Z. Ghahramani and G. E. Hinton. Parameter estimation for linear dynamical systems. University of Toronto Technical Report CRG-TR-96-2, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of operation. A discrete state variable controls the switching between the various linear regimes. This model allows us to capture the nonlinear process behaviour and disturbance transient responses in a very simple way. The parameters of each regime are identied off line with the EM algorithm [5]. Once the stationary parameters have been identied, real time Rao Blackwellised particle ltering (RBPF) algorithms are used to estimate the continuous and discrete states of the system on line [2, 3, 4, 6] and identify the most probable operating condition. A standard PID controller is adjusted ....

....[8] The discrete time state space representation, consisting of matrices (A(z t ) C(z t ) F (z t ) G(z t ) was generated by a standard procedure in control engineering [7] The matrix G(z t ) was null in our application. In the second stage, we applied a maximum likelihood (EM) algorithm [5] to rene the estimates of (A(z t ) C(z t ) F (z t ) and to compute the noise matrices (B(z t ) D(z t ) This algorithm consisted of two steps. In the E step, a Rauch Tung Striebel Kalman smoother was used to compute the sufcient statistics of the Gaussian states. In the M step, we updated the ....

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Z Ghahramani and G E Hinton. Parameter estimation for linear dynamical system. Technical Report CRG-TR-962, Department of Computer Science, University of Toronto, Toronto, 1996.

....we adopt a jump Markov linear Gaussian (JMLG) model to describe an industrial heat exchanger with different linear regimes of operation. A discrete state variable controls the switching between the various linear regimes. The parameters of each regime are identified off line with the EM algorithm [3]. Once the stationary parameters have been identified, real time Rao Blackwellised particle filtering (RBPF) algorithms are used to estimate the continuous and discrete states of the system on line [4, 5] These estimates are used to determine to control policy of a PID controller. The paper is ....

....discrete time state space representation, consisting of matrices I . O . M . A, I O O , was generated by a standard procedure in control engineering [7] The matrix , I O was null in our application. In the second stage, we apply a maximum likelihood (EM) algorithm [3] to refine the estimates of Y O . M . A, I O O and to compute the noise matrices O . M ( O . This algorithm consisted of two steps. In the E step, a Rauch Tung Striebel Kalman smoother was used to compute the sufficient statistics of the Gaussian ....

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Z Ghahramani and G E Hinton. Parameter estimation for linear dynamical system. Technical Report CRG-TR-96-2, Department of Computer Science University of Toronto, Toronto, 1996.

....: arg naxE[logp(x:v,y:vl 9) 5) This converges to a local or global maximum depending on the initial parameter estimate 90. Refer to [8] for more details. EM can thus be applied to the stochastic state space model of equations 1 and 2 to determine optimal parameters O. An explanation is given in [3]. The EM algorithm applied to the SS system consists of two stages per iteration. Firstly, given current parameter estimates, states are estimated using a Kalman smoother. Secondly, given these states, new parameters are estimated by maximising the expected log likelihood function. We employ the ....

....ms in duration, shifted 7.5 ms each frame. Speech is modelled as detailed in section 1. All models are order 8. A frame is assumed silent and no analysis done when the mean energy per sample is less than an empirically defined threshold. For the EM algorithm, a modified version of the software in [3] is used. The initial state vector and covariance matrix are set to zero and identity respectively, and 50 iterations are applied. Q is updated by taking its diagonal only in the M step for numerical stability (see [3] In these experiments, three schemes are compared at initialising parameters ....

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Ghahramani, Z. & Hinton, G. (1996) Parameter Estimation for Linear Dynamical Systems, Tech. rep. CRG-TR-96-2, Dept. of Computer Science, Univ. of Toronto. Software at www. gatsby. ucl. ac. uk/oubin/software. html.

....DSD CSD MIX CSD DSD Figure 3: Dynamic linear Gaussian models and how they relate to some of the static models. Dynamic linear Gaussian models and the corresponding static models are illustrated in Figure 3. Dynamic models with factor analysis observation process include linear dynamical systems [6, 7, 16, 32, 34, 38], mixture of linear dynamical systems and switching state space model [17, 33] as well as di erent variations of factor analysed HMMs presented later in this paper and its restricted version in [40] The linear discriminant observation process is illustrated in case of HMM based [11, 21, 29] and ....

Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996. 35

....is that there is not an analytical solution for composing the optimal mixture of Gaussians for given data, but the mixture has to be found using an iterative search called the expectation maximization (EM) algorithm. The complete definition of the EM algorithm can be found in [RJ93] and [GH96] 2.4.4 Feature selection The aim of feature selection is to pick the essential features and discard the redundant and adverse features from the total set of implemented features [LM98] The process of feature selection involves repeatedly taking a subset of all features for testing and ....

