Integer, Intent (In)	::	m, n, l, ldx, isv(l), ldy
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p, x(ldx,l), y(ldy,l)
Real (Kind=nag_wp), Intent (Inout)	::	sx(l), sy(*), d(m,n)
Character (1), Intent (In)	::	update, scal, stype

C Header Interface

#include <nag.h>

void

g03ebf_ (const char *update, const char *scal, const char *stype, const double *p, const Integer *m, const Integer *n, const Integer *l, const double x[], const Integer *ldx, const Integer isv[], const double y[], const Integer *ldy, double sx[], double sy[], double d[], Integer *ifail, const Charlen length_update, const Charlen length_scal, const Charlen length_stype)

The routine may be called by the names g03ebf or nagf_mv_distance_mat_2.

3 Description

Given two sets of observations on

l

variables, a distance matrix is such that the

i j

th element represents how far apart or how dissimilar the

i

th observation from the first set and the

j

th observation from the second set are.

Let

X

and

Y

m \times l

and

n \times l

data matrices of

m

and

n

observations, respectively, on

l

variables. The distance between observation

i

from

X

and observation

j

from

Y

d_{i j}

, is most commonly defined in terms of the scaled Minkowski

p

-norm:

d_{i j} = {\sum_{k = 1}^{p} {(| x_{i k} / {sx}_{i} - y_{j k} / {sy}_{j} |)}^{p}}^{1 / p},

where

x_{i k}

and

y_{j k}

are the

i k

th and

j k

th elements of

X

and

Y

respectively,

{sx}_{i}

is a standardization for the

i

th variable in

X

{sy}_{j}

is a standardization for the

j

th variable in

Y

, and

p

is the order of the Minkowski norm.

Three standardizations (scalings) for the variables are available.

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

In addition to the Minkowski measure 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 g03ebf 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.

g03ebf returns the full rectangular 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: $scal$ – 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 sx (and posibly sy).
$scal ='U'$: Unscaled.

Constraint:

scal ='S'

'R'

'G'

'U'

3: $stype$ – Character(1) Input

On entry: indicates how the standardization of the variables treats the two sets of observations.

$stype ='A'$: Amalgamated.
$stype ='I'$: Independent.
$stype ='X'$: Standardization is based purley on observations in x.

Constraint:

stype ='A'

'I'

'X'

4: $p$ – Real (Kind=nag_wp) Input

On entry: the order

p

of the Minkowski distance metric.

Constraint:

p \geq 1.0

5: $m$ – Integer Input

On entry:

m

, the number of observations in the data array x.

Constraint:

m \geq 1

6: $n$ – Integer Input

On entry:

n

, the number of observations in the data array y.

Constraint:

n \geq 1

7: $l$ – Integer Input

On entry: l, the total number of variables in arrays x and y.

Constraint:

l > 0

8: $x (ldx, l)$ – Real (Kind=nag_wp) array Input

On entry:

x (i, k)

must contain the value of the

k

th variable for the

i

th observation in the first set of observations, for

i = 1, 2, \dots, m

and

k = 1, 2, \dots, l

9: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which g03ebf is called.

Constraint:

ldx \geq m

10: $isv (l)$ – Integer array Input

On entry:

isv (j)

indicates whether or not the

j

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

isv (k) = 0

the

j

th variable is not included., for

k = 1, 2, \dots, l

isv (k) \geq 1

the

j

th variable is included, for

k = 1, 2, \dots, l

Constraint:

isv (j) > 0

for at least one

j

, for

k = 1, 2, \dots, l

11: $y (ldy, l)$ – Real (Kind=nag_wp) array Input

On entry:

y (j, k)

must contain the value of the

k

th variable for the

j

th observation in the second set of observations, for

j = 1, 2, \dots, n

and

k = 1, 2, \dots, l

12: $ldy$ – Integer Input

On entry: the first dimension of the array y as declared in the (sub)program from which g03ebf is called.

Constraint:

ldy \geq n

13: $sx (l)$ – Real (Kind=nag_wp) array Input/Output

On entry: if

scal ='G'

and

isv (k) > 0

then

sx (k)

must contain the scaling for variable

k

, for

k = 1, 2, \dots, l

Constraint: if

scal ='G'

and

isv (k) > 0

sx (k) > 0.0

, for

k = 1, 2, \dots, l

On exit: if

scal ='S'

and

isv (k) > 0

then

sx (k)

contains the standard deviation of the variable in the

k

th column of x.

scal ='R'

and

isv (k) > 0

sx (k)

contains the range of the variable in the

j

th column of x.

scal ='U'

and

isv (k) > 0

sx (k) = 1.0

scal ='G'

, sx is unchanged.

14: $sy (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension of the array sy must be at least

l

stype ='I'

and

scal \neq'U'

, and at least

0

otherwise.

On entry: if

scal ='G'

and

stype ='I'

and

isv (k) > 0

then

sy (k)

must contain the scaling for variable

k

, for

k = 1, 2, \dots, l

stype \neq'I'

, or

scal ='U'

then sy is not referenced and need not be set.

Constraint: if

scal ='G'

and

stype ='I'

and

isv (k) > 0

sy (k) > 0.0

, for

k = 1, 2, \dots, l

On exit: if

scal ='S'

and

stype ='I'

and

isv (k) > 0

then

sy (k)

contains the standard deviation of the variable in the

k

th column of x.

scal ='R'

and

stype ='I'

and

isv (k) > 0

sy (k)

contains the range of the variable in the

j

th column of x.

scal ='U'

, sy is unchanged.

scal ='G'

, sy is unchanged.

15: $d (m, n)$ – Real (Kind=nag_wp) array Input/Output

On entry: the

m \times n

distance matrix

D

update ='U'

, d need not be set.

On exit: the (possibly updated) distance matrix

D

16: $ifail$ – 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, $update = ⟨ value ⟩$ .
Constraint: $update ='U'$ or $'I'$ .

$ifail = 2$: On entry, $scal = ⟨ value ⟩$ .
Constraint: $scal ='S'$ , $'R'$ , $'G'$ or $'U'$ .

$ifail = 3$: On entry, $stype = ⟨ value ⟩$ .
Constraint: $stype ='A'$ , $'I'$ or $'X'$ .

$ifail = 4$: On entry, $p = ⟨ value ⟩$ .
Constraint: $p \geq 1.0$ .

$ifail = 5$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

$ifail = 6$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 7$: On entry, $l = ⟨ value ⟩$ .
Constraint: $l \geq 1$ .

$ifail = 8$: Variable $⟨ value ⟩$ is constant.

$ifail = 9$: On entry, $ldx = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldx \geq m$ .

$ifail = 10$: On entry, isv does not contain a positive element.

$ifail = 12$: On entry, $ldy = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldy \geq n$ .

$ifail = 13$: On entry, at least one element of $sx \leq 0.0$ .

On entry, at least one element of $sx \leq 0.0$ or $sy \leq 0.0$ .

$ifail = 15$: On entry, at least one element of $d < 0.0$ .

$ifail = - 99$: 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.

g03ebf 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

When there is a large number of observations and these do not require scaling factors to be computed internally, then the distance matrix for all observations can be computed in stages by using g03ebf to systematically calculate the distance matrix for pairs of row blocks of observations. When there are a large number of variables, the distance matrix can be updated in stages using observations on blocks of variables at each update.

10 Example

Two data matrices of five and three observations on three variables is read in and a distance matrix is calculated from variables

2

and

3

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

g03eb: FL CL CPP AD PY MB

NAG FL Interfaceg03ebf (distance_​mat_​2)

▸▿ 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
g03ebf (distance_mat_2)