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

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.

💡 Research Summary

This paper introduces a novel framework for classifying complex dynamical and stochastic systems by leveraging “recurrence microstates” as physics-informed feature vectors for machine learning algorithms. The core innovation lies in transforming raw time series data into a lower-dimensional space defined solely by the recurrence properties of the system’s trajectory, thereby capturing essential dynamical characteristics in a compact and computationally efficient form.

The methodology begins with the concept of Recurrence Plots (RPs), which are binary matrices indicating when a system’s state revisits a neighborhood in phase space. The authors extend this idea to Recurrence Microstates, which are all possible N x N submatrices of an RP. Each microstate represents a specific local recurrence pattern of a short data sequence. For a given time series, the probability distribution of these microstates is computed, forming a feature vector whose dimension is at most 2^(N^2). A key scientific choice is the selection of the recurrence threshold (ε), which is not arbitrary but is determined by maximizing the Shannon entropy of the microstate distribution. This ensures the selected threshold yields the most informative and diverse set of patterns.

The study evaluates this approach on a wide array of systems: five discrete chaotic maps (βx mod 1, Logistic, Gauss, Hénon, Ikeda), two continuous chaotic flows (Lorenz, Rössler), and various types of colored noise (with a 1/f^α power spectrum). For each system, time series are generated across a range of control parameters. The task for the machine learning classifiers is to identify the correct parameter (or noise color) based solely on the recurrence microstate probability distribution of the time series.

Ten established machine learning algorithms—including Decision Tree, Random Forest, K-Nearest Neighbors (KNN), Support Vector Classifier (SVC), Gaussian Naive Bayes, Gradient Boosting, Multi-Layer Perceptron (MLP), and Logistic Regression—are trained and tested using these feature vectors. The results demonstrate that most algorithms, particularly ensemble methods like Random Forest and Gradient Boosting, achieve high classification accuracy (often above 90-95%) across the different systems when using recurrence microstate features.

A critical comparative analysis is presented where the same machine learning models are applied directly to the raw time series data without the recurrence microstate preprocessing. This baseline experiment shows significantly lower classification accuracy and substantially higher computational cost and memory usage. This contrast powerfully highlights the dual advantage of the proposed method: it not only enhances model performance by providing a feature space rich in dynamical information but also drastically reduces the dimensionality of the problem, leading to faster training and lower resource consumption.

In conclusion, the paper successfully establishes recurrence microstate probabilities as a highly effective, physics-based feature extraction technique for time series classification. It provides a principled way to reduce data complexity while preserving dynamical signatures, making it a potent tool for analyzing complex systems in fields ranging from physics and biology to finance and engineering. The method’s independence from stationarity assumptions further broadens its applicability to real-world, non-stationary signals.