At the heart of fuzzy logic lies the theory of fuzzy sets, presented in a series of works by L. Zadeh in 1965-1973. Fuzzy sets and fuzzy logic are generalizations of classical set theory and classical formal logic. The main reason for the emergence of a new theory was the presence of fuzzy and approximate reasoning when a person describes processes, systems, objects.

L. Zadeh, formulating this main property of fuzzy sets, was based on the works of his predecessors. In the early 1920s, the Polish mathematician Lukashevich was working on the principles of multivalued mathematical logic, in which the values ​​of predicates could be more than just “true” or “false”. In 1937, another American scientist M. Black first applied Lukashevich's multivalued logic to lists as sets of objects and called such sets indefinite.

Fuzzy logic as a scientific direction was not easy to develop, and it did not escape accusations of pseudoscience. Even in 1989, when there were dozens of examples of successful application of fuzzy logic in defense, industry and business, the US National Science Society discussed the issue of excluding materials on fuzzy sets from institute textbooks.

The first period of development of fuzzy systems (late 60s - early 70s) is characterized by the development of the theoretical apparatus of fuzzy sets. In 1970, Bellman, together with Zadeh, developed a theory of decision making in fuzzy conditions.

In the 70-80s (the second period), the first practical results appeared in the field of fuzzy control of complex technical systems (a steam generator with fuzzy control). I. Mamdani in 1975 designed the first controller operating on the basis of the Zade algebra to control a steam turbine. At the same time, attention began to be paid to the creation of expert systems based on fuzzy logic, the development of fuzzy controllers. Fuzzy expert systems for decision support have found wide application in medicine and economics.

Finally, in the third period, which lasts from the end of the 80s and continues at the present time, software packages for constructing fuzzy expert systems appear, and the fields of application of fuzzy logic are significantly expanding. It is used in the automotive, aerospace and transportation industries, household appliances, finance, analysis and management decision making and many others. In addition, a significant role in the development of fuzzy logic was played by the proof of the famous FAT (Fuzzy Approximation Theorem) by B. Cosco, which stated that any mathematical system can be approximated by a system based on fuzzy logic.

Information systems based on fuzzy sets and fuzzy logic are called fuzzy systems.

Dignity fuzzy systems:

· Functioning under conditions of uncertainty;

· Operating with qualitative and quantitative data;

· Use of expert knowledge in management;

· Construction of models of approximate reasoning of a person;

· Stability under all possible disturbances acting on the system.

Disadvantages fuzzy systems are:

· Lack of a standard methodology for designing fuzzy systems;

· Impossibility of mathematical analysis of fuzzy systems by existing methods;

· The use of a fuzzy approach in comparison with the probabilistic approach does not lead to an increase in the accuracy of calculations.

The theory of fuzzy sets. The main difference between the theory of fuzzy sets and the classical theory of clear sets is that if for crisp sets the result of calculating the characteristic function can be only two values ​​- 0 or 1, then for fuzzy sets this number is infinite, but limited by the range from zero to one.

Fuzzy set. Let U be the so-called universal set, from the elements of which all other sets considered in the given class of problems are formed, for example, the set of all integers, the set of all smooth functions, etc. The characteristic function of a set is a function whose values ​​indicate whether it is an element of the set A:

In the theory of fuzzy sets, the characteristic function is called a membership function, and its value is the degree of membership of an element x in a fuzzy set A.

More strictly: a fuzzy set A is a collection of pairs

where is the membership function, that is

Let, for example, U = (a, b, c, d, e),. Then the element a does not belong to the set A, the element b belongs to it to a small extent, the element c more or less belongs, the element d belongs to a significant extent, e is an element of the set A.

Example. Let the universe U be the set of real numbers. A fuzzy set A, denoting a set of numbers close to 10, can be specified by the following membership function (Fig.21.1):

,

Example "Hot tea" X = 0 CC; C = 0/0; 0/10; 0/20; 0.15 / 30; 0.30 / 40; 0.60 / 50; 0.80 / 60; 0, 90/70; 1/80; 1/90; 1/100.

Intersection of two fuzzy sets (fuzzy "AND"): MF AB (x) = min (MF A (x), MF B (x)). The union of two fuzzy sets (fuzzy "OR"): MF AB (x) = max (MF A (x), MF B (x)).

According to Lotfi Zadeh, a linguistic variable is a variable whose values ​​are words or sentences of a natural or artificial language. The values ​​of a linguistic variable can be fuzzy variables, i.e. the linguistic variable is at a higher level than the fuzzy variable.

