NAG Library Routine Document

G02HLF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G02HLF calculates a robust estimate of the covariance matrix for user-supplied weight functions and their derivatives.

2 Specification

SUBROUTINE G02HLF (

UCV, RUSER, INDM, N, M, X, LDX, COV, A, WT, THETA, BL, BD, MAXIT, NITMON, TOL, NIT, WK, IFAIL)

INTEGER	INDM, N, M, LDX, MAXIT, NITMON, NIT, IFAIL
REAL (KIND=nag_wp)	RUSER(), X(LDX,M), COV(M(M+1)/2), A(M(M+1)/2), WT(N), THETA(M), BL, BD, TOL, WK(2M)
EXTERNAL	UCV

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.

G02HLF 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)).

G02HLF 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.

G02HLF 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 Parameters

1: UCV – SUBROUTINE, supplied by the user.External Procedure

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:

SUBROUTINE UCV (

T, RUSER, U, UD, W, WD)

REAL (KIND=nag_wp)

T, RUSER(*), U, UD, W, WD

1: T – REAL (KIND=nag_wp)Input: On entry: the argument for which the functions $u$ and $w$ must be evaluated.
2: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: UCV is called with the parameter RUSER as supplied to G02HLF. You are free to use the array RUSER to supply information to UCV as an alternative to using COMMON global variables.
3: U – REAL (KIND=nag_wp)Output: On exit: the value of the $u$ function at the point T.
Constraint: $U \geq 0.0$ .
4: UD – REAL (KIND=nag_wp)Output: On exit: the value of the derivative of the $u$ function at the point T.
5: W – REAL (KIND=nag_wp)Output: On exit: the value of the $w$ function at the point T.
Constraint: $W \geq 0.0$ .
6: WD – REAL (KIND=nag_wp)Output: On exit: the value of the derivative of the $w$ function at the point T.

UCV must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G02HLF is called. Parameters denoted as Input must not be changed by this procedure.

2: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

RUSER is not used by G02HLF, but is passed directly to UCV and may be used to pass information to this routine as an alternative to using COMMON global variables.

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(LDX,M) – REAL (KIND=nag_wp) arrayInput

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: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02HLF is called.

Constraint:

LDX \geq N

8: COV( $M \times (M + 1) / 2$ ) – REAL (KIND=nag_wp) arrayOutput

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))

i \leq j

9: A( $M \times (M + 1) / 2$ ) – REAL (KIND=nag_wp) arrayInput/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 = 1, 2, \dots, m

On exit: the lower triangular elements of the inverse of the matrix

A

, stored row-wise.

10: WT(N) – REAL (KIND=nag_wp) arrayOutput

On exit:

WT (i)

contains the weights,

{wt}_{i} = u ({‖z_{i}‖}_{2})

, for

i = 1, 2, \dots, n

11: THETA(M) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an initial estimate of the location parameter,

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 G07DAF.

On exit: contains the robust estimate of the location parameter,

θ_{j}

, for

j = 1, 2, \dots, m

12: BL – REAL (KIND=nag_wp)Input

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 – REAL (KIND=nag_wp)Input

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.

When printing occurs the output is directed to the current advisory message unit (see X04ABF).

16: TOL – REAL (KIND=nag_wp)Input

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

17: NIT – INTEGEROutput

On exit: the number of iterations performed.

18: WK( $2 \times M$ ) – REAL (KIND=nag_wp) arrayWorkspace

19: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$N \leq 1$ ,
or	$M < 1$ ,
or	$N < M$ ,
or	$LDX < N$ .

$IFAIL = 2$

On entry,	$TOL \leq 0.0$ ,
or	$MAXIT \leq 0$ ,
or	diagonal element of $A = 0.0$ ,
or	$BL \leq 0.0$ ,
or	$BD \leq 0.0$ .

$IFAIL = 3$: A column of X has a constant value.

$IFAIL = 4$: Value of U or W returned by $UCV < 0$ .

$IFAIL = 5$: The routine has failed to converge in MAXIT iterations.

$IFAIL = 6$: One of the following is zero: $D_{1}$ , $D_{2}$ or $D_{3}$ .

This may be caused by the functions $u$ or $w$ being too strict for the current estimate of $A$ (or $C$ ). You should 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)	$d 1 =$ 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 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.

9 Example

A sample of

10

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

A

and THETA and parameter values for the

u

and

w

functions,

c_{u}

and

c_{w}

. The covariance matrix computed by G02HLF 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 Routine DocumentG02HLF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G02HLF