nag_correg_coeffs_zero (g02bd) computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data.

Syntax

[xbar, std, sspz, rz, ifail] = g02bd(x, 'n', n, 'm', m)

[xbar, std, sspz, rz, ifail] = nag_correg_coeffs_zero(x, 'n', n, 'm', m)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 22:

n was made optional

Description

The input data consists of

n

observations for each of

m

variables, given as an array

[x_{i j}], i = 1, 2, \dots, n (n \geq 2), j = 1, 2, \dots, m (m \geq 2),

where

x_{i j}

is the

i

th observation on the

j

th variable.

The quantities calculated are:

(a)

Means:

{\bar{x}}_{j} = \frac{1}{n} \sum_{i = 1}^{n} x_{i j}, j = 1, 2, \dots, m .

(b)

Standard deviations:

s_{j} = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i j} - {\bar{x}}_{j})}^{2}}, j = 1, 2, \dots, m .

(c)

Sums of squares and cross-products about zero:

{\tilde{S}}_{j k} = \sum_{i = 1}^{n} x_{i j} x_{i k}, j, k = 1, 2, \dots, m .

(d)

Correlation-like coefficients:

{\tilde{R}}_{j k} = \frac{{\tilde{S}}_{j k}}{\sqrt{{\tilde{S}}_{j j} {\tilde{S}}_{k k}}}, j, k = 1, 2, \dots, m .

{\tilde{S}}_{j j}

{\tilde{S}}_{k k}

is zero,

{\tilde{R}}_{j k}

is set to zero.

References

None.

Parameters

Compulsory Input Parameters

1: $x (ldx, m)$ – double array: ldx, the first dimension of the array, must satisfy the constraint $ldx \geq n$ .
$x (i, j)$ must be set to the value of $x_{i j}$ , the $i$ th observation on the $j$ th variable, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array x.
$n$ , the number of observations or cases.

Constraint: $n \geq 2$ .
2: $m$ – int64int32nag_int scalar: Default: the second dimension of the array x.
$m$ , the number of variables.

Constraint: $m \geq 2$ .

Output Parameters

1: $xbar (m)$ – double array: $xbar (j)$ contains the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
2: $std (m)$ – double array: The standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
3: $sspz (ldsspz, m)$ – double array: $sspz (j, k)$ is the cross-product about zero, ${\tilde{S}}_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
4: $rz (ldrz, m)$ – double array: $rz (j, k)$ is the correlation-like coefficient, ${\tilde{R}}_{j k}$ , between the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

n < 2

$ifail = 2$

On entry,

m < 2

$ifail = 3$

On entry,	$ldx < n$ ,
or	$ldsspz < m$ ,
or	$ldrz < m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_correg_coeffs_zero (g02bd) does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large

n

Further Comments

The time taken by nag_correg_coeffs_zero (g02bd) depends on

n

and

m

The function uses a two-pass algorithm.

Example

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

Open in the MATLAB editor: g02bd_example

function g02bd_example


fprintf('g02bd example results\n\n');

x = [ 2,  3, 3;
      4,  6, 4;
      9,  9, 0;
      0, 12, 2;
     12, -1, 5];
[n,m] = size(x);
fprintf('Number of variables (columns) = %d\n', m);
fprintf('Number of cases     (rows)    = %d\n\n', n);
disp('Data matrix is:-');
disp(x);

[xbar, std, sspz, rz, ifail] = ...
  g02bd(x);

fprintf('Variable   Mean     St. dev.\n');
fprintf('%5d%11.4f%11.4f\n',[[1:m]' xbar std]');
fprintf('\nSums of squares and cross-products about zero\n');
disp(sspz)
fprintf('Correlation-like coefficients\n');
disp(rz);

g02bd example results

Number of variables (columns) = 3
Number of cases     (rows)    = 5

Data matrix is:-
     2     3     3
     4     6     4
     9     9     0
     0    12     2
    12    -1     5

Variable   Mean     St. dev.
    1     5.4000     4.9800
    2     5.8000     5.0695
    3     2.8000     1.9235

Sums of squares and cross-products about zero
   245    99    82
    99   271    52
    82    52    54

Correlation-like coefficients
    1.0000    0.3842    0.7129
    0.3842    1.0000    0.4299
    0.7129    0.4299    1.0000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox