naginterfaces.library.correg.robustm

naginterfaces.library.correg.robustm(indw, ipsi, isigma, indc, x, y, cpsi, h1, h2, h3, cucv, dchi, theta, sigma, tol=5e-05, maxit=50, nitmon=0, io_manager=None)

robustm performs bounded influence regression ($M$-estimates). Several standard methods are available.

For full information please refer to the NAG Library document for g02ha

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02haf.html

Parameters

indwint

Specifies the type of regression to be performed.

$indw<0$

Mallows type regression with Maronna’s proposed weights.

$indw=0$

Huber type regression.

$indw>0$

Schweppe type regression with Krasker–Welsch weights.

ipsiint

Specifies which $\psi$ function is to be used.

$ipsi=0$

$\psi \left(t\right)=t$, i.e., least squares.

$ipsi=1$

Huber’s function.

$ipsi=2$

Hampel’s piecewise linear function.

$ipsi=3$

Andrew’s sine wave.

$ipsi=4$

Tukey’s bi-weight.

isigmaint

Specifies how $\sigma$ is to be estimated.

$isigma<0$

$\sigma$ is estimated by median absolute deviation of residuals.

$isigma=0$

$\sigma$ is held constant at its initial value.

$isigma>0$

$\sigma$ is estimated using the $\chi$ function.

indcint

If $indw\ne 0$, $indc$ specifies the approximations used in estimating the covariance matrix of $\hat{\theta }$.

$indc=1$

Averaging over residuals.

$indc\ne 1$

Replacing expected by observed.

$indw=0$

$indc$ is not referenced.

xfloat, array-like, shape $(n,m)$

The values of the $X$ matrix, i.e., the independent variables. $x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}\text{j}$th element of $X$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

If $indw<0$, then during calculations the elements of $x$ will be transformed as described in Notes.

Before exit the inverse transformation will be applied.

As a result there may be slight differences between the input $x$ and the output $x$.

yfloat, array-like, shape $\left(n\right)$

The data values of the dependent variable.

$y[\text{i}-1]$ must contain the value of $y$ for the $\text{i}$th observation, for $\text{i}=1,2,\dots ,n$.

If $indw<0$, then during calculations the elements of $y$ will be transformed as described in Notes.

Before exit the inverse transformation will be applied.

As a result there may be slight differences between the input $y$ and the output $y$.

cpsifloat

If $ipsi=1$, $cpsi$ must specify the parameter, $c$, of Huber’s $\psi$ function.

If $ipsi\ne 1$ on entry, $cpsi$ is not referenced.

h1float

If $ipsi=2$, $h1$, $h2$, and $h3$ must specify the parameters $h_1$, $h_2$, and $h_3$, of Hampel’s piecewise linear $\psi$ function. $h1$, $h2$, and $h3$ are not referenced if $ipsi\ne 2$.

h2float

If $ipsi=2$, $h1$, $h2$, and $h3$ must specify the parameters $h_1$, $h_2$, and $h_3$, of Hampel’s piecewise linear $\psi$ function. $h1$, $h2$, and $h3$ are not referenced if $ipsi\ne 2$.

h3float

If $ipsi=2$, $h1$, $h2$, and $h3$ must specify the parameters $h_1$, $h_2$, and $h_3$, of Hampel’s piecewise linear $\psi$ function. $h1$, $h2$, and $h3$ are not referenced if $ipsi\ne 2$.

cucvfloat

If $indw<0$, must specify the value of the constant, $c$, of the function $u$ for Maronna’s proposed weights.

If $indw>0$, must specify the value of the function $u$ for the Krasker–Welsch weights.

If $indw=0$, is not referenced.

dchifloat

$d$, the constant of the $\chi$ function. $dchi$ is not referenced if $ipsi=0$, or if $isigma\le 0$.

thetafloat, array-like, shape $\left(m\right)$

Starting values of the parameter vector $\theta$. These may be obtained from least squares regression. Alternatively if $isigma<0$ and $sigma=1$ or if $isigma>0$ and $sigma$ approximately equals the standard deviation of the dependent variable, $y$, then $theta[\text{i}-1]=0.0$, for $\text{i}=1,2,\dots ,m$ may provide reasonable starting values.

sigmafloat

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.

tolfloat, optional

The relative precision for the calculation of $A$ (if $indw\ne 0$), the estimates of $\theta$ and the estimate of $\sigma$ (if $isigma\ne 0$). Convergence is assumed when the relative change in all elements being considered is less than $tol$.

If $indw<0$ and $isigma<0$, $tol$ is also used to determine the precision of ${\beta }_1$.

