The identification of critical fluctuations and phase transitions in short term and coarse-grained time series—a method for the real-time monitoring of human change processes

Abstract

We introduce two complementary measures for the identification of critical instabilities and fluctuations in natural time series: the degree of fluctuations F and the distribution parameter D. Both are valid measures even of short and coarse-grained data sets, as demonstrated by artificial data from the logistic map (Feigenbaum-Scenario). A comparison is made with the application of the positive Lyapunov exponent to time series and another recently developed complexity measure-the Permutation Entropy. The results justify the application of the measures within computer-based real-time monitoring systems of human change processes. Results from process-outcome research in psychotherapy and functional neuroimaging of psychotherapy processes are provided as examples for the practical and scientific applications of the proposed measures.