....model. Cemgil et al. estimate the beat trajectory by applying a Kalman filter to the output of a local periodicity data they call the tempogram. The Kalman filter optimizes the parameters of a linear dynamical system, where the beat position and the logarithm of beat period are hidden variables [GH96] As a result, estimates of beat positions are produced. The tempogram periodicity data represents energy as a function of period in a local timeframe. The function resembles an IOI histogram with memory but tolerates deviations in onset times in a configurable amount. The model processes MIDI ....

Zoubin Ghahramani and Geoffrey E. Hinton. Parameter estimation for linear dynamical systems. Tech. report CRG-TR-96-2, Univ. of Toronto, Toronto, Canada, 1996.

....DSD CSD MIX CSD DSD Figure 3: Dynamic linear Gaussian models and how they relate to some of the static models. Dynamic linear Gaussian models and the corresponding static models are illustrated in Figure 3. Dynamic models with factor analysis observation process include linear dynamical systems [6, 7, 16, 32, 34, 38], mixture of linear dynamical systems and switching state space model [17, 33] 6 as well as di erent variations of factor analysed HMMs presented later in this paper and its restricted version in [40] The linear discriminant observation process is illustrated in case of HMM based [11, 21, 29] ....

Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996.

.... algorithms for the state inference and parameter learning of a discrete time time invariant linear dynamical system are compared: expectation maximisation (EM) and subspace state space system identi cation (4SID) Particular implementations used are the EM algorithm due to Ghahramani and Hinton [8], and a 4SID algorithm due to Van Overschee and De Moor (algorithm3, chapter 3, 26] The report sets this within a probabilistic framework, noting the inherent phase and state space basis ambiguities, and the requirements for positive realness of the estimated covariance sequence. The EM and ....

....sequence is nite or the data generating process is not a true time invariant LDS. 2. 4 Probability Framework In this section, a probabilistic framework is applied to the joint problems of state inference and parameter learning as described by de Freitas and Niranjan [3] and Ghahramani and Hinton [8]. The initial state distribution is assumed Gaussian, as are the noise processes w t and v t . p(x 1 ) N ( 1 ; Q 1 ) 2 ) k 2 jQ 1 j 1 2 exp 1 2 [x 1 1 ] 0 Q 1 1 [x 1 1 ] 10) Due to the properties of Gaussians, all future states and observations are also Gaussian ....

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Z. Ghahramani and G.E. Hinton. Parameter Estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2, Dept. of Computer Science, University of Toronto, Canada, 1996. http://www.gatsby.ucl.ac.uk/ zoubin/papers.html.

....This converges to a local or global maximum depending on initial parameter estimates. Theta k 1 = arg Theta maxE[log p(x; yj Theta k ) 11) This EM algorithm can be applied to the stochastic state space model to determine optimal parameters Theta. Detailed explanations are given in [2] [6] This EM algorithm consists of two stages per iteration. Firstly, given current parameter estimates, states are estimated using a Kalman smoother. Secondly, given these states, new parameters are estimated. In this paper we use Zoubin Ghahramani s software (www.gatsby.ucl.ac.uk zoubin ) ....

Ghahramani, Z. & Hinton, G. (1996) Parameter estimation for linear dynamical systems, Tech. Rep. CRG-TR-96-2, Dept of Computer Science, University of Toronto, http://www.gatsby.ucl.ac.uk/ zoubin/papers.html.

.... t t x P Z inductively and is formulated as 1 1 1 1 1 ) t t t t t t t t t t t dx x P x x P x z P x P x z P x P Z Z Z (2) Using this formula, the well known Kalman filter that computes ) x E Z for a Gaussian process can be derived [2]. When multiple objects present, if the number of objects is fixed and the posterior distribution of each object is Gaussian, similar solution in analytic form is obtained. If the number of objects may change over time, data association method such as multiple hypothesis tracking (MHT) 3] has to ....

Z. Ghahramani and G. E. Hinton, "Parameter estimation for linear dynamical systems," Technical Report CRG-TR-96-2, Univ. of Toronto, 1996. http://www.cs.utoronto.ca/~zoubin/.

.... problem) In the context of real time control and other applications where learning must be on line, numerical maximization of the likelihood can be performed recursively with a second order method which requires only gradients [129] For off line applications, the EM algorithm can also be used [130], with a backward pass that is equivalent to the Rauch equations. 7.1 Hybrids of Discrete and Continuous State One disadvantage of the discrete representation of the state is that it is an inefficient representation in comparison to a distributed representation with multiple state variables. ....

Z. Ghahramani and G. Hinton, "Parameter estimation for linear dynamical systems," Tech. Rep. Technical Report CRG-TR-91-1, University of Toronto, 1996.

