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XXIII International Conference on Soft Computing and Measurement (SCM'2020)

Journal of Physics: Conference Series 1703 (2020) 012042

IOP Publishing

doi:10.1088/1742-6596/1703/1/012042

1

Combining Bayesian and logical-probabilistic approaches for

fuzzy inference systems implementation

G I Kozhomberdieva1 and D P Burakov2

1 Department of Information and Computing Systems, Emperor Alexander I

St. Petersburg State Transport University, Moskovsky pr., 9, Saint-Petersburg,

190031, Russia

2 Department of Informatics and Information Security, Emperor Alexander I St.

Petersburg State Transport University, Moskovsky pr., 9, Saint-Petersburg, 190031,

Russia

E-mail: kgi-liizht@yandex.ru

Abstract. In the paper, an original way to implement the fuzzy inference systems is considered.

The proposed fuzzy inference model is based on usage of conjunction of probabilistic logic and

Bayes’ formula at inference stage. For that preconditions of all the specified fuzzy production

rules are transformed to the set of probabilistic functions. As the input probabilities, the values

of membership functions of input linguistic variables terms are used. At that given set of

production rules is reduced in the way that number of rules will be equal to number of terms of

output linguistic variables. For that logic expressions of production preconditions are

transformed to orthogonal DNFs. Calculated probability values are used as conditional

probabilities in determining the posterior distributions on the set of hypotheses corresponding to

values of the output linguistic variables. To determine the posterior probability distribution, we

use the formula based on the Bayes’ formula. These posterior distributions are used at

defuzzification stage for determining final output variable values, because they are calculated as

expected values of corresponding random variables. The program implementation has allowed

to estimate the proposed model effectiveness and to compare its results with results of

traditionally used Mamdani and Sugeno fuzzy inference algorithms.

1. Introduction

In present days, researches in the field of fuzzy sets and fuzzy inferences, fuzzy models and fuzzy control

systems are still relevant. The research area started by proceedings of L. A. Zadeh (see, for example,

[1]) has begun to development by implementation of fuzzy controllers and its embedding to Japanese

home appliances which now widespread. [2–6]. Developers of systems with artificial intelligence

elements are continuously “retreating from precision in the face of overpowering complexity” [1, p.

201].

Authors of the papers had experience in using fuzzy logic inference and Bayesian approach for

solving estimation tasks in conditions of incompleteness (inaccuracy) of information [7−9]. Both these

approaches can use as source data for decision making the expert estimates that are inaccuracy

(incompletely defined), this factor causes the methods similarity. Based on this assumption we consider

new approach to fuzzy logic inference task. Main elements of the approach presented at XXII

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

Journal of Physics: Conference Series 1703 (2020) 012042

IOP Publishing

doi:10.1088/1742-6596/1703/1/012042

2

International conference on soft computing and measurements SCM'2019 (https://scm.etu.ru/2019/),

then we have composed paper [10] containing joined and refined results.

Purpose of fuzzy logic inference, is determining values of output linguistic variables (LV) basing on

information about crisp input variable values and taking into account set of specified fuzzy production

rules (FPR).

We suggest that when the fuzzy inference is performed, there is possible [10]:

1. To consider set of values of each input LV as the source for set of Bayesian hypotheses about

some output LV takes any value from its set of terms (term-set).

2. To interpret estimated “crisp” values at scales of input LVs, as evidence for all these

Bayesian hypotheses.

3. To interpret values of membership function (MF) of fuzzy sets of corresponding terms of

input LVs as arguments of some probability logic functions (PLF).

4. To transform set of FPRs to a set of PLFs, arguments of which are degrees of membership

of the estimates-evidence to fuzzy sets, and calculated values are used as conditional

probabilities (degrees of Bayesian hypotheses trueness, i.e. likelihood of them [11]) when

determining the posterior distribution on the set of hypotheses corresponding to the values

of the output LV.

5. To use obtained for each output LV the Bayesian probability distribution to determine

defuzzified value of the variable as expected value of a random variable.

