used for the past several decades in various areas of application including bioinformatics, finance and engineering [1—3]. A standard HMM uses a hidden state at time n to summarize all the information it had before n and thus the ob- servation at time n depends only on the hidden state at time n. Also the hidden state sequence over time in an HMM is a finite state Markov chain. The field of speech recognition has used the theory of HMM with great suc- cess. At the same time there is now a wide perception in the speech research community that new ideas are needed to continue improvements in performance. There has recently been some interest in exploring possible extensions to HMM in several directions. These include [4—11] among others. The factorial HMM arises by forming a state model composed of several layers. The observation process depends upon the current state in each of the layers. In the first model discussed in the sections 2 and 3 the states of the system are divided into layers with independent dynamics observed through a fractional Gaussian noise. However, the probability of an observation at each time depends ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 3 59 ÈÍÔÎÐÌÀÖÈÎÍÍÛÅ ÒÅÕÍÎËÎÃÈÈ upon the current state of all the layers. In many applications, multiple sequences are interacting with one another. Therefore, a more general model is proposed in section 4 where some dependence between the layers is introduced. 2. Dynamics with fractional Gaussian noise. Let Z denote the set of inte- gers, and Z� denote the set of non-negative integers. Following [ 6, 7] we define a set of functions �on Z� with values in �. 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 � the convolution product f f1 2 * is defined by ( * ) ( )f f n f f f f i n i i i n n i i 1 2 0 2 2 0 2 2� � � � � � �� . In this set of functions, consider the function u, which is defined as u u u� �( , ,...)0 1 � ( , ,...)1 1 . The convolution powers of u are: u0 1 0 0� ( , , , ...), u u u2 1 2 3� �* ( , , , ...), u u u3 2 1 3 6� �* ( , , , ...), . . . . . . . . . . . . . . . . . . . . . . . . . . . . u r r r r r rr � � � �� 1 1 1 2 1 2 3 , ! , ( ) ! , ( )( ) ! , ... . Note that for any f in �, f u u f f* * 0 0� � and for any s, r in �, u u ur s s r * .� � In particular u u ur r * � � 0. For more details and proofs see [6, 7]. Let ( � �� P) be a probability space upon which{ }wn , n�� are independent and identically distributed (i.i.d.) Gaussian random variables, having zero means and variances 1 (N (0, 1)). Then [6, 7], the fractional Gaussian noise is defined as w u w n u wn r r i n i r n i� � � �� ( * )( ) 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 n i r( ) ( )� � � 0 2, Cov w w u un r n r i n n i r n i r( , )� � � � � �� 1 0 1 1 1. Lakhdar Aggoun 60 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 3 A system is considered whose state is described by a set of finite-state, ho- mogeneous, discrete-time Markov chains X n m, m M�1, ..., , n��. We suppose X X X M 0 0 1 0�{ , ..., } is given, or its distribution known. If the state space of X n m, m