APPLICATION OF NEURAL NETWORKS IN THE IDENTIFICATION OF MORPHOLOGICAL TYPES

APPLICATION OF NEURAL NETWORKS IN THE
IDENTIFICATION OF MORPHOLOGICAL TYPES


Franjo Prot and Ksenija Bosnar

University of Zagreb


Ankica Ho{ek and Konstantin Momirovi}

Institute of criminological and sociological research


A sample of 737 healthy males, 1
9 to 27 years
old, fairly representative for the Yugoslav
population of this age and gender, was
described over a set of 23 morphological
characteristic selected so to assess factors of
longitudinal and transversal dimensions of
skeleton, muscular mass and

fat tissue. An
algorithm for a neural network for cluster
analysis with coded name Triatlon was applied
in order to detect the morphological types. The
essence of the applied clustering algorithm is
a taxonomic neural network based on adaptive
multilayer
perceptron as a core engine working
on the basis of starting classification
obtained by a rational method of fuzzy
clustering of variables, and then of fuzzy
clustering of objects described on fuzzy
clusters of variables. Triatlon conclude that
five cluste
rs are necessary and sufficient for
the taxonomic description of this data set, and
that by only three hidden neurons can produce
an acceptable classification of objects. After
15 iteration Triatlon produce an excellent
fuzzy classification of variable, bu
t initial
fuzzy clustering of objects is obtained after
71 iteration. However, multilayer perceptron
consider this classification as good, but not
satisfactory, and start learning process in
order to obtain a better classification. The
final classification

is obtained after 24
learning attempts. However, coefficient of
efficacy of Triatlon in this case was only
0.920, markedly lower then in applications of
this program in other taxonomic problems. In
spite of complex position of types in the space
of manife
st morphological characteristics and
not always clear pattern and structure of
discriminant factors obtained types can be
identified as follows:

(1) Typus asthenicus, defined by low
development of skeleton, low muscular mass and
low fat tissue;

(2) Typus s
thenicus, defined by strong
development of skeleton, high amount of
muscular mass and above average fat tissue due
to the high amount of fat cells;

(3) Typus gracilis, defined primarily by small
measures of transversal dimensions of skeleton;

(4) Typus dis
harmonicus, defined by
inconvergent development of morphological
characteristics and low fat tissue;

(5) Typus leptomorphicus, defined by above
average development of longitudinal dimensions
of skeleton.


KEY WORDS

morphological types / neural networks / c
luster analysis




1. INTRODUCTION



In a previous paper (Momirovi}, Ho{ek, Prot and
Bosnar, 2002) a sample of 737 healthy males, 19 to 27
years old, was described, by a procedure which minimize
error of measurement, by 23 anthropometric variables
Morphol
ogical types were determined by neural network
SIMTAX. The algorithm implemented in this network
classify objects in the standardized image space by
iterative application of Lebart's multilayer perceptron.
Initial classification was obtained on the basis o
f
position of objects on the envelope of hyperelipsoid
defined by Orthoblique transformation of principal
components of data matrix, also transformed to
standardized image space. Dimensionality of latent, and
in the same time taxonomic space was determined

by number
of spectral values greater then inflection point of their
distribution. Three taxon were obtained, with
classification efficacy of 0.991 in image and 0.986 in
real space. First taxon, of 35% of examines, was
identified as sthenomorphia, second t
axon, of 29% of
examines, as asthenomorphia, and third taxon, of 36% of
examines, as picnomorphia. Obtained taxons were similar,
but not identical, with taxons K, M and R obtained by a
method of fuzzy clustering applied by A. Ho{ek (1978) on
a set of 200 e
xamines described by the same set of
anthropometric measurements, but not to the taxons
obtained by Zlobec (1975) by concurrent application of a
simple fuzzy clustering method and to taxons obtained by
Ward's method of hierarchical clustering and Friedman
and
Rubin method of local optimization.


The aim of this paper is to present results of an
alternative attempt to solve the old and at yet unsolved
problem of morphological types by an other taxonomic
neural network who analyze objects in real space on the

basis of results obtained by an initial fuzzy
classification similar to classification methods applied
in works of Zlobec (1975) and Ho{ek (1978).


