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Kalman Filtering and Neural Networks P6

LEARNING NONLINEAR DYNAMICAL SYSTEMS USING THE EXPECTATION– MAXIMIZATION ALGORITHM Sam Roweis and Zoubin Ghahramani Gatsby Computational Neuroscience Unit, University College London, London U.K. (zoubin@gatsby.ucl.ac.uk) 6.1 LEARNING STOCHASTIC NONLINEAR DYNAMICS Since the advent of cybernetics, dynamical systems have been an important modeling tool in fields ranging from engineering to the physical and social sciences. Most realistic dynamical systems models have two essential features. First, they are stochastic – the observed outputs are a noisy function of the inputs, and the dynamics itself may be driven by some unobserved noise process. Second, they can be characterized by Kalman Filtering and Neural Networks, Edited by Simon Haykin. | Kalman Filtering and Neural Networks Edited by Simon Haykin Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-36998-5 Hardback 0-471-22154-6 Electronic 6 LEARNING NONLINEAR DYNAMICAL SYSTEMS USING THE EXPECTATIONMAXIMIZATION ALGORITHM Sam Roweis and Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London London U.K. zoubin@gatsby.ucl.ac.uk 6.1 LEARNING STOCHASTIC NONLINEAR DYNAMICS Since the advent of cybernetics dynamical systems have been an important modeling tool in fields ranging from engineering to the physical and social sciences. Most realistic dynamical systems models have two essential features. First they are stochastic - the observed outputs are a noisy function of the inputs and the dynamics itself may be driven by some unobserved noise process. Second they can be characterized by 175 176 6 LEARNING NONLINEAR DYNAMICAL SYSTEMS USING EM some finite-dimensional internal state that while not directly observable summarizes at any time all information about the past behavior of the process relevant to predicting its future evolution. From a modeling standpoint stochasticity is essential to allow a model with a few fixed parameters to generate a rich variety of time-series outputs.1 Explicitly modeling the internal state makes it possible to decouple the internal dynamics from the observation process. For example to model a sequence of video images of a balloon floating in the wind it would be computationally very costly to directly predict the array of camera pixel intensities from a sequence of arrays of previous pixel intensities. It seems much more sensible to attempt to infer the true state of the balloon its position velocity and orientation and decouple the process that governs the balloon dynamics from the observation process that maps the actual balloon state to an array of measured pixel intensities. Often we are able to write down equations governing these dynamical systems directly based on prior knowledge of the problem