NAG FL Interface
g02hlf (robustm_​corr_​user_​deriv)

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1 Purpose

g02hlf calculates a robust estimate of the covariance matrix for user-supplied weight functions and their derivatives.

2 Specification

Fortran Interface
Subroutine g02hlf ( ucv, ruser, indm, n, m, x, ldx, cov, a, wt, theta, bl, bd, maxit, nitmon, tol, nit, wk, ifail)
Integer, Intent (In) :: indm, n, m, ldx, maxit, nitmon
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: nit
Real (Kind=nag_wp), Intent (In) :: x(ldx,m), bl, bd, tol
Real (Kind=nag_wp), Intent (Inout) :: ruser(*), a(m*(m+1)/2), theta(m)
Real (Kind=nag_wp), Intent (Out) :: cov(m*(m+1)/2), wt(n), wk(2*m)
External :: ucv
C Header Interface
#include <nag.h>
void  g02hlf_ (
void (NAG_CALL *ucv)(const double *t, double ruser[], double *u, double *ud, double *w, double *wd),
double ruser[], const Integer *indm, const Integer *n, const Integer *m, const double x[], const Integer *ldx, double cov[], double a[], double wt[], double theta[], const double *bl, const double *bd, const Integer *maxit, const Integer *nitmon, const double *tol, Integer *nit, double wk[], Integer *ifail)
The routine may be called by the names g02hlf or nagf_correg_robustm_corr_user_deriv.

3 Description

For a set of n observations on m variables in a matrix X, a robust estimate of the covariance matrix, C, and a robust estimate of location, θ, are given by:
C=τ2(ATA)-1,  
where τ2 is a correction factor and A is a lower triangular matrix found as the solution to the following equations.
zi=A(xi-θ)  
1n i= 1nw(zi2)zi=0  
and
1ni=1nu(zi2)zi ziT -v(zi2)I=0,  
where xi is a vector of length m containing the elements of the ith row of X,
zi is a vector of length m,
I is the identity matrix and 0 is the zero matrix,
and w and u are suitable functions.
g02hlf covers two situations:
  1. (i)v(t)=1 for all t,
  2. (ii)v(t)=u(t).
The robust covariance matrix may be calculated from a weighted sum of squares and cross-products matrix about θ using weights wti=u(zi). In case (i) a divisor of n is used and in case (ii) a divisor of i=1nwti is used. If w(.)=u(.), then the robust covariance matrix can be calculated by scaling each row of X by wti and calculating an unweighted covariance matrix about θ.
In order to make the estimate asymptotically unbiased under a Normal model a correction factor, τ2, is needed. The value of the correction factor will depend on the functions employed (see Huber (1981) and Marazzi (1987)).
g02hlf finds A using the iterative procedure as given by Huber.
Ak=(Sk+I)Ak-1  
and
θjk=bjD1+θjk- 1,  
where Sk=(sjl), for j=1,2,,m and l=1,2,,m, is a lower triangular matrix such that:
sjl={ -min[max(hjl/D3,-BL),BL], j>l -min[max((hjj/(2D3-D4/D2)),-BD),BD], j=l ,  
where g02hlf is based on routines in ROBETH; see Marazzi (1987).