2. METHODS



A sample of 737 healthy males, 19 to 27 years old,
fairly representative for the Yugoslav po
pulation of this
age and gender, was described over a set of 23
morphological characteristic, defined by the following
variables:



CODED NAME

VARIABLE

WEIGHT

Body mass

HEIGHT

Body height

LLENGTH

Leg length

BIACRO

Biacromial span

BICRIS

Bicristal span

TRISKIN

Triceps skinfold

SCAPSKIN

Subcapular
skinfold

AXSKIN

Axilar skinfold

CRUPARM

Upper arm
circumference

CRLWARM

Lower arm
circumference

CRUPLG

Upper leg
circumference

CRLWLG

Lower leg
circumference

HANDLG

Hand length

HANDDM

Hand diameter

AB
DSKIN

Abdominal
skinfold

LWLSKIN

Lower leg
skinfold

CHCIRC

Chest
circumference

DIWRIST

Diameter of
wrist

DIAEL

Diameter of
elbow

DIAKNE

Diameter of knee

FOOTL

Foot length

FOOTDM

Diameter of foot

ARMLG

Arm length




An algorithm for a neural networ
k for cluster
analysis with coded name Triatlon was applied in order to
detect the morphological types. The essence of the
applied clustering algorithm is a taxonomic neural
network based on adaptive multilayer perceptron as a core
engine working on the ba
sis of starting classification
obtained by a rational method of fuzzy clustering of
variables, and then of fuzzy clustering of objects
described on fuzzy clusters of variables.
1










3. RESULTS



Triatlon conclude that five clusters are necessary
and s
ufficient for the taxonomic description of this data
set, and that by only three hidden neurons can produce an
acceptable classification of objects. After 15 iteration
Triatlon produce an excellent fuzzy classification of
variable, but initial fuzzy cluste
ring of objects is
obtained after 71 iteration. However, multilayer
perceptron consider this classification as good, but not
satisfactory, and start learning process in order to
obtain a better classification. The final classification
is obtained after 24
learning attempts. The whole process
is presented, in an abbreviated form, in the following
tables.


Table 1. Starting input to hidden layer axons




f1

f2

f3

WEIGHT

.313

-
.91
9

.23
0

HEIGHT

-
.471

-
.11
6

-
.44
5

LLENGTH

-
.021

.62
0

.57
9

BIACR
O

-
.250

-
.14
7

-
.03
1



1

Some other taxonomic neural networks were also applied. Hopfield
neural network
Hoptax

produces unsatisfactory clustering with
coefficient of efficacy of only .882. A perfect coefficient of
efficacy was obtai
ned by neural network
Dualtax
, but with very
difficult identification of taxons defined in principal component
space. Similar results to these obtained by Triatlon were obtained by
Intruder
, a very simple neural network, but identification structures
obtai
ned by Triatlon are more informative then the structures
obtained by Intruder due to the intermediary fuzzy clustering of both
variables and subjects. Of course, both hierarchical methods and
classic methods of local optimization produce quite unsatisfacto
ry
results; Ward method produces five clusters with coefficient of
efficacy of only .822, and McQueen's method produces five clusters
with a relatively good coefficient of efficacy (.917), but not
clearly defined in morphological space.

