void	nag_robust_corr_estim (Integer n, Integer m, const double x[], Integer tdx, double eps, double cov[], double theta[], Integer max_iter, Integer print_iter, const char outfile, double tol, Integer iter, NagError *fail)

3

Description

For a set

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} - 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_corr_estim (g02hkc) uses weight functions:

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

and

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

for constants

a_{u}

b_{u}

and

c_{w}

These functions solve a minimax problem considered by Huber (1981).

The values of

a_{u}

b_{u}

and

c_{w}

are calculated from the expected fraction of gross errors,

ε

(see Huber (1981) and Marazzi (1987)). The expected fraction of gross errors is the estimated proportion of outliers in the sample.

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

τ^{2}

, is calculated, (see Huber (1981) and Marazzi (1987)).

Initial estimates of

θ_{j}

, for

j = 1, 2, \dots, m

, are given by the median of the

j

th column of

X

and the initial value of

A

is based on the median absolute deviation (see Marazzi (1987)). nag_robust_corr_estim (g02hkc) 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: $n$ – IntegerInput

On entry: the number of observations,

n

Constraint:

n > 1

2: $m$ – IntegerInput

On entry: the number of columns of the matrix

X

, i.e., number of independent variables,

m

Constraint:

1 \leq m \leq n

3: $x [n \times tdx]$ – const doubleInput

On entry:

x [(i - 1) \times tdx + j - 1]

must contain the

i

th observation for the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: $tdx$ – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

5: $eps$ – doubleInput

On entry: the expected fraction of gross errors expected in the sample,

ε

Constraint:

0.0 \leq eps < 1.0

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

On exit: the

m \times (m + 1)

/2 elements of cov contain the upper triangular part of the covariance matrix. They are stored packed by column, i.e.,

C_{i j}

j \geq i

, is stored in

cov [j (j + 1) / 2 + i]

, for

i = 0, 1, \dots, m - 1

and

j = i, \dots, m - 1

7: $theta [m]$ – doubleOutput

On exit: the robust estimate of the location arguments

θ_{j}

, for

j = 1, 2, \dots, m

8: $max_iter$ – IntegerInput

On entry: the maximum number of iterations that will be used during the calculation of the covariance matrix.

Suggested value:

max_iter = 150

Constraint:

max_iter > 0

9: $print_iter$ – IntegerInput

On entry: indicates if the printing of information on the iterations is required and the rate at which printing is produced.

$print_iter \leq 0$: No iteration monitoring is printed.
$print_iter > 0$: The value of $A$ , $θ$ and $δ$ (see Section 9) will be printed at the first and every print_iter iterations.

10: $outfile$ – const char *Input

On entry: a null terminated character string giving the name of the file to which results should be printed. If outfile is NULL or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.

11: $tol$ – doubleInput

On entry: the relative precision for the final estimates of the covariance matrix.

Constraint:

tol > 0.0

12: $iter$ – Integer *Output

On exit: the number of iterations performed.

13: $fail$ – 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_2_INT_ARG_GT: On entry, $m = 〈value〉$ while $n = 〈value〉$ . These arguments must satisfy $m \leq n$ .
NE_2_INT_ARG_LT: On entry, $tdx = 〈value〉$ while $m = 〈value〉$ . These arguments must satisfy $tdx \geq m$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_C_ITER_UNSTABLE: The iterative procedure to find $C$ has become unstable. This may happen if the value of eps is too large.
NE_CONST_COL: On entry, column $〈value〉$ of array x has constant value.
NE_INT_ARG_LE: On entry, max_iter must not be less than or equal to 0: $max_iter = 〈value〉$ .
NE_INT_ARG_LT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_REAL_ARG_GE: On entry, eps must be not be greater than or equal to 1.0: $eps = 〈value〉$ .
NE_REAL_ARG_LE: On entry, tol must not be less than or equal to 0.0: $tol = 〈value〉$ .
NE_REAL_ARG_LT: On entry, eps must not be less than 0.0: $eps = 〈value〉$ .
NE_TOO_MANY: Too many iterations( $〈value〉$ ).
The iterative procedure to find the co-variance matrix $C$ , has failed to converge in max_iter iterations.

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 the absolute relative change in $A$
(ii)	$d 2 =$ the maximum absolute change in $u ({‖z_{i}‖}_{2})$
(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_corr_estim (g02hkc) is not threaded in any implementation.

9

Further Comments

The existence of

A

, and hence

c

, 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 and the robust estimate of the covariance matrix is computed assuming 10% gross errors are to be expected. The robust covariance is then printed.

NAG Library Function Document

nag_robust_corr_estim (g02hkc)

▸▿ 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