naginterfaces.library.correg.linregm_​update

naginterfaces.library.correg.linregm_update(q, rss, p, tol=1e-06)

linregm_update calculates the regression parameters for a general linear regression model. It is intended to be called after linregm_obs_edit(), linregm_var_add() or linregm_var_del().

For full information please refer to the NAG Library document for g02dd

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g02/g02ddf.html

Parameters

qfloat, array-like, shape $(:,\text{ip}+1)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $rss\le 0.0$: $n$; otherwise: $\text{ip}$.

Must be the array $q$ as output by linregm_obs_edit(), linregm_var_add(), linregm_var_del() or linregm_fit_onestep(). If on entry $rss\le 0.0$ then all $\text{n}$ elements of $c$ are needed. This is provided by functions linregm_var_add(), linregm_var_del() or linregm_fit_onestep().

rssfloat

Either the residual sum of squares or a value less than or equal to $0.0$ to indicate that the residual sum of squares is to be calculated by the function.

pfloat, array-like, shape $(\text{ip}\times \text{ip}+2\times \text{ip})$

Must be array $p$ as output by linregm_fit(), linregm_var_add() or linregm_fit_onestep(), or a previous call to linregm_update.

tolfloat, optional

The value of $tol$ is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of $tol$ the stricter the criterion for selecting the singular value decomposition. If $tol=0.0$, the singular value decomposition will never be used, this may cause run time errors or inaccuracies if the independent variables are not of full rank.

Returns

rssfloat

If $rss\le 0.0$ on entry, then on exit $rss$ will contain the residual sum of squares as calculated by linregm_update.

If $rss$ was positive on entry, it will be unchanged.

idfint

The degrees of freedom associated with the residual sum of squares.

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

The estimates of the $p$ parameters, $\hat{\beta }$.

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

The standard errors of the $p$ parameters given in $b$.

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

The upper triangular part of the variance-covariance matrix of the $p$ 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 $cov[j\times (j-1)/2+i-1]$.

svdbool

If a singular value decomposition has been performed, $svd=True$, otherwise $svd=False$.

irankint

The rank of the independent variables.

If $svd=False$, $irank=\text{ip}$.

If $svd=True$, $irank$ is an estimate of the rank of the independent variables.

$irank$ is calculated as the number of singular values greater than $tol\times$ (largest singular value).

It is possible for the SVD to be carried out but $irank$ to be returned as $\text{ip}$.

pfloat, ndarray, shape $(\text{ip}\times \text{ip}+2\times \text{ip})$

Contains details of the singular value decomposition if used.

If $svd=False$, $p$ is not referenced.

If $svd=True$, the first $\text{ip}$ elements of $p$ are unchanged, the next $\text{ip}$ values contain the singular values.

The following $\text{ip}\times \text{ip}$ values contain the matrix $P^{\ast }$ stored by columns.

Raises

NagValueError

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol\ge 0.0$.

(errno $1$)

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

Constraint: if $rss\le 0.0$, $\text{ldq}\ge n$.

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

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

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

The degrees of freedom for error are less than or equal to $0$. In this case the estimates of $\beta$ are returned but not the standard errors or covariances.

(errno $3$)

SVD solution failed to converge.

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.

A general linear regression model fitted by linregm_fit() may be adjusted by adding or deleting an observation using linregm_obs_edit(), adding a new independent variable using linregm_var_add() or deleting an existing independent variable using linregm_var_del(). Alternatively a model may be constructed by a forward selection procedure using linregm_fit_onestep(). These functions compute the vector $c$ and the upper triangular matrix $R$. linregm_update takes these basic results and computes the regression coefficients, $\hat{\beta }$, their standard errors and their variance-covariance matrix.

If $R$ is of full rank, then $\hat{\beta }$ is the solution to

$$R\hat{\beta }=c_1\text{,}$$

where $c_1$ is the first $p$ elements of $c$.

If $R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of $R$,

$$\begin{array}{c}R=Q_{\ast }\left(\begin{array}{cc}D& 0\\ 0& 0\end{array}\right)P^T\text{,}\end{array}$$

where $D$ is a $k\times k$ diagonal matrix with nonzero diagonal elements, $k$ being the rank of $R$, and $Q_{\ast }$ and $P$ are $p\times p$ orthogonal matrices. This gives the solution

$$\hat{\beta }=P_1{D}^{-1}{Q}_{{\ast }_1}^T{c}_1\text{.}$$

$P_1$ being the first $k$ columns of $P$, i.e., $P=\left(P_1{P}_0\right)$, and $Q_{{\ast }_1}$ being the first $k$ columns of $Q_{\ast }$.

Details of the SVD are made available in the form of the matrix $P^{\ast }$:

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

This will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the parameters. These solutions can be obtained by calling linregm_constrain() after calling linregm_update. Only certain linear combinations of the parameters will have unique estimates; these are known as estimable functions. These can be estimated using linregm_estfunc().

The residual sum of squares required to calculate the standard errors and the variance-covariance matrix can either be input or can be calculated if additional information on $c$ for the whole sample is provided.

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