BICRIS

.327

-
.12
6

-
.03
0

TRISKIN

.038

-
.10
3

-
.02
4

SCAPSKIN

.042

-
.01
5

.21
9

AXSKIN

.092

-
.19
7

-
.00
1

CRUPARM

.040

-
.34
1

-
.08
4

CRLWARM

.201

-
.03
6

.12
8

CRUPLG

-
.600

-
.14
5

.06
5

CRLWLG

-
.058

-
.18
1

.00
1

HANDLG

-
.306

.28
4

.25
3

HANDDM

.
322

.38
7

.31
8

ABDSKIN

-
.070

.26
5

.15
2

LWLSKIN

.195

-
.03
4

-
.01
2

CHCIRC

-
.074

-
.23
6

-
.18
3

DIWRIST

-
1.05
4

-
.04
9

.12
5

DIAEL

-
.025

.00
4

.13
5

DIAKNE

1.05
3

.24
2

.02
5

FOOTL

.137

.31
9

.24
8

FOOTDM

.286

-
.01
9

.15
6

ARMLG

.113

.19
9

.30
6


Table 2. Starting hid
den layer to output axons



g1

g2

g3

g4

g5

f1

.44
9

.23
6

.17
2

-
.81
7

-
.21
4

f2

.33
1

-
.54
4

.09
3

-
.15
2

.75
0

f3

.20
1

.52
1

-
.73
6

.00
6

.38
2


Table 3. Initial and classification in first iteration



g1

g2

g3

g4

g5

g1

54

15

10

0

25

g2

18

150

1

14

9

g3

12

2

15
9

26

15

g4

1

1

0

116

15

g5

6

0

0

12

76



Table 4. Number of objects and accordance of starting
classifications




numbe
r

progno
sis

accorda
nce

g1

104

54

.519

g2

192

150

.781

g3

214

159

.743

g4

133

116

.872

g5

94

76

.809



Table 5. Final input to hi
dden layer axons




g1

g2

g3

WEIGHT

.767

1.91
6

-
.83
4

HEIGHT

.007

.785

.24
2

LLENGTH

-
.118

-
.958

-
.09
7

BIACRO

.015

.062

-
.54
3

BICRIS

.011

.898

.15
5

TRISKIN

.019

.631

-
.05
7

SCAPSKIN

.316

-
.936

-
.63
5

AXSKIN

-
.469

-
.029

-
.18
0

CRUPARM

-
.024

-
.098

-
.34
7

CRLWARM

-
.352

.258

.26
7

CRUPLG

.257

-
.993

-
.60
3

CRLWLG

-
.112

.188

-
.02
5

HANDLG

.070

.061

-
.44
2

HANDDM

-
.250

-
1.11
0

.43
9

ABDSKIN

-
.261

-
.086

.92
4

LWLSKIN

.168

.178

-
.03
8

CHCIRC

.074

-
.474

.28
0

DIWRIST

.946

-
.578

-
.21
9

DIAEL

.192

.346

-
.09
2

DIAKNE

-
1.68
4

-
.263

-
.28
8

FOOTL

-
.023

-
.070

.20
7

FOOTDM

-
.020

-
.078

.24
5

ARMLG

-
.064

-
1.49
3

.34
1



Table 6. Final hidden layer to output axons



g1

g2

g3

g4

g5

g1

-
.35
-
.12
-
.25
.87
6

-
.15
3

5

8

9

g2

-
.05
8

-
.00
1

.69
8

.05
3

-
.71
2

g3

.60
0

-
.76
1

.12
8

.18
7

.09
1




Fisherian discriminant analysis in the whole
variable space
2
, incorporated in program, gives the
following identification structures:




Table 7. Centroids of final taxons




g1

g2

g3

g4

g5

WEIGHT

-
.69
2

.860

-
.43
2

-
.075

.07
0

HEIGHT

-
.26
6

.390

-
.38
6

-
.14
7

.29
4

LLENGTH

-
.16
9

.356

-
.54
6

-
.185

.45
7

BIACRO

-
.64
1

.677

-
.34
7

-
.037

.11
5

BICRIS

-
.12
0

.332

.15
9

-
.248

-
.20
8

TRISKIN

-
.34
4

.889

-
.07
5

-
.566

-
.12
4

SCAPSKIN

-
.45
4

1.00
9

-
.26
1

-
.480

-
.06
9

AXSKIN

-
.24
5

.841

-
.19
7

-
.639

.04
6

CRUPARM

-
.70
4

.913

-
.21
5

-
.193

-
.09
0

CRLWARM

-
.48
0

.733

-
.20
7

-
.315

.04
9



2

Identification st
ructures in taxonomic algorithms must be, at least
initially, defined in the whole space of variables because necessary
inversion operations in intrataxon space can produce, in the case of
perfect or almost perfect classification, very unstable results due

to the possible weak conditionality of matrix of intrataxon
dispersion.

