Residuals \(\left( \boldsymbol{\hat{\varepsilon}} = \mathbf{My} \right)\)

Source: R/epsilonhat.R

Calculates residuals using $$ \boldsymbol{\hat{\varepsilon}} = \mathbf{My} . $$ where $$ \mathbf{M} = \mathbf{I} - \mathbf{P} \\ = \mathbf{I} - \mathbf{X} \left( \mathbf{X}^{T} \mathbf{X} \right)^{-1} \mathbf{X}^{T} . $$

.My(y, M = NULL, X = NULL, P = NULL)

Arguments

y

Numeric vector of length n or n by 1 matrix. The vector \(\mathbf{y}\) is an \(n \times 1\) vector of observations on the regressand variable.
M

n by n numeric matrix. The \(n \times n\) residual maker matrix \(\left( \mathbf{M} \right)\).
X

n by k numeric matrix. The data matrix \(\mathbf{X}\) (also known as design matrix, model matrix or regressor matrix) is an \(n \times k\) matrix of \(n\) observations of \(k\) regressors, which includes a regressor whose value is 1 for each observation on the first column.
P

n by n numeric matrix. The \(n \times n\) projection matrix \(\left( \mathbf{P} \right)\).

Value

Returns an \(n \times 1\) matrix of residuals \(\left( \boldsymbol{\hat{\varepsilon}} \right)\), that is, the difference between the observed \(\left( \mathbf{y} \right)\) and predicted \(\left( \mathbf{\hat{y}} \right)\) values of the regressand variable \(\left( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right)\).

Details

If M = NULL, the M matrix is computed using M() with X as a required argument and P as an optional argument. If M is provided, X and P are not needed.

References

Wikipedia: Errors and Residuals

See also

Other residuals functions: .tepsilonhat(), .yminusyhat(), My(), epsilonhat(), tepsilonhat(), yminusyhat()

Author

Ivan Jacob Agaloos Pesigan