A Fuzzy Approach to Discriminant Analysis based on the Results of
an Iterative Fuzzy k-Means Method
Francesco Campobasso and Annarita Fanizzi
Department of Economics and Mathematics, University of Bari, Via C. Rosalba 53, 70100 Bari, Italy
Keywords: Fuzzy Discriminant Analysis, Non Linear Regression Models, Iterative Fuzzy k-Means Method,
Entrepreneurial Propensity.
Abstract: The common classification techniques are designed for a rigid (even if probabilistic) allocation of each unit
into one of several groups. Nevertheless the dissimilarity among combined units often leads to consider the
opportunity of assigning each of them to more than a single group with different degrees of membership. In
previous works we proposed a fuzzy approach to discriminant analysis, structured by linearly regressing the
degrees of membership of each unit to every groups on the same variables used in a preliminary clustering.
In this work we show that non-linear regression models can be used more profitably than linear ones. The
applicative case concerns the entrepreneurial propensity of provinces in Central and Southern Italy, even if
our methodological proposal was initially conceived to assign new customers to defined groups for
Customer Relationship Management (CRM) purposes.
1 INTRODUCTION
The statistical literature proposes two main lines of
investigation on fuzzy discriminant analysis: one is
focused on the estimation of the coefficients
associated to Fisher’s linear function by maximizing
the so called ratio of fuzzy variances (Watada et al.,
1986); the other, although particularly complex from
a computational point of view, is based on the kernel
method and captures non-linear structures of clusters
(Wu and Zhou, 2006). There have been several
successful applications and developments of such
two lines of investigation in their respective fields of
application (for example (Zhao et al., 2012); (Song
et al., 2010); (Heo and Gader 2011)).
Whatever the fuzzy clustering from which the
discriminant analysis starts, each unit of the starting
collective is in any case attributed to more than one
group with different degrees of membership ranging
in the interval [0,1]: in particular, values of the latter
closer to 1 indicate a greater similarity of such an
unit to the other elements in the group (Campobasso
et al., 2008).
We have recently used the aforesaid values in an
attempt to attribute a new observation to previously
identified fuzzy groups (Campobasso and Fanizzi,
2013). Our proposal was to linearly regress the
degrees of membership to each group on the same
variables used in a preliminary clustering operation
of the sampled units.
In this work we show that non-linear regression
models may better fit such degrees of membership.
The application case deals with economic and
demographic data extracted from the Italian Atlas of
the competitiveness of provinces and regions edited
by UnionCamere, which capture some aspects of the
entrepreneurship provincial economy in Central and
Southern Italy in 2009. In particular, after
classifying with a fuzzy approach some of the
sampled provinces in order to identify homogenous
groups for entrepreneurial propensity, the remaining
ones are assigned to such groups by means of the
estimated discriminant model. In this case the
preliminary clustering is carried out by the fuzzy k-
means method, which appears more robust than
hierarchical ones, in the sense that it is less affected
by the presence of measurement errors or other
spurious sources of variance (and, consequently,
allows us to identify more cohesive groups).
Since, however, there might be no a priori
information about the number k of groups in which
the collective ought to be allocated, it is appropriate
to implement an iterative procedure of the fuzzy k-
means method, which can suggest such a number as
a function of appropriate indices of efficiency in
grouping.
257
Campobasso F. and Fanizzi A..
A Fuzzy Approach to Discriminant Analysis based on the Results of an Iterative Fuzzy k-Means Method.
DOI: 10.5220/0004553802570264
In Proceedings of the 5th International Joint Conference on Computational Intelligence (FCTA-2013), pages 257-264
ISBN: 978-989-8565-77-8
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
The reliability of the proposed discriminant
model is verified by comparing the obtained results
with the ones of our previous suggestion (based on
linear regression models) and also of a classical
discriminant analysis.
Note that such a fuzzy algorithm was initially
designed for an automotive dealer, where new
customers had to be assigned to defined groups for a
Customer Relationship Management (CRM)
application.
2 FUZZY DISCRIMINANT
MODEL
In a recent work we proposed to evaluate, after a
preliminary fuzzy clustering, the weight that each of
the used p variables has had in determining the
degrees of membership of the sampled units to the k
identified groups. For this purpose, we estimated as
many linear regression functions as the considered
groups, thus explaining the degrees of membership
ikig2i1i
...,,...,,,
of the i.th unit to them:
.x...xx
...
,x...xx
....
,x...xx
ikipkp2i2k1i1k0kik
igipgp2i2g1i1g0gig
1iipp12i121i11101i
In so doing, we could define a discriminant model
that allows to estimate the degree of awarding of any
other not sampled unit to the previous k groups. In
other word such a model establishes a hierarchy of
assignment of a generic new observation, that can be
used as a weighting system for further analysis. It is
possible to demonstrate that, as well as the sum of
the degrees of membership to k groups of the i.th
sampled unit equals 1, also the sum of the estimated
degrees of membership of each not sampled unit
equals 1.
Some of the estimated degrees of membership of
the i.th element may be less than zero, expressing a
lack of membership to the considered group. In this
case the other degrees of the i.th unit greater than
zero can be normalized, after setting each of the
negative ones equal to zero. In particular, assuming
that
2i1i
ˆ
,
ˆ
0 in the case of four groups, it would
be sufficient to ensure
0,
*
2i
*
1i
and to normalize
the remaining degrees of membership in the
following way
)
ˆˆ
(
ˆ
4i3i3i
*
3i
,
)
ˆˆ
(
ˆ
4i3i4i
*
4i
,
which leaves the total sum equal to one.
The same discriminant logic can be defined
considering non-linear regression functions,
according to which the marginal effect of a single
independent variable on the degree of membership
ig
of the unit i.th to the g.th group is not constant.
In particular we take into account the r.th order
polynomial functions, still linear in parameters but
no longer in independent variables, which take the
following form:
....x...x...x...
...
,...x...x...x...
....
,...x...x...x...
ik
r
ij)rr(pjk
2
ijpjkijkj0kik
ig
r
ij)1r(pjg
2
ijpjgijjg0gig
1i
r
ij)1r(pj1
2
ijpj1ijj1101i
Such a model better fits the degrees of membership
ikig2i1i
...,,...,,,
, which could increase as well
as decrease at rates which actually vary depending
on the values assumed by the independent variables
ikig2i1i
x...,,x...,,x,x
. In other words it may
happen that the awarding of any unit to one of the k
groups becomes more marked for certain values of
ipig2i1i
x...,,x...,,x,x
than for others.
Note that a higher order r makes the regression
function more flexible and, therefore, allows us to
express more precisely the behavior of the degrees
of membership at different levels of the independent
variables. On the other hand, a lower value of r
allows to have a more parsimonious model, which is
also easier to interpret.
In summary our proposal, framed in the context
of the relationships between clustering variables but
not focused on the fuzzy (within and between)
variances, is to express the degrees of membership
of any unit to each group as non linear functions of
the same variables used in a preliminary clustering.
2
coefficient of
determination provides a measure of the residual
sum of squares of the dependent variable in relative
terms (i.e. with respect to the total deviance of such
a variable) for each regression function,
characterized by its own specification, then an index
of reliability of the discriminant model is given by
the simple average of the adjusted R
2
coefficients
associated to the k estimated regression functions.
Note that the simple average is allowed because the
number of the observations does not differ from one
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
258
function to another (whereas the number of
independent variables, although variable, is taken
into account by means of the adjustment of the R
2
coefficients of determination).
3 AN APPLICATION CASE:
THE LEVEL OF
ENTREPRENEURIAL
PROVINCIAL PROPENSITY IN
CENTRAL AND SOUTHERN
ITALY
The entrepreneurial capacity of a territory is
recognized more and more widely as an important
vehicle for economic development, or as a major
stimulus to growth in terms of productivity,
innovation and employment. In the proposed
application case we seek to analyze the provincial
level of such a capacity in Central and Southern
Italy, during the dark year 2009 of the economic
crisis (National Institute for Foreign Trade, Italy in
the International Economy - Summary Report for
2008-2009)
For this purpose, we extract specific indicators
from the Italian atlas of the competitiveness of
provinces and regions edited by UnionCamere,
which can capture some aspects of provincial
entrepreneurship for the year under review.
The business density, measured by the ratio
between the number of active companies and the
resident population (in percentage terms), and the
density of the local units, measured by the ratio
between the total number of local units and the
considered area in square kilometers (in percentage
terms), are two indicators that describe the
entrepreneurial propensity of a territory.
These indicators do not provide qualitative
information on the economy of a territory, but are
certainly useful in comparisons between
geographical areas, and between the various sectors
of an economic activity.
Another demographic indicator that can be used
in our analysis is the rate of entrepreneurial
evolution, measured by the difference between the
corresponding birth and death rates of businesses.
The first one, representative of the productive
renewal, is defined as the number of new companies
(except for farms) registered in the records kept by
the Chambers of commerce, industry, agriculture
and crafts, with respect to those existing at the
beginning of the period (in percentage terms); the
second one, instead representative of the productive
obsolescence, is defined as the number of companies
(except for farms) deleted from such records, with
respect to those existing at the beginning of the
period (in percentage terms).
The last indicator used to evaluate the territorial
dynamism is the export propensity, defined as the
ratio between the total amount of exports in one year
and the produced added value in the same time
frame (in percentage terms).
On the basis of these indicators, which are
weakly correlated with one another (Table 1), we
randomly select the 80% of the 61 provinces in
Central and Southern Italy, afterwards we identify
homogeneous groups with a fuzzy approach, and
finally we estimate the proposed discriminant model,
in order to assign the remaining 20% of the
provinces to such groups.
Table 1: Correlation coefficients between the considered
indicators.
density
Density of the
local units
Rate of
evolution
Export
propensity
density
- -0.15 0.19 0.03
Density of the
local units
- 0.18 0.07
Rate of
evolution
- 0.17
Export
propensity
-
3.1 Identification of Homogeneous
Fuzzy Groups of the Provinces
3.1.1 Fuzzy k-Means Method
In a hierarchical clustering method any improper
aggregation of units carried out in early stages of the
iterative process can undermine the aggregations in
subsequent stages (Campobasso and Fanizzi, 2013).
On the contrary, in the context of a non-hierarchical
clustering method, the allocation of every unit into
groups is modified until the classification process
does not reach convergence.
Among the non-hierarchical methods, also
defined as optimization techniques, we choose the
fuzzy k-means one (Kaufman and Rousseau, 1990),
which is a generalization of the homonymous classic
method. The corresponding iterative process
minimizes a function of the Cartesian distance
between the elements of each group, once weighted
by the degree of membership to the group itself, and
their centroid.
In particular, let
ig
be the degree of
AFuzzyApproachtoDiscriminantAnalysisbasedontheResultsofanIterativeFuzzyk-MeansMethod
259
membership of the i-th element (i = 1, 2, …, n) to
the g-th cluster (g = 1, 2, .., k) underlying the
following two constraints: 0
ig
1 and
1
k
1g
ig
.
Then the procedure estimates the values of
ig
minimizing the object function


n
1i
k
1g
2
ig
2
ig
d)v,U(J
, with respect to the matrix U
of the degrees of membership
ig
and to the
centroids vector v = (v
1
, v
2
, …, v
k
). The generally
used metric is the Euclidean one on standardized
variables, but it is possible to adopt other metrics
with appropriate precautions.
The main limitation of this procedure, as well as
of any non-hierarchical method, is the need to know
the number of groups in which the collective should
be shared. For this purpose we propose an iterative
process of the fuzzy k-means method for increasing
values of k.
3.1.2 An Iterative Process of the Fuzzy
k-Means Method
The iterations of the fuzzy k-means method cannot
exceed a certain limit, as it would be difficult to
analyze a number of groups too high compared to
the available observations.
Therefore, after the completion of the iterations,
it is necessary to determine such a number,
depending on the actual effectiveness of the fuzzy
clustering.
The latter is more effective when it is able to
capture the degree of fuzziness of the observed units
in belonging to different groups, without causing
difficulties in the interpretation of the final
classification.
In particular the
partition coefficient (Bezdek,
1981) of a fuzzy classification in k groups


k
1g
n
1i
2
g,i
k
n
F
, measures such a degree of
fuzziness and is included in [1/k,1]. A normalized
version of it, which takes values in [0,1], is
represented by
1
k
1kF
'F
k
k
. The higher the value of
'F
k
, the more efficient the partition is.
The efficiency of a fuzzy classification in k
groups can also be expressed by the
index of entropy
(Bezdek, 1981)


k
1g
g,i
n
1i
g,ik
)log(
n
1
H
,
which varies from 0 up to log (k). The minimum
value is obtained in the case of hard clustering,
whereas the maximum one in the case of maximum
uncertainty (in which all the degrees of membership
equal 1/k).
