From one weekly offer undertaken from NHLBI (2010) that is consisted from 7 daily offers based on 4 dishes (breakfast, lunch, snacks, and dinner) was possible to gain 2401 daily offers but using the optimisation approach it was shown that it is not possible to combine all dishes in new a daily offers because the nutrient composition will not satisfy needs of the

From possible 2401 daily offers (new daily menus) the final menu set was reduced on 63 daily offers, almost 2 months. Each weekly plan that has been proposed consisted of 4 meals distributed during the day as breakfast, lunch, snack and dinner. This indicates the limitation of the optimisation approach regarding the input data set, the weekly offer undertaken from NHLBI (2010). This example has shown that when the importance of the objective function is reduced (equal for all offers), this is a deficiency of the LP in meal planning and is no competition at all to the application of fuzzy logic in meal planning.

intake when dishes from different daily offers are combined in an inadequate way.

implications in terms of food choices (Ferguson et al., 2006).

are in accordance with set limits and the goal function.

daily offers (almost 35%) from the set.

Wirsam and Hahn (1999) pointed out.

identification of deviations in order to improve the daily intake.

planning as well as in the demanding cases as DASH diet planning.

target group, man aged 30-51 with the emphasis to hypertension.

**6. Conclusion**

Application of Fuzzy Logic in Diet Therapy – Advantages of Application 63

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**4**

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**Fuzzy Logic and Neuro-Fuzzy Networks for**

Pollution and management of the environment are serious problems which concern the entire planet; the main responsibility should be attributed to human activities that contribute significantly to damage the environment, leading to an imbalance of natural ecosystems. In recent years, numerous studies focused on the three environmental compartments: soil, water and air. The pollution of groundwater is a widespread problem. The causes of pollution are often linked to human activities, including waste disposal.

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Various statistical and mathematical models can be used for a qualitative risk assessment. The planning of a program for environmental remediation and restoration can be supported by expeditious methodologies that allow us to obtain a hierarchical classification of contaminated sites. The literature offers some expeditious and qualitative methods including fuzzy logic (Zadeh, 1965), neural networks and neuro-fuzzy networks, which are more objective methods. The three artificial intelligence systems differ among themselves in some respects: fuzzy inference system learns knowledge of data only through the fuzzy rules; neural network is able to learn knowledge of data using the weights of synaptic connections; neuro-fuzzy systems are able to learn knowledge of neural data with neural

Fuzzy logic was founded in 1965 by Zadeh. The first applications date back to the nineties. They were mainly used to control industrial processes, household electrical appliances and means of transport. Later, this approach was used in several fields including the

paradigm and represent it in the form of fuzzy rules.

**1. Introduction**

precision.

**Environmental Hazard Assessment**

*Department of Engineering and Physics of the Environment,*

Ignazio M. Mancini, Salvatore Masi, Donatella Caniani and Donata S. Lioi

 *University of Basilicata,*

### **Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment**

Ignazio M. Mancini, Salvatore Masi, Donatella Caniani and Donata S. Lioi *Department of Engineering and Physics of the Environment, University of Basilicata, Italy*

#### **1. Introduction**

64 Fuzzy Logic – Emerging Technologies and Applications

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Pollution and management of the environment are serious problems which concern the entire planet; the main responsibility should be attributed to human activities that contribute significantly to damage the environment, leading to an imbalance of natural ecosystems. In recent years, numerous studies focused on the three environmental compartments: soil, water and air. The pollution of groundwater is a widespread problem. The causes of pollution are often linked to human activities, including waste disposal.

Solid waste management has become an important environmental issue in industrialized countries. The most serious problems are related to solid waste disposal. Landfill is still the most used disposal technique but not the safest. In fact, a breakdown of containment elements could easily occur even in controlled landfills. This breakdown could cause contamination of aquifer that is environmental pollution. Such contamination can be mitigated by performing remediation and environmental restoration. The assessment of environmental pollution risk can be performed with different degrees of detail and precision.

Various statistical and mathematical models can be used for a qualitative risk assessment. The planning of a program for environmental remediation and restoration can be supported by expeditious methodologies that allow us to obtain a hierarchical classification of contaminated sites. The literature offers some expeditious and qualitative methods including fuzzy logic (Zadeh, 1965), neural networks and neuro-fuzzy networks, which are more objective methods. The three artificial intelligence systems differ among themselves in some respects: fuzzy inference system learns knowledge of data only through the fuzzy rules; neural network is able to learn knowledge of data using the weights of synaptic connections; neuro-fuzzy systems are able to learn knowledge of neural data with neural paradigm and represent it in the form of fuzzy rules.

Fuzzy logic was founded in 1965 by Zadeh. The first applications date back to the nineties. They were mainly used to control industrial processes, household electrical appliances and means of transport. Later, this approach was used in several fields including the

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 67

"Defuzzification" can be done using different methods, the most widely used one is the center of gravity, which calculates the center of gravity of the final fuzzy set and returns the

Fuzzification is a procedure through which the input variables are turned into fuzzy measures of their membership to given classes. Such a conversion from deterministic sizes to fuzzy sizes is performed through the membership functions pre-set for those classes. A membership function (Fig.2) is a function which associates a value (usually numerical) with the level of membership to the set. By convention, the real number which represents the level of membership [µ (x)] takes a 0 value when the element does not belong to the set, and

Fig. 1. Flow chart of the developed Fuzzy Inference System.

value of abscissa.

**1.1.1 Fuzzification**

1 when it belongs to it completely.

Fig. 2. Membership function.

environment. In fact, it could be used for assessing environmental risk related to contamination of groundwater. The fuzzy approach is advantageous because it allows us a quick assessment of the risk, but is disadvantageous because of the increasing complexity in the definition of fuzzy rules along with the increasing of the number of parameters. In many situations, when the number of parameters are considered high in the analysis, application of these techniques is cumbersome and complex and could be used for neuro-fuzzy models. These models reduce the complexity because they use training data. The neuro-fuzzy model was supported by a sensitivity analysis in order to address the problem of subjectivity and uncertainty of model input data.

#### **1.1 Fuzzy logic**

Fuzzy logic is a binary logic, which is inspired by Buddhist philosophy, which considers the world as something continuous. The fuzzy logic theory derives from the Persian-American engineer, Lotfi Zadeh, who theorized it in 1965 in an article entitled "Information and Control". In traditional logic, Aristotelian principles of non-contradiction and the excluded middle are valid. The principle of non-contradiction states that if X is a generic set and x a generic element, then x may belong to the whole X or not. The fuzzy systems deal with data and their manipulation with greater flexibility than traditional systems. The binary logic (or classical) is only concerned with what is completely true and, as a result, with what is completely false. Fuzzy logic instead extends its interest even to what is not completely true, what is probable or uncertain. The fuzzy logic is based on a linguistic approach, in which words or phrases of natural language are used instead of numbers. This approach simplifies complex situations and concepts which may use the traditional logic. In particular, the fuzzy logic operates on mathematical entities that are fuzzy sets. Fuzzy sets obey rules, structures and axioms which are very similar to those of classical sets; the difference is that an object can simultaneously belong to several subsets, in contrast to the classical theory. In the fuzzy world, membership to a subset is associated to a degree of membership. The set of deduction rules to be applied to a given system to achieve results through the use of fuzzy logic is defined the fuzzy inference process. Main phases of the fuzzy approach (Fig. 1) are the following: definition of membership functions, fuzzification, inference and fuzzy output.

Definition of membership functions is the main step on which all the other subsequent operations are based. Such functions, representing the fuzzy sets, can take different shapes (trapezoidal, triangular, Gaussian, etc.) according to the situations, and by convention can take values included between 0 and 1.

The fuzzification is the process by which input variables are converted to fuzzy measures belonging to certain classes such as Very Low, Low, Medium, High, Very High. This operation normalizes all the data in the interval [0-1]. In this way, comparisons between different amounts, measured in different scales are also possible.

The inference is the phase in which rules of combination of fuzzy sets are applied and it is possible to deduce a result. The rules are linguistic expressions that are translated into a mathematical formalism with the exspression "if ... then" of the logic itself.

The output is a fuzzy membership value that can be used "pure" as a qualitative property or "defuzzificated", as a real number compatible with non-fuzzy approaches (Silvert, 2000).

"Defuzzification" can be done using different methods, the most widely used one is the center of gravity, which calculates the center of gravity of the final fuzzy set and returns the value of abscissa.

Fig. 1. Flow chart of the developed Fuzzy Inference System.

#### **1.1.1 Fuzzification**

66 Fuzzy Logic – Emerging Technologies and Applications

environment. In fact, it could be used for assessing environmental risk related to contamination of groundwater. The fuzzy approach is advantageous because it allows us a quick assessment of the risk, but is disadvantageous because of the increasing complexity in the definition of fuzzy rules along with the increasing of the number of parameters. In many situations, when the number of parameters are considered high in the analysis, application of these techniques is cumbersome and complex and could be used for neuro-fuzzy models. These models reduce the complexity because they use training data. The neuro-fuzzy model was supported by a sensitivity analysis in order to address the problem of subjectivity and

Fuzzy logic is a binary logic, which is inspired by Buddhist philosophy, which considers the world as something continuous. The fuzzy logic theory derives from the Persian-American engineer, Lotfi Zadeh, who theorized it in 1965 in an article entitled "Information and Control". In traditional logic, Aristotelian principles of non-contradiction and the excluded middle are valid. The principle of non-contradiction states that if X is a generic set and x a generic element, then x may belong to the whole X or not. The fuzzy systems deal with data and their manipulation with greater flexibility than traditional systems. The binary logic (or classical) is only concerned with what is completely true and, as a result, with what is completely false. Fuzzy logic instead extends its interest even to what is not completely true, what is probable or uncertain. The fuzzy logic is based on a linguistic approach, in which words or phrases of natural language are used instead of numbers. This approach simplifies complex situations and concepts which may use the traditional logic. In particular, the fuzzy logic operates on mathematical entities that are fuzzy sets. Fuzzy sets obey rules, structures and axioms which are very similar to those of classical sets; the difference is that an object can simultaneously belong to several subsets, in contrast to the classical theory. In the fuzzy world, membership to a subset is associated to a degree of membership. The set of deduction rules to be applied to a given system to achieve results through the use of fuzzy logic is defined the fuzzy inference process. Main phases of the fuzzy approach (Fig. 1) are the following: definition of membership functions, fuzzification, inference and fuzzy output. Definition of membership functions is the main step on which all the other subsequent operations are based. Such functions, representing the fuzzy sets, can take different shapes (trapezoidal, triangular, Gaussian, etc.) according to the situations, and by convention can

The fuzzification is the process by which input variables are converted to fuzzy measures belonging to certain classes such as Very Low, Low, Medium, High, Very High. This operation normalizes all the data in the interval [0-1]. In this way, comparisons between

The inference is the phase in which rules of combination of fuzzy sets are applied and it is possible to deduce a result. The rules are linguistic expressions that are translated into a

The output is a fuzzy membership value that can be used "pure" as a qualitative property or "defuzzificated", as a real number compatible with non-fuzzy approaches (Silvert, 2000).

different amounts, measured in different scales are also possible.

mathematical formalism with the exspression "if ... then" of the logic itself.

uncertainty of model input data.

take values included between 0 and 1.

**1.1 Fuzzy logic**

Fuzzification is a procedure through which the input variables are turned into fuzzy measures of their membership to given classes. Such a conversion from deterministic sizes to fuzzy sizes is performed through the membership functions pre-set for those classes. A membership function (Fig.2) is a function which associates a value (usually numerical) with the level of membership to the set. By convention, the real number which represents the level of membership [µ (x)] takes a 0 value when the element does not belong to the set, and 1 when it belongs to it completely.

Fig. 2. Membership function.

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 69

<sup>2</sup>

1 *<sup>b</sup> f xabc*

Despite their simplicity, such functions cannot be used to represent asymmetry, which is

In order to face a possible asymmetry, we can use another type of function, such as the sigmoid function (Fig. 8), which may have left or right asymmetry and a horizontal

In addition to this function, we have further asymmetric functions, the Dsigm and Psigm membership functions represented in Fig. 9 and10 and described by Eq. (6), depending on

<sup>1</sup> *ax c f xac e*

<sup>1</sup> *ax c f xac e*

; ,

; ,

1

1

;,,

Fig. 5. Gauss membership function.

Fig. 6. Gauss2 membership function.

important in some applications.

four parameters a1, c1, a2 and c2.

