Z. Ghahramani and G. E. Hinton, "Switching statespace models," 6 King's College Road, Toronto M5S 3H5, Canada, Tech. Rep., 1998. [Online]. Available: citeseer.nj.nec.com/ghahramani96switching.html  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with an AR model for each state of the HMM. The techniques are trained off line and must be seeded with reasonable guesses of the HMM in order for the training algorithm to converge correctly. Another class of techniques which are trained off line is switching state space models. See Gharamani [13] for an overview of these techniques. Kohlmorgen and Lemm [19] have developed a technique for automatically segmenting and clustering a time series. The input sequence is projected into a higher dimensional space by including in each sensor reading delayed values of previous sensor readings. A ....

..... 15 Table 6: Miscellaneous Techniques Description On line Segmentation HMM AR Switching state space models Vision disparity Change Detection Reference [18] 33] [13] [9] 1] First Author Kohlmorgen Penny Ghahramani Dima Basseville . # . # . # . # . Multiple time scales . # ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

....data with an AR model for each state of the HMM. The techniques are trained off line and must be seeded with reasonable guesses of the HMM in order for the training algorithm to converge correctly. Another class of techniques which are trained offline is switching state space models. See Gharamani [5] for an overview of these techniques. Kohlmorgen and Lemm [6] have developed a technique for automatically segmenting and clustering a time series. The input sequence is projected into a higher dimensional space by including in each sensor reading delayed values of previous sensor readings. A ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

....type of local approximation in general DBNs has emerged that provides conditions for it to be globally optimal. 4. Previous Work SLDS models and their equivalents have been studied in statistics, time series modeling, and target tracking since early 1970 s. See [16, 13] for a review. Ghahramani [7] introduced a DBN framework for learning and approximate inference in one class of SLDS models. His underlying model differs from ours in assuming the presence of S independent, white noise driven LDSs whose measurements are selected by the Markov switching process. An alternative input switching ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. submitted for publication, 1998.

....type of local approximation in general DBNs has emerged that provides conditions for it to be globally optimal. 4 Previous Work SLDS models and their equivalents have been studied in statistics, time series modeling, and target tracking since early 1970 s. See [13, 12] for a review. Ghahramani [6] introduced a DBN framework for learning and approximate inference in one class of SLDS models. His underlying model differs from ours in assuming the presence of S independent, white noise driven LDSs whose measurements are selected by the Markov switching process. A switching model framework for ....

Z. Ghahramani and G. E. Hinton, "Switching state-space models." submitted for publication, 1998.

....can introduce a nonlinear saturation that prevents the instability scenario which might occur in HMM ARs. 3 of the training complexity, which in practice requires the application of a variational approximation (see below) A different generalization of the HMM AR architecture was suggested by (Ghahramani and Hinton, 1998). In their switching state space model (SSM) the parameter vector w k (that is, the vector of AR coefficients associated with the kth state S t = k) is treated as a continuous hidden state. This generalizes the HMM AR concept, which constitutes a dynamic mixture of static AR models, to a dynamic ....

....k) is treated as a continuous hidden state. This generalizes the HMM AR concept, which constitutes a dynamic mixture of static AR models, to a dynamic mixture of Kalman filters 6 . Again, a price is to pay for this increased model complexity, as it turns out that the E step becomes intractable. (Ghahramani and Hinton, 1998) studied a variational approach, which aims to maximize a lower bound on the log likelihood. The idea is that since expectations with respect to the true posterior distribution P (SjD) are intractable, rather than setting Q(S) P (SjD) in the Estep, a tractable distribution Q is used which ....

[Article contains additional citation context not shown here]

Ghahramani, Z. and Hinton, G. (1998). Switching State-Space Models. Technical report, Department of Computer Science, University of Toronto, 6 King's College Road, Toronto, Canada M5S 3H5.

....let us suppose that the error introduced by this step is at most . Then the results in [BK98b, BK98a] show that for a hybrid DBN, the total error will be a function of and , the mixing rate of the Markov chain, but independent of t. An alternative approach to learning hybrid DBNs, taken in [GH96], is to maximize an exact lower bound on the likelihood, produced by considering a tractable approximation to the original structure. 7.2 Partially observable case If we do not observe the value of every node in each training case, there is no longer a closed form expression for the MLE. In this ....