Each linguistic variable consists of: name; the set of its values, which is also called the base term set T. Elements of the base term set are the names of fuzzy variables; universal set X; syntactic rule G, according to which new terms are generated using words of a natural or formal language; semantic rule P, which assigns to each value of a linguistic variable a fuzzy subset of the set X.

Description of the linguistic variable "Stock price" X = Basic term-set: "Low", "Moderate", "High"

Description of the linguistic variable "Age"

Soft computing fuzzy logic, artificial neural networks, probabilistic reasoning, evolutionary algorithms

Building the network (after choosing the input variables) Select the initial network configuration Conduct a series of experiments with different configurations, remembering the best network (in the sense of a checkout error). Several experiments should be performed for each configuration. If in the next experiment there is underfitting (the network does not produce a result of an acceptable quality), try adding additional neurons to the intermediate layer (s). If that doesn't work, try adding a new intermediate layer. If overfitting takes place (the control error began to grow), try removing several hidden elements (and possibly layers).

Data Mining Problems Solved Using Neural Networks Classification (supervised learning) Prediction Clustering (unsupervised learning) text recognition, speech recognition, personality identification find the best approximation of a function given by a finite set of input values ​​(training examples, the problem of information compression by decreasing the data dimension

The task "Whether to issue a loan to a client" in the analytical package Deductor (BaseGroup) Training set - a database containing information about clients: - Loan amount, - Loan term, - Lending purpose, - Age, - Gender, - Education, - Private property, - Apartment, - Area of ​​the apartment. It is necessary to build a model that will be able to give an answer whether the Client who wants to get a loan is in the risk group of loan default, i.e. the user should receive an answer to the question "Should I issue a loan?" The task belongs to the group of classification tasks, i.e. learning with a teacher.

Let's consider some of the methods of "soft" computing that are not yet widely used in business. The algorithms and parameters of these methods are much less deterministic than traditional ones. The emergence of the concepts of "soft" computing was caused by attempts at simplified modeling of intellectual and natural processes, which are largely random in nature.

Neural networks use the modern understanding of the structure and functioning of the brain. It is believed that the brain consists of simple elements - neurons, connected by synapses, through which they exchange signals.

The main advantage of neural networks is the ability to learn by example. In most cases, learning is the process of changing the weighting coefficients of synapses according to a specific algorithm. This usually requires many examples and many training cycles. Here you can draw an analogy with the reflexes of Pavlov's dog, in which salivation on call also did not begin to appear immediately. We only note that the most complex models of neural networks are many orders of magnitude simpler than the dog's brain; and much more training cycles are needed.

The use of neural networks is justified when it is impossible to build an accurate mathematical model of the object or phenomenon under study. For example, sales in December are usually higher than in November, but there is no formula by which to calculate how much more they will be this year; to predict the volume of sales, you can train a neural network using examples from previous years.

Among the disadvantages of neural networks are: long training time, a tendency to adjust to training data, and a decrease in generalizing abilities with increasing training time. In addition, it is impossible to explain how the network comes to this or that solution of the problem, that is, neural networks are black box systems, because the functions of neurons and the weights of synapses have no real interpretation. Nevertheless, there are a lot of neural network algorithms in which these and other disadvantages are somehow leveled.

In forecasting, neural networks are used most often according to the simplest scheme: as input data, preprocessed information about the values ​​of the predicted parameter for several previous periods is fed into the network, at the output the network issues a forecast for the next periods - as in the above example with sales. There are also less trivial ways to get a forecast; Neural networks are a very flexible tool, so there are many finite models of the networks themselves and their applications.

Another method is genetic algorithms. They are based on directed random search, that is, an attempt to simulate evolutionary processes in nature. In the basic version, genetic algorithms work like this:

1. The solution to the problem is presented in the form of a chromosome.

2. A random set of chromosomes is created - this is the initial generation of solutions.

3. They are processed by special operators of reproduction and mutation.

4. The solutions are evaluated and selected based on the suitability function.

5. A new generation of solutions is displayed and the cycle repeats.

As a result, more perfect solutions are found with each epoch of evolution.

When using genetic algorithms, the analyst does not need a priori information about the nature of the initial data, about their structure, etc. The analogy here is transparent - the color of the eyes, the shape of the nose and the thickness of the hair on the legs are encoded in our genes by the same nucleotides.

In forecasting, genetic algorithms are rarely used directly, since it is difficult to come up with a criterion for evaluating a forecast, that is, a criterion for selecting decisions - at birth it is impossible to determine who a person will become - an astronaut or an alkonaut. Therefore, usually genetic algorithms serve as an auxiliary method - for example, when training a neural network with non-standard activation functions, in which it is impossible to use gradient algorithms. Here, as an example, we can name MIP-networks, which successfully predict seemingly random phenomena - the number of spots on the sun and the intensity of the laser.

Another method is fuzzy logic that simulates thinking processes. Unlike binary logic, which requires precise and unambiguous formulations, fuzzy offers a different level of thinking. For example, formalizing the statement “last month's sales were low” within traditional binary or “boolean” logic requires a clear distinction between low (0) and high (1) sales. For example, sales equal to or greater than 1 million shekels are high, less sales are low.

The question arises: why sales at the level of 999,999 shekels are already considered low? Obviously, this is not entirely correct statement. Fuzzy logic operates with softer concepts. For example, sales of NIS 900,000 would be considered high with a rank of 0.9 and low with a rank of 0.1.

In fuzzy logic, tasks are formulated in terms of rules consisting of sets of conditions and results. Examples of the simplest rules: "If customers were given a modest loan term, then sales will be so-so", "If customers are offered a decent discount, then sales will be good."

After setting the problem in terms of the rules, the clear values ​​of the conditions (loan term in days and the discount amount in percent) are converted into a fuzzy form (large, small, etc.). Then they are processed using logical operations and the inverse transformation to numeric variables (the predicted level of sales in units of production).

Compared to probabilistic methods, fuzzy ones can drastically reduce the amount of calculations performed, but usually do not increase their accuracy. Among the shortcomings of such systems can be noted the absence of a standard design methodology, the impossibility of mathematical analysis by traditional methods. In addition, in classical fuzzy systems, an increase in the number of input quantities leads to an exponential increase in the number of rules. To overcome these and other disadvantages, as in the case of neural networks, there are many modifications of fuzzy-logical systems.

Within the framework of soft computing methods, so-called hybrid algorithms can be distinguished, which include several different components. For example, fuzzy-logical networks, or the already mentioned neural networks with genetic learning.

In hybrid algorithms, as a rule, there is a synergistic effect, in which the disadvantages of one method are compensated by the advantages of others, and the final system shows a result that is inaccessible to any of the components separately.

Title: Fuzzy logic and artificial neural networks.

As you know, the apparatus of fuzzy sets and fuzzy logic has been successfully used for a long time (more than 10 years) for solving problems in which the initial data are unreliable and poorly formalized. The strengths of this approach:
-description of the conditions and method for solving the problem in a language close to natural;
-universality: according to the famous FAT (Fuzzy Approximation Theorem), proved by B.Kosko in 1993, any mathematical system can be approximated by a system based on fuzzy logic;

At the same time, certain disadvantages are characteristic of fuzzy expert and control systems:
1) the initial set of postulated fuzzy rules is formulated by a human expert and may turn out to be incomplete or contradictory;
2) the type and parameters of the membership functions describing the input and output variables of the system are chosen subjectively and may not fully reflect the reality.
To eliminate, at least partially, the indicated shortcomings, a number of authors proposed to implement fuzzy expert and control systems with adaptive ones - adjusting, as the system works, both the rules and parameters of membership functions. Among several variants of such adaptation, one of the most successful, apparently, is the method of the so-called hybrid neural networks.
A hybrid neural network is formally identical in structure to a multilayer neural network with training, for example, according to the error backpropagation algorithm, but the hidden layers in it correspond to the stages of the fuzzy system functioning. So:
The -1st layer of neurons performs the function of introducing fuzziness based on the given membership functions of the inputs;
-2nd layer displays a set of fuzzy rules;
- The 3rd layer has the function of sharpening.
Each of these layers is characterized by a set of parameters (parameters of membership functions, fuzzy decision rules, active
functions, weights of connections), the adjustment of which is performed, in essence, in the same way as for conventional neural networks.
The book examines the theoretical aspects of the components of such networks, namely, the apparatus of fuzzy logic, the foundations of the theory of artificial neural networks and hybrid networks proper in relation to the problems of control and decision-making in conditions of uncertainty.
Particular attention is paid to the software implementation of the models of these approaches using the tools of the MATLAB 5.2 / 5.3 mathematical system.

Previous articles:

Fuzzy sets and fuzzy logic are generalizations of classical set theory and classical formal logic. These concepts were first proposed by the American scientist Lotfi Zadeh in 1965. The main reason for the emergence of a new theory was the presence of fuzzy and approximate reasoning when a person describes processes, systems, objects.

