﻿ g02bt Method
g02bt 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.

Syntax

C#
```public static void g02bt(
string mean,
int m,
double wt,
double[] x,
int incx,
ref double sw,
double[] xbar,
double[] c,
out int ifail
)```
Visual Basic
```Public Shared Sub g02bt ( _
mean As String, _
m As Integer, _
wt As Double, _
x As Double(), _
incx As Integer, _
ByRef sw As Double, _
xbar As Double(), _
c As Double(), _
<OutAttribute> ByRef ifail As Integer _
)```
Visual C++
```public:
static void g02bt(
String^ mean,
int m,
double wt,
array<double>^ x,
int incx,
double% sw,
array<double>^ xbar,
array<double>^ c,
[OutAttribute] int% ifail
)```
F#
```static member g02bt :
mean : string *
m : int *
wt : float *
x : float[] *
incx : int *
sw : float byref *
xbar : float[] *
c : float[] *
ifail : int byref -> unit
```

Parameters

mean
Type: System..::..String
On entry: indicates whether g02bt 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 $\text{"Z"}$.
m
Type: System..::..Int32
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
wt
Type: System..::..Double
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.
x
Type: array<System..::..Double>[]()[][]
An array of size [${\mathbf{m}}×{\mathbf{incx}}$]
On entry: ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{incx}}\right]$ must contain the value of the $j$th variable for the current observation, $j=1,2,\dots ,m$.
incx
Type: System..::..Int32
On entry: the increment of x.
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: ${\mathbf{incx}}>0$.
sw
Type: System..::..Double%
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, ${W}_{i}$.
xbar
Type: array<System..::..Double>[]()[][]
An array of size [m]
On entry: if ${\mathbf{sw}}=0.0$, xbar is initialized, otherwise ${\mathbf{xbar}}\left[\mathit{j}-1\right]$ 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: ${\mathbf{xbar}}\left[\mathit{j}-1\right]$ contains the weighted mean of the $\mathit{j}$th variable, ${\stackrel{-}{x}}_{\mathit{j}}\left(\mathit{i}\right)$, for $\mathit{j}=1,2,\dots ,m$.
c
Type: array<System..::..Double>[]()[][]
An array of size [$\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2$]
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 ${\mathbf{c}}\left[k×\left(k-1\right)/2+j-1\right]$.
On exit: the update sums of squares and cross-products stored as on input.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

g02bt is an adaptation of West's WV2 algorithm; see West (1979). This method 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 $i$.

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

Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}<1$, or ${\mathbf{incx}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{sw}}<0.0$.
${\mathbf{ifail}}=3$
 On entry, $\left({\mathbf{sw}}+{\mathbf{wt}}\right)<0.0$, the current weight causes the sum of weights to be less than $0.0$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{mean}}\ne \text{"M"}$ or $\text{"Z"}$.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

Accuracy

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

Parallelism and Performance

None.

g02bt may be used to update the results returned by g02bu.
g02bw 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 not in this release) f06fd to produce a variance-covariance matrix.

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.

Example program (C#): g02bte.cs

Example program data: g02bte.d

Example program results: g02bte.r