Tree structured non-linear signal modeling and prediction

Abstract

Abstract-In this paper, we develop a regression tree approach to identification and prediction of signals that evolve according to an unknown nonlinear state space model. In this approach, a tree is recursively constructed that partitions the p-dimensional state space into a collection of piecewise homogeneous regions utilizing a 2 p -ary splitting rule with an entropy-based node impurity criterion. On this partition, the joint density of the state is approximately piecewise constant, leading to a nonlinear predictor that nearly attains minimum mean square error. This process decomposition is closely related to a generalized version of the thresholded AR signal model (ART), which we call piecewise constant AR (PCAR). We illustrate the method for two cases where classical linear prediction is ineffective: a chaotic "doublescroll" signal measured at the output of a Chua-type electronic circuit and a second-order ART model. We show that the prediction errors are comparable with the nearest neighbor approach to nonlinear prediction but with greatly reduced complexity.Index Terms-Chaotic signal analysis, nonlinear and nonparametric modeling and prediction, piecewise constant AR models, recursive partitioning, regression trees.

Adaptive registration using local information measures

“…The smallest subblock size may be determined by the fundamental inability to estimate entropy/MI from a very small number of voxels. Entropy estimation based on minimal spanning trees shows promise for estimating entropy/MI from sparse histograms (Michel and Hero, 1999). Further efforts are needed to incorporate that estimation method.…”

Nonlinear turbo equalization using context trees

“…Note that if K is large and the regions are dense such that W(q) (and F(q)) can be considered constant in V k , say equal to W(q k ) for some q k in region V k , then if the adaptation method used in each region converges successfully, this yields w k (t) → W(q k ) and f k (t) → F(q k ) as t → ∞. Hence, if these regions are dense and there is enough data to learn the corresponding models in each region, then this piecewise model can approximate any smoothly varying W(q) and F(q) [22].…”

“…Hence, if these regions are dense and there is enough data to learn the corresponding models in each region, then this piecewise model can approximate any smoothly varying W(q) and F(q) [22].…”

Nonlinear regression using second order methods

“…Dogrusal olma y an baglamm problerni, otomatik ogrenme ve sin y al i�leme literatUrUnde en onemli c;:ah�ma alanlanndan birisidir ve sin y al modelleme, finansal market, y onelim analizi, tavsi y e gibi birc;:ok farkh gerc;:ek ha y at u y gulamasmm temelini olu�turur [1], [2]. Fakat geleneksel baglamm teknikleri, bU y Uk veri ic;:eren gerc;:ek ha y at u y gulamalarmda, elde edilen verinin konvansi y onel sin y al i�leme metotlan ile etkili bir �ekilde i�lenememesi, depolama konusundaki sorunlar ve geleneksel metotlann veri istatistigine oldukc;:a bagh olmasl dola y lSl y la, y eterli performansl gosterememektedir [3].…”