# NAG Library Routine Document

## 1Purpose

g02btf updates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean, for a new observation. The data may be weighted.

## 2Specification

Fortran Interface
 Subroutine g02btf ( mean, m, wt, x, incx, sw, xbar, c,
 Integer, Intent (In) :: m, incx Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: wt, x(m*incx) Real (Kind=nag_wp), Intent (Inout) :: sw, xbar(m), c((m*m+m)/2) Character (1), Intent (In) :: mean
#include <nagmk26.h>
 void g02btf_ (const char *mean, const Integer *m, const double *wt, const double x[], const Integer *incx, double *sw, double xbar[], double c[], Integer *ifail, const Charlen length_mean)

## 3Description

g02btf is an adaptation of West's WV2 algorithm; see West (1979). This routine updates the weighted means of variables and weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean for observations on $m$ variables ${X}_{j}$, for $j=1,2,\dots ,m$. 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:
 $Wi=Wi-1+wi, x-ji=x-ji-1+wiWixj-x-ji-1, j=1,2,…,m$
and
 $cjki=cjki- 1+wiWixj-x-ji- 1xk-x-ki- 1Wi- 1, j= 1,2,…,m;k=j,j+ 1,2,…,m.$
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 $i$.

## 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 g02btf 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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathbf{wt}$ – Real (Kind=nag_wp)Input
On entry: the weight to use for the current observation, ${w}_{i}$.
For unweighted means and cross-products set ${\mathbf{wt}}=1.0$. The use of a suitable negative value of wt, e.g., $-{w}_{i}$ will have the effect of deleting the observation.
4:     $\mathbf{x}\left({\mathbf{m}}×{\mathbf{incx}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\left(j-1\right)×{\mathbf{incx}}+1\right)$ must contain the value of the $j$th variable for the current observation, $j=1,2,\dots ,m$.
5:     $\mathbf{incx}$ – IntegerInput
On entry: the increment of x. Two situations are common.
If ${\mathbf{incx}}=1$, the data values are to be found in consecutive locations in x, i.e., in a column.
If ${\mathbf{incx}}=\mathit{ldx}$, for some positive integer $\mathit{ldx}$, the data values are to be found as a row of an array with first dimension $\mathit{ldx}$.
Constraint: ${\mathbf{incx}}>0$.
6:     $\mathbf{sw}$ – Real (Kind=nag_wp)Input/Output
On entry: the sum of weights for the previous observations, ${W}_{i-1}$.
${\mathbf{sw}}=0.0$
The update procedure is initialized.
${\mathbf{sw}}+{\mathbf{wt}}=0.0$
All elements of xbar and c are set to zero.
Constraint: ${\mathbf{sw}}\ge 0.0$ and ${\mathbf{sw}}+{\mathbf{wt}}\ge 0.0$.
On exit: contains the updated sum of weights, ${W}_{i}$.
7:     $\mathbf{xbar}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{sw}}=0.0$, xbar is initialized, otherwise ${\mathbf{xbar}}\left(\mathit{j}\right)$ must contain the weighted mean of the $\mathit{j}$th variable for the previous $\left(\mathit{i}-1\right)$ observations, ${\stackrel{-}{x}}_{\mathit{j}}\left(\mathit{i}-1\right)$, for $\mathit{j}=1,2,\dots ,m$.
On exit: ${\mathbf{xbar}}\left(\mathit{j}\right)$ contains the weighted mean of the $\mathit{j}$th variable, ${\stackrel{-}{x}}_{\mathit{j}}\left(\mathit{i}\right)$, for $\mathit{j}=1,2,\dots ,m$.
8:     $\mathbf{c}\left(\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{sw}}\ne 0.0$, c must contain the upper triangular part of the matrix of weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean. It is stored packed form by column, i.e., the cross-product between the $j$th and $k$th variable, $k\ge j$, is stored in ${\mathbf{c}}\left(k×\left(k-1\right)/2+j\right)$.
On exit: the update sums of squares and cross-products stored as on input.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{incx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{incx}}\ge 1$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{sw}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sw}}\ge 0.0$.
${\mathbf{ifail}}=3$
On entry, $\left({\mathbf{sw}}+{\mathbf{wt}}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\left({\mathbf{sw}}+{\mathbf{wt}}\right)\ge 0.0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{mean}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.
${\mathbf{ifail}}=-99$
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 method see Chan et al. (1982) and West (1979).

## 8Parallelism and Performance

g02btf is not threaded in any implementation.

g02btf may be used to update the results returned by g02buf.
g02bwf may be used to calculate the correlation matrix from the matrix of sums of squares and cross-products of deviations about the mean and the matrix may be scaled using f06edf (dscal) or f06fdf to produce a variance-covariance matrix.

## 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 (g02btfe.f90)

### 10.2Program Data

Program Data (g02btfe.d)

### 10.3Program Results

Program Results (g02btfe.r)