A normalized version of it, which takes values in
n/k1
)klog(
,0
is represented by
n/k1
H
'H
k
k
. The
less the value of
'H
k
, the more efficient the
partition is. In correspondence of k = 2,
'H
k
often
assumes a relative minimum value, to be compared
with any other.
The so far mentioned indices do not involve the
observed values, but only the membership of the
units, apart from the form taken by the obtained
clusters. Therefore they can be only used to compare
groupings obtained for different values of k through
the same clustering method.
In the present case the values assumed by two
indices
'F
k
and
'H
k
are calculated for increasing
values of k by means of Matlab Editor. Their
graphical representation leads to divide the
provinces of Central and Southern Italy in three
groups; in correspondence of k=3, in fact,
'F
k
reaches a relative maximum value and
'H
k
a
relative minimum value.
Figure 1: Indices of efficiency in clustering for increasing
values of k.
In order to determine the "goodness" of the
clustering results, we use the so called
separation
coefficient
kh
hk
gh,gk
d
rr
maxg
, where
)d(maxr
k,ik,i
ni1
k
is the radius of the k.th group.
Such a radius is defined as the maximum distance
between its centroid and the n units of the entire
collective, weighted by the respective degrees of
membership.
The coefficient g is a functional validation index,
deliberately distinct from those used for the choice
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of k, and it measures the compactness of the clusters
of a fuzzy partition, by comparing their radius with
the distance between the corresponding centroids.
Such a coefficient is obtained as the maximum ratio
of the sum of the radii of the two groups on the
distance between their centroids.
If it is less than one, then the structure of the
groupings is robust, in the sense that the minimum
distance between groups is greater than the
one, then the clusters are tangent, i.e. the minimum
distance is equal to the maximum sum of the radii.
Finally, if it is greater than one, then the clusters
are intersecting, i.e. the minimum distance is less
than the maximum sum of the radii.
Since the separation coefficient of the identified
fuzzy partition shows a value of 0.042, then the
structure of the latter is robust and plausible.
3.2 The Identified Groups
With reference to the application case, the
preliminary analysis of the data set suggests the
subdivision of the collective into three groups.
Since the fuzzy classification generates groups to
which the elements do not belong separately, we
suggest to define the profile of each group by
calculating the weighted average of the values of the
indicators of all the sampled provinces by the
corresponding degrees of membership to all the
groups (Table 2).
Table 2: Average values of the considered indicators
weighted by the degrees of membership of each province,
by group
density
Density of the
local units
Rate of
evolution
Export
propensity
Group 1 8.70 17.50 -0.41 9.94
Group 2 9.81 15.17 0.39 10.28
Group 3 9.41 21.40 0.27 21.36
The first group identifies a low level of
entrepreneurial propensity, on average characterized
by a negative
rate of evolution (equal to -0.41), but
also by a
business density and an export propensity
(respectively equal to 8.45 and 9.94) which are both
lower than the other two groups.
The second group identifies an
intermediate level
of entrepreneurial propensity
, on average
characterized by a positive
rate of evolution (equal
to 0.39), but also by a
propensity
(respectively equal to 9.81 and 10.28)
which are both slightly higher than the first group.
The last group identifies a
high level of
entrepreneurial propensity
, on average characterized
by an
export propensity and a density of the local
units which are both by far the highest (respectively
equal to 21.36 and 21.40).
Actually the provinces characterized by a
predominant degree of membership (greater than
0.5) to the group with a
low level of entrepreneurial
propensity are all located in the South and Islands,
while those to the group with a high level of
entrepreneurial propensity are all located in Central
Italy.
Note that a sharper clustering might be obtained
by assigning each sampled province to the group
with respect to which its degree of membership is
the highest among the three ones.
3.3 Estimation of the Fuzzy
Discriminant Model
Once the three fuzzy groups are identified, it is
possible to determine the weight of the clustering
variables in the explanation of the corresponding
degrees of membership for each sampled province.
For this purpose, as extensively discussed above, we
first estimate a linear regression model in which the
latter ones are function of the former ones (Table 3).
Table 3: Estimated coefficients of the linear regression
models, by group.
