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ON THE LONG TIME BEHAVIOR OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH ASYMPTOTICALLY SMALL DISSIPATION
, 710
"... ABSTRACT. We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H (S) ¨x(t) + a(t) ˙x(t) + ∇G(x(t)) = 0, t ≥ 0, where the map a: R+ → R+ is non increasing and the potential G: H → R is of class C1. If the coefficient a(t) is constant and posi ..."
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ABSTRACT. We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H (S) ¨x(t) + a(t) ˙x(t) + ∇G(x(t)) = 0, t ≥ 0, where the map a: R+ → R+ is non increasing and the potential G: H → R is of class C1. If the coefficient a(t) is constant and positive, we recover the socalled “Heavy Ball with Friction ” system. On the other hand, when a(t) = 1/(t + 1) we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function G is convex, the condition ∫ ∞ 0 a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∫ ∞ 0 e − ∫ t 0 a(s) ds dt < ∞ is necessary to obtain the convergence of the trajectories of (S) toward some minimum point of G. In the onedimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function G. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense. 1.
Selection of Biologically Relevant Genes with a Wrapper Stochastic Algorithm
"... Abstract We investigate an important issue of a metaalgorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is ..."
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Abstract We investigate an important issue of a metaalgorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is then inferred which exhibits important genes for the studied biological process. Theory and application with the SVM classifier were presented in Gadat and Younes, 2007 and we extend this method with CART. The classification error rates are computed on three famous public databases (Leukemia, Colon and Prostate) and compared with those from other wrapper methods (RFE, lo norm SVM, Random Forests). This allows the assessment of the statistical relevance of the proposed algorithm. Furthermore, a biological interpretation with the Ingenuity Pathway Analysis software outputs clearly shows that the gene selections from the different wrapper methods raise very relevant biological information, compared to a classical filter gene selection with Ttest. KEYWORDS: gene selection, classification, stochastic algorithm, cancer databases Author Notes: We are grateful to ACI IMPBio (ENVSTATEXP) who supported this research. We would also like to thank "Projet Calcul en MIdiPyrenées" (CALMIP) for the intensive computations and the anonymous reviewers for their helpful comments on the manuscript.
Variance Sensitivity Analysis of Parameters for Pruning of a Multilayer Perceptron: Application to a Sawmill Supply Chain Simulation Model
, 2013
"... Simulation is a useful tool for the evaluation of a Master Production/Distribution Schedule (MPS). The goal of this paper is to propose a new approach to designing a simulation model by reducing its complexity. According to the theory of constraints, a reduced model is built using bottlenecks and a ..."
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Simulation is a useful tool for the evaluation of a Master Production/Distribution Schedule (MPS). The goal of this paper is to propose a new approach to designing a simulation model by reducing its complexity. According to the theory of constraints, a reduced model is built using bottlenecks and a neural network exclusively. This paper focuses on one step of the network model design: determining the structure of the network. This task may be performed by using the constructive or pruning approaches. The main contribution of this paper is twofold; it first proposes a new pruning algorithm based on an analysis of the variance of the sensitivity of all parameters of the network, and then uses this algorithm to reduce the simulation model of a sawmill supply chain. In the first step, the proposed pruning algorithm is tested with two simulation examples and compared with three classical pruning algorithms from the literature. In the second step, these four algorithms are used to determine the optimal structure of the network used for the complexityreduction design procedure of the simulation model of a sawmill supply chain. Index Terms — multilayer perceptron, pruning, reduced model, simulation. I.
a Wrapper Stochastic Algorithm ∗
"... We investigate an important issue of a metaalgorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is then infer ..."
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We investigate an important issue of a metaalgorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is then inferred which exhibits important genes for the studied biological process. Theory and application with the SVM classifier were presented in Gadat and Younes, 2007 and we extend this method with CART. The classification error rates are computed on three famous public databases (Leukemia, Colon and Prostate) and compared with those from other wrapper methods (RFE, lo norm SVM, Random Forests). This allows the assessment of the statistical relevance of the proposed algorithm. Furthermore, a biological interpretation with the Ingenuity Pathway Analysis software outputs clearly shows that the gene selections from the different wrapper methods raise very relevant biological information, compared to a classical filter gene selection with Ttest.
(2008)"
, 2008
"... Elagage d’un perceptron multicouches: utilisation de l’analyse de la variance de la sensibilité des paramètres. ..."
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Elagage d’un perceptron multicouches: utilisation de l’analyse de la variance de la sensibilité des paramètres.
(2008)"
, 2008
"... Sélection de la structure d’un perceptron multicouches pour la réduction d’un modèle de simulation d’une scierie. ..."
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Sélection de la structure d’un perceptron multicouches pour la réduction d’un modèle de simulation d’une scierie.
Multivariate approaches and post analysis (Open Access publication)
, 2007
"... Analysis of the real EADGENE data set: ..."
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Multiclass classification and gene selection with a stochastic algorithm
"... Microarray technology allows for the monitoring of thousands of gene expressions in various biological conditions, but most of these genes are irrelevant for classifying these conditions. Feature selection is consequently needed to help reduce the dimension of the variable space. We start from the a ..."
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Microarray technology allows for the monitoring of thousands of gene expressions in various biological conditions, but most of these genes are irrelevant for classifying these conditions. Feature selection is consequently needed to help reduce the dimension of the variable space. We start from the application of the stochastic algorithm ”Optimal Feature Weighting ” (OFW) for selecting features in various classification problems. This method does not depend on the classification method. Gadat and Younes (2007) who established the theoretical part of the model, applied SVM in the framework of binary pattern recognition data sets. The application with CART was performed in Lê Cao et al. (2007) who made a comparative study with other binary wrapper methods in the context of microarray data and emphasized on the biological interpretation. In this study, we focus on the multiclass problem that wrapper methods rarely handle. From a computational point of view, one of the main difficulties comes from the commonly unbalanced classes situation when dealing with microarray data. From a theoretical point of view, very few methods have been developped to minimize any classification criterion, compared to the 2class situation (e.g SVM, l0SVM, RFE...). In this paper, we first develop the OFW approach to handle multiclass problems using CART and onevsone SVM as classifiers. We then compare our results with those obtained with other multiclass selection algorithm (Random Forests and the filter method Ftest), on four public microarray data sets. We assess the statistical relevancy of the results by measuring and comparing their performances. The aim of this study is to heuristically evaluate which method would be the best to select genes distinguishing minority classes. Application and biological interpretation are then given in the case of a real world microarray data set.