Algorithms: computation of synchrony and attractors

18.4.2018/Tschacher


SUSY (Surrogate Synchrony)
SUSY computes synchrony as windowed cross-correlation based on time series in a text file you can upload. SUSY works as described in Tschacher, Rees & Ramseyer (2014): Cross-correlations are computed up to a specific lag in seconds <Maxlag>, then aggregated within a chosen <Segment> of e.g. 30s. Aggregation is performed by transforming correlations to Fisher's Z, then computing mean Z in each segment, then across all segments of the time series. Segment shuffling is used to create surrogate time series, on which the same computations are run. This provides effect sizes <ES>. <File>: The time series are the columns of the file, variable names can be in the header line. If <Automatic> is clicked, the synchrony is computed of all adjacent pairs of columns in the file. If <Automatic> is unclicked, you may choose the two columns to be analyzed for synchrony, and two plots are additionally prepared. SUSY provides two different synchrony measures of each twin time series: mean Z and ES of mean Z / mean absolute_Z and ES of mean absolute_Z. SUSY was validated and gives identical values as the SAS code previously used by Ramseyer & Tschacher (2011). Description.pdf

CI (Concordance Index)
CI computes synchrony defined as correlations of window-wise slopes. All slopes of timeseries in column A and B of a text file are determined in this manner: Define a <Window size> (e.g. 2s) and a <Segment size> (e.g. 10s). Then the slope (using mean squares) is computed inside the window, the window shifted by 1s and again the slope is computed, ..., until all windows in the segment are considered. The slopes in the segment n of times series A are correlated with those in the segment n of B. The procedure is repeated until all segments of A and B are covered. All correlations are transformed to Fisher's Z and the mean Z of the two time series is computed. The CI in a segment is defined by the natural logarithm of the sum of all positive correlations divided by the absolute value of the sum of all negative correlations. Segment shuffling is used to create surrogate time series, on which the same computations are run, as in SuSy. <File>: The time series are the columns of the file, variable names can be in the header line. If more than 2 columns are in the file, the CI is computed of all adjacent pairs of columns. CI provides three different synchrony measures of each twin time series: mean Z and ES of mean Z / mean absolute_Z and ES of mean absolute_Z / concordance index and ES of concordance index. Concordance index as suggested by Marci & Orr (2006). The CI algorithm was coded by David Leander Tschacher by order of Wolfgang Tschacher, partly after Marci & Orr (2006).

FPE (Fokker-Planck Equation)
The FPE is a stochastic differential equation that models a time series of some variable x by a stochastic (diffusion) and a deterministic (drift) term (Haken, 2004; Tschacher & Haken, forthcoming). The application to psychology has been announced by Tschacher, Haken & Kyselo (2015) and will be elaborated and discussed in the upcoming book. The FPE algorithm on this website estimates, on the basis of a 1-D time series, the deterministic forces and the stochasticity, for each x contained in the time series, or in the case of too many single values of the variable, for each bucket of x. Thus, the underlying deterministic attractor landscape of fixed-point attractors of a time series can be approximated.

password: please contact wolfgang (at) tschacher.ch
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Haken H (2004). Synergetics. Introduction and Advanced Topics. Berlin: Springer.

Marci CD & Orr SP (2006). The effect of emotional distance on psychophysiologic concordance and perceived empathy between patient and interviewer. Applied Psychophysiology and Biofeedback, 31, 115-128.

Ramseyer F & Tschacher W (2011). Nonverbal synchrony in psychotherapy: Coordinated body-movement reflects relationship quality and outcome. Journal of Consulting and Clinical Psychology, 79, 284-295. (pdf of free version)

Tschacher W, Rees GM & Ramseyer F (2014). Nonverbal synchrony and affect in dyadic interactions. Frontiers in Psychology, 5, 1323. doi: 10.3389/fpsyg.2014.01323 (pdf)

Tschacher W, Haken H & Kyselo M (2015). Alliance: A common factor of psychotherapy modeled by structural theory. Frontiers in Psychology, 6, 421. doi: 10.3389/fpsyg.2015.00421 (pdf)

Tschacher W & Haken H (forthcoming). The Process of Psychotherapy – Causation and Chance.

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