Before the fuzzy approach to modeling complex systems was recognized all over the world, it took more than a decade since the inception of the theory of fuzzy sets. And on this path of development of fuzzy systems, it is customary to distinguish three periods.

The first period (late 60s – early 70s) is characterized by the development of the theoretical apparatus of fuzzy sets (L. Zade, E. Mamdani, Bellman). In the second period (70s – 80s), the first practical results appeared in the field of fuzzy control of complex technical systems (a steam generator with fuzzy control). At the same time, attention began to be paid to the issues of constructing expert systems based on fuzzy logic, the development of fuzzy controllers. Fuzzy expert systems for decision support are widely used in medicine and economics. Finally, in the third period, which lasts from the end of the 80s and continues at the present time, software packages for constructing fuzzy expert systems appear, and the fields of application of fuzzy logic are significantly expanding. It is used in the automotive, aerospace and transportation industries, household appliances, finance, analysis and management decision making and many others.

The triumphant march of fuzzy logic around the world began after Bartholomew Kosco proved the famous FAT (Fuzzy Approximation Theorem) in the late 80s. In business and finance, fuzzy logic gained acceptance after in 1988 a fuzzy rule-based expert system for predicting financial indicators was the only one predicting a stock market crash. And the number of successful fuzzy applications is currently in the thousands.

Mathematical apparatus

The characteristic of a fuzzy set is the Membership Function. We denote by MF c (x) - the degree of membership in a fuzzy set C, which is a generalization of the concept of the characteristic function of an ordinary set. Then a fuzzy set C is the set of ordered pairs of the form C = (MF c (x) / x), MF c (x). The value MF c (x) = 0 means no membership in the set, 1 - full membership.

Let's illustrate this with a simple example. Let's formalize the imprecise definition of "hot tea". The x (area of ​​reasoning) will be the temperature scale in degrees Celsius. Obviously, it will vary from 0 to 100 degrees. A fuzzy set for hot tea might look like this:

C = (0/0; 0/10; 0/20; 0.15 / 30; 0.30 / 40; 0.60 / 50; 0.80 / 60; 0.90 / 70; 1/80; 1 / 90; 1/100).

So, tea with a temperature of 60 C belongs to the set "Hot" with a degree of belonging to 0.80. For one person, tea at a temperature of 60 C may be hot, for another it may not be too hot. It is in this that the indistinctness of the assignment of the corresponding set manifests itself.

For fuzzy sets, as well as for ordinary ones, the basic logical operations are defined. The most basic ones required for calculations are intersection and union.

Intersection of two fuzzy sets (fuzzy "AND"): A B: MF AB (x) = min (MF A (x), MF B (x)).
The union of two fuzzy sets (fuzzy "OR"): A B: MF AB (x) = max (MF A (x), MF B (x)).

In the theory of fuzzy sets, a general approach to the execution of intersection, union and complement operators has been developed, which is implemented in the so-called triangular norms and conorms. The above implementations of intersection and union operations are the most common cases of t-norm and t-conorm.

To describe fuzzy sets, the concepts of fuzzy and linguistic variables are introduced.

A fuzzy variable is described by a set (N, X, A), where N is the name of the variable, X is a universal set (area of ​​reasoning), A is a fuzzy set on X.
The values ​​of a linguistic variable can be fuzzy variables, i.e. the linguistic variable is at a higher level than the fuzzy variable. Each linguistic variable consists of:

• titles;
• the set of its values, which is also called the basic term-set T. Elements of the basic term-set are the names of fuzzy variables;
• universal set X;
• syntactic rule G, according to which new terms are generated using words of a natural or formal language;
• semantic rule P, which assigns to each value of a linguistic variable a fuzzy subset of the set X.

Consider such a fuzzy concept as "Stock Price". This is the name of the linguistic variable. Let's form a basic term-set for it, which will consist of three fuzzy variables: "Low", "Moderate", "High" and set the area of ​​reasoning in the form X = (units). The last thing left to do is to construct membership functions for each linguistic term from the base term set T.

There are over a dozen typical curve shapes for assigning membership functions. The most widespread are: triangular, trapezoidal and Gaussian membership functions.

