nag_correg_coeffs_pearson (g02ba) computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data.

Syntax

[xbar, std, ssp, r, ifail] = g02ba(x, 'n', n, 'm', m)

[xbar, std, ssp, r, ifail] = nag_correg_coeffs_pearson(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 consist 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 of deviations from means:

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

(d)

Pearson product-moment correlation coefficients:

R_{j k} = \frac{S_{j k}}{\sqrt{S_{j j} S_{k k}}}, j, k = 1, 2, \dots, m .

S_{j j}

S_{k k}

is zero,

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 $x_{i j}$ , 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: 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: $ssp (ldssp, m)$ – double array: $ssp (j, k)$ is the cross-product of deviations $S_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
4: $r (ldr, m)$ – double array: $r (j, k)$ is the product-moment correlation coefficient $R_{j k}$ 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	$ldssp < m$ ,
or	$ldr < m$ .

$ifail = - 99$: 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_pearson (g02ba) 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_pearson (g02ba) 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 of deviations from means, and Pearson product-moment correlation coefficients for all three variables are then calculated and printed.

Open in the MATLAB editor: g02ba_example

function g02ba_example


fprintf('g02ba 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, ssp, r, ifail] = g02ba( ...
                                    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 of deviations\n');
disp(ssp)
fprintf('Correlation coefficients\n');
disp(r);

g02ba 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 of deviations
   99.2000  -57.6000    6.4000
  -57.6000  102.8000  -29.2000
    6.4000  -29.2000   14.8000

Correlation coefficients
    1.0000   -0.5704    0.1670
   -0.5704    1.0000   -0.7486
    0.1670   -0.7486    1.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox