Matrix Approach to Linear Regression Dr. Frank Wood Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 1 Random Vectors and Matrices • Let’s say we have a vector consisting of three random variables The expectation of a random vector is defined Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 2 Expectation of a Random Matrix • The expectation of a random matrix is defined similarly Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 3 Covariance Matrix of a Random Vector • The collection of variances and covariances of and between the elements of a random vector can be collection into a matrix called the covariance matrix remember so the covariance matrix is symmetric Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 4 Derivation of Covariance Matrix • In vector terms the covariance matrix is defined by because verify first entry Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 5 Regression Example • Take a regression example with n=3 with constant error terms σ {ǫi} = σ and are ≠ uncorrelated so that σ {ǫi, ǫj} = 0 for all i j • The covariance matrix for the random vector ǫǫǫ is which can be written as Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 6 Basic Results • If A is a constant matrix and Y is a random matrix then is a random matrix Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 7 Multivariate Normal Density • Let Y be a vector of p observations • Let µ be a vector of p means for each of the p observations Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 8 Multivariate Normal Density • Let § be the covariance matrix of Y • Then the multivariate normal density is given by Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 9 Example 2d Multivariate Normal Distribution 0.04 0.02 0 10 mvnpdf([0 0], [10 2;2 2]) 8 6 4 2 10 0 8 6 -2 4 -4 2 0 -6 -2 -4 y -8 -6 Run multivariate_normal_plots.m -8 -10 x -10 Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 10 Matrix Simple Linear Regression • Nothing new – only matrix formalism for previous results • Remember the normal error regression model • This implies Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 11 Regression Matrices • If we identify the following matrices • We can write the linear regression equations in a compact form Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 12 Regression Matrices • Of course, in the normal regression model the expected value of each of the ǫi’s is zero, we can write • This is because Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 13 Error Covariance • Because the error terms are independent and have constant variance σ Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 14 Matrix Normal Regression Model • In matrix terms the normal regression model can be written as where and Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 15 Least Squares Estimation • Starting from the normal equations you have derived we can see that these equations are equivalent to the following matrix operations with demonstrate this on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 16 Estimation • We can solve this equation (if the inverse of X’X exists) by the following and since we have Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 17 Least Squares Solution • The matrix normal equations can be derived directly from the minimization of w.r.t. to βββ Do this on board. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 18 Fitted Values and Residuals • Let the vector of the fitted values be in matrix notation we then have Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 19 Hat Matrix – Puts hat on Y • We can also directly express the fitted values in terms of only the X and Y matrices and we can further define H, the “hat matrix” • The hat matrix plans an important role in diagnostics for regression analysis. write H on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 20 Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. demonstrate on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 21 Residuals • The residuals, like the fitted values of \hat{Y_i} can be expressed as linear combinations of the response variable observations Y i Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 22 Covariance of Residuals • Starting with we see that but which means that and since I-H is idempotent (check) we have we can plug in MSE for σ as an estimate Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 23 ANOVA • We can express the ANOVA results in matrix form as well, starting with where leaving J is matrix of all ones, do 3x3 example Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 24 SSE • Remember • We have derive this on board and this b • Simplified Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 25 SSR • It can be shown that – for instance, remember SSR = SSTO-SSE write these on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 26 Tests and Inference • The ANOVA tests and inferences we can perform are the same as before • Only the algebraic method of getting the quantities changes • Matrix notation is a writing short-cut, not a computational shortcut Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 27 Quadratic Forms • The ANOVA sums of squares can be shown to be quadratic forms. An example of a quadratic form is given by • Note that this can be expressed in matrix notation as (where A is a symmetric matrix) do on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 28 Quadratic Forms • In general, a quadratic form is defined by A is the matrix of the quadratic form. • The ANOVA sums SSTO, SSE, and SSR are all quadratic forms. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 29 ANOVA quadratic forms • Consider the following rexpression of b’X’ • With this it is easy to see that Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 30 Inference • We can derive the sampling variance of the β vector estimator by remembering that where A is a constant matrix which yields Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 31 Variance of b • Since is symmetric we can write and thus Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 32 Variance of b • Of course this assumes that we know σ. If we don’t, we, as usual, replace it with the MSE. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 33 Mean Response • To estimate the mean response we can create the following matrix • The fit (or prediction) is then since Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 34 Variance of Mean Response • Is given by and is arrived at in the same way as for the variance of \beta • Similarly the estimated variance in matrix notation is given by Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 35 Wrap-Up • Expectation and variance of random vector and matrices • Simple linear regression in matrix form • Next: multiple regression Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 36.