ABSTRACT
Nonlinear dynamics is a capsule term referring to a variety of fields of research, incIuding nonequilibrium thermodynamics (Nicolis & Prigogine, 1977), catastrophe theory (Thom, 1975), synergism (Haken, 1983), soliton theory (Toda, 1989), artificial neural network modeling (Grossberg, 1982), and chaos theory (e.g., Ütt, 1993). These fields of research have provided us with profound new insights in the self-organizing capabilities and emergent pattern formation of temporal and spatiotemporal processes. Many concepts and techniques of nonlinear dynamics have found their way to the social sciences and have led to novel ways of conceptualizing the temporal organization of behavior. Among the many innovative applications of nonlinear dynamics in psychology we can discern a number of dedicated attempts to construct explicit causal models of time-or age-dependent processes. Examples are the predator-prey model of human cerebral development of Thatcher (chapter 5, this volume), the neural network model relating cortical and cognitive development proposed by Been (chapter 8, this volume), the catastrophe model of stagewise cognitive development of Van der Maas and Molenaar (1992; Van der Maas, chapter 7, this volume), the nonlinear growth models of Van Geert (1991; chapter 6, this volume), and the model of phase transitions in human hand movements of Haken, Kelso, and Bunz (1985). In each of these applications a nonlinear dynamical model has been specified that can explain the self-organizing tendencies of a particular psychological or developmental process. In this chapter the
The Linear State-Space Model
y(t) = HcX(t) + v(t) (3)
where y(t) is a manifest p-variate process and Hc is a (p,q)-dimensional matrix with possibly time-varying elements. It is assumed that v(t) is a p-variate zero mean Gaussian white noise process:
E[v(t),v(t + u)'] = ö(u)V u = 0, ±1, ... (4)
The Kaiman Filter
The Extended Kaiman Filter (EKF)
where x(t + 110 = ft+![x(tl 0]. Notice that 8At+! and 8Ht+! have the same dimensions as At+! and Ht+! in Equations 1 and 3, respectively. Substitution of 8At+! for At+! and 8Ht+! for Ht+! in the expressions for K(t + 110, X(t + 110 and X(t + 11 t + 1) of the KF given by Equation 6 then defines the analogous expressions of the EKF. The remaining expression for x(t + 11 t + 1) in the EKF reads
Implementation of the EKF
Compared with a nonrecursive batch implementation in whieh a complete (historie) time series y(t), t = 1, ... , T is used at once, a recursive filtering implementation of a given estimator is always suboptimal. This is because at each time t < T the state estimate obtained by filtering only is based on the observed series up to time t, whereas the analogous batch estimate is based on the complete observed series. To get a recursive implementation that makes use of the same information as its batch analogue, it should yield state estimates xCtl T) for all times t = 1, ... , T. Such a recursive estimator is called a smoother. Heuristieally speaking, a smoother consists of a regular filtering of an observed time series up to
time T, followed by a filtering backward in time from time T to time 1. In fact, this is only one version of smoothing. Far alternative types of smoothers the reader is referred to, far example, Sage and Melsa (1971, sections 8.3 and 9.5).