CRUPLG

-
.78
1

.869

-
.31
0

-
.109

.03
6

CRLWLG

-
.57
5

.701

-
.19
6

-
.137

-
.02
2

HANDLG

-
.38
9

.180

-
.54
7

.056

.59
5

HANDDM

-
.05
2

.099

-
.63
2

-
.174

.73
3

ABDSKIN

-
.13
9

.247

-
.22
5

-
.039

.09
2

LWLSKIN

-
.25
2

.537

.01
0

-
.246

-
.19
0

CHCIRC

-
.57
1

.736

-
.38
5

-
.053

.04
7

DIWRIST

-
.60
7

.172

-
.86
8

.945

.25
4

DIAEL

-
.43
6

.455

-
.28
3

.081

.03
1

DIAKNE

.17
2

.753

.11
9

-
1.55
7

.37
7

FOOTL

-
.19
7

.301

-
.49
6

-
.122

.43
0

FOOTDM

.00
9

.105

-
.34
6

.040

.18
8

ARMLG

-
.04
6

.250

-
.63
9

-
.179

.57
0



Table 8. Discriminant coefficients




g1

g2

g3

g4

g5

WEIGHT

-
.45
0

.79
6

.711

.671

-
1.90
8

HEIGHT

.05
0

-
.21
4

.612

.087

-
.499

LLENGTH

-
.04
3

.04
0

-
.589

-
.183

.759

BIACRO

-
.38
6

.38
1

.007

-
.091

-
.055

BICRIS

-
.01
0

-
.14
9

.679

.080

-
.589

TRISKIN

-
.09
9

.02
7

.444

.037

-
.440

SCAPSKIN

-
.29
5

.53
0

-
.922

.126

.443

AXSKIN

.14
2

.24
6

.016

-
.436

.012

CRUPARM

-
.20
3

.26
2

-
.101

-
.092

.049

CRLWARM

.18
8

-
.20
9

.366

-
.255

-
.038

CRUPLG

-
.77
3

.20
2

-
.555

.013

.914

CRLWLG

-
.02
7

.00
8

.188

-
.098

-
.086

HANDLG

-
.61
1

.13
8

.203

-
.058

.160

HANDDM

.41
3

-
.30
4

-
.652

-
.196

.873

ABDSKIN

.61
8

-
.69
1

.151

-
.064

.214

LWLSKIN

-
.04
8

.03
5

.043

.155

-
.193

CHCIRC

.18
5

-
.21
3

-
.326

.094

.339

DIWRIST

-
.38
2

.07
8

-
.712

.763

.202

DIAEL

-
.13
2

.05
3

.172

.171

-
.294

DIAKNE

.39
9

.40
8

.242

-
1.54
9

.4
58

FOOTL

.12
8

-
.16
0

-
.010

.014

.080

FOOTDM

.32
-
-
.044

-
0

.08
7

.138

.048

ARMLG

.53
0

-
.12
1

-
1.14
3

-
.045

.931


Table 9. Structure of discriminant functions




g1

g2

g3

g4

g5

WEIGHT

-
.53
0

.64
1

-
.31
0

-
.04
1

.05
2

HEIGHT

-
.20
3

.29
1

-
.27
7

-
.07
9

.21
8

LLENGTH

-
.12
9

.26
5

-
.39
2

-
.10
0

.33
9

BIACRO

-
.49
0

.50
5

-
.24
9

-
.02
0

.08
5

BICRIS

-
.09
2

.24
7

.11
4

-
.13
5

-
.15
4

TRISKIN

-
.26
3

.66
3

-
.05
4

-
.30
7

-
.09
2

SCAPSKIN

-
.34
8

.75
2

-
.18
7

-
.26
0

-
.05
1

AXSKIN

-
.18
8

.62
7

-
.14
1

-
.34
6

.03
4

CRUPARM

-
.53
8

.68
1

-
.15
4

-
.10
4

-
.06
7

CRLWA
RM

-
.36
7

.54
6

-
.14
9

-
.17
1

.03
7

CRUPLG

-
.59
8

.64
8

-
.22
2

-
.05
9

.02
7

CRLWLG

-
.44
0

.52
3

-
.14
1

-
.07
4

-
.01
6

HANDLG

-
.29
8

.13
4

-
.39
2

.03
0

.44
2

HANDDM

-
.03
9

.07
4

-
.45
3

-
.09
4

.54
4

ABDSKIN

-
.10
.18
4

-
.16
-
.02
.06
8

6

2

1

LWLSKIN

-
.19
3

.40
1

.00
7

-
.13
3

-
.14
1

CHCIRC

-
.43
7

.54
9

-
.27
6

-
.02
9

.03
5

DIWRIST

-
.46
4

.12
8

-
.62
3

.51
2

.18
9

DIAEL

-
.33
3

.33
9

-
.20
3

.04
4

.02
3

DIAKNE

.13
1

.56
2

.08
