Fig 1.
Exemplary time series with pattern transitions that were produced by a model of psychotherapeutic change.
The simulated time series were used to optimize and validate the PTDA. Some transitions are characterized by a change of the mean, some by a change of the variance, and others by a change of different features in the domain of frequency or complexity. X-axis: 100 iterations of the simulation runs, corresponding to the segment between iteration 100 and 200 as shown in Fig 2A, lower part. Y-axis: Intensities of the simulated dynamics. The range of the variables produced by the theoretical computational model is [-1, +1] for E, P, S and [0, +1] for I and M.
Fig 2.
Generation of the time series.
A) Simulations with 400 iterations (data points) were generated. The first 100 data points were discarded to allow for saturation effects (transient period). A transition point (dashed line) was launched at T = 250 in the original time series, corresponding to T = 150 in the adjusted time series. Depending on where the transition point should be, the corresponding interval of 100 data points (brackets) was chosen and assessed with the algorithm. The upper bracket shows the interval where the transition point is located at the middle of the time series, the bracket in the middle the interval where the transition is at the end, and the lower bracket the interval without a transition point (to assess the rate of false positive results). B) The effects of prolonged periods of changes were investigated by linearly increasing the control parameters over several time points (brackets).
Fig 3.
From top to bottom: original time series, Dynamic Complexity, Recurrence Plot, Time Frequency Distribution. The left column shows an example of a time series where all methods find a transition at a converging point (T = 52). The blue circles show points where a change of the mean was detected, the green circles where a change of the variance was detected, and the red circles where a change of the linear trend was detected by the CPA (in this example, the points are identical; note that the vertical placement of the points is irrelevant and adjusted for visual recognition only). The right column shows an example where no change point was found by the CPA (i.e., applied to the original time series), but where the other methods convergently found a transition point at T = 53. X-axis: Time series length (see Fig 2A for explanation). Y-axis: From top to down: Value range of the simulated time series; values of the Dynamic Complexity measure; rows of the Recurrence Plot (time x time); frequency spectrum of the Time Frequency Distribution.
Fig 4.
Flow chart of the Pattern Transition Detection Algorithm (PTDA).
CP: change point; CPA: change point analysis.
Fig 5.
A) Different positions of the simulated transition point (TP) at T = 20 (left), T = 50 (middle) and T = 80 (right). B) Different levels of noise were added to the original time series in order to investigate the effect on the PTDA. Left: Original time series without noise (deterministic). Middle: original time series with noise with a variance of +50% of the variance of the original time series. Right: original time series with noise with a variance of +90% of the variance of the original time series. C) Prolonged transitions. The upper rows show the time series resulting from the parameter increases (lower row). Left: original time series with a sudden pattern transition induced by an instantaneous increase of the control parameter from one to the next iteration. Middle and right: Time series and linear parameter increases in intervals of 10, 20, and 30 iterations (time points). X-axis: time series length. Y-axis: Rage of the time series values. C lower row: drift of parameter values.
Table 1.
Comparison of the PTDA when (A) the 5 five time series per simulation are assessed simultaneously, i.e., the whole system is taken into consideration, when (B) the time series are assessed independently, and (C) when the common CPA algorithm (with respect to the change of the mean) is applied to the single time series.
Based on the 300 simulated time series from 60 simulation runs with an induced change at T = 50, the table shows the mean change point of the algorithms, the standard deviation (SD), the precision, the false negative rate, and the false positive rate. Both algorithms are able to correctly detect the TP in most cases, but the percentages of false negative and false positive findings are much lower in the PTDA.
Table 2.
Results of the performance of the PTDA under three real-world conditions.
Noise: different levels of noise were added to the original time series. The numbers in the second column denote the strength of noise, e.g., 50 refers to added noise with a variance of 50% of the original time series. Position: The transition point was shifted along the time series. The numbers in the second column denote where the change was induced, e.g., 50 refers to a transition point at 50% (the middle) of the original time series. Increase: The transition was not induced abruptly but over a longer period of time. The numbers in the second column denote the range of the time series where the change occurred, e.g., 20 denotes that the change stretched over 20% of the time series (see also Fig 5).
Table 3.
Results of the PTDA for time series (TS) of 20 to 100 time points.
The mean (M) and the standard deviation (SD) of the transition points (TP) of the 300 simulated time series are given along with further indicators of the performance: the precision (Prec.), which gives the rate of identifying a TP within +/- 5 time points around the real TP, and the rates of false negative (FN) and false positive (FP) results.
Fig 6.
PTDA applied to four different psychological time series.
A) Time Series of the depression subscore of the SCL-90 from a open access dataset (see Methods). B) Time series of the “child-state” of a patient with dissociative identity disorder during the psychotherapy process (Schiepek et al., 2016). C and D) Time series of the psychotherapeutic process. The change point is marked by the vertical black line. The bar below gives an impression of the convergence of the CPs over all methods. X-axis: Measurement points (daily assessments): Y-axis: Intensities of the values (A, B, C); z-transformed values of the factor “motivation for change” (D).
Fig 7.
Evolution of DET (Determinism) and RTE (Recurrence Time Entropy) applied to a simulated time series from our validation set.
(A) The time series. (B) Evolution of DET depending on the width of the sliding window (left: 25, middle: 45, right: 65). (C) Evolution of the RET depending on the width of the sliding window (left: 25, middle: 45, right: 65).