NAG Library Function Document

nag_sum_sqs_update (g02btc)

+− 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

nag_sum_sqs_update (g02btc) 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

#include <nag.h>

#include <nagg02.h>

void	nag_sum_sqs_update (Nag_SumSquare mean, Integer m, double wt, const double x[], Integer incx, double sw, double xbar[], double c[], NagError fail)

3 Description

nag_sum_sqs_update (g02btc) is an adaptation of West's WV2 algorithm; see West (1979). This function 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 Arguments

1: mean – Nag_SumSquareInput

On entry: indicates whether nag_sum_sqs_update (g02btc) is to calculate sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.

$mean = Nag_AboutMean$: The sums of squares and cross-products of deviations about the mean are calculated.
$mean = Nag_AboutZero$: The sums of squares and cross-products are calculated.

Constraint:

mean = Nag_AboutMean

Nag_AboutZero

2: m – IntegerInput

On entry:

m

, the number of variables.

Constraint:

m \geq 1

3: wt – doubleInput

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[ $m \times incx$ ] – const doubleInput

On entry:

x [(j - 1) \times incx]

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.

Constraint:

incx > 0

6: sw – double *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] – doubleInput/Output

On entry: if

sw = 0.0

, xbar is initialized, otherwise

xbar [j - 1]

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 - 1]

contains the weighted mean of the

j

th variable,

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

, for

j = 1, 2, \dots, m

8: c[ $(m \times m + m) / 2$ ] – doubleInput/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 - 1]

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

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $incx = ⟨value⟩$ .
Constraint: $incx \geq 1$ .
On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
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.
NE_REAL: On entry, $sw = ⟨value⟩$ .
Constraint: $sw \geq 0.0$ .
NE_SUM_WEIGHT: On entry, $(sw + wt) = ⟨value⟩$ .
Constraint: $(sw + wt) \geq 0.0$ .

7 Accuracy

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

8 Parallelism and Performance

Not applicable.

9 Further Comments

nag_sum_sqs_update (g02btc) may be used to update the results returned by nag_sum_sqs (g02buc).

nag_cov_to_corr (g02bwc) may be used to calculate the correlation matrix from the matrix of sums of squares and cross-products of deviations about the mean .

10 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 Function Documentnag_sum_sqs_update (g02btc)

+− 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_sum_sqs_update (g02btc)