Hat Matrix Elements

Along the way I present the proo.

Hat matrix elements. The diagonal elements of the projection matrix are the leverages which describe the influence each response value has on the fitted value for that. H X XTX 1XT and determines the fitted or predicted values since The diagonal elements of H hii are called leverages and satisfy. I know that for diagonal elements we have.

Let be the hat matrix corresponding to the linear regression problem with design matrix X of full rank. In these lecture notes. The hat matrix is also known as the projection matrix because it projects the vector of observations y onto the vector of predictions thus putting the hat on y.

Every square and idempotent matrix is a projection matrix. It describes the influence each response value has on each fitted value. When I multiply things out I get 1 nSxx nj 1x2j 2nˉxxi nx2i.

Similarly I H can also be shown to be idempotent I HI H I 2H HH I H. The matrix H is the projection matrix onto the column space of X. The matrix elements of the instrument operator modeling the infrared camera used in the experiment are given by Eq.

The diagonal elements of the hat matrix are the leverages which describe the influence each observed value has on the fitted value for that same observation. From the elements of the hat matrix some important relations can be derived eg. The hat matrix H is defined in terms of the data matrix X.

The experiment was conducted by focusing the camera on the source. For things to be true the terms inside the parenthesis can be rearranged to be Sxx nxi ˉx2. It follows then that the trace sum of diagonal elements - in this case sum of 1 s will be the rank of the column space while therell be as many zeros as the dimension of the null space.