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. Strengths 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;
The 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 under 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:

While the engineers who worked in the field automatic control, were engaged in the transition from traditional electromechanical and analog control technologies to digital mechatronic control systems that combine computerized algorithms for analysis and decision-making, new Computer techologies that can bring about even more significant changes. Neural networks and fuzzy logic have already found widespread use and will soon be able to change the ways of building and programming automatic control systems.

Traditional computers have von Neumann architecture, which is based on sequential processing and execution of explicitly given instructions. Artificial neural networks (ANNs) are based on a different architecture. They are assembled from very simple processor units combined into a system with a high level of parallelism. This system executes implicit instructions based on pattern recognition on data inputs from external sources.

Fuzzy logic also turns traditional beliefs upside down. Instead of the results of accurate measurements that establish the position of the value on a given scale (for example, "temperature 23 ° C"), fuzzy information indicates the degree of belonging to fuzzy overlapping sets ("on the colder side of the warmer").

Definitions

Computers (or, more accurately, "inference machines") using these concepts are capable of solving complex problems that are beyond the power of traditional control systems.

An artificial neural network (ANN), according to Wikipedia, is "an interconnected collection of artificial 'neurons' that uses a mathematical or computational model to process information based on the connectedness of calculators."

In most cases, ANN is an adaptive system that changes its structure under the influence of external or internal information passing through the network. Instead of calculating numerical results from numerical input data, ANNs model complex relationships between inputs and outputs or discover patterns in the data.

Elementary nodes (also called "neurons", "neurodes", "processing elements" or "blocks") are connected together and form a network of nodes. The beneficial effect of their application stems from the ability to implement inference algorithms that alter forces or weights. network connections to obtain the required signal flow.

In this example of an artificial neural network, the variable h representing the 3D vector depends on the input variable x. Next, g, a two-dimensional vector variable, depends on h, and finally, the output variable f depends on g.

The most interesting is the possibility of learning, which in practice means the optimization of a certain value, often called "price", which shows the correctness of the result in the context of the problem being solved.

For example, the price in the classic traveling salesman problem is the time required to completely go around the territory of a trade with stops at all required points and arrive at the starting point. A shorter route gives a better solution.

To solve this problem, von Neumann computers must establish all possible routes, then check each route in turn, adding the time delays to determine the total delay for that route. After calculating the sums for all possible routes, the computer simply chooses the shortest one.

In contrast, ANNs consider all routes in parallel in order to find configurations that minimize the total route time. Using these configurations minimizes the resulting route. The training consists of identifying configurations that, based on previous experience, provide route optimization strategies.

Fuzzy logic (again according to Wikipedia) is derived from fuzzy set theory, which deals with reasoning that is more approximate than exact. Truth in fuzzy logic shows belonging to fuzzy sets. In fuzzy logic, decisions can be made based on imprecise, but nevertheless very important characteristics. Fuzzy logic allows changing the values ​​of membership in a set in the range of 0 to 1, inclusive, as well as the use of such vague concepts as "a little", "to some extent" and "very". This makes it possible to implement partial membership in a set in a special way.

The main application can be described by continuous variable subranges. For example, the temperature range of an anti-lock braking system may have several separate accessory functions that define the temperature ranges needed to properly control the brakes. Each function displays whether a temperature value is a truth value in the range of 0 to 1. These truth values ​​can then be used to select a control method for the brake system.

