Statistical recognition methods. Recognition methods “Physical meaning” and terminology

a decision on diagnosis is not made (refusal to recognize) and additional information is required.

The decision-making process in the Bayes method when calculating on a computer occurs quite quickly. For example, making a diagnosis for 24 conditions with 80 multi-digit signs takes only a few minutes on a computer with a speed of 10 - 20 thousand operations per second.

As indicated, the Bayes method has some disadvantages, for example, errors in recognizing rare diagnoses. In practical calculations, it is advisable to carry out diagnostics for the case of equally probable diagnoses, putting

P(D i) = l/n (5.25)

Then the diagnosis will have the largest posterior probability value D i, for which R (K* /D i) maximum:

K* D i,If P( K*/D i) > P( K*/D j)(j = 1, 2,..., n; i ≠ j). (5.26)

In other words, a diagnosis is made D i if this set of symptoms is more common during diagnosis D i than with other diagnoses. This decision rule corresponds maximum likelihood method. It follows from the previous that this method is a special case of the Bayes method with the same prior probabilities of diagnoses. In the maximum likelihood method, “common” and “rare” diagnoses have equal rights.

For recognition reliability, condition (5.26) must be supplemented with a threshold value

P(K */D i) ≥ P i ,(5.27)

Where P i- pre-selected recognition level for diagnosis D i .

To date, a large number of methods have been developed, the use of which makes it possible to recognize the type of technical condition of the diagnosed object. This paper discusses only some of them, the most widely used in diagnostic practice.

Bayes method

The diagnostic method based on the application of the Bayes formula refers to statistical recognition methods.

Probability of event A, which can occur only when one of the incompatible events 2 occurs? 1? IN 2 ,..., In p, equal to the sum of the products of the probabilities of each of these events by the corresponding probability of the event A:

This formula is called the total probability formula. The corollary of the multiplication theorem and the total probability formula is the so-called hypothesis theory. Let's assume that the event A can occur only when one of the incompatible events occurs IN, AT 2 , ..., In p, but since it is not known in advance which of them will occur, they are called hypotheses. The probability of the occurrence of an event A is determined using the total probability formula (1.5), and the conditional probability R A (V/) according to the formula

Substituting the value R(L), we get

Formula (1.6) is called Bayes' formula. It allows the probabilities of hypotheses to be reestimated after the results of the trial in which the event occurred are known. A.

Identifying the magnitude of the conditional probabilities of a trait's occurrence is key to using Bayes' formula to diagnose a condition. The Bayesian approach is widely used in control science, signal detection and pattern recognition theory, and medical and technical diagnostics.

Let us consider the essence of the method in relation to the diagnostic task. The mathematical side of the issue is presented in detail in work Ts3]. During operation, any object can be in one of the possible states TVj, ...,Nj(in the simplest case - “norm”, “refusal”), to which the hypotheses (diagnoses) Z)j,...,Z) are assigned; . During the operation of the facility, parameters (signs) are monitored To, ..., kj. Probability of the joint presence of state Z)- and the attribute in an object kj determined

Where Р(Dj)- probability of diagnosis DJ, determined by statistical data:

Where P- number of objects surveyed;

Nj- number of states;

P(kj/Dj) kj for objects with state Dj. If among P objects with a diagnosis DJ, showed a sign kj, That

P(cr- probability of occurrence of a sign kj in all objects, regardless of the condition (diagnosis) of the object. Let from the total number P objects sign kj was found in rij objects, then

P(Dj/kj) - probability of diagnosis Z); after it has become known that the object in question has the characteristic To-.

The generalized Bayes formula applies to the case when the survey is carried out according to a set of characteristics TO, including signs (ku, k p). Each of the signs kj It has rrij ranks (, To d,

kj2 , ..., kj s, ..., k jm). As a result of the examination, it becomes known

implementation of the characteristic k.-k. and the whole complex of signs TO. In-

deke means the specific meaning of a feature. The Bayes formula for a set of features has the form

Where P(Dj/A*) - probability of diagnosis? D after the results of an examination based on a set of signs become known TO;

P(Dj)- preliminary probability of diagnosis Dj.

It is assumed that the system is in only one of the indicated states, i.e.

