nag_sum_sqs_combine (g02bzc) (PDF version)
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g02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sum_sqs_combine (g02bzc)

## 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 #include
 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 ${\mu }_{x}$ 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 ${\mu }_{x}$
 $Cx= X-e⁢ μxT T ⁢ Dx ⁢ X-e⁢ μxT$
or the sums of squares and cross-products, in which case
 $Cx = XT ⁢ Dx ⁢X$
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 ${\mu }_{y}$, ${C}_{y}$ and ${W}_{y}$ denote the same quantities for the second dataset.
Given ${\mu }_{x},{\mu }_{y},{C}_{x},{C}_{y},{W}_{x}$ and ${W}_{y}$ nag_sum_sqs_combine (g02bzc) calculates ${\mu }_{z}$, ${C}_{z}$ 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
 $Z = X Y .$
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.
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:    $\mathbf{mean}$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.
${\mathbf{mean}}=\mathrm{Nag_AboutMean}$
Sums of squares and cross-products of deviations about the mean have been supplied.
${\mathbf{mean}}=\mathrm{Nag_AboutZero}$
Sums of squares and cross-products have been supplied.
Constraint: ${\mathbf{mean}}=\mathrm{Nag_AboutMean}$ or $\mathrm{Nag_AboutZero}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
3:    $\mathbf{xsw}$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: ${\mathbf{xsw}}\ge 0$.
4:    $\mathbf{xmean}\left[{\mathbf{m}}\right]$doubleInput/Output
On entry: ${\mu }_{x}$, the sample means for the first set of data, $X$.
On exit: ${\mu }_{z}$, the sample means for the combined data, $Z$.
5:    $\mathbf{xc}\left[\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right]$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, $k\ge j$, is stored in ${\mathbf{xc}}\left[k×\left(k-1\right)/2+j-1\right]$.
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:    $\mathbf{ysw}$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: ${\mathbf{ysw}}\ge 0$.
7:    $\mathbf{ymean}\left[{\mathbf{m}}\right]$const doubleInput
On entry: ${\mu }_{y}$, the sample means for the second set of data, $Y$.
8:    $\mathbf{yc}\left[\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right]$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, $k\ge j$, is stored in ${\mathbf{yc}}\left[k×\left(k-1\right)/2+j-1\right]$.
No check is made that ${C}_{y}$ is a valid cross-products matrix.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL
On entry, ${\mathbf{xsw}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xsw}}\ge 0.0$.
On entry, ${\mathbf{ysw}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ysw}}\ge 0.0$.

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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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.

### 10.1  Program Text

Program Text (g02bzce.c)

### 10.2  Program Data

Program Data (g02bzce.d)

### 10.3  Program Results

Program Results (g02bzce.r)

nag_sum_sqs_combine (g02bzc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual