g02bdf computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data.

2 Specification

Fortran Interface

Subroutine g02bdf (

n, m, x, ldx, xbar, std, sspz, ldsspz, rz, ldrz, ifail)

Integer, Intent (In)	::	n, m, ldx, ldsspz, ldrz
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(ldx,m)
Real (Kind=nag_wp), Intent (Inout)	::	sspz(ldsspz,m), rz(ldrz,m)
Real (Kind=nag_wp), Intent (Out)	::	xbar(m), std(m)

C Header Interface

#include <nag.h>

void	g02bdf_ (const Integer n, const Integer m, const double x[], const Integer ldx, double xbar[], double std[], double sspz[], const Integer ldsspz, double rz[], const Integer ldrz, Integer ifail)

The routine may be called by the names g02bdf or nagf_correg_coeffs_zero.

3 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:

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

If ${\tilde{S}}_{j j}$ or ${\tilde{S}}_{k k}$ is zero, ${\tilde{R}}_{j k}$ is set to zero.

4 References

None.

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of observations or cases.

Constraint: $n \geq 2$ .
2: $m$ – Integer Input: On entry: $m$ , the number of variables.

Constraint: $m \geq 2$ .
3: $x (ldx, m)$ – Real (Kind=nag_wp) array Input: On entry: $x (i, j)$ 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 $j = 1, 2, \dots, m$ .
4: $ldx$ – Integer Input: On entry: the first dimension of the array x as declared in the (sub)program from which g02bdf is called.

Constraint: $ldx \geq n$ .
5: $xbar (m)$ – Real (Kind=nag_wp) array Output: On exit: $xbar (j)$ contains the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
6: $std (m)$ – Real (Kind=nag_wp) array Output: On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
7: $sspz (ldsspz, m)$ – Real (Kind=nag_wp) array Output: On exit: $sspz (j, k)$ is the cross-product about zero, ${\tilde{S}}_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
8: $ldsspz$ – Integer Input: On entry: the first dimension of the array sspz as declared in the (sub)program from which g02bdf is called.

Constraint: $ldsspz \geq m$ .
9: $rz (ldrz, m)$ – Real (Kind=nag_wp) array Output: On exit: $rz (j, k)$ is the correlation-like coefficient, ${\tilde{R}}_{j k}$ , between the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
10: $ldrz$ – Integer Input: On entry: the first dimension of the array rz as declared in the (sub)program from which g02bdf is called.

Constraint: $ldrz \geq m$ .
11: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 2$ .

$ifail = 3$: On entry, $ldrz = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldrz \geq m$ .

On entry, $ldsspz = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldsspz \geq m$ .

On entry, $ldx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldx \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

g02bdf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large

n

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02bdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by g02bdf depends on

n

and

m

The routine uses a two-pass algorithm.

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 27: The algorithm underlying this routine was altered to improve efficiency for large problem sizes on a multi-threaded system.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 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.

g02bd: FL CL CPP AD PY MB

NAG FL Interfaceg02bdf (coeffs_​zero)

▸▿ Contents

1 Purpose