Fast fuzzy logic for real-time control Despite the fact that any microcontroller or computer can implement fuzzy logic algorithms in programmatically, this may turn out to be ineffective due to low performance and the need for large volume memory. Jim Sibigtroth, an automotive systems engineer for the Transportation and Standard Products Group microcontroller division of Freescale Semiconductor, says the company's HC12 and HCS12 microcontrollers are very effective at solving this problem by adding four instructions specifically designed to implement the main parts of the fuzzy inference engine. logic. "The main program for the general-purpose inference engine that processes unweighted rules is approximately 57 bytes of object code (approximately 24 lines of assembly code)," he says. Sibigtrot notes that the HCS12 25 MHz model can complete a complete output sequence for two inputs and one output parameter with seven labels for each input and output in about 20 μs. An equivalent program for the MC68HC11 at 8 MHz (no fuzzy logic instructions) would take approximately 250 bytes of object code and approximately 750 μs of time. Even if the MC68HC11 could process the program at the same speed as the HCS12, the fuzzy logic instructions reduce the program by 4 times and reduce the execution time by 12 times. Such short recognition intervals allow the use of fuzzy logic algorithms in real-time control systems without expensive computer equipment or large programs.

Image processing

By making decisions in ANN based on fuzzy logic, a powerful control system can be created. Obviously, these two concepts work well together: an inference algorithm with three fuzzy states (e.g. cold, warm, hot) could be implemented in hardware by using truth values ​​(0.8, 0.2, 0.0) as inputs for three neurons, each of which represents one of three sets. Each neuron processes the input value in accordance with its function and receives an output value, which will then be the input value for the second layer of neurons, etc.

For example, a neurocomputer for image processing can remove numerous restrictions on video recording, lighting, and hardware settings. This degree of freedom becomes possible due to the fact that the neural network allows you to build a recognition mechanism by studying examples. As a result, the system can be trained to recognize good and defective products under strong and low light, when they are located at different angles, etc. The inference engine begins by “evaluating” the lighting conditions (in other words, it sets the degree of similarity to other lighting conditions under which the system knows how to proceed). The system then makes a decision about the content of the image using criteria based on the given lighting conditions. Since the system considers lighting conditions as fuzzy concepts, the inference engine easily determines new conditions from known examples.

The more examples the system examines, the more experience the image processing engine gains. This learning process can be quite easily automated, for example, by pre-sorting into groups of parts with similar properties for training in areas of similarities and differences. These observed similarities and differences can further provide information to the ANN, whose task is to sort the incoming parts into these categories. Thus, the success of the system does not depend on the cost of the equipment, but on the number of images required for training and building a reliable inference engine.

The neurocomputer for image processing is suitable for applications where diagnostics relies on the experience and expert judgment of the operator rather than models and algorithms. The processor can build a recognition engine from simple image comments made by the operator, then extract characteristics or feature vectors from the annotated objects and transmit them to the neural network. Feature vectors describing visible objects can be as simple as pixel row values, histogram or intensity distribution, intensity distribution profiles, or gradients along the corresponding axes. More complex features can include elements of the wavelet transform and the fast Fourier transform.

Generalizations

After learning by examples, the neural network is capable of generalization and can classify situations that have never been observed before, linking them to similar situations from examples. On the other hand, if the system is prone to excessive freedom and generalization of situations, its behavior at any time can be corrected by teaching opposite examples.

From the point of view of a neural network, this operation is to reduce the areas of influence of existing neurons to accommodate new examples that conflict with the existing mapping of the decision space.

An important factor determining the recognition of ANN is self-directed and adaptive learning. This means that the device must be able to study the object with minimal or no operator intervention. In the future, dolls, for example, might recognize the face of a child unfolding them for the first time and ask for their name. Self-study for cell phone could be studying the fingerprint of its first owner. Owner identification can also be enhanced by combining face, fingerprint and speech recognition in a single device.

In a self-taught setting, the device must build its own recognition engine that will function best in its working environment. For example, a smart doll must recognize its original owner regardless of hair and skin color, location, or season.

At first, the engine must use all the feature extraction techniques that it knows. This will lead to the formation of a number of intermediate mechanisms, each of which is designed to identify the same categories of objects, but based on the observation of different features (color, graininess, contrast, border thickness, etc.). The general mechanism can then evaluate the performance of the intermediate mechanisms, choosing those that give the best performance and / or accuracy.

Fish sorting example

PiscesVMK manufactures fish processing equipment on board and offshore. The firm's clients are fish processing vessels that catch various types of fish year-round in the North Sea and the Atlantic Ocean. These consumers want to fill their holds with catch as quickly as possible. the highest quality with a minimum number of employees.

