4. Time Series Linear and Nonlinear Models

Modeling time series includes the three steps of identification, parameter estimation and diagnostic checking. As far as linear models are concerned model building has been extensively studied and well established both theory and practice allow the user to proceed along reliable guidelines. Ergodicity, stationarity and Gaussianity properties are generally assumed to ensure that the structure of a stochastic process may be estimated safely enough from an observed time series. We will limit in this chapter to discrete parameter stochastic processes, that is a collection of random variables indexed by integers that are given the meaning of time. Such stochastic process may be called time series though we shall denote a finite single realization of it as a time series as well. Real time series data are often found that do not conform to our hypotheses. Then we have to model non stationary and non Gaussian time series that require special assumptions and procedures to ensure that identification and estimation may be performed, and special statistics for diagnostic checking. Several devices are available that allow such time series to be handled and remain within the domain of linear models. However there are features that prevent us from building linear models able to explain and predict the behavior of a time series correctly. Examples are asymmetric limit cycles, jump phenomena and dependence between amplitude and frequency that cannot be modeled accurately by linear models. Nonlinear models may account for time series irregular behavior by allowing the parameters of the model to vary with time. This characteristic feature means by itself that the stochastic process is not stationary and cannot be reduced to stationarity by any appropriate transform. As a consequence, the observed time series data have to be used to fit a model with varying parameters. These latter may influence either the mean or the variance of the time series and according to their specification different classes of nonlinear models may be characterized. Linear models are defined by a single structure while nonlinear models may be specified by a multiplicity of different structures. So classes of nonlinear models have been introduced each of which may be applied successfully to real time series data sets that are commonly observed in well delimited application fields. Contributions of evolutionary computing techniques will be reviewed in this chapter for linear models, as regards identification stage and subset models, and to a rather larger extent for some classes of nonlinear models, concerned with identification and parameter estimation. Beginning with the popular autoregressive moving-average linear models, we shall outline the relevant applications of evolutionary computing to the domains of threshold models, including piecewise linear, exponential and autoregressive conditional heteroscedastic structures, bilinear models and artificial neural networks.