M�1,..., , has N m elements it can be identified without loss of generality, with the set S e e X m N m m m� { , ..., }1 , where ei m are unit vectors in � N m with unity as the ith element and zeros elsewhere. Write �n m n mX X0 0��{ , ..., , m M�1, ..., }, for the �-field generated by X m 0 , ... ..., X n m, m M�1, ..., , and{ }�n for the complete filtration generated by the �n 0; this augments �n 0 by including all subsets of events of probability zero. Here we shall assume that P X e P X e Xn m j m n n m j m n m( \| ) ( \| )� �� 1 1� . Write a P X e X eji m n m j m n m i m� � ��( \| )1 , A am ji m� ( ). Define V X A Xn m n m m n m � �� 1 1: . So that X A X Vn m m n m n m � �� 1 1, (1) { }Vn m , m M�1,..., , n��, are sequences of martingale increments. (See [1, 2] for more details) . Let � � �m m N m Nc c m m ( ,..., )1 � , m M�1, ..., and �� ( , ..., )1 m . The state processes X m, m M�1, ..., , are not observed directly. We suppose that our obser- vations have the form y X c w X wn m M n m m n r n n r� � � � � � � 1 , , , (2) where X X Xn n n M�{ , ..., }1 is an N N M1 � �... -dimensional vector of unit vec- tors. Let z u y nn r� �( * )( ). Therefore z u X n w h X X wn r n n r n n� � � � ��( * , )( ) ( ,..., ) � 0 , (3) where w is now a sequence of i.i.d. N (0, 1).Write{ }�n , n�� for the complete filtration generated by { , , ..., }z z zn0 1 . We shall write { }�n for the complete fil- tration generated by X m, m M�1, ..., , and z. 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 in (1) and (3) will hold. Suppose that under P: 1) { }X n m , n�� are Markov chains with semimartingale representations given in (1). Factorial fractional hidden Markov models ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 3 61 2){ }zn , n�� is an i.i.d. N (0, 1) sequence with density function � � ( ) /z e z� �1 2 2 2. Let � � � � � � � ( ( )) ( ) ( ( )) ( ) z h X z y h X y r 0 0 0 0 0 0 0 0 � � � and for l = 1, 2, … � � � � l l l r l l z h X X z ( ( ,..., )) ( ) � 0 . (4) Set �n l l n � � � � 0 . (5) Define P on� � �}by setting the restriction of the Radon-Nykodim deriva- tive dP dP to �n to �n . It can be shown that on � � �}and under P,{ }wn , n�� are i.i.d. N (0, 1) sequences of random variables, where w zn n� �� h Xn r ( ,...0 ..., )X n . Recall that for a �-adapted sequence{ }� n , E E E n n n n n n n [ \| ] [ \| ] [ \| ] � � � � � � � � . Write q j jn M( , ..., )1 , 1 1 1� �j N , ..., 1� �j NM M , n��, for the unnormalized, conditional probability distribution such that E X e q j jn M n j n n M� � � � � � � � � � � ! 1 1, \| ( ,..., )� � . Now i N n m i m m X e � � � 1 1, , so j N j N n M n j N j N M M M M q j j E 1 1 1 1 1 1 1 1 1� � � � � � ��... ( ,..., ) ...� � � � � � � � � ! !� � � � 1 M n j n n nX e E, \| [ \| ]� �� . Therefore the normalized conditional probability distribution p j j E X en M M n j n( ,..., ) , \|1 1 � � � � � ! � � � � � � � Lakhdar Aggoun 62 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 3 is given by p j j q j j n M n M i N i N n M n M ( ,..., ) ( ,..., ) ... 