RU2565494C2 - Method of measurement of fuzzy information - Google Patents

Abstract

FIELD: instrumentation.

SUBSTANCE: invention relates to metrology and can be used for measurement and processing of fuzzy information. The method involves: for measurement of fuzzy information a conditional scale of fuzzy reference objects - terms on a base set is formed by appointment of standard representatives and then all values of the base set are phased by means of membership functions with semantics of initial incomplete measuring information; then on the basis of this conditional scale the incomplete measuring information is phased which is used for preliminary measurement of fuzzy information by determination of its fuzzy inclusion into standard terms; meanwhile the term with the greatest inclusion is considered as a preliminary result of measurement of fuzzy information; the corrections are added to the preliminary result, using operations of its dephasing taking into account values of membership functions in all terms of the scale and all its standard values; the secondary amendment is added by repeated phasing of the named initial information and its subsequent dephasing.

EFFECT: improvement of accuracy.

Description

The proposed method for measuring fuzzy information requires preliminary explanations of the concepts of "fuzzy information" and "measurement of fuzzy information."

First, touch on the concept of "information."

A sufficiently complete and general definition of the essence of information is given in the classical work of C. Shannon [1]. According to this work, information is a sequence of messages, it can be represented by letters, a set of functions that can be quite accurately determined by the statistical (probabilistic) method.

In [2] one can find a concretization of the concept of information by its nature: deterministic and probabilistic. The first is causal; it can be determined a priori. Probabilistic information allows us to judge the statistical characteristics of the studied objects, i.e. here we have in mind the case of repeated tests (measurements). The researcher considers this information in three aspects: practical - from the point of view of achieving specific goals, semantic - from the point of view of semantic content and its correct use, syntactic (technological) - from the point of view of the method and technique of transmitting information. The most common of these is the first aspect.

Another definition of information is also known - as the difference in the entropies of the laws of distribution of a random variable before and after measurement.

The latter assumes the presence of statistical data in the basis, i.e. measurements of the studied quantities and, on their basis, indirect measurements (calculations) of information.

In accordance with [3] (clause 5.18) we have the definition: "Measuring information - information on the values of physical quantities."

Analyzing this definition, we can see that the measuring (clear) information is determined by the values of physical quantities, i.e. measuring information and the values of physical quantities are synonyms: to obtain the value of a physical quantity and means to obtain information about a physical quantity.

This combination allows you to use the phrase - measurement (evaluation) of information (values of physical quantities).

Fuzzy information as an object of study (measurement) was widely used earlier and is currently used [4, 5].

Consider the difference between fuzzy information and clear information.

In the above sense, fuzzy information is difficult to interpret given its features. These features are expressed by the specific presentation of fuzzy information: membership functions of the values of physical quantities to fuzzy sets.

In them, by definition, the characteristics of the task of linguistic variables are reflected: name, set of meanings (terms), syntactic and semantic procedures that allow the formation of syntagmas of a certain meaning.

Thus, fuzzy information is represented by a set of fuzzy values of physical quantities, expressed as membership functions of these values (elements) of fuzzy sets. Thus, these elements can be the values of physical quantities - parameters of the studied, operated objects (devices) in difficultly controlled or difficultly controlled conditions, where ordinary measurements are difficult or impossible, or limited in volume.

From this explanation it follows that fuzzy information is a set of fuzzy numbers, the values of the measured parameters of the studied objects (processes), represented by a set of membership functions.

“Evaluate (measure) fuzzy information” means first measure in a limited amount in a known manner in accordance with the concept of “measurement” [3] the value of a parameter (physical quantity), or evaluate its value expertly, and then determine (evaluate) the membership function. The values of membership functions obtained in this way for all parameters must be compared with conditional scales of fuzzy parameters using algebraic operations. The comparison results will also mean the results of the assessment (measurement) of fuzzy information. The fuzzy information obtained in this way can be used in diagnosing (predicting) the state of objects characterized by appropriate conditions, circumstances of use, operation, storage, etc.

We introduced the concept of “conditional scales of fuzzy parameters”, which can be extended to “conditional scales of fuzzy situations”. The very notion of a conditional scale is given in [3], and we present a fuzzy situation: a fuzzy situation is a set of parameter values and the corresponding values of membership functions at a certain point in time. In a particular case, the situation may be characterized by the value of one parameter and the corresponding value of the membership function.

Let us explain their discrepancy with the known scales [3]: names, order, intervals, relations.

The first classifies sizes according to signs or equivalence, or identity, or equality. Its essence is represented by sets of signs: "=" or "≠".

The second ranks the sizes of the parameters in increasing (decreasing) degree, while the sizes are characterized qualitatively: “more” or “less”. The essence of the scale expresses a set of characters: "=", "≠", "<", ">".

The last two (from the list above) - scales of intervals and relations - metrological. Measurements based on them are associated with comparing the measured physical quantity with a measure (unit of physical quantity).

All of the above scales suggest one or another comparison of similar objects (according to parameters, characteristics, etc.).