5

-
.84
4

.28
0

FOOTL

-
.15
0

.22
4

-
.35
6

-
.06
6

.31
9

FOOTDM

.00
7

.07
8

-
.24
9

.02
2

.14
0

ARMLG

-
.03
5

.18
7

-
.45
8

-
.09
7

.42
3



Table 10. Patter
n of discriminant functions




g1

g2

g3

g4

g5

WEIGHT

-
.17
2

.48
4

-
.27
9

-
.04
4

-
.06
5

HEIGHT

-
.11
2

.19
2

-
.16
3

-
.07
2

.13
2

LLENGTH

.02
5

.23
6

-
.30
7

-
.06
7

.12
4

BIACRO

-
.39
2

.24
9

-
.05
1

-
.06
7

.15
3

BICRIS

.09
7

.25
5

-
.04
8

-
.07
6

-
.20
8

TRISKIN

.27
9

.70
0

-
.36
6

-
.16
4

-
.39
5

SCAPSKIN

.33
9

.81
9

-
.53
0

-
.11
9

-
.45
9

AXSKIN

.34
6

.69
6

-
.43
-
.18
-
.34
6

4

0

CRUPARM

-
.22
4

.48
4

-
.14
9

-
.09
7

-
.09
5

CRLWARM

-
.15
5

.38
8

-
.12
4

-
.13
8

-
.01
1

CRUPLG

-
.45
4

.34
0

-
.03
3

-
.10
2

.12
2

CRLWLG

-
.26
2

.32
5

-
.06
7

-
.08
5

.01
0

HANDLG

-
.71
3

-
.24
8

.20
4

-
.09
3

.71
3

HANDDM

-
.23
0

-
.07
1

-
.10
0

-
.10
9

.49
6

ABDSKIN

.09
8

.21
4

-
.22
0

.00
4

-
.08
7

LWLSKIN

.17
9

.43
7

-
.22
2

-
.05
8

-
.31
4

CHCIRC

-
.06
7

.45
7

-
.30
2

-
.02
0

-
.12
0

DIWRIST

-
.15
7

.10
9

-
.45
6

.35
0

.01
8

DIAEL

-
.13
4

.24
3

-
.16
1

.01
8

-
.02
7

DIAKNE

.11
5

.44
1

.03
1

-
.58
9

.15
3

FOOTL

-
.09
5

.14
2

-
.20
0

-
.06
2

.20
1

FOOTDM

.30
0

.22
9

-
.38
2

.06
4

-
.16
1

ARMLG

.14
1

.22
6

-
.39
6

-
.04
8

.12
1


Table 11. Correlations of discriminant functions



g1

g2

g3

g4

g5

g1

1.00
0

-
.615

.221

-
.411

.251

g2

-
1.00
-
-
-
.615

0

.106

.354

.094

g3

.221

-
.1
06

1.00
0

-
.308

-
.693

g4

-
.411

-
.354

-
.308

1.00
0

-
.273

g5

.251

-
.094

-
.693

-
.273

1.00
0



Table 12. Standardized discriminant coefficients




g1

g2

g3

g4

g5

WEIGHT

-
.34
4

.59
3

.51
0

.36
4

-
1.41
7

HEIGHT

.03
8

-
.16
0

.43
9

.04
7

-
.371

LLENGTH

-
.03
3

.03
0

-
.42
3

-
.09
9

.563

BIACRO

-
.29
5

.28
4

.00
5

-
.04
9

-
.041

BICRIS

-
.00
8

-
.11
1

.48
7

.04
3

-
.437

TRISKIN

-
.07
6

.02
0

.31
9

.02
0

-
.326

SCAPSKIN

-
.22
6

.39
5

-
.66
2

.06
9

.329

AXSKIN

.10
9

.18
3

.01
2

-
.23
6

.009

CRUPARM

-
.15
5

.19
6

-
.07
2

-
.05
0

.036

CRLWARM

.14
4

-
.15
5

.26
3

-
.13
8

-
.028

CRUPLG

-
.59
2

.15
0

-
.39
9

.00
7

.678

CRLWLG

-
.02
1

.00
6

.13
5

-
.05
3

-
.064

HANDLG

-
.46
8

.10
3

.14
6

-
.03
1

.118

HANDDM

.31
6

-
.22
-
.46
-
.10
.648

6

8

6

ABDSKIN

.47
3

-
.51
5

.10
8

-
.03
5

.159

LWLSKIN

-
.03
7

.02
6

.03
1

.08
4

-
.143

CHCIRC

.14
2

-
.15
9

-
.23
4

.05
1

.252

DIWRIST

-
.29
2

.05
8

-
.51
1

.41
3

.150

DIAEL

-
.10
1

.03
9

.12
3

.09
2

-
.218

DIAKNE

.30
5

.30
4

.17
4

-
.83
9

.340

FOOTL

.09
8

-
.11
9

-
.00
7

.00
7

.059

FOOTDM

.24
5

-
.06
5

-
.09
9

