the real data are present, albeit at low levels of cross-correlations. Figure 9.33 displays the stochastic function of the respiratory cross-correlations. The function Q(x) in thisfigure may be interpreted to reflect the reduced stochasticfluctuations in the attractor at cross-correlation values of approximately 0.095 and a possible separatrix at approximately 0.105.
using the cross-correlation function and surrogate controls in the SUSY approach can distinguish significantly synchronized from mutually uninfluenced processes.
The hypothesized presence of coupling can be supported or rejected, and the phase shift of an interaction can be quantified, all based on naturalistic empirical time series. In the case of cardiac time series (example 4), thefindings suggested rejection of the hypothesis of coupled dynamics.
We are aware that further steps must follow. The functionsK(x) andQ(x) of a system may be normalized by the same factor, so that signatures of causation and chance can be meaningfully compared between time series with different properties (duration, sampling rate, amplitude). Some issues of the methodology are not yet finally clarified, such as how the stochasticity is best represented. We chose the standard error (not the standard deviation) to illustrate the stochasticity at each location of state space, because it is normalized by the number of observations at that location. This may however cause erraticfindings when there are few observa- tions in some regions of state space x and/or when highly varying numbers of observations are present per region of state space.
More empirical work also needs to be done in order to connect the SUSY approach and the concordance index (CI) approach to our theoretical treatment of oscillations in the elaboration of the minimal model (Info-Box8.5). We have shown that it is in principle possible to link the empirical cross-correlations to phase shiftsϑ and oscillation periods ω, and compute from these the coupling constants of the minimal model. We stated in the previous chapter that one can determineϑandωby a bestfit of the empirical cross-correlations as, e.g., shown in Fig.9.27. We found empirical phase shifts in examples 1 and 6 of the present chapter by visual inspection of the cross-correlation functions. Yet in naturalistic psychotherapy and social interaction, it is not straightforward to view the behavior as oscillatory when phenomenologically no regular oscillations are observable. Other than under labo- ratory conditions, where oscillations can be prescribed (e.g., Haken, Kelso, & Bunz, 1985), natural conversations have no regular rhythms apart from linguistic turn- takings. Therefore the integration of frequency-based models and correlation-based models in naturalistic data remains to be investigated.
We have emphasized our special interest in dense time series with very high sampling rates above one Hertz, thus“big data.”Having said this, it is also feasible to apply the developed tools to conventional time series. Psychotherapy research generates time series data when administrating questionnaires repeatedly (such as session reports after each therapy session, e.g., Ramseyer et al., 2014) or by ecological assessment (experience sampling) methods (Reisch et al.,2008). It is in principle possible to compute the functionsK(x) andQ(x) even in such more coarse- grained data. It is also worth considering the application of SUSY and CI when repeated self-report data are available from people interacting, e.g., therapist’s and client’s independent assessments of the therapeutic alliance in session reports.
The goal of the present chapter was to demonstrate that the mathematical ideas of the Fokker-Planck approach can be applied to real process data. The computational tools are thus available, and they allow for the estimation of causation and chance, in one-dimensional as well as two-dimensional systems. Especially, the algorithms
150 9 Modeling Empirical Time Series
point to the locations of interest in the respective state spaces of the processes. We have thus paved the way for more systematic empirical studies, e.g., datasets containing process data of samples of psychotherapy sessions measured in compa- rable contexts and under experimental conditions.
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Outlook
In this concluding chapter, we wish to suggest topics on which psychotherapy research should focus in more detail in the future. Thefirst three topics are direct implications and consequences of the Fokker–Planck approach that we have intro- duced throughout this book. The remaining topics, which cover embodiment, free energy, and affordances, are of a more general nature. In sum, this chapter offers a discussion of current research from our point of view and makes suggestions for new directions of research on psychotherapy.