As an argument for the possibility to use Bayesian logical-probabilistic approach in the fuzzy logic

inference we take a remark by L. Zadeh himself about the analogy that can be drawn between the

concepts of fuzziness and probability [1, p. 204]. In the example given by L. Zadeh, it is said about

membership degree of a specific numerical value to a certain LV term. It is possible to mean this

membership value as degree of correspondence of given evidence (estimated “crisp” value) to

assumption about trueness of hypothesis that LV takes some value from its term-set, i.e. understand it

as subjective or Bayesian probability, defined as degree of certainty that some assertion is true [12].

We should note that there is not only L. Zadeh had marked the similarity between fuzziness and

probability: other researches also discussed their similarity (see, for example [13]); moreover, some of

them also name fuzziness as “masked probability”. Therefore, usage of the known apparatus of

probabilistic logic in the Bayesian logical-probabilistic model of fuzzy inference seems understandable

and justified.

In the paper, we present results were obtained by combining of Bayesian and logical-probabilistic

approaches for implementation of the new fuzzy inference model.

This year, undergraduate student of the Department of Information and Computing Systems of our

university G. А. Khamchichev, under the guidance of one of the authors, develops a software

implementation of the model, which allowed us to compare our model with the traditionally used

algorithms of Mamdani and Sugeno and confirmed its effectiveness.

2. Fuzzy logic inference based on the Bayes’ formula and probabilistic logic

2.1. Fuzzy logic inference stages

The fuzzy logic inference consists of the following stages [2, 6]:

1. Preparation stage, at which input

m

XXX ~

,...,

~

,

~21

and output

s

YYY ~

,...,

~

,

~21

LVs are defined and

set of FPRs

t

RRR ~

,...,

~

,

~21

are formed.

2. Fuzzification stage, at which input values are fuzzyfied, i.e. for each measured value xi is

determined membership value

)(μi

Tx

i

j

for each term

i

j

T

of corresponding input LV

ii njmiX ,1,.1,

~==

.

3. Conclusions inference stage, at which, using given FPRs, trueness degrees of each possible

conclusion for each output LV are determined;

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

Journal of Physics: Conference Series 1703 (2020) 012042

IOP Publishing

doi:10.1088/1742-6596/1703/1/012042

3

4. Defuzzification stage, at which final “crisp” values

**

2

*

1,...,, s

yyy

at scales

s

YYY ,...,,21

of each

output LV

s

YYY ~

,...,

~

,

~21

are calculated.

5. Proposed logical-probabilistic approach of fuzzy inference affects only the conclusions

inference and defuzzification stages. We'll describe it without loss of generality for a case

with single output LV

Y

~

.

2.2. Preparatory definitions and transformations

Denote term-set of input LV

i

X

~

as

},...,,{)

~

(21 i

n

ii

ii

TTTXT =

, and term-set of output LV

Y

~

as

},...,,{)

~

(21 N

TTTYT =

. Each FPR

tjRj,1,

~=

, in a fuzzy inference system (FIS) can be denoted as follow:

( )

l

k

i

vi

n

k

jTYTXR =

=

=

~

THEN

~

IF:

~

1

, (1)

where

},,{

is elementary logical operation,

i

X

~

is i-th input LV,

Y

~

is output LV,

)

~

(i

i

vXTT

and

)

~

(YTTl

are terms of corresponding LVs,

Nlmi ,1,,1 ==

.

In FIS, for each FPR

j

R

~

is assigned a real number wj[0, 1], named rule weight. The weight is

understood as measure of certainty that conclusion assertion in the FPR

j

R

~

derived by precondition

assertion.

If we denote k-th fuzzy logic assertion

( )

k

i

vi TX =

~

in (1) as Boolean variable zk, precondition of FPR

j

R

~

can be presented by the Boolean function (BF)

tjzzzF k

n

k

nBj,1,),...,(1

1== =

.

Main proposed idea of the Bayesian logic-probability approach to fuzzy inference involves transition

from BFs in preconditions of FPRs to PLFs, calculate PLF values and use them as conditional

probabilities for determining posterior probability distribution on a set of Bayesian hypotheses

corresponding to terms of output LV

Y

~

.

