naginterfaces.library.correg.robustm_​wts

naginterfaces.library.correg.robustm_wts(ucv, x, a, bl=0.9, bd=0.9, tol=5e-05, maxit=50, nitmon=0, data=None, io_manager=None)

robustm_wts finds, for a real matrix $X$ of full column rank, a lower triangular matrix $A$ such that ${\left(A^{T}A\right)}^{-1}$ is proportional to a robust estimate of the covariance of the variables. robustm_wts is intended for the calculation of weights of bounded influence regression using robustm_user().

For full information please refer to the NAG Library document for g02hb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02hbf.html

Parameters

ucvcallable retval = ucv(t, data=None)

$ucv$ must return the value of the function $u$ for a given value of its argument.

The value of $u$ must be non-negative.

Parameters

tfloat

The argument for which $ucv$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $u\left(t\right)$ evaluated at $t$.

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

The real matrix $X$, 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 $\text{i}=1,2,\dots ,n$.

afloat, array-like, shape $(m\times (m+1)/2)$

An initial estimate of the lower triangular real matrix $A$. Only the lower triangular elements must be given and these should be stored row-wise in the array.

The diagonal elements must be $\ne 0$, although in practice will usually be $>0$.

If the magnitudes of the columns of $X$ are of the same order the identity matrix will often provide a suitable initial value for $A$.

If the columns of $X$ are of different magnitudes, the diagonal elements of the initial value of $A$ should be approximately inversely proportional to the magnitude of the columns of $X$.

blfloat, optional

The magnitude of the bound for the off-diagonal elements of $S_k$.

bdfloat, optional

The magnitude of the bound for the diagonal elements of $S_k$.

tolfloat, optional

The relative precision for the final value of $A$. Iteration will stop when the maximum value of $\left|s_{jl}\right|$ is less than $tol$.

maxitint, optional

The maximum number of iterations that will be used during the calculation of $A$.

A value of $maxit=50$ will often be adequate.

nitmonint, optional

Determines the amount of information that is printed on each iteration.

$nitmon>0$

The value of $A$ and the maximum value of $\left|s_{jl}\right|$ will be printed at the first and every $nitmon$ iterations.

$nitmon\le 0$

No iteration monitoring is printed.

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

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

afloat, ndarray, shape $(m\times (m+1)/2)$

The lower triangular elements of the matrix $A$, stored row-wise.

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

The value ${\parallel z_{\text{i}}\parallel }_2$, for $\text{i}=1,2,\dots ,n$.

nitint

The number of iterations performed.

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: $n\ge m$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $2$)

On entry, $bd=⟨value⟩$.

Constraint: $bd>0.0$.

(errno $2$)

On entry, $bl=⟨value⟩$.

Constraint: $bl>0.0$.

(errno $2$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit>0$.

(errno $2$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $2$)

On entry, $i=⟨value⟩$ and the $i$th diagonal element of $A$ is $0$.

Constraint: all diagonal elements of $A$ must be non-zero.

(errno $3$)

Value returned by $ucv$ function $<0$: $u(⟨value⟩)=⟨value⟩$.

The value of $u$ must be non-negative.

(errno $4$)

Iterations to calculate weights failed to converge in $maxit$ iterations: $maxit=⟨value⟩$.

Notes

In fitting 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,
	$\theta$ is a vector of length $m$ of unknown parameters,
and	$ϵ$ is a vector of length $n$ of unknown errors,

it may be desirable to bound the influence of rows of the $X$ matrix. This can be achieved by calculating a weight for each observation. Several schemes for calculating weights have been proposed (see Hampel et al. (1986) and Marazzi (1987)). 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

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

and

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

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

The weights for use with robustm_user() may then be computed using

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

for a suitable user-supplied function $f$.

robustm_wts finds $A$ using the iterative procedure

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

where $S_k=\left(s_{jl}\right)$, for $\text{l}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$, is a lower triangular matrix such that

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

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

and $\text{BD}$ and $\text{BL}$ are suitable bounds.

In addition the values of ${\parallel z_i\parallel }_2$, for $i=1,2,\dots ,n$, are calculated.

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