.02
4

-
.036

ARMLG

.40
6

-
.09
0

-
.82
0

-
.02
4

.691



Table 13. Contingency matrix of Neural netw
ork and
Fisherian classification



g1

g2

g3

g4

g5

g1

110

0

3

0

3

g2

0

162

4

0

5

g3

14

4

136

0

0

g4

1

4

0

137

2

g5

17

2

0

0

133



Therefore, classification of entities in
morphological space is really a hard task for any
clustering algorithm, as can b
e seen from the measure of
efficacy of final classification obtained by Triatlon, a
taxonomic neural network with almost perfect behavior in
the classification of objects in other fields of
anthropological space (Momirovi}, 2002).




Table 14. Number of ob
jects and efficacy of final
classification



numbe
r

progno
sis

erro
r

g1

116

110

6

g2

171

162

9

g3

154

136

18

g4

144

137

7

g5

152

133

19



Coefficient of efficacy of Triatlon in this case was
only 0.920, markedly lower then in other applications of
thi
s program in biochemistry, physiology, psychology,
sociology and criminology.


However, in spite of complex position of types in
the space of manifest morphological characteristics and
not always clear pattern and structure of discriminant
factors obtained

types can be identified as follows:


(1) Typus asthenicus
, defined by low development of
skeleton, low muscular mass and low fat tissue;

(2) Typus sthenicus
, defined by strong development of
skeleton, high amount of muscular mass and above average
fat tis
sue due to the high amount of fat cells;

(3) Typus gracilis
, defined primarily by small measures
of transversal dimensions of skeleton;

(4) Typus disharmonicus
, defined by inconvergent
development of morphological characteristics and low fat
tissue;

(5) Ty
pus leptomorphicus
, defined by above average
development of longitudinal dimensions of skeleton.



Therefore, morphological types in real space are
differnt of those obtained in image space by Momirovi},
Ho{ek, Prot and Bosnar (2002) but are similar, altou
gh
not identical, with taxons obtained by method of fuzzy
clustering applied by A. Ho{ek (1978).



4. DISCUSSION



It seems, in any case, that manifes morphological
space is not an ideal envinronment for the determination
of morphological types. Some reaso
ns for this apparently
paradoxical statment is the true nature of anthropometric
variables. Namely, most of them are partially included
one to other to an udetermined manner, and most of them,
altough manifestly different, have the same genetical
origin. T
his results in very uncertain position of
objects in the space of manifest morphological
characteristics, partly because of latent degeneration of
this space due to the near singularity of some of
segments spanned by specific morphological vectors.