It is advisable for $tol$ to be greater than $100\times \text{machine precision}$.

maxitint, optional

The maximum number of iterations that should be used in the calculation of $A$ (if $indw\ne 0$), and of the estimates of $\theta$ and $\sigma$, and of ${\beta }_1$ (if $indw<0$ and $isigma<0$).

A value of $maxit=50$ should be adequate for most uses.

nitmonint, optional

The amount of information that is printed on each iteration.

$nitmon=0$

No information is printed.

$nitmon\ne 0$

The current estimate of $\theta$, the change in $\theta$ during the current iteration and the current value of $\sigma$ are printed on the first and every $abs\left(nitmon\right)$ iterations.

Also, if $indw\ne 0$ and $nitmon>0$, then information on the iterations to calculate $A$ is printed.

This is the current estimate of $A$ and the maximum value of $S_{ij}$ (see Notes).

When printing occurs the output is directed to the file object associated with the advisory I/O unit (see FileObjManager).

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

xfloat, ndarray, shape $(n,m)$

Unchanged, except as described above.

yfloat, ndarray, shape $\left(n\right)$

Unchanged, except as described above.

thetafloat, ndarray, shape $\left(m\right)$

$theta[\text{i}-1]$ contains the M-estimate of ${\theta }_{\text{i}}$, for $\text{i}=1,2,\dots ,m$.

sigmafloat

Contains the final estimate of $\sigma$ if $isigma\ne 0$ or the value assigned on entry if $isigma=0$.

cfloat, ndarray, shape $(m,m)$

The diagonal elements of $c$ contain the estimated asymptotic standard errors of the estimates of $\theta$, i.e., $c[i-1,i-1]$ contains the estimated asymptotic standard error of the estimate contained in $theta[i-1]$.

The elements above the diagonal contain the estimated asymptotic correlation between the estimates of $\theta$, i.e., $c[i-1,j-1]$, $1\le i<j\le m$ contains the asymptotic correlation between the estimates contained in $theta[i-1]$ and $theta[j-1]$.

The elements below the diagonal contain the estimated asymptotic covariance between the estimates of $\theta$, i.e., $c[i-1,j-1]$, $1\le j<i\le m$ contains the estimated asymptotic covariance between the estimates contained in $theta[i-1]$ and $theta[j-1]$.

rsfloat, ndarray, shape $\left(n\right)$

The residuals from the model evaluated at final value of $theta$, i.e., $rs$ contains the vector $(y-X\hat{\theta })$.

wgtfloat, ndarray, shape $\left(n\right)$

The vector of weights. $wgt[\text{i}-1]$ contains the weight for the $\text{i}$th observation, for $\text{i}=1,2,\dots ,n$.

statfloat, ndarray, shape $\left(4\right)$

The following values are assigned to $stat$:

$stat\left[0\right]={\beta }_1$ if $isigma<0$, or $stat\left[0\right]={\beta }_2$ if $isigma>0$.

$stat\left[1\right]=$ number of iterations used to calculate $A$.

$stat\left[2\right]=$ number of iterations used to calculate final estimates of $\theta$ and $\sigma$.

$stat\left[3\right]=k$, the rank of the weighted least squares equations.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$ and $\text{ldx}=⟨value⟩$.

Constraint: $\text{ldx}\ge n$.

(errno $1$)

On entry, $n=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $n>m$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry: $ipsi=⟨value⟩$.

Constraint: $ipsi=0$, $1$, $2$, $3$ or $4$.

(errno $3$)

On entry, $cucv=⟨value⟩$.

Constraint: $cucv\ge m$.

(errno $3$)

On entry, $cucv=⟨value⟩$.

Constraint: $cucv\ge \sqrt{m}$.

(errno $3$)

On entry: $h1$, $h2$, $h3$ incorrectly set.

Constraint: $0.0\le h1\le h2\le h3$ and $h3>0.0$.

(errno $3$)

On entry: $cpsi=⟨value⟩$.

Constraint: $cpsi>0.0$.

(errno $3$)

On entry, $dchi=⟨value⟩$.

Constraint: $dchi>0.0$.

(errno $3$)

On entry, $sigma=⟨value⟩$.

Constraint: $sigma>0.0$.

(errno $4$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit>0$.

(errno $4$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $5$)

The number of iterations required to calculate the weights exceeds $maxit$. (Only if $indw\ne 0$.)

(errno $6$)

The number of iterations required to calculate ${\beta }_1$ exceeds $maxit$. (Only if $indw<0$ and $isigma<0$.)

