Machine learning approach to detect dynamical states from recurrence measures

TL;DR

This study integrates machine learning approaches with nonlinear time series analysis, specifically utilizing recurrence measures to classify various dynamical states emerging from time series, and illustrates how the trained algorithms can successfully predict the dynamical states of two variable stars.

Abstract

We integrate machine learning approaches with nonlinear time series analysis, specifically utilizing recurrence measures to classify various dynamical states emerging from time series. We implement three machine learning algorithms Logistic Regression, Random Forest, and Support Vector Machine for this study. The input features are derived from the recurrence quantification of nonlinear time series and characteristic measures of the corresponding recurrence networks. For training and testing we generate synthetic data from standard nonlinear dynamical systems and evaluate the efficiency and performance of the machine learning algorithms in classifying time series into periodic, chaotic, hyper-chaotic, or noisy categories. Additionally, we explore the significance of input features in the classification scheme and find that the features quantifying the density of recurrence points are the most relevant. Furthermore, we illustrate how the trained algorithms can successfully predict the dynamical states of two variable stars, SX Her and AC Her from the data of their light curves.

Figures (7)

• Figure 1: Schematic of the methodology developed for the classification of the time series data using recurrence quantification and machine learning algorithms. From the given (a) nonlinear time series, we reconstruct the attractor (b) using time delay embedding. Following this, we generate (c) recurrence plots and (d) recurrence networks (nodes are shown in red color and edges are shown in blue color). Then, we extract (e) six recurrence measures from the recurrence plots and two measures from the recurrence networks. These measures serve as features for the machine learning algorithms (f) Logistic Regression, Random Forest and Support Vector Machine. The final step(g) classifies the time series into four classes: periodic, chaotic, hyperchaotic, or noise.
• Figure 2: Confusion matrix for multi-class classification of time series data into Periodic, Chaotic, Hyperchaotic or Noise. To find the overall accuracy of the classifiers, we use the multi-class confusion matrix shown in the left panel (a), and to find performance for each class, we use the binary confusion matrix as shown in the right panel (b) for the periodic class for reference.
• Figure 3: Data sets generated for training and testing of the classification scheme. The four types of nonlinear time series data (a)Periodic time series obtained from Rössler system with parameters $a=0.2$, $b=0.2$, $c=1$; (b) Chaotic time series obtained from Rössler system with parameters $a=0.2$, $b=0.2$, $c=7$; (c)Hyperchaotic time series obtained from Chen system with parameters $a=35$, $b=4.9$, $c=25$, $d=5$, $e=35$ and $k=24$; and (d)White Noise obtained with Gaussian distribution with mean=0 and standard deviation=1. The re-constructed phase-space structure of the attractors (e-h), recurrence plots (i-l) and recurrence networks (m-p) corresponding to each time series
• Figure 4: Ranges of the recurrence measures evaluated from the time series of dynamical states, periodic (green), chaotic (red), hyperchaotic (orange), and noisy data (blue)
• Figure 5: Performance analysis of the three machine learning algorithms, namely Logistic Regression (blue), Random Forest (orange), and Support Vector Machine (grey). The left panel corresponds to choice recurrence threshold $\epsilon_1$ and the right panel corresponds to $\epsilon_2$. The performance of each algorithm is evaluated in terms of accuracy, sensitivity, specificity, precision, and F1 score for all the classes of patterns. We can see the performance evolution of all machine learning algorithms for periodic patterns (a,e), chaotic patterns (b,f), hyperchaotic patterns (c,g), and for noise patterns (d,h).