G02BUF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

G02BUF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

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.

2  Specification

SUBROUTINE G02BUF ( MEAN, WEIGHT, N, M, X, LDX, WT, SW, WMEAN, C, IFAIL)
INTEGER  N, M, LDX, IFAIL
REAL (KIND=nag_wp)  X(LDX,M), WT(*), SW, WMEAN(M), C((M*M+M)/2)
CHARACTER(1)  MEAN, WEIGHT

3  Description

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 Xj, for j=1,2,,m. The algorithm makes a single pass through the data.
For the first i-1 observations let the mean of the jth variable be x-ji-1, the cross-product about the mean for the jth and kth variables be cjki-1 and the sum of weights be Wi-1. These are updated by the ith observation, xij, for j=1,2,,m, with weight wi 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 x-j1=x1j, the first observation, and cij1=0.0.
For the unweighted case wi=1 and Wi=i for all i.
Note that only the upper triangle of the matrix is calculated and returned packed by column.

4  References

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

5  Arguments

1:     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.
MEAN='M'
The sums of squares and cross-products of deviations about the mean are calculated.
MEAN='Z'
The sums of squares and cross-products are calculated.
Constraint: MEAN='M' or 'Z'.
2:     WEIGHT – CHARACTER(1)Input
On entry: indicates whether the data is weighted or not.
WEIGHT='U'
The calculations are performed on unweighted data.
WEIGHT='W'
The calculations are performed on weighted data.
Constraint: WEIGHT='W' or 'U'.
3:     N – INTEGERInput
On entry: n, the number of observations in the dataset.
Constraint: N1.
4:     M – INTEGERInput
On entry: m, the number of variables.
Constraint: M1.
5:     XLDXM – REAL (KIND=nag_wp) arrayInput
On entry: Xij must contain the ith observation on the jth variable, for i=1,2,,n and j=1,2,,m.
6:     LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which G02BUF is called.
Constraint: LDXN.
7:     WT* – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array WT must be at least N if WEIGHT='W', and at least 1 otherwise.
On entry: the optional weights of each observation.
If WEIGHT='U', WT is not referenced.
If WEIGHT='W', WTi must contain the weight for the ith observation.
Constraint: if WEIGHT='W', WTi0.0, for i=1,2,,n.
8:     SW – REAL (KIND=nag_wp)Output
On exit: the sum of weights.
If WEIGHT='U', SW contains the number of observations, n.
9:     WMEANM – REAL (KIND=nag_wp) arrayOutput
On exit: the sample means. WMEANj contains the mean for the jth variable.
10:   CM×M+M/2 – REAL (KIND=nag_wp) arrayOutput
On exit: the cross-products.
If MEAN='M', C contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.
If MEAN='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 jth and kth variable, kj, is stored in Ck×k-1/2+j.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ 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​ 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 -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry 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:
IFAIL=1
On entry,M<1,
orN<1,
orLDX<N.
IFAIL=2
On entry,MEAN'M' or 'Z'.
IFAIL=3
On entry,WEIGHT'W' or 'U'.
IFAIL=4
On entry, WEIGHT='W', and a value of WT<0.0.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
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.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

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

8  Parallelism and Performance

G02BUF is not threaded in any implementation.

9  Further Comments

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.

10  Example

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.

10.1  Program Text

Program Text (g02bufe.f90)

10.2  Program Data

Program Data (g02bufe.d)

10.3  Program Results

Program Results (g02bufe.r)


G02BUF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016