nag_robust_m_corr_user_fn_no_derr (g02hmc) (PDF version)
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nag_robust_m_corr_user_fn_no_derr (g02hmc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_robust_m_corr_user_fn_no_derr (g02hmc) computes a robust estimate of the covariance matrix for user-supplied weight functions. The derivatives of the weight functions are not required.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_robust_m_corr_user_fn_no_derr (Nag_OrderType order,
void (*ucv)(double t, double *u, double *w, Nag_Comm *comm),
Integer indm, Integer n, Integer m, const double x[], Integer pdx, double cov[], double a[], double wt[], double theta[], double bl, double bd, Integer maxit, Integer nitmon, const char *outfile, double tol, Integer *nit, Nag_Comm *comm, NagError *fail)

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
where τ2 is a correction factor and A is a lower triangular matrix found as the solution to the following equations.
1n i= 1nwzi2zi=0  
1ni=1nuzi2zi ziT -vzi2I=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.
nag_robust_m_corr_user_fn_no_derr (g02hmc) covers two situations:
(i) vt=1 for all t,
(ii) vt=ut.
The robust covariance matrix may be calculated from a weighted sum of squares and cross-products matrix about θ using weights wti=uzi. 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)).
nag_robust_m_corr_user_fn_no_derr (g02hmc) finds A using the iterative procedure as given by Huber; see Huber (1981).
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= -minmaxhjl/D2,-BL,BL, j>l -minmax12hjj/D2-1,-BD,BD, j=l ,  
where and BD and BL are suitable bounds.
The value of τ may be chosen so that C is unbiased if the observations are from a given distribution.
nag_robust_m_corr_user_fn_no_derr (g02hmc) 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:     order Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     ucv function, supplied by the userExternal Function
ucv must return the values of the functions u and w for a given value of its argument.
The specification of ucv is:
void  ucv (double t, double *u, double *w, Nag_Comm *comm)
1:     t doubleInput
On entry: the argument for which the functions u and w must be evaluated.
2:     u double *Output
On exit: the value of the u function at the point t.
Constraint: u0.0.
3:     w double *Output
On exit: the value of the w function at the point t.
Constraint: w0.0.
4:     comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to ucv.
userdouble *
iuserInteger *
The type Pointer will be void *. Before calling nag_robust_m_corr_user_fn_no_derr (g02hmc) you may allocate memory and initialize these pointers with various quantities for use by ucv when called from nag_robust_m_corr_user_fn_no_derr (g02hmc) (see Section in the Essential Introduction).
3:     indm IntegerInput
On entry: indicates which form of the function v will be used.
4:     n IntegerInput
On entry: n, the number of observations.
Constraint: n>1.
5:     m IntegerInput
On entry: m, the number of columns of the matrix X, i.e., number of independent variables.
Constraint: 1mn.
6:     x[dim] const doubleInput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×m when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
Where Xi,j appears in this document, it refers to the array element
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On entry: Xi,j must contain the ith observation on the jth variable, for i=1,2,,n and j=1,2,,m.
7:     pdx IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
8:     cov[m×m+1/2] doubleOutput
On exit: 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), that is Cij is returned in cov[j×j-1/2+i-1], ij.
9:     a[m×m+1/2] doubleInput/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=0,1,,m-1.
On exit: the lower triangular elements of the inverse of the matrix A, stored row-wise.
10:   wt[n] doubleOutput
On exit: wt[i-1] contains the weights, wti=uzi2, for i=1,2,,n.
11:   theta[m] doubleInput/Output
On entry: an initial estimate of the location argument, 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 nag_median_1var (g07dac).
On exit: contains the robust estimate of the location argument, θj, for j=1,2,,m.
12:   bl doubleInput
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 doubleInput
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 IntegerInput
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 IntegerInput
On entry: indicates the amount of information on the iteration that is printed.
The value of A, θ and δ (see Section 7) will be printed at the first and every nitmon iterations.
No iteration monitoring is printed.
16:   outfile const char *Input
On entry: a null terminated character string giving the name of the file to which results should be printed. If outfile=NULL or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.
17:   tol doubleInput
On entry: the relative precision for the final estimate of the covariance matrix. Iteration will stop when maximum δ (see Section 7) is less than tol.
Constraint: tol>0.0.
18:   nit Integer *Output
On exit: the number of iterations performed.
19:   comm Nag_Comm *
The NAG communication argument (see Section in the Essential Introduction).
20:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in the Essential Introduction for further information.
On entry, argument value had an illegal value.
Column value of x has constant value.
Iterations to calculate weights failed to converge.
u value returned by ucv<0.0: uvalue=value.
w value returned by ucv<0.0: wvalue=value.
On entry, m=value.
Constraint: m1.
On entry, maxit=value.
Constraint: maxit>0.
On entry, n=value.
Constraint: n>1.
On entry, pdx=value.
Constraint: pdx>0.
On entry, m=value and n=value.
Constraint: 1mn.
On entry, n=value and m=value.
Constraint: nm.
On entry, pdx=value and m=value.
Constraint: pdxm.
On entry, pdx=value and n=value.
Constraint: pdxn.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
Cannot close file value.
Cannot open file value for writing.
On entry, bd=value.
Constraint: bd>0.0.
On entry, bl=value.
Constraint: bl>0.0.
On entry, tol=value.
Constraint: tol>0.0.
On entry, diagonal element value of a is 0.0.
Sum of u's (D2) is zero.
Sum of w's (D1) is zero.

7  Accuracy

On successful exit the accuracy of the results is related to the value of tol; see Section 5. At an iteration let
(i) d1= the maximum value of sjl
(ii) d2= the maximum absolute change in wti
(iii) d3= the maximum absolute relative change in θj
and let δ=maxd1,d2,d3. Then the iterative procedure is assumed to have converged when δ<tol.

8  Parallelism and Performance

nag_robust_m_corr_user_fn_no_derr (g02hmc) is not threaded by NAG in any implementation.
nag_robust_m_corr_user_fn_no_derr (g02hmc) 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 function. 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.
If derivatives of the u and w functions are available then the method used in nag_robust_m_corr_user_fn (g02hlc) will usually give much faster convergence.

10  Example

A sample of 10 observations on three variables is read in along with initial values for A and θ and argument values for the u and w functions, cu and cw. The covariance matrix computed by nag_robust_m_corr_user_fn_no_derr (g02hmc) is printed along with the robust estimate of θ.
ucv computes the Huber's weight functions:
ut=1, if  tcu2 ut= cut2, if  t>cu2  
wt= 1, if   tcw wt= cwt, if   t>cw.  

10.1  Program Text

Program Text (g02hmce.c)

10.2  Program Data

Program Data (g02hmce.d)

10.3  Program Results

Program Results (g02hmce.r)

nag_robust_m_corr_user_fn_no_derr (g02hmc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015