Typically, fish are brought on board by nets and unloaded into containers on a conveyor belt that carries them through cleaning, slicing and filleting machines. Possible deviations include unsuitable variety, damaged fish, more than one fish in the tank and incorrect position before entering the slicing machine. Implementation of such control by traditional means of image processing is difficult, since the size, shape and volume are difficult to describe mathematically. In addition, these parameters may vary depending on the sailing location and the season.

Pisces has installed over 20 systems based on Matrox's Iris smart camera and General Vision's CogniSight recognition engine. The camera is mounted above the conveyor so that the fish pass underneath just before entering the filleting machine. The camera is connected with the Siemens Simatic S7-224 controller (PLC) and with local area network(LAN). A stroboscopic light source, mounted next to the camera, is triggered every time a new container appears in the field of view. Connecting the camera to a local network is necessary to perform three operations: adjusting the transducer to ensure focus and proper image contrast, training the recognition engine, and accessing statistics that continuously report the number of conditioned and substandard fish.

The transducer is set up only once when the camera is installed in a waterproof housing. Training is done at the start of each swim using fish samples from the first catch or by loading an existing file.

Once the camera receives the knowledge base, it can start to recognize fish autonomously, without communication with a personal computer. ANN sorts it into categories "accepted", "rejected", "for processing" or "empty". This signal goes to the PLC, which controls two brushes that guide the respective fish into containers for disposal or processing. The PLC is also connected to a magnetic sensor that generates a start signal every time a container of fish passes under the camera.

Pisces has now installed over 20 systems in 5 different fishing fleets in Norway, Iceland, Scotland and Denmark. The system estimates 360 conveyor belts per minute on herring lines, but it can run even faster.

For a network of 80 neurons, 98% accuracy was achieved when classifying 16 tonnes of fish. Fishermen are happy with the system because of its reliability, flexibility and ease of use. Benefits: shorter sailing times, better catch quality and better income distributed among fewer fishermen.

In discrete manufacturing tools, neural networks have found application in vehicle control, pattern recognition in radar systems, personality recognition, object recognition, handwriting, gestures, and speech.

Fuzzy logic is already being used to control the car and other vehicle subsystems such as ABS and cruise control, as well as air conditioning, cameras, digital image processing, artificial intelligence computer games and pattern recognition in remote sensing systems.

Similar soft computing technologies are also used to create reliable charger for breathing apparatus batteries. In the continuous and batch industries, fuzzy logic and neural networks are the backbone of some self-adjusting controllers. Some microcontrollers and microprocessors are optimized for fuzzy logic so that systems can run even faster (see “Fast Fuzzy Logic for Real Time Control” below).

Fuzzy logic and neural networks

Introduction

Fuzzy logic- a branch of mathematics, which is a generalization of classical logic and set theory, based on the concept of a fuzzy set, first introduced by Lotfi Zadeh in 1965 as an object with an element belonging to a set function that takes any values ​​in the interval , and not just 0 or 1. On the basis of this concept, various logical operations over fuzzy sets, the concept of a linguistic variable is formulated, the values ​​of which are fuzzy sets.

The subject of fuzzy logic is the study of reasoning in conditions of fuzzy, fuzzy, similar to reasoning in the usual sense, and their application in computing systems.

Directions of Fuzzy Logic Research

Currently, there are at least two main lines of research in the field of fuzzy logic:

Fuzzy logic in a broad sense (theory of approximate calculations);

Fuzzy logic in the narrow sense (symbolic fuzzy logic).

Symbolic fuzzy logic

Symbolic fuzzy logic is based on the concept t-norms... After choosing some t-norm (and it can be introduced by several different ways) it becomes possible to define the basic operations on propositional variables: conjunction, disjunction, implication, negation, and others.

It is not difficult to prove the theorem that distributivity, which is present in classical logic, holds only in the case when Gödel's t-norm is chosen as the t-norm.

