nag_correg_corrmat (g02bx) calculates the sample means, the standard deviations, the variance-covariance matrix, and the matrix of Pearson product-moment correlation coefficients for a set of data. Weights may be used.

Syntax

[xbar, std, v, r, ifail] = g02bx(x, 'nonzwt', nonzwt, 'n', n, 'm', m, 'wt', wt)

[xbar, std, v, r, ifail] = nag_correg_corrmat(x, 'nonzwt', nonzwt, 'n', n, 'm', m, 'wt', wt)

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

At Mark 23:	nonzwt was added to the interface; weight was removed from the interface; wt was made optional
At Mark 22:	n was made optional

Description

For

n

observations on

m

variables the one-pass algorithm of West (1979) as implemented in nag_correg_ssqmat (g02bu) is used to compute the means, the standard deviations, the variance-covariance matrix, and the Pearson product-moment correlation matrix for

p

selected variables. Suitables weights may be used to indicate multiple observations and to remove missing values. The quantities are defined by:

(a) The means

{\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i} x_{i j}}{\sum_{i = 1}^{n} w_{i}} j = 1, \dots, p

(b) The variance-covariance matrix

C_{j k} = \frac{\sum_{i = 1}^{n} w_{i} (x_{i j} - {\bar{x}}_{j}) (x_{i k} - {\bar{x}}_{k})}{\sum_{i = 1}^{n} w_{i} - 1} j, k = 1, \dots, p

s_{j} = \sqrt{C_{j j}} j = 1, \dots, p

(d) The Pearson product-moment correlation coefficients

R_{j k} = \frac{C_{j k}}{\sqrt{C_{j j} C_{k k}}} j, k = 1, \dots, p

where

x_{i j}

is the value of the

i

th observation on the

j

th variable and

w_{i}

is the weight for the

i

th observation which will be 1 in the unweighted case.

Note that the denominator for the variance-covariance is

\sum_{i = 1}^{n} w_{i} - 1

, so the weights should be scaled so that the sum of weights reflects the true sample size.

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

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 contain the $i$ th observation for the $j$ th variable, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$ .

Optional Input Parameters

1: $nonzwt$ – string (length ≥ 1): Default: $'W'$
The variance calculation uses a divisor which is either the number of weights or the number of nonzero weights.

Constraint: $nonzwt ='W'$ or $'V'$ .
2: $n$ – int64int32nag_int scalar: Default: the first dimension of the array x.
The number of data observations in the sample.

Constraint: $n > 1$ .
3: $m$ – int64int32nag_int scalar: Default: the second dimension of the array x.
The number of variables.

Constraint: $m \geq 1$ .
4: $wt (n)$ – double array: The dimension of the array wt must be at least $n$ if $weight ='W'$ or $'V'$ , and at least $1$ otherwise

$w$ , the optional frequency weighting for each observation, with $wt (i) = w_{i}$ . Usually $w_{i}$ will be an integral value corresponding to the number of observations associated with the $i$ th data value, or zero if the $i$ th data value is to be ignored. If $weight ='U'$ , $w_{i}$ is set to $1$ for all $i$ and wt is not referenced.

Constraint: if $weight ='W'$ or $'V'$ , $\sum_{i = 1}^{n} wt (i) > 1.0$ , $wt (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ .

Output Parameters

1: $xbar (m)$ – double array: The sample means. $xbar (j)$ contains the mean of the $j$ th variable.
2: $std (m)$ – double array: The standard deviations. $std (j)$ contains the standard deviation for the $j$ th variable.
3: $v (ldv, m)$ – double array: The variance-covariance matrix. $v (j, k)$ contains the covariance between variables $j$ and $k$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
4: $r (ldv, m)$ – double array: The matrix of Pearson product-moment correlation coefficients. $r (j, k)$ contains the correlation coefficient between variables $j$ and $k$ .
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_correg_corrmat (g02bx) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$m < 1$ ,
or	$n \leq 1$ ,
or	$ldx < n$ ,
or	$ldv < m$ .

$ifail = 2$

On entry,

weight \neq'U'

'V'

'W'

$ifail = 3$

On entry,

weight ='W'

'V'

and a value of

wt < 0.0

$ifail = 4$: $weight ='W'$ and the sum of weights is not greater than $1.0$ , or $weight ='V'$ and fewer than $2$ observations have nonzero weights.

W $ifail = 5$: A variable has a zero variance. In this case v and std are returned as calculated but r will contain zero for any correlation involving a variable with zero variance.

$ifail = - 99$: 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

For a discussion of the accuracy of the one pass algorithm see Chan et al. (1982) and West (1979).

Further Comments

None.

Example

The data are some of the results from 1988 Olympic Decathlon. They are the times (in seconds) for the 100m and 400m races and the distances (in metres) for the long jump, high jump and shot. Twenty observations are input and the correlation matrix is computed and printed.

Open in the MATLAB editor: g02bx_example

function g02bx_example


fprintf('g02bx example results\n\n');

x = [11.25  48.9  7.43  2.270  15.48;
     10.87  47.7  7.45  1.971  14.97;
     11.18  48.2  7.44  1.979  14.20;
     10.62  49.0  7.38  2.026  15.02;
     11.02  47.4  7.43  1.974  12.92;
     10.83  48.3  7.72  2.124  13.58;
     11.18  49.3  7.05  2.064  14.12;
     11.05  48.2  6.95  2.001  15.34;
     11.15  49.1  7.12  2.035  14.52;
     11.23  48.6  7.28  1.970  15.25;
     10.94  49.9  7.45  1.974  15.34;
     11.18  49.0  7.34  1.942  14.48;
     11.02  48.2  7.29  2.063  12.92;
     10.99  47.8  7.37  1.973  13.61;
     11.03  48.9  7.45  1.974  14.20;
     11.09  48.8  7.08  2.039  14.51;
     11.46  51.2  6.75  2.008  16.07;
     11.57  49.8  7.00  1.944  16.60;
     11.07  47.9  7.04  1.947  13.41;
     10.89  49.6  7.07  1.798  15.84];

[xbar, std, v, r, ifail] = g02bx(x);

disp('   Means');
disp(xbar');
disp('   Standard deviations');
disp(std');
mtitle = '  Correlation matrix:';
matrix = 'Upper';
diag   = 'Non-unit';

[ifail] = x04ca( ...
                 matrix, diag, r, mtitle);

g02bx example results

   Means
   11.0810   48.7900    7.2545    2.0038   14.6190

   Standard deviations
    0.2132    0.9002    0.2349    0.0902    1.0249

   Correlation matrix:
          1       2       3       4       5
 1   1.0000  0.4416 -0.5427  0.0696  0.3912
 2           1.0000 -0.5058 -0.0678  0.7057
 3                   1.0000  0.2768 -0.4352
 4                           1.0000 -0.1494
 5                                   1.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox