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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_robustm_user (g02hd)

## Purpose

nag_correg_robustm_user (g02hd) performs bounded influence regression ($M$-estimates) using an iterative weighted least squares algorithm.

## Syntax

[x, y, wgt, theta, k, sigma, rs, nit, ifail] = g02hd(chi, psi, psip0, beta, indw, isigma, x, y, wgt, theta, sigma, 'n', n, 'm', m, 'tol', tol, 'eps', eps, 'maxit', maxit, 'nitmon', nitmon)
[x, y, wgt, theta, k, sigma, rs, nit, ifail] = nag_correg_robustm_user(chi, psi, psip0, beta, indw, isigma, x, y, wgt, theta, sigma, 'n', n, 'm', m, 'tol', tol, 'eps', eps, 'maxit', maxit, 'nitmon', nitmon)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 23: nitmon, tol, maxit and eps were made optional

## Description

For the linear regression model
 $y=Xθ+ε,$
 where $y$ is a vector of length $n$ of the dependent variable, $X$ is a $n$ by $m$ matrix of independent variables of column rank $k$, $\theta$ is a vector of length $m$ of unknown arguments, and $\epsilon$ is a vector of length $n$ of unknown errors with var $\left({\epsilon }_{i}\right)={\sigma }^{2}$,
nag_correg_robustm_user (g02hd) calculates the M-estimates given by the solution, $\stackrel{^}{\theta }$, to the equation
 $∑i=1nψri/σwiwixij=0, j=1,2,…,m,$ (1)
 where ${r}_{i}$ is the $i$th residual, i.e., the $i$th element of the vector $r=y-X\stackrel{^}{\theta }$, $\psi$ is a suitable weight function, ${w}_{i}$ are suitable weights such as those that can be calculated by using output from nag_correg_robustm_wts (g02hb), and $\sigma$ may be estimated at each iteration by the median absolute deviation of the residuals $\stackrel{^}{\sigma }={\mathrm{med}}_{i}\left[\left|{r}_{i}\right|\right]/{\beta }_{1}$
or as the solution to
 $∑i=1nχri/σ^wiwi2=n-kβ2$
for a suitable weight function $\chi$, where ${\beta }_{1}$ and ${\beta }_{2}$ are constants, chosen so that the estimator of $\sigma$ is asymptotically unbiased if the errors, ${\epsilon }_{i}$, have a Normal distribution. Alternatively $\sigma$ may be held at a constant value.
The above describes the Schweppe type regression. If the ${w}_{i}$ are assumed to equal $1$ for all $i$, then Huber type regression is obtained. A third type, due to Mallows, replaces (1) by
 $∑i=1nψri/σwixij=0, j=1,2,…,m.$
This may be obtained by use of the transformations
 $wi* ←wi yi* ←yiwi xij* ←xijwi, j= 1,2,…,m$
(see Marazzi (1987)).
The calculation of the estimates of $\theta$ can be formulated as an iterative weighted least squares problem with a diagonal weight matrix $G$ given by
 $Gii= ψri/σwi ri/σwi , ri≠0 ψ′0, ri=0. .$
The value of $\theta$ at each iteration is given by the weighted least squares regression of $y$ on $X$. This is carried out by first transforming the $y$ and $X$ by
 $y~i =yiGii x~ij =xijGii, j=1,2,…,m$
and then using nag_linsys_real_gen_solve (f04jg) . If $X$ is of full column rank then an orthogonal-triangular ($QR$) decomposition is used; if not, a singular value decomposition is used.
Observations with zero or negative weights are not included in the solution.
Note:  there is no explicit provision in the function for a constant term in the regression model. However, the addition of a dummy variable whose value is $1.0$ for all observations will produce a value of $\stackrel{^}{\theta }$ corresponding to the usual constant term.
nag_correg_robustm_user (g02hd) is based on routines in ROBETH, see Marazzi (1987).

## References

Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley
Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust and bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{chi}$ – function handle or string containing name of m-file
If ${\mathbf{isigma}}>0$, chi must return the value of the weight function $\chi$ for a given value of its argument. The value of $\chi$ must be non-negative.
[result] = chi(t)

Input Parameters

1:     $\mathrm{t}$ – double scalar
The argument for which chi must be evaluated.