....Kalman Filter models can be represented in a unified way, leading to a single, general purpose inference algorithm. We then show how to find approximate Maximum Likelihood Estimates of the parameters using the EM algorithm, extending previous results on learning using EM in the non switching case [DRO93, GH96a] and in the switching, but fully observed, case [Ham90] 1 Introduction Dynamical systems are often assumed to be linear and subject to Gaussian noise because it is mathematically convenient. In particular, we can then use the well known Kalman filter to efficiently perform inference. i.e. to ....

.... V t 1jt Delta J 0 t V t 1;tjT = V t 1;tjt 1 Gamma V t 1jT Gamma V t 1jt 1 Delta V Gamma1 t 1jt 1 V t 1;tjt 1 We now present an alternative way to compute the smoothed estimates of the cross variance terms, V t;t Gamma1jT , which does not require the corresponding filtered terms [SS91, GH96a]. The Smooth operator (x tjT ; V tjT ; V t;t Gamma1jT ) Smooth 0 (x t 1jT ; V t 1jT ; V t 1;tjT ; x tjt ; V tjt ; V t Gamma1jt Gamma1 ; F t 1 ; Q t 1 ; F t ; Q t ) is defined as above, except V t;t Gamma1jT = V tjt J 0 t Gamma1 J t (V t 1;tjT Gamma F t 1 V tjt )J 0 t Gamma1 where the ....

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Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Dept. Comp. Sci., Univ. Toronto, 1996.

....on data up to time t whereas the HMMs were trained in batch mode on the entire data sequence. This is, however, not a fundamental difference as it is possible to train Kalman filters on the entire data sequence either recursively, using a Rauch smoother [19] or in batch mode using an EM algorithm [20]. Our approach consisted of embedding an AR model into an HMM. This approach has also been taken by Fraser et al. in the context of time series prediction [15] The method is also readily extendible to Multivariate Autoregressive (MAR) models which are used for cross spectral analysis. This can be ....

Z. Ghahramani and G.E. Hinton. Parameter Estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996.

.... problem) In the context of real time control and other applications where learning must be on line, numerical maximization of the likelihood can be performed recursively with a second order method which requires only gradients [115] For off line applications, the EM algorithm can also be used [116], with a backward pass that is equivalent to the Rauch equations. 7.1 Hybrids of Discrete and Continuous State One disadvantage of the discrete representation of the state is that it is an inefficient representation in comparison to a distributed representation with multiple state variables. ....

Z. Ghahramani and G. Hinton, "Parameter estimation for linear dynamical systems," Tech. Rep. Technical Report CRG-TR-91-1, University of Toronto, 1996.

....Kalman Filter models can be represented in a unified way, leading to a single, general purpose inference algorithm. We then show how to find approximate Maximum Likelihood Estimates of the parameters using the EM algorithm, extending previous results on learning using EM in the non switching case [DRO93, GH96a] and in the switching, but fully observed, case [Ham90] 1 Introduction Dynamical systems are often assumed to be linear and subject to Gaussian noise. This model, called the Linear Dynamical System (LDS) model, can be defined as x t = A t x t Gamma1 v t y t = C t x t w t where x t is the ....

....at time t, y t is the observation at time t, and v t N(0; Q t ) and w t N(0; R t ) are independent Gaussian noise sources. Typically the parameters of the model Theta = f(A t ; C t ; Q t ; R t )g are assumed to be time invariant, so that they can be estimated from data using e.g. EM [GH96a]. One of the main advantages of this model is that there is an efficient algorithm for performing inference (i.e. computing the belief state P (X t jy 1:t ) the well known Kalman filter, and its generalization to the offline case, the Rauch Tung Strieber smoother (for computing P (X t jy 1:T ) ....

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Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Dept. Comp. Sci., Univ. Toronto, 1996.

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Z. Ghahramani and G. E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [ftp://ftp.cs.toronto.edu/pub/zoubin/tr96 -2.ps.gz] , Department of Computer Science, University of Toronto, 1996. 29

....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. (1996a). Parameter estimation for linear dynamical systems (Tech. Rep. CRG-TR-96-2). Toronto: Department of Computer Science, University of Toronto. Available from ftp://ftp.cs.toronto.edu/ pub/zoubin/.

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Ghahramani, Z. and Hinton, G. E. (1996a). Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [ftp://ftp.cs.toronto.edu/pub/zoubin/tr-962. ps.gz] .

.... parameter estimation from data sets with missing or hidden variables [1] EM has been applied to system identi cation in linear statespace models, where the state variables are hidden from the observer and both the state and the parameters of the model have to be estimated simultaneously [2] 3] [4]. Here we generalize the EM algorithm to estimate parameters of nonlinear dynamical state space models. The expectation step makes use of Extended Kalman Smoothing to estimate the state, while the maximization step re estimates the parameters using these uncertain state estimates. In general, ....

