Computation of Three-Dimensional and Two-Dimensional Biphase Spectrum Plots for Time Series Analysis

To effectively analyze nonlinear processes, it is crucial to generate both three-dimensional and two-dimensional visualizations of the time series biphase spectrum. These plots enable comprehensive data interpretation through surface plots (3D) showing amplitude-phase relationships and contour plots (2D) revealing spectral symmetry properties. Implementation typically involves Fast Fourier Transform (FFT) operations combined with bispectral estimation algorithms to compute third-order cumulants. The computational workflow includes: signal preprocessing, biphase calculation using MATLAB's signal processing toolbox or Python's SciPy library, and visualization through matplotlib (3D surface plotting) or seaborn (2D heatmaps). Further statistical analysis can incorporate hypothesis testing for nonlinearity detection using surrogate data methods. These techniques facilitate deeper understanding of system dynamics by revealing phase coupling patterns and nonlinear interactions, potentially leading to novel insights in complex system behavior.