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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(b, mrcomm)
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = nag_stat_summary_onevar_combine(b, mrcomm)

Description

Assume a dataset containing nn observations, denoted by x = {xi : i = 1,2,,n} x = { xi : i=1,2,,n }  and a set of weights, w = {wi : i = 1,2,,n} w = { wi : i=1,2,,n } , has been split into bb blocks, and each block summarised via a call to nag_stat_summary_onevar (g01at). Then nag_stat_summary_onevar_combine (g01au) takes the bb communication arrays returned by nag_stat_summary_onevar (g01at) and returns the mean (xx-), standard deviation (s2s2), coefficients of skewness (s3s3) and kurtosis (s4s4), and the maximum and minimum values for the whole dataset.
For a definition of x,s2,s3x-,s2,s3 and s4s4 see Section [Description] in (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:     b – int64int32nag_int scalar
bb, the number of blocks the full dataset was split into.
Constraint: b1b1.
2:     mrcomm(2020,bb) – double array
The jjth column of mrcomm must contain the information returned in rcomm from one of the runs of nag_stat_summary_onevar (g01at).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     pn – int64int32nag_int scalar
The number of valid observations, that is the number of observations with wi > 0wi>0, for i = 1,2,,ni=1,2,,n.
2:     xmean – double scalar
xx-, the mean.
3:     xsd – double scalar
s2s2, the standard deviation.
4:     xskew – double scalar
s3s3, the coefficient of skewness.
5:     xkurt – double scalar
s4s4, the coefficient of kurtosis.
6:     xmin – double scalar
The smallest value.
7:     xmax – double scalar
The largest value.
8:     rcomm(2020) – 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:     ifail – int64int32nag_int scalar
ifail = 0ifail=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.

  ifail = 11ifail=11
Constraint: b1b1.
  ifail = 21ifail=21
On entry, mrcomm is not in the expected format.
W ifail = 31ifail=31
On entry, the number of valid observations is zero.
W ifail = 51ifail=51
On exit we were unable to calculate xskew or xkurt. A value of 00 has been returned.
W ifail = 52ifail=52
On exit we were unable to calculate xsd, xskew or xkurt. A value of 00 has been returned.

Accuracy

Not applicable.

Further Comments

The order that the bb 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

function nag_stat_summary_onevar_combine_example
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
  pn = int64(0);
  if ( i == 1)
    [pn, xmean, xsd, xskew, xkurt, xmin, xmax, mrcomm(:, 1), ifail] = ...
        nag_stat_summary_onevar(x1, 'wt', wt1);
  else
    [pn, xmean, xsd, xskew, xkurt, xmin, xmax, mrcomm(:, i), ifail] = ...
        nag_stat_summary_onevar(data{i});
  end

  % Display the results for this block
  fprintf('\nSummary for block %d\n', i);
  if (ifail==53)
    fprintf('No valid observations supplied. All weights are zero.\n')
  else
    fprintf('%d valid observations\n', pn);
    fprintf('Mean          %13.2f\n', xmean);
    if (ifail==72)
      fprintf('Unable to calculate the standard deviation, skewness or kurtosis\n');
    else
      fprintf('Std devn      %13.2f\n', xsd);
      if (ifail==71)
        fprintf('Unable to calculate the skewness or kurtosis\n');
      else
        fprintf('Skewness      %13.2f\n', xskew);
        fprintf('Kurtosis      %13.2f\n', xkurt);
      end
    end
    fprintf('Minimum       %13.2f\n', xmin);
    fprintf('Maximum       %13.2f\n', xmax);
  end
end

% Combine the summaries across all the blocks
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = ...
    nag_stat_summary_onevar_combine(int64(3), mrcomm);

% Display the combined results
fprintf('\nSummary for the combined data\n');
if (ifail==53)
  fprintf('No valid observations supplied. All weights are zero.\n')
else
  fprintf('%d valid observations\n', pn);
  fprintf('Mean          %13.2f\n', xmean);
  if (ifail==72)
    fprintf('Unable to calculate the standard deviation, skewness or kurtosis\n');
  else
    fprintf('Std devn      %13.2f\n', xsd);
    if (ifail==71)
      fprintf('Unable to calculate the skewness or kurtosis\n');
    else
      fprintf('Skewness      %13.2f\n', xskew);
      fprintf('Kurtosis      %13.2f\n', xkurt);
    end
  end
  fprintf('Minimum       %13.2f\n', xmin);
  fprintf('Maximum       %13.2f\n', xmax);
end
 

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

function g01au_example
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);
  if (ifail==53)
    fprintf('No valid observations supplied. All weights are zero.\n')
  else
    fprintf('%d valid observations\n', pn);
    fprintf('Mean          %13.2f\n', xmean);
    if (ifail==72)
      fprintf('Unable to calculate the standard deviation, skewness or kurtosis\n');
    else
      fprintf('Std devn      %13.2f\n', xsd);
      if (ifail==71)
        fprintf('Unable to calculate the skewness or kurtosis\n');
      else
        fprintf('Skewness      %13.2f\n', xskew);
        fprintf('Kurtosis      %13.2f\n', xkurt);
      end
    end
    fprintf('Minimum       %13.2f\n', xmin);
    fprintf('Maximum       %13.2f\n', xmax);
  end
end

% Combine the summaries across all the blocks
[pn, xmean, xsd, xskew, xkurt, xmin, xmax, rcomm, ifail] = ...
    g01au(int64(3), mrcomm);

% Display the combined results
fprintf('\nSummary for the combined data\n');
if (ifail==53)
  fprintf('No valid observations supplied. All weights are zero.\n')
else
  fprintf('%d valid observations\n', pn);
  fprintf('Mean          %13.2f\n', xmean);
  if (ifail==72)
    fprintf('Unable to calculate the standard deviation, skewness or kurtosis\n');
  else
    fprintf('Std devn      %13.2f\n', xsd);
    if (ifail==71)
      fprintf('Unable to calculate the skewness or kurtosis\n');
    else
      fprintf('Skewness      %13.2f\n', xskew);
      fprintf('Kurtosis      %13.2f\n', xkurt);
    end
  end
  fprintf('Minimum       %13.2f\n', xmin);
  fprintf('Maximum       %13.2f\n', xmax);
end
 

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


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