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>

void	g02btc (Nag_SumSquare mean, Integer m, double wt, const double x[], Integer incx, double sw, double xbar[], double c[], NagError fail)

The function may be called by the names: g02btc, nag_correg_ssqmat_update or nag_sum_sqs_update.

3 Description

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_SumSquare Input

On entry: indicates whether 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: $m$ – Integer Input

On entry:

m

, the number of variables.

Constraint:

m \geq 1

3: $wt$ – double 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: $x [m \times incx]$ – const double Input

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: $incx$ – Integer Input

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: $xbar [m]$ – double Input/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: $c [(m \times m + m) / 2]$ – double Input/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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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

g02btc is not threaded in any implementation.

9 Further Comments

g02btc may be used to update the results returned by g02buc.

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.

g02bt: FL CL CPP AD

NAG CL Interfaceg02btc (ssqmat_​update)

▸▿ 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 CL Interface
g02btc (ssqmat_update)