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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_lars (g02ma)

## Purpose

nag_correg_lars (g02ma) performs Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO).

## Syntax

[b, fitsum, ifail] = g02ma(mtype, m, d, y, 'pred', pred, 'prey', prey, 'n', n, 'isx', isx, 'mnstep', mnstep, 'ropt', ropt)
[b, fitsum, ifail] = nag_correg_lars(mtype, m, d, y, 'pred', pred, 'prey', prey, 'n', n, 'isx', isx, 'mnstep', mnstep, 'ropt', ropt)

## Description

nag_correg_lars (g02ma) implements the LARS algorithm of Efron et al. (2004) as well as the modifications needed to perform forward stagewise linear regression and fit LASSO and positive LASSO models.
Given a vector of $n$ observed values, $y=\left\{{y}_{i}:i=1,2,\dots ,n\right\}$ and an $n×p$ design matrix $X$, where the $j$th column of $X$, denoted ${x}_{j}$, is a vector of length $n$ representing the $j$th independent variable ${x}_{j}$, standardized such that $\sum _{\mathit{i}=1}^{n}{x}_{ij}=0$, and $\sum _{\mathit{i}=1}^{n}{x}_{ij}^{2}=1$ and a set of model parameters $\beta$ to be estimated from the observed values, the LARS algorithm can be summarised as:
 1 Set $k=1$ and all coefficients to zero, that is $\beta =0$. 2 Find the variable most correlated with $y$, say ${x}_{{j}_{1}}$. Add ${x}_{{j}_{1}}$ to the ‘most correlated’ set $\mathcal{A}$. If $p=1$ go to 8. 3 Take the largest possible step in the direction of ${x}_{{j}_{1}}$ (i.e., increase the magnitude of ${\beta }_{{j}_{1}}$) until some other variable, say ${x}_{{j}_{2}}$, has the same correlation with the current residual, $y-{x}_{{j}_{1}}{\beta }_{{j}_{1}}$. 4 Increment $k$ and add ${x}_{{j}_{k}}$ to $\mathcal{A}$. 5 If $\left|\mathcal{A}\right|=p$ go to 8. 6 Proceed in the ‘least angle direction’, that is, the direction which is equiangular between all variables in $\mathcal{A}$, altering the magnitude of the parameter estimates of those variables in $\mathcal{A}$, until the $k$th variable, ${x}_{{j}_{k}}$, has the same correlation with the current residual. 7 Go to 4. 8 Let $K=k$.
As well as being a model selection process in its own right, with a small number of modifications the LARS algorithm can be used to fit the LASSO model of Tibshirani (1996), a positive LASSO model, where the independent variables enter the model in their defined direction (i.e., ${\beta }_{kj}\ge 0$), forward stagewise linear regression (Hastie et al. (2001)) and forward selection (Weisberg (1985)). Details of the required modifications in each of these cases are given in Efron et al. (2004).
The LASSO model of Tibshirani (1996) is given by
 $minimize α , βk∈ ℝp y- α- XT βk 2 subject to βk1 ≤tk$
for all values of ${t}_{k}$, where $\alpha =\stackrel{-}{y}={n}^{-1}\sum _{\mathit{i}=1}^{n}{y}_{i}$. The positive LASSO model is the same as the standard LASSO model, given above, with the added constraint that
 $βkj ≥ 0 , j=1,2,…,p .$
Unlike the standard LARS algorithm, when fitting either of the LASSO models, variables can be dropped as well as added to the set $\mathcal{A}$. Therefore the total number of steps $K$ is no longer bounded by $p$.
Forward stagewise linear regression is an iterative procedure of the form:
1. Initialize $k=1$ and the vector of residuals ${r}_{0}=y-\alpha$.
2. For each $j=1,2,\dots ,p$ calculate ${c}_{j}={x}_{j}^{\mathrm{T}}{r}_{k-1}$. The value ${c}_{j}$ is therefore proportional to the correlation between the $j$th independent variable and the vector of previous residual values, ${r}_{k}$.
3. Calculate ${j}_{k}=\underset{j}{\mathrm{argmax}}\phantom{\rule{0.25em}{0ex}}\left|{c}_{j}\right|$, the value of $j$ with the largest absolute value of ${c}_{j}$.
4. If $\left|{c}_{{j}_{k}}\right|<\epsilon$ then go to 7.
5. Update the residual values, with
 $rk = rk-1 + δ ⁢ ​ ​ signcjk ⁢ xjk$
where $\delta$ is a small constant and $\mathrm{sign}\left({c}_{{j}_{k}}\right)=-1$ when ${c}_{{j}_{k}}<0$ and $1$ otherwise.
6. Increment $k$ and go to 2.
7. Set $K=k$.
If the largest possible step were to be taken, that is $\delta =\left|{c}_{{j}_{k}}\right|$ then forward stagewise linear regression reverts to the standard forward selection method as implemented in nag_correg_linregm_fit_onestep (g02ee).
The LARS procedure results in $K$ models, one for each step of the fitting process. In order to aid in choosing which is the most suitable Efron et al. (2004) introduced a ${C}_{p}$-type statistic given by
 $Cpk = y- XT βk 2 σ2 -n+2⁢νk,$
where ${\nu }_{k}$ is the approximate degrees of freedom for the $k$th step and
 $σ2 = n-yTyνK .$
One way of choosing a model is therefore to take the one with the smallest value of ${C}_{p}^{\left(k\right)}$.

## References

Efron B, Hastie T, Johnstone I and Tibshirani R (2004) Least Angle Regression The Annals of Statistics (Volume 32) 2 407–499
Hastie T, Tibshirani R and Friedman J (2001) The Elements of Statistical Learning: Data Mining, Inference and Prediction Springer (New York)
Tibshirani R (1996) Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistics Society, Series B (Methodological) (Volume 58) 1 267–288
Weisberg S (1985) Applied Linear Regression Wiley

## 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:     $\mathrm{m}$int64int32nag_int scalar
$m$, the total number of independent variables${\mathbf{m}}=p$ when all variables are included.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathrm{d}\left(\mathit{ldd},:\right)$ – double array
The first dimension of the array d must be at least ${\mathbf{n}}$.
The second dimension of the array d must be at least ${\mathbf{m}}$.
$D$, the data, which along with pred and isx, defines the design matrix $X$. The $\mathit{i}$th observation for the $\mathit{j}$th variable must be supplied in ${\mathbf{d}}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
4:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
$y$, the observations on the dependent variable.

### Optional Input Parameters

1:     $\mathrm{pred}$int64int32nag_int scalar
Default: ${\mathbf{pred}}=3$
Indicates the type of data preprocessing to perform on the independent variables supplied in d to comply with the standardized form of the design matrix.
${\mathbf{pred}}=0$
No preprocessing is performed.
${\mathbf{pred}}=1$
Each of the independent variables, ${x}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$, are mean centered prior to fitting the model. The means of the independent variables, $\stackrel{-}{x}$, are returned in b, with ${\stackrel{-}{x}}_{\mathit{j}}={\mathbf{b}}\left(\mathit{j},\mathbf{nstep}+2\right)$, for $\mathit{j}=1,2,\dots ,p$.
${\mathbf{pred}}=2$
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 ${\mathbf{b}}\left(\mathit{j},\mathbf{nstep}+1\right)$.
${\mathbf{pred}}=3$
As ${\mathbf{pred}}=1$ and $2$, all of the independent variables are mean centered prior to being normalized.
Constraint: ${\mathbf{pred}}=0$, $1$, $2$ or $3$.
2:     $\mathrm{prey}$int64int32nag_int scalar
Default: ${\mathbf{prey}}=1$
Indicates the type of data preprocessing to perform on the dependent variable supplied in y.
${\mathbf{prey}}=0$
No preprocessing is performed, this is equivalent to setting $\alpha =0$.
${\mathbf{prey}}=1$
The dependent variable, $y$, is mean centered prior to fitting the model, so $\alpha =\stackrel{-}{y}$. Which is equivalent to fitting a non-penalized intercept to the model and the degrees of freedom etc. are adjusted accordingly.
The value of $\alpha$ used is returned in ${\mathbf{fitsum}}\left(1,\mathbf{nstep}+1\right)$.
Constraint: ${\mathbf{prey}}=0$ or $1$.
3:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array y and the first dimension of the array d. (An error is raised if these dimensions are not equal.)
$n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathrm{isx}\left(\mathbf{lisx}\right)$int64int32nag_int array
Indicates which independent variables from d 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 d, is included in the design matrix;
${\mathbf{isx}}\left(j\right)=0$
To indicated that the $j$th variable, as supplied in d, is not included in the design matrix;
and $p=\sum _{\mathit{j}=1}^{m}{\mathbf{isx}}\left(\mathit{j}\right)$
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 ${\mathbf{mtype}}=1$, ${\mathbf{mnstep}}={\mathbf{m}}$;
• otherwise ${\mathbf{mnstep}}=200×{\mathbf{m}}$.
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 $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,n\right)$ if ${\mathbf{prey}}=0$, otherwise $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,n-1\right)$.
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}. 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 $100×\mathit{eps}$ 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 $1$, parameter estimates are rescaled before being returned.
If set to $0$, no rescaling is performed.
This argument has no effect when ${\mathbf{pred}}=0$ or $1$.
Default is for the parameter estimates to be rescaled.
${\mathbf{ropt}}\left(4\right)$
If set to $1$, it is assumed that the model contains an intercept during the model fitting process and when calculating the degrees of freedom.
If set to $0$, no intercept is assumed.
This has no effect on the amount of preprocessing performed on y.
Default is to treat the model as having an intercept when ${\mathbf{prey}}=1$ and as not having an intercept when ${\mathbf{prey}}=0$.
${\mathbf{ropt}}\left(5\right)$
As implemented, the LARS algorithm can either work directly with $y$ and $X$, or it can work with the cross-product matrices, ${X}^{\mathrm{T}}y$ and ${X}^{\mathrm{T}}X$. In most cases it is more efficient to work with the cross-product matrices. This flag allows you direct control over which method is used, however, the default value will usually be the best choice.
If ${\mathbf{ropt}}\left(5\right)=1$, $y$ and $X$ are worked with directly.
If ${\mathbf{ropt}}\left(5\right)=0$, the cross-product matrices are used.
Default is $1$ when $p\ge 500$ and $n and $0$ otherwise.
Constraints:
• ;
• ;
• ${\mathbf{ropt}}\left(3\right)=0$ or $1$;
• ${\mathbf{ropt}}\left(4\right)=0$ or $1$;
• ${\mathbf{ropt}}\left(5\right)=0$ or $1$.

### Output Parameters

1:     $\mathrm{b}\left(\mathit{p},:\right)$ – double array
The second dimension of the array b will be $\mathbf{nstep}+2$.
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, $k=1,2,\dots ,\mathbf{nstep}$.
By default, when ${\mathbf{pred}}=2$ or $3$ 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.
If ${\mathbf{pred}}=0$,
$\begin{array}{lll}{\mathbf{b}}\left(j,\mathbf{nstep}+1\right)& =& 1\\ {\mathbf{b}}\left(j,\mathbf{nstep}+2\right)& =& 0\end{array}$
If ${\mathbf{pred}}=1$,
$\begin{array}{lll}{\mathbf{b}}\left(j,\mathbf{nstep}+1\right)& =& 1\\ {\mathbf{b}}\left(j,\mathbf{nstep}+2\right)& =& {\stackrel{-}{x}}_{j}\end{array}$
If ${\mathbf{pred}}=2$,
$\begin{array}{lll}{\mathbf{b}}\left(j,\mathbf{nstep}+1\right)& =& 1/\sqrt{{x}_{j}^{\mathrm{T}}{x}_{j}}\\ {\mathbf{b}}\left(j,\mathbf{nstep}+2\right)& =& 0\end{array}$
If ${\mathbf{pred}}=3$,
$\begin{array}{lll}{\mathbf{b}}\left(j,\mathbf{nstep}+1\right)& =& 1/\sqrt{{\left({x}_{j}-{\stackrel{-}{x}}_{j}\right)}^{\mathrm{T}}\left({x}_{j}-{\stackrel{-}{x}}_{j}\right)}\\ {\mathbf{b}}\left(j,\mathbf{nstep}+2\right)& =& {\stackrel{-}{x}}_{j}\end{array}$
for $j=1,2,\dots ,p$.
The number of parameter estimates, $p$, is the number of columns in d when isx is not supplied, i.e., $p={\mathbf{m}}$. Otherwise $p$ is the number of nonzero values in isx.
2:     $\mathrm{fitsum}\left(6,{\mathbf{mnstep}}+1\right)$ – double array
The second dimension of the array fitsum 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.
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 ${\mathbf{pred}}=2$ or $3$, 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 ${C}_{p}^{\left(k\right)}=\frac{{\mathrm{RSS}}_{k}}{{\sigma }^{2}}-n+2{\nu }_{k}$.
${\mathbf{fitsum}}\left(5,k\right)$
${\stackrel{^}{C}}_{k}$, 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 $y$.
${\mathbf{fitsum}}\left(6,k\right)$
${\stackrel{^}{\gamma }}_{k}$, the step size used at step $k$.
${\mathbf{fitsum}}\left(1,\mathbf{nstep}+1\right)$
$\alpha$, with $\alpha =\stackrel{-}{y}$ if ${\mathbf{prey}}=1$ and $0$ otherwise.
${\mathbf{fitsum}}\left(2,\mathbf{nstep}+1\right)$
${\mathrm{RSS}}_{0}$, the residual sums of squares for the null model, where ${\mathrm{RSS}}_{0}={y}^{\mathrm{T}}y$ when ${\mathbf{prey}}=0$ and ${\mathrm{RSS}}_{0}={\left(y-\stackrel{-}{y}\right)}^{\mathrm{T}}\left(y-\stackrel{-}{y}\right)$ otherwise.
${\mathbf{fitsum}}\left(3,\mathbf{nstep}+1\right)$
${\nu }_{0}$, the degrees of freedom for the null model, where ${\nu }_{0}=0$ if ${\mathbf{prey}}=0$ 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 ${\mathbf{ifail}}={\mathbf{112}}$ 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

Note: nag_correg_lars (g02ma) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

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: ${\mathbf{pred}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=31$
Constraint: ${\mathbf{prey}}=0$ or $1$.
${\mathbf{ifail}}=41$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=51$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=81$
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: ${\mathbf{mnstep}}\ge 1$.
W  ${\mathbf{ifail}}=112$
Fitting process did not finish 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  ${\mathbf{ifail}}=161$
${\sigma }^{2}$ is approximately zero and hence the ${C}_{p}$-type criterion cannot be calculated. All other output is returned as documented.
W  ${\mathbf{ifail}}=162$
${\nu }_{K}=n$, therefore $\sigma$ has been set to a large value. Output is returned as documented.
W  ${\mathbf{ifail}}=163$
Degenerate model, no variables added and $\mathbf{nstep}=0$. Output is returned as documented.
${\mathbf{ifail}}=181$
Constraint: $0\le \mathbf{lropt}\le 5$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

nag_correg_lars (g02ma) returns the parameter estimates at various points along the solution path of a LARS, LASSO or stagewise regression analysis. If the solution is required at a different set of points, for example when performing cross-validation, then nag_correg_lars_param (g02mc) can be used.
For datasets with a large number of observations, $n$, it may be impractical to store the full $X$ matrix in memory in one go. In such instances the cross-product matrices ${X}^{\mathrm{T}}y$ and ${X}^{\mathrm{T}}X$ can be calculated, using for example, multiple calls to nag_correg_ssqmat (g02bu) and nag_correg_ssqmat_combine (g02bz), and nag_correg_lars_xtx (g02mb) called to perform the analysis.
The amount of workspace used by nag_correg_lars (g02ma) depends on whether the cross-product matrices are being used internally (as controlled by ropt). If the cross-product matrices are being used then nag_correg_lars (g02ma) internally allocates approximately $2{p}^{2}+4p+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(np\right)$ elements of real storage compared to ${p}^{2}+3p+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(np\right)+2n+n×p$ elements when $X$ and $y$ are used directly. In both cases approximately $5p$ elements of integer storage are also used. If a forward linear stagewise analysis is performed than an additional ${p}^{2}+5p$ elements of real storage are required.

## Example

This example performs a LARS on a simulated dataset with $20$ observations and $6$ independent variables.
function g02ma_example

fprintf('g02ma example results\n\n');

% Going to be fitting a LAR model via g02ma
mtype = int64(1);

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

% Dependent variable
y = [-46.47; -35.80; -129.22;  -42.44; -73.51;
-26.61; -63.90;  -76.73;  -32.64; -83.29;
-16.31;  -5.82;  -47.75;   18.38; -54.71;
-55.62; -45.28;  -22.76; -104.32; -55.94];

% g02ma 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] = g02ma(mtype,d,y);

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

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

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

% Plot the parameter estimates
fig1 = figure;

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

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

% 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);

title({'{\bf g02ma Example Plot}'; ...
'Estimates for LAR model fitted to simulated dataset'});
xlabel('{\bf \Sigma_j |\beta_{kj} |}');
ylabel('{\bf Parameter Estimates (\beta_{kj})}');

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);


g02ma 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

This example plot shows the regression coefficients (${\beta }_{k}$) plotted against the scaled absolute sum of the parameter estimates (${‖{\beta }_{k}‖}_{1}$).