## Algorithms: computation of synchrony and attractors |
9.11.2023/Tschacher |

SUSY is now also available as an R package

SUSY computes

**mvSUSY (multivariate Surrogate Synchrony)**

mvSUSY is available as an R package **on CRAN**

mvSUSY computes **synchrony** based
on multi-dimensional time series, estimating the synchrony within a dataset that contains more than two time series. mvSUSY works
as described in Meier & Tschacher (2021). Two methods are available to assess mvSUSY: "lambda_max" and "omega". lambda_max is computed by the eigendecomposition of the correlation matrix. The correlation matrix of the m time series of the system can be described by m eigenvalues lambda, the largest of which provides an assessment of multivariate synchrony, i.e. the coupling between the time series. lambda is computed in each segment, then aggregated across all segments. omega is a measure of multivariate synchrony that makes use of the actually measured degree of entropy, a measure of disorder of a system, equivalent to Shannon information. Landsberg suggested to normalize entropy S by the potential entropy Spot possible in a system, providing the measure of omega ("Landsberg order") as omega = 1–S/Spot. The entropies can be computed based on the variance-covariance matrix of the multiple time series (Shiner, Davison & Landsberg, 1999). Again, omega is computed in each segment then aggregated. Both lambda_max and omega are tested against surrogate control conditions using segment-shuffling. Tschacher, Scheier & Grawe (1998) applied these methods in psychotherapy research.

**SUCO (Surrogate Concordance)**

SUCO 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 i of times series A
are correlated with those in the segment i of B. The procedure is repeated until
all segments of A and B are covered. Each segment is thus characterized by a
correlation value. All correlations are transformed to Fisher's Z and the mean
Z of the two time series is computed. The Concordance Index (CI) of the time series 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, SUCO applies computations on all adjacent
pairs of columns. SUCO 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), SUCO extends this by surrogate tests. The SUCO algorithm was coded by David Leander
Tschacher instructed by Wolfgang Tschacher, partly following Marci & Orr
(2006). For applications see Tschacher & Meier (2020).

**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). The application to psychology
has been announced by Tschacher, Haken & Kyselo (2015) and is elaborated
and discussed in the book Tschacher & Haken, 2019. Examples are provided by Tschacher & Haken (2020). The FPE algorithm on
this website estimates, on the basis of a one-dimensional 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 (bin) of *x*.
Thus, the underlying deterministic attractor landscape of fixed-point attractors
of a time series can be approximated. The FPE algorithm was coded by Nikolai Philipp
Tschacher instructed by Wolfgang Tschacher, following ideas of Hermann Haken. FPE-Description.pdf

*password: please contact wolfgang.tschacher (at) unibe.ch*

*acknowledging use of SUSY, SUCO, or FPE: please cite Tschacher & Haken
(2019)
*_______________________________________

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.

Meier D & Tschacher W (2021). Beyond dyadic coupling: The method of multivariate Surrogate Synchrony (mv-SUSY). Entropy, 23, 1385. doi: 10.3390/e23111385 (pdf)

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)

Shiner JS, Davison M, & Landsberg PT (1999). On measures for order and its relation to complexity. In Tschacher W & Dauwalder J-P (Eds). Dynamics, Synergetics, Autonomous Agents. Singapore: World Scientific, pp. 49-63.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 (2019). The Process of Psychotherapy – Causation and Chance. Cham: Springer. doi: 10.1007/978-3-030-12748-0 (chapter abstracts)

Tschacher W & Meier D (2020). Physiological synchrony in psychotherapy sessions. Psychotherapy Research, 30, 558-573. doi:10.1080/10503307.2019.1612114

Tschacher W & Haken H (2020). Causation and chance: Detection of deterministic and stochastic ingredients in psychotherapy processes. Psychotherapy Research, 30, 1075-1087. doi:10.1080/10503307.2019.1685139

Tschacher W, Scheier C, & Grawe K (1998). Order and pattern formation in psychotherapy. Nonlinear Dynamics, Psychology and Life Sciences, 2, 195-215.

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