Municipal Creditworthiness Modelling by means of Fuzzy Inference Systems
and Neural Networks
Petr Hájek (Institute of System Engineering and Informatics, Faculty of Economics
and Administration, University of Pardubice, Czech Republic) - [email protected],
Vladimír Olej (Institute of System Engineering and Informatics, Faculty of Economics
and Administration, University of Pardubice, Czech Republic) [email protected]
The paper presents the design of municipal creditworthiness parameters. Municipal
creditworthiness modelling is realized by fuzzy logic based systems and by
unsupervised methods. Therefore, current designs of hierarchical structures of
Mamdani-type fuzzy inference systems and their analysis are introduced. Cluster
analysis, fuzzy cluster analysis and self-organizing feature maps seem to be
preferable in terms of unsupervised modelling.
Key-Words: Municipal creditworthiness, Mamdani-type fuzzy inference systems,
hierarchical structures of fuzzy inference systems, unsupervised learning, neural
networks, self-organizing feature maps.
586
1 Introduction
Municipal creditworthiness [1] is the ability of a municipality to meet its short-term and
long-term financial obligations. It is assigned based on parameters (factors) relevant to the
assessed object. High municipal creditworthiness shows a low credit risk, while low one
shows a high credit risk.
Municipal creditworthiness evaluation is currently realized by methods combining
mathematical-statistical methods and expert opinion [1,2]. Scoring systems [3], rating [4],
rating-based models [1,5], default models [6] and unsupervised models [2,7] fall into these
methods. Their output is represented either by a score evaluating the municipal
creditworthiness (scoring systems) or by an assignment of the municipalities to the j-th class
Ȧj  ȍ, ȍ={Ȧ1,Ȧ2, … ,Ȧj, … ,Ȧc} according to their creditworthiness (rating, unsupervised
models). Scoring systems are typical for easy calculation, low accuracy and inability to work
with uncertainty and expert knowledge. Rating is an independent expert evaluation based on
complex analysis of all known risk parameters of municipal creditworthiness. It is considered
to be rather subjective. Municipalities are classified into classes Ȧ1,Ȧ2, … ,Ȧc by rating-based
models. The classes are assigned to the municipalities by rating agencies. The models showed
low classification accuracy [1]. The objective of default models is to find the causes of
municipalities default. Low number of municipalities makes the application of default models
impossible. Municipalities are assigned to classes based on selected parameters similarity by
unsupervised methods. Unsupervised methods [8,9] are used in economic fields. The models
of financial analysis [10], bankruptcy prediction [11] and share price [12], etc., are used as
samples. They include statistical methods [8], the neural networks based methods [13] and
fuzzy logic [14,15], etc.
Therefore, the methods capable of processing and learning the expert knowledge, enabling
their user to generalize and properly interpret, have been considered most suitable. The use
of natural language is typical for municipal creditworthiness decision-making process.
Natural language is characterized by semantic vagueness, therefore it cannot be transformed
directly into mathematical formulas. Moreover, precise description of municipal
creditworthiness parameters does not correspond to reality. The presented problem can be
solved by fuzzy logic [16,17,18], enabling its user to model the meaning of natural language
words. For example, fuzzy inference systems [19] are suitable for municipal creditworthiness
evaluation. Evaluation of company and bank client creditworthiness illustrates their possible
application [11,20,21].
Therefore, the paper presents the design of municipal creditworthiness parameters and their
modelling by hierarchical structures of Mamdani-type fuzzy inference systems and by cluster
analysis (K-means algorithm) [22], fuzzy cluster analysis (Gustafson Kessel algorithm) [23]
and neural networks (self-organizing feature maps).
2 Municipal Creditworthiness Parameters Design
Economic, debt, financial and administrative categories [1,24] are listed among the
common categories of parameters. The economic, debt and financial parameters are pivotal
[24]. The differences in creditworthiness evaluation methods are presented in the used
parameters and their weights. Methods in [1,24] assume a high fiscal autonomy of
municipalities. This allows the municipalities to influence their revenues through local
taxation and charges for municipal services. On the other hand, the municipalities in the
Czech Republic have low fiscal autonomy. Therefore, the parameters of evaluation differ
from the methods.
2.1 Economic Parameters Design
587
Economic parameters affect long-term municipal creditworthiness [25]. The municipalities
with more diversified economy and more favourable social and economic conditions are
better prepared for economic recession [1]. The economic growth, however, is able to enlarge
public services, and consequently, to increase the indebtedness. Stable municipal economy
can indicate economic stagnation. There is no synthetic parameter that would quantify the
level of municipal economies. The economic parameters for municipal creditworthiness
evaluation can be designed as follows:
Parameter p1 POr ,
(1)
where POr is population in the r-th year. Higher value of the parameter p1 entails especially
higher municipal tax revenues. Tax revenues depend on the number of inhabitants and on the
coefficient, which indicates the size category of a given municipality. Larger municipalities
have higher share in tax yield, because the more populated municipalities have higher
spending for the infrastructure and other public goods. Higher population guarantees future
municipal revenues for the creditors. At the same time, it decreases the credit risk [16].
Parameter p 2 POr / POr s ,
(2)
where Rr-s is population in the year r-s, and s is the selected period of time. The change in
the number of inhabitants is a good criterion of the economic vitality of a municipality [24].
Economic growth of the municipality leads to the growing number of its inhabitants. Sudden
growth of the parameter should be assessed prudently, because the real trend is not needed.
Parameter p 3 U ,
(3)
where U is the unemployment rate in a municipality. The rate of unemployment evaluates
general economic wealth of the municipality. Economic growth reduces the unemployment in
a given municipality. Therefore, low rate of unemployment indicates good economic
conditions. High unemployment rate entails higher expenses for social services. Jobs
deficiency also reduces the price of real estates, while the budget revenues from the real estate
tax decrease.