Output Parameters

1:     $\mathrm{result}$ – double scalar
The value of the weight function $\chi$ evaluated at t.
If ${\mathbf{isigma}}\le 0$, the actual argument chi may be the string nag_correg_robustm_user_dummy_chi (g02hdz). (nag_correg_robustm_user_dummy_chi (g02hdz) is included in the NAG Toolbox.)
2:     $\mathrm{psi}$ – function handle or string containing name of m-file
psi must return the value of the weight function $\psi$ for a given value of its argument.
[result] = psi(t)

Input Parameters

1:     $\mathrm{t}$ – double scalar
The argument for which psi must be evaluated.

Output Parameters

1:     $\mathrm{result}$ – double scalar
The value of the weight function $\psi$ evaluated at t.
3:     $\mathrm{psip0}$ – double scalar
The value of $\psi \prime \left(0\right)$.
4:     $\mathrm{beta}$ – double scalar
If ${\mathbf{isigma}}<0$, beta must specify the value of ${\beta }_{1}$.
For Huber and Schweppe type regressions, ${\beta }_{1}$ is the $75$th percentile of the standard Normal distribution (see nag_stat_inv_cdf_normal (g01fa)). For Mallows type regression ${\beta }_{1}$ is the solution to
 $1n∑i=1nΦβ1/wi=0.75,$
where $\Phi$ is the standard Normal cumulative distribution function (see nag_specfun_cdf_normal (s15ab)).
If ${\mathbf{isigma}}>0$, beta must specify the value of ${\beta }_{2}$.
 $β2= ∫-∞∞χzϕzdz, in the Huber case; β2= 1n∑i=1nwi∫-∞∞χzϕzdz, in the Mallows case; β2= 1n∑i=1nwi2∫-∞∞χz/wiϕzdz, in the Schweppe case;$
where $\varphi$ is the standard normal density, i.e., $\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-\frac{1}{2}{x}^{2}\right)$.
If ${\mathbf{isigma}}=0$, beta is not referenced.
Constraint: if ${\mathbf{isigma}}\ne 0$, ${\mathbf{beta}}>0.0$.
5:     $\mathrm{indw}$int64int32nag_int scalar
Determines the type of regression to be performed.
${\mathbf{indw}}=0$
Huber type regression.
${\mathbf{indw}}<0$
Mallows type regression.
${\mathbf{indw}}>0$
Schweppe type regression.
6:     $\mathrm{isigma}$int64int32nag_int scalar
Determines how $\sigma$ is to be estimated.
${\mathbf{isigma}}=0$
$\sigma$ is held constant at its initial value.
${\mathbf{isigma}}<0$
$\sigma$ is estimated by median absolute deviation of residuals.
${\mathbf{isigma}}>0$
$\sigma$ is estimated using the $\chi$ function.
7:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
ldx, the first dimension of the array, must satisfy the constraint $\mathit{ldx}\ge {\mathbf{n}}$.
The values of the $X$ matrix, i.e., the independent variables. ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must contain the $\mathit{i}\mathit{j}$th element of ${\mathbf{x}}$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
If ${\mathbf{indw}}<0$, during calculations the elements of x will be transformed as described in Description. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input x and the output x.
8:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The data values of the dependent variable.
${\mathbf{y}}\left(\mathit{i}\right)$ must contain the value of $y$ for the $\mathit{i}$th observation, for $\mathit{i}=1,2,\dots ,n$.
If ${\mathbf{indw}}<0$, during calculations the elements of y will be transformed as described in Description. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input y and the output y.
9:     $\mathrm{wgt}\left({\mathbf{n}}\right)$ – double array
The weight for the $\mathit{i}$th observation, for $\mathit{i}=1,2,\dots ,n$.
If ${\mathbf{indw}}<0$, during calculations elements of wgt will be transformed as described in Description. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input wgt and the output wgt.
If ${\mathbf{wgt}}\left(i\right)\le 0$, the $i$th observation is not included in the analysis.
If ${\mathbf{indw}}=0$, wgt is not referenced.
10:   $\mathrm{theta}\left({\mathbf{m}}\right)$ – double array
Starting values of the argument vector $\theta$. These may be obtained from least squares regression. Alternatively if ${\mathbf{isigma}}<0$ and ${\mathbf{sigma}}=1$ or if ${\mathbf{isigma}}>0$ and sigma approximately equals the standard deviation of the dependent variable, $y$, then ${\mathbf{theta}}\left(\mathit{i}\right)=0.0$, for $\mathit{i}=1,2,\dots ,m$ may provide reasonable starting values.
11:   $\mathrm{sigma}$ – double scalar
A starting value for the estimation of $\sigma$. sigma should be approximately the standard deviation of the residuals from the model evaluated at the value of $\theta$ given by theta on entry.
Constraint: ${\mathbf{sigma}}>0.0$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays y, wgt and the first dimension of the array x. (An error is raised if these dimensions are not equal.)
$n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
2:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array theta and the second dimension of the array x. (An error is raised if these dimensions are not equal.)
$m$, the number of independent variables.
Constraint: $1\le {\mathbf{m}}<{\mathbf{n}}$.
3:     $\mathrm{tol}$ – double scalar
Default: $5e-5$
The relative precision for the final estimates. Convergence is assumed when both the relative change in the value of sigma and the relative change in the value of each element of theta are less than tol.
It is advisable for tol to be greater than .
Constraint: ${\mathbf{tol}}>0.0$.
4:     $\mathrm{eps}$ – double scalar
Default: $5e-6$
A relative tolerance to be used to determine the rank of $X$. See nag_linsys_real_gen_solve (f04jg) for further details.
If  or ${\mathbf{eps}}>1.0$ then machine precision will be used in place of tol.
A reasonable value for eps is $5.0×{10}^{-6}$ where this value is possible.
5:     $\mathrm{maxit}$int64int32nag_int scalar
Default: $50$
The maximum number of iterations that should be used during the estimation.
A value of ${\mathbf{maxit}}=50$ should be adequate for most uses.
Constraint: ${\mathbf{maxit}}>0$.
6:     $\mathrm{nitmon}$int64int32nag_int scalar
Default: $0$
Determines the amount of information that is printed on each iteration.
${\mathbf{nitmon}}\le 0$
No information is printed.
${\mathbf{nitmon}}>0$
On the first and every nitmon iterations the values of sigma, theta and the change in theta during the iteration are printed.
When printing occurs the output is directed to the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).