To determine the probability of diagnosis using the Bayes method, a diagnostic matrix is ​​formed based on preliminary statistical material (Table 1.1). The number of lines corresponds to the number of possible diagnoses. The number of columns is calculated as the sum of the products of the number of features and the corresponding number of digits plus one for the prior probabilities of diagnoses. This table contains the probabilities of character categories for various diagnoses. If recognized

ki are two-digit (simple signs “yes - no”), then in the table it is enough to indicate the probability of occurrence of the sign R(k-/Dj). Probability of missing feature I. More convenient

use a uniform form, assuming, for example, for a two-digit sign. It should be clarified that , Where nij- number of attribute digits kj. The sum of the probabilities of all possible implementations of the attribute is equal to one. The decision rule is the rule according to which the decision about the diagnosis is made. In the Bayes method, an object with a complex of features ft refers to the diagnosis with the highest (posterior) probability ft e Dj, If P(Dj/lt) >

> P(Dj/ft) (J - 1, 2, ..., n i * j). This rule is usually refined by introducing a threshold value for the probability of diagnosis P(Dj/ft) >

>Pj, Where Pj- pre-selected recognition level for diagnosis Dj. In this case, the probability of the closest competing diagnosis is not higher than 1 - Pj. Usually accepted P ( > 0.9. Given that PiD/t?) a decision on diagnosis is not made and additional information is required.

Table 1.1

Diagnostic matrix in the Bayes method

Example. A diesel locomotive is under surveillance. In this case, two signs are checked: To- increase in hourly diesel fuel consumption at the nominal position of the driver’s controller by more than 10% of the rated value, to 2- reduction in the power of the diesel generator set at the nominal position of the driver’s controller by more than 15% of the rated value. Let us assume that the appearance of these signs is associated either with increased wear of parts of the cylinder-piston group (diagnosis /)]), or with a malfunction of the fuel equipment (diagnosis D 2). If the diesel engine is in good condition (diagnosis D 3) sign To not observed, but a sign to 2 observed in 7% of cases. According to statistical data, it has been established that 60% of engines diagnosed with Z) 3 are modified before scheduled repairs. D 2- 30%, with diagnosis Z)j - 10%. It was also found that the sign To j at state Z)| occurs in 10%, and in the condition D 2 - in 40% of cases; sign to 2 under state Z)| occurs in 15%, and in the condition D 2- in 20% of cases. We present the initial information in the form of a table. 1.2.

Table 1.2

Probabilities of conditions and manifestations of symptoms

Let's calculate the probabilities of states for various options for implementing controlled features:

1. Signs To And to 2 found, then:

2. Sign To detected, sign to 2 absent.

Absence of sign k i means the presence of a sign To.(the opposite event), and P(k./D.)-- P(k./D.).

3. Sign To 2 detected, sign To absent:

4. Signs /:| And to 2 missing:

Analysis of the obtained calculation results allows us to draw the following conclusions:

• 1. Presence of two signs k and k 2 s probability 0.942 indicates the condition DJ
• 2. Presence of a sign To with a probability of 0.919 indicates the condition D 2(fuel equipment malfunction).
• 3. Presence of a sign to 2 with a probability of 0.394 indicates the condition D 2(fuel equipment malfunction) and with a probability of 0.459 about state Z) 3 (proper condition). With such a probability ratio, decision making is difficult, so additional examinations are required.
• 4. The absence of both signs with a probability of 0.717 indicates a good condition (Z) 3).

Setting technical diagnostic tasks

Main directions of technical diagnostics

Basics of technical diagnostics

SECTION No. 5

Definitions. The term “diagnosis” comes from the Greek word “diagnosis”, which means recognition, determination.

During the diagnostic process, a diagnosis is established, ᴛ.ᴇ. the condition of the patient (medical diagnostics) or the state of the technical system (technical diagnostics) is determined.

Technical diagnostics is usually called the science of recognizing the state of a technical system.

Objectives of technical diagnostics. Let us briefly consider the main content of technical diagnostics. Technical diagnostics studies methods for obtaining and evaluating diagnostic information, diagnostic models and decision-making algorithms. The purpose of technical diagnostics is to increase the reliability and service life of technical systems.

As is known, the most important indicator of reliability is the absence of failures during the operation (operation) of a technical system. Failure of an aircraft engine during flight conditions, ship machinery during a ship's voyage, or power plants operating under load can lead to serious consequences.

Technical diagnostics, thanks to the early detection of Defects and malfunctions, makes it possible to eliminate such failures during the maintenance process, which increases the reliability and efficiency of operation, and also makes it possible to operate critical technical systems according to their condition.

