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21
LeastSquares Policy Iteration
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2003
"... We propose a new approach to reinforcement learning for control problems which combines valuefunction approximation with linear architectures and approximate policy iteration. This new approach ..."
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We propose a new approach to reinforcement learning for control problems which combines valuefunction approximation with linear architectures and approximate policy iteration. This new approach
Hierarchical regression for epidemiologic analyses of multiple exposures
 Environmental Health Perspective
, 1994
"... Many epidemiologic investigations are designed to study the effects of multiple exposures. Most of these studies are analyzed either by fitting a riskregression model with all exposures forced in the model, or by using a preliminarytesting algorithm, such as stepwise regression, to produce a small ..."
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Many epidemiologic investigations are designed to study the effects of multiple exposures. Most of these studies are analyzed either by fitting a riskregression model with all exposures forced in the model, or by using a preliminarytesting algorithm, such as stepwise regression, to produce a smaller model. Research indicates that hierarchical modeling methods can outperform these conventional approaches. These methods are reviewed and compared to two hierarchical methods, empiricalBayes regression and a variant here called "semiBayes " regression, to fullmodel maximum likelihood and to model reduction by preliminary testing. The performance of the methods in a problem of predicting neonatalmortality rates are compared. Based on the literature to date, it is suggested that hierarchical methods should become part of the standard approaches to multipleexposure studies.Environ Health Perspect 102(Suppl 8):3339 (1994)
Bayesian Prediction Using Adaptive Ridge Estimators
"... The Bayesian linear model framework has become increasingly popular building block in regression problems. It has been shown to produce models with good predictive power and can be used with basis functions that are nonlinear in the data to provide exible estimated regression functions. Further, ..."
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Cited by 4 (0 self)
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The Bayesian linear model framework has become increasingly popular building block in regression problems. It has been shown to produce models with good predictive power and can be used with basis functions that are nonlinear in the data to provide exible estimated regression functions. Further, model uncertainty can be accounted for by Bayesian model averaging. We propose a more simple way to account for model uncertainty that is based on generalized ridge regression estimators. This is shown to predict well and to be much more computationally ecient than standard model averaging methods. Further, we demonstrate how to eciently mix over dierent sets of basis functions, letting the data determine which are most appropriate for the problem at hand. Keywords: Bayesian model averaging, generalized ridge regression, prediction, regression splines, shrinkage. 1
Anomalies in the foundations of ridge regression
 Tracking MSE efficiencies in ridge regression. Advances and Applications in Statistical Sciences (In
, 2008
"... Abstract. Anomalies persist in the foundations of ridge regression as set forth in Hoerl and Kennard (1970) and subsequently. Conventional ridge estimators and their properties do not follow on constraining lengths of solution vectors using LaGrange’s method, as claimed. Estimators so constrained ha ..."
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Cited by 3 (2 self)
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Abstract. Anomalies persist in the foundations of ridge regression as set forth in Hoerl and Kennard (1970) and subsequently. Conventional ridge estimators and their properties do not follow on constraining lengths of solution vectors using LaGrange’s method, as claimed. Estimators so constrained have singular distributions; the proposed solutions are not necessarily minimizing; and heretofore undiscovered bounds are exhibited for the ridge parameter. None of the considerable literature on estimation, prediction, cross–validation, choice of ridge parameter, and related issues, collectively known as ridge regression, is consistent with constrained optimization, nor with corresponding inequality constraints. The problem is traced to a misapplication of LaGrange’s principle, failure to recognize the singularity of distributions, and misplaced links between constraints and the ridge parameter. Other principles, based on condition numbers, are seen to validate both conventional ridge and surrogate ridge regression to be defined. Numerical studies illustrate that ridge analysis often exhibits some of the same pathologies it is intended to redress. 1.
Principal Components Regression with DataChosen Components and Related Methods
, 2002
"... Multiple regression with correlated explanatory variables is relevant to a broad range of problems in the physical, chemical, and engineering sciences. Chemometricians, in particular, have made heavy use of principal components regression and related procedures for predicting a response variable fr ..."
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Multiple regression with correlated explanatory variables is relevant to a broad range of problems in the physical, chemical, and engineering sciences. Chemometricians, in particular, have made heavy use of principal components regression and related procedures for predicting a response variable from a large number of highly correlated variables. In this paper we develop a general theory for selecting principal components that yield estimates of regression coefficients with low mean squared error. Our numerical results suggest that the theory also can be used to improve partial least squares regression estimators and regression estimators based on rotated principal components. Although our work has been motivated by the statistical genetics problem of mapping quantitative trait loci, the results are applicable to any problem where estimation of regression coefficients for correlated explanatory variables is of interest.
