g02bef computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data omitting completely any cases with a missing observation for any variable.

2 Specification

Fortran Interface

Subroutine g02bef (

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

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

C Header Interface

#include <nag.h>

void	g02bef_ (const Integer n, const Integer m, const double x[], const Integer ldx, Integer miss[], double xmiss[], double xbar[], double std[], double sspz[], const Integer ldsspz, double rz[], const Integer ldrz, Integer ncases, Integer *ifail)

The routine may be called by the names g02bef or nagf_correg_coeffs_zero_miss_case.

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. In addition, each of the

m

variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the

j

th variable is denoted by

{xm}_{j}

. Missing values need not be specified for all variables.

Let

w_{i} = 0

if observation

i

contains a missing value for any of those variables for which missing values have been declared, i.e., if

x_{i j} = {xm}_{j}

for any

j

for which an

{xm}_{j}

has been assigned (see also Section 7); and

w_{i} = 1

otherwise, for

i = 1, 2, \dots, n

The quantities calculated are:

(a)Means:

${\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i} x_{i j}}{\sum_{i = 1}^{n} w_{i}}, j = 1, 2, \dots, m .$
(b)Standard deviations:

$s_{j} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i} {(x_{i j} - {\bar{x}}_{j})}^{2}}{\sum_{i = 1}^{n} w_{i} - 1}}, j = 1, 2, \dots, m .$
(c)Sums of squares and cross-products about zero:

${\tilde{S}}_{j k} = \sum_{i = 1}^{n} w_{i} x_{i j} x_{i k}, j, k = 1, 2, \dots, m .$
(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 $x_{i j}$ , the value of 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 g02bef is called.

Constraint: $ldx \geq n$ .
5: $miss (m)$ – Integer array Input/Output: On entry: $miss (j)$ must be set equal to $1$ if a missing value, $x m_{j}$ , is to be specified for the $j$ th variable in the array x, or set equal to $0$ otherwise. Values of miss must be given for all $m$ variables in the array x.

On exit: the array miss is overwritten by the routine, and the information it contained on entry is lost.
6: $xmiss (m)$ – Real (Kind=nag_wp) array Input/Output: On entry: $xmiss (j)$ must be set to the missing value, $x m_{j}$ , to be associated with the $j$ th variable in the array x, for those variables for which missing values are specified by means of the array miss (see Section 7).

On exit: the array xmiss is overwritten by the routine, and the information it contained on entry is lost.
7: $xbar (m)$ – Real (Kind=nag_wp) array Output: On exit: the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
8: $std (m)$ – Real (Kind=nag_wp) array Output: On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
9: $sspz (ldsspz, m)$ – 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 $k = 1, 2, \dots, m$ .
10: $ldsspz$ – Integer Input: On entry: the first dimension of the array sspz as declared in the (sub)program from which g02bef is called.

Constraint: $ldsspz \geq m$ .
11: $rz (ldrz, m)$ – 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 $k = 1, 2, \dots, m$ .
12: $ldrz$ – Integer Input: On entry: the first dimension of the array rz as declared in the (sub)program from which g02bef is called.

Constraint: $ldrz \geq m$ .
13: $ncases$ – Integer Output: On exit: the number of cases actually used in the calculations (when cases involving missing values have been eliminated).
14: $ifail$ – 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: $ldx \geq n$ .

$ifail = 4$: After observations with missing values were omitted, no cases remained.

$ifail = 5$: After observations with missing values were omitted, only one case remained.

$ifail = - 99$: 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

g02bef does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large

n

You are warned of the need to exercise extreme care in your selection of missing values. g02bef treats all values in the inclusive range

(1 \pm {0.1}^{(x02bef - 2)}) \times {x m}_{j}

, where

{xm}_{j}

is the missing value for variable

j

specified in xmiss.

You must, therefore, ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.

8 Parallelism and Performance

g02bef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02bef 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 g02bef depends on

n

and

m

, and the occurrence of missing values.

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 has been 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. Missing values of

0.0

are declared for the first and third variables; no missing value is specified for the second variable. 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, omitting completely all cases containing missing values; cases

3

and

4

are, therefore, eliminated, leaving only three cases in the calculations.

g02be: FL CL CPP AD

NAG FL Interfaceg02bef (coeffs_​zero_​miss_​case)

▸▿ Contents

1 Purpose