Parameter p 4
2
¦ (PZOi /PZ ) ,
k
(4)
i 1
where PZOi is the employed population in a given municipality in the i-th economic sector,
i=1,2, … ,k, PZ is the total number of employed inhabitant, and k is the number of the
economic sectors. Parameter p4 represents the concentration of employment in economic
sectors and presents the measure of municipal economic concentration. Low value of
parameter p4 means a long -term flexibility of the municipal economy and protection against
bankruptcy of one sector. According to [1], the parameter p4 is the most considerable factor of
municipal creditworthiness.
2.2 Debt Parameters Design
Debt parameters include the size and structure of the debt. Ratios are often used to measure
both the debt of the municipality and its ability to pay off a debt service. However, using the
ratios is only effective if the parameters for comparable municipalities are available. The
comparison with the municipalities illustrates the current debt and financial situation of a
given municipality. Based on the aforementioned facts, the debt parameters can be designed
as follows:
Parameter p 5 DS/OP ,
(5)
where p5<0,1>, DS is the debt service and OP stands for the periodical revenues. It is a
crucial debt factor measuring the ability of the municipality to pay off the DS from regular
budget revenues [24]. The debt service includes yearly interest and annuity payment.
Periodical revenues are total revenues minus nonrecurring and capital revenues. The value of
parameter p5 above 0.15 can be considered a signal of the imminent debt trap.
588
Parameter p6 CD/PO , [Czech crowns],
(6)
where CD is a total debt. The indicated parameter measures gross measure of the municipality
indebtedness, i.e. the extent of the accrued debt per an inhabitant. Its absolute value is not
predicative itself. It is necessary to compare the value with those of other municipalities in the
region, or with the whole country [1].
Parameter p7 KD/CD ,
(7)
where p7<0,1> and KD is a short-term debt. It analyses the debt structure. A short-term debt
is designed to meet the short-term engagements resulting from the insufficient cash flow. The
short-term debt should be paid off during a fiscal year. If KD is intended to cover a budget
deficit or to finance the capital projects, it should be considered alarming, since it negatively
influences the credit risk [26]. Interest rates of short-term debt are usually floating rates. This
may result in the inability to pay the debt service.
2.3 Financial Parameters Design
Financial parameters inform about the scope of budget implementation. Their values are
extracted from the municipality budget. Financial parameters for municipal creditworthiness
evaluation can be designed as follows:
Parameter p 8 OP/BV ,
(8)
+
where p8 R and BV are current expenditures. The parameter p8 reports on the quality of the
budget implementation. If it is constantly greater than 1, i.e. current budget is in excess, and at
the same time a growing trend is indicated, the municipality is in good financial state. Good
financial standing enables the municipality to use common surplus to finance its
engagements. On this account, the parameter is regarded a key factor in [26].
Parameter p 9 VP/CP ,
(9)
where p9 <0,1>, VP are own revenues and CP are total revenues. Higher proportion of own
revenues in total revenues entails higher fiscal autonomy of the municipality. Consequently,
higher fiscal autonomy leads to lower indebtedness. According to [26], the size of the fiscal
autonomy affects the municipality decision-making management. Municipal management
chooses a combination of the VP and the debt on public goods financing. The higher is the
fiscal autonomy of the municipality, the smaller the need for the debt as a financing tool.
Parameter p10 KV/CV ,
(10)
where p10 <0,1> and KV are capital expenditures, CV are total expenditures. Higher value
of the parameter indicates capital activity of the municipality and a good common
management enabling its further development [26]. This hypothesis complies with the intergeneration theory of justice, where both the contemporary as well as future users of public
goods should take part in capital expenses.
Parameter p11 IP/CP ,
(11)
where p11 <0,1> and IP are capital revenues. The debt is primarily intended to finance the
capital (investment) expenditures (projects). The higher is the parameter p11, the smaller the
need of the next indebtedness to finance capital projects.
Parameter p12 LM/PO , [Czech crowns],
(12)
where LM is the size of municipal liquid assets. The municipalities control their own assets.
These are often used as bank's credit collateral. The banks grant a credit only on condition,
that the collateral assets are liquid enough, i.e. cashable in a short time. The liquid assets of
the municipality include suitably situated extensive land properties, commercial buildings,
agricultural land properties and assets for commercial use being in possession of the
municipality.
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3 Hierarchical Structures of Fuzzy Inference Systems Design for
Municipal Creditworthiness Evaluation
General structure of fuzzy inference system (FIS) is presented in Fig. 1 [14,17,19]. It
contains the fuzzification process by means of input membership functions, construction of
base rules (BRs) or automatic extraction of rules from the input data, application of operators
(AND, OR, NOT) in rules, implication and aggregation within rules and the defuzzification
process of obtained outputs to the crisp values. Based on the general structure of FIS, three
fundamental types of FIS can be designed, i.e. Mamdani-type [19], Takagi-Sugeno-type [27]
and Tsukamoto-type [15].
Input values
Fuzzification
Application
of operators
Inference
mechanism
Input
membership
function
Implication
within rules
Agregation
Output
membership
function
Defuzzification
Output values
Fig. 1 General structure of fuzzy inference system
Normalisation of the inputs and their transformation to the range of values of the input
membership functions is realised during the fuzzification process. The inference mechanism
is based on the operations of fuzzy logic and implication within rules. Transformation of the
outputs of individual rules to the output fuzzy set is realised on the basis of the aggregation
process. Conversion of fuzzy values to expected crisp values is realised during the
defuzzification process. The most commonly-used defuzzification method is the Centre of
Gravity method, one of the simplest defuzzification methods is the Max Criterion Method,
eventually the Mean of Maxima Method. There exists no universal method for designing
shape, the number and parameters of the input and output membership functions. Triangular,
trapezoidal and other membership functions are used for the design of FIS. The input to the
fuzzification process is the crisp value given by universe of discourse (reference set). The
output of the fuzzification process is the membership function value.
The construction of BRs can be realised by extraction of rules from historical data, if they
are at one’s disposal. Various optimisation methods for number of rules in BRs are presented
in [15,19]. The BRs consists of rules. The rules are used for the construction of conditional
terms, which create the basis of FIS.
