NAG Toolbox: nag_correg_lars_xtx (g02mb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates the type of model to fit.

$\mathbf{mtype}=1$

LARS is performed.

$\mathbf{mtype}=2$

Forward linear stagewise regression is performed.

$\mathbf{mtype}=3$

LASSO model is fit.

$\mathbf{mtype}=4$

A positive LASSO model is fit.

Constraint: $\mathbf{mtype}=1$, $2$, $3$ or $4$.

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

$n$, the number of observations.

Constraint: $\mathbf{n}\ge 1$.

3:     $\mathrm{dtd}$ – double array

The the array dtd must be a vector of length at least $\mathbf{m}\left(\mathbf{m}+1\right)/2$ or a matrix of size at least ($\mathbf{m}$,$\mathbf{m}$)

$D^{\mathrm{T}}D$, the cross-product matrix, which along with isx, defines the design matrix cross-product $X^{\mathrm{T}}X$.

If supplied in compressed format, $\mathbf{dtd}\left(\mathit{i}\times \left(\mathit{i}-1\right)/2+\mathit{j}\right)$ must contain the cross-product of the $\mathit{i}$th and $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{m}$ and $\mathit{j}=1,2,\dots ,\mathbf{m}$. That is the cross-product stacked by columns as returned by nag_correg_ssqmat (g02bu), for example.

Otherwise $\mathbf{dtd}\left(\mathit{i},\mathit{j}\right)$ must contain the cross-product of the $\mathit{i}$th and $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,\mathbf{m}$ and $\mathit{j}=1,2,\dots ,\mathbf{m}$. It should be noted that, even though $D^{\mathrm{T}}D$ is symmetric, the full matrix must be supplied.

The matrix specified in dtd must be a valid cross-products matrix.

4:     $\mathrm{dty}\left(\mathbf{m}\right)$ – double array

$D^{\mathrm{T}}y$, the cross-product between the dependent variable, $y$, and the independent variables $D$.

5:     $\mathrm{yty}$ – double scalar

$y^{\mathrm{T}}y$, the sums of squares of the dependent variable.

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

Optional Input Parameters

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

Default: $2$

Indicates the type of preprocessing to perform on the cross-products involving the independent variables, i.e., those supplied in dtd and dty.

$\mathbf{pred}=0$

No preprocessing is performed.

$\mathbf{pred}=2$

Each independent variable is normalized, with the $j$th variable scaled by $1/\sqrt{x_j^{\mathrm{T}}{x}_j}$. The scaling factor used by variable $j$ is returned in $\mathbf{b}\left(\mathit{j},\mathbf{nstep}+1\right)$.

Constraint: $\mathbf{pred}=0$ or $2$.

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

Default: $1$

Indicates the type of data preprocessing that was perform on the dependent variable, $y$, prior to calling this function.

$\mathbf{intcpt}=0$

No preprocessing was performed.

$\mathbf{intcpt}=1$

The dependent variable, $y$, was mean centered.

Constraint: $\mathbf{intcpt}=0$ or $1$.

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

Default: the second dimension of the array dtd and the dimension of the array dty. (An error is raised if these dimensions are not equal.)

$m$, the total number of independent variables$\mathbf{m}=p$ when all variables are included.

Constraint: $\mathbf{m}\ge 1$.

4:     $\mathrm{isx}\left(\mathbf{lisx}\right)$ – int64int32nag_int array

Indicates which independent variables from dtd will be included in the design matrix, $X$.

If isx is not supplied, all variables are included in the design matrix.

Otherwise,, for $\mathit{j}=1,2,\dots ,\mathbf{m}$ when $\mathbf{isx}\left(j\right)$ must be set as follows:

$\mathbf{isx}\left(j\right)=1$

To indicate that the $j$th variable, as supplied in dtd, is included in the design matrix;

$\mathbf{isx}\left(\mathit{j}\right)=0$

To indicate that the $j$th variable, as supplied in dtd, is not included in the design matrix;

and $p={\displaystyle \sum _{\mathit{j}=1}^m}\mathbf{isx}\left(\mathit{j}\right)$

Constraint: $\mathbf{isx}\left(\mathit{j}\right)=0$ or $1$ and at least one value of $\mathbf{isx}\left(\mathit{j}\right)\ne 0$, for $\mathit{j}=1,2,\dots ,\mathbf{m}$.