....and the observations. In the past, the EM algorithm has been applied to learning linear dynamical systems in speci c cases (such as multiple indicator multiple cause (MIMC) models with a single latent variable [2] or state space models with the observation matrix known [3] as well more generally [4]. This chapter is an extension of our earlier work [35] since then, there has been other similar work applying EM to nonlinear dynamical systems [36] 37] Whereas other work uses sampling for the E step and gradient M steps, our algorithm uses the RBF networks to obtain a computationally ecient ....

Zoubin Ghahramani and Georey Hinton, \Parameter estimation for linear dynamical systems," Tech. Rep. CRG-TR-962, Dept. of Computer Science, University of Toronto, February 1996.

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Ghahramani, Z. and Hinton, G. E. (1996a). Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-2.ps.gz] .

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Ghahramani, Z. and Hinton, G. (1996a). Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Dept. of Computer Science, University of Toronto.

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Ghahramani, Z. and Hinton, G. (1996a). Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [ftp://ftp.cs.toronto.edu/pub/zoubin/], Dept. of Computer Science, University of Toronto.

....constant for Q. Both Q and P de ne distributions in the exponential family. As a consequence, the zeros of the derivatives of KL with respect to the variational parameters can be obtained simply by equating derivatives of hHi and hHQ i with respect to corresponding su cient statistics (Ghahramani, 1997) hHQ Hi hS (m) t i = 0 (43) hHQ Hi hX (m) t i = 0 (44) hHQ Hi hP (m) t i = 0 (45) where P (m) t = hX (m) t X (m) t 0 i hX (m) t ihX (m) t i 0 is the covariance of X (m) t under Q. Many terms cancel when we subtract the two hamiltonians HQ H= M X m=1 T X t=1 1 2 h (m) t S (m) ....

Ghahramani, Z. and Hinton, G. E. (1996a). Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [http://www.gatsby.ucl.ac.uk/zoubin/papers/tr-96-2.ps.gz], Department of Computer Science, University of Toronto.

.... to the posterior distribution of X by hf(X)i, hf(X)i = Z X f(X) P (XjY; k ) dX: 20) Then, the M step for C is C X t Y t hX t i 0 X t hX t X 0 t i Gamma1 : Similar M steps can be derived for all the other parameters by taking derivatives of the expected log probability [47, 11, 15]. 7 In general we require all terms of the kind hX t i, hX t X 0 t i and hX t X 0 t Gamma1 i. These terms can be computed using the Kalman smoothing algorithm. 4.4 Kalman smoothing The Kalman smoother solves the problem of estimating the state at time t of a linear Gaussian state space ....

Z. Ghahramani and G. E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2 [ftp://ftp.cs.toronto.edu/pub/zoubin/tr96 -2.ps.gz] , Department of Computer Science, University of Toronto, 1996.

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Z. Ghahramani and G. E. Hinton. Parameter estimation for linear dynamical systems. University of Toronto Technical Report CRG-TR-96-2, 1996.

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Z. Ghahramani, and G.E. Hinton, Parameter Estimation for Linear Dynamical Systems, Technical Report CRG-TR-96-2, University of Toronto, 1996

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Zoubin Ghahramani and Geoff E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, University of Toronto, 1996.

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Z. Ghahramani and G. E. Hinton, "Parameter estimation for linear dynamical systems. (crg-tr-96-2)," University of Totronto. Dept. of Computer Science., Tech. Rep., 1996.

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Z. Ghahramani and G. E. Hinton, "Parameter estimation for linear dynamical systems. (crg-tr-96-2)," University of Totronto. Dept. of Computer Science., Tech. Rep., 1996.

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Z. Ghahramani and G.E. Hinton, "Parameter Estimation for Linear Dynamical Systems," Technical Report: CRG-TR-96-2, Univ. of Toronto 1996.

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Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996. 35

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Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996. Available at http://www.gatsby.ucl.ac.uk/ zoubin/papers.html.

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Zoubin Ghahramani and Georey Hinton. Parameter Estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2, University of Toronto, 1996.

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Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto, 1996.

No context found.

Zoubin Ghahramani and Geo#rey Hinton. Parameter Estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2, University of Toronto, 1996.

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Z. Ghahramani and G.E.Hinton, "Parameter estimation for linear dynamical system," University of Toronto Technical Report CRG-TR-96-2 (1996).

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Z. Ghahramani and G.E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, Dept of Computer Science, University of Toronto, 1996. http://www.gatsby.ucl.ac.uk/ zoubin/papers.html. 47

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Zoubin Ghahramani and Georey E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, University of Toronto, 1996. http://www.cs.utoronto.ca/~zoubin/.

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Zoubin Ghahramani and Geoffrey E. Hinton. Parameter estimation for linear dynamical systems. Technical Report CRG-TR-96-2, University of Toronto, 1996. http://www.cs.utoronto.ca/~zoubin/.

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