NAG C Library Function Document

nag_robust_m_corr_user_fn_no_derr (g02hmc)

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
C=τ2ATA-1,  
where τ2 is a correction factor and A is a lower triangular matrix found as the solution to the following equations.
zi=Axi-θ  
1n i= 1nwzi2zi=0  
and
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).
Ak=Sk+IAk-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= -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 3.3.1.3 in How to Use the NAG Library and its Documentation 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 *
pPointer 
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 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: ucv should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_robust_m_corr_user_fn_no_derr (g02hmc). If your code inadvertently does return any NaNs or infinities, nag_robust_m_corr_user_fn_no_derr (g02hmc) is likely to produce unexpected results.
3:     indm IntegerInput
On entry: indicates which form of the function v will be used.
indm=1
v=1.
indm1
v=u.
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.
Constraints:
  • 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 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 nag_median_1var (g07dac).
On exit: contains the robust estimate of the location parameter, 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.
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.
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 3.3.1.1 in How to Use the NAG Library and its Documentation).
20:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONST_COL
On entry, a variable has a constant value, i.e., all elements in column value of x are identical.
NE_CONVERGENCE
Iterations to calculate weights failed to converge.
NE_FUN_RET_VAL
u value returned by ucv<0.0: uvalue=value.
Constraint: u0.0.
w value returned by ucv<0.0: wvalue=value.
Constraint: w0.0.
NE_INT
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.
NE_INT_2
On entry, m=value and n=value.
Constraint: 1mn.
On entry, n=value and m=value.
Constraint: n or m.
On entry, pdx=value and m=value.
Constraint: pdxm.
On entry, pdx=value and n=value.
Constraint: pdx or n.
NE_INTERNAL_ERROR
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_CLOSE_FILE
Cannot close file value.
NE_NOT_WRITE_FILE
Cannot open file value for writing.
NE_REAL
On entry, bd=value.
Constraint: bd>0.0.
On entry, bl=value.
Constraint: bl>0.0.
On entry, tol=value.
Constraint: tol>0.0.
NE_ZERO_DIAGONAL
On entry, i=value and the ith diagonal element of A is 0.
Constraint: all diagonal elements of A must be non-zero.
NE_ZERO_SUM
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.

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) 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 parameter 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  
and
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)