Hat Matrix

Properties and Interpretation Week 5 Lecture 1 1 Hat Matrix 11 From Observed to Fitted Values The OLS estimator was found to be given by the p 1 vector b XT X 1XT y.

Hat matrix. For multiple regression models the formula for calculating the hat matrix diagonal elements h i requires the use of matrix algebra and is. Check out the video for scenes from the launch party and a tour of the store. The Hat Matrix Elements h i In Section 138 h i was defined for the simple linear regression model when constructing the confidence interval estimate of the mean response.

The hat matrix in regression is just another name for the projection matrix. The predicted values ybcan then be written as by X b XXT X 1XT y. Denote it by uThis implies that Hu u because a projection matrix is idempotent.

Hat matrix is a n n symmetric and idempotent matrix with many special properties play an important role in diagnostics of regression analysis by transforming the vector of observed responses Y into the vector of fitted responses Y. The hat matrix H is defined in terms of the data matrix X. HatMatrix is an n -by- n matrix in.

It follows that the hat matrix His symmetric too. So you have actually fitted d i s t β 0 β 1 s p e e d ϵ. Where H XXT X 1XT is an n nmatrix which puts the hat on y and is therefore.

The least-squares estimate hatbeta XTX-1XTy. The diagonal elements of H hii are called leverages and satisfy. You rarely want to drop the intercept term but if you did you can do lm dist 0 speed cars or even lm dist speed - 1 cars.

Analysis of elements of the projection hat matrix plays an important role in regression diagnostics because the diagonal elements of this matrix H ii x i X T X 1 x T i indicate the presence of leverage points which are not detected by analysis of residuals. Y X y XX0X 1X0y y Hy H XX0X 1X0 so called the hat matrix because it transforms y to y The diagonal elements of the hat matrix the h is are proportional to the distance between X i from X i Hence h i is a simple measure of the leverage of Y i. Let Hbe a symmetric idempotent real valued matrix.