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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_summary_onevar_combine (g01au)

## Purpose

nag_stat_summary_onevar_combine (g01au) combines sets of summaries produced by nag_stat_summary_onevar (g01at).

## Syntax

[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = g01au(mrcomm, 'b', b)
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = nag_stat_summary_onevar_combine(mrcomm, 'b', b)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 25: b was made optional

## Description

Assume a dataset containing $n$ observations, denoted by $x=\left\{{x}_{i}:i=1,2,\dots ,n\right\}$ and a set of weights, $w=\left\{{w}_{i}:i=1,2,\dots ,n\right\}$, has been split into $b$ blocks, and each block summarised via a call to nag_stat_summary_onevar (g01at). Then nag_stat_summary_onevar_combine (g01au) takes the $b$ communication arrays returned by nag_stat_summary_onevar (g01at) and returns the mean ($\stackrel{-}{x}$), standard deviation (${s}_{2}$), coefficients of skewness (${s}_{3}$) and kurtosis (${s}_{4}$), and the maximum and minimum values for the whole dataset.
For a definition of $\stackrel{-}{x},{s}_{2},{s}_{3}$ and ${s}_{4}$ see Description in nag_stat_summary_onevar (g01at).

## References

West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{mrcomm}\left(20,{\mathbf{b}}\right)$ – double array
The $j$th column of mrcomm must contain the information returned in rcomm from one of the runs of nag_stat_summary_onevar (g01at).

### Optional Input Parameters

1:     $\mathrm{b}$int64int32nag_int scalar
Default: $1$
$b$, the number of blocks the full dataset was split into.
Constraint: ${\mathbf{b}}\ge 1$.

### Output Parameters

1:     $\mathrm{pn}$int64int32nag_int scalar
The number of valid observations, that is the number of observations with ${w}_{i}>0$, for $\mathit{i}=1,2,\dots ,n$.
2:     $\mathrm{xmean}$ – double scalar
$\stackrel{-}{x}$, the mean.
3:     $\mathrm{xsd}$ – double scalar
${s}_{2}$, the standard deviation.
4:     $\mathrm{xskew}$ – double scalar
${s}_{3}$, the coefficient of skewness.
5:     $\mathrm{xkurt}$ – double scalar
${s}_{4}$, the coefficient of kurtosis.
6:     $\mathrm{xmin}$ – double scalar
The smallest value.
7:     $\mathrm{xmax}$ – double scalar
The largest value.
8:     $\mathrm{rcomm}\left(20\right)$ – double array
An amalgamation of the information held in mrcomm. This is in the same format as rcomm from nag_stat_summary_onevar (g01at).
9:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=11$
Constraint: ${\mathbf{b}}\ge 1$.
${\mathbf{ifail}}=21$
On entry, mrcomm is not in the expected format.
W  ${\mathbf{ifail}}=31$
On entry, the number of valid observations is zero.
W  ${\mathbf{ifail}}=51$
On exit we were unable to calculate xskew or xkurt. A value of $0$ has been returned.
W  ${\mathbf{ifail}}=52$
On exit we were unable to calculate xsd, xskew or xkurt. A value of $0$ has been returned.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

The order that the $b$ communication arrays are stored in mrcomm is arbitrary. Different orders can lead to slightly different results due to numerical accuracy of floating-point calculations.
Both nag_stat_summary_onevar_combine (g01au) and nag_stat_summary_onevar (g01at) consolidate results from multiple summaries. Whereas the former can only be used to combine summaries calculated sequentially, the latter combines summaries calculated in an arbitrary order allowing, for example, summaries calculated on different processing units to be combined.

## Example

This example summarises some simulated data. The data is supplied in three blocks, the first consisting of $21$ observations, the second $51$ observations and the last $28$ observations. Summaries are produced for each block of data separately and then an overall summary is produced.
```function g01au_example

fprintf('g01au example results\n\n');

x1 = [-0.62; -1.92; -1.72; -6.35;  2.00;  7.65;  6.15;
3.81;  4.87; -0.51;  6.88; -5.85; -0.72;  0.66;
2.23; -1.61; -0.15; -1.15; -8.74; -3.94;  3.61];
wt1 = [4.91;  0.25;  3.90;  3.75;  1.17;  3.19;  2.66;
0.02;  3.59;  3.63;  4.83;  3.72;  1.72;  0.78;
4.74;  1.72;  3.94;  1.33;  0.51;  2.40;  3.90];
x2 = [-0.66; -2.39; -6.25;  1.23;  2.27; -2.27; 10.12;
8.29; -2.99;  8.71; -0.74;  0.02;  1.22;  1.70;
4.30;  2.99; -0.83; -1.00;  6.57;  2.32; -3.47;
-1.41; -5.26;  0.53;  1.80;  4.79; -3.04;  1.20;
-3.21; -3.75;  0.86;  1.27; -5.95; -5.27;  1.63;
3.59; -0.01; -1.38; -4.71; -4.82;  3.55;  0.46;
2.57;  1.76; -4.05;  1.23; -1.99;  3.20; -0.65;
8.42; -6.01];
x3 = [ 1.13; -8.86;  5.92; -1.71; -3.99;  6.57; -2.01;
-2.29; -1.11;  7.14;  4.84; -4.44; -3.32; 10.25;
-2.11;  8.02; -7.31;  2.80; -1.20;  1.01;  1.37;
-2.28;  1.28; -3.95;  3.43; -0.61; 4.85; -0.11];
data = {x1; x2; x3};

mrcomm = zeros(20,3);
% Initialise the number of valid observations processed so far
for i =1:3
% Summarise this block of data
if (i == 1)
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, mrcomm(:, 1), ifail] = ...
g01at(x1, 'wt', wt1);
else
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, mrcomm(:, i), ifail] = ...
g01at(data{i});
end

% Display the results for this block
fprintf('\nSummary for block %d\n', i);
fprintf('%d valid observations\n', pn);
fprintf('Mean          %13.2f\n', xmean);
fprintf('Std devn      %13.2f\n', xsd);
fprintf('Skewness      %13.2f\n', xskew);
fprintf('Kurtosis      %13.2f\n', xkurt);
fprintf('Minimum       %13.2f\n', xmin);
fprintf('Maximum       %13.2f\n', xmax);
end

% Combine the summaries across all the blocks
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = ...
g01au(mrcomm);

% Display the combined results
fprintf('\nSummary for the combined data\n');
fprintf('%d valid observations\n', pn);
fprintf('Mean          %13.2f\n', xmean);
fprintf('Std devn      %13.2f\n', xsd);
fprintf('Skewness      %13.2f\n', xskew);
fprintf('Kurtosis      %13.2f\n', xkurt);
fprintf('Minimum       %13.2f\n', xmin);
fprintf('Maximum       %13.2f\n', xmax);

```
```g01au example results

Summary for block 1
21 valid observations
Mean                   0.73
Std devn               4.40
Skewness              -0.05
Kurtosis              -1.00
Minimum               -8.74
Maximum                7.65

Summary for block 2
51 valid observations
Mean                   0.28
Std devn               3.96
Skewness               0.46
Kurtosis              -0.16
Minimum               -6.25
Maximum               10.12

Summary for block 3
28 valid observations
Mean                   0.48
Std devn               4.65
Skewness               0.19
Kurtosis              -0.58
Minimum               -8.86
Maximum               10.25

Summary for the combined data
100 valid observations
Mean                   0.51
Std devn               4.24
Skewness               0.18
Kurtosis              -0.59
Minimum               -8.86
Maximum               10.25
```