5:     $\mathrm{mnstep}$ – int64int32nag_int scalar

Default:

• if $\mathbf{mtype}=1$, $\mathbf{m}$;
• otherwise $200\times \mathbf{m}$.

The maximum number of steps to carry out in the model fitting process.

If $\mathbf{mtype}=1$, i.e., a LARS is being performed, the maximum number of steps the algorithm will take is $\min \left(p,n\right)$ if $\mathbf{intcpt}=0$, otherwise $\min \left(p,n-1\right)$.

If $\mathbf{mtype}=2$, i.e., a forward linear stagewise regression is being performed, the maximum number of steps the algorithm will take is likely to be several orders of magnitude more and is no longer bound by $p$ or $n$.

If $\mathbf{mtype}=3$ or $4$, i.e., a LASSO or positive LASSO model is being fit, the maximum number of steps the algorithm will take lies somewhere between that of the LARS and forward linear stagewise regression, again it is no longer bound by $p$ or $n$.

Constraint: $\mathbf{mnstep}\ge 1$.

6:     $\mathrm{ropt}\left(\mathbf{lropt}\right)$ – double array

Optional parameters to control various aspects of the LARS algorithm.

The default value will be used for $\mathbf{ropt}\left(i\right)$ if $\mathbf{lropt}<i$. The default value will also be used if an invalid value is supplied for a particular argument, for example, setting $\mathbf{ropt}\left(i\right)=-1$ will use the default value for argument $i$.

$\mathbf{ropt}\left(1\right)$

The minimum step size that will be taken.

Default is $100\times \mathit{eps}$ is used, where $\mathit{eps}$ is the machine precision returned by nag_machine_precision (x02aj).

$\mathbf{ropt}\left(2\right)$

General tolerance, used amongst other things, for comparing correlations.

Default is $\mathbf{ropt}\left(1\right)$.

$\mathbf{ropt}\left(3\right)$

If set to $1$ then parameter estimates are rescaled before being returned. If set to $0$ then no rescaling is performed. This argument has no effect when $\mathbf{pred}=0$.

Default is for the parameter estimates to be rescaled.

Constraints:

• $\mathbf{ropt}\left(1\right)>\mathit{machine precision}$;
• $\mathbf{ropt}\left(2\right)>\mathit{machine precision}$.

Output Parameters

1:     $\mathrm{b}\left(\mathit{p},:\right)$ – double array

The second dimension of the array b will be $\mathbf{nstep}+1$.

Note: nstep is equal to $K$, the actual number of steps carried out in the model fitting process. See Description for further information.

$\beta$ the parameter estimates, with $\mathbf{b}\left(j,k\right)={\beta }_{kj}$, the parameter estimate for the $j$th variable, $j=1,2,\dots ,p$ at the $k$th step of the model fitting process, $k=1,2,\dots ,\mathbf{nstep}$.

The number of parameter estimates, $p$, is the number of variables in dtd when isx is not supplied, i.e., $p=\mathbf{m}$. Otherwise $p$ is the number of nonzero values in isx.

By default, when $\mathbf{pred}=2$ the parameter estimates are rescaled prior to being returned. If the parameter estimates are required on the normalized scale, then this can be overridden via ropt.

The values held in the remaining part of b depend on the type of preprocessing performed.

$\begin{array}{llll}\text{If }\mathbf{pred}=0& \mathbf{b}\left(j,\mathbf{nstep}+1\right)& =& 1\text{,}\\ \\ \text{if }\mathbf{pred}=2& \mathbf{b}\left(j,\mathbf{nstep}+1\right)& =& 1/\sqrt{x_j^{\mathrm{T}}{x}_j}\text{,}\end{array}$

for $j=1,2,\dots p$.