Fig. 7. Generalized bell (Gbell) membership function.

asymptote. This function is ruled by parameters a and c (Eq. (5))

1

*x c a*

(4)

(5)

(6)

Membership functions can be of several types: the simplest are made up of straight lines, while the most used are the triangular (Fig. 3) and trapezoidal (Fig. 4) functions; the former are characterized by a triangular trend while the latter by a trapezoidal one. The advantage of these functions is in their simplicity. The triangular membership function depends on three scalar parameters a, b and c and is given by the following expression:

$$f(x;a,b,c) = \max\left(\min\left(\frac{x-a}{b-a}, \frac{c-x}{c-b}\right) 0\right) \tag{1}$$

while the trapezoidal one depends on four scalar parameters (a, b, c and d), as shown in the following formula:

$$f(x;a,b,c,d) = \max\left(\min\left(\frac{x-a}{b-a}, 1, \frac{d-x}{d-c}\right)0\right) \tag{2}$$

$$\sum\_{a=1}^{n} \left(\frac{x-a}{b-a}, \frac{x-a}{b-a}\right) \tag{3}$$

Fig. 3. Triangular membership function.

Fig. 4. Trapezoidal membership function.

There are other more complex functions, i.e. the Gauss function made up of a simple Gaussian curve (Fig. 5) which depends on parameters r and c (Eq. (3)); and the Gauss2 function (Fig. 6) given by the fusion of two different Gaussian functions and depending on four parameters: r1 and c1, which define the shape of the function in the left part, and r2 e c2, which define the shape of the function in the right part. Moreover, between these types of functions, there is the bell membership function (Gbell) (Fig. 7) which is a hybrid of the Gaussian function; it is mainly used to manage non-fuzzy sets and depends on three parameters: a, b and c (Eq. (4))

$$f\left(\mathbf{x}; \sigma, \mathbf{c}\right) = e^{\frac{-\left(\mathbf{x} - \mathbf{c}\right)^2}{2\sigma^2}}\tag{3}$$

Fig. 5. Gauss membership function.

68 Fuzzy Logic – Emerging Technologies and Applications

Membership functions can be of several types: the simplest are made up of straight lines, while the most used are the triangular (Fig. 3) and trapezoidal (Fig. 4) functions; the former are characterized by a triangular trend while the latter by a trapezoidal one. The advantage of these functions is in their simplicity. The triangular membership function depends on

while the trapezoidal one depends on four scalar parameters (a, b, c and d), as shown in the

There are other more complex functions, i.e. the Gauss function made up of a simple Gaussian curve (Fig. 5) which depends on parameters r and c (Eq. (3)); and the Gauss2 function (Fig. 6) given by the fusion of two different Gaussian functions and depending on four parameters: r1 and c1, which define the shape of the function in the left part, and r2 e c2, which define the shape of the function in the right part. Moreover, between these types of functions, there is the bell membership function (Gbell) (Fig. 7) which is a hybrid of the Gaussian function; it is mainly used to manage non-fuzzy sets and depends on three

*fx c e*

<sup>2</sup> <sup>2</sup> ; ,

 2

(3)

*x c*

0,,1,minmax,,,; *cd*

0,,minmax,,; *bc*

*ab*

*ab*

>

*xd*

*ax dcbaxf* (2)

*xc*

*ax cbaxf* (1)

three scalar parameters a, b and c and is given by the following expression:

following formula:

Fig. 3. Triangular membership function.

Fig. 4. Trapezoidal membership function.

parameters: a, b and c (Eq. (4))

Fig. 6. Gauss2 membership function.

Fig. 7. Generalized bell (Gbell) membership function.

Despite their simplicity, such functions cannot be used to represent asymmetry, which is important in some applications.

In order to face a possible asymmetry, we can use another type of function, such as the sigmoid function (Fig. 8), which may have left or right asymmetry and a horizontal asymptote. This function is ruled by parameters a and c (Eq. (5))

$$f\left(\mathbf{x}; a, c\right) = \frac{1}{1 + e^{-a\left(\mathbf{x} - c\right)}}\tag{5}$$

In addition to this function, we have further asymmetric functions, the Dsigm and Psigm membership functions represented in Fig. 9 and10 and described by Eq. (6), depending on four parameters a1, c1, a2 and c2.

$$f\left(\mathbf{x}; a, c\right) = \frac{1}{1 + c^{-a\left(\mathbf{x} - c\right)}}\tag{6}$$

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 71

After the definition of the fuzzy data which comes from the fuzzification process, it is necessary to insert in the decisional engine the rules which supply the fuzzy output. The rules are usually made up of an if–then–else structure, which in its turn is made up of an antecedent which defines the conditions, and a consequent which defines the action. For each input variable of the model, in the antecedent we have a clause of the type (x is L) where L is a linguistic label revealing a fuzzy set. In this way, the antecedent supplies a characterization of the condition of the system we want to model, namely its description in quantitative terms. Usually the antecedent includes a conjunction of clauses, one for each observed variable, while the condition of the consequent determines the condition of

In conclusion, a fuzzy system can be considered as a non-linear function which transforms a certain number of input variables into output ones through a set of fuzzy rules. In the application of rules, some of them often lead to the same consequence with different levels of strength: in these cases the common custom is choosing the highest value. Following this phase, which is defined fuzzy inference, it is necessary to turn the data coming from the evaluation of rules into real numerical data: this process is the opposite of input

Defuzzification consists in drawing the output deterministic value from the fuzzy model. A careful analysis of the problem is at the basis of a correct defuzzification: it can be linguistic, when the output is a predicate to which a level of membership is associated, or numerical, of ''crisp" type (non-fuzzy) (used in fuzzy control). Many criteria of defuzzification exist: often in engineering the choice depends on computational simplicity. The most used defuzzification

fuzzification, in fact it is called either output fuzzification or defuzzification.

Fig. 12. Pi membership function.

Fig. 13. S membership function.

**1.1.2 Fuzzy inference**

**1.1.3 Defuzzification**

methods are the following:

outputs.

Fig. 8. Sigmoidale (Sig) membership function.

Fig. 9. Dsig membership function.

Fig. 10. Psig membership function.

Three more membership functions correlated with them are functions Z (Eq. (7)), S and Pi. The first one is an asymmetric function open to the left, the second is open to the right, while the third one is asymmetric but closed at both ends (Fig. 11-13).

Fig. 11. Z membership function.

Fig. 12. Pi membership function.

70 Fuzzy Logic – Emerging Technologies and Applications

2

12 , <sup>2</sup>

*bx ab x b*

Three more membership functions correlated with them are functions Z (Eq. (7)), S and Pi. The first one is an asymmetric function open to the left, the second is open to the right,

while the third one is asymmetric but closed at both ends (Fig. 11-13).

*xa ab a x b a*

(7)

2 , <sup>2</sup>

*b a x b*

*x a*

1,

0,

Fig. 8. Sigmoidale (Sig) membership function.

Fig. 9. Dsig membership function.

Fig. 10. Psig membership function.

Fig. 11. Z membership function.

Fig. 13. S membership function.

#### **1.1.2 Fuzzy inference**

After the definition of the fuzzy data which comes from the fuzzification process, it is necessary to insert in the decisional engine the rules which supply the fuzzy output. The rules are usually made up of an if–then–else structure, which in its turn is made up of an antecedent which defines the conditions, and a consequent which defines the action. For each input variable of the model, in the antecedent we have a clause of the type (x is L) where L is a linguistic label revealing a fuzzy set. In this way, the antecedent supplies a characterization of the condition of the system we want to model, namely its description in quantitative terms. Usually the antecedent includes a conjunction of clauses, one for each observed variable, while the condition of the consequent determines the condition of outputs.

In conclusion, a fuzzy system can be considered as a non-linear function which transforms a certain number of input variables into output ones through a set of fuzzy rules. In the application of rules, some of them often lead to the same consequence with different levels of strength: in these cases the common custom is choosing the highest value. Following this phase, which is defined fuzzy inference, it is necessary to turn the data coming from the evaluation of rules into real numerical data: this process is the opposite of input fuzzification, in fact it is called either output fuzzification or defuzzification.

#### **1.1.3 Defuzzification**

Defuzzification consists in drawing the output deterministic value from the fuzzy model. A careful analysis of the problem is at the basis of a correct defuzzification: it can be linguistic, when the output is a predicate to which a level of membership is associated, or numerical, of ''crisp" type (non-fuzzy) (used in fuzzy control). Many criteria of defuzzification exist: often in engineering the choice depends on computational simplicity. The most used defuzzification methods are the following:

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 73

( ) *out*

*<sup>i</sup> out*

*y*

*y*

Whereas, the formula for discrete function is:

not consider the distribution of membership function.

other functions can be used, such as Dsigm, Psigm and Pi.

**2.1 Architecture of a neuro-fuzzy network**

output.

**2. Fuzzy neural network**

( )

*y dy*

( )

*y y*

( ) *i i*

*y*

*i*

(10)

(11)

*y y dy*

*i*

In *Largest of maximum,* the precise value of the variable output is one of which the fuzzy subset has the maximum truth value. The main disadvantage of this method is that it does

However, the *smallest of maximum method* obtained the minimum value in fuzzy set as

The fuzzy neural network essentially fills the gaps in the fuzzy systems as well as other neurals. The fuzzy inference requires heuristics and does not acquire knowledge from inputoutput relationships as do neural networks. The advantage of a fuzzy neural network compared to a neural structure is that it can be represented by "linguistic rules". The nodes that form a neuro-fuzzy network have weights which do not commonly occur in a system based on a neural network. The network training is done using back-propagation algorithms. The Anfis models (Zimmermann 1991) acquire knowledge from data using algorithms typical of neural networks. This is represented using fuzzy rules. Substantially, neural networks are structured on different levels, starting from the input and output related systems which generate fuzzy rules that guide the process of construction output. As in fuzzy logic, the end result is linked to the fuzzy rules and membership functions. The membership functions can be of various types. The simplest consists of straight lines, while the most used functions are triangular and trapezoidal. There are more complex functions such as the Gauss function which consists of a simple Gaussian curve and function Gauss2 formed by the merger of two different Gaussian functions. In addition, among the functions of this kind, there is a bell membership function (Gbell) which is a hybrid of Gaussian function, and is used primarily to manage non-fuzzy sets. In order to meet any asymmetry,

The scientific literature has various applications of fuzzy neural network from the classical management of Humanoid Robots that will replace humans in dangerous jobs in the medical field or in the field of services (Dusko Katic et al., 2003). The fuzzy neural network was also used in the study of time series of solar activity (Abdel-Fattah Attia et al., 2005), in the assessment of noise in the workplace (Zaheeruddin Garima, 2006) and many others.

The proposed forecasting model for assessing the environmental risk of contamination of aquifers is based on an Adaptive Neural Network Fuzzy Inference System (ANFIS)

Among the methods found in literature, the most common are the centroid and maximum methods.

In the middle of the maximum method, output is obtained as the arithmetic mean of the values of "y" where fuzzy set height is maximum.

B' is the fuzzy set inferred by rules and

$$\text{Mogt}(B') = \left\{ y \, \middle| \, \mu\_{\mathbb{B}^\circ}(y) = \sup\_{y \in B^\circ} \mu\_{\mathbb{B}^\circ}(y) \right\} \tag{8}$$

is the set value of "y" for which height "µB'(y)" is maximum.

Therefore, it has

$$\mathcal{Y}\_{out} = \frac{\int\_{\text{bigt}(\mathcal{B'})} y dy}{\int\_{\text{bigt}(\mathcal{B'})} dy} \tag{9}$$

whose geometric significance is shown in Figure 14:

Fig. 14. Geometric significance of the Middle of maximum defuzzification method.

The output of the *Centroid method* is obtained as the abscissa of the center of gravity inferred from the rules in the space of fuzzy sets of algorithm output. The formula in the case of continuous function is:

$$y\_{out} = \frac{\int y \mu(y) dy}{\int \mu(y) dy} \tag{10}$$

Whereas, the formula for discrete function is:

$$y\_{out} = \frac{\sum\_{i} y\_i \mu(y\_i)}{\sum\_{i} \mu(y\_i)} \tag{11}$$

In *Largest of maximum,* the precise value of the variable output is one of which the fuzzy subset has the maximum truth value. The main disadvantage of this method is that it does not consider the distribution of membership function.

However, the *smallest of maximum method* obtained the minimum value in fuzzy set as output.

