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 = τ^{2} {(A^{T} A)}^{- 1},

where

τ^{2}

is a correction factor and

A

is a lower triangular matrix found as the solution to the following equations.

z_{i} = A (x_{i} - θ)

\frac{1}{n} \sum_{i = 1}^{n} w ({‖z_{i}‖}_{2}) z_{i} = 0

and

\frac{1}{n} \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i} z_{i}^{T} - v ({‖z_{i}‖}_{2}) I = 0,

where	$x_{i}$ is a vector of length $m$ containing the elements of the $i$ th row of $X$ ,
	$z_{i}$ 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 (g02hlc) covers two situations:

(i)	$v (t) = 1$ for all $t$ ,
(ii)	$v (t) = u (t)$ .

The robust covariance matrix may be calculated from a weighted sum of squares and cross-products matrix about

θ

using weights

{wt}_{i} = u (‖z_{i}‖)

. In case (i) a divisor of

n

is used and in case (ii) a divisor of

\sum_{i = 1}^{n} {wt}_{i}

is used. If

w (.) = \sqrt{u (.)}

, then the robust covariance matrix can be calculated by scaling each row of

X

\sqrt{{wt}_{i}}

and calculating an unweighted covariance matrix about

θ

In order to make the estimate asymptotically unbiased under a Normal model a correction factor,

τ^{2}

, 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 (g02hlc) finds

A

using the iterative procedure as given by Huber.

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

and

θ_{j_{k}} = \frac{b_{j}}{D_{1}} + θ_{j_{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} / D_{3}, - BL), BL], & j > l \\ - \min [\max ((h_{j j} / (2 D_{3} - D_{4} / D_{2})), - BD), BD], & j = l \end{cases},

where

$D_{1} = \sum_{i = 1}^{n} \{w ({‖z_{i}‖}_{2}) + \frac{1}{m} w^{'} ({‖z_{i}‖}_{2}) {‖z_{i}‖}_{2}\}$
$D_{2} = \sum_{i = 1}^{n} \{\frac{1}{p} (u^{'} ({‖z_{i}‖}_{2}) {‖z_{i}‖}_{2} + 2 u ({‖z_{i}‖}_{2})) {‖z_{i}‖}_{2} - v^{'} ({‖z_{i}‖}_{2})\} {‖z_{i}‖}_{2}$
$D_{3} = \frac{1}{m + 2} \sum_{i = 1}^{n} \{\frac{1}{m} (u^{'} ({‖z_{i}‖}_{2}) {‖z_{i}‖}_{2} + 2 u ({‖z_{i}‖}_{2})) + u ({‖z_{i}‖}_{2})\} {‖z_{i}‖}_{2}^{2}$
$D_{4} = \sum_{i = 1}^{n} \{\frac{1}{m} u ({‖z_{i}‖}_{2}) {‖z_{i}‖}_{2}^{2} - v ({‖z_{i}‖}_{2}^{2})\}$
$h_{j l} = \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) z_{i j} z_{i l}$ , for $j > l$
$h_{j j} = \sum_{i = 1}^{n} u ({‖z_{i}‖}_{2}) (z_{i j}^{2} - {‖z_{i j}‖}_{2}^{2} / m)$
$b_{j} = \sum_{i = 1}^{n} w ({‖z_{i}‖}_{2}) (x_{i j} - b_{j})$
and $BD$ and $BL$ are suitable bounds.

nag_robust_m_corr_user_fn (g02hlc) 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 2.3.1.3 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 values of the functions

u

and

w

and their derivatives for a given value of its argument.

The specification of ucv is:

void	ucv (double t, double u, double ud, double w, double wd, 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:

u \geq 0.0

3: $ud$ – double *Output

On exit: the value of the derivative of the

u

function at the point t.

4: $w$ – double *Output

On exit: the value of the

w

function at the point t.

Constraint:

w \geq 0.0

5: $wd$ – double *Output

On exit: the value of the derivative of the

w

function at the point t.

6: $comm$ – 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_corr_user_fn (g02hlc) you may allocate memory and initialize these pointers with various quantities for use by ucv when called from nag_robust_m_corr_user_fn (g02hlc) (see Section 2.3.1.1 in How to Use the NAG Library and its Documentation).

3: $indm$ – IntegerInput

On entry: indicates which form of the function

v

will be used.

$indm = 1$: $v = 1$ .
$indm \neq 1$: $v = u$ .

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:

1 \leq m \leq n

6: $x [\dim]$ – 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:

X (i, j)

must contain the

i

th observation on the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, 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 $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

8: $cov [m \times (m + 1) / 2]$ – doubleOutput

On exit: contains 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),

C_{i j}

is returned in

cov [(j \times (j - 1) / 2 + i) - 1]

i \leq j

9: $a [m \times (m + 1) / 2]$ – 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

, 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 \times (j - 1) / 2 + j] \neq 0.0

, for

j = 0, 1, \dots, 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,

{wt}_{i} = u ({‖z_{i}‖}_{2})

, for

i = 1, 2, \dots, n

11: $theta [m]$ – doubleInput/Output

On entry: an initial estimate of the location argument,

θ_{j}

, for

j = 1, 2, \dots, m

In many cases an initial estimate of

θ_{j} = 0

, for

j = 1, 2, \dots, 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,

θ_{j}

, for

j = 1, 2, \dots, m

12: $bl$ – doubleInput

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

S_{k}

B L

Suggested value:

bl = 0.9

Constraint:

bl > 0.0

13: $bd$ – doubleInput

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

S_{k}

B D

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.
$nitmon \leq 0$: 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 estimates 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 2.3.1.1 in How to Use the NAG Library and its Documentation).

20: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 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: Column $〈value〉$ of x has constant value.
NE_CONVERGENCE: Iterations to calculate weights failed to converge.
NE_FUN_RET_VAL: $u$ value returned by $ucv < 0.0$ : $u (〈value〉) = 〈value〉$ .

$w$ value returned by $ucv < 0.0$ : $w (〈value〉) = 〈value〉$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

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, $n = 〈value〉$ and $m = 〈value〉$ .
Constraint: $n \geq m$ .

On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .

On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq n$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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 $0.0$ .
NE_ZERO_SUM: The sum $D1$ is zero.

The sum $D2$ is zero.

The sum $D3$ 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)	$d 1 =$ the maximum value of $\|s_{j l}\|$
(ii)	$d 2 =$ the maximum absolute change in $w t (i)$
(iii)	$d 3 =$ the maximum absolute relative change in $θ_{j}$

and let

δ = \max (d 1, d 2, d 3)

. Then the iterative procedure is assumed to have converged when

δ < tol

8 Parallelism and Performance

nag_robust_m_corr_user_fn (g02hlc) 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.

10 Example

A sample of

10

observations on three variables is read in along with initial values for

A

and theta and argument values for the

u

and

w

functions,

c_{u}

and

c_{w}

. The covariance matrix computed by nag_robust_m_corr_user_fn (g02hlc) is printed along with the robust estimate of

θ

. ucv computes the Huber's weight functions:

\begin{array}{l} u (t) = 1, & if t \leq c_{u}^{2} \\ u (t) = \frac{c_{u}}{t^{2}}, & if t > c_{u}^{2} \end{array}

and

\begin{array}{l} w (t) = 1, & if t \leq c_{w} \\ w (t) = \frac{c_{w}}{t}, & if t > c_{w} \end{array}

and their derivatives.

NAG Library Function Documentnag_robust_m_corr_user_fn (g02hlc)

▸▿ 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

NAG Library Function Document

nag_robust_m_corr_user_fn (g02hlc)