Let x1, x2, … ,xi, … ,xn be the input variables defined in the reference sets X1, X2, … ,Xi,
… ,Xn and let y be the output variable defined in the reference set Y. Then FIS has n input
variables and one output variable. Each set Xi, i=1,2, … ,n, can be divided into pj, j=1,2, …
,m, the fuzzy sets P1(i)(x), P2(i)(x), … ,Ppj(i)(x), … ,Pm(i)(x). Individual fuzzy sets P1(i)(x),
P2(i)(x), … ,Ppj(i)(x), … ,Pm(i)(x), i=1,2, … ,n; j=1,2, … ,m represent the assignment of
linguistic variables relating to sets Xi. The set Y is also divided into pk, k=1,2, … ,o the fuzzy
sets P1(y),P2(y), … ,Ppk(y), … ,Po(y). The fuzzy sets P1(y),P2(y), … ,Ppk(y), … ,Po(y)
represent the assignment of linguistic variables for the set Y. Then the Mamdani-type FIS rule
can be put as follows [15,19]
590
(13)
IF x1 is A1(i) AND x2 is A2(i) AND … AND xn is Apj(i) THEN y is B,
where: - i=1,2, … ,n, j=1,2, … ,m,
- A1(i), A2(i), … ,Apj(i) represent linguistic variables corresponding to fuzzy sets P1(i)(x),
P2(i)(x), … ,Ppj(i)(x), … ,Pm(i)(x),
- B represents linguistic variable corresponding to fuzzy sets P1(y), P2(y), … ,Ppk(y),
… ,Po(y), k=1,2, … ,o.
Let’s have a given Mamdani-type FIS. Then the number of rules in this FIS is defined
according to the relation
N PP k m ,
(14)
where NPP is number of rules, k is number of membership functions in FIS and m is number
of input variables. Due to a great number of m, FIS can be ineffective with regard to the
increase of NPP. The design of the FIS hierarchical structures is one of the ways leading to the
decrease of rules number NPP [28,29,30]. The BRs reduction lowers the computational cost
and makes FIS interpretation possible. Fuzzy inference system is easily to interpret if the
following conditions are met [31]:
x FIS has low number of rules NPP with low number of variables in antecedent.
x FIS has low number of membership functions for individual variables k.
x There are no weights assigned to rules.
x The same language expressions are represented by the same membership functions.
The number of rules in hierarchical structure of fuzzy inference system (HSFIS) is defined
§mt · t
N PP ¨
(15)
1 k ,
¹̧
© t 1
where t is number of variables contained in each layer. The fundamental types of HSFIS [28],
i.e. cascade, tree and combination of tree and cascade structure, are used in the design of the
HSFIS for municipal creditworthiness evaluation. The HSFIS model for municipal
creditworthiness evaluation is presented in Fig. 2.
HSFIS1
Design of
parameters
p1,p2, … ,p12
HSFIS2
HSFIS3
Classification of
municipalities
into classes
HSFIS4
Fig. 2 HSFIS model
The decrease of rules number NPP is obtained by the design of this model. In addition to
rules reduction, the model design should reproduce the expert’s decision-making by
municipal creditworthiness evaluation with the intent to consider the similarity and mutual
relations of parameters p1,p2, … ,p12. The designs of HSFIS for municipal creditworthiness
evaluation are presented in Fig. 3 (HSFIS1-cascade structure), Fig. 4 (HSFIS2-tree structure
591
1), Fig. 5 (HSFIS3-tree structure 2) and Fig. 6 (HSFIS4-combination of tree and cascade
structure).
Design of HSFIS1
y*=y11,1
FIS11,1
Layer 11
y10,1
FIS10,1
Layer 10
y9,1
y3,1
FIS3,1
Layer 3
y2,1
FIS2,1
Layer 2
y
Layer 1
1,1
FIS1,1
p1
p2
p3
p4
…
p11
p12
Fig. 3 Design of HSFIS1 for municipal creditworthiness evaluation
Legend: p1,p2, … ,p12 are input variables, y1,1,y2,1, … ,y11,1 are outputs of subsystems
FIS1,1,FIS2,1, … ,FIS11,1, L=11 is the number of HSFIS1 layers.
The HSFIS1 model can be formalized by BRs R
10,1 *
,y ,y of single subsystems HSFIS1 this way:
Layer 1: FIS1,1: R
h1,1
h
h1,1
,R
h
h 2,1
, … ,R
h11,1
: IF p1 is A1 1,1 AND p2 is A 21,1 THEN y1,1 is B
h 2,1
h1,1
h
A 3 2,1
and outputs y1,1,y2,1, …
h1,1
,
h 2,1
Layer 2: FIS2,1: R : IF y1,1 is B AND p3 is
THEN y2,1 is B ,
…
h
h
h
h
Layer 11: FIS11,1: R 11,1 : IF y10,1 is B 10,1 AND p12 is A1211,1 THEN y* is B 11,1 ,
¦y
p
y1,1 (B
h1,1
)
j 1
1,1
j
¦µ
¦y
p
y 2,1 (B
h 2,1
)
u µ Bh1,1 (y1,1
j )
,
p
j 1
j 1
2,1
j
B
h1,1
y1,1
j
(
(17)
)
u µ Bh 2,1 (y 2,1
j )
¦µ
,
p
j 1
(16)
B
h 2,1
y 2,1
j
(
(18)
)
…
592
¦y
p
y * (B
h11,1
¦µ
j 1
)
11,1
j
u µ Bh11,1 (y11,1
j )
,
p
j 1
B
y11,1
j
(
h11,1
(19)
)
where: - p1,p2, … ,p12 are input parameters,
h
h
h
- A1 1,1 , A 21,1 , … , A1211,1 represent linguistic variables corresponding to the fuzzy sets
h
h
h
µ 1 1,1 (p j ) , µ 21,1 ( p j ) , … , µ 1211,1 ( p j ) ,
- B
h1,1
h
h
, B 2,1 , … , B 11,1 represent linguistic variables corresponding to the fuzzy sets
h
h
h
µ 1,1 ( y1,1 ) , µ 2,1 ( y 2,1 ) , … , µ 11,1 ( y*) ,
2,1
11,1
- µ Bh1,1 (y1,1
j ) , µ Bh 2,1 (y j ) , … , µ Bh11,1 (y j ) are membership functions values of the
2,1
11,1
aggregated fuzzy set for values y1,1
from the reference sets.