#### **2. Fuzzy neural network**

72 Fuzzy Logic – Emerging Technologies and Applications

Centroid method: the chosen numerical value for the output is calculated as the centre

Bisector method: the output is the abscissa of the bisector of the area subtended to the

Middle of maximum method: the output value is determined as the average of

Largest of maximum method: the output numerical value is calculated as the maximum

Smallest of maximum method: the output value is represented by the output minimum

Among the methods found in literature, the most common are the centroid and maximum

In the middle of the maximum method, output is obtained as the arithmetic mean of the

( ') ( ) ( ) *B B*

( ')

*hgt B*

*<sup>y</sup> dy*

*out*

Fig. 14. Geometric significance of the Middle of maximum defuzzification method.

The output of the *Centroid method* is obtained as the abscissa of the center of gravity inferred from the rules in the space of fuzzy sets of algorithm output. The formula in the case of

( ')

*hgt B*

*hgt B y y Sup y*

' ' '

*ydy*

*y B*

(8)

(9)

of mass of the fuzzy set.

maximum values (Mom: middle of maximum).

of the maximum (Lom: Largest of maximum).

value (Som: Smallest of maximum).

B' is the fuzzy set inferred by rules and

values of "y" where fuzzy set height is maximum.

is the set value of "y" for which height "µB'(y)" is maximum.

whose geometric significance is shown in Figure 14:

fuzzy data set.

methods.

Therefore, it has

continuous function is:

The fuzzy neural network essentially fills the gaps in the fuzzy systems as well as other neurals. The fuzzy inference requires heuristics and does not acquire knowledge from inputoutput relationships as do neural networks. The advantage of a fuzzy neural network compared to a neural structure is that it can be represented by "linguistic rules". The nodes that form a neuro-fuzzy network have weights which do not commonly occur in a system based on a neural network. The network training is done using back-propagation algorithms. The Anfis models (Zimmermann 1991) acquire knowledge from data using algorithms typical of neural networks. This is represented using fuzzy rules. Substantially, neural networks are structured on different levels, starting from the input and output related systems which generate fuzzy rules that guide the process of construction output. As in fuzzy logic, the end result is linked to the fuzzy rules and membership functions. The membership functions can be of various types. The simplest consists of straight lines, while the most used functions are triangular and trapezoidal. There are more complex functions such as the Gauss function which consists of a simple Gaussian curve and function Gauss2 formed by the merger of two different Gaussian functions. In addition, among the functions of this kind, there is a bell membership function (Gbell) which is a hybrid of Gaussian function, and is used primarily to manage non-fuzzy sets. In order to meet any asymmetry, other functions can be used, such as Dsigm, Psigm and Pi.

The scientific literature has various applications of fuzzy neural network from the classical management of Humanoid Robots that will replace humans in dangerous jobs in the medical field or in the field of services (Dusko Katic et al., 2003). The fuzzy neural network was also used in the study of time series of solar activity (Abdel-Fattah Attia et al., 2005), in the assessment of noise in the workplace (Zaheeruddin Garima, 2006) and many others.

#### **2.1 Architecture of a neuro-fuzzy network**

The proposed forecasting model for assessing the environmental risk of contamination of aquifers is based on an Adaptive Neural Network Fuzzy Inference System (ANFIS)

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 75

Groundwater intrinsic vulnerability has been assessed by a method of zoning for homogeneous areas: the GNDCI-CNR method (M. Civita, 1990). However, landfills hazard has been determined through the use of fuzzy logic and neuro-fuzzy parameters (input data) by considering morphological, hydrological and environmental parameters for each site, such as: water table depth, leachate production, volume and type of waste, landfill coverage, landfill activity and proximity to river. Some of these were obtained using GIS

**LANDFILL HAZARD GROUNDWATER**

**GROUNDWATER POLLUTION RISK**

For the simple management algorithm, the parameters previously indicated were used to define three different fuzzy inferences, as shown in the conceptual scheme in Figure 17. The results obtained through the first two fuzzy inferences, defined as *site vulnerability* and *landfill potentiality* respectively, were then aggregated to the crisp parameter called *landfill*

As shown in Figure 17, *site vulnerability*, defined through acclivity, depth to water table and watercourse proximity, allowed us to obtain the site's predisposition to suffer from contamination, namely the site's propensity to be contaminated because of a possible leachate seepage. The increase in vulnerability is favoured by low slopes, proximity to surface watercourses (meant as index, so the higher the index the higher the site vulnerability) and reduced depth of the water table. Thus, the values of the three parameters are low when the trend to undergo contamination is high. On the contrary, the *landfill potential* evaluates the potential of a landfill to release contaminants by virtue of the waste volume and the leachate production. Therefore, with the increase of these two factors such potential will increase. The procedure to determine the *landfill hazard index* combines the results obtained by the two previous fuzzy diagrams with the addition of the *landfill conditions*. The array of required training algorithm has been constructed, considering that the increase in values of the three parameters in the subset result in an increase in the hazard of landfills. The end result of the neuro-fuzzy process has been achieved through the training data which provided a numerical value between 0 and 1 representing the hazard index. In addition to training data that facilitates the definition of fuzzy rules, it is necessary to determine for each of the three fuzzy inferences the following: the type of membership function and the classes (very high, high, medium, low, very low). In conclusion, the results relating to site vulnerability, the

**FUZZY LOGIC/NEURO-FUZZY SYSTEM**

Fig. 16. Fuzzy and neuro-fuzzy models for the groundwater pollution risk.

*conditions*, by obtaining the hazard index of each landfill.

landfill potential and the landfill hazard have been obtained.

**SENSITIVITY ANALYSIS**

**VULNERABILITY**

applications.

(Zimmermann 1991; Jang. 1993). ANFIS algorithm allowed us to calibrate membership functions of the fuzzy inference training the Artificial Neural Network. In order to perform the training, the definition of a matrix of input parameters, a single output value and the number of times (numbers interpolating the training matrix) was necessary.

However, ANFIS models acquire knowledge from data using the typical neural networks algorithms but represent it using fuzzy rules.

TThese kind of neural networks are basically structured on five different levels which autonomously generate systems of fuzzy rules that guide the process of construction of the outputs, starting from related inputs and outputs.

Each node of the first level integrates the membership function associated with the represented fuzzy term. Variables Xi are the linguistic variables that are associated with terms placed in the nodes (A11 = Low, A21 = High etc.).

Fig. 15. Architecture of a neuro-fuzzy network.

Nodes of the second level incorporate the antecedents of fuzzy rules. Within these nodes, only an AND logical operation between the active inputs is performed.

In the third level, each node calculates the degree of fulfillment of each rule and returns a weighted term which enters as input in the corresponding node of the next level.

The nodes of this layer incorporate the resulting rules instead. Each node accepts the corresponding weight that comes from the previous level in input, in addition to all the input variables to the first level.

The fifth and last node simply performs the sum of all inputs and returns the final output of the system.

#### **3 Case study**

#### **3.1 Fuzzy and neuro-fuzzy models for groundwater pollution risk assessment**

This study proposes two methods for environmental risk assessment: fuzzy logic and fuzzyneural networks. Fuzzy and neuro-fuzzy models have been used to assess environmental risk in landfills, by using groundwater intrinsic vulnerability of landfill hazard (Fig. 17).

(Zimmermann 1991; Jang. 1993). ANFIS algorithm allowed us to calibrate membership functions of the fuzzy inference training the Artificial Neural Network. In order to perform the training, the definition of a matrix of input parameters, a single output value and the

However, ANFIS models acquire knowledge from data using the typical neural networks

TThese kind of neural networks are basically structured on five different levels which autonomously generate systems of fuzzy rules that guide the process of construction of the

Each node of the first level integrates the membership function associated with the represented fuzzy term. Variables Xi are the linguistic variables that are associated with

Nodes of the second level incorporate the antecedents of fuzzy rules. Within these nodes,

In the third level, each node calculates the degree of fulfillment of each rule and returns a

The nodes of this layer incorporate the resulting rules instead. Each node accepts the corresponding weight that comes from the previous level in input, in addition to all the

The fifth and last node simply performs the sum of all inputs and returns the final output of

This study proposes two methods for environmental risk assessment: fuzzy logic and fuzzyneural networks. Fuzzy and neuro-fuzzy models have been used to assess environmental risk in landfills, by using groundwater intrinsic vulnerability of landfill hazard (Fig. 17).

only an AND logical operation between the active inputs is performed.

weighted term which enters as input in the corresponding node of the next level.

**3.1 Fuzzy and neuro-fuzzy models for groundwater pollution risk assessment**

number of times (numbers interpolating the training matrix) was necessary.

algorithms but represent it using fuzzy rules.

outputs, starting from related inputs and outputs.

Fig. 15. Architecture of a neuro-fuzzy network.

input variables to the first level.

the system.

**3 Case study**

terms placed in the nodes (A11 = Low, A21 = High etc.).

Groundwater intrinsic vulnerability has been assessed by a method of zoning for homogeneous areas: the GNDCI-CNR method (M. Civita, 1990). However, landfills hazard has been determined through the use of fuzzy logic and neuro-fuzzy parameters (input data) by considering morphological, hydrological and environmental parameters for each site, such as: water table depth, leachate production, volume and type of waste, landfill coverage, landfill activity and proximity to river. Some of these were obtained using GIS applications.

Fig. 16. Fuzzy and neuro-fuzzy models for the groundwater pollution risk.

For the simple management algorithm, the parameters previously indicated were used to define three different fuzzy inferences, as shown in the conceptual scheme in Figure 17. The results obtained through the first two fuzzy inferences, defined as *site vulnerability* and *landfill potentiality* respectively, were then aggregated to the crisp parameter called *landfill conditions*, by obtaining the hazard index of each landfill.

As shown in Figure 17, *site vulnerability*, defined through acclivity, depth to water table and watercourse proximity, allowed us to obtain the site's predisposition to suffer from contamination, namely the site's propensity to be contaminated because of a possible leachate seepage. The increase in vulnerability is favoured by low slopes, proximity to surface watercourses (meant as index, so the higher the index the higher the site vulnerability) and reduced depth of the water table. Thus, the values of the three parameters are low when the trend to undergo contamination is high. On the contrary, the *landfill potential* evaluates the potential of a landfill to release contaminants by virtue of the waste volume and the leachate production. Therefore, with the increase of these two factors such potential will increase. The procedure to determine the *landfill hazard index* combines the results obtained by the two previous fuzzy diagrams with the addition of the *landfill conditions*. The array of required training algorithm has been constructed, considering that the increase in values of the three parameters in the subset result in an increase in the hazard of landfills. The end result of the neuro-fuzzy process has been achieved through the training data which provided a numerical value between 0 and 1 representing the hazard index. In addition to training data that facilitates the definition of fuzzy rules, it is necessary to determine for each of the three fuzzy inferences the following: the type of membership function and the classes (very high, high, medium, low, very low). In conclusion, the results relating to site vulnerability, the landfill potential and the landfill hazard have been obtained.

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 77

Fig. 18. Location of the uncontrolled landfills in the Basilicata Region.

Frequency

% cumulated

**Groundwater pollution risk classes (Neuro-fuzzy model)**

Fig. 19. Cumulated frequency curve of groundwater pollution risk for neuro-fuzzy model.

**Frequency**

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Among the three output values for each landfill, our attention was mostly focused on the landfill hazard.

Fig. 17. Conceptual diagram of the implemented fuzzy model.

In addition, to address the risk of subjectivity and to overcome the problem of uncertainty, linked to input data and to the developed models, sensitivity analysis has been used, through which different fuzzy and neuro -fuzzy schemes have been compared. The various fuzzy schemes differ in the type of membership functions and defuzzification methods. Then, each fuzzy scheme is characterized by "if-then" fuzzy rules, membership functions and defuzzification method. The fuzzy rules have been defined considering that groundwater pollution risk rises with the increase of groundwater intrinsic vulnerability and landfill hazard.

However, the neuro-fuzzy schemes differ only in the type of membership functions, while the fuzzy rules are automatically generated by the algorithm using the assigned training matrix. The results obtained from the simulations of both models were compared with input data to identify the best fuzzy and neuro-fuzzy scheme.

The proposed algorithms have been applied to some uncontrolled landfills present in the Basilicata Region, detected through the 2002 census ("Corpo Forestale dello Stato [Forest Rangers]" and "Regional Reclamation Plan"), which identified 469 areas needing reclamation actions, environmental recovery and/or safety measures: 315 in the province of Potenza and 204 in the province of Matera. Among these areas, 290 are illegal landfills (Fig. 18): 122 in the province of Matera and 168 in the province of Potenza.

