g02bzf combines two sets of sample means and sums of squares and crossproducts matrices. It is designed to be used in conjunction with
g02buf to allow large datasets to be summarised.
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 crossproducts of deviations from
${\mu}_{x}$
or the sums of squares and crossproducts, in which case
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
${W}_{y}$ g02bzf 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
g02buf, with
$Z$ constructed as
g02bzf has been designed to combine the results from two calls to
g02buf 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, singlepass, parallel statistics algorithms Proceedings of IEEE International Conference on Cluster Computing

1:
$\mathbf{mean}$ – Character(1)
Input

On entry: indicates whether the matrices supplied in
xc and
yc are sums of squares and crossproducts, or sums of squares and crossproducts of deviations about the mean.
 ${\mathbf{mean}}=\text{'M'}$
 Sums of squares and crossproducts of deviations about the mean have been supplied.
 ${\mathbf{mean}}=\text{'Z'}$
 Sums of squares and crossproducts have been supplied.
Constraint:
${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.

2:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of variables.
Constraint:
${\mathbf{m}}\ge 1$.

3:
$\mathbf{xsw}$ – Real (Kind=nag_wp)
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:
$\mathbf{xmean}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) array
Input/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:
$\mathbf{xc}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output
Note: the dimension of the array
xc
must be at least
$\left({\mathbf{m}}\times {\mathbf{m}}+{\mathbf{m}}\right)/2$.
On entry:
${C}_{x}$, the sums of squares and crossproducts matrix for the first set of data,
$X$, as returned by
g02buf.
g02buf, returns this matrix packed by columns, i.e., the crossproduct between the
$j$th and
$k$th variable,
$k\ge j$, is stored in
${\mathbf{xc}}\left(k\times \left(k1\right)/2+j\right)$.
No check is made that ${C}_{x}$ is a valid crossproducts matrix.
On exit:
${C}_{z}$, the sums of squares and crossproducts matrix for the combined dataset,
$Z$.
This matrix is again stored packed by columns.

6:
$\mathbf{ysw}$ – Real (Kind=nag_wp)
Input

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:
$\mathbf{ymean}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: ${\mu}_{y}$, the sample means for the second set of data, $Y$.

8:
$\mathbf{yc}\left(*\right)$ – Real (Kind=nag_wp) array
Input
Note: the dimension of the array
yc
must be at least
$\left({\mathbf{m}}\times {\mathbf{m}}+{\mathbf{m}}\right)/2$.
On entry:
${C}_{y}$, the sums of squares and crossproducts matrix for the second set of data,
$Y$, as returned by
g02buf.
g02buf, returns this matrix packed by columns, i.e., the crossproduct between the
$j$th and
$k$th variable,
$k\ge j$, is stored in
${\mathbf{yc}}\left(k\times \left(k1\right)/2+j\right)$.
No check is made that ${C}_{y}$ is a valid crossproducts matrix.

9:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Not applicable.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
None.
This example illustrates the use of
g02bzf by dividing a dataset into three blocks of
$4$,
$5$ and
$3$ observations respectively. Each block of data is summarised using
g02buf and then the three summaries combined using
g02bzf.
The resulting sums of squares and crossproducts matrix is then scaled to obtain the covariance matrix for the whole dataset.