Projection matrix: Difference between revisions

Revision as of 22:51, 12 January 2008

Revision as of 22:52, 12 January 2008

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The '''hat matrix''', '''H''', is used in [[statistics]] to relate [[errors]] in [[errors and residuals in statistics|residuals]] to [[observational error|experimental errors]]. Suppose that a [[linear least squares]] problem is being addressed. The model can be written as

Delete, moving to 'Hat Matrix'

:<math>\mathbf{y^{calc}=Jp}</math>

where '''J''' is a matrix of coefficients and '''p''' is a vector of parameters. The solution to the un-weighted least-squares equations is given by

:<math>\mathbf{p=\left(J^TJ \right)^{-1} J^Ty^{obs}}</math>

The vector of un-weighted residuals, '''r''', is given by

:<math>\mathbf {r=y^{obs}-y^{calc}=y^{obs}-J \left(J^TJ \right)^{-1} J^Ty^{obs}}</math>

The matrix <math>\mathbf {J \left(J^TJ \right)^{-1} J^T}</math> is known as the hat matrix. Thus, the residuals can be expressed simply as

:<math>\mathbf{r=\left(I-H \right) y^{obs}}</math>

The [[variance-covariance matrix]] of the residuals is, by [[error propagation]], equal to <math>\mathbf{\left(I-H \right)M\left(I-H \right) }</math>, where '''M''' is the variance-covariance matrix of the observations.

The hat matrix is [[idempotent]], that is, <math>\mathbf {HH=H}</math>. Some other useful properties of the hat matrix are summarized in <ref>P. Gans, ''Data Fitting in the Chemical Sciences,'', Wiley, 1992.</ref>

== See also ==

[[Studentized residuals]]

== References ==

<references />

[[Category:Statistics]]