g02df:: Correlation and Regression Analysis (NAG Toolbox)

Description

When selecting a linear regression model it is sometimes useful to drop independent variables from the model and to examine the resulting sub-model. nag_correg_linregm_var_del (g02df) updates the

Q R

decomposition used in the computation of the linear regression model. The

Q R

decomposition may come from nag_correg_linregm_fit (g02da) or nag_correg_linregm_var_add (g02de), or a previous call to nag_correg_linregm_var_del (g02df).

For the general linear regression model with

p

independent variables fitted nag_correg_linregm_fit (g02da) or nag_correg_linregm_var_add (g02de) compute a

Q R

decomposition of the (weighted) independent variables and form an upper triangular matrix

R

and a vector

c

. To remove an independent variable

R

and

c

have to be updated. The column of

R

corresponding to the variable to be dropped is removed and the matrix is then restored to upper triangular form by applying a series of Givens rotations. The rotations are then applied to

c

. Note only the first

p

elements of

c

are affected.

The method used means that while the updated values of

R

and

c

are computed an updated value of

Q

from the

Q R

decomposition is not available so a call to nag_correg_linregm_var_add (g02de) cannot be made after a call to nag_correg_linregm_var_del (g02df).

nag_correg_linregm_update (g02dd) can be used to calculate the parameter estimates,

\hat{β}

, from the information provided by nag_correg_linregm_var_del (g02df).

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

A dataset consisting of

12

observations on four independent variables and one dependent variable is read in. The full model, including a mean term, is fitted using nag_correg_linregm_fit (g02da). The value of indx is read in and that variable dropped from the regression. The parameter estimates are calculated by nag_correg_linregm_update (g02dd) and printed. This process is repeated until indx is

0

function g02df_example


fprintf('g02df example results\n\n');

x = [ 1.0 1.4 0.0 0.0;
      1.5 2.2 0.0 0.0;
      2.0 4.5 0.0 0.0;
      2.5 6.1 0.0 0.0;
      3.0 7.1 0.0 0.0;
      3.5 7.7 0.0 0.0;
      4.0 8.3 1.0 4.0;
      4.5 8.6 1.0 4.5;
      5.0 8.8 1.0 5.0;
      5.5 9.0 1.0 5.5;
      6.0 9.3 1.0 6.0;
      6.5 9.2 1.0 6.5];
y = [ 4.32  5.21  6.49  7.10  7.94  8.53 ...
      8.84  9.02  9.27  9.43  9.68  9.83];
[n,m]  = size(x);
isx    = ones(m,1,'int64');
mean_p = 'M';
ip     = int64(m+1);

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

% Display initial results
if svd
  fprintf('Model not of full rank\n\n', irank);
end
fprintf('Residual sum of squares = %12.4e\n', rss);
fprintf('Degrees of freedom      = %4d\n', idf);

% Variables to drop
indx = int64([2, 4]);
perm = double([1:ip]);

for j = 1:numel(indx)
  % drop j-th variable
  [q, rss, ifail] = g02df( ...
                           q, indx(j), rss, 'ip', ip);

  perm(indx(j):ip-1) = perm(indx(j)+1:ip);
  ip = ip - 1;

  fprintf('\nVariable %2d dropped\n', indx(j));

  % Calculate parameter estimates
  [rss, idf, b, se, covar, svd, irank, p, ifail] = ...
  g02dd(int64(n), ip, q, rss);

  % Display results having dropped indx(j)
  if svd
    fprintf('Model not of full rank, rank = %4d\n\n', irank);
  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 = perm(1:ip)';
  fprintf('%6d%20.4e%20.4e\n',[ivar b se]');
end

g02df example results

Residual sum of squares =   8.4066e-02
Degrees of freedom      =    7

Variable  2 dropped
Residual sum of squares =   2.1239e-01
Degrees of freedom      =    8

Variable   Parameter estimate   Standard error

     1          3.6372e+00          1.5083e-01
     3          6.1264e-01          2.8007e-02
     4         -6.0154e-01          4.2335e-01
     5          1.6709e-01          7.8656e-02

Variable  4 dropped
Residual sum of squares =   3.3220e-01
Degrees of freedom      =    9

Variable   Parameter estimate   Standard error

     1          3.5974e+00          1.7647e-01
     3          6.2088e-01          3.2706e-02
     4          2.4247e-01          1.7235e-01

On entry,	$ip < 1$ ,
or	$ldq < ip$ ,
or	$indx < 1$ ,
or	$indx > ip$ ,
or	$rss < 0.0$ .

NAG Toolbox: nag_correg_linregm_var_del (g02df)

▸▿ Contents

Purpose

Syntax