2:     $\mathrm{fitsum}\left(6,\mathbf{mnstep}+1\right)$ – double array

The second dimension of the array fitsum will be $\mathbf{nstep}+1$.

Summaries of the model fitting process. When $k=1,2,\dots ,\mathbf{nstep}$

$\mathbf{fitsum}\left(1,k\right)$

${‖{\beta }_k‖}_1$, the sum of the absolute values of the parameter estimates for the $k$th step of the modelling fitting process. If $\mathbf{pred}=2$, the scaled parameter estimates are used in the summation.

$\mathbf{fitsum}\left(2,k\right)$

${\mathrm{RSS}}_k$, the residual sums of squares for the $k$th step, where ${\mathrm{RSS}}_k={‖y-X^{\mathrm{T}}{\beta }_k‖}^2$.

$\mathbf{fitsum}\left(3,k\right)$

${\nu }_k$, approximate degrees of freedom for the $k$th step.

$\mathbf{fitsum}\left(4,k\right)$

$C_p^{\left(k\right)}$, a $C_p$-type statistic for the $k$th step, where $C_p^{\left(k\right)}=\frac{{\mathrm{RSS}}_k}{{\sigma }^2}-n+2{\nu }_k$.

$\mathbf{fitsum}\left(5,k\right)$

${\hat{C}}_k$, correlation between the residual at step $k-1$ and the most correlated variable not yet in the active set $\mathcal{A}$, where the residual at step $0$ is $y$.

$\mathbf{fitsum}\left(6,k\right)$

${\hat{\gamma }}_k$, the step size used at step $k$.

In addition

$\mathbf{fitsum}\left(1,\mathbf{nstep}+1\right)$

$0$.

$\mathbf{fitsum}\left(2,\mathbf{nstep}+1\right)$

${\mathrm{RSS}}_0$, the residual sums of squares for the null model, where ${\mathrm{RSS}}_0=y^{\mathrm{T}}y$.

$\mathbf{fitsum}\left(3,\mathbf{nstep}+1\right)$

${\nu }_0$, the degrees of freedom for the null model, where ${\nu }_0=0$ if $\mathbf{intcpt}=0$ and ${\nu }_0=1$ otherwise.

$\mathbf{fitsum}\left(4,\mathbf{nstep}+1\right)$

$C_p^{\left(0\right)}$, a $C_p$-type statistic for the null model, where $C_p^{\left(0\right)}=\frac{{\mathrm{RSS}}_0}{{\sigma }^2}-n+2{\nu }_0$.

$\mathbf{fitsum}\left(5,\mathbf{nstep}+1\right)$

${\sigma }^2$, where ${\sigma }^2=\frac{n-{\mathrm{RSS}}_K}{{\nu }_K}$ and $K=\mathbf{nstep}$.

Although the $C_p$ statistics described above are returned when $\mathbf{ifail}=\mathbf{122}$ they may not be meaningful due to the estimate ${\sigma }^2$ not being based on the saturated model.

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

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

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=11$

Constraint: $\mathbf{mtype}=1$, $2$, $3$ or $4$.

   $\mathbf{ifail}=21$

Constraint: $\mathbf{pred}=0$ or $2$.

   $\mathbf{ifail}=31$

Constraint: $\mathbf{intcpt}=0$ or $1$.

   $\mathbf{ifail}=41$

Constraint: $\mathbf{n}\ge 1$.

   $\mathbf{ifail}=51$

Constraint: $\mathbf{m}\ge 1$.

   $\mathbf{ifail}=61$

The cross-product matrix supplied in dtd is not symmetric.

   $\mathbf{ifail}=62$

Constraint: diagonal elements of $D^{\mathrm{T}}D$ must be positive.