A feature of the approach is the requirement that number of FPRs be equal to number of terms of

output LV. So, if source FPR set contains several FPRs with the same conclusion part

l

TY =

~

, they

replaced with one generalized FPR, containing assertion

l

TY =

~

as conclusion and containing at

precondition part a BF constructed as disjunction of preconditions of generalized FPRs. After that,

reduced FPR set will contain precisely N of FPRs

N

RRR ~

,...,

~

,

~21

, where N is number of terms in

)

~

(YT

,

and, also, number of corresponding Bayesian hypotheses. When several FPRs replaced with one

generalized FPR, weight wo of this FPR is determined as average value of weights of excluded FPRs.

All the BFs that are FPR preconditions, in order to simplification transformation to PLFs, are

preliminarily transformed to orthogonal DNF form (ODNF). This transformation is executed due to

algorithm proposed in [14].

Formal rules of transformation BF described in ODNF form to the corresponding PLF the following:

1. Boolean variables

n

zz ,...,

1

are replaced with the corresponding probabilities

n

pp ,...,

1

.

2. Inversions

i

z

are replaced with

i

p−1

.

3. Conjunction and disjunction signs are replaced with arithmetic multiplication and addition

correspondingly.

In Table 1, examples of transformation elementary BFs to PLFs are presented.

Note that probability logic operations presented in Table 1 are associative, commutative and

monotonic, have segment [0, 1] as the codomain, moreover, they use either 1 or 0 as the neutral element.

Therefore, they meet requirements for t-norm and t-conorm using for conjunction and disjunction

operations implementation in the fuzzy logic [3].

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

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doi:10.1088/1742-6596/1703/1/012042

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Table 1. Elementary BFs and corresponding PLFs.

BF

PLF

iiB zzF =)(

iiP pzF −=1)(

jijiB zzzzF =),(

jijiP ppzzF =),(

jiijijiB zzzzzzzF ==),(

jijijiP ppppzzF −+=),(

So, the proposed model of fuzzy inference assumes that MF values of input LV terms are interpreted

as subjective probabilities

i

T

i

ji

TX njmixTXPp i

j

i

ji ,1,,1),(μ)

~

(

~=====

, then FPR set

}

~

{l

R

is

transformed to the set of PLFs

)},...,({ 1nP zzF l

corresponding to BFs

NlzzF nBl,1)},,...,({ 1=

, presenting

preconditions of these FPRs.

2.3. Conclusions inference stage

A PLF

),...,( 1nP zzF l

value calculated basing on subjective probability values

i

jiTX

p~

, is used as

conditional probability P(e | Hl) that measures how the evidence e, expressed by given “crisp” values on

input LV scales, compliances assumption about trueness of Bayesian hypothesis Hl that

NlTY l,1,

~==

.

Derive equation for calculation the posterior probability distribution {P(Hl | e)}. The equation basing

on the well-known Bayes’ formula [15]:

=

=N

kkkm

llm

ml HPHeeP

HPHeeP

eeHP

11

1

1)()|,...,(

)()|,...,(

),...,|(

, (3)

where N is number of hypotheses, m is number of evidence, and P(Hl) is prior probability of

corresponding hypothesis Hl.

In our case, number of hypotheses equals to number of FPRs (because, if necessary, FPR set is

reduced, as it is described above), at that, for each hypothesis Hl there is only one evidence, i.e. m=1.

Besides that, because prior probabilities are not used at fuzzy logic inference, let P(Hl) = 1/N. Also we

must use FPR weight wl as the factor that decreases P(e | Hl) value as level of compliance of evidence e

to assumption of trueness of Bayesian hypothesis Hl,

Nl ,1=

.

Finally, we have the equation:

=

=N

kkk

ll

lHePw

HePw

eHP

1

)|(

)|(

)|(

. (4)

Using equation (4), we make a posterior distribution of Bayesian probabilities on the set of hypotheses

that the output LV

Y

~

takes each value Tl from its term-set

)

~

(YT

. This distribution we will use for output

LV defuzzification.

2.4. Defuzzification stage

Purpose of the defuzzification stage is to obtain “crisp” value

Yy

*

of output LV

Y

~

, taking into

account results of conclusions inference and forms of MFs

Niy

i

T,1),(μ=

of fuzzy sets – terms of this

LV. It assumes that term-set

)

~

(YT

is defined at some segment Y=[ymin, ymax]R that is union of supports

of all terms of the LV. The defuzzified (“crisp”) value

*

y

must be “characterized value” of some term

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

Journal of Physics: Conference Series 1703 (2020) 012042

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doi:10.1088/1742-6596/1703/1/012042

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of

Y

~

, for example, it can be either one of this term’s MF modal values, or some weighed central point,

where MF’s values are used as weights.