Z. Ghahramani and G. Hinton. Switching state-space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., Univ. Toronto, 1996.

....from the Bayesian network literature can be applied to dynamical systems. In particular, it has been shown that estimation in LDSs and inference in HMMs are special cases of inference in DBNs. The focus of this paper is on a subclass of DBN models called Switching Linear Dynamic Systems [2, 26, 17, 9, 22]. Intuitively, these models attempt to describe a complex nonlinear dynamic system with a succession of linear models that are indexed by a switching variable. While other approaches such as learning weighted combinations of linear models are possible, the switching approach has an appealing ....

....switching state distributions at each time instance, Pr#s t #=# t #i# and no temporal dependency between switching states. Krishnamurthy and Evans [18] imposed Markov dynamics on the switching model. However, they assumed that noisy measurements of the switching states are available. Ghahramani [9] introduced a DBN framework for learning and approximate inference in one class of SLDS models. His underlying model differs from ours in assuming the presence of S independent, white noise driven LDSs whose measurements are selected by the Markov switching process. An alternative ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. submitted for publication, 1998.

....type of local approximation in general DBNs has emerged that provides conditions for it to be globally optimal. 4 Previous Work SLDS models and their equivalents have been studied in statistics, time series modeling, and target tracking since early 1970 s. See [16, 13] for a review. Ghahramani [7] introduced a DBN framework for learning and approximate inference in one class of SLDS models. His underlying model differs from ours in assuming the presence of S independent, white noise driven LDSs whose measurements are selected by the Markov switching process. An alternative ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. submitted for publication, 1998.

....models are training signals for learning. The exponential transform of the errors has been replaced by a more general likelihood model. 1325 D.M. Wolpert, M. Kawato Neural Networks 11 (1998) 1317 1329 series into components, with each component captured by an expert (Pawelzik et al. 1996; Ghahramani and Hinton, 1998). Gomi and Kawato (1993) combined the feedback errorlearning approach and the mixture of experts modular architecture to learn multiple inverse models for multiple manipulated objects. They used both the visual signal of manipulated objects and internal signals, e.g. somatosensory feedback and ....

Ghahramani, Z. & Hinton, G.E. (1998). Switching state-space models. (Submitted).

....of annealing EM satisfies the original optimization problem. Overall, 30 this modified EM procedure tends to lead toward a global minimum of the cost function. At this point we note that the use of annealing EM seems particularly e#ective for GEM algorithms with approximate expectation steps [32]. 31 CHAPTER 3 STOCHASTIC MODELING OF IMAGE WAVELET COEFFICIENTS AND COMMUNICATION CHANNELS There are two essential intellectual steps in designing any signal processing system. The first step comprises the choice of a sound operational model of the signal under investigation. In the second ....

Z. Ghahramani and G. E. Hinton, "Switching state-space models," submitted for publication, 1998.

....particularly appropriate for financial markets. We use a dynamic switching (modelled by a hidden Markov model) combined with a linear dynamical system in a hybrid switching state space model (SSSM) and discuss the practical details of training such models with a variational EM algorithm due to [Ghahramani and Hinton, 1998]. The performance of the SSSM is evaluated on several financial data sets and it is shown to improve on a number of existing benchmark methods. 1 Introduction Most traditional time series models are based on the assumption of stationarity: the underlying generator of the data is assumed to be ....

....is that the complexity of the exact training algorithm grows exponentially with order M T , where M is the number of models and T is the length of the time sequence. Various not completely satisfactory approximations have been proposed during the last decade [Bar Shalom and Li, 1993] Recently, [Ghahramani and Hinton, 1998] reintroduced the SSSM and proposed an efficient and principled approximate algorithm for training these models in a maximum likelihood approach. In this paper we propose to use switching state space models for modelling financial data. The approach is motivated by the fact that market behaviour ....

[Article contains additional citation context not shown here]

Ghahramani, Z. and Hinton, G. E. (1998). Switching state-space models.

