NAG Library Routine Document
G02HBF
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
SUBROUTINE G02HBF ( |
UCV, N, M, X, LDX, A, Z, BL, BD, TOL, MAXIT, NITMON, NIT, WK, IFAIL) |
INTEGER |
N, M, LDX, MAXIT, NITMON, NIT, IFAIL |
REAL (KIND=nag_wp) |
UCV, X(LDX,M), A(M*(M+1)/2), Z(N), BL, BD, TOL, WK(M*(M+1)/2) |
EXTERNAL |
UCV |
|
3 Description
In fitting the linear regression model
where |
y is a vector of length n of the dependent variable, |
|
X is an n by 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
and
where |
xi is a vector of length m containing the elements of the ith row of X, |
|
A is an m by 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
- sjl=
-minmaxhjl/n,-BL,BL,
j>l
-minmax12hjj/n-1,-BD,BD,
j=l
- hjl=∑i=1nuzi2zijzil
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 Parameters
- 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:
- 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. Parameters denoted as
Input must
not be changed by this procedure.
- 2: N – INTEGERInput
On entry: n, the number of observations.
Constraint:
N>1.
- 3: M – INTEGERInput
On entry: m, the number of independent variables.
Constraint:
1≤M≤N.
- 4: X(LDX,M) – REAL (KIND=nag_wp) arrayInput
On entry: the real matrix X, i.e., the independent variables.
Xij must contain the ijth element of X, for i=1,2,…,n and j=1,2,…,m.
- 5: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G02HBF is called.
Constraint:
LDX≥N.
- 6: A(M×M+1/2) – REAL (KIND=nag_wp) arrayInput/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) arrayOutput
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 – INTEGERInput
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 – INTEGERInput
On entry: determines the amount of information that is printed on each iteration.
- NITMON>0
- The value of A and the maximum value of sjl will be printed at the first and every NITMON iterations.
- NITMON≤0
- No iteration monitoring is printed.
When printing occurs the output is directed to the current advisory message unit (see
X04ABF).
- 13: NIT – INTEGEROutput
On exit: the number of iterations performed.
- 14: WK(M×M+1/2) – REAL (KIND=nag_wp) arrayWorkspace
- 15: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
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, | N≤1, |
or | M<1, |
or | N<M, |
or | LDX<N. |
- IFAIL=2
On entry, | TOL≤0.0, |
or | MAXIT≤0, |
or | diagonal element of A=0.0, |
or | BL≤0.0, |
or | BD≤0.0. |
- IFAIL=3
Value returned by UCV<0.
- IFAIL=4
-
The routine has failed to converge in
MAXIT iterations.
7 Accuracy
On successful exit the accuracy of the results is related to the value of
TOL; see
Section 5.
8 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.
9 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.
9.1 Program Text
Program Text (g02hbfe.f90)
9.2 Program Data
Program Data (g02hbfe.d)
9.3 Program Results
Program Results (g02hbfe.r)