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→See also: Moore-Penrose pseudoinverse |
correlated residuals |
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Some other properties of the hat matrix are summarized in <ref>P. Gans, ''Data Fitting in the Chemical Sciences,'', Wiley, 1992.</ref>.
==Correlated residuals==
The above may be generalized to the case of correlated residuals. Suppose that the [[covariance matrix]] of the residuals is <math>\mathbf{A}</math>. Then <math>\hat{\mathbf{\beta}}=\mathbf{X}\left(\mathbf{X}^\top\mathbf{A}^{-1}\mathbf{X}\right)^{-1}\mathbf{X}^\top\mathbf{A}^{-1}\mathbf{y}</math> and the hat matrix is
:<math>\mathbf{H}=
\mathbf{X}\left(\mathbf{X}^\top\mathbf{A}^{-1}\mathbf{X}\right)^{-1}\mathbf{X}^\top\mathbf{A}^{-1}</math>
and again it may be seen that <math>\mathbf{H}^2=\mathbf{H}</math>.
== See also ==
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