Projection matrix: Difference between revisions

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If the vector of [[Response variable|response values]] is denoted by <math>\mathbf{y}</math> and the vector of fitted values by <math>\mathbf{\hat{y}}</math>,

:<math>\mathbf{\hat{y}} = \mathbf{P} \mathbf{y}.</math>

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 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:<ref>Wood, Simon N. Generalized additive models: an introduction with R. chapman and hall/CRC, 2006.</ref>

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

==Application for residuals==

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\end{align}</math>.

Therefore, since <math>\mathbf{Ax}</math> is on the column space of <math>\mathbf{A}</math>, the projection matrix, which maps <math>\mathbf{b}</math> onto <math>\mathbf{x}</math>, is just <math>\mathbf{A}</math>, or <math>\mathbf{A}\left(\mathbf{A}^\textsf{T}\mathbf{A}\right)^{-1}\mathbf{A}^\textsf{T}</math>.

== Linear model ==

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[[Category:Regression analysis]]

[[Category:Matrices (mathematics)]]