j , yj , … , yj
Design of HSFIS2
The HSFIS2 model can be formalized by BRs R
,y5,1,y* of single subsystems HSFIS2 this way:
Layer 1: FIS1,1: R
FIS1,2: R
h1,2
h1,1
h
h1,1
,R
h1,2
, … ,R
h
h 6,1
: IF p1 is A1 1,1 AND p2 is A 21,1 THEN y1,1 is B
h
A 31,2
: IF p3 is
AND p4 is
h
h
A 41,2
1,2
THEN y
h
h
is B
h1,2
and outputs y1,1,y1,2, …
h1,1
,
,
h
Layer 2: FIS2,1: R 2,1 : IF y1,1 is B 1,1 AND y1,2 is B 1,2 THEN y2,1 is B 2,1 ,
h
h
h
h
FIS2,2: R 2,2 : IF p5 is A 5 2,2 AND p6 is A 6 2,2 THEN y2,2 is B 2,2 ,
…
h
h
h
h
Layer 6: FIS6,1: R 6,1 : IF y5,1 is B 5,1 AND y5,2 is B 5,2 THEN y* is B 6,1 ,
¦y
u µ Bh1,1 (y1,1
j )
p
h1,1
y1,1 (B
)
j 1
1,1
j
¦µ
j 1
p
y1,2 (B
h1,2
)
j 1
1,2
j
h 6,1
)
(21)
(
h1,1
,
(22)
,
(23)
)
u µ Bh1,2 (y1,2
j )
¦µ
¦y
p
y* (B
B
y1,1
j
p
j 1
…
,
p
¦y
j 1
6 ,1
j
B
h1,2
y1,2
j
(
)
u µ Bh 6,1 (y 6,1
j )
¦µ
p
j 1
B
h 6,1
y 6,1
j
(
)
where: - p1,p2, … ,p12 are input parameters,
h
h
h
- A1 1,1 , A 21,1 , … , A126,1 represent linguistic variables corresponding to fuzzy sets
h
h
(20)
h
µ 1 1,1 ( p j ) , µ 21,1 ( p j ) , … , µ 126,1 ( p j ) ,
593
- B
h1,1
h
h
, B 1,2 , … , B 6,1 represent linguistic variables corresponding to fuzzy sets
h
h
h1,1
µ ( y1,1 ) , µ 1,2 ( y1,2 ) , … , µ 6,1 ( y*) ,
1,2
6,1
- µ Bh1,1 (y1,1
j ) , µ Bh1,2 (y j ) , … , µ Bh L,1 (y j ) are membership functions values of aggregated
1,2
6,1
fuzzy set for values y1,1
j , y j , … , y j from the reference sets.
y*=y6,1
FIS6,1
Layer 6
y
Layer 5
y5,2
5,1
FIS5,1
FIS5,2
y4,1
FIS4,1
Layer 4
y
FIS4,2
3,1
y3,2
FIS3,1
Layer 3
FIS3,2
y2,2
y2,1
Layer 2
FIS2,2
FIS2,1
y1,1
Layer 1
y
4,2
y1,2
FIS1,1
FIS1,2
p1 p2
p3 p4
p5 p6
p7 p8
p9 p10 p11 p12
Fig. 4 Design of HSFIS2 for municipal creditworthiness evaluation
Legend: p1,p2, … p12 are input variables, y1,1,y1,2, … ,y6,1 are outputs of subsystems
FIS1,1,FIS1,2, … ,FIS6,1, L=6 is the number of HSFIS2 layers.
Design of HSFIS3
y*=y2,1
Layer 2
FIS2,1
y1,1
Layer 1
FIS1,2
FIS1,1
p1 p2 p3 p4
y1,3
y1,2
FIS1,3
p5 p6 p7 p8 p9 p10 p11 p12
Fig. 5 Design of HSFIS3 for municipal creditworthiness evaluation
Legend: p1,p2, … ,p12 are input variables, y1,1,y1,2, … ,y2,1 are outputs of subsystems
FIS1,1,FIS1,2, … ,FIS2,1, L=2 is the number of HSFIS3 layers.
The HSFIS3 model can be formalized by BRs R
,y of single subsystems HSFIS3 this way:
h1,1
,R
h1,2
, … ,R
h 2,1
and outputs y1,1,y1,2, …
*
Layer 1: FIS1,1: R
y1,1 is B
h1,1
h1,1
h
h
h
h
: IF p1 is A1 1,1 AND p2 is A 21,1 AND p3 is A 31,1 AND p4 is A 41,1 THEN
,
594
h
h
h
h
h
FIS1,2: R 1,2 : IF p5 is A 51,2 AND p6 is A 61,2 AND p7 is A 71,2 THEN y1,2 is B 1,2 ,
…
(24)
h 2,1
h1,1
h1,2
h1,3
h 2,1
1,1
1,2
1,3
*
Layer 2: FIS2,1: R : IF y is B AND y is B AND y is B THEN y is B ,
¦y
p
h1,1
y1,1 (B
)
1,1
j
¦µ
j 1
p
¦y
j 1
p
y1,2 (B
h1,2
)
u µ Bh1,1 (y1,1
j )
j 1
1,2
j
B
h 2,1
)
(25)
,
(26)
)
¦µ
p
¦y
p
y * (B
(
,
u µ Bh1,2 (y1,2
j )
j 1
…
h1,1
y1,1
j
j 1
2,1
j
B
y1,2
j
(
h1,2
)
u µ Bh 2,1 (y 2,1
j )
¦µ
,
p
j 1
B
h 2,1
y 2,1
j
(
(27)
)
where: - p1,p2, … ,p12 are input parameters,
h
h
h
- A1 1,1 , A 21,1 , … , A122,1 represent linguistic variables corresponding to fuzzy sets
h
h
h
µ 1 1,1 ( p j ) , µ 21,1 ( p j ) , … , µ 122,1 (p j ) ,
- B
h1,1
h
h
, B 1,2 , … , B 2,1 represent linguistic variables corresponding to fuzzy sets
h1,1
h
h
µ ( y1,1 ) , µ 1,2 ( y1,2 ) , … , µ 2,1 ( y*) ,
1,2
2,1
- µ Bh1,1 (y1,1
j ) , µ Bh1,2 (y j ) , … , µ Bh 2,1 (y j ) are membership functions values of the
1,2
2,1
aggregated fuzzy set for values y1,1
j , y j , … , y j from the reference sets.