The triangular membership function is determined by a triple of numbers (a, b, c), and its value at the point x is calculated according to the expression:

$$ MF \, (x) = \, \ begin (cases) \; 1 \, - \, \ frac (b \, - \, x) (b \, - \, a), \, a \ leq \, x \ leq \, b & \ \\ 1 \, - \, \ frac (x \, - \, b) (c \, - \, b), \, b \ leq \, x \ leq \ , c & \ \\ 0, \; x \, \ not \ in \, (a; \, c) \ \ end (cases) $$

For (b-a) = (c-b), we have the case of a symmetric triangular membership function, which can be uniquely specified by two parameters from the triple (a, b, c).

Similarly, to set the trapezoidal membership function, you need four numbers (a, b, c, d):

$$ MF \, (x) \, = \, \ begin (cases) \; 1 \, - \, \ frac (b \, - \, x) (b \, - \, a), \, a \ leq \, x \ leq \, b & \\ 1, \, b \ leq \, x \ leq \, c & \\ 1 \, - \, \ frac (x \, - \, c) (d \, - \, c), \, c \ leq \, x \ leq \, d & \\ 0, x \, \ not \ in \, (a; \, d) \ \ end (cases) $$

When (b-a) = (d-c), the trapezoidal membership function takes on a symmetric form.

The membership function of the Gaussian type is described by the formula

$$ MF \, (x) = \ exp \ biggl [- \, (\ Bigl (\ frac (x \, - \, c) (\ sigma) \ Bigr)) ^ 2 \ biggr] $$

and operates with two parameters. Parameter c denotes the center of a fuzzy set, and the parameter is responsible for the steepness of the function.

The set of membership functions for each term from the base term set T are usually depicted together on one graph. Figure 3 shows an example of the linguistic variable "Stock Price" described above, and Figure 4 shows the formalization of the imprecise concept of "Human Age". So, for a 48-year-old person, the degree of belonging to the set "Young" is 0, "Average" - 0.47, "Above average" - 0.20.

The number of terms in a linguistic variable rarely exceeds 7.

Fuzzy inference

The basis for the operation of fuzzy logical inference is the rule base containing fuzzy statements in the form "If-then" and membership functions for the corresponding linguistic terms. In this case, the following conditions must be met:

1. There is at least one rule for every linguistic term in the output variable.
2. For any term in an input variable, there is at least one rule in which this term is used as a prerequisite (the left side of the rule).

Otherwise, there is an incomplete fuzzy rule base.

Let the rule base have m rules of the form:
R 1: IF x 1 is A 11 ... AND ... x n is A 1n, THEN y is B 1
…
R i: IF x 1 is A i1 ... AND ... x n is A in, THEN y is B i
…
R m: IF x 1 is A i1 ... AND ... x n is A mn, THEN y is B m,
where x k, k = 1..n - input variables; y - output variable; A ik - given fuzzy sets with membership functions.

The result of fuzzy inference is a clear value of the variable y * based on the given clear values ​​x k, k = 1..n.

In general, the inference mechanism includes four stages: fuzzy introduction (fuzzification), fuzzy inference, composition and reduction to clarity, or defuzzification (see Figure 5).

Fuzzy inference algorithms differ mainly in the type of rules used, logical operations and a kind of defuzzification method. Fuzzy inference models for Mamdani, Sugeno, Larsen, Tsukamoto have been developed.

Let's take a closer look at the fuzzy inference using the Mamdani mechanism as an example. This is the most common inference in fuzzy systems. It uses minimax composition of fuzzy sets. This mechanism includes the following sequence of actions.

1. Fuzzification procedure: the degrees of truth are determined, i.e. the values ​​of the membership functions for the left-hand sides of each rule (prerequisites). For a rule base with m rules, we denote the degrees of truth as A ik (x k), i = 1..m, k = 1..n.

Fuzzy inference. First, the "clipping" levels are determined for the left side of each of the rules:

$$ alfa_i \, = \, \ min_i \, (A_ (ik) \, (x_k)) $$

$$ B_i ^ * (y) = \ min_i \, (alfa_i, \, B_i \, (y)) $$

Composition, or union of the obtained truncated functions, for which the maximum composition of fuzzy sets is used:

$$ MF \, (y) = \ max_i \, (B_i ^ * \, (y)) $$

where MF (y) is the membership function of the final fuzzy set.

Defasification, or reduction to clarity. There are several methods of defuzzification. For example, the middle center method, or the centroid method:
$$ MF \, (y) = \ max_i \, (B_i ^ * \, (y)) $$

The geometric meaning of this value is the center of gravity for the MF (y) curve. Figure 6 graphically shows the Mamdani fuzzy inference process for two input variables and two fuzzy rules R1 and R2.

