Construction of multivariate surrogate sets from nonlinear data using the wavelet transform

The use of surrogate data has become a crucial first step in the study of nonlinearity in time series data. A widely used technique to construct surrogate data is to randomize the phases of the data in the Fourier domain. In this paper, an alternative technique based on the resampling of wavelet coefficients is discussed. This approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data. It proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients. Experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties, removal of nonlinear structure and computational demands. However, the wavelet technique may have specific and distinct advantages in the application to complex data sets, such as numerical analysis of turbulence and experimental brain imaging data, where wavelets give a more parsimonious representation of spatio-temporal patterns than Fourier modes. It is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints.