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Bluelink 1 book for verifiability (refca)) #IABot (v2.0.1) (GreenC bot |
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where <math>\mathbf{\Sigma}</matH> is the [[covariance matrix]] of the error vector (and by extension, the response vector as well). For the case of linear models with [[independent and identically distributed]] errors in which <math>\mathbf{\Sigma} = \sigma^{2} \mathbf{I}</math>, this reduces to:<ref name="Hoaglin1977"/>
:<math>\mathbf{\Sigma}_\mathbf{r} = \left( \mathbf{I} - \mathbf{P} \right) \sigma^{2}</math>.
==Intuition==
[[File:Projection of a vector onto the column space of a matrix.svg|thumb|A matrix, <math>\mathbf{A}</math> has it's column space depicted as the green line. The projection of some vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math> is the vector <math>\mathbf{x}</math>]]
From the figure, it is clear that the closest point from the vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math>, is <math>\mathbf{Ax}</math>, and is one where we can draw a line orthogonal to the column space of <math>\mathbf{A}</math>. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
:<math>\mathbf{A}^{T}(\mathbf{b}-\mathbf{Ax}) = 0</math>
From there, one rearranges, so
:<math>\mathbf{A}^{T}\mathbf{b}-\mathbf{A}^{T}\mathbf{Ax} = 0</math>
:<math>\mathbf{A}^{T}\mathbf{b}=\mathbf{A}^{T}\mathbf{Ax}</math>
:<math>\mathbf{x}=(\mathbf{A}^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{b}</math>
Therefore, since <math>\mathbf{x}</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{Ax}</math>, or <math>\mathbf{A}(\mathbf{A}^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{b}</math>
== Linear model ==
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