The comparison of each fuzzy and neuro-fuzzy scheme has been performed by applying statistical tests to the distributions of output data (environmental risk index). The results show that the best scheme for the fuzzy model is characterized by Gauss2 membership functions and Centroid defuzzification method, and the best scheme of neuro-fuzzy model is marked by Gauss membership function. The final results show the environmental risk index and have been recalculated for a classification of groundwater pollution risk in linguistic terms by using a cumulative frequency distribution curve (Fig. 19 and 20).

Among the three output values for each landfill, our attention was mostly focused on the

In addition, to address the risk of subjectivity and to overcome the problem of uncertainty, linked to input data and to the developed models, sensitivity analysis has been used, through which different fuzzy and neuro -fuzzy schemes have been compared. The various fuzzy schemes differ in the type of membership functions and defuzzification methods. Then, each fuzzy scheme is characterized by "if-then" fuzzy rules, membership functions and defuzzification method. The fuzzy rules have been defined considering that groundwater pollution risk rises with the increase of groundwater intrinsic vulnerability

However, the neuro-fuzzy schemes differ only in the type of membership functions, while the fuzzy rules are automatically generated by the algorithm using the assigned training matrix. The results obtained from the simulations of both models were compared with

The proposed algorithms have been applied to some uncontrolled landfills present in the Basilicata Region, detected through the 2002 census ("Corpo Forestale dello Stato [Forest Rangers]" and "Regional Reclamation Plan"), which identified 469 areas needing reclamation actions, environmental recovery and/or safety measures: 315 in the province of Potenza and 204 in the province of Matera. Among these areas, 290 are illegal landfills (Fig.

The comparison of each fuzzy and neuro-fuzzy scheme has been performed by applying statistical tests to the distributions of output data (environmental risk index). The results show that the best scheme for the fuzzy model is characterized by Gauss2 membership functions and Centroid defuzzification method, and the best scheme of neuro-fuzzy model is marked by Gauss membership function. The final results show the environmental risk index and have been recalculated for a classification of groundwater pollution risk in

linguistic terms by using a cumulative frequency distribution curve (Fig. 19 and 20).

Fig. 17. Conceptual diagram of the implemented fuzzy model.

input data to identify the best fuzzy and neuro-fuzzy scheme.

18): 122 in the province of Matera and 168 in the province of Potenza.

landfill hazard.

and landfill hazard.

Fig. 18. Location of the uncontrolled landfills in the Basilicata Region.

Fig. 19. Cumulated frequency curve of groundwater pollution risk for neuro-fuzzy model.

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 79

Fuzzy model Neuro-fuzzy model

Fig. 21. Distributions of aquifers in classes of risk for fuzzy and neuro-fuzzy models.

We have evaluated the risk indices obtained from the two models for a more appropriate comparison. The assessment of the risk index without the subdivision into classes has demonstrated that the performance of the two distributions is very similar (Fig. 22) as

0 20 40 60 80 100 120

Fig. 22. Variation of the environmental risk index for each site.

Neuro-fuzzy model Fuzzy model

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

confirmed by the box-plots (Fig. 23).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

#### **3.2 Comparison between environmental risk results obtained from the fuzzy and neuro-fuzzy models**

The two designed models applied to some aquifers in Basilicata region have provided different results. For this reason, we next performed a statistical comparison. Visual analysis of histograms representing the percentage of aquifers falling in different classes of risk (fuzzy and neuro-fuzzy models) shows that distributions give different classes of risk (Fig. 21).

**Groundwater pollution risk classes (Fuzzy model)**

Fig. 20. Cumulated frequency curve of groundwater pollution risk for fuzzy model.

**3.2 Comparison between environmental risk results obtained from the fuzzy and**

Fuzzy model Neuro-fuzzy model

The two designed models applied to some aquifers in Basilicata region have provided different results. For this reason, we next performed a statistical comparison. Visual analysis of histograms representing the percentage of aquifers falling in different classes of risk (fuzzy and neuro-fuzzy models) shows that distributions give different classes of risk (Fig. 21).

0

**neuro-fuzzy models**

0%

10%

20%

30%

40%

50%

60%

70%

5

10

15

20

**Frequency**

25

30

35

40

Frequency % cumulated 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Very\_low Low Medium High Very\_high

Fig. 21. Distributions of aquifers in classes of risk for fuzzy and neuro-fuzzy models.

We have evaluated the risk indices obtained from the two models for a more appropriate comparison. The assessment of the risk index without the subdivision into classes has demonstrated that the performance of the two distributions is very similar (Fig. 22) as confirmed by the box-plots (Fig. 23).

Fig. 22. Variation of the environmental risk index for each site.

Fuzzy Logic and Neuro-Fuzzy Networks for Environmental Hazard Assessment 81

The fuzzy and neuro-fuzzy approaches used for the realization of models of environmental risk assessment have been fast, effective and affordable methods and at the same time useful support for decisions. The neuro-fuzzy model is faster in the application because by using training data, it is able to generate the fuzzy rules, which are particularly complex and

In addition, integration of sensitivity analysis in the two models is a positive element because it is able to mitigate the problems of subjectivity and arbitrariness of the evaluation based on fuzzy approaches commonly found in literature, in particular with regards to the choice of membership functions. The case study proposed, in fact, shows that by varying the choice of the membership functions very different results if not contradictory can be

In conclusion, a model can be substituted with the exception of the neuro-fuzzy model that is rapidly applicable in case you have data available for training a neuro-fuzzy network.

The fuzzy method is advantageous because it allows a rapid and efficient risk assessment and is an inexpensive and expeditious planning tool for a program of remediation. However, it is disadvantageous due to the complexity in the definition of fuzzy rules especially when the number of parameters is high. In fact, in many situations when the number of parameters of the analysis is high, the application of these techniques is cumbersome and complex; in these cases, neuro-fuzzy models, that reduce the complexity of the models thanks to the training data, could be used. Using the adaptive methods of fuzzy inference neural networks, you can easily manage fuzzy rules of the analysis and reduce the

The use of this kind of analysis with respect to neural models does not provide very different results, as assessed and analyzed even by Vieira et al. in 2004, but only a different

Abdel-Fattah, Attia, Rabab, Abdel-Hamid, & Maha, Quassim (2005). Prediction of solar

Caniani, D., Lioi, D.S., Mancini, I.M., & Masi, S. (2011). Application of fuzzy logic and

Civita, M.; (1990) Legenda unicata per le Carte della vulnerabilità dei corpi idrici

Skarlatos, Dimitris, Karakasis, Kleomenis, & Trochidis, Athanassios, (2004)Railway wheel fault diagnosis using a fuzzy-logic method. *Applied Acoustics*, 65 (2004) 951–966. Duško, Katic, & Miomir, Vukobratovic (2003). Survey of Intelligent Control Techniques for Humanoid Robots. *Journal of Intelligent and Robotic Systems*, 37-117-141. Hitoshi, Iyatomi, & Masafumi, Hagiwara (2004). Adaptive fuzzy inference neural network.

sensitivity analysis for soil contamination hazard classification. *Waste Management*

sotterranei/ Unified legend for the groundwater pollution vulnerability Maps.

activity based on neuro-fuzzy modeling, *Solar Physics* 227: 177–191.

*Studi sulla Vulnerabilità degli Acquiferi*, 1 Pitagora, Bologna

increases along with the number of parameters assigned to the model.

artifices of fuzzy and neural models (Iyatomi et al., 2004).

training time influenced by the order of the model.

*Pattern Recognition* 37 2049-2057.

**4. Conclusion**

obtained.

**5. References**

31 583–594.

Fig. 23. Box-plots of the distributions of output data for fuzzy and neuro-fuzzy models.

Fig. 24. Scatter plot between the environmental risk index obtained from the fuzzy and neuro-fuzzy models.

In fact, even the scatter plot (Fig. 24) shows a good correlation as evaluated by the coefficient of determination R2,which is the square of the correlation coefficient R = 0.8832. Moreover, similarity between results of the two models can also be inferred from the comparison between variances and standard distributions and F test (Table 1).

Table 1. Statistical indices and test F results.

#### **4. Conclusion**

80 Fuzzy Logic – Emerging Technologies and Applications

Fuzzy model Neuro-fuzzy model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

In fact, even the scatter plot (Fig. 24) shows a good correlation as evaluated by the coefficient of determination R2,which is the square of the correlation coefficient R = 0.8832. Moreover, similarity between results of the two models can also be inferred from the comparison

Environmental risk index

(Fuzzy model)

Fig. 24. Scatter plot between the environmental risk index obtained from the fuzzy and

between variances and standard distributions and F test (Table 1).

Standard deviation 0.136 0.133 Variance 0.018 0.018 R² = 0.780

Environmental risk index (Neuro-fuzzy model)

Fig. 23. Box-plots of the distributions of output data for fuzzy and neuro-fuzzy models.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

neuro-fuzzy models.

F test 0.870

Table 1. Statistical indices and test F results.

The fuzzy and neuro-fuzzy approaches used for the realization of models of environmental risk assessment have been fast, effective and affordable methods and at the same time useful support for decisions. The neuro-fuzzy model is faster in the application because by using training data, it is able to generate the fuzzy rules, which are particularly complex and increases along with the number of parameters assigned to the model.

In addition, integration of sensitivity analysis in the two models is a positive element because it is able to mitigate the problems of subjectivity and arbitrariness of the evaluation based on fuzzy approaches commonly found in literature, in particular with regards to the choice of membership functions. The case study proposed, in fact, shows that by varying the choice of the membership functions very different results if not contradictory can be obtained.

In conclusion, a model can be substituted with the exception of the neuro-fuzzy model that is rapidly applicable in case you have data available for training a neuro-fuzzy network.

The fuzzy method is advantageous because it allows a rapid and efficient risk assessment and is an inexpensive and expeditious planning tool for a program of remediation. However, it is disadvantageous due to the complexity in the definition of fuzzy rules especially when the number of parameters is high. In fact, in many situations when the number of parameters of the analysis is high, the application of these techniques is cumbersome and complex; in these cases, neuro-fuzzy models, that reduce the complexity of the models thanks to the training data, could be used. Using the adaptive methods of fuzzy inference neural networks, you can easily manage fuzzy rules of the analysis and reduce the artifices of fuzzy and neural models (Iyatomi et al., 2004).

The use of this kind of analysis with respect to neural models does not provide very different results, as assessed and analyzed even by Vieira et al. in 2004, but only a different training time influenced by the order of the model.

#### **5. References**

**5**

Yetis Sazi Murat

 *Turkey*

 *Civil Engineering Department, Transportation Division, Denizli,*

Number of Dead people (per 100 million veh-km)

**Fuzzy Clustering Approach**

*Pamukkale University, Faculty of Engineering,*

**for Accident Black Spot Centers Determination**

Traffic accident rates of Turkey are higher than most of the European Union countries and other countries in the world (Table 1). Every year, almost more than 8000 people die by traffic accidents in Turkey. This figure is very high comparing to many countries at same size. There have been many projects conducted by national or international organizations for decreasing these rates. In order to develop sustainable prevention models, accidents

Traffic accident data can be analyzed in different ways, based on amount and types of data. The analysis is not complicated if the data are smooth and not dispersed. But it is not an easy task if the data are scattered. Although there is not a general definition for black spots, locations where at least more than one accident occured are treated as black spots. Based on this definition, the number of black spots can be increased and analysis of them is getting

should be analyzed in detail considering primary and secondary reasons.

100 million veh-km)

USA 58 0.3 England 60 1 Germany 351 1 Japan 111 1.4 France 27 1.9 Türkiye 108 7.5

Several methods can be used for determination of black spots and centers. It can be determined by eye using simple observations. But this simple approach can include subjective perceptions and also results obtained can not be sensitive and scientific. Besides to locations, other specifications of black spots should be taken into consideration for a scientific analysis. Developing countermeasures and classifying by characteristics for black spots that are intensified and covered whole area on the map is not an easy task. Although

Country Number of İnjured people (per

Table 1. Traffic Accident rates (Murat and Sekerler, 2009)

**1. Introduction**

more difficult.