   $\mathbf{ifail}=81$

Constraint: $\mathbf{isx}\left(i\right)=0$ or $1$ for all $i$.

   $\mathbf{ifail}=82$

On entry, all values of isx are zero.
Constraint: at least one value of isx must be nonzero.

   $\mathbf{ifail}=111$

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

   $\mathbf{ifail}=112$

A negative value for the residual sums of squares was obtained. Check the values of dtd, dty and yty.

   $\mathbf{ifail}=121$

Constraint: $\mathbf{mnstep}\ge 1$.

W  $\mathbf{ifail}=122$

Fitting process did not finished in mnstep steps. Try increasing the size of mnstep.
All output is returned as documented, up to step mnstep, however, $\sigma$ and the $C_p$ statistics may not be meaningful.

W  $\mathbf{ifail}=171$

${\sigma }^2$ is approximately zero and hence the $C_p$-type criterion cannot be calculated. All other output is returned as documented.

W  $\mathbf{ifail}=172$

${\nu }_K=n$, therefore sigma has been set to a large value. Output is returned as documented.

W  $\mathbf{ifail}=173$

Degenerate model, no variables added and $\mathbf{nstep}=0$. Output is returned as documented.

   $\mathbf{ifail}=191$

Constraint: $0\le \mathbf{lropt}\le 3$.

   $\mathbf{ifail}=-99$

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: g02mb_example

function g02mb_example


fprintf('g02mb example results\n\n');

% Going to be fitting a LAR model via g02mb
mtype = int64(1);

% Augmented matrix [D y]
dy = [10.28  1.77  9.69 15.58  8.23 10.44  -46.47;
       9.08  8.99 11.53  6.57 15.89 12.58  -35.80;
      17.98 13.10  1.04 10.45 10.12 16.68 -129.22;
      14.82 13.79 12.23  7.00  8.14  7.79  -42.44;
      17.53  9.41  6.24  3.75 13.12 17.08  -73.51;
       7.78 10.38  9.83  2.58 10.13  4.25  -26.61;
      11.95 21.71  8.83 11.00 12.59 10.52  -63.90;
      14.60 10.09 -2.70  9.89 14.67  6.49  -76.73;
       3.63  9.07 12.59 14.09  9.06  8.19  -32.64;
       6.35  9.79  9.40 12.79  8.38 16.79  -83.29;
       4.66  3.55 16.82 13.83 21.39 13.88  -16.31;
       8.32 14.04 17.17  7.93  7.39 -1.09   -5.82;
      10.86 13.68  5.75 10.44 10.36 10.06  -47.75;
       4.76  4.92 17.83  2.90  7.58 11.97   18.38;
       5.05 10.41  9.89  9.04  7.90 13.12  -54.71;
       5.41  9.32  5.27 15.53  5.06 19.84  -55.62;
       9.77  2.37  9.54 20.23  9.33  8.82  -45.28;
      14.28  4.34 14.23 14.95 18.16 11.03  -22.76;
      10.17  6.80  3.17  8.57 16.07 15.93 -104.32;
       5.39  2.67  6.37 13.56 10.68  7.35  -55.94];

% Number of observations in the dataset
n = int64(size(dy,1));

% Calculate the means and cross-product matrix around the mean
mean_p = 'M';
[~,wmean,dtd,ifail] = g02bu( ...
                             dy,'mean_p',mean_p);

% Number of variables
m = int64(size(dy,2) - 1);

% The first pm elements of dtd contain the cross-products of D
% with itself, the next m elements hold the cross-product of D
% and y and the last element holds the cross-product of y with
% itself
pm = m*(m+1)/2;

% g02mb can issue warnings, but return sensible results,
% so save current warning state and turn warnings on
warn_state = nag_issue_warnings();
nag_issue_warnings(true);

[b,fitsum,ifail] = g02mb( ...
                          mtype,n,dtd(1:pm),dtd((pm+1):(pm+m)),dtd(end));

% Reset the warning state to its initial value
nag_issue_warnings(warn_state);

