Detection of nonlinearity in a time-series: by the synthesis of surrogate data using a Kolmogorov-Smirnoff tested, hidden Markov model

Conventional methods of hypothesis testing for nonlinearity in a time-series employ the method of surrogate data which makes use of the Fourier transform (FT). As various authors have shown, this can lead to artifacts in the surrogates and spurious detection of nonlinearity can result. This paper documents a new method to synthesize surrogate data using a 1st order hidden Markov model (HMM) combined with a Kolmogorov-Smirnoff test (KS-test), to determine the required resolution of the HMM. The method provides a way to retain the dynamics of a time-series and impart the null hypothesis (H/sub 0/) onto the synthesized surrogate which avoids the FT and its associated artifact. Significance test results for a sinewave, Henon map and Gaussian noise time-series are presented. It is demonstrated through ´significance testing´ that KS-tested, HMM surrogates can be successfully used to distinguish between a deterministic and stochastic time-series. Then by applying a simple test for linearity, using linear and nonlinear predictors, it is possible to determine the nature of the deterministic class and hence, conclude whether the system is linear deterministic or nonlinear deterministic. Furthermore, it is demonstrated that the method works for periodic functions too, where FT surrogates break down.