### **Fuzzy Clustering Approach for Accident Black Spot Centers Determination**

#### Yetis Sazi Murat

*Pamukkale University, Faculty of Engineering, Civil Engineering Department, Transportation Division, Denizli, Turkey*

#### **1. Introduction**

82 Fuzzy Logic – Emerging Technologies and Applications

Mei-qin, LIU, Sen-lin, Zhang, & Gang-feng, YAN, (2008). A new neural network model for

Miha, Mraz, (2001) The design of intelligent control of a kitchen refrigerator. *Mathematics and*

Saad, R., & Halgamuge, S. K., (2004). Stability of hierarchical fuzzy systems generated by

Silvert, William, (2000) Fuzzy indices of environmental conditions. *Ecological Modelling* 130,

Van der Wal, A.J., (1995), Application of fuzzy logic control in industry. *Fuzzy Sets and*

Vieira, Josè, Fernando, Morgado Dias, & Alexandre, Mota (2004). Artificial neural networks

Zaheeruddi, Garima(2006). A neuro-fuzzy approach for prediction of human work

Zimmermann, H.-J. (1991). Fuzzy set theory and its applications, Dordrecht: Kluwer, second

study, *Engineering Applications of Artificial Intelligence*, 17 265–273.

efficiency in noisy environment. *Applied Soft Computing* 6 283-294.

Zadeh, L.A.; ( 1965) Information and control; *Fuzzy sets* 338-353 (1965).

and neuro-fuzzy systems for modelling and controlling real systems: a comparative

Saltelli, A., Chan, K., & Scott, E.M., (2000) Sensitivity analysis, Wiley & Sons

9(8):1015-1023.

111–119.

edition.

*Systems*, 74 33-41(1995).

*Computers in Simulation*, 56 259–267

Neuro-Fuzzy, *Soft Computing* 8 409–416.

the feedback stabilization of nonlinear systems, Liu et al. / J Zhejiang Univ Sci A

Traffic accident rates of Turkey are higher than most of the European Union countries and other countries in the world (Table 1). Every year, almost more than 8000 people die by traffic accidents in Turkey. This figure is very high comparing to many countries at same size. There have been many projects conducted by national or international organizations for decreasing these rates. In order to develop sustainable prevention models, accidents should be analyzed in detail considering primary and secondary reasons.

Traffic accident data can be analyzed in different ways, based on amount and types of data. The analysis is not complicated if the data are smooth and not dispersed. But it is not an easy task if the data are scattered. Although there is not a general definition for black spots, locations where at least more than one accident occured are treated as black spots. Based on this definition, the number of black spots can be increased and analysis of them is getting more difficult.

Table 1. Traffic Accident rates (Murat and Sekerler, 2009)

Several methods can be used for determination of black spots and centers. It can be determined by eye using simple observations. But this simple approach can include subjective perceptions and also results obtained can not be sensitive and scientific. Besides to locations, other specifications of black spots should be taken into consideration for a scientific analysis. Developing countermeasures and classifying by characteristics for black spots that are intensified and covered whole area on the map is not an easy task. Although

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 85

collision risks of motor vehicles and bicycles is developed by Wang and Nihan (2004). They classified the accidents considering movements of traffic flows such as through, right turning, and left turning. A probability based method is used in the research. Three negative binomial regression models are improved in the study. The study showed that Negative Binomial regression approach can be used instead of Poisson regression approach. Abdel-Aty and Pange (2007) investigated crash data in two levels (i.e.collective and individual level). They focused real time estimation of crash likelihood and discussed advantages and disadvantages of the analysis in two levels. Saploğlu and Karaşahin (2006) examined traffic accidents of Isparta, Turkey using Geographical Information Systems. They determined black spots and found that there is an increase in number of black spots by the years. They also emphasized that most of the accidents are occurred in junctions. Another remarkable study has been accomplished in Singapore. Kamalasudhan et al. (2000) obtained accident density map using digital accident data. They searched black spots or hot spots using the accident density map. The types of accidents by the days, hours, pavement conditions and

Individual analysis of traffic accidents has been taken into consideration in most of the studies given in literature. Besides, analysis of density (i.e. densely recorded area) and determination of black spots are also very important for traffic safety researches. On the other hand, determination of black spots and their center is not an easy task and need to be made many trials. To handle this problem, cluster analysis approach is used in the research. The main objectives of this paper are determination of black spots' center by k-means and

In recent years, cluster analysis has been widely used in the engineering application such as civil engineering, target recognition, medical diagnosis etc. Cluster analysis is an unsupervised method for classifying data, i.e. to divide a given data into a set of classes or

K-means clustering approach is one of the popular methods used in industrial and scientific areas. Euclidian distance is used in k-means clustering algorithm. In this analysis, the desired number of clusters should be determined in the beginning. The following objective

1 1

*j i J xc*

*k n*

<sup>2</sup> ( )

(1)

*<sup>i</sup> x* and the corresponding center ( *<sup>j</sup> c* ).

*j i j*

fuzzy clustering approaches and analysis of these points to reveal main reasons.

vehicle types are analyzed in the study.

**3. Cluster analysis approaches**

**3.1 Conventional (K-Means) clustering**

function is tried to be minimized in this approach.

*<sup>i</sup> <sup>j</sup> x c* is the distance between the data ( )*<sup>j</sup>*

The following steps are used in k-means clustering approach.

clusters.

where,

<sup>2</sup> ( )*<sup>j</sup>*

J is total distance.

some black spots can have common characteristics, they can be located far away from each other. On the other hand, characteristics of black spots that are closely located to each other can be different. Therefore definition and analysis of black spots include uncertainties and conventional approaches can not be used for this purpose. In this study, cluster analysis approach is used for determination of black spots center and definition of the centers. Two types of analysis as k-means and fuzzy c-means are used. The (hard) k-means clustering approach is used as conventional method. In k-means clustering, the boundaries of clusters are determined as crisp. Thus some black spots that belong to a cluster based on their characteristics can be defined in different cluster by k-means clustering approach. But it can be defined in both clusters. To remove this deficiency, fuzzy c-means clustering approach is used. Fuzzy c-means are used for representing uncertainties in belonging to clusters. Thus some black spots are treated as members of two centers with two membership (belonging) values. In addition to determination of black spots centers, associative factors about the black spots are analyzed and discussed in the research.

In this study, first the black spots are determined using the data provided from Local Police Department, after that, the centers where black spots are intensified are revealed. These centers and the black spots around are considered in detail, the reasons of common results and findings are searched.

#### **2. Literature review**

There have been numerous studies about traffic safety. In this study, only GIS based studies and some important researches that include cluster analysis are taken into account and summarized.

Cluster analysis has been used in modeling traffic accident data by many researchers. Wong et al.(2004) used cluster analysis to develop a qualitative assessment methodology. Yannis et al. (2007) proposed a multilevel negative binomial modeling approach for the regional effect of enforcement on road accidents at Greece using cluster analysis. They made geographical and mathematical cluster analysis and reported that alcohol enforcement is the most significant one among the various types of enforcement.

Abdel-Aty and Radwan (2000), used Negative Binomial Distribution for modeling traffic accident occurrence and involvement. The models indicated that young and older drivers experience more accidents than middle aged drivers in heavy traffic volume, and reduced shoulder and median widths. They also obtained that heavy traffic volume, speeding, narrow lane width, larger number of lanes, urban roadway sections, narrow shoulder width and reduced median width increase the likelihood for accident involvement. Ng et al. (2002) aimed at developing an algorithm to estimate the number of traffic accidents and assess the risk of traffic accidents in Hong Kong. They presented an algorithm that involves a combination of mapping technique (Geographical Information System (GIS) techniques) and statistical methods. The results showed that the algorithm improves accident risk estimation when comparing to the estimated risk based on only the historical accident records. Abdel-Aty (2003) analyzed driver injury severity using ordered probit modeling approach. The models consider showed the significance of driver's age, gender, and seat belt use, point of impact, speed, and vehicle type on the injury severity level. An estimation model for

some black spots can have common characteristics, they can be located far away from each other. On the other hand, characteristics of black spots that are closely located to each other can be different. Therefore definition and analysis of black spots include uncertainties and conventional approaches can not be used for this purpose. In this study, cluster analysis approach is used for determination of black spots center and definition of the centers. Two types of analysis as k-means and fuzzy c-means are used. The (hard) k-means clustering approach is used as conventional method. In k-means clustering, the boundaries of clusters are determined as crisp. Thus some black spots that belong to a cluster based on their characteristics can be defined in different cluster by k-means clustering approach. But it can be defined in both clusters. To remove this deficiency, fuzzy c-means clustering approach is used. Fuzzy c-means are used for representing uncertainties in belonging to clusters. Thus some black spots are treated as members of two centers with two membership (belonging) values. In addition to determination of black spots centers, associative factors about the

In this study, first the black spots are determined using the data provided from Local Police Department, after that, the centers where black spots are intensified are revealed. These centers and the black spots around are considered in detail, the reasons of common results

There have been numerous studies about traffic safety. In this study, only GIS based studies and some important researches that include cluster analysis are taken into account and

Cluster analysis has been used in modeling traffic accident data by many researchers. Wong et al.(2004) used cluster analysis to develop a qualitative assessment methodology. Yannis et al. (2007) proposed a multilevel negative binomial modeling approach for the regional effect of enforcement on road accidents at Greece using cluster analysis. They made geographical and mathematical cluster analysis and reported that alcohol enforcement is the most

Abdel-Aty and Radwan (2000), used Negative Binomial Distribution for modeling traffic accident occurrence and involvement. The models indicated that young and older drivers experience more accidents than middle aged drivers in heavy traffic volume, and reduced shoulder and median widths. They also obtained that heavy traffic volume, speeding, narrow lane width, larger number of lanes, urban roadway sections, narrow shoulder width and reduced median width increase the likelihood for accident involvement. Ng et al. (2002) aimed at developing an algorithm to estimate the number of traffic accidents and assess the risk of traffic accidents in Hong Kong. They presented an algorithm that involves a combination of mapping technique (Geographical Information System (GIS) techniques) and statistical methods. The results showed that the algorithm improves accident risk estimation when comparing to the estimated risk based on only the historical accident records. Abdel-Aty (2003) analyzed driver injury severity using ordered probit modeling approach. The models consider showed the significance of driver's age, gender, and seat belt use, point of impact, speed, and vehicle type on the injury severity level. An estimation model for

black spots are analyzed and discussed in the research.

significant one among the various types of enforcement.

and findings are searched.

**2. Literature review**

summarized.

collision risks of motor vehicles and bicycles is developed by Wang and Nihan (2004). They classified the accidents considering movements of traffic flows such as through, right turning, and left turning. A probability based method is used in the research. Three negative binomial regression models are improved in the study. The study showed that Negative Binomial regression approach can be used instead of Poisson regression approach. Abdel-Aty and Pange (2007) investigated crash data in two levels (i.e.collective and individual level). They focused real time estimation of crash likelihood and discussed advantages and disadvantages of the analysis in two levels. Saploğlu and Karaşahin (2006) examined traffic accidents of Isparta, Turkey using Geographical Information Systems. They determined black spots and found that there is an increase in number of black spots by the years. They also emphasized that most of the accidents are occurred in junctions. Another remarkable study has been accomplished in Singapore. Kamalasudhan et al. (2000) obtained accident density map using digital accident data. They searched black spots or hot spots using the accident density map. The types of accidents by the days, hours, pavement conditions and vehicle types are analyzed in the study.

Individual analysis of traffic accidents has been taken into consideration in most of the studies given in literature. Besides, analysis of density (i.e. densely recorded area) and determination of black spots are also very important for traffic safety researches. On the other hand, determination of black spots and their center is not an easy task and need to be made many trials. To handle this problem, cluster analysis approach is used in the research. The main objectives of this paper are determination of black spots' center by k-means and fuzzy clustering approaches and analysis of these points to reveal main reasons.

#### **3. Cluster analysis approaches**

In recent years, cluster analysis has been widely used in the engineering application such as civil engineering, target recognition, medical diagnosis etc. Cluster analysis is an unsupervised method for classifying data, i.e. to divide a given data into a set of classes or clusters.

#### **3.1 Conventional (K-Means) clustering**

K-means clustering approach is one of the popular methods used in industrial and scientific areas. Euclidian distance is used in k-means clustering algorithm. In this analysis, the desired number of clusters should be determined in the beginning. The following objective function is tried to be minimized in this approach.

$$J = \sum\_{j=1}^{k} \sum\_{i=1}^{n} \left\| \mathbf{x}\_i^{(j)} - \mathbf{c}\_j \right\|^2 \tag{1}$$

where, <sup>2</sup> ( )*<sup>j</sup> <sup>i</sup> <sup>j</sup> x c* is the distance between the data ( )*<sup>j</sup> <sup>i</sup> x* and the corresponding center ( *<sup>j</sup> c* ).