Integration with intelligent paradigms

Hybridization of methods of intelligent information processing is the motto under which the 90s have passed among Western and American researchers. As a result of combining several artificial intelligence technologies, a special term appeared - "soft computing", which was introduced by L. Zadeh in 1994. Currently, soft computing unites such areas as: fuzzy logic, artificial neural networks, probabilistic reasoning and evolutionary algorithms. They complement each other and are used in various combinations to create hybrid intelligent systems.

The influence of fuzzy logic turned out to be perhaps the most extensive. Just as fuzzy sets have expanded the scope of classical mathematical set theory, fuzzy logic has "invaded" almost most of the Data Mining methods, endowing them with new functionality. The most interesting examples of such associations are given below.

Fuzzy neural networks

Fuzzy-neural networks carry out inferences based on the apparatus of fuzzy logic, however, the parameters of the membership functions are tuned using the learning algorithms of the neural network. Therefore, to select the parameters of such networks, we will apply the error backpropagation method originally proposed for training a multilayer perceptron. For this, the fuzzy control module is presented in the form of a multilayer network. A fuzzy neural network usually consists of four layers: a fuzzification layer for input variables, a condition activation value aggregation layer, a fuzzy rule aggregation layer, and an output layer.

The most widespread at present are the fuzzy neural network architectures such as ANFIS and TSK. It is proved that such networks are universal approximators.

Fast learning algorithms and interpretability of accumulated knowledge - these factors have made fuzzy neural networks one of the most promising and effective tools for soft computing today.

Adaptive fuzzy systems

Classical fuzzy systems have the disadvantage that for the formulation of rules and membership functions it is necessary to involve experts in a particular subject area, which is not always possible to ensure. Adaptive fuzzy systems solve this problem. In such systems, the selection of parameters of a fuzzy system is carried out in the learning process on experimental data. Learning algorithms for adaptive fuzzy systems are relatively laborious and complex in comparison with learning algorithms for neural networks, and, as a rule, consist of two stages: 1. Generation of linguistic rules; 2. Correction of membership functions. The first problem is an enumeration type problem, the second one is optimization in continuous spaces. In this case, a certain contradiction arises: to generate fuzzy rules, membership functions are needed, and to carry out fuzzy inference, rules. In addition, when automatically generating fuzzy rules, it is necessary to ensure their completeness and consistency.

A significant part of the methods of training fuzzy systems use genetic algorithms. In the English-language literature, this corresponds to a special term - Genetic Fuzzy Systems.

A group of Spanish researchers headed by F. Herrera made a significant contribution to the development of the theory and practice of fuzzy systems with evolutionary adaptation.

Fuzzy queries

Fuzzy queries are a promising trend in modern information processing systems. This tool allows you to formulate queries in natural language, for example: "List low-cost housing offers close to the city center", which is not possible using the standard query mechanism. For this purpose, fuzzy relational algebra and special extensions of the SQL languages ​​for fuzzy queries have been developed. Most of the research in this area belongs to Western European scientists D. Dubois and G. Prade.

Fuzzy association rules

Fuzzy associative rules are a tool for extracting patterns from databases that are formulated in the form of linguistic statements. Special concepts of fuzzy transaction, support and validity of fuzzy association rule are introduced here.

Fuzzy cognitive maps

Fuzzy cognitive maps were proposed by B. Kosko in 1986 and are used to model the causal relationships identified between the concepts of a certain area. Unlike simple cognitive maps, fuzzy cognitive maps are a fuzzy directed graph, the nodes of which are fuzzy sets. The directed edges of the graph not only reflect the causal relationships between concepts, but also determine the degree of influence (weight) of the related concepts. The active use of fuzzy cognitive maps as a means of modeling systems is due to the possibility of a visual representation of the analyzed system and the ease of interpretation of cause-and-effect relationships between concepts. The main problems are associated with the process of building a cognitive map, which does not lend itself to formalization. In addition, it is necessary to prove that the constructed cognitive map is adequate to the real modeled system. To solve these problems, algorithms for the automatic construction of cognitive maps based on data sampling have been developed.

Fuzzy clustering

Fuzzy clustering methods, in contrast to clear-cut methods (for example, Kohonen neural networks), allow the same object to belong simultaneously to several clusters, but with varying degrees. Fuzzy clustering in many situations is more "natural" than clear-cut, for example, for objects located on the border of clusters. The most common: the c-means fuzzy self-organization algorithm and its generalization in the form of the Gustafson-Kessel algorithm.

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