## Switching State-Space Models (1996)

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Venue: | King’s College Road, Toronto M5S 3H5 |

Citations: | 41 - 2 self |

### BibTeX

@TECHREPORT{Ghahramani96switchingstate-space,

author = {Zoubin Ghahramani and Geoffrey E. Hinton},

title = {Switching State-Space Models},

institution = {King’s College Road, Toronto M5S 3H5},

year = {1996}

}

### Years of Citing Articles

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### Abstract

We introduce a statistical model for times series data with nonlinear dynamics which iteratively segments the data into regimes with approximately linear dynamics and learns the parameters of each of those regimes. This model combines and generalizes two of the most widely used stochastic time series models---the hidden Markov model and the linear dynamical system---and is related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network model (Jacobs et al., 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact Expectation Maximization (EM) alogithm cannot be applied. However, we present a variational approximation which maximizes a lower bound on the log likelihood and makes use of both the forward--backward recursio...

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Citation Context ...., 1994), and fault detection (Smyth, 1994). Given an HMM with known parameters and a sequence of observations, two algorithms are commonly used to solve two different forms of the inference problem (=-=Rabiner and Juang, 1986-=-). The first computes the posterior probabilities of the hidden states using a recursive algorithm known as the forward--backward algorithm. The computations in the forward pass are exactly analogous ... |

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Citation Context ...e linear dynamical system---and is related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network model (=-=Jacobs et al., 1991-=-) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and t... |

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Citation Context ...g to explain the observations. With regard to the literature on neural computation, the model presented in this paper is a generalization of the mixtures of experts architecture (Jacobs et al., 1991; =-=Jordan and Jacobs, 1994-=-). 5 Previous dynamical generalizations of the mixture of experts architecture consider the case in which the gating network has Markovian dynamics (Cacciatore and Nowlan, 1994; Kadirkamanathan and Ka... |

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Citation Context ... the resulting posterior is also Gaussian. The special cases of the inference problem for state-space models play a prominent role in the engineering literature: filtering, smoothing, and prediction (=-=Anderson and Moore, 1979-=-; Goodwin and Sin, 1984). The goal of filtering is to compute the probability of the current hidden state X t given the sequence of inputs and outputs up to time t---P (X t jfY g t 1 ; fUg t 1 ). 3 Th... |

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Citation Context ...aussian state distributions via a set of samples which are stochastically propagated and reweighted. This approach has been successfully applied to the problem of contour tracking in computer vision (=-=Isard and Blake, 1996-=-; Blake et al., 1995). We have explored elsewhere the use of the EKF in deriving an EM algorithm for general stochastic nonlinear dynamical systems (Ghahramani and Roweis, in preparation). Switching s... |

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Citation Context ...on matrix P (S t jS t\Gamma1 ). 6 Note that the state vectors could be concatenated into one large state vector with factorized (blockdiagonal) transition matrices (cf. factorial hidden Markov model; =-=Ghahramani and Jordan, 1997-=-). However, this obscures the decoupled structure of the model. 8 X (M) 2 S 2 Y 2 X (M) 3 S 3 Y 3 X (M) 1 S 1 Y 1 X (1) 2 X (1) 3 X (1) 1 (2) t X S Y X (1) X (M) t Figure 2: a) Graphical model represe... |

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Citation Context ... multiple real-valued state vectors. 6 We present a learning algorithm for all of the parameters of the model, including the Markov switching parameters. Using a structured variational approximation (=-=Saul and Jordan, 1996-=-), we show that this algorithm maximizes a strict lower bound on the log likelihood of the data, rather than a heuristically motivated pseudo-likelihood. The resulting algorithm has a simple and intui... |

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Citation Context ...d to allow for input variables, such that it models the conditional distribution of sequences of output observations given sequences of inputs (Cacciatore and Nowlan, 1994; Bengio and Frasconi, 1995; =-=Meila and Jordan, 1996-=-). The approach used in Bengio and Frasconi's Input Output HMMs (IOHMMs) suggests modeling P (S t jS t\Gamma1 ; U t ), where U t is the input, as M separate neural networks, one for each setting of S ... |

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