....and learning techniques from the BN literature can be applied to dynamical systems. In particular, it has been shown that estimation in LDSs and inference in HMMs are special cases of inference in DBNs. The focus of this paper is on a subclass of DBN models called Switching Linear Systems [2, 22, 14, 8, 19]. Intuitively, these models attempt to describe a complex nonlinear dynamic system with a succession of linear models that are indexed by a switching variable. While other approaches such as learning weighted combinations of linear models are possible, the switching approach has an appealing ....

....switching state distributions at each time instance, Pr#s t # = # t #i# and no temporal dependency between switching states. Krishnamurthy and Evans [15] imposed Markov dynamics on the switching model. However, they assumed that noisy measurements of the switching states are available. Ghahramani [8] introduced a DBN framework for learning and approximate inference in one class of SLDS models. His underlying model differs from ours in assuming the presence of S independent, white noise driven LDSs whose measurements are selected by the Markov switching process. An alternative input switching ....

[Article contains additional citation context not shown here]

Z. Ghahramani and G. E. Hinton, "Switching state-space models." submitted for publication, 1998.

....apply a corpus of exact and approximate statistical inference and learning techniques from the BN literature to time series modeling. This has resulted in new approaches to inference and in novel complex temporal models such as factorial HMMs [8] coupled HMMs [2, 13] switching state space models [7], mixtures of DBNs [13] etc. We consider an instance of a complex DBN that arises as a combination of discrete state HMMs and continuous state LDSs. We call such DBNs mixed state DBNs. Namely, a mixed state DBN is a HMM coupled with a LDS (see Figure 1) The output of a HMM is the driving input ....

Z. Ghahramani and G. E. Hinton. Switching state-space models. submitted for publication, 1998.

.... which have been developed for HMMs and many related models, such as hybrids of HMMs with artificial neural networks [1, 2, 3] Input Output HMMs [4, 5, 6, 7] weighted transducers [8, 9, 10, 11] variable length Markov models [12, 13] Markov switching models [14] and switching state space models [15, 16]. Of course, there is a lot more litterature on HMMs and their applications than can be covered here, but this survey wants to be representative of the issues addressed here, mainly concerning learning algorithms and extensions of HMMs and related models. Note that what we call learning here is ....

....have to be done numerically in general. To model both the abrupt and gradual changes in time series, several researchers have in fact proposed hybrids of state space models and discrete state HMMs (or IOHMMs) also known as state space models with switching, or jump linear systems. See [137] and [16] for a review of such models. Many early models assume that some of the parameters of the distribution are known a priori, and others [15] approximate the EM algorithm with a heuristic, because the E step would require exponential computations. Others [138, 139] used expensive Monte Carlo ....

[Article contains additional citation context not shown here]

Z. Ghahramani and G. Hinton, "Switching state-space models," Tech. Rep. Technical Report CRGTR -91-3, University of Toronto, 1996.

....the continuous variables have a jointly Gaussian distribution. In the language of Bayesian networks (graphical models) Pea88] the continuous nodes cannot have discrete children. Formally, the model consists of the following two equations, called the system and measurement equations (see e.g. [Ham84, Har89, BSL93, GH96b, WH97]) x t = A t x t Gamma1 v t y t = C t x t w t where v t N(0; Q t ) and w t N(0; R t ) are independent Gaussian noise sources, and the system s parameters depend on the current state or mode, S t , i.e. A t = A[S t ] C t = C[S t ] Q t = Q[S t ] and R t = R[S t ] 1 This model is ....

....consider a model with Markovian switching observation matrices, and two independent hidden processes, X1 t and X2 t . X1(1) X1(2) X2(1) X2(2) S(1) S(2) Y (1) Y (2) The switch variable in this case can be thought of as selecting one of the sub processes to pass through to the output variable [GH96b], or as choosing a permutation matrix H t to apply, to model the fact that we are uncertain about which process causes which observation [SS91] this is called data association ambiguity [BSF88] 3 A factored model with k components can always be converted into canonical form by combining the ....

[Article contains additional citation context not shown here]

Z. Ghahramani and G. Hinton. Switching state-space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., Univ. Toronto, 1996.

....to a fixed value estimated by passing a static AR model over an initial portion of the time series. This is a simplification that we make for all the simulations described in this paper. An alternative procedure is to use multiple Kalman filter models, each with a different observation noise level [7, 8]. The algorithm we have described for updating the state noise relies on a parameter ff, which smooths the point estimates of the observation noise. Larger values of ff correspond to smoother estimates of state noise. In this paper, ff is set according to the following argument. The first term in ....

