R. J. Elliott, L. Aggoun, and J. B. Moore. Hidden Markov Models: Estimation and Control. Applications of Mathematics. Springer-Verlag, New York, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 2rr where Ak l : tk l tk, an is the center of mass of the bin An and Cj,k = Ck(qbk = an, qbk l = Cj,k depends on the number of class j, since qb k is class specific) Optimal nonlinear filter (6) can be greatly simplified by switching to unnormalized filtering dis tributions [4]. Specifically, one can show that pjk l i(An) jk l i(An) jk l i(An) n:l where the unnormalized filtering distribution jk l i(An) is given by jjk l,l (An) n ln,m k,i(A Pj,k l Pj(m,i,n, wsj,k l j mj m,i with Pj,k l : exp j (an)r lrk l 15. an)r 1 2 j,k l (7) 8) ....

R.J. Elliott, L. Aggoun, and J.B. Moore. Hid- den Markov Models: Estimation and Control. Springer-Verlag, New York, 1995.

....processes. We consider all ML estimation problems that are amenable to finite computer programs, i.e. those for which conditional densities can be finitely parametrized. In addition to Gaussian models, these include processes taking values in a finite set, such as finite state Markov chains [8, 9] or Markov random fields [10, 11, 12] As in [7] an important feature of the approach we employ to formulate recursive ML estimation problems is that we make no distinction between equations describing the model 2 dynamics and measurements, and view all of them as observations . To motivate ....

....depending indirectly on the states. Models of this type are often used in speech processing [8] or in digital communications for decoding convolutional codes or the deconvolution 36 of intersymbol interference [38] 39] 40] For a detailed study of HMMs from a control perspective, see [9]. Let x k be a Markov chain defined for 0 k N , taking values in a finite set X . The joint probability distribution of the chain can be expressed as p(X) q i (x 0 )q e (x N ) N Gamma1 Y k=0 (x k ; x k 1 ) 6:17) This form is a slight variant of the standard expression for Markov ....

R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models: Estimation and Control. New York: Springer--Verlag, 1995.

....which source generated each observation. As soon as this is accomplished, the problem is essentially reduced to the supervised case. An approach to the problem is represented by the Hidden Markov Model methodology (HMM) where each source is mapped to a state of an unobservable Markov chain [6]. However, HMM usually involves relatively simple and static source models for every hidden state. There are extensions of HMM which involve dynamic input output models [12] but usually they operate o#ine. In this paper we are interested in an online solution to the problem. In the sequel we ....

R.J. Elliot, L. Aggoun and J.B. Moore, Hidden Markov Models: Estimation and Control, 1995, Springer.

....To encode the high level evolution of feature configurations we adopted the well known formalism of hidden Markov models. In these models the state 4 form a Markov chain; the only observable quantity is a corrupted version 4 of the state called observation process. Using the notation in [6] we can associate the elements of the finite state space (e w to coordinate versors ( H w P w I L and write the model as 4 ( 44 4 ( 4) P Hk 4 I where 4 is a sequence of martingale increments and 4 is a sequence of i.i.d. ....

....capability of self learning the set of parameters and given a sequence of observations that are supposed to be produced by the system. The algorithm we use is an application of the EM technique that is slightly different from the classical Baum Welch procedure and is based on a one pass [6] iterative update of the matrices: at each loop the entire sequence of data is processed by computing the state estimates. The probabilistic distance between the measurement and each state representative R in the d dimensional observation space H 4 I ( H J 6 I ....

R. J. Elliot, L. Aggoun, and J. B. Moore. Hidden Markov models: estimation and control. 1995.

....3.1 Hidden Markov models At the highest level of the model of action we have a stochastic automaton, or hidden Markov model. The states of the model fX k g form a Markov chain; the only observable quantity is a corrupted version y k of the state called observation process. Using the notation in [7] we can associate the elements of the finite state space X = f1; ng to coordinate versors e i = 0; 0; 1; 0; 0) 2 IR n and write the model as X k 1 = AX k V k 1 y k 1 = CX k diag(W k 1 ) X k where fV k 1 g is a sequence of martingale increments and fW k 1 g is a sequence ....

....the capability of self learning the set of parameters A; C and given a sequence of observations that are supposed to be produced by the system. The algorithm we use is an application of the EM technique that is slightly different from the classical BaumWelch procedure and is based on a one pass [7] iterative update of the matrices: at each loop the entire sequence of data is processed by computing the state estimates. The probabilistic distance between the measurement and each state representative C e j in the d dimensional observation space i (y k 1 ) d Y j=1 g( y j k 1 c ....

R. J. Elliot, L. Aggoun, and J. B. Moore. Hidden Markov models: estimation and control. 1995.

....To encode the high level evolution of feature configurations we adopted the well known formalism of hidden Markov models. In these models the state fX k g form a Markov chain; the only observable quantity is a corrupted version y k of the state called observation process. Using the notation in [6] we can associate the elements of the finite state space X = f1; ng to coordinate versors e i = 0; 0; 1; 0; 0) 2 R n and write the model as X k 1 = AX k V k 1 y k 1 = CX k diag(W k 1 ) X k where fV k 1 g is a sequence of martingale increments and fW k 1 g is a sequence of ....