Group 1 Group 2 Group 3
Intercept 0.32*** 0.40*** 0.28***
Density of the local
units
0.00 -0.03 0.04*
Rate of evolution -0.06*** 0.06*** 0.00
Export propensity -0.05** -0.10** 0.16***
2
0.58*** 0.52*** 0.59***
Index of reliability of the discriminant model 0.56
*** significant at 0.01 level ** significant at 0.05 level
* significant at 0.10 level
2
coefficients of
determination associated to the three linear
regressions are significant and all greater than 0.5,
so that their simple average equals 0.56. Therefore
the discriminant model reaches an acceptable level
of reliability for the collective under examination.
Now we show that the quadratic regressions are
more suitable than the linear regressions in the
estimation of the degrees of membership (Table 4).
2
coefficients of
determination associated to the three quadratic
regressions are significant and all greater than 0.6,
so that their simple average is higher (0.68) than that
obtained using linear regressions.
AFuzzyApproachtoDiscriminantAnalysisbasedontheResultsofanIterativeFuzzyk-MeansMethod
261
Table 4: Estimated coefficients of the quadratic regression
models, by group.
Group 1 Group 2 Group 3
Intercept 0.32*** 0.35*** 0.33***
Density of the local units 0.07 -0.11* 0.04*
Rate of evolution -0.11*** 0.09** 0.02
Export propensity -0.08*** -0.15*** 0.24***
Business density (^2) 0.05** -0.04* -0.01
Density of the local units
(^2)
-0.02* 0.02* -0.01
Rate of evolution (^2) -0.04*** 0.02** 0.01
Export propensity (^2) 0.01 0.03** -0.04***
2
0.71*** 0.62*** 0.71***
Index of reliability of the discriminant model 0. 68
*** significant at 0.01 level ** significant at 0.05 level
* significant at 0.10 level
The simple average of the adjusted R
2
coefficients of determination still grows (0.77) in the
case of cubic regressions, because the latter ones
become even more flexible and allow a better
representation of awarding of the sampled provinces
to the three identified groups (Table 5).
Table 5: Estimated coefficients of the cubic regression
models, by group.
Group 1 Group 2 Group 3
Intercept 0,28*** 0,42*** 0,29***
Density of the local units 0,02 -0,15** 0,13**
Rate of evolution -0,13*** 0,10*** 0,03
Export propensity -0,07** -0,15*** 0,22***
Business density (^2) 0,01 0,00 -0,01
Density of the local units
(^2)
0,02 0,08* -0,09**
Rate of evolution (^2) -0,01 0,00 0,02
Export propensity (^2) 0,07* -0,12*** 0,05
Business density (^3) 0,04** -0,05*** 0,01
Density of the local units
(^3)
0,00 -0,01 0,01**
Rate of evolution (^3) 0,01 -0,01 0,00
Export propensity (^3) -0,01 0,04*** -0,02***
2
0,76*** 0,79*** 0,77***
Index of reliability of the discriminant model 0. 77
*** significant at 0.01 level ** significant at 0.05 level
* significant at 0.10 level
Since the discriminant model defined via cubic
functions reaches a remarkable goodness of fit, we
prefer not to further increase the order of
polynomials, so as to maintain a parsimonious
number of the involved independent variables and to
allow an easier interpretability of the obtained
estimates.
The examination of the estimated regression
coefficients confirms what emerges from an
overview of the average profiles of the groups, since
the variables with a greater discriminating power in
the definition of degrees of membership seem to
correspond to those discussed above. In particular
the
business density and the rate of evolution are the
most significant variables in explaining the
assignment to the first two groups, while the
export
propensity and density of local units to the third
group (actually characterized by high average values
of the latter ones).
Note that the
business density is not significant at
all in determining the degree of membership of a
province to the third group, likely because the
corresponding average value is close to that of the
second one. Similar consideration can be done be
with reference both to the
density of the local units,
which is not significant in determining the degree of
membership to the first group, and to the
rate of
evolution
, which is not significant in determining the
degree of membership to the third group.
In general terms an independent variable,
expressed in cubic terms, helps in discriminating one
group from another when the gap between the
corresponding average values of the same
(standardized) independent variable is remarkable.
For clarity we present the average values of all
the standardized indicators for each group of
provinces in Table 6.
Table 6: Average values of the standardized indicators
weighted by the degrees of membership of each province,
by group.
density
Density of
the local
units
Rate of
evolution
Export
propensity
Group 1 -0,46 0,06 -0,46 -0,20
Group 2 0,51 -0,04 0,20 -0,18
Group 3 0,20 0,23 0,10 0,68
On the basis of such an estimation procedure, it
is possible to determine the degrees of membership
to the three groups for each of the 20% of the
provinces not yet taken into account (Table 7).