Design of HSFIS4
y*=y
5
1
Layer 5
y2,1
FIS5,1
y4,1
FIS4,1
Layer 4
y3,1
y1,3
FIS3,1
Layer 3
y2,2
Layer 2
FIS2,1
Layer 1
FIS2,2
y1,2
y1,1
y1,4
FIS1,1
FIS1,2
FIS1,3
p1 p2
p3 p4
p5 p6 p7
FIS1,4
p8 p9 p11 p10
p12
Fig. 6 Design of HSFIS4 for municipal creditworthiness evaluation
595
Legend: p1,p2, … ,p12 are input variables, y1,1,y1,2, … ,y5,1 are outputs of subsystems
FIS1,1,FIS1,2, … ,FIS5,1, L=5 is the number of HSFIS4 layers.
The HSFIS4 model can be formalized by BRs R
,y* of single subsystems HSFIS4 this way:
Layer 1: FIS1,1: R
h1,1
h
h1,1
,R
h1,2
, … ,R
h
h 5,1
: IF p1 is A 1 1,1 AND p2 is A 21,1 THEN y1,1 is B
h1,2
h
A 41,2
h
A 3 1,2
1,2
and outputs y1,1,y1,2, …
h1,1
,
h1,2
FIS1,2: R : IF p3 is
AND p4 is
THEN y is B ,
…
h
h
h
h
Layer 5: FIS5,1: R 5,1 : IF y2,1 is B 2,1 AND y4,1 is B 4,1 THEN y* is B 5,1 ,
¦y
p
h1,1
y1,1 (B
p
¦y
j 1
p
y1,2 (B
h1,2
u µ Bh1,1 (y1,1
j )
¦µ
j 1
)
1,1
j
j 1
)
1,2
j
B
h 5,1
)
(
,
(29)
,
(30)
)
¦µ
p
¦y
p
y * (B
h1,1
y1,1
j
u µ Bh1,2 (y1,2
j )
j 1
…
j 1
5,1
j
B
y1,2
j
(
h1,2
)
u µ Bh 5,1 (y 5,1
j )
¦µ
,
p
j 1
(28)
B
h 5,1
y 5,1
j
(
(31)
)
where: - p1,p2, … ,p12 are input parameters,
h
h
h
- A 1 1,1 , A 21,1 , … , A 125,1 represent linguistic variables corresponding to fuzzy sets
h
h
h
µ 1 1,1 (p j ) , µ 21,1 (p j ) , … , µ 125,1 ( p j ) ,
- B
h1,1
h
h
, B 1,2 , … , B 5,1 represent linguistic variables corresponding to fuzzy sets
h
h
h1,1
µ ( y1,1 ) , µ 1,2 ( y1,2 ) , … , µ 5,1 ( y*) ,
1,2
5,1
- µ Bh1,1 (y1,1
j ) , µ B h1,2 (y j ) , … , µ B h 5,1 (y j ) are membership functions values of the
1,2
5,1
aggregated fuzzy set for values y1,1
from the reference sets.
j ,yj , … ,yj
The numbers and shapes of input and output membership functions and BRs are defined
for the designed models HSFIS1, HSFIS2, HSFIS3 and HSFIS4.
4 Municipal Creditworthiness Modelling by Unsupervised Methods
Modelling of the municipal creditworthiness is a classification problem. It is generally
possible to define it this way:
Let F(x) be a function defined on a set A, which assigns picture x̂ (the value of the function
from a set B) to each element xA, x̂ =F(x)B,
F : Ao B.
(32)
596
The problem defined this way it is possible to model by the supervised methods (if classes
Ȧ1,Ȧ2, … ,Ȧj, … ,Ȧc of the objects are known) or by unsupervised methods (if these classes
are not known). Several Czech municipalities have assigned the class Ȧ1,Ȧ2, … ,Ȧj, … ,Ȧc by
a rating agency. Therefore, it is appropriate to model the municipal creditworthiness by e.g.
statistical methods (e.g. cluster analysis methods) and neural networks (e.g. self-organizing
feature maps). It is possible to create the classes on the basis of the objects’ similarity by
using these methods.
4.1 Municipal Creditworthiness Modelling by Cluster Analysis
The cluster analysis [9] belongs to the methods which deal with the search for the similarity
among multidimensional data objects and with their classification to classes (clusters). The
classes in cluster analysis are not assigned to data objects. The number of classes or clusters is
unknown, too. The found clusters represent the data structure only with reference to the
selected parameters. This method does not contain a technique capable of distinguishing the
significant and insignificant parameters, it only distinguishes the clusters.
