NAG Toolbox: nag_correg_linregm_fit_newvar (g02dg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{rss}$ – double scalar

The residual sum of squares for the original dependent variable.

Constraint: $\mathbf{rss}>0.0$.

2:     $\mathrm{ip}$ – int64int32nag_int scalar

$p$, the number of independent variables (including the mean if fitted).

Constraint: $1\le \mathbf{ip}\le \mathbf{n}$.

3:     $\mathrm{irank}$ – int64int32nag_int scalar

The rank of the independent variables, as given by nag_correg_linregm_fit (g02da).

Constraint: $\mathbf{irank}>0$, and if $\mathbf{svd}=\mathit{false}$, then $\mathbf{irank}=\mathbf{ip}$, else $\mathbf{irank}\le \mathbf{ip}$.

4:     $\mathrm{covar}\left(\mathbf{ip}\times \left(\mathbf{ip}+1\right)/2\right)$ – double array

The covariance matrix of the parameter estimates as given by nag_correg_linregm_fit (g02da).

5:     $\mathrm{q}\left(\mathit{ldq},\mathbf{ip}+1\right)$ – double array

ldq, the first dimension of the array, must satisfy the constraint $\mathit{ldq}\ge \mathbf{n}$.

The results of the $QR$ decomposition as returned by nag_correg_linregm_fit (g02da).

6:     $\mathrm{svd}$ – logical scalar

Indicates if a singular value decomposition was used by nag_correg_linregm_fit (g02da).

$\mathbf{svd}=\mathit{true}$

A singular value decomposition was used by nag_correg_linregm_fit (g02da).

$\mathbf{svd}=\mathit{false}$

A singular value decomposition was not used by nag_correg_linregm_fit (g02da).

7:     $\mathrm{p}\left(:\right)$ – double array

The dimension of the array p must be at least $\mathbf{ip}$ if $\mathbf{svd}=\mathit{false}$, and at least $\mathbf{ip}\times \mathbf{ip}+2\times \mathbf{ip}$ otherwise

Details of the $QR$ decomposition and SVD, if used, as returned in array p by nag_correg_linregm_fit (g02da).

If $\mathbf{svd}=\mathit{false}$, only the first ip elements of p are used; these contain the zeta values for the $QR$ decomposition (see nag_lapack_dgeqrf (f08ae) for details).

If $\mathbf{svd}=\mathit{true}$, the first ip elements of p contain the zeta values for the $QR$ decomposition (see nag_lapack_dgeqrf (f08ae) for details) and the next $\mathbf{ip}\times \mathbf{ip}+\mathbf{ip}$ elements of p contain details of the singular value decomposition.

8:     $\mathrm{y}\left(\mathbf{n}\right)$ – double array

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

9:     $\mathrm{wk}\left(5\times \left(\mathbf{ip}-1\right)+\mathbf{ip}\times \mathbf{ip}\right)$ – double array

If $\mathbf{svd}=\mathit{true}$, wk must be unaltered from the previous call to nag_correg_linregm_fit (g02da) or nag_correg_linregm_fit_newvar (g02dg).

If $\mathbf{svd}=\mathit{false}$, wk is used as workspace.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the array y and the first dimension of the array q. (An error is raised if these dimensions are not equal.)

$n$, the number of observations.

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

2:     $\mathrm{wt}\left(:\right)$ – double array

The dimension of the array wt must be at least $\mathbf{n}$ if $\mathit{weight}=\text{'W'}$, and at least $1$ otherwise

If provided>, wt must contain the weights to be used in the weighted regression.

If $\mathbf{wt}\left(i\right)=0.0$, 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 $n$.

Constraint: if $\mathit{weight}=\text{'W'}$, $\mathbf{wt}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.

Output Parameters

1:     $\mathrm{rss}$ – double scalar

The residual sum of squares for the new dependent variable.

