Learning Nonlinear Dynamical Systems Using an EM Algorithm
Authors: Zoubin Ghahramani, Sam T. Roweis
Published in: Advances in Neural Information Processing Systems 11  (NIPS 1998) (Main Conference Track), 1998

Abstract: The Expectation-Maximization (EM) algorithm is an iterative pro(cid:173) cedure  for  maximum  likelihood  parameter  estimation  from  data  sets  with  missing  or  hidden  variables  [2].  It has  been  applied  to  system identification in linear stochastic state-space models,  where  the state variables are hidden from the observer and both the state  and  the  parameters  of  the  model  have  to  be  estimated  simulta(cid:173) neously  [9].  We  present  a  generalization  of the  EM  algorithm for  parameter estimation in  nonlinear dynamical systems.  The "expec(cid:173) tation" step makes use of Extended Kalman Smoothing to estimate  the state, while the  "maximization"  step re-estimates the parame(cid:173) ters using these uncertain state estimates. In general, the nonlinear  maximization step is difficult because it requires integrating out the  uncertainty  in  the states.  However,  if Gaussian  radial  basis  func(cid:173) tion  (RBF)  approximators  are  used  to  model  the  nonlinearities,  the integrals  become  tractable  and the  maximization step  can  be  solved  via systems of linear equations. 
1  Stochastic  Nonlinear Dynamical Systems 
We  examine inference and learning in discrete-time dynamical systems with hidden  state  Xt,  inputs  Ut,  and  outputs  Yt. 1  The  state  evolves  according  to  stationary  nonlinear dynamics driven by the inputs  and by additive noise 
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1 All  lowercase  characters  (except indices)  denote  vectors.  Matrices  are represented by 
uppercase characters. 
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Z.  Ghahramani and S.  T  Roweis 
where  w  is  zero-mean  Gaussian noise  with  covariance Q.  2  The outputs  are  non(cid:173) linearly related to the states and inputs by 
Yt  =  g(Xt, Ut)  + v 
(2) 
where  v  is  zero-mean Gaussian noise with  covariance R.  The vector-valued non lin(cid:173) earities f  and 9  are assumed to be  differentiable,  but otherwise arbitrary.  Models  of this  kind  have been examined for  decades  in  various communities.  Most  notably,  nonlinear state-space models form  one of the cornerstones of modern sys(cid:173) tems  and  control  engineering.  In this  paper,  we  examine these  models  within  the  framework  of probabilistic graphical models  and derive  a  novel  learning algorithm  for  them  based  on  EM.  With one exception,3  this  is  to  the  best  of our  knowledge  the first  paper addressing learning of stochastic nonlinear dynamical systems of the  kind  we  have described  within  the framework of the EM algorithm.  The classical approach to system identification treats the parameters as hidden vari(cid:173) ables, and applies the Extended Kalman Filtering algorithm (described in section 2)  to  the  nonlinear  system  with  the  state  vector  augmented  by  the  parameters  [5]. 4  This approach is  inherently on-line, which may be important in certain applications.  Furthermore,  it  provides  an estimate of the  covariance  of the  parameters  at  each  time step.  In contrast, the EM algorithm we  present is  a  batch algorithm and does  not  attempt to estimate the covariance of the parameters. 
There are three important advantages the  EM algorithm has  over the classical ap(cid:173) proach.  First, the EM algorithm provides a straightforward and  principled method  for  handing  missing  inputs  or  outputs.  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.  Third,  whereas  it  is  often  very  difficult  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. 
In the next sections we will describe the basic components of the learning algorithm.  For the expectation step of the algorithm, we infer the conditional distribution of the  hidden states using Extended Kalman Smoothing (section 2).  For the maximization  step  we  first  discuss  the  general  case  (section  3)  and  then  describe  the  particular  case  where  the  nonlinearities  are  represented  using  Gaussian  radial  basis  function  (RBF;  [6])  networks  (section 4). 
2  Extended Kalman  Smoothing 
Given  a  system  described  by  equations  (1)  and  (2),  we  need  to  infer  the  hidden  states from  a  history  of observed  inputs  and  outputs.  The  quantity  at the  heart  of this  inference problem is  the conditional density P(XtIUl,""  UT, Yl,.' "  YT),  for  1 ::;  t  ::;  T,  which  captures the fact  that the system is  stochastic and therefore our  inferences about  x  will  be uncertain. 
2The Gaussian noise assumption is  less restrictive for  nonlinear systems than for  linear 
systems since the nonlinearity can  be  used  to  generate  non-Gaussian  state noise. 
3The authors have just become aware that Briegel and Tresp (this volume) have applied  EM to essentially the same model.  Briegel  and Tresp's method uses multilayer perceptrons  (MLP) to approximate the nonlinearities,  and requires sampling from the hidden states to  fit  the  MLP.  We  use  Gaussian  radial  basis functions  (RBFs)  to  model  the  nonlinearities,  which  can be fit  analytically without sampling  (see section  4) . 
41t  is  important not  to  confuse  this use  of the  Extended Kalman  algorithm,  to simul(cid:173)
taneously  estimate  parameters  and  hidden  states,  with  our use  of EKS,  to  estimate just  the hidden state as  part of the E  step of EM. 
Learning Nonlinear Dynamics Using EM 
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For linear dynamical systems with  Gaussian state evolution and observation noises,  this conditional  density  is  Gaussian and  the recursive algorithm for  computing its  mean  and  covariance is  known  as  Kalman  smoothing [4,  8].  Kalman  smoothing is  directly analogous to the forward-backward algorithm for computing the conditional  hidden  state  distribution  in  a  hidden  Markov  model,  and  is  also  a  special  case  of  the belief propagation algorithm. 5  For  nonlinear systems  this  conditional  density  is  in  general  non-Gaussian and  can  in  fact  be quite  complex.  Multiple  approaches  exist  for  inferring the  hidden state  distribution  of such  nonlinear  systems,  including  sampling  methods  [7]  and  varia(cid:173) tional approximations [3].  We focus  instead in this paper on a classic approach from  engineering,  Extended Kalman  Smoothing  (EKS). 
Extended Kalman Smoothing simply applies Kalman smoothing to a local lineariza(cid:173) tion  of the  nonlinear  system.  At  every  point  x in  x-space,  the  derivatives  of  the  vector-valued functions f  and 9 define the matrices, Ax ==  M I x=x  and ex ==  ~ I x=x'  respectively.  The dynamics  are linearized  about  Xt,  the mean of the Kalman filter  state estimate at time t: 
The output equation  (2)  can be similarly linearized.  If the prior distribution of the  hidden state at t =  1 was  Gaussian, then, in this  linearized system, the conditional  distribution of the hidden state at any time t  given the history of inputs and outputs  will also be Gaussian.  Thus, Kalman smoothing can be used on the linearized system  to infer  this conditional distribution  (see  figure  1,  left  panel). 
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