### Output Parameters

1:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
Unchanged, except as described above.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
Unchanged, except as described above.
3:     $\mathrm{wgt}\left({\mathbf{n}}\right)$ – double array
Unchanged, except as described above.
4:     $\mathrm{theta}\left({\mathbf{m}}\right)$ – double array
The M-estimate of ${\theta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$.
5:     $\mathrm{k}$int64int32nag_int scalar
The column rank of the matrix $X$.
6:     $\mathrm{sigma}$ – double scalar
The final estimate of $\sigma$ if ${\mathbf{isigma}}\ne 0$ or the value assigned on entry if ${\mathbf{isigma}}=0$.
7:     $\mathrm{rs}\left({\mathbf{n}}\right)$ – double array
The residuals from the model evaluated at final value of theta, i.e., rs contains the vector $\left(y-X\stackrel{^}{\theta }\right)$.
8:     $\mathrm{nit}$int64int32nag_int scalar
The number of iterations that were used during the estimation.
9:     $\mathrm{ifail}$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_robustm_user (g02hd) 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.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}\le 1$, or ${\mathbf{m}}<1$, or ${\mathbf{n}}\le {\mathbf{m}}$, or $\mathit{ldx}<{\mathbf{n}}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{beta}}\le 0.0$, and ${\mathbf{isigma}}\ne 0$, or ${\mathbf{sigma}}\le 0.0$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{tol}}\le 0.0$, or ${\mathbf{maxit}}\le 0$.
${\mathbf{ifail}}=4$
A value returned by the chi function is negative.
${\mathbf{ifail}}=5$
During iterations a value of ${\mathbf{sigma}}\le 0.0$ was encountered.
${\mathbf{ifail}}=6$
A failure occurred in nag_linsys_real_gen_solve (f04jg) . This is an extremely unlikely error. If it occurs, please contact NAG.
W  ${\mathbf{ifail}}=7$
The weighted least squares equations are not of full rank. This may be due to the $X$ matrix not being of full rank, in which case the results will be valid. It may also occur if some of the ${G}_{ii}$ values become very small or zero, see Further Comments. The rank of the equations is given by k. If the matrix just fails the test for nonsingularity then the result ${\mathbf{ifail}}={\mathbf{7}}$ and ${\mathbf{k}}={\mathbf{m}}$ is possible (see nag_linsys_real_gen_solve (f04jg)).
${\mathbf{ifail}}=8$
The function has failed to converge in maxit iterations.
${\mathbf{ifail}}=9$
Having removed cases with zero weight, the value of ${\mathbf{n}}-{\mathbf{k}}\le 0$, i.e., no degree of freedom for error. This error will only occur if ${\mathbf{isigma}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy of the results is controlled by tol. For the accuracy of the weighted least squares see nag_linsys_real_gen_solve (f04jg).

