Tree-structured nonlinear signal modeling and prediction

In this paper, we develop a regression tree approach to identification and prediction of signals that evolve according to an unknown nonlinear state s pace model. In this approach, a tree is recursively constructed that partit ions the p-dimensional state space into a collection of piecewise homogeneo us regions utilizing a 2(p)-ary splitting rule with an entropy-based node i mpurity criterion. On this partition, the joint density of the state is app roximately piecewise constant, leading to a nonlinear predictor that nearly attains minimum mean square error. This process decomposition is closely r elated to a generalized version of the thresholded AR signal model (ART), w hich we call piecewise constant AR (PCAR). We illustrate the method for two cases where classical linear prediction is ineffective: a chaotic "double- scroll" signal measured at the output of a Chua-type electronic circuit and a second-order ART model. We show that the prediction errors are comparabl e with the nearest neighbor approach to nonlinear prediction but with great ly reduced complexity.