In practice, the service life of such systems is determined by the “weakest” copies of products. During condition-based operation, each specimen is operated to its limiting condition in accordance with the recommendations of the technical diagnostic system. Condition-based operation can bring benefits equivalent to the cost of 30% of the total vehicle fleet.

The main tasks of technical diagnostics. Technical diagnostics solves a wide range of problems, many of which are related to the problems of other scientific disciplines. The main task of technical diagnostics is to recognize the state of a technical system in conditions of limited information.

Technical diagnostics are sometimes called in-place diagnostics, i.e. diagnostics carried out without disassembling the product. State analysis is carried out under operating conditions in which obtaining information is extremely difficult. Often it is not possible to draw an unambiguous conclusion from the available information and statistical methods have to be used.

The general theory of pattern recognition should be considered the theoretical foundation for solving the main problem of technical diagnostics. This theory, which forms an important section of technical cybernetics, deals with the recognition of images of any nature (geometric, sound, etc.), machine recognition of speech, printed and handwritten texts, etc. Technical diagnostics studies recognition algorithms as applied to diagnostic problems, which can usually be considered classification problems.

Recognition algorithms in technical diagnostics are partly based on diagnostic models that establish a connection between the states of a technical system and their mappings in the space of diagnostic signals. An important part of the recognition problem are decision rules (decision rules).

Solving a diagnostic problem (classifying a product as serviceable or faulty) is always associated with the risk of a false alarm or missing a target. To make an informed decision, it is advisable to use methods of statistical decision theory, developed for the first time in radar.

Solving technical diagnostic problems is always associated with predicting reliability for the next period of operation (until the next technical inspection). Here, decisions must be based on failure models studied in reliability theory.

The second important area of ​​technical diagnostics is the theory of controllability. Controllability is usually called the property of a product to provide a reliable assessment of its

technical condition and early detection of faults and failures. Controllability is created by the design of the product and the adopted technical diagnostic system.

A major task of the theory of control capacity is the study of means and methods for obtaining diagnostic information. Complex technical systems use automated condition monitoring, which involves processing diagnostic information and generating control signals. Methods for designing automated control systems constitute one of the areas of the theory of controllability. Finally, very important tasks of the theory of controllability are associated with the development of fault finding algorithms, the development of diagnostic tests, and minimizing the process of establishing a diagnosis.

Due to the fact that technical diagnostics initially developed only for radio-electronic systems, many authors identify the theory of technical diagnostics with the theory of controllability (fault detection and monitoring), which, of course, limits the scope of application of technical diagnostics.

Structure of technical diagnostics. In Fig. Figure 5.1 shows the structure of technical diagnostics. It is characterized by two interpenetrating and interconnected directions: the theory of recognition and the theory of control ability. Recognition theory contains sections related to the construction of recognition algorithms, decision rules and diagnostic models. The theory of controllability includes the development of tools and methods for obtaining diagnostic information, automated control and troubleshooting. Technical diagnostics should be considered as a section of the general theory of reliability.

Rice. 5.1. Structure of technical diagnostics

Introductory remarks. Let it be necessary to determine the state of the spline connection of the gearbox shafts under operating conditions. With excessive wear of the splines, distortions and fatigue damage appear. Direct inspection of the splines is impossible, since it requires disassembling the gearbox, i.e., stopping operation. A malfunction of the spline connection can affect the vibration spectrum of the gearbox housing, acoustic vibrations, iron content in the oil and other parameters.

The task of technical diagnostics is to determine the degree of spline wear (the depth of the destroyed surface layer) based on measurement data of a number of indirect parameters. As indicated, one of the important features of technical diagnostics is recognition in conditions of limited information, when it is necessary to be guided by certain techniques and rules to make an informed decision.

State of the system is described by a set (set) of its defining parameters (features). Of course, the set of defining parameters (features) should be different, primarily in connection with the recognition task itself. For example, to recognize the state of an engine spline connection, a certain group of parameters is sufficient, but it must be supplemented if other parts are also diagnosed.

System State Recognition- assignment of the system state to one of the possible classes (diagnoses). The number of diagnoses (classes, typical conditions, standards) depends on the characteristics of the problem and the goals of the study.

It is often necessary to select one of two diagnoses (differential diagnosis or dichotomy); for example, “faulty state” and “faulty state”. In other cases, it is extremely important to characterize the faulty condition in more detail, for example, increased wear of splines, increased vibration of blades, etc. In most technical diagnostic tasks, diagnoses (classes) are established in advance, and in these conditions the recognition task is often called a classification task.