In addition, for certain reasons, an operation called residium is most often chosen as an implication (it, generally speaking, also depends on the choice of the t-norm).

The definition of the basic operations listed above leads to the formal definition of basic fuzzy logic, which has much in common with classical Boolean-valued logic (more precisely, with the propositional calculus).

There are three main basic fuzzy logics: Lukasiewicz's logic, Gödel's logic, and product logic. Interestingly, the union of any two of the three logics listed above leads to the classical Boolean-valued logic.

Characteristic function

For the space of reasoning and a given membership function fuzzy set is defined as

The membership function quantitatively grades the belonging of the elements of the fundamental set of the space of reasoning to a fuzzy set. The value means that the element is not included in the fuzzy set, describes a fully included element. Values ​​between and characterize indistinctly included elements.

Fuzzy set and classic, clear ( crisp) a bunch of

Examples of fuzzy sets

1. Let E = {0, 1, 2, . . ., 10}, M =... Fuzzy set "Several" can be defined as follows:

"Several" = 0.5 / 3 + 0.8 / 4 + 1/5 + 1/6 + 0.8 / 7 + 0.5 / 8; its characteristics: height = 1, carrier = {3, 4, 5, 6, 7, 8}, transition points - {3, 8}.

2. Let E = {0, 1, 2, 3,…, n,… ). Fuzzy set "Small" can be defined:

3. Let E= (1, 2, 3,..., 100) and corresponds to the concept "Age", then the fuzzy set "Young" can be determined using

Fuzzy set "Young" on a universal set E "= (IVANOV, PETROV, SIDOROV, ...) is set using the membership function μ Young ( x) on the E =(1, 2, 3,..., 100) (age), called with respect to E " compatibility function, while:

where NS- SIDOROV's age.

4. Let E= (ZAPOROZHETS, ZHIGULI, MERCEDES, ...) - many brands of cars, and E "= - universal set "Cost", then on E " we can define fuzzy sets like:

Rice. 1.1. Examples of membership functions

“For the poor”, “For the middle class”, “Prestigious”, with membership functions like fig. 1.1.

Having these functions and knowing the cost of cars from E in this moment time, we thereby define on E " fuzzy sets with the same names.

So, for example, the fuzzy set "For the poor", given on the universal set E =(ZAPOROZHETS, ZHIGULI, MERCEDES, ...), looks as shown in fig. 1.2.

Rice. 1.2. An example of defining a fuzzy set

Similarly, you can define the fuzzy set "High-speed", "Medium", "Slow", etc.

5. Let E- a set of integers:

E= {-8, -5, -3, 0, 1, 2, 4, 6, 9}.

Then a fuzzy subset of numbers, by absolute value close to zero, you can define, for example, like this:

A ={0/-8 + 0,5/-5 + 0,6/-3 +1/0 + 0,9/1 + 0,8/2 + 0,6/4 + 0,3/6 + 0/9}.

Logical operations

Inclusion. Let be BUT and IN- fuzzy sets on a universal set E. They say that BUT contained in IN, if

Designation: BUT⊂ IN.

The term is sometimes used domination, those. in the case when BUT⊂ IN, they say that IN dominates BUT.

Equality. A and B are equal if

Designation: A = B.

Addition. Let be M = , BUT and IN- fuzzy sets defined on E. A and IN complement each other if

Designation:

It's obvious that (addition is defined for M=, but it is obvious that it can be defined for any ordered M).

Intersection. BUT⋂ IN is the largest fuzzy subset contained simultaneously in BUT and IN:

Union.A∪IN- the smallest fuzzy subset, including both BUT, and so IN, with membership function:

Difference. with membership function:

Disjunctive sum

BUT ⊕ IN = (A - B) ∪ (B - A) = (A ⋂ ̅ B) ∪ (̅A ⋂ B)

with membership function:

Examples. Let be

Here:

1) A ⊂ IN, i.e., A is contained in B or B dominates BUT WITH incomparable not with A nor with IN, those. couples ( A, C) and ( A, C) are pairs of non-dominated fuzzy sets.

