NAG Library Routine Document

G02BTF

+− 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

G02BTF updates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean, for a new observation. The data may be weighted.

2 Specification

SUBROUTINE G02BTF (

MEAN, M, WT, X, INCX, SW, XBAR, C, IFAIL)

INTEGER	M, INCX, IFAIL
REAL (KIND=nag_wp)	WT, X(MINCX), SW, XBAR(M), C((MM+M)/2)
CHARACTER(1)	MEAN

3 Description

G02BTF is an adaptation of West's WV2 algorithm; see West (1979). This routine updates the weighted means of variables and weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean for observations on

m

variables

X_{j}

, for

j = 1, 2, \dots, m

. For the first

i - 1

observations let the mean of the

j

th variable be

{\bar{x}}_{j} (i - 1)

, the cross-product about the mean for the

j

th and

k

th variables be

c_{j k} (i - 1)

and the sum of weights be

W_{i - 1}

. These are updated by the

i

th observation,

x_{i j}

, for

j = 1, 2, \dots, m

, with weight

w_{i}

as follows:

W_{i} = W_{i - 1} + w_{i}, {\bar{x}}_{j} (i) = {\bar{x}}_{j} (i - 1) + \frac{w_{i}}{W_{i}} (x_{j} - {\bar{x}}_{j} (i - 1)), j = 1, 2, \dots, m

and

c_{j k} (i) = c_{j k} (i - 1) + \frac{w_{i}}{W_{i}} (x_{j} - {\bar{x}}_{j} (i - 1)) (x_{k} - {\bar{x}}_{k} (i - 1)) W_{i - 1}, j = 1, 2, \dots, m; k = j, j + 1, 2, \dots, m .

The algorithm is initialized by taking

{\bar{x}}_{j} (1) = x_{1 j}

, the first observation and

c_{i j} (1) = 0.0

For the unweighted case

w_{i} = 1

and

W_{i} = i

for all

i

4 References

Chan T F, Golub G H and Leveque R J (1982) Updating Formulae and a Pairwise Algorithm for Computing Sample Variances Compstat, Physica-Verlag

West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

5 Parameters

1: MEAN – CHARACTER(1)Input

On entry: indicates whether G02BTF is to calculate sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.

$MEAN ='M'$: The sums of squares and cross-products of deviations about the mean are calculated.
$MEAN ='Z'$: The sums of squares and cross-products are calculated.

Constraint:

MEAN ='M'

'Z'

2: M – INTEGERInput

On entry:

m

, the number of variables.

Constraint:

M \geq 1

3: WT – REAL (KIND=nag_wp)Input

On entry: the weight to use for the current observation,

w_{i}

For unweighted means and cross-products set

WT = 1.0

. The use of a suitable negative value of WT, e.g.,

- w_{i}

will have the effect of deleting the observation.

4: X( $M \times INCX$ ) – REAL (KIND=nag_wp) arrayInput

On entry:

X ((j - 1) \times INCX + 1)

must contain the value of the

j

th variable for the current observation,

j = 1, 2, \dots, m

5: INCX – INTEGERInput

On entry: the increment of X. Two situations are common.

INCX = 1

, the data values are to be found in consecutive locations in X, i.e., in a column.

INCX = ldx

, for some positive integer

ldx

, the data values are to be found as a row of an array with first dimension

ldx

Constraint:

INCX > 0

6: SW – REAL (KIND=nag_wp)Input/Output

On entry: the sum of weights for the previous observations,

W_{i - 1}

$SW = 0.0$: The update procedure is initialized.
$SW + WT = 0.0$: All elements of XBAR and C are set to zero.

Constraint:

SW \geq 0.0

and

SW + WT \geq 0.0

On exit: contains the updated sum of weights,

W_{i}

7: XBAR(M) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

SW = 0.0

, XBAR is initialized, otherwise

XBAR (j)

must contain the weighted mean of the

j

th variable for the previous

(i - 1)

observations,

{\bar{x}}_{j} (i - 1)

, for

j = 1, 2, \dots, m

On exit:

XBAR (j)

contains the weighted mean of the

j

th variable,

{\bar{x}}_{j} (i)

, for

j = 1, 2, \dots, m

8: C( $(M \times M + M) / 2$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

SW \neq 0.0

, C must contain the upper triangular part of the matrix of weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean. It is stored packed form by column, i.e., the cross-product between the

j

th and

k

th variable,

k \geq j

, is stored in

C (k \times (k - 1) / 2 + j)

On exit: the update sums of squares and cross-products stored as on input.

9: 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,	$M < 1$ ,
or	$INCX < 1$ .

$IFAIL = 2$

On entry,

SW < 0.0

$IFAIL = 3$

On entry,

(SW + WT) < 0.0

, the current weight causes the sum of weights to be less than

0.0

$IFAIL = 4$

On entry,

MEAN \neq'M'

'Z'

7 Accuracy

For a detailed discussion of the accuracy of this method see Chan et al. (1982) and West (1979).

8 Further Comments

G02BTF may be used to update the results returned by G02BUF.

G02BWF may be used to calculate the correlation matrix from the matrix of sums of squares and cross-products of deviations about the mean and the matrix may be scaled using F06EDF (DSCAL) or F06FDF to produce a variance-covariance matrix.

9 Example

A program to calculate the means, the required sums of squares and cross-products matrix, and the variance matrix for a set of

3

observations of

3

variables.

NAG Library Routine DocumentG02BTF

+− 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

G02BTF