1 1 1 11 1 1 1 � � �� q i in n M n( ,..., )1 1 1� � . We have the following result. Theorem 1. The unnormalized, conditional probability distribution qn is given by q j jn M i i i i iM M n ( , ..., ) ... , ..., , ..., , ... 1 1 0 0 1 1 1 1 2 � � � � , , ..., , ... i i i M i i M n n M n a � � � � � � �2 1 1 1 1 0 1� � � � ... ( ( )) ( ) ( , , a a z h z z h i i j i i r n n n � � � � � � � � � � � 1 2 1 00 0 1� � �e ( , )) ( ) ... e e i i z 0 1 1� ... ( ( )) ( ) ( , , ..., � � z h z p i i n r i i j n n� �e e e0 1 0 1 0 2 0 , ..., , iM 0 ). P r o o f. Let us denote e j j j M= e e M ( , ..., ) 1 1 , e i i i M k k M k= e e( , ..., ) 1 1 and i ik M k 1 , ..., � � � � � � i N i N k M k M 1 1 1 1 ... . In view of (4) and (5) q j j E z h X X z Xn M n n n r n n M ( ,..., ) ( ( , ..., )) ( ) 1 1 0 1 � � � � �� n j ne� � � , \|� � � � � ! � � � � � ��E X X z h X X l n l l n n r n j� � � � 0 1 1 0 0 1 ( ,..., ) ( ( ,..., , ))e ( ) , \| z X e n M n j n � � � � � � � � � � � ! !1 � � � � � � � � � ��E X X z h X X l n l l n n r n j� � � 0 1 1 0 0 1 ( ,..., ) ( ( ,..., ,e )) ( )� zn " " � � �� i i M n i M n n M n nX e A X e 1 1 1 1 1 1 1 1 , ..., � � � � � � � , , j n � � \|� � ! � � � � � � � � �E X X z h X l n l l n n r in � � � 0 1 1 0 0 ( ) ( ( , ..., , ..., e � " 1 , e j nz )) ( )� " � � � �� i i M n i M i n M n n nX e A e 1 1 1 1 1 1 1 1, ..., , � � � � � � � � � � , e j n\|� � ! . Factorial fractional hidden Markov models ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 3 63 Now recall that X A X Vk k k � � � �� 1 , and i i M k i k M k kX e 1 1 1 , ..., ,� � � � � � � � . Therefore q j j E X X z h n M l n l l n n r ( ,..., ) ( ) ( ( 1 0 1 1 0� � � � � � � �� , ..., X z i j n n0 1, ..., ,e e� " )) ( )� " � � � �� i i M n i M j i n M n n nX e a 1 1 1 1 1 1 1 1, ..., � � � � � � � , , � \|�n � ! � � � � � � � � � � E X X z h X l n l l n n r in � � � 0 1 1 0 0 ( ) ( ( , ..., , ..., e 1 ,e j nz )) ( )� " " � � �� i i M n i M j i n M n n nA X e a 1 1 1 1 1 2 1, ..., � � � � � � � � , , � � !1 � \|�n � � � � � � � � � � E X X z h X l n l l n n r in � � � 0 1 1 0 0 ( ) ( ( , ..., , ..., e 1 ,e j nz )) ( )� " " � � �� i i M n i M j i n M n n nA X e a 1 1 1 1 1 2 1, ..., � � � � � � � � , , � � � �� !1 1 2 2 2 1 2 � � � � � i i M n i n n M n nX e , ..., , \|� � � � � � � � � � � E X X z h X l n l l n n r in � � � 0 1 1 0 0 ( ) ( ( , ..., , ..., e 1 ,e j nz )) ( )� " " � � � � �� i i M i i M j i n M n n n na a 1 1 1 1 2 1 1, ..., � � � � � � � , , 1 1 2 2 2 1 2 � � � � � i i M n i n n M n nX e � � �� ! , ..., , \|� � ………………….…………………………………… � � � � � � �i i i i i i iM M n M n n 1 0 0 1 1 1 1 2 2 1 1, ..., , ..., , ..., , ... ..., , ... i M i i M n a � � � �1 1 0 1� � � � ... ( ( )) ( ) ( , , a a z h z z h i i j i i r n n n � � � � � � � � � � � 1 2 1 00 0 1� � �e ( , )) ( ) ... e e i i z 0 1 1� ... ( ( )) ( ) ( , � � z h z p i i n r i i j n n� �e e e0 1 0 1 0 2 0 , ..., , , ..., iM 0 ) . Which finishes the proof. Lakhdar Aggoun 64 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 3 3. Approximate recursions. In this section, as in [6, 7] we give recursive approximate estimates of the hidden states. The