(errno $7$)

Iterations to calculate estimates of $theta$ failed to converge in $maxit$ iterations: $maxit=⟨value⟩$.

(errno $7$)

The number of iterations required to calculate $\theta$ and $\sigma$ exceeds $maxit$.

(errno $12$)

Error degrees of freedom $n-k\le 0$, where $n=⟨value⟩$ and the rank of $x$, $k=⟨value⟩$.

(errno $12$)

Estimated value of $sigma$ is zero.

Warns

NagAlgorithmicWarning

(errno $8$)

Weighted least squares equations not of full rank: rank $=⟨value⟩$.

(errno $9$)

Failure to invert matrix while calculating covariance.

(errno $10$)

Factor for covariance matrix $=0$, uncorrected ${\left(X^{T}X\right)}^{-1}$ given.

(errno $11$)

Variance of an element of $theta\le 0.0$, correlations set to $0$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

For the linear regression model

$$y=X\theta +ϵ\text{,}$$

where	$y$ is a vector of length $n$ of the dependent variable,
	$X$ is an $n\times m$ matrix of independent variables of column rank $k$,
	$\theta$ is a vector of length $m$ of unknown parameters,
and	$ϵ$ is a vector of length $n$ of unknown errors with $var\left({ϵ}_i\right)={\sigma }^2$,

robustm calculates the M-estimates given by the solution, $\hat{\theta }$, to the equation

$$\sum _{i=1}^n\psi \left(r_i/\left(\sigma w_i\right)\right)w_i{x}_{ij}=0\text{,}j=1,2,\dots ,m\text{,}$$

where	$r_i$ is the $i$th residual, i.e., the $i$th element of $r=y-X\hat{\theta }$,
	$\psi$ is a suitable weight function,
	$w_i$ are suitable weights,
and	$\sigma$ may be estimated at each iteration by the median absolute deviation of the residuals $\hat{\sigma }={med}_i\left(\left[\left|r_i\right|\right]/{\beta }_1\right)$

or as the solution to

$$\sum _{i=1}^n\chi \left(r_i/\left(\hat{\sigma }{w}_i\right)\right)w_i^2=(n-k){\beta }_2$$

for 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, ${ϵ}_i$, 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

$$\sum _{i=1}^n\psi \left(r_i/\sigma \right)w_i{x}_{ij}=0\text{,}j=1,2,\dots ,m\text{.}$$

This may be obtained by use of the transformations

$$\begin{array}{c}\begin{array}{cc}{w}_i^{\ast }\leftarrow \sqrt{w_i}& \\ y_i^{\ast }\leftarrow y_i\sqrt{w_i}& \\ x_{ij}^{\ast }\leftarrow x_{ij}\sqrt{w_i}\text{,}& j=1,2,\dots ,m\end{array}\end{array}$$

(see Section 3 of Marazzi (1987a)).

For Huber and Schweppe type regressions, ${\beta }_1$ is the 75th percentile of the standard Normal distribution. For Mallows type regression ${\beta }_1$ is the solution to

$$\frac{1}{n}\sum _{i=1}^n\Phi \left({\beta }_1/\sqrt{w_i}\right)=0.75\text{,}$$

where $\Phi$ is the standard Normal cumulative distribution function (see specfun.cdf_normal).

${\beta }_2$ is given by

$$\begin{array}{c}\begin{array}{cc}{\beta }_2={\int }_{-\infty }^{\infty }\chi \left(z\right)\varphi \left(z\right)dz& \text{in the Huber case;}\\ & \\ & \\ {\beta }_2=\frac{1}{n}{\sum }_{i=1}^n{w}_i{\int }_{-\infty }^{\infty }\chi \left(z\right)\varphi \left(z\right)dz& \text{in the Mallows case;}\\ & \\ & \\ {\beta }_2=\frac{1}{n}{\sum }_{i=1}^n{w}_i^2{\int }_{-\infty }^{\infty }\chi \left(z/w_i\right)\varphi \left(z\right)dz& \text{in the Schweppe case;}\end{array}\end{array}$$

where $\varphi$ is the standard Normal density, i.e., $\frac{1}{\sqrt{2\pi }}exp(-\frac{1}{2}{x}^2)\text{.}$

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

$$\begin{array}{c}{G}_{ii}=\{\begin{array}{cc}\frac{\psi \left(r_i/\left(\sigma w_i\right)\right)}{\left(r_i/\left(\sigma w_i\right)\right)}\text{,}& r_i\ne 0\\ & \\ {\psi }^{\prime }\left(0\right)\text{,}& r_i=0\end{array}\text{,}\end{array}$$

where ${\psi }^{\prime }\left(t\right)$ is the derivative of $\psi$ at the point $t$.

