# NAG Library Routine Document

## 1Purpose

g02buf calculates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations from the mean, in a single pass for a set of data. The data may be weighted.

## 2Specification

Fortran Interface
 Subroutine g02buf ( mean, n, m, x, ldx, wt, sw, c,
 Integer, Intent (In) :: n, m, ldx Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(ldx,m), wt(*) Real (Kind=nag_wp), Intent (Out) :: sw, wmean(m), c((m*m+m)/2) Character (1), Intent (In) :: mean, weight
#include nagmk26.h
 void g02buf_ (const char *mean, const char *weight, const Integer *n, const Integer *m, const double x[], const Integer *ldx, const double wt[], double *sw, double wmean[], double c[], Integer *ifail, const Charlen length_mean, const Charlen length_weight)

## 3Description

g02buf is an adaptation of West's WV2 algorithm; see West (1979). This routine calculates the (optionally weighted) sample means and (optionally weighted) sums of squares and cross-products or sums of squares and cross-products of deviations from the (weighted) mean for a sample of $n$ observations on $m$ variables ${X}_{j}$, for $\mathit{j}=1,2,\dots ,m$. The algorithm makes a single pass through the data.
For the first $i-1$ observations let the mean of the $j$th variable be ${\stackrel{-}{x}}_{j}\left(i-1\right)$, the cross-product about the mean for the $j$th and $k$th variables be ${c}_{jk}\left(i-1\right)$ and the sum of weights be ${W}_{i-1}$. These are updated by the $i$th observation, ${x}_{ij}$, for $\mathit{j}=1,2,\dots ,m$, with weight ${w}_{i}$ as follows:
 $Wi = Wi-1 + wi x-j i = x-j i-1 + wiWi xj - x-j i-1 , j=1,2,…,m$
and
 $cjk i = cjk i- 1 + wi Wi xj - x-j i- 1 xk - x-k i-1 Wi-1 , j=1,2,…,m ​ and ​ k=j,j+ 1,…,m .$
The algorithm is initialized by taking ${\stackrel{-}{x}}_{j}\left(1\right)={x}_{1j}$, the first observation, and ${c}_{ij}\left(1\right)=0.0$.
For the unweighted case ${w}_{i}=1$ and ${W}_{i}=i$ for all $i$.
Note that only the upper triangle of the matrix is calculated and returned packed by column.

## 4References

Chan T F, Golub G H and Leveque R J (1982) Updating Formulae and a Pairwise Algorithm for Computing Sample Variances Compstat, Physica-Verlag
West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

## 5Arguments

1:     $\mathbf{mean}$ – Character(1)Input
On entry: indicates whether g02buf is to calculate sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.
${\mathbf{mean}}=\text{'M'}$
The sums of squares and cross-products of deviations about the mean are calculated.
${\mathbf{mean}}=\text{'Z'}$
The sums of squares and cross-products are calculated.
Constraint: ${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.
2:     $\mathbf{weight}$ – Character(1)Input
On entry: indicates whether the data is weighted or not.
${\mathbf{weight}}=\text{'U'}$
The calculations are performed on unweighted data.
${\mathbf{weight}}=\text{'W'}$
The calculations are performed on weighted data.
Constraint: ${\mathbf{weight}}=\text{'W'}$ or $\text{'U'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations in the dataset.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
5:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must contain the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
6:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02buf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
7:     $\mathbf{wt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array wt must be at least ${\mathbf{n}}$ if ${\mathbf{weight}}=\text{'W'}$.
On entry: the optional weights of each observation.
If ${\mathbf{weight}}=\text{'U'}$, wt is not referenced.
If ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(i\right)$ must contain the weight for the $i$th observation.
Constraint: if ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.
8:     $\mathbf{sw}$ – Real (Kind=nag_wp)Output
On exit: the sum of weights.
If ${\mathbf{weight}}=\text{'U'}$, sw contains the number of observations, $n$.
9:     $\mathbf{wmean}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the sample means. ${\mathbf{wmean}}\left(j\right)$ contains the mean for the $j$th variable.
10:   $\mathbf{c}\left(\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the cross-products.
If ${\mathbf{mean}}=\text{'M'}$, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.
If ${\mathbf{mean}}=\text{'Z'}$, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products.
These are stored packed by columns, i.e., the cross-product between the $j$th and $k$th variable, $k\ge j$, is stored in ${\mathbf{c}}\left(k×\left(k-1\right)/2+j\right)$.
11:   $\mathbf{ifail}$ – IntegerInput/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 3.4 in How to Use the NAG Library and its Documentation 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{mean}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{weight}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{weight}}=\text{'W'}$ or $\text{'U'}$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{wt}}\left(〈\mathit{\text{value}}〉\right)<0.0$.
Constraint: ${\mathbf{wt}}\left(i\right)\ge 0.0$, for $i=1,2,\dots ,n$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For a detailed discussion of the accuracy of this algorithm see Chan et al. (1982) or West (1979).

## 8Parallelism and Performance

g02buf is not threaded in any implementation.

g02bwf may be used to calculate the correlation coefficients from the cross-products of deviations about the mean. The cross-products of deviations about the mean may be scaled using f06edf (dscal) or f06fdf to give a variance-covariance matrix.
The means and cross-products produced by g02buf may be updated by adding or removing observations using g02btf.
Two sets of means and cross-products, as produced by g02buf, can be combined using g02bzf.

## 10Example

A program to calculate the means, the required sums of squares and cross-products matrix, and the variance matrix for a set of $3$ observations of $3$ variables.

### 10.1Program Text

Program Text (g02bufe.f90)

### 10.2Program Data

Program Data (g02bufe.d)

### 10.3Program Results

Program Results (g02bufe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017