4 References

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 Subroutine, supplied by the user. External Procedure
ucv must return the values of the functions u and w and their derivatives for a given value of its argument.
The specification of ucv is:
Fortran Interface
Subroutine ucv ( t, ruser, u, ud, w, wd)
Real (Kind=nag_wp), Intent (In) :: t
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: u, ud, w, wd
C Header Interface
void  ucv (const double *t, double ruser[], double *u, double *ud, double *w, double *wd)
1: t Real (Kind=nag_wp) Input
On entry: the argument for which the functions u and w must be evaluated.
2: ruser(*) Real (Kind=nag_wp) array User Workspace
ucv is called with the argument ruser as supplied to g02hlf. You should use the array ruser to supply information to ucv.
3: u Real (Kind=nag_wp) Output
On exit: the value of the u function at the point t.
Constraint: u0.0.
4: ud Real (Kind=nag_wp) Output
On exit: the value of the derivative of the u function at the point t.
5: w Real (Kind=nag_wp) Output
On exit: the value of the w function at the point t.
Constraint: w0.0.
6: wd Real (Kind=nag_wp) Output
On exit: the value of the derivative of the w function at the point t.
ucv must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which g02hlf 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 g02hlf. If your code inadvertently does return any NaNs or infinities, g02hlf is likely to produce unexpected results.
2: ruser(*) Real (Kind=nag_wp) array User Workspace
ruser is not used by g02hlf, but is passed directly to ucv and may be used to pass information to this routine.
3: indm Integer Input
On entry: indicates which form of the function v will be used.
indm=1
v=1.
indm1
v=u.
4: n Integer Input
On entry: n, the number of observations.
Constraint: n>1.
5: m Integer Input
On entry: m, the number of columns of the matrix X, i.e., number of independent variables.
Constraint: 1mn.
6: x(ldx,m) Real (Kind=nag_wp) array Input
On entry: x(i,j) must contain the ith observation on the jth variable, for i=1,2,,n and j=1,2,,m.
7: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02hlf is called.
Constraint: ldxn.
8: cov(m×(m+1)/2) Real (Kind=nag_wp) array Output
On exit: contains a robust estimate of the covariance matrix, C. The upper triangular part of the matrix C is stored packed by columns (lower triangular stored by rows), Cij is returned in cov((j×(j-1)/2+i)), ij.
9: 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, and 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.
Constraint: a(j×(j-1)/2+j)0.0, for j=1,2,,m.
On exit: the lower triangular elements of the inverse of the matrix A, stored row-wise.
10: wt(n) Real (Kind=nag_wp) array Output
On exit: wt(i) contains the weights, wti=u(zi2), for i=1,2,,n.
11: theta(m) Real (Kind=nag_wp) array Input/Output
On entry: an initial estimate of the location parameter, θj, for j=1,2,,m.
In many cases an initial estimate of θj=0, for j=1,2,,m, will be adequate. Alternatively medians may be used as given by g07daf.
On exit: contains the robust estimate of the location parameter, θj, for j=1,2,,m.
12: bl Real (Kind=nag_wp) Input
On entry: the magnitude of the bound for the off-diagonal elements of Sk, BL.
Suggested value: bl=0.9.
Constraint: bl>0.0.
13: bd Real (Kind=nag_wp) Input
On entry: the magnitude of the bound for the diagonal elements of Sk, BD.
Suggested value: bd=0.9.
Constraint: bd>0.0.
14: maxit Integer Input
On entry: the maximum number of iterations that will be used during the calculation of A.
Suggested value: maxit=150.
Constraint: maxit>0.
15: nitmon Integer Input
On entry: indicates the amount of information on the iteration that is printed.
nitmon>0
The value of A, θ and δ (see Section 7) will be printed at the first and every nitmon iterations.
nitmon0
No iteration monitoring is printed.
When printing occurs the output is directed to the current advisory message unit (see x04abf).
16: tol Real (Kind=nag_wp) Input
On entry: the relative precision for the final estimates of the covariance matrix. Iteration will stop when maximum δ (see Section 7) is less than tol.
Constraint: tol>0.0.
17: nit Integer Output
On exit: the number of iterations performed.
18: wk(2×m) Real (Kind=nag_wp) array Workspace
19: 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:
ifail=1
On entry, ldx=value and n=value.
Constraint: ldxn.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n2.
On entry, n=value and m=value.
Constraint: nm.
ifail=2
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.
ifail=3
On entry, a variable has a constant value, i.e., all elements in column value of x are identical.
ifail=4
u value returned by ucv<0.0: u(value)=value.
Constraint: u0.0.
w value returned by ucv<0.0: w(value)=value.
Constraint: w0.0.
ifail=5
Iterations to calculate weights failed to converge.
ifail=6
The sum D1 is zero. Try either a larger initial estimate of A or make u and w less strict.
The sum D2 is zero. Try either a larger initial estimate of A or make u and w less strict.
The sum D3 is zero. Try either a larger initial estimate of A or make u and w less strict.
ifail=-99
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.
ifail=-399
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.
ifail=-999
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. At an iteration let
  1. (i)d1= the maximum value of |sjl|
  2. (ii)d2= the maximum absolute change in wt(i)
  3. (iii)d3= the maximum absolute relative change in θj
and let δ=max(d1,d2,d3). Then the iterative procedure is assumed to have converged when δ<tol.

8 Parallelism and Performance

g02hlf 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 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

A sample of 10 observations on three variables is read in along with initial values for A and theta and parameter values for the u and w functions, cu and cw. The covariance matrix computed by g02hlf is printed along with the robust estimate of θ. ucv computes the Huber's weight functions:
u(t)=1, if  tcu2 u(t)= cut2, if  t>cu2  
and
w(t)= 1, if   tcw w(t)= cwt, if   t>cw  
and their derivatives.

10.1 Program Text

Program Text (g02hlfe.f90)

10.2 Program Data

Program Data (g02hlfe.d)

10.3 Program Results

Program Results (g02hlfe.r)