2) A≠ B ≠ C

3) ̅A = 0,6/x 1 + 0,8/x 2 + 1/x 3 + 0/x 4 ; ̅B = 0,3/x 1 + 0,1/x 2 + 0,9/x 3 +0/x 4 .

4) BUT⋂ B = 0,4/x 1 + 0,2/x 2 + 0/x 3 + 1 /NS 4 .

5) A∪ IN= 0.7 / x 1+ 0,9/x 2 + 0,1/x 3 + 1/x 4 .

6) A - B= BUT⋂̅В = 0,3/x 1 + 0, l / x 2 + 0/x 3 + 0/x 4 ;

IN- A = ̅A⋂ IN= 0,6/x 1 + 0,8/x 2 + 0, l / x 3 + 0/x 4 .

7) BUT⊕ B = 0,6/x 1 + 0,8/x 2 + 0,1/x 3 + 0/x 4 .

Visual representation of logical operations on fuzzy sets. For fuzzy sets, you can build a visual representation. Consider a rectangular coordinate system, on the ordinate axis of which the values μ BUT(NS), the elements on the abscissa are in random order E(we have already used such a representation in examples of fuzzy sets). If E is ordered by its nature, it is desirable to preserve this order in the arrangement of elements on the abscissa axis. This representation makes clear simple logical operations on fuzzy sets (see Fig. 1.3).

Rice. 1.3. Graphic interpretation of logical operations:
α - fuzzy set BUT; b- fuzzy set ̅А, в - BUT⋂BUT; G-A∪ BUT

In fig. 1.3α the shaded part corresponds to a fuzzy set BUT and, to be precise, it depicts the range of values BUT and all fuzzy sets contained in BUT. In fig. 1.3 b, c, d are given A, A ⋂ ̅A,A U BUT.

Operation properties ∪ and ⋂

Let be A, B, C- fuzzy sets, then the following properties are fulfilled:

Unlike crisp sets, for fuzzy sets in general

A ⋂ ̅A ≠ ∅, A ∪ ̅A ≠ E

(which, in particular, is illustrated above in the example of a visual representation of fuzzy sets).

Comment ... The above operations on fuzzy sets are based on the use of the max and min operations. In the theory of fuzzy sets, questions are being developed to construct generalized, parameterized operators of intersection, union and complement, allowing to take into account various semantic shades of the corresponding connectives "and", "or", "not".

Triangular norms and conorms

One approach to the intersection and union operators is to define them in class of triangular norms and conorms.

Triangular norm (t-norm) called a binary operation (two-place real function)

1. Limitation:.

2. Monotony:.

3. Commutability:.

4. Associativity:.

Examples of triangular norms

min ( μ A,μ B)

work μ Aμ B

max (0, μ A +μ B - 1).

Triangular shape(abbreviated -conorm) is a double real function

satisfying the following conditions:

1. Limitation:.

2. Monotony:.

3. Commutability:.

4. Associativity:.

Triangular conorm is an Archimedean if it is continuous
and for anyone fuzzy set performed inequality .

It is called strict if function strictly decreases in both arguments.

Examples of t-conorms

max ( μ A,μ B)

μ A + μ B - μ A μ B

min (1, μ A +μ B).

Examples of triangular conorms are the following operators:

Triangular norm T and the triangular conorm S are called additional binary operations if

T ( a,b) + S(1 − a,1 − b) = 1

The most popular in Zadeh's theory are three pairs of additional triangular norms and conorms.

1) Intersection and union by Zade:

T Z(a,b) = min ( a,b}, S Z(a,b) = max ( a,b}.

2) Intersection and union according to Lukasiewicz:

3) Probabilistic intersection and union:

Complement Operators

In theory fuzzy sets the complement operator is not unique.

Besides the well-known

exist whole a set of complement operators fuzzy set.

