Correntropy as a novel measure for nonlinearity tests

Nonlinearity tests have become an essential step in system analysis and modeling due to the computational demands and complexity of analysis involved in nonlinear modeling. Standard nonlinear measures are either too complicated to estimate accurately (such as Lyapunov exponents and correlation dimension), or not able to capture sufficient but not necessary conditions of nonlinearity (such as time asymmetry). Correntropy is a kernel-based similarity measure which contains the information of both statistical and temporal structure of the underlying dataset. The capability of preserving nonlinear characteristics makes correntropy a suitable candidate as a measure for determining nonlinear dynamics. Moreover, since correntropy makes use of kernel methods, its estimation is computationally efficient. Using correntropy as the test statistic, nonlinearity tests based on the null hypothesis that signals of interest are realizations of linear Gaussian stochastic processes are carried out via surrogate data methods. Experiments performed on linear Gaussian, linear non-Gaussian, and nonlinear systems with varying in-band noise levels, data lengths, and kernel sizes confirm that correntropy can be employed as a discriminative measure for detecting nonlinear characteristics in time series. Results of tests performed on data collected from natural systems are in agreement with findings in time series analysis literature.