The traditional fuzzy inference approach [2, 3] forms an unified term TR on Y – the scale of LV

Y

~

.

A MF

)(μy

R

T

of this term is constructed as a combination of MFs

Niy

i

T,1),(μ=

, of source terms of

this LV, defined before fuzzy inference. Within the combination process, modified MFs of activated

terms are unified to aggregated MF

)(μy

R

T

using some aggregation rule (sum or max). Modification of

source MFs

Niy

i

T,1),(μ=

, is defined by some selected composition rule (min or prod).

In Figure 1, there is an example of obtaining

)(μy

R

T

for final term TR of output LV

Y

~

having three

terms T1, T2 and T3, defined at scale Y=[a, d]. Here R(Ti) is trueness degree of corresponding FPR’s

precondition. Hypograph of the MF

)(μy

R

T

of final term TR is filled up in Figure 1.

Figure 1. Forming of unified term TR.

To obtain a defuzzified value of output LV, is selected at the scale Y an abscissa either of any modal

value of MF

)(μy

R

T

(it can be left, right or middle modal value), or of centre of area or of gravity of the

resulting hypograph [2, 3].

Feature of the proposed approach is that at conclusions inference stage the generalized final term has

not formed. Instead of this, we define probability distribution

NiTP i,1)},({ =

on the set of output LV

terms. Thus, defuzzified value

*

y

must take into account MFs forms of output LV terms and given

probability distribution.

In simple defuzzification case, it is enough to use boundary points of terms at scale of output LV,

weigh them by given Bayesian probabilities

NiTP i,1)},({ =

and then sum obtained values.

At that, easy to spot that term-set

)

~

(YT

is part of number segment Y, on

)

~

(YT

there is defined

discrete probability distribution

NiTP i,1)},({ =

, so we can obtain value

*

y

as the expected value [15]

of a random variable y:

=

= N

iii TPyyE

1

)(

~

)(

, (5)

where

i

y

~

is “characterized value” of term

)

~

(YTTi

, and P(Ti) is probability that output LV

Y

~

equals

to value Ti,

NiYyi,1,

~=

.

A “characterized value” of term Ti in this case is any internal value

],[

~max,min, iii TTy

, where segment

[Ti,min, Ti,max] is such part of scale Y, where MF

)(μy

i

T

dominates MFs

)(μy

j

T

of other terms:

m(y)

a d y

mT2(y)

R(T1)

R(T3)

R(T2)

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

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.,,1

)},(μ)(μ|{sup

)},(μ)(μ|{inf

max,

min,

jiNj

yyYyT

yyYyT

ji

ji

TTi

TTi

=

=

=

(6)

As we can see, scale of output LV is covered by sequence of segments [Ti,min, Ti,max],

Ni ,1=

. For

example, in Figure 2, LV

Y

~

has three terms T1, T2 and T3, defined on scale [a, d]. According to rule (6),

T1,min=a, T1,max=b; T2,min=b, T2,max=c; T3,min=c, T3,max=d.

Figure 2. Segments of terms on LV scale.

As an internal point of segment [Ti,min, Ti,max], there is possible to take the middle point:

2max,min, ii

iTT

y+

=

. (7)

But equation (7) does not take into account MF

)(μy

i

T

values on segment [Ti,min, Ti,max], that

determine degree of membership for values at this segment to the corresponding fuzzy set. To consider

MF values, is necessary to select as the “characterized value” of term Ti the weighed center of the

segment, using equation:

=max,

min,

max,

min,

)(μ

)(μ

ˆi

i

i

i

i

i

T

T

T

T

T

T

i

dyy

dyyy

y

. (8)

Note that if

)(μy

i

T

is symmetric at [Ti,min, Ti,max],

i

y

and

i

y

ˆ

, calculated by (7) and (8), be equal.

Obtained set of pairs

))}(,

~

{( ii TPy

, where either

ii yy =

~

or

Niyy ii ,1,

ˆ

~==

, is discrete random

variable, its expected value (5) is the defuzzified value

Yy

*

.