REFER
ENCES


Ho{ek, A. (1978):

Povezanost morfolo{kih taksona sa manifestnim i latentnim
dimenzijama koordinacije.

Disertacija, Fakultet za fizi~ku kulturu Sveu~ili{ta u
Zagrebu.


Ho{ek, A. (2002):

O odre|ivanju antropolo{kih taksona.

Glasnik Antropolo{kog dru{t
va Jugoslavije,
37
:147
-
156.


Momirovi}, K.; Zakraj{ek, E. (1973):

Odre|ivanje taksonomskih skupina direktnom oblimin
transformacijom ortogonaliziranih originalnih i latentnih
varijabli.

Kineziologija
,
3
, 1: 83
-
92.


Momirovi}, K. (1978):

XTQ procedures for
the determination of polar taxonomic
variables.

Informatika 78,

3
, 104.


Momirovi}, K.; Zakraj{ek, E.; Ho{ek, A.; Stojanovi}, M.
(1979):

Comparative evaluation of some taxonomic algorithms for
the determination of morphological types.

Collegium Antropologi
cum
,
3
: 59
-
65.


Momirovi}, K. (1981):

A class of algorithms for the determination of polar
taxons.

In
Multidimensional data analysis,
475
-
491.

Le Chesney: SRCE, INRIA et ISDUN.


Momirovi}, K.; Gredelj, M. (1982):

Jednostavan postupak za detekciju konzisten
tnih rojeva.

Zbornik radova VI simpozija iz informatike 'Jahorina 82',
282: 1
-
7.

Sarajevo: Elektrotehni~Ki fakultet.


Momirovi}, K. (1986):

COMTAX: Algoritam i program za detekciju i komparaciju
polarnih i distinktnih taksona.

Statisti~ka revija
,
36
, 3
-
4:
141
-
149.


Momirovi}, K. (1993):

O jednom taksonomskom algoritmu u parcijalnom image
prostoru.

Zbornik radova 6 i 7 Sekcije za klasifikacije Saveza
statisti~kih dru{tava Jugoslavije,
22
-
30.

Beograd: savezni zavod za statistiku.


Momirovi}, K.; Ho{ek, A.; Pr
ot, F.; Bosnar, K. (2002):

O morfolo{kim tipovima mladih odraslih mu{karaca
(Morphological types of adult young men).

Technical report, Institute of criminological and
sociological research, Belgrade.


Momirovi}, K.; Ho{ek, A.; Popovi}, D. A.; Boli. E.
(20
02):

Cluster analysis by neural networks.

Proceedings of 10
th

International Congress of Physical
Education and Sport.

Komotini: Democritus University of Trace.



Momirovi}, K. (2002):

A taxonomic neural network.

Technical report, Institute of criminologica
l and
sociological research, Belgrade.


Popovi}, D. A.; Momirovi}, K. (2002):

Taksonomske neuronske mre`e.

Tehni~ki izve{taj, Fakultet za fizi~ku kulturu
Univerziteta u Pri{tini.


Stojanovi}, M.; Solari}, S.; Momirovi}, K.;
Vukosavljevi}, R. (1975):

Pouzd
anost antropometrijskih mjerenja.

Kineziologija
,
5
, 1
-
2:91
-
122.



Szirovicza, L.; Gredelj, M.; Momirovi}, K. (1978):

MORPHOTAX: Algoritam i program za taksonomsku analizu u
prostoru multivarijatno raspore|enih varijabli.

Informatica 78
,
7
, 105.


[talec,

J.; Momirovi}, K. (1971):

Ukupna koli~ina valjane varijance kao osnov
kriterija za odre|ivanje broja zna~ajnih glavnih
komponenata.

Kineziologija
,
1
, 1: 83
-
90.


Zlobec, L. (1975):

Komparativna analiza nekih taksonomskih algoritama.

Magistarsk
i rad, Elektrotehni~ki fakultet u Zagrebu.