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

$$\begin{array}{c}\begin{array}{cc}{\tilde{y}}_i=y_i\sqrt{G_{ii}}& \\ {\tilde{x}}_{ij}=x_{ij}\sqrt{G_{ii}}\text{,}& j=1,2,\dots ,m\end{array}\end{array}$$

and then solving the associated least squares problem. If $X$ is of full column rank then an orthogonal-triangular ($QR$) decomposition is used; if not, a singular value decomposition is used.

The following functions are available for $\psi$ and $\chi$ in robustm.

1. Unit Weights

   $$\psi \left(t\right)=t\text{,}\chi \left(t\right)=\frac{t^2}{2}\text{.}$$

   This gives least squares regression.

2. Huber’s Function

   $$\begin{array}{c}\psi \left(t\right)=max(-c,min(c,t))\text{,}\chi \left(t\right)=\{\begin{array}{cc}\frac{t^2}{2}\text{,}& \left|t\right|\le d\\ & \\ \frac{d^2}{2}\text{,}& \left|t\right|>d\end{array}\end{array}$$

3. Hampel’s Piecewise Linear Function

   $$\begin{array}{c}{\psi }_{h_1,h_2,h_3}\left(t\right)=-{\psi }_{h_1,h_2,h_3}(-t)=\{\begin{array}{cc}t\text{,}& 0\le t\le h_1\\ & \\ h_1\text{,}& h_1\le t\le h_2\\ & \\ h_1(h_3-t)/(h_3-h_2)\text{,}& h_2\le t\le h_3\\ & \\ 0\text{,}& h_3<t\end{array}\end{array}$$
   $$\begin{array}{c}\chi \left(t\right)=\{\begin{array}{cc}\frac{t^2}{2}\text{,}& \left|t\right|\le d\\ & \\ \frac{d^2}{2}\text{,}& \left|t\right|>d\end{array}\end{array}$$

4. Andrew’s Sine Wave Function

   $$\begin{array}{c}\psi \left(t\right)=\{\begin{array}{cc}\sin \left(t\right)\text{,}& -\pi \le t\le \pi \\ & \\ 0\text{,}& \left|t\right|>\pi \end{array}\chi \left(t\right)=\{\begin{array}{cc}\frac{t^2}{2}\text{,}& \left|t\right|\le d\\ & \\ \frac{d^2}{2}\text{,}& \left|t\right|>d\end{array}\end{array}$$

5. Tukey’s Bi-weight

   $$\begin{array}{c}\psi \left(t\right)=\{\begin{array}{cc}t{(1-t^2)}^2\text{,}& \left|t\right|\le 1\\ & \\ 0\text{,}& \left|t\right|>1\end{array}\chi \left(t\right)=\{\begin{array}{cc}\frac{t^2}{2}\text{,}& \left|t\right|\le d\\ & \\ \frac{d^2}{2}\text{,}& \left|t\right|>d\end{array}\end{array}$$

where $c$, $h_1$, $h_2$, $h_3$, and $d$ are given constants.

Several schemes for calculating weights have been proposed, see Hampel et al. (1986) and Marazzi (1987a). As the different independent variables may be measured on different scales, one group of proposed weights aims to bound a standardized measure of influence. To obtain such weights the matrix $A$ has to be found such that:

$$\frac{1}{n}\sum _{i=1}^{n}u\left({\parallel z_i\parallel }_2\right)z_i{z}_i^T=I$$

and

$$z_i=A{x}_i\text{,}$$

where	$x_i$ is a vector of length $m$ containing the $i$th row of $X$,
	$A$ is an $m\times m$ lower triangular matrix,
and	$u$ is a suitable function.

The weights are then calculated as

$$w_i=f\left({\parallel z_i\parallel }_2\right)$$

for a suitable function $f$.

robustm finds $A$ using the iterative procedure

$$A_k=(S_k+I)A_{k-1}\text{,}$$

where $S_k=\left(s_{jl}\right)$,

$$\begin{array}{c}{s}_{jl}=\{\begin{array}{cc}-min[max(h_{jl}/n,-BL),BL]\text{,}& j>l\\ & \\ -min[max(\frac{1}{2}(h_{jj}/n-1),-BD),BD]\text{,}& j=l\end{array}\end{array}$$

and

$$h_{jl}=\sum _{i=1}^{n}u\left({\parallel z_i\parallel }_2\right)z_{ij}{z}_{il}$$

and $BL$ and $BD$ are bounds set at $0.9$.

