naginterfaces.library.correg.linregm_​fit_​newvar

naginterfaces.library.correg.linregm_fit_newvar(rss, irank, cov, q, svd, p, y, wk, wt=None)

linregm_fit_newvar calculates the estimates of the parameters of a general linear regression model for a new dependent variable after a call to linregm_fit().

For full information please refer to the NAG Library document for g02dg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02dgf.html

Parameters

rssfloat

The residual sum of squares for the original dependent variable.

irankint

The rank of the independent variables, as given by linregm_fit().

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

The covariance matrix of the parameter estimates as given by linregm_fit().

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

The results of the $QR$ decomposition as returned by linregm_fit().

svdbool

Indicates if a singular value decomposition was used by linregm_fit().

$svd=True$

A singular value decomposition was used by linregm_fit().

$svd=False$

A singular value decomposition was not used by linregm_fit().

pfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $not\left(svd\right)$: $\text{ip}$; otherwise: $\text{ip}\times \text{ip}+2\times \text{ip}$.

Details of the $QR$ decomposition and SVD, if used, as returned in array $p$ by linregm_fit().

If $svd=False$, only the first $\text{ip}$ elements of $p$ are used; these contain the zeta values for the $QR$ decomposition (see lapackeig.dgeqrf for details).

If $svd=True$, the first $\text{ip}$ elements of $p$ contain the zeta values for the $QR$ decomposition (see lapackeig.dgeqrf for details) and the next $\text{ip}\times \text{ip}+\text{ip}$ elements of $p$ contain details of the singular value decomposition.

yfloat, array-like, shape $\left(n\right)$

The new dependent variable, $y_{\text{new}}$.

wkfloat, array-like, shape $(5\times (\text{ip}-1)+\text{ip}\times \text{ip})$

If $svd=True$, $wk$ must be unaltered from the previous call to linregm_fit() or linregm_fit_newvar.

If $svd=False$, $wk$ is used as workspace.

wtNone or float, array-like, shape $\left(n\right)$, optional

If provided $wt$ must contain the weights to be used with the model.

If $wt[i-1]=0.0$, the $i$th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights.

If $wt$ is not provided the effective number of observations is $n$.

Returns

rssfloat

The residual sum of squares for the new dependent variable.

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 $cov[(j\times (j-1)/2+i)-1]$.

qfloat, ndarray, shape $(n,\text{ip}+1)$

The first column of $q$ contains the new values of $c$, the remainder of $q$ will be unchanged.

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

The least squares estimates of the parameters of the regression model, $\hat{\beta }$.

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

The standard error of the estimates of the parameters.

resfloat, ndarray, shape $\left(n\right)$

The residuals for the new regression model.

Raises

NagValueError

(errno $1$)

On entry, $irank=⟨value⟩$.

Constraint: $irank>0$.

(errno $1$)

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

Constraint: if $svd=False$, $irank=\text{ip}$.

(errno $1$)

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

Constraint: if $svd=True$, $irank\le \text{ip}$.

(errno $1$)

On entry, $\text{weight}=⟨value⟩$.

Constraint: $\text{weight}=\text{'W'}$ or $\text{'U'}$.

(errno $1$)

On entry, $rss=⟨value⟩$.

Constraint: $rss>0.0$.

(errno $1$)

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

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

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

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

(errno $2$)

On entry, $wt[⟨value⟩]<0.0$.

Constraint: $wt[i-1]\ge 0.0$, for $i=1,2,\dots ,n$.

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_fit_newvar uses the results given by linregm_fit() to fit the same set of independent variables to a new dependent variable.

linregm_fit() computes a $QR$ decomposition of the matrix of $p$ independent variables and also, if the model is not of full rank, a singular value decomposition (SVD). These results can be used to compute estimates of the parameters for a general linear model with a new dependent variable. The $QR$ decomposition leads to the formation of an upper triangular $p\times p$ matrix $R$ and an $n\times n$ orthogonal matrix $Q$. In addition the vector $c=Q^{T}y$ (or $Q^T{W}^{1/2}y$) is computed. For a new dependent variable, $y_{new}$, linregm_fit_newvar computes a new value of $c=Q^T{y}_{\text{new}}$ or $Q^T{W}^{1/2}{y}_{\text{new}}$.

If $R$ is of full rank, then the least squares parameter estimates, $\hat{\beta }$, are 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, then linregm_fit() will have computed an 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 by linregm_fit() in the form of the matrix $P^{\ast }$:

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

The matrix $Q_{\ast }$ is made available through the workspace of linregm_fit().

In addition to parameter estimates, the new residuals are computed and the variance-covariance matrix of the parameter estimates are found by scaling the variance-covariance matrix for the original regression.

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