Classifying Complex Dynamical and Stochastic Systems via Physics-Based Recurrence Features

TL;DR

The paper tackles the challenge of classifying parameters in chaotic, continuous, and stochastic systems from time series. It introduces recurrence microstate probabilities and entropy-based thresholding to create a physics-informed, compact feature space, then evaluates a suite of ML classifiers on this representation. The key finding is that recurrence microstate features dramatically improve classification accuracy and reduce computation compared with raw data, with Random Forest and MLP frequently excelling (e.g., Lorenz with $N=4$ achieves 100% accuracy). This approach offers a practical, scalable means to extract dynamical signatures from time series and is applicable to real-world data domains like neuroscience and climatology. ${S(\varepsilon) = -\sum_{i=1}^{2^{N^{2}}} P_i(\varepsilon)\ln P_i(\varepsilon)}$ and similar recurrence-analytic constructs underpin the method's effectiveness, enabling robust discrimination of dynamical regimes in a compact feature space.

Abstract

In this study, we employ the recently developed recurrence microstate probabilities as features to improve accuracy of several well-established machine learning (ML) algorithms. These algorithms are applied to classify discrete and continuous dynamical systems, as well as colored noise. We demonstrate that the dynamical characteristics quantified by this method are effectively captured in the recurrence microstate space, a space defined solely by the recurrence properties of the signal. This space change reduces dimensions, which also reduces the necessary time to perform calculations and obtain relevant information about the underlying system. Here, we also demonstrate that a few optimal machine learning (ML) algorithms are particularly effective for classification when combined with recurrence microstates. Furthermore, these new machine learning vectors significantly reduce memory usage and computational complexity, outperforming the direct analysis of raw data.

Figures (7)

• Figure 1: Example of microstates embedded in a $(K\times K)$ RP. A short data sequence of size $N < K$ is translated into a recurrence microstate, a $(N\times N)$ matrix encoding recurrence relations of short sequences of the data. In the figure, the $2\times2$ microstates are highlighted in red. Three such microstates can be observed, with those along the diagonal corresponding to the same configuration.
• Figure 2: Confusion matrices for predictions by various machine learning models classifying parameters of selected dynamical maps. Each column corresponds to a specific classification model, while each row represents a dynamical map. The horizontal axis shows predicted labels, and the vertical axis shows true labels. The color scale indicates prediction frequencies, with yellow shades representing higher counts. The colorbar below provides the absolute counts for each cell in the matrices. Values along the main diagonal correspond to correct classifications, indicating the number of times each class was accurately predicted by the model.
• Figure 3: Mean Accuracy achieved by various machine learning algorithms for predicting each dynamic system. Results are displayed for N=2, 3 and 4, where $N$ represents the size of the recurrence microstates. Each bar represents the mean accuracy obtained by running all 20 ordered train–test permutations across five distinct generated data sets. Error bars indicate the standard deviation.
• Figure 4: Machine learning accuracy for classifying two distinct continuous and chaotic dynamical systems.
• Figure 5: Machine learning accuracy for classifying color noises using the raw data and using the microstates analysis for $N=2$, $N=3$ and $N=4$.