naginterfaces.library.correg.linregm_​constrain

naginterfaces.library.correg.linregm_constrain(p, c, b, rss, idf)[source]

linregm_constrain calculates the estimates of the parameters of a general linear regression model for given constraints from the singular value decomposition results.

For full information please refer to the NAG Library document for g02dk

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g02/g02dkf.html

Parameters
pfloat, array-like, shape

As returned by linregm_fit() and linregm_update().

cfloat, array-like, shape

The constraints stored by column, i.e., the th constraint is stored in the th column of .

bfloat, array-like, shape

The parameter estimates computed by using the singular value decomposition, .

rssfloat

The residual sum of squares as returned by linregm_fit() or linregm_update().

idfint

The degrees of freedom associated with the residual sum of squares as returned by linregm_fit() or linregm_update().

Returns
bfloat, ndarray, shape

The parameter estimates of the parameters with the constraints imposed, .

sefloat, ndarray, shape

The standard error of the parameter estimates in .

covfloat, ndarray, shape

The upper triangular part of the variance-covariance matrix of the parameter estimates given in . They are stored packed by column, i.e., the covariance between the parameter estimate given in and the parameter estimate given in , , is stored in .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

does not give a model of full rank.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

linregm_constrain computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to linregm_fit() or linregm_update().

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates, , and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix :

as described by linregm_fit() and linregm_update().

Alternative solutions can be formed by imposing constraints on the parameters. If there are parameters and the rank of the model is , then constraints will have to be imposed to obtain a unique solution.

Let be a matrix of constraints, such that

then the new parameter estimates are given by

where is the identity matrix, and the variance-covariance matrix is given by

provided exists.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Hammarling, S, 1985, The singular value decomposition in multivariate statistics, SIGNUM Newsl. (20(3)), 2–25

Searle, S R, 1971, Linear Models, Wiley