2.5. Defuzzification method requirements

It should be noted that defuzzification method used for fuzzy inference, must have such qualities as

continuity, uniqueness and likelihood [4], at same time, this method must have low computational

complexity because that is important for embedding to fuzzy regulators and microcontrollers.

Continuity assumes that low input values variation should perform low variation of defuzzified value

*

y

, uniqueness means that for any input values combination the defuzzification method should return

only single value

*

y

, and likelihood means that the defuzzified value

*

y

every time is “characterized

value” of any output LV term. Finally, low computational complexity defines, how computationally

intensive is process of finding defuzzified value

*

y

basing on information about source terms of output

LV and obtained final term.

m(y)

a b c d y

mT2(y)

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

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Note that with traditional fuzzy inference, defuzzified value can be obtained by five different ways,

so uniqueness and likelihood of the result can be broken, therefore constructed FIS must be verified and

maybe debugged [2–4]. Besides that, obtaining defuzzified value using centroid methods requires

implementation of numerical integration methods, which often are computationally complex, especially

if nonlinear MFs (Gaussian, sigmoid, etc.) there are used.

Proposed defuzzification method based on expected value E(y) calculation, meets continuity and

likelihood conditions according to features of expected value: defuzzified value

*

y

, depending on

probabilities in distribution

NiTP i,1)},({ =

, shifts towards “characteristic value” of the most probable

output LV term. Uniqueness of the method is not worse of traditional way, because it has less

arbitrariness when “characterized values” of terms are chosen using equation (7) or (8). Moreover, the

proposed approach does not require construction final term TR of output LV, this feature decreases

computational complexity because in case using equation (7) to find “characterized values” of terms Ti,

the obtained solution

*

y

is weakly different from defuzzified value obtained by equation (8).

3. Examples

3.1. Problem “Dinner for two”

The well-known problem “Dinner for two” (included in MATLAB’s guide [6]) is used as an illustrative

example. The problem described by FIS containing FPRs, presented in Table 2. Weights of all these

FPRs are equal to 1.00.

Table 2. FPR Set.

#

FPR

1

IF (Service is poor) OR (Food is rancid) THEN (Tip is cheap)

2

IF (Service is good) THEN (Tip is average)

3

IF (Service is excellent) AND (Food is delicious) THEN (Tip is generous)

MFs of terms of LVs Service, Food and Tip are presented in Figures 3, 4.

Service

Food

Figure 3. MFs of input LVs terms.

Figure 4. MFs of output LV Tip terms.

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

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Tables 3 and 4 containing tabulated MF values of input LVs terms.

Table 3. Linguistic variable Service.

LV terms

Scale values (scores)

1

2

3

4

5

6

7

8

9

10

poor

1.0

0.9

0.7

0.4

0.2

0.0

0.0

0.0

0.0

0.0

good

0.0

0.1

0.4

0.7

1.0

0.7

0.4

0.1

0.0

0.0

excellent

0.0

0.0

0.0

0.2

0.4

0.5

0.8

0.9

1.0

1.0

Fuzzy sets

Membership degrees

Table 4. Linguistic variable Food.

LV terms

Scale values (scores)

1

2

3

4

5

6

7

8

9

10

rancid

1.0

0.9

0.7

0.4

0.2

0.0

0.0

0.0

0.0

0.0

delicious

0.0

0.0

0.1

0.3

0.4

0.5

0.7

0.9

1.0

1.0

Fuzzy sets

Membership degrees

Define PLFs according to three FPRs above:

.),()|(

,)()|(

,),()|(

213

12

211

3

2

1

FDSEP

SGP

FRSPFRSPP

ppzzFHeP

pzFHeP

ppppzzFHeP

==

==

−+==

where

i

jiTX

p~

are probabilities

)

~

(i

ji TXP =

, for example, pSP is subjective probability that LV Service

has “poor” value.

Execute the fuzzy logic inference according to approach described above, and define tips amount for

following cases:

1. Service is evaluated for 3, and food is evaluated for 8 scores.

2. Both service and food are evaluated for 5 scores.

Inference results presented in Tables 5 and 6 correspondingly. To obtain “characterized values” of

terms of output LV Tip, use the equation (7) because MFs of these terms are symmetric. Note that tip

value in Table 5 almost same as the result of traditionally used fuzzy inference.

Table 5. Obtaining tip value for first case.

Hypothesis No, l

1

2

3

Probabilities P(e | Hl)

FPR 1, P(e | H1)

0.7

0.7 + 0 – 0.70 = 0.7

FPR 2, P(e | H2)

0.4

0.4

FPR 3, P(e | H3)

0.0

00.9 = 0.0

Distribution P(Hl | e)

0.64

0.36

0.00

Result (tip value):

2.5 0.64 + 12.50.36 + 22.50.0 = 6.1%

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

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Table 6. Obtaining tip value for second case.

Hypothesis No, l

1

2

3

Probabilities P(e | Hl)

FPR 1, P(e | H1)

0.36

0.2 + 0.2 – 0.20.2 = 0.36

FPR 2, P(e | H2)

1.0

1.0

FPR 3, P(e | H3)

0.16

0.40.4 = 0.16

Distribution P(Hl | e)

0.24

0.65

0.11

Result (tip value):

2.50.24 + 12.50.65 + 22.50.11 = 11.2%

3.2. Problem “Road behavior”

The example above as it is very simple and demonstrative, did not require reduction of FPR set.

Moreover, because MFs of output LV terms are symmetric, so it is possible to use segment middle points

(7). At most common case, as noted above, can be required to use preliminary reduction of FPR set

before conclusions inference and at defuzzification stage, weighed segment centers (8) should be used.

Example of such FIS which emulates behavior of a car driver exceeds the car speed before road

policeman, is descriped in details at [10]. Here we only compare result given by proposed approach with

the traditionally used Mamdani algorithm results. Determined defuzzified output LV value is y*=29.57,

and traditionally used fuzzy inference results (depending on specified way of defuzzification) are

following for the same input data:

428,522,716,329,829 ***** .=y.=y.=y.=y.=y RMMMLMCACG

.

Note that defuzzification results with the traditional fuzzy inference approach significantly depend

on chosen defuzzification method, and the defuzzification of the Bayesian logical-probabilistic fuzzy

inference result is closest to the defuzzification results given by center of gravity (

*

CG

y

) and by center

of area (

*

CA

y

) methods, which are most often used in the traditional model of fuzzy inference.

4. Software Implementation

In order to assess practical applicability of the proposed approach, as well as in order to assess

complexity of its implementation, under the guidance of one of the authors, a software implementation

of the above-considered Bayesian logical-probabilistic model of fuzzy inference was performed.

The program was implemented using the Java programming language. During the development

process, class packages were implemented for the main elements of the proposed model. It is shown that

due to the fact that the integration is used for simple term MFs, an acceptable accuracy of calculations

is achieved using simple methods of numerical integration.

The developed program was verified by comparing the obtained results with the results of Mamdani

and Sugeno algorithms implemented in MATLAB system. Comparison showed that software

implementation of the proposed model gives results that are insignificantly differ from the results

obtained using classical methods of fuzzy inference.

5. Conclusion

The article describes the Bayesian logical-probabilistic model of fuzzy inference, which differs from the

traditionally used model by the content of the stages of fuzzy conclusions inference and defuzzification.

It is based on the application of the apparatus of probabilistic logic and Bayes’ formula. Here, set of

FPRs is transformed into a set of PLFs, the arguments of which are the MF values of the input LVs

terms, and the calculated values are used as conditional probabilities to determine the posterior Bayesian

distribution on a set of hypotheses corresponding to the output LV terms. Resulting probability

distribution is used to defuzzify the output LV value.

XXIII International Conference on Soft Computing and Measurement (SCM'2020)

Journal of Physics: Conference Series 1703 (2020) 012042

IOP Publishing

doi:10.1088/1742-6596/1703/1/012042

10

Defuzzification is performed by calculating the expected value of a discrete random variable, the

values of which are the average points of the intervals of the output LV terms, and the distribution law

is the probability distribution obtained at the stage of inference of fuzzy conclusions on the set of

hypotheses that the output variable will receive a term value from its term-set.

Effectiveness of the application of the proposed model is shown by examples and confirmed by

software implementation, carried out under the guidance of one of the authors.

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