In cases when ${\mathbf{isigma}}\ne 0$ it is important for the value of sigma to be of a reasonable magnitude. Too small a value may cause too many of the winsorized residuals, i.e., $\psi \left({r}_{i}/\sigma \right)$, to be zero, which will lead to convergence problems and may trigger the ${\mathbf{ifail}}={\mathbf{7}}$ error.
By suitable choice of the functions chi and psi this function may be used for other applications of iterative weighted least squares.
For the variance-covariance matrix of $\theta$ see nag_correg_robustm_user_varmat (g02hf).

## Example

Having input $X$, $Y$ and the weights, a Schweppe type regression is performed using Huber's $\psi$ function. The function BETCAL calculates the appropriate value of ${\beta }_{2}$.
```function g02hd_example

fprintf('g02hd example results\n\n');

global dchi
dchi = 1.5;

x = [1, -1, -1;
1, -1,  1;
1,  1, -1;
1,  1,  1;
1,  0,  3];
y   = [10.5;    11.3;     12.6;     13.4;     17.1   ];
wgt = [0.4039;   0.5012;   0.4039;   0.5012;   0.3862];

[n,m] = size(x);

% Calculate beta
[beta] = betcal(wgt);

% Control Parameters
psip0 = 1;
indw = int64(1);
isigma = int64(1);

% Intial values
sigma = 1;
theta = zeros(m,1);

% Perform bounded influence regression
[x, y, wgt, theta, k, sigma, rs, nit, ifail] = ...
g02hd( ...
@chi, @psi, psip0, beta, indw, isigma, x, y, wgt, theta, sigma);

fprintf(' iterations to convergence = %4d\n', nit);
fprintf('                         k = %4d\n',k);
fprintf('                     sigma = %9.4f\n',sigma);
fprintf('Theta:\n');
disp(theta');
fprintf('\n  Weights  Residuals\n');
fprintf('%9.4f%9.4f\n', [wgt rs]');

function [result] = chi(t)
global dchi

if (abs(t) < dchi)
ps=t;
else
ps=dchi;
end
result = ps*ps/2;

function [result] = psi(t)
global dchi

if t < -dchi
result = -dchi;
elseif abs(t) < dchi
result = t;
else
result = dchi;
end;

function [beta] = betcal(wgt)
%  Calculate beta for Schweppe type regression
global dchi

n = numel(wgt);
amaxex = -log(x02ak);
anormc = sqrt(2*pi);
d2 = dchi*dchi;
beta = 0;
for i = 1:n
w2 = wgt(i)*wgt(i);
dw = wgt(i)*dchi;
[pc, ifail] = s15ab(dw);
dw2 = dw*dw;
if dw2<amaxex
dc = exp(-dw2/2)/anormc;
else
dc = 0;
end
b = (-dw*dc+pc-0.5)/w2 + (1-pc)*d2;
beta = b*w2/n + beta;
end
```
```g02hd example results

iterations to convergence =    5
k =    3
sigma =    2.7783
Theta:
12.2321    1.0500    1.2464

Weights  Residuals
0.4039   0.5643
0.5012  -1.1286
0.4039   0.5643
0.5012  -1.1286
0.3862   1.1286
```