For fuzzy information, such a comparison is difficult (almost impossible). Therefore, it is proposed to use conditional scales, based on which it is proposed to use an ordered set (sets) of terms on the basic sets of parameter values (characteristics), which are mathematized using the indicated scales.

In order to be able to operate with fuzzy information, conditional scales must be converted into a new form - into algebra-scales, which will be used as a measure of fuzzy information.

The following is a detailed explanation of the claims.

Нижнее нулевое значение получается путем элементарного преобразования: путем вычитания нижнего предметного значения из текущих предметных значений параметра.The expert method (heuristically) assigns the number of terms and their typical representatives on the base set. The basic set can be defined by the boundaries of the controlled (measured) parameter on the subject scale, for example [0; X _b ], where X _b is the upper value of the base set, 0 is the lower value, or on a universal scale like this: [0; l], where fuzzy values are evaluated as a ratio

The lower zero value is obtained by an elementary transformation: by subtracting the lower subject value from the current subject value of the parameter.

After this operation, the identified expression of the membership function is selected, which would take into account all the parameters that allow them to adapt to changing semantics.

For these purposes, you can use various expressions. For example, the authors propose using the following expression of membership functions of elements (parameter values) to a fuzzy set

- функция принадлежности значений xⁱ нечеткому множеству

;Where

- the membership function of the values of x ⁱ fuzzy set

;

i is the number of the measured (investigated) parameter;

- порог нечеткого включения, принимается равным ординате точки пересечения функций принадлежности двух соседних термов, например, с номерами $$j_iиj_{i+1}\cdot t_{n_i}\in$$

(0,5;1); $$t_{n_i}$$

- the threshold of fuzzy inclusion, is taken equal to the ordinate of the point of intersection of the membership functions of two adjacent terms, for example, with numbers $$j_i\mathrm{and}{j}_{i+\mathrm{one}}\cdot t_{n_i}\in$$

(0.5; 1);

- xⁱ параметра;j _i - term number on the base set $$X_{b_i}$$

- x ⁱ parameter;

- базовое множество xⁱ -параметра; $$X_{b_i}$$

- base set of x ⁱ -parameter;

(без учета нулевого) для упрощения (1);J _i - the number of terms selected on the base set $$X_{b_i}$$

(excluding zero) to simplify (1);

- параметр функции принадлежности, соответствующий степени нарушения комплементарности нечетких множеств $${\tilde{X}}_{j_i},j_i\in \left\{\mathrm{0,1,}\dots ,j_i\right\}.$$

- parameter of the membership function corresponding to the degree of violation of the complementarity of fuzzy sets $${\tilde{X}}_{j_i},j_i\in \left\{\mathrm{0,1}...,j_i\right\}.$$

When changing the semantics of terms, expression (1) is adapted by using the appropriate display functions, which shift the internal terms on the base set towards smaller or larger values.

The first bias is associated, for example, with tightening the requirements for the parameter: what was “average” became “large”, and vice versa. In the second case, the “average” became “small”.

Here is an example to explain: [0; 100] - the basic set of temperature values; the terms “low”, “medium”, “high” temperature with typical values are used $$x_{j=0}^{{}^i}=0^{\circ }C;x_{j=\mathrm{one}}^{{}^i}={\mathrm{fifty}}^{\circ }C;x_{j=2}^{{}^i}={\mathrm{one\; hundred}}^{\circ }C.$$

Circumstances have changed for which the “average” temperature of 50 ° C has become “low” and its value has been changed to 65 ° C. Then the term membership function will also change. When moving to a universal scale (from x to δ), the membership function with a linear subject scale (as indicated at the beginning of the example) takes a different form

- линейная функция отображения: осуществляет переход к универсальной шкале без ее деформации;Where

- linear display function: makes the transition to a universal scale without deformation;

In order to adapt the temperature scale to the new requirements, it must be converted using the display function. To reduce the requirements, the average term is shifted towards large values specified in the example. Therefore, you can use the display function of the form

where γ ⁱ is the value of the i-th parameter on the adapted universal scale;

δ ⁱ is the value of the i-th parameter on the non-adapted (linear) universal scale;

k = ln (γ ⁱ ) / ln (δ ⁱ ) is the parameter of the mapping function, which determines its convexity (concavity), corresponding to the displacement of the average term. In our example, an offset of 15%. This will correspond to 50 ° C + 15 ° C = 65 ° C on the subject adapted scale, on the universal adapted scale: 0.5 + 0.15 = 0.65.

, что соответствует показателю некомплементарности, равному 0,226, методика его оценивания известна, выражение (2) примет видAt t _n = 0.6 and $$g_0^{j_i}=\mathrm{1,5}$$

, which corresponds to a non-complementarity index of 0.226, the methodology for its assessment is known, expression (2) takes the form

- новое значение j_i-го терма i-го параметра;Where $${\gamma }^i{}_{j_i}={\delta }^i{}_{j_i}+{\Delta }^i{}_{j_i}$$

- the new value of the j _i -th term of the i-th parameter;

- смещение j_i-го терма i-го параметра. $${\Delta }^i{}_{j_i}$$

- offset j of the _i- th term of the i-th parameter.

