﻿ g02bk Method
g02bk computes means and standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for selected variables.

# Syntax

C#
```public static void g02bk(
int n,
int m,
double[,] x,
int nvars,
int[] kvar,
double[] xbar,
double[] std,
double[,] sspz,
double[,] rz,
out int ifail
)```
Visual Basic
```Public Shared Sub g02bk ( _
n As Integer, _
m As Integer, _
x As Double(,), _
nvars As Integer, _
kvar As Integer(), _
xbar As Double(), _
std As Double(), _
sspz As Double(,), _
rz As Double(,), _
<OutAttribute> ByRef ifail As Integer _
)```
Visual C++
```public:
static void g02bk(
int n,
int m,
array<double,2>^ x,
int nvars,
array<int>^ kvar,
array<double>^ xbar,
array<double>^ std,
array<double,2>^ sspz,
array<double,2>^ rz,
[OutAttribute] int% ifail
)```
F#
```static member g02bk :
n : int *
m : int *
x : float[,] *
nvars : int *
kvar : int[] *
xbar : float[] *
std : float[] *
sspz : float[,] *
rz : float[,] *
ifail : int byref -> unit
```

#### Parameters

n
Type: System..::..Int32
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
m
Type: System..::..Int32
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
x
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge {\mathbf{n}}$
On entry: ${\mathbf{x}}\left[\mathit{i}-1,\mathit{j}-1\right]$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
nvars
Type: System..::..Int32
On entry: $p$, the number of variables for which information is required.
Constraint: $2\le {\mathbf{nvars}}\le {\mathbf{m}}$.
kvar
Type: array<System..::..Int32>[]()[][]
An array of size [nvars]
On entry: ${\mathbf{kvar}}\left[\mathit{j}\right]$ must be set to the column number in x of the $\mathit{j}$th variable for which information is required, for $\mathit{j}=0,1,\dots ,p-1$.
Constraint: $1\le {\mathbf{kvar}}\left[\mathit{j}\right]\le {\mathbf{m}}$, for $\mathit{j}=0,1,\dots ,p-1$.
xbar
Type: array<System..::..Double>[]()[][]
An array of size [nvars]
On exit: the mean value, ${\stackrel{-}{x}}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,p$.
std
Type: array<System..::..Double>[]()[][]
An array of size [nvars]
On exit: the standard deviation, ${s}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,p$.
sspz
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, nvars]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge {\mathbf{nvars}}$
On exit: ${\mathbf{sspz}}\left[\mathit{j}-1,\mathit{k}-1\right]$ is the cross-product about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for the variables specified in ${\mathbf{kvar}}\left[\mathit{j}-1\right]$ and ${\mathbf{kvar}}\left[\mathit{k}-1\right]$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
rz
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, nvars]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge {\mathbf{nvars}}$
On exit: ${\mathbf{rz}}\left[\mathit{j}-1,\mathit{k}-1\right]$ is the correlation-like coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the variables specified in ${\mathbf{kvar}}\left[\mathit{j}-1\right]$ and ${\mathbf{kvar}}\left[\mathit{k}-1\right]$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
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

The input data consists of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,nn≥2,j=1,2,…,mm≥2,$
where ${x}_{ij}$ is the $i$th observation on the $j$th variable, together with the subset of these variables, ${v}_{1},{v}_{2},\dots ,{v}_{p}$, for which information is required.
The quantities calculated are:
(a) Means:
 $x-j=∑i=1nxijn, j=v1,v2,…,vp.$
(b) Standard deviations:
 $sj=1n-1∑i=1nxij-x-j2, j=v1,v2,…,vp.$
(c) Sums of squares and cross-products about zero:
 $S~jk=∑i=1nxijxik, j,k=v1,v2,…,vp.$
(d) Correlation-like coefficients:
 $R~jk=S~jkS~jjS~kk, j,k=v1,v2,…,vp.$
If ${\stackrel{~}{S}}_{jj}$ or ${\stackrel{~}{S}}_{kk}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero.

None.

# Error Indicators and Warnings

Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDX, LDSSPZ, LDRZ) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{nvars}}<2$, or ${\mathbf{nvars}}>{\mathbf{m}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{kvar}}\left[j-1\right]<1$, or ${\mathbf{kvar}}\left[j-1\right]>{\mathbf{m}}$ for some $j=1,2,\dots ,{\mathbf{nvars}}$.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

g02bk does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.

# Parallelism and Performance

None.

The time taken by g02bk depends on $n$ and $p$.
The method uses a two-pass algorithm.

# Example

This example reads in a set of data consisting of five observations on each of four variables. The means, standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for the fourth, first and second variables are then calculated and printed.

Example program (C#): g02bke.cs

Example program data: g02bke.d

Example program results: g02bke.r