Projection matrix: Difference between revisions

Revision as of 08:46, 18 December 2020

Revision as of 08:56, 18 December 2020

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As <math>\mathbf{\hat{y}}</math> is usually pronounced "y-hat", the projection matrix <math>\mathbf{P}</math> is also named ''hat matrix'' as it "puts a [[circumflex|hat]] on <math>\mathbf{y}</math>". The formula for the vector of [[errors and residuals in statistics|residual]]s <math>\mathbf{r}</math> can also be expressed compactly using the projection matrix:

:<math>\mathbf{r} = \mathbf{y} - \mathbf{\hat{y}} = \mathbf{y} - \mathbf{P} \mathbf{y} = \left( \mathbf{I} - \mathbf{P} \right) \mathbf{y}.</math>

where <math>\mathbf{I}</math> is the [[identity matrix]]. The matrix <math>\mathbf{M} \equiv \left( \mathbf{I} - \mathbf{P} \right)</math> is sometimes referred to as the '''residual maker matrix'''. Moreover, the element in the ''i''th row and ''j''th column of <math>\mathbf{P}</math> is equal to the [[covariance]] between the ''j''th response value and the ''i''th fitted value, divided by the [[variance]] of the former:

:<math>p_{ij} = \frac{\operatorname{Cov}\left[ \hat{y}_i, y_j \right]}{\operatorname{Var}\left[y_j \right]}</math>