Let some display

.

This is display will be called the negation operator in theory fuzzy sets if the following conditions are met:

If, in addition, the following conditions are met:

(3) - strictly decreasing function

(4) - continuous function

then it is called strict denial.

Function called strong denial or involution if, along with conditions (1) and (2), it is valid for it:

(5) .

Here are some examples of the negation function:

Classic negation:.

Quadratic negation: .

Denial of Sugeno:.

Threshold type addition: .

We will call any meaning, for which , equilibrium point... For any continuous negation, there is a single equilibrium point.

Fuzzy numbers

Fuzzy numbers- fuzzy variables defined on the numerical axis, i.e. a fuzzy number is defined as a fuzzy set BUT on the set of real numbers ℝ with the membership function μ A(NS) ϵ, where NS- real number, i.e. NS ϵ ℝ.

Fuzzy number It's okay if max μ A(x) = 1; convex, if for any NS ≤ at ≤ z performed

μ A (x) ≥ μ A(at) ˄ μ A(z).

A bunch of α -level fuzzy number BUT defined as

Aα = {x/μ α (x) ≥ α } .

Subset S A⊂ ℝ is called the support of a fuzzy number BUT, if

S A = { x / μ A (x)> 0 }.

Fuzzy number And unimodal if condition μ A(NS) = 1 is only valid for one point on the real axis.

Convex fuzzy number BUT called fuzzy zero, if

μ A (0) = sup ( μ A(x)).

Fuzzy number And positively, if ∀ xϵ S A, x> 0 and negatively, if ∀ NS ϵ S A, x< 0.

Fuzzy numbers (L-R) -Type

Fuzzy numbers (L-R) -type are a kind of fuzzy numbers special kind, i.e. set according to certain rules in order to reduce the amount of calculations during operations with them.

Membership functions of fuzzy numbers (L-R) -type are specified using functions of a real variable L ( x) and R ( x) satisfying the properties:

a) L (- x) = L ( x), R (- x) = R ( x);

b) L (0) = R (0).

Obviously, the class of (L-R) -functions includes functions whose graphs have the form shown in Fig. 1.7.

Rice. 1.7. Possible view(L-R) -functions

Examples of analytical task (L-R) -functions can be

Let L ( at) and R ( at) - (L-R) -type functions (specific). Unimodal fuzzy number BUT with fashion a(i.e. μ A(but) = 1) using L ( at) and R ( at) is set as follows:

where a is the fashion; α > 0, β > 0 - left and right fuzzy coefficients.

Thus, for given L ( at) and R ( at) a fuzzy number (unimodal) is given by a triple BUT = (but, α, β ).

Tolerant fuzzy number is set, respectively, by four parameters BUT = (a 1 , but 2 , α, β ), where but 1 and but 2 - limits of tolerance, i.e. in the interim [ a 1 , but 2], the value of the membership function is 1.

Examples of graphs of membership functions of fuzzy numbers (L-R) -type are shown in Fig. 1.8.

Rice. 1.8. Examples of graphs of membership functions of fuzzy numbers (L-R) -type

Note that in specific situations the functions L (y), R (y), and also parameters but, β fuzzy numbers (but, α, β ) and ( a 1 , but 2 , α, β ) should be selected in such a way that the result of an operation (addition, subtraction, division, etc.) is exactly or approximately equal to a fuzzy number with the same L (y) and R (y), and the parameters α" and β" the result did not go beyond the limitations on these parameters for the initial fuzzy numbers, especially if the result will later participate in operations.

Comment... Solving problems of mathematical modeling of complex systems using the apparatus of fuzzy sets requires performing a large amount of operations on various kinds of linguistic and other fuzzy variables. For the convenience of performing operations, as well as for input-output and data storage, it is desirable to work with membership functions of a standard type.

Fuzzy sets that have to be operated in most problems are, as a rule, unimodal and normal. One of possible methods approximation of unimodal fuzzy sets is approximation by means of (L-R) -type functions.

