is proportional to a robust estimate of the covariance of the variables. nag_robust_m_regsn_wts (g02hbc) is intended for the calculation of weights of bounded influence regression using nag_robust_m_regsn_user_fn (g02hdc).

2

Specification

#include <nag.h>

#include <nagg02.h>

void

nag_robust_m_regsn_wts (Nag_OrderType order,

double

(*ucv)(double t, Nag_Comm *comm),

Integer n, Integer m, const double x[], Integer pdx, double a[], double z[], double bl, double bd, double tol, Integer maxit, Integer nitmon, const char *outfile, Integer *nit, Nag_Comm *comm, NagError *fail)

3

Description

In fitting the linear regression model

y = X θ + ε,

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 arguments,
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

\frac{1}{n} \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i} z_{i}^{T} = I ​ (I ​ is the identity matrix)

and

z_{i} = A x_{i},

where	$x_{i}$ is a vector of length $m$ containing the elements of the $i$ th row of $X$ ,
	$A$ is an $m$ by $m$ lower triangular matrix,
	$z_{i}$ is a vector of length $m$ ,
and	$u$ is a suitable function.

The weights for use with nag_robust_m_regsn_user_fn (g02hdc) may then be computed using

w_{i} = f ({‖z_{i}‖}_{2})

for a suitable user-supplied function

f

nag_robust_m_regsn_wts (g02hbc) finds

A

using the iterative procedure

A_{k} = (S_{k} + I) A_{k - 1},

where

S_{k} = (s_{j l})

, for

j = 1, 2, \dots, m

and

l = 1, 2, \dots, m

, is a lower triangular matrix such that

$s_{j l} = \{\begin{cases} - \min [\max (h_{j l} / n, - BL), BL], & j > l \\ - \min [\max (\frac{1}{2} (h_{j j} / n - 1), - BD), BD], & j = l \end{cases}$
$h_{j l} = \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i j} z_{i l}$

and

BD

and

BL

are suitable bounds.

In addition the values of

{‖z_{i}‖}_{2}

, for

i = 1, 2, \dots, n

, are calculated.

nag_robust_m_regsn_wts (g02hbc) 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

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

Nag_ColMajor

2: $ucv$ – function, supplied by the userExternal Function

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:

double

ucv (double t, Nag_Comm *comm)

1: $t$ – doubleInput

On entry: the argument for which ucv must be evaluated.

2: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to ucv.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_robust_m_regsn_wts (g02hbc) you may allocate memory and initialize these pointers with various quantities for use by ucv when called from nag_robust_m_regsn_wts (g02hbc) (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_regsn_wts (g02hbc). If your code inadvertently does return any NaNs or infinities, nag_robust_m_regsn_wts (g02hbc) is likely to produce unexpected results.

3: $n$ – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n > 1

4: $m$ – IntegerInput

On entry:

m

, the number of independent variables.

Constraint:

1 \leq m \leq n

5: $x [\dim]$ – const doubleInput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

Where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the real matrix

X

, i.e., the independent variables.

X (i, j)

must contain the

i j

th element of

x

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

6: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

7: $a [m \times (m + 1) / 2]$ – 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

\neq 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.

8: $z [n]$ – doubleOutput

On exit: the value

{‖z_{i}‖}_{2}

, for

i = 1, 2, \dots, n

9: $bl$ – doubleInput

On entry: the magnitude of the bound for the off-diagonal elements of

S_{k}

Suggested value:

bl = 0.9

Constraint:

bl > 0.0

10: $bd$ – doubleInput

On entry: the magnitude of the bound for the diagonal elements of

S_{k}

Suggested value:

bd = 0.9

Constraint:

bd > 0.0

11: $tol$ – doubleInput

On entry: the relative precision for the final value of

A

. Iteration will stop when the maximum value of

|s_{j l}|

is less than tol.

Constraint:

tol > 0.0

12: $maxit$ – 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

13: $nitmon$ – 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 $|s_{j l}|$ will be printed at the first and every nitmon iterations.
$nitmon \leq 0$: No iteration monitoring is printed.

14: $outfile$ – 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.

15: $nit$ – Integer *Output

On exit: the number of iterations performed.

16: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

17: $fail$ – 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_CONVERGENCE: Iterations to calculate weights failed to converge in maxit iterations: $maxit = 〈value〉$ .
NE_FUN_RET_VAL: Value returned by ucv function $< 0$ : $u (〈value〉) = 〈value〉$ .
NE_INT: 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: $1 \leq m \leq n$ .

On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .
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, diagonal element $〈value〉$ of a is $0$ .

7

Accuracy

On successful exit the accuracy of the results is related to the value of tol; see Section 5.

8

Parallelism and Performance

nag_robust_m_regsn_wts (g02hbc) 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 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.

10

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.

NAG Library Function Document

nag_robust_m_regsn_wts (g02hbc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results