Projection matrix: Difference between revisions

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Revision as of 23:03, 25 November 2018

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As <math>\mathbf{\hat{y}}</math> is usually pronounced "y-hat", the projection matrix 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{u}</math> can also be expressed compactly using the projection matrix:

:<math>\mathbf{u} = \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 '''annihilatorresidual 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>

\begin{align}