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The '''hat matrix''', '''H''', is used in [[statistics]] to relate [[errors]] in [[residuals]] to [[experimental errors]]. Suppose that a [[linear least squares]] problem is being addressed. The model can be written as
:<math>\mathbf{y^{calc}=Jp}</math>
where '''J''' is a matrix of cefficients and '''p''' is a vector of parameters. The solution to the un-weighted least-squares equations is given by
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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 hat matrix is [[idempotent]], that is, <math>\mathbf {HH=H}</math>. Using this property it can easily be shown by [[error propagation]] that the [[variance-covariance matrix]] of the residuals is equal to <math>\mathbf{I-H }</math
== References ==
*P. Gans, ''Data Fitting in the Chemical Sciences,'', Wiley, 1992
{{statistics-stub}}
[[Category:Statistics]]
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