Projection matrix: Difference between revisions

Revision as of 13:04, 18 March 2009

Revision as of 14:19, 18 March 2009

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==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 matrixsince is

:<math> \hat{\mathbf{\beta}} = \mathbf{X}\left(\mathbf{X}^\top\mathbf{A}^{-1}\mathbf{X}\right)^{-1}

:<math>\mathbf{H}=

\mathbf{X}\left( \mathbf{X}^\top\mathbf{A}^{-1}\mathbf{X}\right)^{-1},\mathbf{X}^\top\mathbf{A}^{-1y}</math>,

</math>

the hat matrix is thus

and again it may be seen that <math>\mathbf{H}^2=\mathbf{H}</math>.

:<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 '''H'''<mathsup>\mathbf{H}^2=\mathbf{H}</mathsup> = '''H'''.

== See also ==