The goal of the municipal creditworthiness evaluation is classification of the objects
(municipalities) to the classes Ȧ1,Ȧ2, … ,Ȧj, … ,Ȧc. In terms of definition of the cluster
analysis scope, the data are standardized by normalization of each of the parameters to its Zscore. The standardization facilitates mutual comparison of parameters’ values (their average
is 0 and the standard deviation is 1). The positive values are then above-average and the
negative are below the average. All the parameters are of quantitative type. Therefore, the
distance measures can be used. Further, it is necessary to choose the clustering algorithm and
to resolve upon the expected number c of the clusters. Both the mentioned decisions have
influence on the results interpretation. There are two basic algorithms of clustering, namely
hierarchical and partitioning algorithms [9]. The hierarchical algorithms construct a tree
structure of the clusters, so-called dendrogram [9]. These algorithms are not suitable for the
analysis of extensive samples, the results are affected by outlying objects and the undesirable
preceding combinations persist in the analysis. In the partitioning algorithms [9] (K-modes,
K-means algorithms, etc.) the objects are assigned to the number c of clusters given in
advance. First step is the setting of initial cluster centres and all the objects situated inside the
given distance to a cluster centre are assigned to this cluster. The choice of the initial cluster
centres is crucial. The K-means algorithm is used for the analysis of the municipal
creditworthiness. A disadvantage of this algorithm is the dependence of the clustering results
on the initial cluster centres. Therefore the initial cluster centres are set up by the hierarchical
algorithm (Ward’s method). The results of the cluster analysis are negatively affected by the
existence of outlying objects and by multicolinearity of the parameters. The outlying objects
are identified by the Mahalanobis distance and removed in consequence. The multicolinearity
has not been noticed. The unknown number c of clusters is the next problem of the cluster
analysis. This number is set by empirical experience and Dunn's index [32].
4.2 Municipal Creditworthiness Modelling by Self-organizing Feature Maps
Self-organizing feature maps (SOFM) are neural networks models based on competitive
learning strategy [13]. Output neurons of the network compete for their activity. The selforganizing feature maps are based on unsupervised learning. The aim of the unsupervised
learning is to approximate the probability density p(x) of the real input vectors xRn by the
finite number of representatives (codebook vectors) wiRn, where i=1,2, … ,h [13]. When the
codebook vectors are identified, the representative wi* (winner, the best matching unit, BMU)
is assigned to each vector x, for which
i* = argmin __x- wi__,
(33)
597
where i* is the index of the winning neuron. The SOFM is a two-layer neural network with
the completely connected units among the layers. The input layer is formed by n neurons,
which serves the distribution of the input values x. The units in the output layer serve as the
representatives. They are organized into topological structure (most often a two-dimensional
grid), which designates the neighbouring network units.
On the learning process it is necessary to define the concept of neighbourhood function,
which determines the range of cooperation among the neurons, i.e. how many weight vectors
belonging to neurons in the neighbourhood of the BMU will be adapted, and to what degree.
Gaussian neighbourhood function is in common use, which is defined this way
h(i * , i ) e
(
d E2 (i* ,i )
)
O2 (t )
(34)
where h(i*,i) is neighbourhood function, d E(i*,i) is Euclidean distance of neurons i and i* in
the grid, O(t) is the size of the neighbourhood in time t. After the BMUs are found the
adaptation of weights (learning) follows. The sequential and batch learning algorithm are
available. The principle of the learning algorithm is, that the weight vectors of the BMU and
its topological neighbours move towards the actual input vector according to the relation
,
2
wi(t+1) = wi(t) + D(t).h(i*,i).[x(t) - wi(t)],
(35)
where D(t)(0,1) is learning rate. The batch learning algorithm of the SOFM is a variant of
the sequential algorithm. The difference consists in the fact that the whole training set passes
through the network only once and only then the weights are adapted. The adaptation is
realized by replacing the weight vector with the weighted average of the input vectors. The
weight factors are represented by the values of the neighbourhood function
¦ h(i*, i)(t)x
n
wi(t+1) =
¦ h(i*, i)(t)
j 1
n
j
,
(36)
j 1
where j is index of an input value, n is the number of the input values. The learning algorithm
proceeds in two phases: rough phase with high values of neighbourhood radius and high value
of learning rate and fine tune phase with low values of neighbourhood radius and learning
rate.
The created SOFM is clustered by K-means algorithm [22]. The number of clusters is
determined by the Davies- Bouldin's index minimisation [32].
4.3 Municipal Creditworthiness Modelling by Fuzzy Cluster Analysis
Clusters are disjoint subsets of data objects’ set in cluster analysis. The clusters can overlap
in fuzzy cluster analysis (FCA). Let RqŽO, where Rq are fuzzy sets, O is set of data objects,
O={o1,o2, … ,on} and q is the cluster index, then
0 d R q (oi ) d 1 ,
¦ R q ( oi )
(37)
c
j 1
0
1,
(38)
¦ R q ( oi ) 1 ,
n
(39)
i 1
598
where Rq(oi) is the membership degree of the i-th object into the q-th cluster, c is number of
the clusters, oi is the i-th object, i=1,2, … ,n.
Each object must belong at least to one cluster (37). The sum of all membership degrees of
the i-th object into all c clusters equals to 1. All the objects of the set O might not belong to
one cluster with maximum membership degree and each cluster might not be empty (39).
Modelling municipal rating is realized by Gustafson Kessel algorithm (GKA) [23]. The
algorithm is a modification of fuzzy C-means algorithm (FCM) [23]. Contrary to the FCM
algorithm, by which it is possible to detect the clusters of spherical shape only, the clusters of
different shapes and orientations can be detected by the GKA algorithm, because each cluster
is described by its centre and by special matrix Hq, q=1,2, … ,c in the GKA algorithm.
Covariance matrix Sq is applied in calculation of the matrix Hq according to the following
¦ ( Rq(a) (u i )) b (u i s (qa) )T (u i s (qa) )
n
Sq
¦
i 1
n
,
(40)
( Rq( a ) (u i )) b
i 1
where a is the order of fuzzy pseudo-partition, ui is the vector of membership degrees of the ith object, b is a parameter specified in advance, sq is vector of coordinates of the q-th centre.
The matrix Hq is calculated this way
Hq
(det( S q
1
)) n
S q 1 .
(41)
Rules Pq are derived to interpret the found clusters. The rules have the following form
Pq: X1 je Aq1 AND … AND Xj je Aqj AND … AND Xm je Aqm,
(42)
where q=1,2, … ,c, c is the number of clusters, m is the number of parameters, Aqj is the fuzzy
set for the q-th cluster and j-th parameter. Fuzzy set Aqj is calculated as a projection on the jth parameter this way
Aqj (u ij )
V
up
u pj uij
Rq (u p ) ,
(43)
where uij is the membership degree of the i-th object to the q-th cluster for the j-th parameter,
up is the column vector of the membership degrees of objects to the j-th parameter, p=1,2, …
,n. The acquired fuzzy sets can be approximated by triangular or trapezoidal membership
functions, for instance [23].
