Transcription of Gaussian Linear Models - MIT OpenCourseWare
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Gaussian Linear Models - MIT OpenCourseWare
Gaussian Linear Models Gaussian Linear Models MIT Dr. Kempthorne Spring 2016 1 MIT Gaussian Linear Models Gaussian Linear Models Linear Regression: Overview Ordinary Least Squares (OLS) distribution Theory: normal Regression Models Maximum Likelihood Estimation Generalized M Estimation Outline 1 Gaussian Linear Models Linear Regression: Overview Ordinary Least Squares (OLS) distribution Theory: normal Regression Models Maximum Likelihood Estimation Generalized M Estimation 2 MIT Gaussian Linear Models Gaussian Linear Models Linear Regression: Overview Ordinary Least Squares (OLS) distribution Theory: normal Regression Models Maximum Likelihood Estimation Generalized M Estimation General Linear Model: For each case i, the conditional distribution [yi | xi ] is given by yi = yi + Ei where y i = 1xi,1 + 2xi,2 + + i,pxi,p = ( 1, 2,.., p)T are p regression parameters (constant over all cases) Ei Residual (error) variable (varies over all cases) Extensive breadth of possible Models Polynomial approximation (xi,j = (xi )j , explanatory variables are different powers of the same variable x = xi ) Fourier Series: (xi,j = sin(jxi ) or cos(jxi ), explanatory variables are different sin/cos terms of a Fourier series expansion) Time series regressions: time indexed by i, and explanatory variables
Distribution Theory: Normal Regression Models Maximum Likelihood Estimation Generalized M Estimation. Steps for Fitting a Model (1) Propose a model in terms of Response variable Y (specify the scale) ... Multivariate Normal with mean µ and covariance Σ ...
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