Projection matrix

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

${\displaystyle \mathbf {y^{calc}=Jp} }$

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

${\displaystyle \mathbf {p=\left(J^{T}J\right)^{-1}J^{T}y^{obs}} }$

The vector of un-wieghted residuals, r, is given by

${\displaystyle \mathbf {r=y^{obs}-y^{calc}=y^{obs}-J\left(J^{T}J\right)^{-1}J^{T}y^{obs}} }$

The matrix ${\displaystyle \mathbf {J\left(J^{T}J\right)^{-1}J^{T}} }$ is known as the hat matrix. Thus the residuals can be expressed simply as

${\displaystyle \mathbf {r=\left(I-H\right)y^{obs}} }$

The hat matrix is idempotent, that is, ${\displaystyle \mathbf {HH=H} }$. Using this property it can easily be shown by error propagation that the variance-covariance matrix of the residuals is equal to ${\displaystyle \mathbf {I-H} }$.

References

• P. Gans, Data Fitting in the Chemical Sciences,, Wiley, 1992