naginterfaces.library.correg.glm_​constrain

naginterfaces.library.correg.glm_constrain(v, c, b, s)

glm_constrain calculates the estimates of the parameters of a generalized linear model for given constraints from the singular value decomposition results.

For full information please refer to the NAG Library document for g02gk

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g02/g02gkf.html

Parameters

vfloat, array-like, shape $(\text{ip},\text{ip}+7)$

The array $v$ as returned by glm_normal(), glm_binomial(), glm_poisson() or glm_gamma().

cfloat, array-like, shape $(\text{ip},\text{iconst})$

Contains the $\text{iconst}$ constraints stored by column, i.e., the $i$th constraint is stored in the $i$th column of $c$.

bfloat, array-like, shape $\left(\text{ip}\right)$

The parameter estimates computed by using the singular value decomposition, ${\hat{\beta }}_{\text{svd}}$.

sfloat

The estimate of the scale parameter.

For results from glm_normal() and glm_gamma() then $s$ is the scale parameter for the model.

For results from glm_binomial() and glm_poisson() then $s$ should be set to $1.0$.

Returns

bfloat, ndarray, shape $\left(\text{ip}\right)$

The parameter estimates of the parameters with the constraints imposed, ${\hat{\beta }}_c$.

sefloat, ndarray, shape $\left(\text{ip}\right)$

The standard error of the parameter estimates in $b$.

covfloat, ndarray, shape $(\text{ip}\times (\text{ip}+1)/2)$

The upper triangular part of the variance-covariance matrix of the $\text{ip}$ parameter estimates given in $b$. They are stored packed by column, i.e., the covariance between the parameter estimate given in $b[i-1]$ and the parameter estimate given in $b[j-1]$, $j\ge i$, is stored in $cov[(j\times (j-1)/2+i)-1]$.

Raises

NagValueError

(errno $1$)

On entry, $\text{iconst}=⟨value⟩$.

Constraint: $\text{iconst}>0$.

(errno $1$)

On entry, $s=⟨value⟩$.

Constraint: $s>0.0$.

(errno $1$)

On entry, $\text{iconst}=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Constraint: $\text{iconst}<\text{ip}$.

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 1$.

(errno $2$)

$c$ 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.

glm_constrain computes the estimates given a set of linear constraints for a generalized linear model which is not of full rank. It is intended for use after a call to glm_normal(), glm_binomial(), glm_poisson() or glm_gamma().

In the case of a model not of full rank the functions use a singular value decomposition to find the parameter estimates, ${\hat{\beta }}_{\text{svd}}$, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix $P^{\ast }$:

$$\begin{array}{c}{P}^{\ast }=\left(\begin{array}{c}{D}^{-1}{P}_1^T\\ P_0^T\end{array}\right)\end{array}$$

as described by glm_normal(), glm_binomial(), glm_poisson() and glm_gamma(). Alternative solutions can be formed by imposing constraints on the parameters. If there are $p$ parameters and the rank of the model is $k$ then $n_c=p-k$ constraints will have to be imposed to obtain a unique solution.

Let $C$ be a $p\times n_c$ matrix of constraints, such that

$$C^T\beta =0\text{,}$$

then the new parameter estimates ${\hat{\beta }}_c$ are given by:

$$\begin{array}{c}\begin{array}{cc}{\hat{\beta }}_c& =A{\hat{\beta }}_{svd}\\ & =(I-P_0{\left(C^T{P}_0\right)}^{-1}){\hat{\beta }}_{svd}\text{, where}I\text{is the identity matrix,}\end{array}\end{array}$$

and the variance-covariance matrix is given by

$$A{P}_1{D}^{-2}{P}_1^T{A}^T$$

provided ${\left(C^T{P}_0\right)}^{-1}$ exists.

References

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

McCullagh, P and Nelder, J A, 1983, Generalized Linear Models, Chapman and Hall

Searle, S R, 1971, Linear Models, Wiley