void	nag_mv_distance_mat (Nag_MatUpdate update, Nag_DistanceType dist, Nag_VarScaleType scale, Integer n, Integer m, const double x[], Integer tdx, const Integer isx[], double s[], double d[], NagError *fail)

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

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 nag_mv_distance_mat (g03eac):

(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:

1.	Standard deviation: $s_{i} = \sqrt{\sum_{j = 1}^{n} {(x_{j i} - \bar{x})}^{2} / (n - 1)}$
2.	Range: $s_{i} = \max (x_{1 i}, x_{2 i}, \dots, x_{n i}) - \min (x_{1 i}, x_{2 i}, \dots, x_{n i})$
3.	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 nag_mv_distance_mat (g03eac) 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 the g02 Chapter Introduction and the g11 Chapter Introduction respectively.

nag_mv_distance_mat (g03eac) 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: $update$ – Nag_MatUpdateInput

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

$update = Nag_MatUp$: The matrix $D$ is updated and distances are added to $D$ .
$update = Nag_NoMatUp$: The matrix $D$ is initialized to zero before the distances are added to $D$ .

Constraint:

update = Nag_MatUp

Nag_NoMatUp

2: $dist$ – Nag_DistanceTypeInput

On entry: indicates which type of distances are computed.

$dist = Nag_DistAbs$: Absolute distances.
$dist = Nag_DistEuclid$: Euclidean distances.
$dist = Nag_DistSquared$: Euclidean squared distances.

Constraint:

dist = Nag_DistAbs

Nag_DistEuclid

Nag_DistSquared

3: $scale$ – Nag_VarScaleTypeInput

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

$scale = Nag_VarScaleStd$: Standard deviation.
$scale = Nag_VarScaleRange$: Range.
$scale = Nag_VarScaleUser$: Standardizations given in array $S$ .
$scale = Nag_NoVarScale$: Unscaled.

Constraint:

scale = Nag_VarScaleStd

Nag_VarScaleRange

Nag_VarScaleUser

Nag_NoVarScale

4: $n$ – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n \geq 2

5: $m$ – IntegerInput

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

Constraint:

m > 0

6: $x [n \times tdx]$ – const doubleInput

On entry:

x [(i - 1) \times tdx + j - 1]

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: $tdx$ – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

8: $isx [m]$ – const IntegerInput

On entry:

isx [j - 1]

indicates whether or not the

j

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

isx [j - 1] > 0

the

j

th variable is included, for

j = 1, 2, \dots, m

; otherwise it is not referenced.

Constraint:

isx [j - 1] > 0

for at least one

j

, for

j = 1, 2, \dots, m

9: $s [m]$ – doubleInput/Output

On entry: if

scale = Nag_VarScaleUser

and

isx [j - 1] > 0

then

s [j - 1]

must contain the scaling for variable

j

, for

j = 1, 2, \dots, m

Constraint: if

scale = Nag_VarScaleUser

and

isx [j - 1] > 0

s [j - 1] > 0.0

, for

j = 1, 2, \dots, m

On exit: if

scale = Nag_VarScaleStd

and

isx [j - 1] > 0

then

s [j - 1]

contains the standard deviation of the variable in the

j

th column of x.

scale = Nag_VarScaleRange

and

isx [j - 1] > 0

then

s [j - 1]

contains the range of the variable in the

j

th column of x.

scale = Nag_NoVarScale

and

isx [j - 1] > 0

then

s [j - 1] = 1.0

and if

scale = Nag_VarScaleUser

then s is unchanged.

10: $d [n \times (n - 1) / 2]$ – doubleInput/Output

On entry: if

update = Nag_MatUp

then 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 - 1]

i > j

must contain

d_{i j}

Constraint: if

update = Nag_MatUp

d [j - 1] \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 - 1]

i > j

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdx = 〈value〉$ while $m = 〈value〉$ . These arguments must satisfy $tdx \geq m$ .
NE_BAD_PARAM: On entry, argument dist had an illegal value.

On entry, argument scale had an illegal value.

On entry, argument update had an illegal value.
NE_IDEN_ELEM_COND: On entry, $scale = Nag_VarScaleRange$ or $scale = Nag_VarScaleStd$ , and $x [(i - 1) \times tdx + j - 1] = x [(i) \times tdx + j - 1]$ , for $i = 1, 2, \dots, n - 1$ , for some $j$ with $isx [i - 1] > 0$ .
NE_INT_ARG_LE: On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
NE_INTARR: On entry, $isx [〈value〉] = 〈value〉$ .
Constraint: $isx [i - 1] > 0$ , for at least one $i, i = 1, 2, \dots, m$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REALARR: On entry, $d [〈value〉] = 〈value〉$ .
Constraint: $d [i - 1] \geq 0$ , for $i = 1, 2, \dots, n \times (n - 1) / 2$ , when $update = Nag_MatUp$ .

On entry, $s [〈value〉] = 〈value〉$ .
Constraint: $s [j - 1] > 0$ , for $j = 1, 2, \dots, m$ , when $scale = Nag_VarScaleUser$ and $isx [j - 1] > 0$ .

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

nag_mv_distance_mat (g03eac) is not threaded in any implementation.

9

Further Comments

nag_mv_hierar_cluster_analysis (g03ecc) 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.

NAG Library Function Document

nag_mv_distance_mat (g03eac)

▸▿ 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

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