....the capability of self learning the set of parameters A; C and given a sequence of observations that are supposed to be produced by the system. The algorithm we use is an application of the EM technique that is slightly different from the classical Baum Welch procedure and is based on a one pass [6] iterative update of the matrices: at each loop the entire sequence of data is processed by computing the state estimates. The probabilistic distance between the measurement and each state representative C e j in the d dimensional observation space i (y k 1 ) d Y j=1 g( y j k 1 c ....

R. J. Elliot, L. Aggoun, and J. B. Moore. Hidden Markov models: estimation and control. 1995.

....algorithm in [5] that it allows for data aggregation over a fixed number of instants in between two successive 2 updates of the parameter for better performance. Moreover, the algorithm in [5] was only for ordinary Markov processes and not hidden Markov models (which is a more general setting) [14] considered here. We prove the convergence of both of these schemes and numerically demonstrate the algorithms on a feedback queueing network with high dimensional parameters. These schemes are found to converge orders of magnitude faster than their (N 1) Simulation analogues in [4] and [5] and ....

R. J. Elliott, L. Aggoun, and J. B. Moore. Hidden Markov Models: Estimation and Control. Springer-Verlag, New York, 1995.

....k;1 ] 3. 2) where the ijth element of J k is the number of transitions from state j to state i#theith diagonal elementof O k is the number of times in state i (O k is a diagonal matrix)# and the ith elementofT k is the summation of y k when in state i (finite dimensional filters are given in [3]) Under certain persistence of excitation conditions (see [4] for details) the following theorem holds (whichisa restatement of a result established in [4] Theorem 1 Consider the HMM representation of the linear system and assume the system is persistently exciting. lim k 1 A k # C k = ....

R.J. Elliott, L. Aggoun, and J.B. Moore, Hidden Markov Models: Estimation and Control, Springer, New York, 1995.

....the risk sensitive estimation can be re formulated as x k 2 argmin # E#L k exp #f k,1 X l=0 l#x l ; x l # l#x k ;##gjY k # (2.4) where L k = Q k l=0 exp##Hx k # 0 y k , 1 2 #Hx k # 0 #HX k ##. For details on this particular application of change of probability measure technique, see [22](discrete time) and [23] 20] continuous time) 3 Discrete time risk sensitive estimation with Gaussian initial condition In this section, we present the risk sensitive estimation results for discrete time linear Gaussian systems with Gaussian initial conditions and study the asymptotic ....

R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models: Estimation and Control. Springer-Verlag, 1994.

....estimation problem is re formulated as x t 2 argmin # E#L t exp #f Z t 0 l#x s ; x s #ds l#x t ;##gjY t # (5) where L t = exp# R t 0 #Hx s # # dy s , 1 2 R t 0 #Hx s # # #Hx s #ds#. For details on this particular application of change of probability measure technique, see [15](discrete time) and [4] 11] continuous time) 3. Risk sensitive estimation with Gaussian initial condition. In this section, we present the risk sensitive estimation results for linear Gaussian systems with Gaussian initial conditions and study the asymptotic forgetting property of the ....

R. J. ELLIOTT,L.AGGOUN, AND J. B. MOORE, Hidden Markov Models: Estimation and Control, SpringerVerlag, 1994.

....produced e.g. by on going processes. In these cases it is desirable to update the model parameters on line, i.e. with each incoming data point. The importance of on line learning for HMMs has been appreciated recently in a variety of approaches, each with speci c de ciencies and advantages (see [10][11] 12] and refs. therein) Our contribution to this continuously developing subject is a new method, which is conceptually very simple and easy to implement. In contrast to previous schemes we do not reestimate the HMM transition probabilities directly, but lifted parameters, which contain ....

....eventually allows us to solve the on line learning problem. We introduce jV j time dependent tensorial quantities N t; ijk (y) of dimension jSj 3 , N t; ijk (y) t 1 i a t 1 ij (y(t) t; j;k y;y(t) 1) which can be regarded as smoothed estimates in the nomenclature of [10]: Apart from normalization N t; ijk (y) is the probability of making at time t a transition from state i to j with symbol y and to be in state k at time t (in a uctuating environment with yet undetermined time dependent transition matrices a t ) Its weighted time average N ijk ....

R.J. Elliott, L. Aggoun, J.B. Moore, Hidden Markov Models: Estimation and Control. New York: Springer, 1995.

....11, 305 345 (1999) c # 1999 Massachusetts Institute of Technology 306 Sam Roweis and Zoubin Ghahramani Kalman filtering. In this article we unify many of the disparate observations made by previous authors (Rubin Thayer, 1982; Delyon, 1993; Digalakis et al. 1993; Hinton et al. 1995; Elliott, Aggoun, Moore, 1995; Ghahramani Hinton, 1996a,b, 1997; Hinton Ghahramani, 1997) and present a review of all these algorithms as instances of a single basic generative model. This unified view allows us to show some interesting relations between previously disparate algorithms. For example, factor analysis and ....

