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A Unifying Review of Linear Gaussian Models
, 1999
"... Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observa ..."
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Cited by 348 (18 self)
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Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observations and derivations made by many previous authors and introducing a new way of linking discrete and continuous state models using a simple nonlinearity. Through the use of other nonlinearities, we show how independent component analysis is also a variation of the same basic generative model. We show that factor analysis and mixtures of gaussians can be implemented in autoencoder neural networks and learned using squared error plus the same regularization term. We introduce a new model for static data, known as sensible principal component analysis, as well as a novel concept of spatially adaptive observation noise. We also review some of the literature involving global and local mixtures of the basic models and provide pseudocode for inference and learning for all the basic models.
Observable Operator Models for Discrete Stochastic Time Series
, 1999
"... A widely used class of models for stochastic systems is Hidden Markov models. Systems which can be modeled by hidden Markov models are a proper subclass of linearly dependent processes, a class of stochastic systems known from mathematical investigations carried out over the last four decades. This ..."
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Cited by 97 (8 self)
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A widely used class of models for stochastic systems is Hidden Markov models. Systems which can be modeled by hidden Markov models are a proper subclass of linearly dependent processes, a class of stochastic systems known from mathematical investigations carried out over the last four decades. This article provides a novel, simple characterization of linearly dependent processes, called observable operator models. The mathematical properties of observable operator models lead to a constructive learning algorithm for the identification of linearly dependent processes. The core of the algorithm has a time complexity of O(N + nm³), where N is the size of training data, n is the number of distinguishable outcomes of observations, and m is model state space dimension.
Risksensitive Generalizations of Minimum Variance Estimation and Control
, 1997
"... We define here risksensitive filtering as minimising the expected value of the exponential of an estimation error (quadratic) cost scaled by a risksensitive parameter. Such filtering is a generalization of standard riskneutral filtering in that as the risksensitive parameter approaches zero, ris ..."
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Cited by 13 (6 self)
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We define here risksensitive filtering as minimising the expected value of the exponential of an estimation error (quadratic) cost scaled by a risksensitive parameter. Such filtering is a generalization of standard riskneutral filtering in that as the risksensitive parameter approaches zero, riskneutral (minimum error variance) filtering is achieved. Also taking small noise limits, a differential game associated with H1 filtering results. In this paper, the risksensitive nonlinear stochastic filtering problem is studied in both continuous and discretetime for quite general finitedimensional signal models, including also discrete state hidden Markov models (HMMs). The risksensitive estimates are expressed in terms of the socalled information state of the model given by the Zakai equation which is linear. In the linear Gaussian signal model case, the risksensitive (minimum exponential variance) estimates are identical to the minimum variance Kalman filter state estimates, and ...
Norm observable operator models
, 2007
"... Hidden Markov models (HMMs) are one of the most popular and successful statistical models for time series. Observable operator models (OOMs) are generalizations of HMMs which exhibit several attractive advantages. In particular, a variety of highly efficient, constructive and asymptotically correc ..."
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Cited by 3 (1 self)
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Hidden Markov models (HMMs) are one of the most popular and successful statistical models for time series. Observable operator models (OOMs) are generalizations of HMMs which exhibit several attractive advantages. In particular, a variety of highly efficient, constructive and asymptotically correct learning algorithms are available for OOMs. However, the OOM theory suffers from the negative probability problem (NPP): a given, learnt OOM may sometimes predict negative “probabilities ” for certain events. It was recently shown that it is undecidable whether a given OOM will eventually produce such negative values. We propose a novel variant of OOMs, called norm observable operator models (NOOMs), which avoid the NPP by design. Like OOMs, NOOMs use a set of linear operators to update system states. But differing from OOMs, they represent probabilities by the square of the norm of system states, thus precluding negative “probability ” values. While being free of the NPP, NOOMs retain most advantages of OOMs. For example, NOOMs also capture (some) processes that cannot be modelled by HMMs. More importantly, in principle NOOMs can be learnt from data in a constructive way; and the learnt models are asymptotically correct. We also prove that NOOMs capture all Markov chain (MC) describable processes. This contribution presents the mathematical foundations of NOOMs, discusses the expressiveness of the model class, and explains how a NOOM can be estimated from data constructively. 1
The error controlling algorithm for learning OOMs
 International University Bremen
, 2007
"... Observable operator models (OOMs) are matrix models for describing stochastic processes. In this report we proposed a novel algorithm for learning OOMs from empirical data. Like other learning algorithms of OOMs, the presented algorithm is based upon the learning equation, a linear equation on obser ..."
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Cited by 1 (0 self)
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Observable operator models (OOMs) are matrix models for describing stochastic processes. In this report we proposed a novel algorithm for learning OOMs from empirical data. Like other learning algorithms of OOMs, the presented algorithm is based upon the learning equation, a linear equation on observable operators involving two adjustable matrices P and Q. The algorithm designs P,Q so that an upper bound of the error between the estimated model and the “real ” model is minimized. So we call it the error controlling (EC) algorithm. One important feature of the EC algorithm is that it can be used online, where it updates the estimation of observable operators on each new observation via a set of RLSlike formulas 1. By the linearity of the learning equation, we are able to prove the asymptotic correctness of the online learning procedure. The main numerical problem of the EC algorithm is to control the condition of the learning equation online, leading to a special optimization problem. For this special problem we proposed an efficient method that searches for the global optimum alternatively on two orthogonal directions and call it the orthogonal
RISK SENSITIVE GENERALIZATION OF MINIMUM VARIANCE ESTIMATION AND CONTROL *
"... Abstract. In this paper, the risksensitive nonlinear stochastic filtering problem is addressed in both continuous and discretetime for quite general finitedimensional signal models, including also discrete state hidden Markov models (HMMs). The risk sensitive estimates are expressed in terms of t ..."
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Abstract. In this paper, the risksensitive nonlinear stochastic filtering problem is addressed in both continuous and discretetime for quite general finitedimensional signal models, including also discrete state hidden Markov models (HMMs). The risk sensitive estimates are expressed in terms of the socalled information state of the model given by the Zakai equation which is linear. In the linear Gaussian signal model case, the risksensitive (minimum exponential variance) estimates are identical to the minimum variance Kalman filter state estimates, and are thus given by a finite dimensional estimator. The estimates are also finite dimensional for discretestate HMMs, but otherwise, in general, are infinite dimensional. In the small noise limit, these estimates (including the minimum variance estimates) have an interpretation in terms of a worst case deterministic noise estimation problem given from a differential game. The related control task, that is the risksensitive generalization of minimumvariance control is studied for the discretetime models. This is motivated by the need for robustness in the widely used (risk neutral) minimum variance control, including adaptive control, of systems which are minimum phase, that is having stable inverses.
EQUIVALENT REPRESENTATIONS OF HIDDEN MARKOV MODELS
"... Abstract. In this article, we classify the class of hidden Markov models through the laws of the observation processes, since the Markov chains are not observable. Here, we also present some properties regarding this classification. ..."
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Abstract. In this article, we classify the class of hidden Markov models through the laws of the observation processes, since the Markov chains are not observable. Here, we also present some properties regarding this classification.