recursion is initialized by the as- sumption that X X X X M 0 0 0 1 0� � � ~ { ~ ,..., ~ }is known, and for n #1we define: ~ ( ,..., ) , \|q j j E X en M n M n j n1 1 � � � � � ! � �� E z h X X X z X en n r n n m n M n j� 1 0 1 1 � � ( ( ~ ,..., ~ , ~ )) ( ) , � � � � \|�n � � � � � ! ! � � � � � � �E z h X X z n n r n j n i , ..., iM � 1 0 1 1 � � ( ( ~ ,..., ~ , )) ( ) e � 1 1 1 1 M n i M n j nX e A X e� �� ! ! � � � � � � � � , \|, � � � �� E z h X X z n n r n j n i N � 1 0 1 11 1� � ( ( ~ ,..., ~ , )) ( ) .. e . , \| , i N M n i M j i n M M X e a � � � � �� ! 1 1 1 1 1 � � � � � � � � � � � � � � � � � � ( ( ~ ,..., ~ , )) ( ) , ..., z h X X z a n r n j n i i M j M 0 1 1 1 e � � , ~ ( ,..., ) i n Mq i i � � �1 1 . Here ~ ( )p ik m m is the approximate marginal conditional distribution of X k m, and ~ ( ~ \| ) ~ ( )X e P X e e p ik m i N i m k m i m k i N i m k m m m m m � � � � � � � 1 1 � . To summarize we have. Theorem 2. The approximate unnormalized conditional joint distribution at time n of the unobserved Markov chains{ ,..., }X Xn n M1 is given by the recursion: ~ ( ,..., ) ( ( ~ ,..., ~ , )) ( ) , . q j j z h X X z n M n r n j n i 1 0 1 1 � � �� e .., , ~ ( ,..., ) i M j i n M M a q i i� � � � � � � � 1 1 1 . 4. A model with coupled layers. In this section we assume that the dynam- ics of X 1 are not changed but for m = 2, …, M P X e P X e X Xn m j m n n m j m n m n m( \| ) ( \| , )� � � � � � � 1 1 1 . Write b P X e X e X ej ih m n m j m n m i m n m h m , ( \| , )� � � �� 1 1 1 1 , B bm j ih m� ( ), , so that X B X X Vn m m n m n m n m� $ �� 1 1 1 . Factorial fractional hidden Markov models ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 3 65 The observed process is as defined in (2) and (3) and P is defined on � � �}in exactly the same manner as in the previous section. Write %n Mj j( ,..., )1 , 1 1 1� �j N , ..., 1� �j NM M , n��, for the unnorma- lized, conditional probability distribution such that E X e j jn M n j n n M� � � � � � � � � � � ! 1 1, \| ( ,..., )� �% . Theorem 3. The unnormalized, conditional probability distribution %n is given by %n M i i i i i j j M M n ( ,..., ) ... , ..., , ..., , ... 1 1 0 0 1 1 1 1 2 � � � � , , ..., , , ... i i i i i i i j M n n M n n na a a � � � � �� 2 1 1 1 1 1 1 0 1 1 1 2 1 1 , i n 1 1� " " � � � � � � � � � � � � � � � � � � 2 1 0 1 0 1 2 1 2 M i i i i i i j i b b bn n n n, , , ... � � �1 1 1i n � � � � � � ( ( )) ( ) ( ( , )) ( ) ... z h z z h z i r i i0 0 1 1 0 0 1� �e e e ... ( ( )) ( ) ( , , ..., � � z h z p i i n r i i j n n� �e e e0 1 0 1 0 2 0 , ..., , iM 0 ) . P r o o f. Again using (4), (5) and the same notation as in Theorem 1 %n Mj j( )1,..., � � � � � �E X X z h X X zl n l l n n r n n � � � � 0 1 1 0 0( ) ( ( )) ( ) ,..., , ..., � � � �� ! 1 M n j nX e, \|� � � � � � � � � ��E X X z h X X l n l l n n r n� � � 0 1 1 0 0 1 ( ) ( ( ,..., , ..., , e j nz )) ( )� " " � � �� i i M n i n j M n M n nX e A X e 1 1 1 1 1 1 1 1 1 1 2, ..., � � � � � , , � � � �$ � !B X X en n j n � � � � � 1 1 1 , \|� � � � � � � � � ��E X X z h X X l n l l n n r n� � � 0 1 1 0 0 1 ( ) ( ( ,..., , ..., , e j nz )) ( )� " " � � � �� i i M n i j i M n M n n n X e a b 1 1 1 1 1 1 1 1 1 2, ..., � � � � � , , j i i nn n � � � � , \|� � � � !1 1 1 � � ………………….