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Weighted least squares

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression. WLS is also a specialization of generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors, are null. Formulation The fit of a model to a data point is measured by its residual, r_i , defined as the difference between a measured value of the dependent variable, y_i and the value predicted by the model, f(x_i, \boldsymbol\beta): r_i(\boldsymbol\beta) = y_i - f(x_i, \boldsymbol\beta). If the errors are uncorrelated and have equal variance, then the function S(\boldsymbol\beta) = \sum_i r_i(\boldsymbol\beta)^2, is minimised at \boldsymbol\hat\beta, such that \frac{\pa

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A Weighted Reduced Basis Method For Elliptic Partial Differential Equations With Random Input Data

Peng Chen, Alfio Quarteroni, Gianluigi Rozza

In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.

Siam Publications2013

Chargement

A weighted empirical interpolation method: a priori convergence analysis and applications

Peng Chen, Alfio Quarteroni, Gianluigi Rozza

We extend the classical empirical interpolation method to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work. We apply our method to geometric Brownian motion, exponential Karhunen-Loeve expansion and reduced basis approximation of non-ane stochastic elliptic equations. We demonstrate its improved accuracy and eciency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

EDP Sciences2014

Chargement

Méthode des moindres carrés ordinaire

vignette|Graphique d'une régression linéaire La méthode des moindres carrés ordinaire (MCO) est le nom technique de la régression mathématique en statistiques, et plus particulièrement de la régressio

Régression linéaire

En statistiques, en économétrie et en apprentissage automatique, un modèle de régression linéaire est un modèle de régression qui cherche à établir une relation linéaire entre une variable, dite exp

Théorème de Gauss-Markov

En statistiques, le théorème de Gauss–Markov, nommé ainsi d'après Carl Friedrich Gauss et Andrei Markov, énonce que dans un modèle linéaire dans lequel les erreurs ont une espérance nulle, sont non co

MATH-341: Linear models

Regression modelling is a fundamental tool of statistics, because it describes how the law of a random variable of interest may depend on other variables. This course aims to familiarize students with linear models and some of their extensions, which lie at the basis of more general regression model

CS-401: Applied data analysis

This course teaches the basic techniques, methodologies, and practical skills required to draw meaningful insights from a variety of data, with the help of the most acclaimed software tools in the data science world (pandas, scikit-learn, Spark, etc.)

FIN-407: Financial econometrics

This course aims to give an introduction to the application of machine learning to finance. These techniques gained popularity due to the limitations of traditional financial econometrics methods tackling big data. We will review and compare traditional methods and machine learning algorithms.