5 Analysis of the Results
The comparison of given models according to the number of rules is presented in Table 1.
The designed models are compared to Mamdani-type FIS. As the results show, the HSFIS1
and HSFIS2 models contain the lowest number of rules. The input parameters p1,p2, … ,p12
form the following common categories, the economic (p1,p2,p3,p4), debt (p5,p6,p7) and
financial (p8,p9,p10,p11,p12) parameters. In the design of HSFIS model, it is possible to take
into account the membership of the parameters to the categories mentioned herein.
Consequently, the interpretability of the model is improved. The HSFIS1 and HSFIS2 models
do not reflect the affiliation of these parameters. Therefore, it is impossible to reproduce
expert decision-making in the field of municipal creditworthiness evaluation. The parameters
599
affiliation to categories (economic, debt and financial) was reflected in the design of HSFIS3
and HSFIS4 models. As the number of membership functions k increases, the design of
HSFIS3 model starts to be too complicated due to increasing number of rules NPP. The
HSFIS4 model contains low number of rules NPP even for great number of membership
functions k. As a consequence, this model is suitable for municipal creditworthiness
modelling.
Table 1 Comparison of HSFIS models
FIS
HSFIS1 HSFIS2 HSFIS3 HSFIS4
Number of FIS
1
11
11
4
9
Number of layers L
1
11
6
2
5
NPP for k=3
0.53x106
99
99
378
117
NPP for k=4
6
176
176
1 408
240
6
275
275
4 000
425
9
396
396
9 504
684
NPP for k=5
NPP for k=6
16.8x10
244 x10
2.18x10
Number of municipalities
Classification of municipalities into classes Ȧ1,Ȧ2, … ,Ȧ6 by HSFIS models and their
frequency is presented in Fig. 7 (HSFIS1), Fig. 8 (HSFIS2), Fig. 9 (HSFIS3) and Fig. 10
(HSFIS4). In term of creditworthiness, the best municipalities are assigned to class Ȧ1, the
worst ones to class Ȧ6. The models HSFIS1, HSFIS2, HSFIS3 and HSFIS4 classify the
municipalities so that the classes Ȧ3 and Ȧ4 have the highest frequencies. The average
municipalities dominate in the data sample. The designed fuzzy sets and BRs of the models
HSFIS1, HSFIS2, HSFIS3 a HSFIS4 make the interpretation of the generated classes
possible.
152
160
140
124
120
90
100
80
60
53
32
40
20
0
1
Ȧ1
Ȧ1
Ȧ
2
Ȧ2
ȦȦ3
3
Ȧ
Ȧ4
Ȧ4
Ȧ
Ȧ55
ȦȦ6
6
Class
Fig. 7 Classification of municipalities into classes by HSFIS1
600
Number of municipalities
200
180
160
140
120
100
80
60
40
20
0
181
127
96
36
12
Ȧ1
Ȧ1
0
Ȧ
Ȧ22
Ȧ3
Ȧ3
Ȧ4
Ȧ4
Ȧ5
Ȧ5
Ȧ
Ȧ6
6
Class
Number of municipalities
Fig. 8 Classification of municipalities into classes by HSFIS2
170
180
160
130
140
120
100
80
68
80
60
40
20
0
2
2
Ȧ1
Ȧ1
ȦȦ22
Ȧ
Ȧ33
Ȧ
4
Ȧ4
Ȧ5
Ȧ5
Ȧ
6
Ȧ6
Class
Fig. 9 Classification of municipalities into classes by HSFIS3
Number of municipalities
140
124
113
112
120
93
100
80
60
40
20
0
8
Ȧ1
Ȧ1
2
Ȧ2
Ȧ2
Ȧ
Ȧ33
Ȧ4
Ȧ4
Ȧ
Ȧ
5
Ȧ5
Ȧ
6
Ȧ6
Class
Fig. 10 Classification of municipalities into classes by HSFIS4
The K-means algorithm assigns every object to a single cluster. The clusters can be
interpreted on the basis of the centroids (i.e. the average points in the multidimensional space
defined for each cluster), a group of distant points, decision trees or logical rules [9]. The
logical rules are designed in terms of the known classification of the objects to the clusters.
601
For each cluster r, where r{1,2, … ,9}, logical rules Vr,k were created, where k is the
sequence of a rule for the r- th cluster. The algorithm PART (partial decision trees) was used
for the rules creation [33]. The algorithm is a combination of a method generating the
decision trees and a method based on separate and conquer paradigm. The rules are presented
in Table 2.
Table 2 Logical rules
V1,1 IF p10 ” 0.324 AND p7 > 0.861 AND p3 ” 13.58 AND p5 ” 0.206 AND p2 >
0.896
V1,2 IF p6 ” 1021.1 AND p10 ” 0.332 AND p7 > 0.25 AND p1 ” 1513 AND p5 ”
0.311 AND p3 ” 18.627 AND p6 > 87.43
V1,3 IF p5 ” 0.071 AND p3 ” 9.15 AND p3 ” 8.387
V1,4 IF p4 ” 0.249 AND p10 ” 0.266 AND p5 ” 0.006
V2,1 IF p7 ” 0.438 AND p8 ” 0.511 AND p2 > 0.917
V2,2 IF p1 > 2570
V3,1 IF p8 > 1.623 AND p9 ” 0.263 AND p5 ” 0.224
V3,2 IF p1 ” 3321 AND p10 > 0.376 AND p8 ” 1.573AND p5 ” 0.008 AND p11 ”
0.309
…
V8,1 IF p7 ” 0.438 AND p5 > 0.255 AND p6 ” 17059.46 AND p5 > 0.326
V9,1 IF p10 > 0.376 AND p6 > 12353.67 AND p6 > 17059.46
THEN r=1
THEN r=1
THEN r=1
THEN r=1
THEN r=2
THEN r=2
THEN r=3
THEN r=3
THEN r=8
THEN r=9
The representatives of the SOFM were clustered by the K-means algorithm (Fig. 11).
