g02hbf finds, for a real matrix of full column rank, a lower triangular matrix such that 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.
The routine may be called by the names g02hbf or nagf_correg_robustm_wts.
3Description
In fitting the linear regression model
where
is a vector of length of the dependent variable,
is an matrix of independent variables,
is a vector of length of unknown parameters,
and
is a vector of length of unknown errors,
it may be desirable to bound the influence of rows of the 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 has to be found such that
and
where
is a vector of length containing the elements of the th row of ,
is an lower triangular matrix,
is a vector of length ,
and
is a suitable function.
The weights for use with g02hdf may then be computed using
for a suitable user-supplied function .
g02hbf finds using the iterative procedure
where , for and , is a lower triangular matrix such that
and and are suitable bounds.
In addition the values of , for , are calculated.
g02hbf is based on routines in ROBETH; see Marazzi (1987).
4References
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
5Arguments
1: – real (Kind=nag_wp) Function, supplied by the user.External Procedure
ucv must return the value of the function for a given value of its argument. The value of must be non-negative.
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: – IntegerInput
On entry: , the number of observations.
Constraint:
.
3: – IntegerInput
On entry: , the number of independent variables.
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
On entry: the real matrix , i.e., the independent variables.
must contain the th element of , for and .
5: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02hbf is called.
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput/Output
On entry: an initial estimate of the lower triangular real matrix . Only the lower triangular elements must be given and these should be stored row-wise in the array.
The diagonal elements must be , although in practice will usually be . If the magnitudes of the columns of are of the same order the identity matrix will often provide a suitable initial value for . If the columns of are of different magnitudes, the diagonal elements of the initial value of should be approximately inversely proportional to the magnitude of the columns of .
On exit: the lower triangular elements of the matrix , stored row-wise.
7: – Real (Kind=nag_wp) arrayOutput
On exit: the value
, for .
8: – Real (Kind=nag_wp)Input
On entry: the magnitude of the bound for the off-diagonal elements of .
Suggested value:
.
Constraint:
.
9: – Real (Kind=nag_wp)Input
On entry: the magnitude of the bound for the diagonal elements of .
Suggested value:
.
Constraint:
.
10: – Real (Kind=nag_wp)Input
On entry: the relative precision for the final value of . Iteration will stop when the maximum value of is less than tol.
Constraint:
.
11: – IntegerInput
On entry: the maximum number of iterations that will be used during the calculation of .
A value of will often be adequate.
Constraint:
.
12: – IntegerInput
On entry: determines the amount of information that is printed on each iteration.
The value of and the maximum value of 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: – IntegerOutput
On exit: the number of iterations performed.
14: – Real (Kind=nag_wp) arrayWorkspace
15: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, and the th diagonal element of is .
Constraint: all diagonal elements of must be non-zero.
On entry, .
Constraint: .
On entry, .
Constraint: .
Value returned by ucv function : . The value of must be non-negative.
Iterations to calculate weights failed to converge in maxit iterations: .
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.
7Accuracy
On successful exit the accuracy of the results is related to the value of tol; see Section 5.
8Parallelism 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.
9Further Comments
The existence of will depend upon the function ; (see Hampel et al. (1986) and Marazzi (1987)), also if is not of full rank a value of will not be found. If the columns of are almost linearly related then convergence will be slow.
10Example
This example reads in a matrix of real numbers and computes the Krasker–Welsch weights (see Marazzi (1987)). The matrix and the weights are then printed.