Fokker-Planck Terms of Two-Dimensional Time Series

correlations. Real and surrogate cross-correlations show a clear difference in that there is significant synchrony at a phase shift of +3 s and at3 s. In addition to the default procedure of SUSY, which computes synchrony on the basis of absolute cross-correlation values, we additionally computed synchrony without taking the absolute values (Fig.9.27right panel). The effect size without taking the absolutes was very high, ESsy¼7.44. We may thus assume that respiratory coupling was present in this therapy session. The time series of patient and client were significantly coupled with a phase shift of about 3 s—the client was leading by 3 s in this session.

Two cross-correlation series have resulted from the application of SUSY: Z_real andZsurr—Fisher’sZ-transformed cross-correlations of original (“real”) and surro- gate (“surr”) movement time series, ordered by increasing lagsL(cf. Sect.9.4). In the present case, the cross-correlations span a window of 10 s, which yields, due to the sampling rate of 10 Hz, series of n ¼ 101 correlations for Z_real and Z_surr, respectively. All values are positive because SUSY transforms all correlations to positiveZvalues.

We estimated the functionsK(x) that describe the deterministic part of the Fokker- Planck equation ofZ_realandZ_surr. Here the values of the“state”variablexare given by correlation values, ranging in this example from 0.15 to 0.45. Bucket size was chosen to span 0.01. Figure9.28shows the potential functions of both the real cross- correlations and the surrogate cross-correlations.

Figure9.28illustrates that there is a clear deterministic signature in the cross- correlationsZ_real, likely indicating two attractors, the stronger one at a correlation value of approximately 0.21, and a smaller attractor at 0.32. This signature is clearly distinguishable from the deterministic signature ofZ_surrwhere no signs of attracting behavior are visible in the potential function. Determinism inZsurris of course not to be expected, asZsurrstems from shuffled surrogates that were generated as random controls by SUSY. One may note that the potential minimum of the“small”attractor ofZ_realis not significantly different from the surrogate values. This supplements the results shown in Fig.9.10.

Fig. 9.28 Movement coupling: Potential functions, i.e., deterministic signatures of the cross- correlations of a 5-min interaction. Original correlations (Z_real, green graph) and surrogate correla- tions (Z_surr, blue graph).Z_realsuggests the presence of two attractors

The stochastic functionsQ(x) of theZ_realandZ_surrseries are shown in Fig.9.29.

The normalized value for Z_real was computed by the mean of all standard errors divided by the sampling rate, it isQ_norm¼21.4, and for theZ_surr:Q_norm¼1.73. Thus stochasticity in the surrogate cross-correlations is lower, by about one order of magnitude, than in the real cross-correlations—this appears counter-intuitive but is explained by the generation of surrogate controls in SUSY by averaging over many runs of segment-wise shuffled time series (cf. Sect.9.4). The highest values ofQ(x) of real cross-correlations are observed between 0.25 and 0.3, which is again the region of state space where the separatrix between the two attractor basins of Fig.9.28is approximately located.

9.6.2 Example 4*: Two Simultaneously Monitored Electrocardiograms

The second two-dimensional dataset consisted of simultaneously monitored heart activity of a female therapist and her female client (cf. Sect.9.5.4). The sampling rate was 80 Hz, the units ofXare microvolts, and 2 min at the beginning of a therapy session are covered (see Fig.9.17for an illustration of the time series). The cross- correlation series were generated by SUSY, where Z_real denotes the cross- correlations of the actual cardiac time series andZ_surr the mean cross-correlations of the surrogates. Both series consist of 801 values (10 s sampled at 80 Hz plus one Fig. 9.29 Movement coupling: Stochastic signatures of the cross-correlations of 5 min of interac- tion. Ordinate: Standard errors (SE) of original correlations (SE_real, blue graph) and of surrogate correlations (SE_surr, red graph)

146 9 Modeling Empirical Time Series

correlation at lagL ¼0). Z_realranges between values 0.007 and 0.045, and Z_surr between 0.01 and 0.034. These numbers already suggest that the coupling of the cardiac time series is likely nonsignificant.

We estimated the functionsK(x) that describe the deterministic part of the Fokker- Planck equation ofZ_realandZ_surr(Fig.9.30). The stochastic functions are shown in Fig.9.31.

The potential functions are located in an area of very small cross-correlations, indicating that no coupling was present. The shape of the functions appears parab- oloid, which indicates the presence of Gaussian noise, a premise of the Fokker- Planck approach. The stochastic functions are shown in Fig.9.31. The normalized value forZ_real, computed by the mean of all standard errors divided by the sampling rate, isQ_norm¼10.67 andQ_norm¼4.83 for theZ_surr.

9.6.3 Example 6*: Two Simultaneously Monitored Respiration Time Series

The third two-dimensional dataset consisted of simultaneously monitored respiration of client and therapist (see example 6 in Sect.9.5.6). The potential function of the cross-correlations of the real data point to attractor-like stability at cross-correlation Fig. 9.30 Cardiac coupling: Potential functions, i.e., deterministic signatures of the cross- correlations of 2 min of interaction, with ranges of abscissa and ordinate chosen as in Fig.9.28to facilitate comparison with the amount of coupling of example 1*. Original correlations (Z_real, green graph) and surrogate correlations (Z_surr, red graph)

values of approximately 0.1, whereas the shuffled surrogate data produce no clear signature of an attractor (Fig.9.32). The potential values of the real data are much lower than those of the surrogate data, demonstrating that deterministic dynamics in Fig. 9.31 Cardiac coupling: Stochastic signatures of the cross-correlations of 2 min of interaction.

Ordinate: Standard errors of original correlations (SE_real, blue graph) and of surrogate correlations (SE_surr, red graph)

Fig. 9.32 Respiratory coupling: Potential functions, i.e., deterministic signatures of the cross- correlations of 50 min of therapy. Left: With ranges of abscissa, yet not ordinate, chosen as in Fig.9.28to facilitate comparison with the amounts of coupling in examples 1* and 4*. Right: Same data with adapted range of abscissa, for better legibility. Original correlations (Z_real, green graph) and surrogate correlations (Z_surr, red graph)

148 9 Modeling Empirical Time Series

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.