…………………………………… � � � � � � �i i i i i i iM M n M n n 1 0 0 1 1 1 1 2 2 1 1, ..., , ..., , ..., , ... ..., , , , ... i i i i i j i M n n n n a a a � � � �� " 1 1 1 1 0 1 1 1 2 1 1 1 1 Lakhdar Aggoun 66 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 3 " � � � � � � � � � � � � � � � � � � 2 1 0 1 0 1 2 1 2 M i i i i i i j i b b bn n n n, , , ... � � � � � 1 1 1 0 0 10 0 1 1 i i r i i n z h z z h z� � � � � � ( ( )) ( ) ( ( , )) ( ) . e e e .. ... ( ( )) ( ) ( , , ..., � � z h z p i i n r i i j n n� �e e e0 1 0 1 0 2 0 , ..., , iM 0 ), which finishes the proof. Çàãàëüíîâ³äîì³ ìàðêîâñüê³ ìîäåë³ ïðèõîâóâàííÿ ³íôîðìàö³¿ ÿâëÿþòü ñîáîþ ëàíöþã Ìàð- êîâà, îòðèìàíèé çà äîïîìîãîþ ë³í³éíîãî ïåðåòâîðåííÿ, âèêðèâëåíîãî àäèòèâíèì ãàóññî- âèì øóìîì. Ìåíüøå â³äîìèì ðîçøèðåííÿì öüîãî êëàñó ìîäåëåé º òàê çâàíà ôàêòîð³àëüíà ìîäåëü ïðèõîâóâàííÿ ³íôîðìàö³¿ (FHMM), ÿêà òàêîæ øèðîêî çàñòîñîâóºòüñÿ, íàïðèêëàä, ïðè íàâ÷àíí³ ìàøèí è ðîçï³çíàâàíí³ ìîâè. Ðîçãëÿíóòî FHMM ç àäèòèâíèì äðîáîâèì ãàóññîâèì øóìîì ó ïðîöåñ³, ùî ñïîñòåð³ãàºòüñÿ. 1. Aggoun L., Elliott R. J. Measure Theory and Filtering: Introduction with Applications. Cam- bridge Series In Statistical and Probabilistic Mathematics. — 2004. 2. Elliott R. J., Aggoun L., Moore J. B. Hidden Markov Models:Estimation and Control. New York: Springer-Verlag, 1995. —Applications of Mathematics. — N 29. 3. MacDonald I. L., Zucchini W. Hidden Markov and Other Model for Discrete-Valued Time Series. — London : Chapman and Hall, 1997. 4. Aggoun L. Partially observed discrete-valued time series in fractional Gaussian noise// Elec- tronic Modeling. — 2011. — 33, N 3. 5. Brand M. Coupled hidden Markov models for modeling interacting processes MIT Media Lab Perceptual Computing/Learning and Common Sense Techincal Report 405 (Revised). — June, 1997. 6. Elliott R. J., Deng J. A filter for a state space model with fractional Gaussian noise// Automatica. — 2010. — 46. — Ð. 1689—1695. 7. Elliott R. J., Deng J. A filter for a hidden Markov chain observed in fractional Gaussian noise// Systems and Control Letters. — 2011. — 60. — P. 93—100. 8. Ghahramani Z., Jordan M. Factorial Hidden Markov Models. Computational Cognitive Sci- ence Technical Report 9502 (Revised). — July, 1996. 9. Logan B., Pedro J. M. Factorial Hidden Markov Models for Speech Recognition: Prelimi- nary Experiments.— Cambridge Research Laboratory, September, 1997. 10. Malcolm W. P., Quadrianto N., Aggoun L. State estimation schemes for independent compo- nent coupled hidden Markov models// Stochastic Analysis and Appl. — 2010. — 28 (3). — P. 430—446. 11. Zhong S. , Ghosh J. Coupled Hidden Markov Models. — Tech. Report. — June, 2001. Submitted 15.02.12 Factorial fractional hidden Markov models ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. 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Factorial Fractional Hidden Markov Models

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