Two weights are available in robustm:

1. Krasker–Welsch Weights

   $$u\left(t\right)=g_1\left(\frac{c}{t}\right)\text{,}$$

   where	$g_1\left(t\right)=t^2+(1-t^2)(2\Phi \left(t\right)-1)-2t\varphi \left(t\right)$,
   	$\Phi \left(t\right)$ is the standard Normal cumulative distribution function,
   	$\varphi \left(t\right)$ is the standard Normal probability density function,
   and	$f\left(t\right)=\frac{1}{t}$.

   These are for use with Schweppe type regression.

2. Maronna’s Proposed Weights

   $$\begin{array}{c}\begin{array}{c}u\left(t\right)=\{\begin{array}{cc}\frac{c}{t^2}& \left|t\right|>c\\ 1& \left|t\right|\le c\end{array}\\ f\left(t\right)=\sqrt{u\left(t\right)}\text{.}\end{array}\end{array}$$

   These are for use with Mallows type regression.

Finally the asymptotic variance-covariance matrix, $C$, of the estimates $\theta$ is calculated.

For Huber type regression

$$C=f_H{\left(X^{T}X\right)}^{-1}{\hat{\sigma }}^2\text{,}$$

where

$$f_H=\frac{1}{n-m}\frac{{\sum }_{i=1}^n{\psi }^2\left(r_i/\hat{\sigma }\right)}{{\left(\frac{1}{n}{\sum }_{i=1}^n{\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right)\right)}^2}{\kappa }^2$$

$${\kappa }^2=1+\frac{m}{n}\frac{\frac{1}{n}{\sum }_{i=1}^n{({\psi }^{\prime }\left(r_i/\hat{\sigma }\right)-\frac{1}{n}{\sum }_{i=1}^n{\psi }^{\prime }\left(r_i/\hat{\sigma }\right))}^2}{{\left(\frac{1}{n}{\sum }_{i=1}^n{\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right)\right)}^2}\text{.}$$

See Huber (1981) and Marazzi (1987b).

For Mallows and Schweppe type regressions $C$ is of the form

$${\frac{\hat{\sigma }}{n}}^2{S}_1^{-1}{S}_2{S}_1^{-1}\text{,}$$

where $S_1=\frac{1}{n}{X}^{T}DX$ and $S_2=\frac{1}{n}{X}^{T}PX$.

$D$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^{\prime }\left(r_i/\left(\sigma w_i\right)\right)\right)$ in the Schweppe case and $E\left({\psi }^{\prime }\left(r_i/\sigma \right)w_i\right)$ in the Mallows case.

$P$ is a diagonal matrix such that the $i$th element approximates $E\left({\psi }^2\left(r_i/\left(\sigma w_i\right)\right)w_i^2\right)$ in the Schweppe case and $E\left({\psi }^2\left(r_i/\sigma \right)w_i^2\right)$ in the Mallows case.

Two approximations are available in robustm:

1. Average over the $r_i$

   $$\begin{array}{c}\begin{array}{cc}\text{Schweppe}& \text{Mallows}\\ & \\ D_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^{\prime }\left(\frac{r_j}{\hat{\sigma }{w}_i}\right)\right)w_i& D_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^{\prime }\left(\frac{r_j}{\hat{\sigma }}\right)\right)w_i\\ & \\ P_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^2\left(\frac{r_j}{\hat{\sigma }{w}_i}\right)\right)w_i^2& P_i=\left(\frac{1}{n}{\sum }_{j=1}^n{\psi }^2\left(\frac{r_j}{\hat{\sigma }}\right)\right)w_i^2\end{array}\end{array}$$

2. Replace expected value by observed

   $$\begin{array}{c}\begin{array}{cc}\text{Schweppe}& \text{Mallows}\\ & \\ D_i={\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }{w}_i}\right)w_i& D_i={\psi }^{\prime }\left(\frac{r_i}{\hat{\sigma }}\right)w_i\\ & \\ P_i={\psi }^2\left(\frac{r_i}{\hat{\sigma }{w}_i}\right)w_i^2& P_i={\psi }^2\left(\frac{r_i}{\hat{\sigma }}\right)w_i^2\end{array}\text{.}\end{array}$$

See Hampel et al. (1986) and Marazzi (1987b).

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 $\hat{\theta }$ corresponding to the usual constant term.

robustm is based on routines in ROBETH; see Marazzi (1987a).

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, Weights for bounded influence regression in ROBETH, Cah. Rech. Doc. IUMSP, No. 3 ROB 3, Institut Universitaire de Médecine Sociale et Préventive, Lausanne

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