NAG Library Routine Document

G02BDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

SUBROUTINE G02BDF (

N, M, X, LDX, XBAR, STD, SSPZ, LDSSPZ, RZ, LDRZ, IFAIL)

INTEGER	N, M, LDX, LDSSPZ, LDRZ, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), XBAR(M), STD(M), SSPZ(LDSSPZ,M), RZ(LDRZ,M)

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 .

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

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

is zero,

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

is set to zero.

4 References

None.

5 Parameters

1: $N$ – INTEGERInput: On entry: $n$ , the number of observations or cases.

Constraint: $N \geq 2$ .
2: $M$ – INTEGERInput: On entry: $m$ , the number of variables.

Constraint: $M \geq 2$ .
3: $X (LDX, M)$ – REAL (KIND=nag_wp) arrayInput: 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: $LDX$ – INTEGERInput: 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) arrayOutput: 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) arrayOutput: 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) arrayOutput: 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: $LDSSPZ$ – INTEGERInput: 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) arrayOutput: 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: $LDRZ$ – INTEGERInput: 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: $IFAIL$ – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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 < 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.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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

G02BDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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.

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.

NAG Library Routine DocumentG02BDF