Bispectral-based methods for clustering time series

Distinguishing among linear and nonlinear time series or between nonlinear time series generated by different underlying processes is challenging, as second-order properties are generally insufficient for the task. Different nonlinear processes have different nonconstant bispectral signatures, whereas the bispectral density function of a Gaussian or linear time series is constant. Based on this, we propose a procedure to distinguish among various nonlinear time series and between nonlinear and linear time series through application of a hierarchical clustering algorithm based on distance measures computed from the square modulus of the estimated normalized bispectra. We find that clustering using a distance measure computed by averaging the ratio of normalized bispectral periodogram ordinates over the intersection of the principle domain of each pair of time series provides good performance, subject to trimming of extreme bispectral values prior to taking the ratios. Additionally, we show through simulation studies that the distance procedure performs better than a significance test that we derive. Moreover, it is robust with respect to the choice of smoothing parameter in estimating the bispectrum. As an example, we apply the method to a set of time series of intensities of gamma-ray bursts, some of which exhibit nonlinear behavior; this enables us to identify gamma-ray bursts that may be emanating from the same type of astral event.