Lecture

Linear Regression: Mean-square-error Inference

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In course

EE-566: Adaptation and learning

In this course, students learn to design and master algorithms and core concepts related to inference and learning from data and the foundations of adaptation and learning theories with applications.

Description

This lecture covers the mean-square-error (MSE) problem of estimating a random variable from observations of another random variable, focusing on linear regression models. It addresses the challenge of determining the MSE estimator by limiting the mapping function to affine functions of the input variable. The lecture explains the optimal estimate, the linear least-mean-square-error estimator, and the orthogonality principle. It also discusses data fusion methods and the application of linear models in estimating target properties. The concepts are illustrated through scatter diagrams and regression hyperplanes.

Instructor

Ali H. Sayed

Ali H. Sayed is Dean of Engineering at EPFL, Switzerland, where he also leads the Adaptive Systems Laboratory. He has also served as Distinguished Professor and Chairman of Electrical Engineering at UCLA. He is recognized as a Highly Cited Researcher and is a member of the US National Academy of Engineering. He is also a member of the World Academy of Sciences and served as President of the IEEE Signal Processing Society during 2018 and 2019. Dr. Sayed is an author/co-author of over 570 scholarly publications and six books. His research involves several areas including adaptation and learning theories, data and network sciences, statistical inference, and multiagent systems. His work has been recognized with several major awards including the 2022 IEEE Fourier Award, the 2020 Norbert Wiener Society Award and the 2015 Education Award from the IEEE Signal Processing Society, the 2014 Papoulis Award from the European Association for Signal Processing, the 2013 Meritorious Service Award and the 2012 Technical Achievement Award from the IEEE Signal Processing Society, the 2005 Terman Award from the American Society for Engineering Education, the 2005 Distinguished Lecturer from the IEEE Signal Processing Society, the 2003 Kuwait Prize, and the 1996 IEEE Donald G. Fink Prize. His publications have been awarded several Best Paper Awards from the IEEE (2002, 2005, 2012, 2014) and EURASIP (2015). He is a Fellow of IEEE, EURASIP, and the American Association for the Advancement of Science (AAAS); the publisher of the journal Science.

Official source

Ontological neighbourhood

Statistics

Data analysis: Regression analysis

Related concepts (32)

General linear model

The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as where Y is a matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors (noise).

Linear model

In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term is also used in time series analysis with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related statistical theory is possible.

Bayesian linear regression

Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often labelled ) conditional on observed values of the regressors (usually ).

Generalized linear model

In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression.

Segmented regression

Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning the various independent variables. Segmented regression is useful when the independent variables, clustered into different groups, exhibit different relationships between the variables in these regions.

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