Classification of the representatives to the clusters is evident in Fig. 11. The clustering is in
contradistinction to cluster analysis realized only after SOFM has been made, which keeps the
structure of the original data.
Fig. 11 K-means algorithm in SOFM
q
The clusters can be interpreted on the basis of values of the parameters for the
representatives of the SOFM (Fig. 12). The values of parameters for individual clusters can be
derived this way.
602
…
Fig. 12 Values of parameters for SOFM representatives
Legend: p1,p2, … ,p12 are parameters of rating, d is a scale of parameters’ values.
Rules R1 to R8 for clusters q=1,2, … ,8 in form (42) are created to interpret the clusters
found by GKA. Fuzzy sets Aqj are calculated as projections on the j-th parameter according to
(43). The acquired fuzzy sets Aqj are approximated by trapezoidal membership functions. The
peaks of the membership functions indicate the parameters values of individual centres, i.e.
the points with maximum membership degree to clusters. The concrete forms of these rules
are given in Fig. 13.
603
...
Fig. 13 Rules for individual clusters in the form of membership functions
Legend: horizontal axis: values of parameters p1,p2, … ,p12, vertical axis: membership
degrees of these values to the q-th cluster.
p8
p8
p2
Fig. 14 Shapes of clusters made by K-means algorithm
Modelling by K-means algorithm is computationally fast and easily applicable. This
approach enables the user to detect clusters of spherical shapes only (Fig. 14). The results of
604
clustering are influenced by the outlying objects and depend on initial partition of data object
set. Other negative feature is the convergence of the criterion function to local minimum.
Modelling by GKA algorithm enables the user to find clusters of different shapes and
orientations (Fig. 15). The clusters are fuzzy sets of points that may overlap. The algorithm is
computationally more demanding than the previous one.
Fig. 15 Shapes of clusters made by GKA algorithm
Modelling by self-organizing feature maps preserves topological structure of the data in 2dimensional space, for example. These maps do not directly support the clustering. For this
purpose, it is necessary to complete them by clustering (Fig. 11). The objects are surrounded
with similar objects in the grid, but similar objects are not always located nearby. Sometimes
the cluster of similar objects is divided, so it is recommended to create more maps to reach
good results. The result of the SOFM, i.e. the clusters’ interpretation, is an assignment of
clusters to classes Ȧ1,Ȧ2, … ,Ȧ6. Their description is presented in Table 3.
Table 3 Description of classes Ȧ1,Ȧ2, … ,Ȧ6
Class
Description of belonging municipalities
Booming,
good
implementation
of budget, no problems with indebtedness
Ȧ1
Ȧ2 Excellent economic environment, no investment development
Ȧ3
Ȧ4
Ȧ5
Ȧ6
Average debt, good economic environment, good investment development
Big liquid assets, average debt, signals of economic recession
High debts, good investment development
High debt service, bad economic environment, good investment development
Comparison of the classification by models HSFIS1, HSFIS2, HSFIS3, HSFIS4 and
SOFM is presented in Table 4. In Table 5, there is comparison of the classification by these
models, where the one class deviation is allowed.
Table 4 Comparison of classification [%] by models HSFIS and SOFM
HSFIS1 HSFIS2 HSFIS3 HSFIS4 SOFM
605
HSFIS1
HSFIS2
HSFIS3
HSFIS4
SOFM
100.00
30.09
26.77
21.02
23.01
30.09
100.00
38.50
40.93
25.44
26.77
38.50
100.00
48.67
22.57
21.02
40.93
48.67
100.00
25.22
23.01
25.44
22.57
25.22
100.00
Table 5 Comparison of classification [%] by models HSFIS and SOFM with one class
deviation
HSFIS1
HSFIS2
HSFIS3
HSFIS4
SOFM
HSFIS1 HSFIS2 HSFIS3 HSFIS4
100.00 64.82
72.79
63.05
64.82 100.00 89.60
93.14
72.79
89.60 100.00 84.51
63.05
93.14
84.51 100.00
54.87
71.24
67.26
74.12
SOFM
54.87
71.24
67.26
74.12
100.00
6 Conclusion
The paper presents the design of municipal creditworthiness parameters. Further, the
HSFIS1, HSFIS2, HSFIS3 and HSFIS4 models are designed to realize the municipal
creditworthiness evaluation. The designed HSFIS models are suitable instruments for
municipal creditworthiness evaluation due to their effectiveness and easy interpretation. The
lowest number of rules has been achieved by HSFIS1 and HSFIS2 models. The parameters
p1,p2, … ,p12 affiliation to categories (economic, debt and financial) was reflected in the
design of HSFIS3 and HSFIS4 models. As a result of this procedure, the effectiveness of this
model decreases. The HSFIS2 is considered to be the most suitable of the presented models,
because it contains a low number of rules NPP and, at the same time, it models an expert’s
decision-making process in a given field.
The municipalities are assigned to classes Ȧ1,Ȧ2, … ,Ȧ6 by unsupervised methods on the
basis of selected parameters similarity. The selection of parameters is crucial in the
modelling. If selected parameters do not influence the creditworthiness or there are other
parameters missing, the results of the modelling are inaccurate.
In further research, it would be suitable to deal with the models that classify accurately, are
able to generalize, can be easily interpreted and, at the same time, they can competently work
with the expert knowledge. The combination of Mamdani-type fuzzy inference system and
feed-forward neural network is feasible.
The HSFIS models were carried out in Matlab Simulink in MS Windows XP operation
system. The unsupervised models were carried out in Statistica 6.0 and Statgraphics Plus 5.1
(cluster analysis), Weka 3.4.7 (decision trees), MATLAB 5.3 - SOM toolbox (self-organizing
feature maps) in operation system Windows XP and FCluster (fuzzy cluster analysis) in
operation system Linux.
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