An Unscented Kalman Filter Approach to the Estimation of Nonlinear Dynamical Systems Models

Abstract

In the past several decades, methodologies used to estimate nonlinear relationships among latent variables have been developed almost exclusively to fit cross-sectional models. We present a relatively new estimation approach, the unscented Kalman filter (UKF), and illustrate its potential as a tool for fitting nonlinear dynamic models in two ways: (1) as a building block for approximating the log–likelihood of nonlinear state–space models and (2) to fit time-varying dynamic models wherein parameters are represented and estimated online as other latent variables. Furthermore, the substantive utility of the UKF is demonstrated using simulated examples of (1) the classical predator-prey model with time series and multiple–subject data, (2) the chaotic Lorenz system and (3) an empirical example of dyadic interaction.

ACKNOWLEDGMENTS

We thank Jack McArdle, Ellen Bass, Howard Epstein, Fumiaki Hamagami and a few anonymous reviewers for their valuable comments on earlier versions of this article. This study was supported by a National Institute on Aging grant 5 R01 AG18330 awarded to John R. Nesselroade while the first author was completing her Ph.D. at the University of Virginia. Support from the Humboldt Foundation and the Max Planck Institute for Human Development is also gratefully acknowledged. Matlab codes used for model fitting can be downloaded from Sy-Miin Chow's website at http://www.nd.edu/∼schow. Correspondence concerning this article should be addressed to Sy-Miin Chow, Department of Psychology, 108 Haggar Hall, University of Notre Dame, Notre Dame, IN 46556. Electronic mail may be sent to [email protected].

Notes

¹Closely related but not considered here are Kalman smoothers that are used to make predictions by incorporating information from all time points (Dolan & Molenaar, 1991; Harvey, 1989; Oud, Jansen, van Leeuwe, Aarnoutse, & Voeten, 1999; Shumway & Stoffer, 2004).

²Although less prevalent, maximum likelihood–based methods and their sequential counterparts have also been considered in various engineering applications (e.g., Mehra, 1972).

³Note that the person and time indices in p _it are used to indicate that incomplete observations at the individual level can be accomodated flexibly in the context of the prediction error decomposition function. The dimension for P _{y t} , however, should remain the same across all individuals and time points.

⁴The subject index in F _it and H _it is used to indicate that the associated Jacobian matrices have different numerical values because they are evaluated at each subject's respective current state estimates, not that the dynamic or measurement functions are subject–dependent.

⁵Note that half of the sigma points are placed to the left of the previous estimate η _t−1|t−1, and the other half to its right. Thus, the sigma–point index k goes from 1 … w, and w+1 … 2w.

^aEstimation procedure took 7.52 minutes of CPU time on a 2 GHz IBM laptop with 1 GB RAM.

^bEstimation procedure took 14.54 hours of CPU time on a 2 GHz IBM laptop with 1 GB RAM.

⁶This is the property that slight differences in initial conditions could lead to cascading divergence in later dynamics.

⁷Chaotic systems are known to be nonlinear dynamical systems that are deterministic and yet manifest seemingly random behaviors.

⁸Alternatively, a separate UKF chain can be run to estimate the parameters, again by treating them as state variables. This is the dual Kalman filter approach mentioned earlier.

^aEstimation procedure took 8.97 minutes of CPU time on a 2 GHz IBM laptop with 1 GB RAM.