A new approach to piecewise linear modeling of time series

Due to the inherent non-linearity and non-stationarity of a wide class of time series, nonlinear models have been the object of an increasing interest over the past years. Piecewise linear models, in which a linear sub-model is associated with each region of a state-space decomposition, have been proposed as an attractive alternative to threshold autoregressive models. However, it is still unclear how this type of models can be actually estimated. We show how a new combinatorial optimization approach, which we devised for the general problem of piecewise linear model estimation, can be successfully applied to piecewise linear modeling of time series. The idea is to focus on the inconsistent linear system that arises when considering a simple linear model and to partition it into a minimum number of consistent subsystems (MIN PCS). Although the resulting problem (MIN PCS) is NP-hard, satisfactory approximate solutions can be obtained using simple variants of the perceptron algorithm studied in the artificial neural network literature. Simulation results for two well-known chaotic time series are reported.