NAG C Library Function Document

nag_sum_sqs (g02buc) calculates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations from the mean, in a single pass for a set of data. The data may be weighted.

2

Specification

#include <nag.h>

#include <nagg02.h>

void	nag_sum_sqs (Nag_OrderType order, Nag_SumSquare mean, Integer n, Integer m, const double x[], Integer pdx, const double wt[], double sw, double wmean[], double c[], NagError fail)

3

Description

nag_sum_sqs (g02buc) is an adaptation of West's WV2 algorithm; see West (1979). This function calculates the (optionally weighted) sample means and (optionally weighted) sums of squares and cross-products or sums of squares and cross-products of deviations from the (weighted) mean for a sample of

n

observations on

m

variables

X_{j}

, for

j = 1, 2, \dots, m

. The algorithm makes a single pass through the data.

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:

\begin{matrix} 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 \end{matrix}

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 ​ and ​ k = j, j + 1, \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

Note that only the upper triangle of the matrix is calculated and returned packed by column.

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: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mean$ – Nag_SumSquareInput

On entry: indicates whether nag_sum_sqs (g02buc) 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

3: $n$ – IntegerInput

On entry:

n

, the number of observations in the dataset.

Constraint:

n \geq 1

4: $m$ – IntegerInput

On entry:

m

, the number of variables.

Constraint:

m \geq 1

5: $x [\dim]$ – const doubleInput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

Where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

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

6: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

7: $wt [\dim]$ – const doubleInput

Note: the dimension, dim, of the array wt must be at least

n

On entry: the optional weights of each observation. If weights are not provided then wt must be set to NULL, otherwise

wt [i - 1]

must contain the weight for the

i

th observation.

Constraint: if

wt is not NULL

wt [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n

8: $sw$ – double *Output

On exit: the sum of weights.

wt is NULL

, sw contains the number of observations,

n

9: $wmean [m]$ – doubleOutput

On exit: the sample means.

wmean [j - 1]

contains the mean for the

j

th variable.

10: $c [(m \times m + m) / 2]$ – doubleOutput

On exit: the cross-products.

mean = Nag_AboutMean

, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.

mean = Nag_AboutZero

, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products.

These are stored packed by columns, 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]

11: $fail$ – 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .

On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq$ or $n$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARRAY_ELEM_CONS: On entry, $wt [〈value〉] < 0.0$ .
Constraint: $wt [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ .

7

Accuracy

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

8

Parallelism and Performance

nag_sum_sqs (g02buc) is not threaded in any implementation.

9

Further Comments

nag_cov_to_corr (g02bwc) may be used to calculate the correlation coefficients from the cross-products of deviations about the mean. The cross-products of deviations about the mean may be scaled to give a variance-covariance matrix.

The means and cross-products produced by nag_sum_sqs (g02buc) may be updated by adding or removing observations using nag_sum_sqs_update (g02btc).

Two sets of means and cross-products, as produced by nag_sum_sqs (g02buc), can be combined using nag_sum_sqs_combine (g02bzc).

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 C Library Function Document

nag_sum_sqs (g02buc)

▸▿ 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