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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_zero_subset (g02bk)

## Purpose

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

## Syntax

[xbar, std, sspz, rz, ifail] = g02bk(x, kvar, 'n', n, 'm', m, 'nvars', nvars)
[xbar, std, sspz, rz, ifail] = nag_correg_coeffs_zero_subset(x, kvar, 'n', n, 'm', m, 'nvars', nvars)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22: n has been made optional
.

## Description

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

None.

## Parameters

### Compulsory Input Parameters

1:     x(ldx,m) – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
x(i,j)${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to xij${x}_{\mathit{i}\mathit{j}}$, the value of the i$\mathit{i}$th observation on the j$\mathit{j}$th variable, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ and j = 1,2,,m$\mathit{j}=1,2,\dots ,m$.
2:     kvar(nvars) – int64int32nag_int array
nvars, the dimension of the array, must satisfy the constraint 2nvarsm$2\le {\mathbf{nvars}}\le {\mathbf{m}}$.
kvar(j)${\mathbf{kvar}}\left(\mathit{j}\right)$ must be set to the column number in x of the j$\mathit{j}$th variable for which information is required, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$.
Constraint: 1kvar(j)m$1\le {\mathbf{kvar}}\left(\mathit{j}\right)\le {\mathbf{m}}$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array x.
n$n$, the number of observations or cases.
Constraint: n2${\mathbf{n}}\ge 2$.
2:     m – int64int32nag_int scalar
Default: The second dimension of the array x.
m$m$, the number of variables.
Constraint: m2${\mathbf{m}}\ge 2$.
3:     nvars – int64int32nag_int scalar
Default: The dimension of the array kvar.
p$p$, the number of variables for which information is required.
Constraint: 2nvarsm$2\le {\mathbf{nvars}}\le {\mathbf{m}}$.

ldx ldsspz ldrz

### Output Parameters

1:     xbar(nvars) – double array
The mean value, xj${\stackrel{-}{x}}_{\mathit{j}}$, of the variable specified in kvar(j)${\mathbf{kvar}}\left(\mathit{j}\right)$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$.
2:     std(nvars) – double array
The standard deviation, sj${s}_{\mathit{j}}$, of the variable specified in kvar(j)${\mathbf{kvar}}\left(\mathit{j}\right)$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$.
3:     sspz(ldsspz,nvars) – double array
ldsspznvars$\mathit{ldsspz}\ge {\mathbf{nvars}}$.
sspz(j,k)${\mathbf{sspz}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product about zero, jk${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for the variables specified in kvar(j)${\mathbf{kvar}}\left(\mathit{j}\right)$ and kvar(k)${\mathbf{kvar}}\left(\mathit{k}\right)$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$ and k = 1,2,,p$\mathit{k}=1,2,\dots ,p$.
4:     rz(ldrz,nvars) – double array
ldrznvars$\mathit{ldrz}\ge {\mathbf{nvars}}$.
rz(j,k)${\mathbf{rz}}\left(\mathit{j},\mathit{k}\right)$ is the correlation-like coefficient, jk${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the variables specified in kvar(j)${\mathbf{kvar}}\left(\mathit{j}\right)$ and kvar(k)${\mathbf{kvar}}\left(\mathit{k}\right)$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$ and k = 1,2,,p$\mathit{k}=1,2,\dots ,p$.
5:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 2${\mathbf{n}}<2$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, nvars < 2${\mathbf{nvars}}<2$, or ${\mathbf{nvars}}>{\mathbf{m}}$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, ldx < n$\mathit{ldx}<{\mathbf{n}}$, or ldsspz < nvars$\mathit{ldsspz}<{\mathbf{nvars}}$, or ldrz < nvars$\mathit{ldrz}<{\mathbf{nvars}}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, kvar(j) < 1${\mathbf{kvar}}\left(j\right)<1$, or kvar(j) > m${\mathbf{kvar}}\left(j\right)>{\mathbf{m}}$ for some j = 1,2, … ,nvars$j=1,2,\dots ,{\mathbf{nvars}}$.

## Accuracy

nag_correg_coeffs_zero_subset (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$n$.

The time taken by nag_correg_coeffs_zero_subset (g02bk) depends on n$n$ and p$p$.
The function uses a two-pass algorithm.

## Example

```function nag_correg_coeffs_zero_subset_example
x = [3, 3, 1, 2;
6, 4, -1, 4;
9, 0, 5, 9;
12, 2, 0, 0;
-1, 5, 4, 12];
kvar = [int64(4);1;2];
[xbar, std, sspz, rz, ifail] = nag_correg_coeffs_zero_subset(x, kvar)
```
```

xbar =

5.4000
5.8000
2.8000

std =

4.9800
5.0695
1.9235

sspz =

245    99    82
99   271    52
82    52    54

rz =

1.0000    0.3842    0.7129
0.3842    1.0000    0.4299
0.7129    0.4299    1.0000

ifail =

0

```
```function g02bk_example
x = [3, 3, 1, 2;
6, 4, -1, 4;
9, 0, 5, 9;
12, 2, 0, 0;
-1, 5, 4, 12];
kvar = [int64(4);1;2];
[xbar, std, sspz, rz, ifail] = g02bk(x, kvar)
```
```

xbar =

5.4000
5.8000
2.8000

std =

4.9800
5.0695
1.9235

sspz =

245    99    82
99   271    52
82    52    54

rz =

1.0000    0.3842    0.7129
0.3842    1.0000    0.4299
0.7129    0.4299    1.0000

ifail =

0

```