Time series nonlinearity modeling: A Giannakis formula type approach

Abstract

Listen

Translate article

In this paper, we propose a method for identifying the coefficients of a simplified Second Order Volterra Model (SOVM) driven by a normal i.i.d. white noise. The interest of estimating the coefficients of such a model is to easily model nonlinear time series by identifying a linear spectrum and a nonlinear spectrum. In fact, the nonlinear spectrum is the spectrum of output data of a quadratic system (squarer) driven by a normal i.i.d. white noise while the linear spectrum is the output data spectrum of a linear system driven by the same noise. Consequently, by estimating the linear and nonlinear spectrum components, the proposed algorithm locates (in the Fourier domain) and quantifies the nonlinear artifacts in an observed time series, this observed time series being the output of a nonlinear system and the input data of this system not being available. The method for estimating the model coefficients is quite simple and is based on the ratio of products of Higher Order Cumulants. For this reason, the method of identification is close to Giannakis’ formula which identifies the coefficients of a linear system driven by a non symmetric noise and also uses the ratio of cumulants. In this paper, we also address the question of order selection of both parts of the simplified SOVM (i.e. the linear and quadratic parts) based on hypothesis testing, the order of each part interfering strongly in the final results. Finally, we propose a method for verifying that the higher order statistics (HOS) of the observed time series are matched with the HOS derived from the estimated coefficients, thus proving that the time series is well modeled by the estimated nonlinear parametric model.