Regression Trees for Non Parametric Modeling and Time Series Prediction. Arbres de Régression Modélisation non Paramétrique et Analyse des Séries Temporelles

Regression Trees for Non Parametric Modeling and Time Series Prediction

Arbres de Régression Modélisation non Paramétrique et Analyse des Séries Temporelles

Anne-Emmanuelle Badel Olivier Michel  Alfred Hero 

Laboratoire de Physique (URA 1 325 CNRSI, École Normale Supérieurede Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA

Page: 
117-133
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Received: 
19 June 1996
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Accepted: 
N/A
| | Citation

OPEN ACCESS

Abstract: 

We present a non-parametric approach to nonlinear modeling and prediction based on adaptive partitioning of the reconstructed phase space associated with the process. The partitioning method is implementedwith a recursive tree-structured algorithm which successively refines the partition by binary splitting where the splitting threshold is determined by a penalized maximum entropy criterion. An analysis of the statistical behavior of the splitting rule suggests a criterion for determining the depth of the tree. The effectiveness of this method is illustrated through comparisons with classical approaches for nonlinear system analysis on the basis of reconstruction error and computational complexity. An importantrelation between our tree-structured model for the process and generalized non-linear thresholded AR model (ART) is established. We illustrate our method for cases where classical linear prediction is known to be rather ineffective : chaotic signals (measured at the output of a Chua-type electronic circuit), and second order ART signal.

Résumé 

Nous présentons une approche non linéaire non paramétrique pour la modélisation et la prédiction de signaux, basée sur une méthode de partition récursive de l'espace des phases reconstruit, associé au système sur lequel le signal est prélevé. La partition de l'espace des phases est obtenue par un algorithme récursif de partition binaire . Les seuils de partition sont déterminés à l'aide d'un critère de maximum d'entropie. Une courte analyse statistique du comportement de ces seuils permet de définir un critère simple d'arrêt de la partition récursive. L'intérêt de cette méthode est illustré par la comparaison avec des méthodes classiques dans le cadre de l'analyse de systèmes non linéaires, ainsi que du point de vue du coût de calcul . Nous présentons un lien important entre cette méthode reposant sur une partition hiérarchique (en arbre) et les modèles non linéaires auto-régressifs à seuils (ART). Dans ce contexte, la méthode présentée est appliquée dans des cas pour lesquels les méthodes linéaires échouent en général :les signaux de chaos (séries expérimentales mesurées sur des circuits électroniques de type Chua), ainsi que sur des séries numériques ART d'ordre deux. 

Keywords: 

Modeling, Non linear, Non parametric, Statistics, Dynamical Systems, Regression Trees, Phase Space, Prediction .

Mots clés 

Modélisation, Non linéaire, Non paramétrique, Statistique, Systèmes dynamiques, Arbres de régression, Espace des phases, Prédiction.

1. Introduction
2. Description du Problème
3. Estimation d'une Distribution de Probabilité sur L'Espace des Phases
4. Application à la Prédiction
5. Lien avec les Modèles Auto-Régressifs a Seuils
6. Conclusions et Perspectives
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