....originally derived by Shumway and Stoffer (1982) and recently reintroduced (and extended) in the neural computation field by Ghahramani and Hinton (1996a,b) Digalakis et al. 1993) made a similar reintroduction and extension in the speech processing community. Once again we mention the book by Elliott et al. 1995), which also covers learning in this context. The basis of all the learning algorithms presented by these authors is the powerful EM algorithm (Baum Petrie, 1966; Dempster, Laird, Rubin, 1977) The objective of the algorithm is to maximize the likelihood of the observed data (equation 4.1) in ....

Elliott, R. J., Aggoun, L., & Moore, J. B. (1995). Hidden Markov models: Estimation and control, New York: Springer-Verlag.

....Forschungsgemeinschaft (Ho 496 4 2) 2 I. Introduction Hidden Markov models (HMM) were successfully applied in various fields of time series analysis, e.g. in speech recognition [1] ion channel analysis [2] 3] 4] protein and nucleic acid sequence analysis [5] 6] communication technology [7] or econometrics [8] Sometimes, the basic concept of an unobserved process consisting of transitions between so called states or regimes reflects a real system producing state dependent output. In other cases, the HMM is just used because it performs a good description of the data. However, with ....

R.J. Elliott, L. Aggoun, and J.B. Moore, Hidden Markov models: estimation and control, Applications of mathematics. Springer, 1995.

....ble to rewrite (2) as log p(X T 1 ) T X t=1 log N X i=1 p(x t ; s t = ijX t Gamma1 1 ) # = T X t=1 log N X i=1 f i (x t )OE t (i) # (3) where OE t (i) P (s t = ijX t Gamma1 1 ) denotes the state prediction filter. OE t (i) can be updated recursively using (see [11]) OE 1 (j) p(s 1 = j) OE t 1 (j) 1 c t N X i=1 f i (x t )OE t (i) Pi ij (t 1) 4) The normalization factors c t are given by c t = N X k=1 f k (x t )OE t (k) 5) Except for the fact that it is normalized, 4) bear close resemblance with the formulas used for updating the forward ....

....The most obvious constraint is that Pi must be a stochastic matrix, where each line contains positive elements whichsum to one. Usual solutions to this problem include the reparameterization of each line of the transition matrix as a point on the R N hypersphere represented by N Gamma 1 angles [11]. In the following section, we use the computationallysimpler solution which consists in using the natural transition matrix parameterization and (i) projecting the gradient on the subspace orthogonal to the linear constraints P N j=1 Pi ij = 1 (1 i N ) ii) take into account the parameters ....

J.B. Moore R.J. Elliot, L. Aggoun. Hidden Markov models: Estimation and control. Springer-Verlag, New York, 1994.

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R.J. Elliott, L. Aggoun and J.B. Moore. Hidden Markov Models: Estimation and Control. Applications of Mathematices 29. Springer-Verlag, Berlin-Heidelberg-New York, December 1994.

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R.J. Elliott, L. Aggoun and J.B. Moore. Hidden Markov Models: Estimation and Control. Applications of Mathematics 29. Springer-Verlag, Berlin-Heidelberg-New York, December 1994.

....stochastic modelling. For example, Hidden Markov Models have become widely used in Language Engineering applications because they are well understood and computationally tractable (e.g. Young and Bloothooft 1997, Manning and Schutze 1999, Jurafsky and Martin 2000, Huang 1990, MacDonald 1997, Elliott et al. 1995, Woodward 1997) Although (Chomsky 1957) famously demonstrated that a finite state model is a theoretically inadequate approximation for certain aspects of language modelling, Language Engineers have come to realise that HMMs can be adapted to work most of the time, and that the theoretically ....

Elliott R, Lakhdar A, Aggoun J, Moore R 1995 Hidden Markov models: estimation and control. London, Springer-Verlag.

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R. J. Elliott, L. Aggoun, and J. B. Moore. Hidden Markov Models: Estimation and Control. Applications of Mathematics. Springer-Verlag, New York, 1995.

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Elliott, R. J., Aggoun, L., Moore, J. B. 1995. Hidden Markov models: Estimation and Control. New York: Springer-Verlag.

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Elliott, R. J., Aggoun, L., Moore, J. B. (1995) Hidden Markov Models: Estimation and Control. Springer, New York.

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Elliott, R.J., Hidden Markov Models: Estimation and Control, Springer, 1995.

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Elliott, R.J., Hidden Markov Models: Estimation and Control, Springer, 1995.

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Elliott, R. J., Aggoun, L., Moore, J. B. (1995) Hidden Markov Models: Estimation and Control. Springer, New York.

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R. J. Elliott, L. Aggoun and J.B. Moore. Hidden Markov Models: Estimation and Control (1994) Applications of Mathematics 29, Springer-Verlag.

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R. Elliot, L. Aggoun, and J. Moore, Hidden Markov models: estimation and control, 1995.

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