NAG Library Function Document

nag_sum_sqs_combine (g02bzc)

+− 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_combine (g02bzc) combines two sets of sample means and sums of squares and cross-products matrices. It is designed to be used in conjunction with nag_sum_sqs (g02buc) to allow large datasets to be summarised.

2 Specification

#include <nag.h>

#include <nagg02.h>

void	nag_sum_sqs_combine (Nag_SumSquare mean, Integer m, double xsw, double xmean[], double xc[], double ysw, const double ymean[], const double yc[], NagError fail)

3 Description

Let

X

and

Y

denote two sets of data, each with

m

variables and

n_{x}

and

n_{y}

observations respectively. Let

μ_{x}

denote the (optionally weighted) vector of

m

means for the first dataset and

C_{x}

denote either the sums of squares and cross-products of deviations from

μ_{x}

C_{x} = {(X - e μ_{x}^{T})}^{T} D_{x} (X - e μ_{x}^{T})

or the sums of squares and cross-products, in which case

C_{x} = X^{T} D_{x} X

where

e

is a vector of

n_{x}

ones and

D_{x}

is a diagonal matrix of (optional) weights and

W_{x}

is defined as the sum of the diagonal elements of

D

. Similarly, let

μ_{y}

C_{y}

and

W_{y}

denote the same quantities for the second dataset.

Given

μ_{x}, μ_{y}, C_{x}, C_{y}, W_{x}

and

W_{y}

nag_sum_sqs_combine (g02bzc) calculates

μ_{z}

C_{z}

and

W_{z}

as if a dataset

Z

, with

m

variables and

n_{x} + n_{y}

observations were supplied to nag_sum_sqs (g02buc), with

Z

constructed as

Z = (\begin{matrix} X \\ Y \end{matrix}) .

nag_sum_sqs_combine (g02bzc) has been designed to combine the results from two calls to nag_sum_sqs (g02buc) allowing large datasets, or cases where all the data is not available at the same time, to be summarised.

4 References

Bennett J, Pebay P, Roe D and Thompson D (2009) Numerically stable, single-pass, parallel statistics algorithms Proceedings of IEEE International Conference on Cluster Computing

5 Arguments

1: mean – Nag_SumSquareInput

On entry: indicates whether the matrices supplied in xc and yc are sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.

$mean = Nag_AboutMean$: Sums of squares and cross-products of deviations about the mean have been supplied.
$mean = Nag_AboutZero$: Sums of squares and cross-products have been supplied.

Constraint:

mean = Nag_AboutMean

Nag_AboutZero

2: m – IntegerInput

On entry:

m

, the number of variables.

Constraint:

m \geq 1

3: xsw – double *Input/Output

On entry:

W_{x}

, the sum of weights, from the first set of data,

X

. If the data is unweighted then this will be the number of observations in the first dataset.

On exit:

W_{z}

, the sum of weights, from the combined dataset,

Z

. If both datasets are unweighted then this will be the number of observations in the combined dataset.

Constraint:

xsw \geq 0

4: xmean[m] – doubleInput/Output

On entry:

μ_{x}

, the sample means for the first set of data,

X

On exit:

μ_{z}

, the sample means for the combined data,

Z

5: xc[ $(m \times m + m) / 2$ ] – doubleInput/Output

On entry:

C_{x}

, the sums of squares and cross-products matrix for the first set of data,

X

, as returned by nag_sum_sqs (g02buc).

nag_sum_sqs (g02buc), returns this matrix packed by columns, i.e., the cross-product between the

j

th and

k

th variable,

k \geq j

, is stored in

xc [k \times (k - 1) / 2 + j - 1]

No check is made that

C_{x}

is a valid cross-products matrix.

On exit:

C_{z}

, the sums of squares and cross-products matrix for the combined dataset,

Z

This matrix is again stored packed by columns.

6: ysw – doubleInput

On entry:

W_{y}

, the sum of weights, from the second set of data,

Y

. If the data is unweighted then this will be the number of observations in the second dataset.

Constraint:

ysw \geq 0

7: ymean[m] – const doubleInput

On entry:

μ_{y}

, the sample means for the second set of data,

Y

8: yc[ $(m \times m + m) / 2$ ] – const doubleInput

On entry:

C_{y}

, the sums of squares and cross-products matrix for the second set of data,

Y

, as returned by nag_sum_sqs (g02buc).

nag_sum_sqs (g02buc), returns this matrix packed by columns, i.e., the cross-product between the

j

th and

k

th variable,

k \geq j

, is stored in

yc [k \times (k - 1) / 2 + j - 1]

No check is made that

C_{y}

is a valid cross-products matrix.

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, $m = ⟨value⟩$ .
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, $xsw = ⟨value⟩$ .
Constraint: $xsw \geq 0.0$ .
On entry, $ysw = ⟨value⟩$ .
Constraint: $ysw \geq 0.0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_sum_sqs_combine (g02bzc) is not threaded by NAG in any implementation.

nag_sum_sqs_combine (g02bzc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example illustrates the use of nag_sum_sqs_combine (g02bzc) by dividing a dataset into three blocks of

4

5

and

3

observations respectively. Each block of data is summarised using nag_sum_sqs (g02buc) and then the three summaries combined using nag_sum_sqs_combine (g02bzc).

The resulting sums of squares and cross-products matrix is then scaled to obtain the covariance matrix for the whole dataset.

NAG Library Function Documentnag_sum_sqs_combine (g02bzc)

+− 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_combine (g02bzc)