Since technical diagnostics is associated with the processing of a large amount of information, decision-making (recognition) is often carried out using electronic computers (computers).

The set of sequential actions in the recognition process is usually called recognition algorithm. An essential part of the recognition process is selection of parameters, describing the state of the system. They must be sufficiently informative so that, given the selected number of diagnoses, the process of separation (recognition) can be carried out.

Mathematical formulation of the problem. In diagnostic tasks, the state of the system is often described using a set of signs

K=(k l , k 2 ,..., k j,..., k v), (5.1)

Where k j- a sign that has m j discharges.

Let, for example, a sign k j is a three-digit sign ( m j= 3), characterizing the gas temperature behind the turbine: reduced, normal, increased. Each digit (interval) of the sign k j denoted by k js, for example, increased temperature behind the turbine k j h. In fact, the observed state corresponds to a certain implementation of the characteristic, which is indicated by the superscript *. For example, at elevated temperatures, the implementation of the trait k*j = k j h.

In general, each instance of the system corresponds to some implementation of a set of features:

K* = (k 1 * , k 2 * ,..., k j *,..., kv*). (5.2)

In many recognition algorithms it is convenient to characterize the system with parameters x j, forming v- dimensional vector or point at v-dimensional space:

X =(x l, x 2 , x j,,xv). (5.3)

In most cases the parameters x j have a continuous distribution. For example, let x j- a parameter expressing the temperature behind the turbine. Let us assume that the correspondence between the parameter x j(°C) and three-digit sign k j is this:

< 450 to j l

450 - 550 to j 2

> 500 to j 3

IN in this case, using the sign k j a discrete description is obtained, whereas the parameter x j gives a continuous description. Note that with a continuous description, a much larger amount of preliminary information is usually required, but the description is more accurate. If, however, the statistical laws of distribution of the parameter are known, then the required amount of preliminary information is reduced.

It is clear from the previous that there are no fundamental differences when describing a system using features or parameters, and both types of description will be used in the future.

As indicated, in technical diagnostics problems the possible states of the system - diagnoses D i- are considered famous.

There are two basic approaches to the recognition problem: probabilistic and deterministic. Formulation of the problem with probabilistic recognition methods this is the case. There is a system that is in one of the random states D i. A set of signs (parameters) is known, each of which characterizes the state of the system with a certain probability. It is required to construct a decision rule with the help of which the presented (diagnosed) set of signs would be assigned to one of the possible conditions (diagnoses). It is also advisable to assess the reliability of the decision made and the degree of risk of an erroneous decision.

With deterministic recognition methods, it is convenient to formulate the problem in geometric language. If the system is characterized v-dimensional vector X , then any state of the system is a point in the v-dimensional space of parameters (features). It is assumed that diagnosis D corresponds to some region of the considered feature space. It is required to find a decision rule according to which the presented vector X * (the object being diagnosed) will be assigned to a specific area of ​​diagnosis. Thus, the task comes down to dividing the feature space into diagnostic areas.

With a deterministic approach, the domains of diagnoses are usually considered ʼʼnon-overlappingʼʼ, ᴛ.ᴇ. the probability of one diagnosis (in the area of ​​which the point falls) is equal to one, the probability of others is equal to zero. Similarly, it is assumed that each symptom is either present with a given diagnosis or absent.

Probabilistic and deterministic approaches have no fundamental differences. Probabilistic methods are more general, but they often require a much larger amount of preliminary information. Deterministic approaches more briefly describe the essential aspects of the recognition process, are less dependent on redundant, low-value information, and are more consistent with the logic of human thinking.

The following chapters outline the basic recognition algorithms for technical diagnostic problems.

Among technical diagnostic methods, the method based on the generalized Bayes formula occupies a special place due to its simplicity and efficiency.

Of course, the Bayes method has disadvantages: a large amount of preliminary information, “suppression” of rare diagnoses, etc.
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Moreover, in cases where the volume of statistical data allows the use of the Bayes method, it is advisable to use it as one of the most reliable and effective methods.

Basics of the method. The method is based on a simple Bayes formula. If there is a diagnosis D i and a simple sign k j , occurring with this diagnosis, then the probability of the joint occurrence of events (the presence of the condition in the object D i and sign k j)

P (D i k j) = P (D i) P ( k j/D i) = P ( k j)P(Di/ k j). (5.4)

Bayes’ formula follows from this equality (see Chapter 11)

P(D i / k j) = P(D i) P( k i /D i)/P( k j) (5.5)

It is very important to determine the exact meaning of all quantities included in this formula.