Examples of (L-R) -representations of some linguistic variables are given in table. 1.2.

Table 1.2. Possible (L-R) -representation of some linguistic variables

Fuzzy relationship

Fuzzy relationship play a fundamental role in the theory of fuzzy systems. Apparatus of theory fuzzy relationships is used in the construction of the theory of fuzzy automata, in modeling the structure of complex systems, in the analysis of decision-making processes.

Basic definitions

Theory fuzzy relationships finds also Appendix in tasks in which the theory of ordinary (clear) relations is traditionally applied. As a rule, the apparatus of the theory of clear relations is used in the qualitative analysis of the relationships between the objects of the studied system, when the connections are of a dichotomous nature and can be interpreted in terms of " connection present "," connection is absent ", or when the methods of quantitative analysis of relationships for some reason are inapplicable and relationships are artificially brought to a dichotomous form. For example, when the value of the relationship between objects takes values ​​from the rank scale, the choice of the threshold for the strength of the relationship allows you to transform connection to the required form. However, a similar approach, allowing for high-quality analysis systems, leads to the loss of information about the strength of connections between objects, or requires calculations at different thresholds for the strength of connections. Data analysis methods based on the theory fuzzy relationships that allow for high-quality analysis systems, taking into account the difference in the strength of connections between the objects of the system.

Plain unblurred - articulation defined as subset cartesian product of sets

Like a fuzzy set fuzzy attitude can be specified using its membership function

where in the general case we will assume that is a complete distributive lattice. Thus, is a partially ordered set in which any non-empty subset has the largest bottom and smallest top facets and intersection operations and unions in satisfy the laws of distributiveness. Everything operations over fuzzy relationships are determined using these operations from. For example, if we take as a bounded set of real numbers, then the operations of intersection and union in will be, respectively, operations and, and these operations will determine and operations over fuzzy relationships.

If multitudes and are finite, fuzzy attitude between and can be represented using it relationship matrices, the first row and the first column of which are assigned the elements of the sets and, and an element is placed at the intersection of the row and column (see Table 2.1).

Table 2.1.
			0,5	0,8
	0,7		0,6	0,3
		0,7		0,4

In the case when multitudes and match, fuzzy attitude are called fuzzy relation on the set X.

In the case of finite or countable universal sets obvious fuzzy relation interpretation as weighted graph, in which each pair of vertices from is connected by an edge with a weight.

Example... Let be and then fuzzy graph shown in fig. 2.1, sets some fuzzy attitude .

Rice. 2.1.

Fuzzy relationship properties

Various types fuzzy relationships are determined using properties analogous to those of ordinary relations, and for fuzzy relationships you can specify various ways to generalize these properties.

1. Reflexivity:

2. Weak reflexivity:

3. Strong reflexivity:

4. Anti-reflectiveness:

5. Weak antireflexivity:

6. Strong anti-reflectiveness:

7. Symmetry:

8. Antisymmetry:

9. Asymmetry:

10. Strong linearity:

11. Weak linearity:

12. Transitivity:

Fuzzy relationship projections

An important role in the theory of fuzzy sets is played by the concept fuzzy ratio projections... Let's give definition binary fuzzy ratio projection.

Let be - fuzzy membership function in . Projection and the relationship is on and - is multitudes in and with a membership function of the form

Conditional projection of fuzzy relation on, for an arbitrary fixed, is called a set with a membership function of the form.

The conditional projection on given:

From this definition it is seen that the projections and do not affect the conditional projections and, respectively. Let's give further definition, which takes into account their relationship.

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 worked 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 appear in the field of fuzzy management of complex technical systems(fuzzy steam generator). 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, 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 applied in the automotive, aerospace and transportation industries, in the field of products household appliances, in the field of 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.

Advantages 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 crisp 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 large 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 more high level than a 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, while remembering the best network(in the sense of a check 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).

Tasks Data mining solved using neural networks Classification (supervised learning) Forecasting 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 reducing the dimension of data

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.