2:     $\mathrm{covar}\left(\mathbf{ip}\times \left(\mathbf{ip}+1\right)/2\right)$ – double array

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

3:     $\mathrm{q}\left(\mathit{ldq},\mathbf{ip}+1\right)$ – double array

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

4:     $\mathrm{b}\left(\mathbf{ip}\right)$ – double array

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

5:     $\mathrm{se}\left(\mathbf{ip}\right)$ – double array

The standard error of the estimates of the parameters.

6:     $\mathrm{res}\left(\mathbf{n}\right)$ – double array

The residuals for the new regression model.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{ip}<1$,
or	$\mathbf{n}<\mathbf{ip}$,
or	$\mathbf{irank}\le 0$,
or	$\mathbf{svd}=\mathit{false}$ and $\mathbf{irank}\ne \mathbf{ip}$,
or	$\mathbf{svd}=\mathit{true}$ and $\mathbf{irank}>\mathbf{ip}$,
or	$\mathit{ldq}<\mathbf{n}$,
or	$\mathbf{rss}\le 0.0$,
or	$\mathit{weight}\ne \text{'U'}$ or $\text{'W'}$.

   $\mathbf{ifail}=2$

On entry,

$\mathit{weight}=\text{'W'}$ and a value of $\mathbf{wt}<0.0$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g02dg_example

function g02dg_example


fprintf('g02dg example results\n\n');

x = [1, 0, 0, 0;
     0, 0, 0, 1;
     0, 1, 0, 0;
     0, 0, 1, 0;
     0, 0, 0, 1;
     0, 1, 0, 0;
     0, 0, 0, 1;
     1, 0, 0, 0;
     0, 0, 1, 0;
     1, 0, 0, 0;
     0, 0, 1, 0;
     0, 1, 0, 0];
y = [33.63;     39.62;     38.18;     41.46;     38.02;     35.83;
     35.99;     36.58;     42.92;     37.80;     40.43;     37.89];
ynew = [63; 69; 68; 71; 68; 65; 65; 66; 72; 67; 70; 67];

[n,m]  = size(x);
isx    = ones(m,1,'int64');
mean_p = 'M';
ip     = int64(m+1);

% Fit general linear regression model to y
[rss, idf, b, se, covar, res, h, q, svd, irank, p, wk, ifail] = ...
  g02da(mean_p, x, isx, ip, y);

% Display results for y
fprintf('Results for original y-variable using g02da\n\n');
if svd
  fprintf('Model not of full rank\n\n');
end
fprintf('Residual sum of squares = %12.4e\n', rss);
fprintf('Degrees of freedom      = %4d\n', idf);
fprintf('\nVariable   Parameter estimate   Standard error\n\n');
ivar = double([1:ip]');
fprintf('%6d%20.4e%20.4e\n',[ivar b se]');

% Fit same model to ynew
[rss, covar, q, b, se, res, ifail] = ...
    g02dg( ...
           rss, ip, irank, covar, q, svd, p, ynew, wk);

% Display results for ynew
fprintf('\nResults for second y-variable using g02dg\n\n');
fprintf('Residual sum of squares = %12.4e\n', rss);
fprintf('Degrees of freedom      = %4d\n', idf);
fprintf('\nVariable   Parameter estimate   Standard error\n\n');
ivar = double([1:ip]');
fprintf('%6d%20.4e%20.4e\n',[ivar b se]');

g02dg example results

Results for original y-variable using g02da

Model not of full rank

Residual sum of squares =   2.2227e+01
Degrees of freedom      =    8

Variable   Parameter estimate   Standard error

     1          3.0557e+01          3.8494e-01
     2          5.4467e+00          8.3896e-01
     3          6.7433e+00          8.3896e-01
     4          1.1047e+01          8.3896e-01
     5          7.3200e+00          8.3896e-01

Results for second y-variable using g02dg

Residual sum of squares =   2.4000e+01
Degrees of freedom      =    8

Variable   Parameter estimate   Standard error

     1          5.4067e+01          4.0000e-01
     2          1.1267e+01          8.7178e-01
     3          1.2600e+01          8.7178e-01
     4          1.6933e+01          8.7178e-01
     5          1.3267e+01          8.7178e-01