Surrogate Testing by Estimating the Local Dispersions in Phase Space

Abstract

In this report, a clustering approach is presented to detect dynamical nonlinearity in a stationary time series. The one-dimensional time series sampled from a dynamical system is mapped on to the m-dimensional phase space by the method of delays. The vectors in the phase space are partitioned by k-means clustering technique. The local trajectory matrix for the vectors in each of the clusters is determined. The eigenvalues of the local trajectory matrices represent the variation along the principal directions and are obtained by singular value decomposition (SVD). The product of the eigenvalues represents the generalized variance and is a measure of the local dispersion in the phase space. The sum of the local dispersions (gT) is used as the discriminant statistic to classify data sets obtained from deterministic and stochastic settings. The surrogates were generated by the Iterated Amplitude Adjusted Fourier Transform (IAAFT) technique.