Such degrees are compared with the analogue
posterior probabilities of assignment, deriving from
a classical discriminant analysis based on Bayes'
theorem, in order to evaluate their reliability. Note
that the used prior probabilities coincide with the
proportions of sampled provinces included in each
of the three groups within a classification procedure
based on the found maximum degree of
membership.
Before conducting such a classical discriminant
analysis, we have assessed – by means of the
coefficient of kurtosis defined by Mardia - the actual
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262
multivariate normality of the indicators observed in
the three groups.
Table 7: Estimated degrees of membership to the three
identified groups, by not sampled province in Central and
Southern Italy (2009).
Sampled provinces Group 1 Group 2 Group 3
Pisa 0,22 0,32 0,47
Livorno 0,41 0,19 0,40
L'aquila 0,21 0,52 0,27
Isernia 0,31 0,46 0,24
Avellino 0,23 0,52 0,26
Brindisi 0,42 0,31 0,27
Ogliastra 0,35 0,51 0,13
Rieti 0,43 0,45 0,12
Palermo 0,50 0,41 0,09
Reggio Calabria 0,68 0,20 0,11
Enna 0,58 0,36 0,06
Medio Campidano 0,64 0,30 0,06
The achieved comparison confirms the validity
of the estimation procedure of the degrees of
membership, by means of which we assigned the
sampled provinces into homogeneous groups, since
the group with the highest degree of membership is
also the more likely one (Table 8).
Some slight discrepancies between the results of
the method proposed by us and those of the classical
discriminant analysis can happen, in the case of an
estimated not predominant membership to one of the
three groups.
See, for example, the provinces of Isernia,
Oristaglia, and Rieti, which show a degree of
membership to the first group slightly lower
(respectively equal to 0.31 0.35 and 0.43) than the
second group (respectively equal to 0.46 0.51 and
0.46), but a probability of assignment to the first
group higher than the second group. See, also, the
province of Medio Campidano, which shows a high
degree of membership to the first group (0.64), even
if the probability of assignment to the second group
is higher than the first one.
Note that the measure of probability of a
classical discriminant analysis has been used only to
assess the actual validity of the proposed model,
because such a measure represents in any case a
magnitude not directly comparable with a degree of
membership (which evokes, by definition, the
concept of a not exclusive assignment).
The latter, in particular, does not refer to the
occurrence or not of bivalent random events, but
represents the measure of a deterministic but
“vague” fact that occurs to a certain extent.
Table 8: Probability of assignment to the identified three
groups, estimated by means of a classical discriminant
analysis, by not sampled province in Central and Southern
Italy (2009).
Sampled provinces Group 1 Group 2 Group 3
Pisa 0.08 0.21 0.71
Livorno 0.51 0.16 0.33
L'Aquila 0.39 0.44 0.17
Isernia 0.68 0.30 0.02
Avellino 0,22 0.60 0.18
Brindisi 0.93 0.07 0.00
Ogliastra 0.51 0.44 0.04
Rieti 0.84 0.15 0.00
Palermo 0.69 0.22 0.10
Reggio Calabria 0.98 0.01 0.00
Enna 0.97 0,03 0.00
Medio Campidano 0.23 0.64 0.13
4 CONCLUSIONS
In this work we advance a proposal for a fuzzy
discriminant analysis, not focused on (within and
between) variances, but framed in the context of the
relationships between clustering variables.
In particular, in an attempt to regress the degrees
of membership of each unit to more groups on the
same variables used in a preliminary fuzzy
clustering, we show that polynomial functions are
more profitably than linear ones. This is because
such degrees of membership could increase as well
as decrease at rates which actually vary depending
on the values assumed by the clustering variables.
A higher order of polynomials provides more
precision in the corresponding estimates; on the
other hand, a lower value allows to have a more
parsimonious model, which is also easier to
interpret.
The reliability of our proposal is measured by the
simple average of the adjusted R
2
coefficients of
determination associated to the estimated regression
functions, since the number of the observations does
not differ from one function to another (whereas the
number of independent variables, although variable,
is taken into account by means of the adjustment of
the R
2
coefficients of determination).
As an application case we analyse the
entrepreneurial propensity in the provinces of
Central and Southern Italy on the basis of specific
indicators extracted from the Italian atlas of the
competitiveness of provinces and regions.
