Integer, Intent (In)	::	n, m, ldx, isx(m)
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(ldx,m)
Real (Kind=nag_wp), Intent (Inout)	::	s(m), d(n*(n-1)/2)
Character (1), Intent (In)	::	update, dist, scal

C Header Interface

#include <nag.h>

void	g03eaf_ (const char update, const char dist, const char scal, const Integer n, const Integer m, const double x[], const Integer ldx, const Integer isx[], double s[], double d[], Integer *ifail, const Charlen length_update, const Charlen length_dist, const Charlen length_scal)

The routine may be called by the names g03eaf or nagf_mv_distance_mat.

3 Description

Given

n

objects, a distance or dissimilarity matrix is a symmetric matrix with zero diagonal elements such that the

i j

th element represents how far apart or how dissimilar the

i

th and

j

th objects are.

Let

X

be an

n \times p

data matrix of observations of

p

variables on

n

objects, then the distance between object

j

and object

k

d_{j k}

, can be defined as:

d_{j k} = {\sum_{i = 1}^{p} D (x_{j i} / s_{i}, x_{k i} / s_{i})}^{α},

where

x_{j i}

and

x_{k i}

are the

j i

th and

k i

th elements of

X

s_{i}

is a standardization for the

i

th variable and

D (u, v)

is a suitable function. Three functions are provided in g03eaf.

(a)Euclidean distance: $D (u, v) = {(u - v)}^{2}$ and $α = \frac{1}{2}$ .
(b)Euclidean squared distance: $D (u, v) = {(u - v)}^{2}$ and $α = 1$ .
(c)Absolute distance (city block metric): $D (u, v) = | u - v |$ and $α = 1$ .

Three standardizations are available.

(a)Standard deviation: $s_{i} = \sqrt{\sum_{j = 1}^{n} {(x_{j i} - \bar{x})}^{2} / (n - 1)}$
(b)Range: $s_{i} = \max (x_{1 i}, x_{2 i}, \dots, x_{n i}) - \min (x_{1 i}, x_{2 i}, \dots, x_{n i})$
(c)User-supplied values of $s_{i}$ .

In addition to the above distances there are a large number of other dissimilarity measures, particularly for dichotomous variables (see Krzanowski (1990) and Everitt (1974)). For the dichotomous case these measures are simple to compute and can, if suitable scaling is used, be combined with the distances computed by g03eaf using the updating option.

Dissimilarity measures for variables can be based on the correlation coefficient for continuous variables and contingency table statistics for dichotomous data, see Chapters G02 and G11 respectively.

g03eaf returns the strictly lower triangle of the distance matrix.

4 References

Everitt B S (1974) Cluster Analysis Heinemann

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

5 Arguments

1: $update$ – Character(1) Input

On entry: indicates whether or not an existing matrix is to be updated.

$update ='U'$: The matrix $D$ is updated and distances are added to $D$ .
$update ='I'$: The matrix $D$ is initialized to zero before the distances are added to $D$ .

Constraint:

update ='U'

'I'

2: $dist$ – Character(1) Input

On entry: indicates which type of distances are computed.

$dist ='A'$: Absolute distances.
$dist ='E'$: Euclidean distances.
$dist ='S'$: Euclidean squared distances.

Constraint:

dist ='A'

'E'

'S'

3: $scal$ – Character(1) Input

On entry: indicates the standardization of the variables to be used.

$scal ='S'$: Standard deviation.
$scal ='R'$: Range.
$scal ='G'$: Standardizations given in array s.
$scal ='U'$: Unscaled.

Constraint:

scal ='S'

'R'

'G'

'U'

4: $n$ – Integer Input

On entry:

n

, the number of observations.

Constraint:

n \geq 2

5: $m$ – Integer Input

On entry: the total number of variables in array x.

Constraint:

m > 0

6: $x (ldx, m)$ – Real (Kind=nag_wp) array Input

On entry:

x (i, j)

must contain the value of the

j

th variable for the

i

th object, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

7: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which g03eaf is called.

Constraint:

ldx \geq n

8: $isx (m)$ – Integer array Input

On entry:

isx (j)

indicates whether or not the

j

th variable in x is to be included in the distance computations.

isx (j) > 0

the

j

th variable is included, for

j = 1, 2, \dots, m

; otherwise it is not referenced.

Constraint:

isx (j) > 0

for at least one

j

, for

j = 1, 2, \dots, m

9: $s (m)$ – Real (Kind=nag_wp) array Input/Output

On entry: if

scal ='G'

and

isx (j) > 0

then

s (j)

must contain the scaling for variable

j

, for

j = 1, 2, \dots, m

Constraint: if

scal ='G'

and

isx (j) > 0

s (j) > 0.0

, for

j = 1, 2, \dots, m

On exit: if

scal ='S'

and

isx (j) > 0

then

s (j)

contains the standard deviation of the variable in the

j

th column of x.

scal ='R'

and

isx (j) > 0

s (j)

contains the range of the variable in the

j

th column of x.

scal ='U'

and

isx (j) > 0

s (j) = 1.0

scal ='G'

, s is unchanged.

10: $d (n \times (n - 1) / 2)$ – Real (Kind=nag_wp) array Input/Output

On entry: if

update ='U'

, d must contain the strictly lower triangle of the distance matrix

D

to be updated.

D

must be stored packed by rows, i.e.,

d ((i - 1) (i - 2) / 2 + j)

i > j

must contain

d_{i j}

update ='I'

, d need not be set.

Constraint: if

update ='U'

d (j) \geq 0.0

, for

j = 1, 2, \dots, n (n - 1) / 2

On exit: the strictly lower triangle of the distance matrix

D

stored packed by rows, i.e.,

d_{i j}

is contained in

d ((i - 1) (i - 2) / 2 + j)

i > j

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

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, $dist = ⟨ value ⟩$ .
Constraint: $dist ='A'$ , $'E'$ or $'S'$

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

On entry, $m = ⟨ value ⟩$ .
Constraint: $m > 0$ .

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

On entry, $scal = ⟨ value ⟩$ .
Constraint: $scal ='S'$ , $'R'$ , $'G'$ or $'U'$ .

On entry, $update = ⟨ value ⟩$ .
Constraint: $update ='U'$ or $'I'$ .

$ifail = 2$: On entry, at least one element of $d < 0.0$ .

On entry, at least one element of $s \leq 0.0$ .

On entry, isx does not contain a positive element.

Variable $⟨ value ⟩$ is constant.

$ifail = - 99$: 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

The computations are believed to be stable.

8 Parallelism and Performance

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

g03eaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g03eaf 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

g03ecf can be used to perform cluster analysis on the computed distance matrix.

10 Example

A data matrix of five observations and three variables is read in and a distance matrix is calculated from variables

2

and

3

using squared Euclidean distance with no scaling. This matrix is then printed.

g03ea: FL CL CPP AD PY MB

NAG FL Interfaceg03eaf (distance_​mat)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

NAG FL Interface
g03eaf (distance_mat)