Z. Ghahramani and G. Hinton. Switching state-space models. Technical report, Department of Computer Science, University of Toronto, 1998.

....in cases in which tractable substructure can be identified in the graph. This approach was first presented by Saul and Jordan (1996) as a refined version of mean field theory for Markov random fields, and has been developed further in a number of recent studies (e.g. Ghahramani Jordan, 1997; Ghahramani Hinton, 1996; Jordan, et al. 1997) In the block approach, we begin by identifying a substructure in the graph of interest that we know is amenable to exact inference methods (or, more generally, to efficient approximate inference methods) For example, we might pick out a tree or a set of chains in the ....

Ghahramani, Z., & Hinton, G. E. (1996). Switching state-space models. (Technical Report CRGTR -96-3). Toronto: Department of Computer Science, University of Toronto.

....One advantage of the Gaussian case over the discrete case is that marginalizing the posterior over two slices is an efficient operation. Ultimately we wish to tackle the case of hybrid DPNs, with both discrete and continuous variables. The advantage of hybrid DPNs over switching state space models [16] is that the state variables can be represented in factored form. For example, in the driving domain, we can have separate variables for the continuous observations (such as speed and position) and for the discrete hidden states (such as want to change lane or want to overtake ) The question ....

Z. Ghahramani and G. Hinton. Switching State-Space Models. Submitted for publication, 1998.

.... which have been developed for HMMs and many related models, such as hybrids of HMMs with artificial neural networks [1, 2, 3] Input Output HMMs [4, 5, 6, 7] weighted transducers [8, 9, 10] variable length Markov models [11, 12] Markov switching models [13] and switching state space models [14, 15]. Of course, there is a lot more litterature on HMMs and their applications than can be covered here, but this survey wants to be representative of the issues addressed here, mainly concerning learning algorithms and extensions of HMMs and related models. Note that what we call learning here is ....

....reasons of computational tractability. To model both the abrupt and gradual changes in time series, several researchers have in fact proposed hybrids of state space models and discrete state HMMs (or IOHMMs) also known as state space models with switching, or jump linear systems. See [118] and [15] for a review of such models. Many early models assume that some of the parameters of the distribution are known a priori, and others [14] approximate the EM algorithm with a heuristic, because the E step would require exponential computations. Others [119, 120] used expensive Monte Carlo ....

[Article contains additional citation context not shown here]

Z. Ghahramani and G. Hinton, "Switching state-space models," Tech. Rep. Technical Report CRGTR -91-3, University of Toronto, 1996. 25

....in cases in which tractable substructure can be identified in the graph. This approach was first presented by Saul and Jordan (1996) as a refined version of mean field theory for Markov random fields, and has been developed further in a number of recent studies (e.g. Ghahramani Jordan, 1997; Ghahramani Hinton, 1996; Jordan, et al. 1997) In the block approach, we begin by identifying a substructure in the graph of interest that we know is amenable to exact inference methods (or, more generally, to efficient approximate inference methods) For example, we might pick out a tree or a set of chains in the ....

Ghahramani, Z., & Hinton, G. E. (1996). Switching state-space models. University of Toronto Technical Report CRG-TR-96-3, Department of Computer Science.

....us suppose that the error introduced by this step is at most ffl. Then the results in [BK98b, BK98a] show that for a hybrid DBN, the total error will be a function of ffl and fl, the mixing rate of the Markov chain, but independent of t. An alternative approach to learning hybrid DBNs, taken in [GH96], is to maximize an exact lower bound on the likelihood, produced by considering a tractable approximation to the original structure. 6.2 Partially observable case If we do not observe the value of every node in each training case, there is no longer a closed form expression for the MLE. In this ....

Z. Ghahramani and G. Hinton. Switching state-space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., Univ. Toronto, 1996.

....be a very broad covariance (e.g. approximately uniform) The prior probability of S t reflects how often we expect outliers to occur. This is a widely used technique for making linear regression more robust, see e.g. PG88] and for modelling sensor failure [Wil76] Recently, Ghahrhamani et al. [GH96b] have proposed the model shown in Figure 1(c) This also has switching observations, but the interpretation is different. The switch variable in this case can be thought of as selecting one of the sub processes to pass through to the output variable or as choosing a permutation matrix C t to ....