P(D i) - probability of diagnosis D i, determined from statistical data ( prior probability of diagnosis). So, if previously examined N objects and N i objects had a condition D i, That

P(D i) = N i/N. (5.6)

P(k j/D i) - k j for objects with state D i. In case among N i objects with a diagnosis D i, y N ij a sign appeared k j , That

P(k j/D i) = N ij /N i. (5.7)

P(k j) - probability of occurrence of a sign k j in all objects, regardless of the state (diagnosis) of the object. Let the total number N objects sign k j was discovered N j objects, then

P( k j ) = N j/N. (5.8)

To establish a diagnosis, a special calculation P(kj) not required. As will be clear from what follows , values P(D i)And P(k j/ D i), known for all possible states, determine the value P(k j).

Equality (3.2) P(D i/k j)- probability of diagnosis D i after it has become known that the object in question has the characteristic k j (posterior probability of diagnosis).

Generalized Bayes formula. This formula applies to the case when the examination is carried out according to a set of signs TO, including signs k 1 , k 2 , ..., k v. Each of the signs k j It has m j ranks ( k j l, k j 2 , ..., k js, ..., ). As a result of the examination, the implementation of the characteristic becomes known

k j *= k js(5.9)

and the whole complex of signs K*. Index *, as before, means the specific meaning (realization) of the attribute. The Bayes formula for a complex of features has the form

P(D i/TO* )= P(D i)P(TO */D i)/P(TO* )(i= 1, 2, ..., n), (5.10)

Where P(D i/TO* ) - probability of diagnosis D i after the results of the examination on a set of signs became known TO, P(D i) - preliminary probability of diagnosis D i(according to previous statistics).

Formula (5.10) applies to any of n possible states (diagnoses) of the system. It is assumed that the system is in only one of the specified states and therefore

In practical problems, the possibility of the existence of several states is often allowed A 1 , ..., A r, and some of them may occur in combination with each other. Then, as various diagnoses D i individual conditions should be considered D 1 = A 1 , ..., D r= A r and their combinations D r +1 = A 1 ^ A 2 , ... etc.

Let's move on to the definition P(TO*/ D i). If the complex of signs consists of v signs, then

P(TO*/ D i) = P( k 1 */ D i)P(k 2 */k 1* D i)...P(k v*/k l*...k*v- 1 D i), (5.12)

Where k j* = k js- category of a sign revealed as a result of the examination. For diagnostically independent signs

P(TO*/ D i) = P(k 1 */ D i) P(k 2 */ D i)... P(kv*/ D i). (5.13)

In most practical problems, especially with a large number of features, it is possible to accept the condition of independence of features even in the presence of significant correlations between them.

Probability of appearance of a complex of signs TO*

P(TO *)= P(D s)P(TO */D s). (5.14)

The generalized Bayes formula should be written like this :

P(D i/K* ) (5.15)

Where P(TO*/ D i)is determined by equality (5.12) or (5.13). From relations (5.15) it follows

P(D i/TO *)=l , (5.16)

which, of course, should be the case, since one of the diagnoses is necessarily realized, and the realization of two diagnoses at the same time is impossible.

It should be noted that the denominator of the Bayes formula is the same for all diagnoses. This allows us to first determine the probabilities of co-occurrence i-th diagnosis and given implementation of the complex of signs

P(D iTO *) = P(D i)P(TO */D i) (5.17)

and then the posterior probability of diagnosis

P(D i/TO *) = P(D i TO *)/P(D s TO *). (5.18)

Note that sometimes it is advisable to use preliminary logarithm of formula (5.15), since expression (5.13) contains products of small quantities.

If the implementation of a certain set of features TO * is determining for diagnosis Dp, then this complex does not occur in other diagnoses:

Then, by virtue of equality (5.15)

(5.19)

However, the deterministic logic of diagnosis is a special case of probabilistic logic. Bayes' formula can also be used in the case when some of the features have a discrete distribution, and the other part has a continuous distribution. It is worth saying that for continuous distribution, distribution densities are used. Moreover, in the calculation plan, the specified difference in characteristics is insignificant if the continuous curve is specified using a set of discrete values.