J is total distance.

The following steps are used in k-means clustering approach.

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 87

function of feature vector, *m*[1 ∞] is weight exponent for each fuzzy membership and it determines the fuzziness of the clusters and controls the extent of membership shared among the fuzzy clusters. *U*, which is given in equation (7), is the fuzzy partition matrix which contains the membership of each feature vector in each fuzzy cluster. It should be

(,)(,) (,) *<sup>T</sup>*

11 1 1

*uu u*

*i ik iN*

*uu u*

i. Initialize fuzzy partition matrix U or Fuzzy cluster centroid matrix V using a random

ii. If the FCM algorithm is initialized with fuzzy partition matrix, the initial memberships

*u for i c k N*

iii. If the FCM algorithm is initialized with fuzzy cluster centroid matrix containing the fuzzy cluster centroid, memberships belonging to cluster is determined using equation

> 1 1

<sup>1</sup> ( ) (,) 1 ,1 <sup>1</sup> ( ) (,)

*dxv u for <sup>i</sup> <sup>c</sup> <sup>k</sup> <sup>N</sup>*

The steps (iii) and (iv) are repeated until the change in the value of memberships between

The main problem in fuzzy clustering is that the number of clusters (*c*) must be specified beforehand. Selections of a different number of initial clusters result in different clustering

( ) ( )

*u*

*u x*

*N m ik k k <sup>i</sup> <sup>N</sup> <sup>m</sup> ik k*

1 ,1

(8)

(9)

(10)

.....

*k N*

.....

.....

*c ck cN cxN*

1

*U uu u*

The procedure of FCM based on iterative optimization (Bezdek, 1981) can be given as;

..

.

1

*ki ki iki d x v x v Ax v* (6)

(7)

noted that, sum of membership values for a cluster must be equal to 1.

2

belonging to cluster is adjusted using equation (8).

iv. *<sup>i</sup> v* fuzzy centroid is computed by equation (9),

2

2 1

two iterations is sufficiently small level.

*dxv*

1

*u*

*v*

1/( 1)

*m*

1/( 1)

v. The fuzzy membership ( *uik* ) is updated by equation (10),

*k i ik <sup>c</sup> <sup>m</sup> i k i*

*u*

*initial ik ik <sup>c</sup> initial ik i*

number generator.

(9).

**3.1 Validation**

One of the main disadvantages of this approach is determination of number of clusters in the beginning. Another disadvantage is sensitivity of the algorithm to the outliers.

#### **3.2 Fuzzy C-Means clustering**

Fuzzy C-Means (FCM) clustering algorithm has been widely used and applied in different areas. The description of the original fuzzy clustering algorithm based on objective function dates back to 1973 (Bezdek, 1973; Dunn, 1974). This algorithm was conceived in 1973 by Dunn (1974) and further generalized by Bezdek (1973). Among the existing fuzzy clustering methods, the Fuzzy c-means (FCM) algorithm proposed by Bezdek (1981) is the simplest and is the most popular technique of clustering. It is an extension of the hard K-means algorithm to fuzzy framework. Grubesic (2006) explored the use of a generalized partitioning method known as fuzzy clustering for crime hot-spot detection.

FCM algorithm is extension of Hard K-means with advantage of fuzzy set theory and contrary to the K-means method the FCM is more flexible because it shows those objects that have some interface with more than one cluster in the partition. In traditional clustering algorithms such as Hard K-Means, an element belongs fully to a cluster or not (i.e. 0 or 1). On the other hand, in Fuzzy clustering, each element can belongs to several clusters with different membership degrees. The main goal of any clustering algorithm is to determine appropriate the partition matrix *U(X)* of a given data set *X* consisting of patterns ( *X xx x* 1 2 , ........ *<sup>N</sup>* ) and to find the appropriate number of clusters. The objective function and constraints can be defined as;

Objective function

$$f(X; \mathcal{U}, V) = \sum\_{i=1}^{c} \sum\_{k=1}^{N} (\mu\_{ik})^m d^2(x\_{k'} v\_i) \tag{2}$$

$$V = \left[v\_1, v\_2, \dots, v\_c\right], \ v\_i \in \mathbb{R}^n \tag{3}$$

Constraints

$$\sum\_{i=1}^{c} \mu\_{ik} = 1 \quad \forall \, k \in \{1, \dots, N\} \tag{4}$$

$$0 \prec \sum\_{i=1}^{c} \mu\_{ik} \prec N \quad \forall \ i \in \{1, \ldots, c\} \tag{5}$$

Where, *c* is the number of cluster, *<sup>i</sup> v* is the centroid, *d* is the Euclidian distance between rescaled feature vector and centroid of cluster, *uik* [0,1] denotes the degree of membership

One of the main disadvantages of this approach is determination of number of clusters in

Fuzzy C-Means (FCM) clustering algorithm has been widely used and applied in different areas. The description of the original fuzzy clustering algorithm based on objective function dates back to 1973 (Bezdek, 1973; Dunn, 1974). This algorithm was conceived in 1973 by Dunn (1974) and further generalized by Bezdek (1973). Among the existing fuzzy clustering methods, the Fuzzy c-means (FCM) algorithm proposed by Bezdek (1981) is the simplest and is the most popular technique of clustering. It is an extension of the hard K-means algorithm to fuzzy framework. Grubesic (2006) explored the use of a generalized

FCM algorithm is extension of Hard K-means with advantage of fuzzy set theory and contrary to the K-means method the FCM is more flexible because it shows those objects that have some interface with more than one cluster in the partition. In traditional clustering algorithms such as Hard K-Means, an element belongs fully to a cluster or not (i.e. 0 or 1). On the other hand, in Fuzzy clustering, each element can belongs to several clusters with different membership degrees. The main goal of any clustering algorithm is to determine appropriate the partition matrix *U(X)* of a given data set *X* consisting of patterns ( *X xx x* 1 2 , ........ *<sup>N</sup>* ) and to find the appropriate number of clusters. The objective function

> 1 1 (;,) ( ) ( ,) *c N <sup>m</sup>*

1 1,.......

*u Ni c*

*ukN*

0 1,.......

Where, *c* is the number of cluster, *<sup>i</sup> v* is the centroid, *d* is the Euclidian distance between rescaled feature vector and centroid of cluster, *uik* [0,1] denotes the degree of membership

*i k*

*JXUV*

1

1

*c ik i*

*c ik i*

2

*dxv*

*ik k i*

(2)

1 2 , ,..... , *<sup>n</sup> V vv v v R c i* (3)

(4)

(5)

the beginning. Another disadvantage is sensitivity of the algorithm to the outliers.

partitioning method known as fuzzy clustering for crime hot-spot detection.

Select the number of cluster centers (k)

**3.2 Fuzzy C-Means clustering**

and constraints can be defined as;

Objective function

Constraints

Assign each object to the nearest cluster center group

Repeat 2nd and 3rd steps till the cluster centers are fixed

After assigning all objects, recalculate locations of cluster center

function of feature vector, *m*[1 ∞] is weight exponent for each fuzzy membership and it determines the fuzziness of the clusters and controls the extent of membership shared among the fuzzy clusters. *U*, which is given in equation (7), is the fuzzy partition matrix which contains the membership of each feature vector in each fuzzy cluster. It should be noted that, sum of membership values for a cluster must be equal to 1.

$$d^2(\mathbf{x}\_k, \upsilon\_i) = (\mathbf{x}\_k, \upsilon\_i)^T A\_i(\mathbf{x}\_k, \upsilon\_i) \tag{6}$$

$$\mathbf{U} = \begin{bmatrix} u\_{11} & u\_{1k} & \dots & u\_{1N} \\ \vdots & & \\ u\_{i1} & u\_{ik} & \dots & u\_{iN} \\ \vdots & & \\ u\_{c1} & u\_{ck} & \dots & u\_{cN} \end{bmatrix}\_{\text{ccN}} \tag{7}$$

The procedure of FCM based on iterative optimization (Bezdek, 1981) can be given as;

$$\mu\_{ik} = \frac{\mathbf{u}\_{ik}^{initial}}{\sum\_{i=1}^{c} \mathbf{u}\_{ik}^{initial}} \qquad \text{for } 1 \le i \le c, \ 1 \le k \le N \tag{8}$$

$$\upsilon\_i = \frac{\sum\_{k=1}^{N} (\mu\_{ik})^m x\_k}{\sum\_{k=1}^{N} (\mu\_{ik})^m} \tag{9}$$

v. The fuzzy membership ( *uik* ) is updated by equation (10),

$$u\_{ik} = \frac{(\frac{1}{d^2(\mathbf{x}\_{k'}, \upsilon\_i)})^{1/(m-1)}}{\sum\_{i=1}^c (\frac{1}{d^2(\mathbf{x}\_{k'}, \upsilon\_i)})^{1/(m-1)}} \qquad \text{for } 1 \le i \le c, \ 1 \le k \le N \tag{10}$$

The steps (iii) and (iv) are repeated until the change in the value of memberships between two iterations is sufficiently small level.

#### **3.1 Validation**

The main problem in fuzzy clustering is that the number of clusters (*c*) must be specified beforehand. Selections of a different number of initial clusters result in different clustering

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 89

min ( , ) ( ) min min , max max ( , )

In this study, Denizli that is a medium sized city (current population is about 700000) of Turkey is considered. Traffic accident records for the years of 2004, 2005 and 2006 are used in analyzing accidents. The accident reports are provided by Local Police Department. All of the data and documents are taken from an ongoing research project. Following information

*dxy DI c i k*

Road and environmental conditions (signal, pavement, policeman, obstacle etc.)

Crossings (school crossing, pedestrian crossing, railroad crossing)

are analyzed using k-means and fuzzy clustering approaches.

 Information about vehicles (type, model, damage condition, speed etc) Information about drivers (age, sex, alcohol, usage of safety belt etc )

Information about passengers and pedestrians (age, sex, alcohol, usage of safety belt etc )

All of these data given above are recorded using first MS Excel. Then, coordinates of each accident point are determined using street definition system in MAPINFO software. The data from Excel data base including coordinates of the accident points are transferred to MAPINFO data base. This data base is constituted for accident analysis by inquiring different attribution of each accident. Thus, traffic accident can be evaluated from different points of view and the relations about reasons and results of accidents can be revealed by

The data related to coordinates of accidents (locations) are used in cluster analysis. The Figure 1 shows sample processed data on Denizli city GIS map (Murat et al, 2008). The data

*ic kc*

**4. Study area and available data**

Location of accident (coordinates)

Accident type for occurrence

 Problems based on roadway Presence of warning sign Vertical route conditions Horizontal route conditions Intersection conditions

Vehicle insurance conditions

these analysis.

Road direction type (one or two way)

Accident type for participating vehicle number

Pavement surface conditions (dry, wet, icy etc.)

Other factors (narrow road, bridge, tunnel)

are collected from the reports:

Type of accident

Weather condition

Type of pavement

 Date Time  , ,

*dxy*

(16)

*<sup>i</sup> x C y Ck*

*k C xy C*

partitions. Therefore, it is necessary to validate each of fuzzy partitions after the cluster analysis. Cluster validity refers to the problem whether a given fuzzy partition fits to the data all. The clustering algorithm always tries to find the best fit for a fixed number of clusters and the parameterized cluster shapes. However this does not mean that even the best fit is meaningful at all. Either the number of clusters might be wrong or the cluster shapes might not correspond to the groups in the data, if the data can be grouped in a meaningful way at all. In this study, several clustering indexes were used and tested for different values of both cluster number (*c*) and to examine their adequacy in analyzing of traffic accidents. These indexes are *Partition Coefficient (PC), Classification Entropy (CE), Partition Index (SC), Separation Index (S), Xie and Beni's Index (XB) and Dunn's Index (DI)*.

*Partition Coefficient (PC)* measures the amount of "overlapping" between two Fuzzy clusters (Bezdek, 1981). The disadvantage of this index is lack of direct connection to properties of the data. The optimal number of cluster is at the maximum value and the range of this index is [1/c, 1].

$$PC(c) = \frac{1}{N} \sum\_{i=1}^{c} \sum\_{k=1}^{N} \left( u\_{ik} \right)^2 \tag{11}$$

*Classification Entropy (CE)* measures the fuzziness of the cluster partition. The range of CE is [0, log *a c*( ) ] and optimal number of cluster is at minimum value.

$$CE(c) = -\frac{1}{N} \sum\_{i=1}^{c} \sum\_{k=1}^{N} \mu\_{ik} \log\_{a} a u\_{ik} \tag{12}$$

*Partition Index (SC)* is the ratio of the sum compactness and separation of the clusters. It is a sum of individual cluster validity measures normalized through division by the fuzzy cardinality of each cluster. Comparing different partitions having equal of clusters, SC is useful index and a lower value of this index demonstrates a better partition.

$$\text{SCf}(\mathbf{c}) = \sum\_{i=1}^{c} \frac{\sum\_{k=1}^{N} (\mu\_{ik})^m \left\| \mathbf{x}\_k - \boldsymbol{\upsilon}\_i \right\|^2}{N\_i \sum\_{k=1}^{c} (\mu\_{ik})^m \left\| \boldsymbol{\upsilon}\_k - \boldsymbol{\upsilon}\_i \right\|^2} \tag{13}$$

*Separation Index (S)* uses a minimum distance separation for partition validity.

$$\text{SC}(c) = \frac{\sum\_{i=1}^{c} \sum\_{k=1}^{N} (u\_{ik})^2 \left\| \mathbf{x}\_k - \mathbf{v}\_i \right\|^2}{N \min\_{i,k} \left\| \mathbf{v}\_k - \mathbf{v}\_i \right\|^2} \tag{14}$$

*Xie and Beni's Index (XB)* aims to quantify the ratio of the total variation within clusters and separation of clusters. The optimal value of cluster is at minimum value of this index.

$$\text{SCf}(\mathbf{c}) = \frac{\sum\_{i=1}^{c} \sum\_{k=1}^{N} (u\_{ik})^m \left\| \mathbf{x}\_k - \boldsymbol{v}\_i \right\|^2}{N \min\_{i,k} \left\| \mathbf{x}\_k - \boldsymbol{v}\_i \right\|^2} \tag{15}$$

*Dunn's Index (DI)* is proposed to use the identification of compactness and separated cluster.

$$DI(\mathbf{c}) = \min\_{i \in \mathcal{C}} \left\{ \min\_{k \in \mathcal{C}} i \neq k \left| \frac{\min\_{\mathbf{x} \in \mathbb{C}\_{i, y \in \mathcal{C}} d} d(\mathbf{x}, y)}{\max\_{k \in \mathcal{C}} \left\{ \max\_{\mathbf{x}, y \in \mathcal{C}} d(\mathbf{x}, y) \right\}} \right| \right\} \tag{16}$$

#### **4. Study area and available data**

In this study, Denizli that is a medium sized city (current population is about 700000) of Turkey is considered. Traffic accident records for the years of 2004, 2005 and 2006 are used in analyzing accidents. The accident reports are provided by Local Police Department. All of the data and documents are taken from an ongoing research project. Following information are collected from the reports:

88 Fuzzy Logic – Emerging Technologies and Applications

partitions. Therefore, it is necessary to validate each of fuzzy partitions after the cluster analysis. Cluster validity refers to the problem whether a given fuzzy partition fits to the data all. The clustering algorithm always tries to find the best fit for a fixed number of clusters and the parameterized cluster shapes. However this does not mean that even the best fit is meaningful at all. Either the number of clusters might be wrong or the cluster shapes might not correspond to the groups in the data, if the data can be grouped in a meaningful way at all. In this study, several clustering indexes were used and tested for different values of both cluster number (*c*) and to examine their adequacy in analyzing of traffic accidents. These indexes are *Partition Coefficient (PC), Classification Entropy (CE), Partition Index (SC), Separation Index (S), Xie and Beni's Index (XB) and Dunn's Index (DI)*.

*Partition Coefficient (PC)* measures the amount of "overlapping" between two Fuzzy clusters (Bezdek, 1981). The disadvantage of this index is lack of direct connection to properties of the data. The optimal number of cluster is at the maximum value and the range of this index

> 1 1 <sup>1</sup> () ( ) *c N*

*i k PC c u N*

*Classification Entropy (CE)* measures the fuzziness of the cluster partition. The range of CE is

1 1 <sup>1</sup> ( ) log *c N*

*i k CE c u au N*

*Partition Index (SC)* is the ratio of the sum compactness and separation of the clusters. It is a sum of individual cluster validity measures normalized through division by the fuzzy cardinality of each cluster. Comparing different partitions having equal of clusters, SC is

1

*<sup>c</sup> <sup>N</sup> <sup>m</sup>*

1 1

( ) ( ) min

*c N*

,

,

*N xv*

*Xie and Beni's Index (XB)* aims to quantify the ratio of the total variation within clusters and separation of clusters. The optimal value of cluster is at minimum value of this index.

> *c N m ik k i i k ik k i*

*Dunn's Index (DI)* is proposed to use the identification of compactness and separated cluster.

1 1

( ) ( ) min

*N vv*

*ik k i i k ik k i*

( )

*N u vv*

*ik k i k <sup>c</sup> <sup>m</sup> <sup>i</sup> i ik k i <sup>k</sup>*

*u xv*

<sup>1</sup> <sup>1</sup> ( ) ( )

useful index and a lower value of this index demonstrates a better partition.

*Separation Index (S)* uses a minimum distance separation for partition validity.

[0, log *a c*( ) ] and optimal number of cluster is at minimum value.

*SC c*

*SC c*

*SC c*

2

(11)

(12)

2

2 2

*u xv*

*u xv*

2

2

2

2

(13)

(14)

(15)

*ik*

*ik a ik*

is [1/c, 1].

All of these data given above are recorded using first MS Excel. Then, coordinates of each accident point are determined using street definition system in MAPINFO software. The data from Excel data base including coordinates of the accident points are transferred to MAPINFO data base. This data base is constituted for accident analysis by inquiring different attribution of each accident. Thus, traffic accident can be evaluated from different points of view and the relations about reasons and results of accidents can be revealed by these analysis.

The data related to coordinates of accidents (locations) are used in cluster analysis. The Figure 1 shows sample processed data on Denizli city GIS map (Murat et al, 2008). The data are analyzed using k-means and fuzzy clustering approaches.

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 91

**Objective Function**

Fig. 2. Variation value of objective function of FCM with change in the number of clusters

**Dead+Injured Type Accidents**

0246

**Number of Clusters**

Fig. 3. Variation value of objective function of FCM with change in the number of clusters

**ED+Injured Type Accidents**

0 5 10 15 **Number of Clusters**

**ED Type Accidents**

0 2 4 6 8 10 12 **Number of Clusters**

for ED and ED+Injured type accidents.

0 0,5 1 1,5 2 2,5 3 3,5 4

**Objective Function**

for Dead and Injured accidents.

**Objective Function**

Fig. 1. Sample processed traffic accident data on Denizli city map.

#### **5. Analysis of traffic accidents**

In clustering analysis, the observed scales of the variables must be transformed so that their ranges are comparable because the clustering methods are sensitive to scale differences. Therefore, the variables were rescaled between 0 and 1 using equation (18).

$$\text{Latitude}\tag{18}$$

$$\text{Latitude}\tag{19}$$

$$Y = \begin{pmatrix} X - X\_{\text{min}} \end{pmatrix} / \begin{pmatrix} X\_{\text{max}} - X\_{\text{min}} \end{pmatrix}\tag{10}$$

Longitude min max min *Y XX X X* ( )/( )

Where *Y* the feature at site is, *X and X* min max are the maximum and minimum of the feature within the data set. These rescaled characteristics were employed as the basis for classifying the traffic accidents. To determine the optimum the cluster number, sensitivity of the results from FCM algorithm to variation in the value of cluster numbers (*c*) is varied from 2 to 9 with increment of 1. The variations of objective function of Fuzzy C-means algorithm for Economically Damaged (ED) and ED+ injured accidents with change in the number of cluster ranging from 2 to 11 are shown in Figure 2. On the other hand, for the Dead and Injured accidents, Figure 3 shows the variations of objective function of FCM algorithm with change in the number of cluster ranging from 2 to 5.

It is seen in Figure 2 and 3 that the values of objective functions of FCM algorithm, in generally, decrease with increase in the number cluster. The optimal number of clusters in the data set is identified by using objective function and fuzzy cluster validation indexes. The variations of cluster indexes for Economically Damaged accidents given in section 2 with change in the number of cluster were calculated and given in Figure 4.

In clustering analysis, the observed scales of the variables must be transformed so that their ranges are comparable because the clustering methods are sensitive to scale differences.

Latitude min max min *Y XX X X* ( )/( ) (18)

Where *Y* the feature at site is, *X and X* min max are the maximum and minimum of the feature within the data set. These rescaled characteristics were employed as the basis for classifying the traffic accidents. To determine the optimum the cluster number, sensitivity of the results from FCM algorithm to variation in the value of cluster numbers (*c*) is varied from 2 to 9 with increment of 1. The variations of objective function of Fuzzy C-means algorithm for Economically Damaged (ED) and ED+ injured accidents with change in the number of cluster ranging from 2 to 11 are shown in Figure 2. On the other hand, for the Dead and Injured accidents, Figure 3 shows the variations of objective function of FCM

It is seen in Figure 2 and 3 that the values of objective functions of FCM algorithm, in generally, decrease with increase in the number cluster. The optimal number of clusters in the data set is identified by using objective function and fuzzy cluster validation indexes. The variations of cluster indexes for Economically Damaged accidents given in section 2

Fig. 1. Sample processed traffic accident data on Denizli city map.

Therefore, the variables were rescaled between 0 and 1 using equation (18).

algorithm with change in the number of cluster ranging from 2 to 5.

with change in the number of cluster were calculated and given in Figure 4.

**5. Analysis of traffic accidents**

Longitude min max min *Y XX X X* ( )/( )

Fig. 2. Variation value of objective function of FCM with change in the number of clusters for ED and ED+Injured type accidents.

Fig. 3. Variation value of objective function of FCM with change in the number of clusters for Dead and Injured accidents.

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 93

The data are also analyzed by conventional K-means clustering approach. In this analysis, seven clusters are obtained. Figure 6 depicts results of k-means clustering approach. Table 3 shows the coordinates of cluster centers and the number of accidents for the clusters

> 1 29,082 37,782 1303 2 29,07 37,774 648 3 29,089 37,787 1389 4 29,086 37,772 1349 5 29,097 37,779 1253 6 29,107 37,802 501 7 29,08 37,795 509 8 29,037 37,773 248 9 29,099 37,792 605 10 29,089 37,756 662 11 29,101 37,763 631 TOTAL 9098

Table 2. Coordinates of Cluster centers given by Fuzzy C-means clustering analysis

Table 3. Coordinates of Cluster centers given by conventional k-means analysis

It can be said that analysis of the fuzzy clustering approach has more details than the

1 29,085 37,787 2480 2 29,041 37,772 302 3 29,102 37,798 973 4 29,072 37,778 835 5 29,096 37,780 1589 6 29,095 37,760 1274 7 29,086 37,772 1645 TOTAL 9098

**Coordinates Number of X Y Accidents**

**Coordinates Number of Accidents x Y**

determined by conventional k-means clustering analysis.

**Cluster No**

**Cluster No**

conventional k-means clustering approach.

Fig. 4. Variations of cluster indexes with change in the number of cluster

The main drawback of PC is the monotonic decreasing with *c* and the lack of direct connection to the data. CE has the same problems: monotonic increasing with *c* and hardly detectable connection to the data structure. It is seen in Figure 4 that SC and S decrease with increase in the number of cluster. On the other hand, SC, S and XB indexes reaches the optimal value of number of cluster *c* = 10 and 11. For Economically Damaged(ED) traffic accident analysis, eleven clusters were chosen as the optimal number of cluster according to optimal values of objective function and validity indexes given in Figure 3 and 4. Table 2 exhibits the coordinates of cluster centers for ED and injured type accidents obtained from fuzzy cluster analysis (Murat and Sekerler, 2009). Similar procedure was carried out for ED+ Injured and Dead and Injured type accidents. Figure 5 shows the location of the corresponding fuzzy clusters for traffic accidents in Denizli city.

CE

S

0.20 0.40 0.60 0.80 1.00 1.20

1.0E-04 3.0E-04 5.0E-04 7.0E-04 9.0E-04

1.00E-06 8.01E-04 1.60E-03 2.40E-03 3.20E-03

DI

The main drawback of PC is the monotonic decreasing with *c* and the lack of direct connection to the data. CE has the same problems: monotonic increasing with *c* and hardly detectable connection to the data structure. It is seen in Figure 4 that SC and S decrease with increase in the number of cluster. On the other hand, SC, S and XB indexes reaches the optimal value of number of cluster *c* = 10 and 11. For Economically Damaged(ED) traffic accident analysis, eleven clusters were chosen as the optimal number of cluster according to optimal values of objective function and validity indexes given in Figure 3 and 4. Table 2 exhibits the coordinates of cluster centers for ED and injured type accidents obtained from fuzzy cluster analysis (Murat and Sekerler, 2009). Similar procedure was carried out for ED+ Injured and Dead and Injured type accidents. Figure 5 shows the location of the

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

0.40 0.50 0.60 0.70 0.80

0.60 1.20 1.80 2.40 3.00 3.60 4.20 4.80

8.0 12.0 16.0 20.0 24.0 28.0 32.0

XB

SC

PC

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

2 3 4 5 6 7 8 9 10 11 Number of Cluster (c)

Fig. 4. Variations of cluster indexes with change in the number of cluster

corresponding fuzzy clusters for traffic accidents in Denizli city.

The data are also analyzed by conventional K-means clustering approach. In this analysis, seven clusters are obtained. Figure 6 depicts results of k-means clustering approach. Table 3 shows the coordinates of cluster centers and the number of accidents for the clusters determined by conventional k-means clustering analysis.

Table 2. Coordinates of Cluster centers given by Fuzzy C-means clustering analysis

Table 3. Coordinates of Cluster centers given by conventional k-means analysis

It can be said that analysis of the fuzzy clustering approach has more details than the conventional k-means clustering approach.

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 95

As seen on Figure 6, the centers obtained are similar to that obtained by fuzzy clustering approach. But fuzzy clustering approach provided four more clusters comparing to conventional k-means clustering approach. One of the important cluster (center named Ucgen) that has the biggest number of accident is not defined by k-means clustering

Total numbers of accidents are defined in both clustering analysis approaches. But the distributions of accidents are different for k-means and fuzzy c-means clustering approaches. This is come from the number of clusters and difference about the analysis.

Using cluster analysis, different types of traffic accidents are analyzed and three types of clusters are carried out as four, seven and eleven clusters respectively. Following table shows the common points that can be considered as black spots for three types of clusters. These points are also determined as the center of each cluster. As seen on Table 3, the

Location of black spots has an importance in analysis. It should be considered as urban sections and rural areas. But it is difficult to determine a strict line for clustering accidents. Some accidents location can be defined in more than one region or cluster. Therefore, fuzzy clustering approach is preferred. The results show that, Fuzzy clustering approach provided four more black spot centers comparing to k-means clustering approach. Three of them are

approach. But it is defined by fuzzy c-means clustering approach.

Fig. 6. Clusters obtained by K-means clustering approach.

number of accidents certified the black spots centers determined.

located in urban areas and one of them is in rural areas.

**6. Results and discussion**

Fig. 5. The location of the corresponding fuzzy clusters for traffic accidents in Denizli city (a) for 11 clusters, (b) for 4 clusters.

Fig. 5. The location of the corresponding fuzzy clusters for traffic accidents in Denizli city (a)

for 11 clusters, (b) for 4 clusters.

As seen on Figure 6, the centers obtained are similar to that obtained by fuzzy clustering approach. But fuzzy clustering approach provided four more clusters comparing to conventional k-means clustering approach. One of the important cluster (center named Ucgen) that has the biggest number of accident is not defined by k-means clustering approach. But it is defined by fuzzy c-means clustering approach.

Total numbers of accidents are defined in both clustering analysis approaches. But the distributions of accidents are different for k-means and fuzzy c-means clustering approaches. This is come from the number of clusters and difference about the analysis.

Fig. 6. Clusters obtained by K-means clustering approach.

#### **6. Results and discussion**

Using cluster analysis, different types of traffic accidents are analyzed and three types of clusters are carried out as four, seven and eleven clusters respectively. Following table shows the common points that can be considered as black spots for three types of clusters. These points are also determined as the center of each cluster. As seen on Table 3, the number of accidents certified the black spots centers determined.

Location of black spots has an importance in analysis. It should be considered as urban sections and rural areas. But it is difficult to determine a strict line for clustering accidents. Some accidents location can be defined in more than one region or cluster. Therefore, fuzzy clustering approach is preferred. The results show that, Fuzzy clustering approach provided four more black spot centers comparing to k-means clustering approach. Three of them are located in urban areas and one of them is in rural areas.

Fuzzy Clustering Approach for Accident Black Spot Centers Determination 97

One of the different black spot is determined as 25. Cadde. This area is an industrial area that has many roads and intersections. The traffic composition includes heavy and light vehicle traffic flows are seen in this area. Most of the accidents are seen at intersections and all of them are controlled by isolated systems. One of the serious problems is related to

One of the main advantages of fuzzy cluster analysis is to handle the problem in an easiest and practical way. The conventional approaches take very long time and require many trials

Actually, most of reasons of problems occurred in all of these black spots are similar. These are high traffic density, non-optimized signal timing, inconvenient pedestrian crossings, and aggressive driving (caused by non-optimized control), geometric design faults (for weaving areas, taper design etc), inconvenient lane use and lane changing, excessive speed etc. Solutions for these problems can be achieved by developing some policies and intervene the system considering international standards for design. But sustainable solution can be found

The fuzzy clustering approach presented in this study can be used in determination of black spots instead of conventional approaches and traffic safety policies can be developed considering detailed analysis of these black spots. The fuzzy clustering approach can also be used for analyzing other related effective factors on traffic accidents and very interesting

The findings of this research can be used for investigation of different effects on traffic accident occurrence considering characteristics of black spots and centers. Safety levels of black spots can be determined by this way. Some countermeasures can be developed using safety levels of black spots. On the other hand, the risk analysis can be made in detail in these centers and priorities of investment planning can be defined considering level of safety risk. The fuzzy clustering analysis can be extended considering other characteristics (such as accident type, occurrence type etc.) of black spots. Therefore, definition of black spots can be

This research was supported by The Scientific and Technical Research Council of Turkey (TUBITAK) under project number 105G090. Part of the analysis presented is made by Mr. Alper Şekerler during preparation of his MSc. dissertation. These supports are appreciated.

Abdel-Aty M., Radwan E, (2000). Modelling traffic accident occurrence and involvement,

Abdel-Aty M.,Pange, A., (2007). Crash data analysis: Collective vs. individual crash level

design of intersections especially some uncontrolled intersections.

by generalizing and improving "*traffic concept*" in the society.

*Accident Analysis and Prevention* 32, 633-642.

approach*, Journal of Safety Research,* 38, 581–587

results may be obtained based on these analysis.

for the analysis.

**7. Conclusion**

reviewed after these analyses.

**8. Acknowledgement**

**9. References**

The black spots that are determined by cluster analysis are examined in detail regarding types of accident occurrence. The geometric and physical conditions of black spots are also examined in detail and the results obtained are summarized as follow.

Table 3. The common black spots determined by cluster analysis

The first black spot, Ucgen Intersection, is one of the most important intersections in Denizli city center. The main arterials from Ankara, İzmir and Antalya cities are connected at this point. Average Annual Daily Traffic volume of this intersection is very high and congested traffic conditions are seen in peak hours. The Antalya road arm has some problems occurred by geometric design. There is not enough weaving area from the previous intersection for the vehicles that wants to change the lanes in this arm. Most of accidents are occurred because of this fault. Another important problem is related to pedestrian crossings. The pedestrians do not obey the rules while crossing the street.

 The main problem of this intersection is also related to geometric design. This intersection has five approaches. Therefore optimizing signal timing is very hard. Main reasons of accidents in this intersection are related to high speed and sight distance. The speed reduction and monitoring systems can reduce number of accidents and accident severity.

Cnar is another important intersection of the city. This intersection is served most of the daily traffic circulating in the city. There are some problems about optimizing signal timings. On the other hand, some drivers ride as aggressive drivers and cause traffic accidents. There is serious pedestrian traffic in this area and traffic signal is designed as filter for vehicles and pedestrians. Traffic safety problems are seen, because of the design and can be solved by re-calculating the intersection signal timings and changing phase plans.

The main problems in Kiremitci intersection is related to intersection geometry and aggressive drivers. The problem can be solved by re-designing and optimizing signal timings.

One of the different black spot is determined as 25. Cadde. This area is an industrial area that has many roads and intersections. The traffic composition includes heavy and light vehicle traffic flows are seen in this area. Most of the accidents are seen at intersections and all of them are controlled by isolated systems. One of the serious problems is related to design of intersections especially some uncontrolled intersections.

One of the main advantages of fuzzy cluster analysis is to handle the problem in an easiest and practical way. The conventional approaches take very long time and require many trials for the analysis.

Actually, most of reasons of problems occurred in all of these black spots are similar. These are high traffic density, non-optimized signal timing, inconvenient pedestrian crossings, and aggressive driving (caused by non-optimized control), geometric design faults (for weaving areas, taper design etc), inconvenient lane use and lane changing, excessive speed etc. Solutions for these problems can be achieved by developing some policies and intervene the system considering international standards for design. But sustainable solution can be found by generalizing and improving "*traffic concept*" in the society.

The fuzzy clustering approach presented in this study can be used in determination of black spots instead of conventional approaches and traffic safety policies can be developed considering detailed analysis of these black spots. The fuzzy clustering approach can also be used for analyzing other related effective factors on traffic accidents and very interesting results may be obtained based on these analysis.

#### **7. Conclusion**

96 Fuzzy Logic – Emerging Technologies and Applications

The black spots that are determined by cluster analysis are examined in detail regarding types of accident occurrence. The geometric and physical conditions of black spots are also

Ucgen 1 229 17 1 Karayollar 46 4 2 Cinar 1 86 1 3 Kiremitci 25 0 4 Yeni Adliye 27 6 5 İstasyon 1 107 12 6 Sevindik 93 8 7 Emniyet 1 76 9 8 Ulus 94 4 9 Hastane-M.Efendi 50 1 10 25. cadde 1 200 26 11

Type of Accident Cluster Dead+Injured No

ED+Injured Accidents

Economically Damaged

(ED) Accidents

examined in detail and the results obtained are summarized as follow.

Total 5 1033 88

monitoring systems can reduce number of accidents and accident severity.

The first black spot, Ucgen Intersection, is one of the most important intersections in Denizli city center. The main arterials from Ankara, İzmir and Antalya cities are connected at this point. Average Annual Daily Traffic volume of this intersection is very high and congested traffic conditions are seen in peak hours. The Antalya road arm has some problems occurred by geometric design. There is not enough weaving area from the previous intersection for the vehicles that wants to change the lanes in this arm. Most of accidents are occurred because of this fault. Another important problem is related to pedestrian crossings. The

Karayollar Intersection is determined as the second black spot. The main problem of this intersection is also related to geometric design. This intersection has five approaches. Therefore optimizing signal timing is very hard. Main reasons of accidents in this intersection are related to high speed and sight distance. The speed reduction and

Cnar is another important intersection of the city. This intersection is served most of the daily traffic circulating in the city. There are some problems about optimizing signal timings. On the other hand, some drivers ride as aggressive drivers and cause traffic accidents. There is serious pedestrian traffic in this area and traffic signal is designed as filter for vehicles and pedestrians. Traffic safety problems are seen, because of the design and can be solved by re-calculating the intersection signal timings and changing phase

The main problems in Kiremitci intersection is related to intersection geometry and aggressive drivers. The problem can be solved by re-designing and optimizing signal

Table 3. The common black spots determined by cluster analysis

pedestrians do not obey the rules while crossing the street.

Accidents

Intersection or Road Section (Black Spot)

plans.

timings.

The findings of this research can be used for investigation of different effects on traffic accident occurrence considering characteristics of black spots and centers. Safety levels of black spots can be determined by this way. Some countermeasures can be developed using safety levels of black spots. On the other hand, the risk analysis can be made in detail in these centers and priorities of investment planning can be defined considering level of safety risk. The fuzzy clustering analysis can be extended considering other characteristics (such as accident type, occurrence type etc.) of black spots. Therefore, definition of black spots can be reviewed after these analyses.

#### **8. Acknowledgement**

This research was supported by The Scientific and Technical Research Council of Turkey (TUBITAK) under project number 105G090. Part of the analysis presented is made by Mr. Alper Şekerler during preparation of his MSc. dissertation. These supports are appreciated.

#### **9. References**

Abdel-Aty M., Radwan E, (2000). Modelling traffic accident occurrence and involvement, *Accident Analysis and Prevention* 32, 633-642.