More specifically, we first identify a partition of
the sampled provinces in three clusters, through an
iterative procedure of the fuzzy k-means method,
AFuzzyApproachtoDiscriminantAnalysisbasedontheResultsofanIterativeFuzzyk-MeansMethod
263
which is more efficient and robust than hierarchical
methods; after we determine the degrees of awarding
of the not sampled provinces to each of the three
clusters, by means of regression cubic functions. The
latter ones show a good level of fit to the data in
The estimated degrees are compared with the
analogue posterior probabilities of assignment
deriving from a classical discriminant analysis, in
order to evaluate their reliability. Such a
discriminant analysis moves from a classification of
the sampled provinces based on their maximum
degree of membership to one of the three clusters.
The comparison confirms the validity of the
proposed procedure, since the group characterized
by the highest degree of awarding is also the more
likely one, albeit in general terms.
Note that the degrees of membership represent
measures of deterministic facts which occur to a
certain extent, while the posterior probabilities of
assignment refer to the occurrence or not of random
events; therefore the two aforesaid magnitudes are
heterogeneous, even if they are compared with each
other in order to demonstrate the actual cogency of
our proposal.
The same algorithm can be applied for a
Customer Relationship Management (CRM)
application, because of the similarity of the problem
homogeneous groups. Actually such an algorithm
was initially conceived in the case of an Italian
automotive dealer.
ACKNOWLEDGEMENTS
The innovative contents described in this paper are
disclosed after the permission of MILLENIA S.r.l.
company, which committed to Laboratorio KAD3
S.c.a.r.l. the research project called “Data Integration
and Statistical Models per la Business Intelligence
(DISMoBI)”, co-funded by Italian Government
(Legge 12 luglio 2011, n. 106 – Credito di imposta
per le imprese che finanziano progetti di ricerca in
Università o enti pubblici di ricerca).
Although the work is the result of joint reflections by
the authors, chapters 3, 3.2, 3.3 and 4 are to be
attributed to Francesco Campobasso, and chapters
1, 2 and 3.1 to Annarita Fanizzi.
REFERENCES
Bezdek, J. C., 1981. Pattern Recognition with Fuzzy
Objective Function Algorithms. Plenum, New York.
Campobasso, F., Fanizzi, A., 2013. A Proposal for a
Discriminant Analysis Based on the Results of a
Preliminary Fuzzy Clustering. In Computational
Science and Its Applications – ICCSA 2013, LNCS,
vol. 7974, pp. 444-456. Springer, Heidelberg.
Campobasso, F., Fanizzi, A., 2013. A fuzzy approach to
Ward’s method of classification: an application case to
the Italian university system. In Statistical Methods for
spatial planning and monitoring, pp.31-46. Springer-
Verlag, Berlin Heidelberg.
Campobasso, F., Fanizzi, A., Perchinunno, P., 2008.
Homogenous urban poverty clusters within the city of
Bari. In Computational Science and its Applications -
ICCSA 2008, Part I. LNCS, vol. 5072, pp. 232-244.
Springer, Heidelberg.
Kaufman, L., Rousseau, P. J., 1990. Finding Groups in
Data - An Introduction to Cluster Analysis. John
Wiley and Sons, New York.
UnionCamera, Italain Atlas of the competitiveness of
provinces and regions, http://www.unioncamere.gov.
it/Atlante/.
Watada J., H. Tanaka, K. Asai. 1986. Fuzzy discriminant
analysis in fuzzy groups. In Fuzzy Sets and Systems,
vol. 19, pp. 261–271. Elsevier, The Netherlands.
Wu H. X., J. J. Zhou. 2006. Fuzzy discriminant analysis
with kernel methods. Pattern Recognition, vol. 39, pp.
2236-2239. Elsevier, The Netherlands.
Song X., X. Yang, J. Yang, X. Wu, Y. Zheng. 2010.
Discriminant analysis approach using fuzzy fourfold
subspaces model. In Neurocomputing, vol. 73, pp.
255–2265. Elsevier, The Netherlands.
Zhao M., T. W. S. Chow, Z. Zhang. 2012. Random walk-
based fuzzy linear discriminant analysis for
dimensionality reduction. In Soft Computing, vol. 16,
pp. 1393-1409. Springer-Verlag, Berlin, Heidelberg.
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