Modeling Empirical Time Series

Abstract

The relevant parameters of the one-dimensional Fokker-Planck equation can be estimated from empirical time series of the state variable x. The deterministic forces that derive from attractors in state space and the production of stochasticity are represented by the functions K(x) and Q(x), respectively, which express how force k and diffusion Q vary with different values of state x. In empirical data, the functions can be derived from the (deterministic) slopes at each x and the (stochastic) standard error of the slopes at each x. From K(x) we compute the potential function V(x) by integration, and we can thus depict the attractor landscape that is inherent in the time series. In two-dimensional time series, which may represent both the therapist’s and client’s behavior, we are interested in the coupling (synchrony) of their behavior streams. We therefore compute the synchrony of therapist and client using the application SUSY (surrogate synchrony), which is based on windowed cross-correlation controlled by surrogate tests. An alternative application to estimate synchrony is the concordance index, which focuses on the correlations of window-wise slopes of therapist-client time series. Finally, we also compute V(x) of the cross-correlations to detect possible attractors. In this chapter, we conduct time series analyses of exemplary behavioral and physiological datasets sampled at high frequency (one-dimensional systems, body movement, respiration, electrocardiogram, simulated Markov process; two-dimensional systems, body movements, respiration, electrocardiograms of two persons in interaction). We find that the applications yield the deterministic and stochastic signatures of the empirical time series as well as the synchrony and entrainment of the two-dimensional data.