% Print the results
ip = size(b,1);
nstep = size(b,2) - 1;

fprintf('  Step %s Parameter Estimate\n ',repmat(' ',1,max(ip-2,0)*5));
fprintf(repmat('-',1,5+ip*10));
fprintf('\n');
for k = 1:nstep
  fprintf('  %3d',k);
  for j = 1:ip
    fprintf(' %9.3f',b(j,k));
  end
  fprintf('\n');
end
fprintf('\n');
fprintf(' alpha: %9.3f\n', wmean(m+1));
fprintf('\n');
fprintf('  Step     Sum      RSS       df       Cp       Ck     Step Size\n ');
fprintf(repmat('-',1,64));
fprintf('\n');
for k = 1:nstep
  fprintf('  %3d %9.3f %9.3f %6.0f  %9.3f %9.3f %9.3f\n', ...
          k,fitsum(1,k),fitsum(2,k),fitsum(3,k), ...
          fitsum(4,k),fitsum(5,k),fitsum(6,k));
end
fprintf('\n');
fprintf(' sigma^2: %9.3f\n', fitsum(5,nstep+1));

% Plot the parameter estimates
fig1 = figure;
ip = size(b,1);
nstep = size(b,2) - 2;

% Extract the sum of the absolute parameter estimates
xpos = transpose(repmat(fitsum(1,1:nstep),ip,1));

% Extract the parameter estimates
ypos = transpose(b(1:ip,1:nstep));

% Start both xpos and ypos at zero
xpos = [zeros(1,ip);xpos];
ypos = [zeros(1,ip);ypos];

% Get min and max for X and Y
xmin = min(min(xpos));
xmax = max(max(xpos));
ymin = min(min(ypos));
ymax = max(max(ypos));

% Get a range that is 10% past the data
ext = 1 + [-0.1 0.1];
xrng = [min(xmin*ext),max(xmax*ext)];
yrng = [min(ymin*ext),max(ymax*ext)];

% Extend the end of the lines we plot to cover this range
xpos = [xpos;xrng(2)*ones(1,ip)];
ypos = [ypos;ypos(end,:)];

% Produce the plot
plot(xpos,ypos);

% Change the axis limits
xlim(xrng);
ylim(yrng);

% Add title and labels
title({'{\bf g02mb Example Plot}'; ...
       'Estimates for LAR model fitted to simulated dataset'});
xlabel('{\bf \Sigma_j |\beta_{kj} |}');
ylabel('{\bf Parameter Estimates (\beta_{kj})}');

% Add legend
label = [repmat('\beta_{k',ip,1) num2str(transpose(linspace(1,ip,ip))) ...
         repmat('}',ip,1)];
h = legend(label,'Location','SouthOutside','Orientation','Horizontal');
set(h,'FontSize',get(h,'FontSize')*0.8);

g02mb example results

  Step                      Parameter Estimate
 -----------------------------------------------------------------
    1     0.000     0.000     3.125     0.000     0.000     0.000
    2     0.000     0.000     3.792     0.000     0.000    -0.713
    3    -0.446     0.000     3.998     0.000     0.000    -1.151
    4    -0.628    -0.295     4.098     0.000     0.000    -1.466
    5    -1.060    -1.056     4.110    -0.864     0.000    -1.948
    6    -1.073    -1.132     4.118    -0.935    -0.059    -1.981

 alpha:   -50.037

  Step     Sum      RSS       df       Cp       Ck     Step Size
 ----------------------------------------------------------------
    1    72.446  8929.855      2     13.355   123.227    72.446
    2   103.385  6404.701      3      7.054    50.781    24.841
    3   126.243  5258.247      4      5.286    30.836    16.225
    4   145.277  4657.051      5      5.309    19.319    11.587
    5   198.223  3959.401      6      5.016    12.266    24.520
    6   203.529  3954.571      7      7.000     0.910     2.198

 sigma^2:   304.198