EFFICIENT APPROXIMATE POLICY ITERATION METHODS for Sequential Decision Making . . .
, 2003
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AN ACCURATE APPROXIMATION TO THE DISTRIBUTION OF A LINEAR COMBINATION OF NONCENTRALCHISQUARE RANDOMVARIABLES
"... • This paper provides an accessible methodology for approximating the distribution of a general linear combination of noncentral chisquare random variables. Attention is focused on the main application of the results, namely the distribution of positive definite and indefinite quadratic forms in n ..."
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• This paper provides an accessible methodology for approximating the distribution of a general linear combination of noncentral chisquare random variables. Attention is focused on the main application of the results, namely the distribution of positive definite and indefinite quadratic forms in normal random variables. After explaining that the moments of a quadratic form can be determined from its cumulants by means of a recursive formula, we propose a momentbased approximation of the density function of a positive definite quadratic form, which consists of a gamma density function that is adjusted by a linear combination of Laguerre polynomials or, equivalently, by a single polynomial. On expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, explicit representations of approximations to its density and distribution functions are obtained in terms of confluent hypergeometric functions. The proposed closed form expressions converge rapidly and provide accurate approximations over the entire support of the distribution. Additionally, bounds are derived for the integrated squared and absolute truncation errors. An easily implementable algorithm is provided and several illustrative numerical examples are
OPTIMAL PREDICTION IN LINEAR REGRESSION ANALYSIS
, 1987
"... Expressions are derived for generalized ridge (GR), ordinary ridge (OR) and shrunken least squares (SLS) predictors that are optimal for predicting the response at a single or at multiple future observations. As in the case of biased estimation, these predictors depend on the true (population) regre ..."
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Expressions are derived for generalized ridge (GR), ordinary ridge (OR) and shrunken least squares (SLS) predictors that are optimal for predicting the response at a single or at multiple future observations. As in the case of biased estimation, these predictors depend on the true (population) regression coefficient values and the true variance of the underlying linear regression model. Hence, we propose operational predictors where the unknown parameters in the biased predictors are estimated from the data. Using the Mean Squared Error of Prediction (MSEP) criterion, we compare the proposed predictors with the OLS predictor. Several traditional biased predictors, including the predictors based on the
Multivariate Flattening for Better Predictions
, 2000
"... Multivariate regression with p responses as opposed to p multiple regressions is getting increasingly more attention, especially in the context of prediction. Multivariate flattening methods are investigated as a way to obtain improved predictions over ordinary leastsquares. With respect to sum of ..."
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Multivariate regression with p responses as opposed to p multiple regressions is getting increasingly more attention, especially in the context of prediction. Multivariate flattening methods are investigated as a way to obtain improved predictions over ordinary leastsquares. With respect to sum of squares of prediction error, or SPE risk, an unbiased estimate of the risk is derived for two recent prediction methods, OPT and GCV, proposed by Breiman & Friedman (1997). Expressions for the exact SPE risk of OPT and GCV are derived generally for p # 1 and evaluated numerically for p =1. R ESUM E Les methodes de prevision de p variables utilisant la regression multivariee, plutot que p regression multiples, sont de plus en plus utilisees. Le retrecissement multivarieestetudie comme methode de prevision pour ameliorer la methode des moindres carres. On obtient l'estimateur sans biais du risque SPE de deux methodes recentes de prevision, la methode OPT et celle du GCV de Breiman et Friedm...
Nonlinear empirical modeling using local PLS models
, 1994
"... This thesis proposes some new iterative local modeling algorithms for the multivariate approximation problem (mapping from R P to R). Partial Least Squares Regression (PLS) is used as the local linear modeling technique. The local models are interpolated by means of normalized Gaussian weight func ..."
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This thesis proposes some new iterative local modeling algorithms for the multivariate approximation problem (mapping from R P to R). Partial Least Squares Regression (PLS) is used as the local linear modeling technique. The local models are interpolated by means of normalized Gaussian weight functions, providing a smooth total nonlinear model. The algorithms are tested on both artiøcial and real world set of data, yielding good predictions compared to other linear and nonlinear techniques. Preface This thesis a completes my work for the degree Candidatus Scientarum at the Department of Informatics, University of Oslo. It has been carried out at SINTEF, Oslo during two years from August 1992 to August 1994. My supervisors have been Tom Kavli, at SINTEF, and Nils Christophersen, at the Department of Informatics. I thank them both for valuable guidance and assistance into the world of empirical modeling. I will also thank Glenn Lines for intense and fruitful discussions, Irene R#d...