In our example $${\delta }^i{}_{j_i}+{\Delta }^i{}_{j_i}=0.5+0.15=\mathrm{0.65.}{J}_i=2.$$

всех параметров и всех термов параметров, для получения требуемой точности нечеткой информации вводятся логические операции, позволяющие сравнить фактические ситуации с эталонными. К числу таких операций относятся:Further, using the results of fuzzification, the values

all parameters and all terms of parameters, to obtain the required accuracy of fuzzy information, logical operations are introduced that allow you to compare the actual situation with the reference. These operations include:

- fuzzy inclusion of fuzzy situations

- fuzzy equality of fuzzy situations (objects)

- типовой (эталонный) объект (ситуация, мера);Where

- typical (reference) object (situation, measure);

- нечеткий текущий объект (ситуация);

- fuzzy current object (situation);

→ - implication;

& - conjunction.

To detail these operations, private operations are used, expressed in terms of membership functions (to illustrate, we express them in a general way: through membership functions on a subject scale).

- negations (additions) of a fuzzy set $${\tilde{X}}_i$$

- conjunctions

- clauses

- implications

с учетом адаптации (перехода на шкалы δ или γ) вполне очевидно.Their use in the assessment

taking into account adaptation (transition to the scales of δ or γ) is quite obvious.

с мерой - с типовыми объектами (нечеткими ситуациями

Thus, operations on scales reflecting the sets of fuzzy objects (elements) are expressed through the membership functions of these elements to fuzzy sets that are specific in nature or (2) or (4) and allow you to implement a measurement method - comparing a fuzzy object (situation

with a measure - with typical objects (fuzzy situations

или

измеряемый объект (параметр, ситуация) приравнивают той

, при которой

или

оказались max. Так, косвенным путем получают предварительный результат измерения нечеткой информации.After identifying all

or

the measured object (parameter, situation) is equated with

, with which

or

turned out to be max. So, indirectly, a preliminary result of measuring fuzzy information is obtained.

In order to improve the accuracy of the preliminary measurement result of fuzzy information, amendments are introduced into them by using the defuzzification operation, using the rule

- уточненное значение параметра, соответствующее моменту времени t_k Where

- the adjusted parameter value corresponding to the time t _k

- более краткое обозначение нечеткого множества

- a shorter designation of a fuzzy set

- типовое значение j_i-го терма i-го параметра.

- the typical value of the j _i -th term of the i-th parameter.

выражает вес (значимость) нечеткой информации в каждом терме условной шкалы, т.е. шкалирует нечеткую информацию в каждом терме путем оценивания ее веса.Attitude

expresses the weight (significance) of fuzzy information in each term of the conditional scale, i.e. scales fuzzy information in each term by estimating its weight.

Thus, rule (7) convolves all the fuzzy information in all terms about the investigated (measured) parameter γ ⁱ corresponding to the time t _k , and thereby amend $${\gamma }^i{}_{t_k}$$

, выполняя процедуры правила свертки (7).Using the refined fuzzy information, the re-refined measurement information is then determined

performing the convolution rule procedures (7).

The criterion for stopping this process are inequalities of the form

- допустимое значение погрешности оценивания параметра γⁱ (назначается);Where

- the permissible value of the error in estimating the parameter γ ⁱ (assigned);

0 <α <1 - coefficient of reduction of the permissible error. Expression (8) takes into account, as an example, the first and second refinements. Similarly, for example, m and (m + 1) refinements are taken into account.

Literature

1. Shannon K. Works on information theory and cybernetics. M .: Foreign literature. 1963.829 s.

2. Mathematics and cybernetics in economics. Dictionary dictionary. M .: Economics. 1975.699 p.

3. RMG 29-99. Metrology. Key terms and definitions.

4. Krugloe V.V., Dli II, Golunov R.Yu. Fuzzy logic and artificial neural networks. M .: Fizmatgiz. 2001.225 s.

5. Melikhov A.N., Bernstein L.S., Korovin S.Ya. Situational advisory systems with fuzzy logic. M .: Science. 1990.272 p.

Claims (1)

A method for measuring fuzzy information involves the use of scales (intervals, relationships) to obtain incomplete measurement information on the base set, characterized in that for measuring fuzzy information, a conditional scale of fuzzy reference objects is formed - terms on the base set by assigning typical representatives and subsequently fuzzy all values basic set by using functions, accessories with semantics of initial incomplete measuring information; then, based on this conditional scale, incomplete measurement information is fuzzified, which is used for preliminary measurement of fuzzy information by determining its fuzzy inclusion in standard terms; while the term with the largest inclusion is considered a preliminary result of measuring fuzzy information; Corrections are introduced into the preliminary result; for this, operations of its defuzzification are used taking into account the values of membership functions in all terms of the scale and all its typical values; the secondary amendment is introduced by repeated fuzzification of the updated initial information and its subsequent defuzzification.