....of the best matrix at each step [SM80] ffl Variational: essentially we break all the vertical links in the model, but introduce new variational parameters to couple them together in as tight a way as possible. Using EM with such a model will maximize a lower bound on the likelihood: see [GH96b] for details. In this paper, we focus on the collapsing approximation. One worry is that the errors introduced at each time step by approximating the posterior might accumulate over time, leading to very poor performance. However, as shown in [BK98b, BK98a] the stochasticity of the process ....

[Article contains additional citation context not shown here]

Z. Ghahramani and G. Hinton. Switching state-space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., Univ. Toronto, 1996.

....e j , y . P k i=1 P x . e i , y . 6.6a) x . j = N C j , R y . P x . e j P k i=1 N (C i , R) y . P (x . e i ) 12 As in the continuous static case, we again dispense with any special treatment of the initial state. 322 Sam Roweis and Zoubin Ghahramani C WTA### x # w # v # y # 0 x w # v # y C Figure 4: Static generative model (discrete state) The WTA[ block implements the winner take all nonlinearity. The covariance matrix of the input noise w is Q and the covariance matrix of the output noise v is R. In the network model below, the smaller circles represent noise ....

Ghahramani, Z., & Hinton, G. (1996b). Switching state-space models (Tech. Rep.

....application of the EM algorithm to learning of partially unknown linear dynamical systems [3] Second, EM generalizes readily to more complex models with combinations of discrete and real valued hidden variables. For example, one can formulate EM for a mixture of nonlinear dynamical systems [38], 39] Third, whereas it is often very dicult to prove or analyze stability within the classical on line approach, the EM algorithm is always attempting to maximize the likelihood, which acts as a Lyapunov function for stable learning. Fourth, the EM framework facilitates Bayesian extensions to ....

Zoubin Ghahramani and Georey Hinton, \Switching statespace models," Tech. Rep. CRG-TR-96-3, Dept. of Computer Science, University of Toronto, July 1996.

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Ghahramani, Z. and Hinton, G. (1996b). Switching state-space models. Technical Report CRG-TR-96-3, Dept. of Computer Science, University of Toronto (Submitted for Publication).

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Z. Ghahramani and G. E. Hinton, "Switching statespace models," 6 King's College Road, Toronto M5S 3H5, Canada, Tech. Rep., 1998. [Online]. Available: citeseer.nj.nec.com/ghahramani96switching.html

No context found.

Zoubin Ghahramani and Geoffrey E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

No context found.

Ghahramani, Z., & Hinton, G. E. (1998). Switching state-space models (Technical Report). 6 King's College Road, Toronto M5S 3H5, Canada.

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Zoubin Ghahramani and Geoffrey E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

No context found.

Z. Ghahramani and G. E. Hinton, "Switching State-Space Models," Dept. Comp. Sci., Univ. Toronto, Toronto, ON, Canada, Tech. Rep. CRG-TR-96-3, 1996.

No context found.

Z. Ghahramani and G. E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

No context found.

Z. Ghahramani and G. Hinton. Switching state-space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., Univ. Toronto, 1996.

No context found.

Z. Ghahramani and G. E. Hinton. Switching state space models. Technical Report CRG-TR-96-3, Dept. Comp. Sci., University of Toronto, 1996.

No context found.

Z. Ghahramani and G. E. Hinton, "Switching statespace models," 6 King's College Road, Toronto M5S 3H5, Canada, Tech. Rep., 1998. [Online]. Available: citeseer.nj.nec.com/ghahramani96switching.html

No context found.

Zoubin Ghahramani and Geoffrey E. Hinton. Switching state-space models. Technical report, 6 King's College Road, Toronto M5S 3H5, Canada, 1998.

No context found.

Z. Ghahramani and G. E. Hinton, "Switching state-space models." 1998.

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Z. Ghahramani and G. E. Hinton, "Switching state-space models," Dept. Comp. Sci., Univ. Toronto, Tech. Rep. CRG-TR-96-3, 1996.

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