Diagnostic matrix. To determine the probability of diagnoses using the Bayesian method, it is extremely important to create a diagnostic matrix (Table 5.1), which is formed on the basis of preliminary statistical material. This table contains the probabilities of character categories for various diagnoses.

Table 5.1

Diagnostic matrix in the Bayes method

If the signs are two-digit (simple signs “yes - no”), then in the table it is enough to indicate the probability of occurrence of the sign P (k i /D i). Probability of missing feature R( /D,-) = 1 - P (k i /D i).

In this case, it is more convenient to use a uniform form, assuming, for example, for a two-digit attribute R (k j/D i)= R(k i 1 /D i); R( /D,) = P (k i 2 /D i).

Note that P(k js/Di)= 1, where T, - number of attribute digits k j. The sum of the probabilities of all possible implementations of the attribute is equal to one.

The diagnostic matrix includes a priori probabilities of diagnoses. The learning process in the Bayes method consists of forming a diagnostic matrix. It is important to provide for the possibility of clarifying the table during the diagnostic process. To do this, not only values ​​should be stored in the computer memory P(k js/Di), but also the following quantities: N- the total number of objects used to compile the diagnostic matrix; N i- number of objects with diagnosis D i; N ij- number of objects with diagnosis D i, examined based on k j. If a new object with a diagnosis arrives Dμ, then the previous a priori probabilities of diagnoses are adjusted as follows:

(5.20)

Next, corrections are introduced to the probabilities of the features. Let the new object with the diagnosis Dμ discharge detected r sign k j. In this case, for further diagnostics, new values ​​of the probability intervals of the feature are accepted k j upon diagnosis Dμ:

(5.21)

Conditional probabilities of signs for other diagnoses do not require adjustment.

Example. Let us explain the Bayes method. Let two signs be checked when observing a gas turbine engine: k 1 - increase in gas temperature behind the turbine by more than 50 °C and k 2- increase in time to reach maximum speed by more than 5 s. Let us assume that for this type of engine the appearance of these symptoms is associated either with a malfunction of the fuel regulator (condition D 1 ,), or with an increase in the radial clearance in the turbine (state D 2).

When the engine is in normal condition (condition D 3) sign k 1 is not observed, but a sign k 2 is observed in 5% of cases. Based on statistical data, it is known that 80% of engines produce a service life in normal condition, 5% of engines have a condition D 1 and 15% - condition D2. It is also known that the sign k 1 occurs in the condition D 1 in 20%, and in case of condition D 2 in 40% of cases; sign k 2 in condition D 1 occurs in 30%, and in the condition D 2- in 50% of cases. Let's summarize these data in a diagnostic table (Table 5.2).

Let us first find the probabilities of engine states when both signs are detected k 1 and k 2 . To do this, considering the signs to be independent, we apply formula (5.15).

State probability

Similarly we get P (D 2 /k 1 k 2) = 0,91; P (D 3 /k 1 k 2)= 0.

Let us determine the probability of engine conditions if the examination shows that there is no increase in temperature (sign k 1 2 are different from zero, since the characteristics under consideration are not determining for them. From the calculations carried out, it can be established that if there are signs k 1 And k 2 with probability 0.91 there is a condition in the engine D1,ᴛ.ᴇ. increase in radial clearance. In the absence of both signs, the most likely condition is normal (probability 0.92). In the absence of a sign k 1 and the presence of a sign k 2 state probabilities D 2 And D 3 approximately the same (0.46 and 0.41) and additional examinations are required to clarify the condition of the engine.

Table 5.2

Feature probabilities and prior state probabilities

Decisive rule- the rule according to which the decision on diagnosis is made. In the Bayes method, an object with a complex of features TO * refers to the diagnosis with the highest (posterior) probability

K*D i,If P(D i / K*) > P(D j / K*) (j = 1, 2,..., n; i ≠ j). (5.22)

Symbol , used in functional analysis, means belonging to a set. Condition (5.22) indicates that an object possessing a given implementation of a complex of features TO * or, in short, implementation TO * belongs to the diagnosis (condition) D i . Rule (5.22) is usually refined by introducing a threshold value for the probability of diagnosis:

P (D i /K *) ≥ P i, (5.23)

Where Pi.- pre-selected recognition level for diagnosis D i. In this case, the probability of the closest competing diagnosis is not higher than 1 – P i. Usually accepted P i≥ 0.9. Given that

P(D i /K *)

(5.24)