NAG FL Interface
g02hbf (robustm_​wts)

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1 Purpose

g02hbf finds, for a real matrix X of full column rank, a lower triangular matrix A such that (ATA)−1 is proportional to a robust estimate of the covariance of the variables. g02hbf is intended for the calculation of weights of bounded influence regression using g02hdf.

2 Specification

Fortran Interface
Subroutine g02hbf ( ucv, n, m, x, ldx, a, z, bl, bd, tol, maxit, nitmon, nit, wk, ifail)
Integer, Intent (In) :: n, m, ldx, maxit, nitmon
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: nit
Real (Kind=nag_wp), External :: ucv
Real (Kind=nag_wp), Intent (In) :: x(ldx,m), bl, bd, tol
Real (Kind=nag_wp), Intent (Inout) :: a(m*(m+1)/2)
Real (Kind=nag_wp), Intent (Out) :: z(n), wk(m*(m+1)/2)
C Header Interface
#include <nag.h>
void  g02hbf_ (
double (NAG_CALL *ucv)(const double *t),
const Integer *n, const Integer *m, const double x[], const Integer *ldx, double a[], double z[], const double *bl, const double *bd, const double *tol, const Integer *maxit, const Integer *nitmon, Integer *nit, double wk[], Integer *ifail)
The routine may be called by the names g02hbf or nagf_correg_robustm_wts.

3 Description

In fitting the linear regression model
where y is a vector of length n of the dependent variable,
X is an n×m matrix of independent variables,
θ 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
1ni=1nu(zi2)zi ziT =I​  (I​ is the identity matrix)  
where xi is a vector of length m containing the elements of the ith row of X,
A is an m×m lower triangular matrix,
zi is a vector of length m,
and u is a suitable function.
The weights for use with g02hdf may then be computed using
for a suitable user-supplied function f.
g02hbf finds A using the iterative procedure
where Sk=(sjl), for j=1,2,,m and l=1,2,,m, is a lower triangular matrix such that and BD and BL are suitable bounds.
In addition the values of zi2, for i=1,2,,n, are calculated.
g02hbf is based on routines in ROBETH; see Marazzi (1987).

4 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

5 Arguments

1: ucv real (Kind=nag_wp) Function, supplied by the user. External Procedure
ucv must return the value of the function u for a given value of its argument. The value of u must be non-negative.
The specification of ucv is:
Fortran Interface
Function ucv ( t)
Real (Kind=nag_wp) :: ucv
Real (Kind=nag_wp), Intent (In) :: t
C Header Interface
double  ucv (const double *t)
1: t Real (Kind=nag_wp) Input
On entry: the argument for which ucv must be evaluated.
ucv must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which g02hbf is called. Arguments denoted as Input must not be changed by this procedure.
Note: ucv should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g02hbf. If your code inadvertently does return any NaNs or infinities, g02hbf is likely to produce unexpected results.
2: n Integer Input
On entry: n, the number of observations.
Constraint: n>1.
3: m Integer Input
On entry: m, the number of independent variables.
Constraint: 1mn.
4: x(ldx,m) Real (Kind=nag_wp) array Input
On entry: the real matrix X, i.e., the independent variables. x(i,j) must contain the ijth element of x, for i=1,2,,n and j=1,2,,m.
5: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02hbf is called.
Constraint: ldxn.
6: a(m×(m+1)/2) Real (Kind=nag_wp) array Input/Output
On entry: 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 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.
On exit: the lower triangular elements of the matrix A, stored row-wise.
7: z(n) Real (Kind=nag_wp) array Output
On exit: the value zi2, for i=1,2,,n.
8: bl Real (Kind=nag_wp) Input
On entry: the magnitude of the bound for the off-diagonal elements of Sk.
Suggested value: bl=0.9.
Constraint: bl>0.0.
9: bd Real (Kind=nag_wp) Input
On entry: the magnitude of the bound for the diagonal elements of Sk.
Suggested value: bd=0.9.
Constraint: bd>0.0.
10: tol Real (Kind=nag_wp) Input
On entry: the relative precision for the final value of A. Iteration will stop when the maximum value of |sjl| is less than tol.
Constraint: tol>0.0.
11: maxit Integer Input
On entry: the maximum number of iterations that will be used during the calculation of A.
A value of maxit=50 will often be adequate.
Constraint: maxit>0.
12: nitmon Integer Input
On entry: determines the amount of information that is printed on each iteration.
The value of A and the maximum value of |sjl| will be printed at the first and every nitmon iterations.
No iteration monitoring is printed.
When printing occurs the output is directed to the current advisory message unit (see x04abf).
13: nit Integer Output
On exit: the number of iterations performed.
14: wk(m×(m+1)/2) Real (Kind=nag_wp) array Workspace
15: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n2.
On entry, n=value and ldx=value.
Constraint: ldxn.
On entry, n=value and m=value.
Constraint: nm.
On entry, bd=value.
Constraint: bd>0.0.
On entry, bl=value.
Constraint: bl>0.0.
On entry, i=value and the ith diagonal element of A is 0.
Constraint: all diagonal elements of A must be non-zero.
On entry, maxit=value.
Constraint: maxit>0.
On entry, tol=value.
Constraint: tol>0.0.
Value returned by ucv function <0: u(value)=value.
The value of u must be non-negative.
Iterations to calculate weights failed to converge in maxit iterations: maxit=value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

On successful exit the accuracy of the results is related to the value of tol; see Section 5.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02hbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The existence of A will depend upon the function u; (see Hampel et al. (1986) and Marazzi (1987)), also if X is not of full rank a value of A will not be found. If the columns of X are almost linearly related then convergence will be slow.

10 Example

This example reads in a matrix of real numbers and computes the Krasker–Welsch weights (see Marazzi (1987)). The matrix A and the weights are then printed.

10.1 Program Text

Program Text (g02hbfe.f90)

10.2 Program Data

Program Data (g02hbfe.d)

10.3 Program Results

Program Results (g02hbfe.r)