MATLAB Implementation of Phase Space Reconstruction with Code Explanations

Phase space reconstruction is a crucial technique for recovering system dynamics from time series data, widely applied in nonlinear dynamics and chaotic system studies. In MATLAB implementation, the key challenge involves selecting appropriate embedding dimensions and time delays to accurately reconstruct the phase space trajectory of the system. The fundamental concept of phase space reconstruction involves expanding a one-dimensional time series into a higher-dimensional space using the delay coordinate method. Given an original time series x(t), by choosing suitable time delay τ and embedding dimension m, we can construct reconstructed phase space vectors. Each reconstructed point can be represented as X(t) = [x(t), x(t+τ), ..., x(t+(m-1)τ)]. This transforms the one-dimensional time series into an m-dimensional phase space representation. In MATLAB implementation, the phase space reconstruction process typically involves these steps: First, preprocess the time series through standardization or denoising using functions like zscore or smoothdata to improve reconstruction accuracy. Then determine the optimal time delay τ using mutual information method (implemented with custom functions calculating mutual information) or autocorrelation function (using xcorr). Next, identify the appropriate embedding dimension m using false nearest neighbors method (FNN algorithm implementation). Finally, generate phase space points using the delay coordinate method and create phase space plots. Phase space visualization can be achieved using MATLAB's 3D plotting functions like plot3 or scatter3, which effectively display system trajectories in the reconstructed phase space. For two-dimensional phase space representations, the plot function can directly visualize delay coordinate plots. By analyzing these phase space plots, researchers can identify system dynamic characteristics such as periodicity, chaotic behavior, or random patterns using functions like lyapunovExponent for quantitative analysis. Phase space reconstruction finds extensive applications in signal processing, fault diagnosis, and physiological signal analysis. MATLAB's powerful matrix operations (vectorized computations) and advanced visualization capabilities make it an ideal platform for implementing phase space reconstruction algorithms, with specialized toolboxes like Signal Processing Toolbox providing additional support for time series analysis.