Combining Bayesian and logical-probabilistic approaches for fuzzy inference systems implementation

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.

Full text

Journal of Physics: Conference Series
PAPER • OPEN ACCESS
Combining Bayesian and logical-probabilistic approaches for fuzzy
inference systems implementation
To cite this article: G I Kozhomberdieva and D P Burakov 2020 J. Phys.: Conf. Ser. 1703 012042
View the article online for updates and enhancements.
This content was downloaded from IP address 181.215.207.31 on 02/02/2021 at 01:27

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
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)
Journal of Physics: Conference Series 1703 (2020) 012042
IOP Publishing
doi:10.1088/1742-6596/1703/1/012042
4
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
IOP Publishing
doi:10.1088/1742-6596/1703/1/012042
5
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)
Journal of Physics: Conference Series 1703 (2020) 012042
IOP Publishing
doi:10.1088/1742-6596/1703/1/012042
6
.,,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)
Journal of Physics: Conference Series 1703 (2020) 012042
IOP Publishing
doi:10.1088/1742-6596/1703/1/012042
7
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)
Journal of Physics: Conference Series 1703 (2020) 012042
IOP Publishing
doi:10.1088/1742-6596/1703/1/012042
8
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 – 0.70 = 0.7
FPR 2, P(e | H2)
0.4
0.4
FPR 3, P(e | H3)
0.0
00.9 = 0.0
Distribution P(Hl | e)
0.64
0.36
0.00
Result (tip value):
2.5 0.64 + 12.50.36 + 22.50.0 = 6.1%

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
9
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 – 0.20.2 = 0.36
FPR 2, P(e | H2)
1.0
1.0
FPR 3, P(e | H3)
0.16
0.40.4 = 0.16
Distribution P(Hl | e)
0.24
0.65
0.11
Result (tip value):
2.50.24 + 12.50.65 + 22.50.11 = 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.
References
[1] Zadeh L A 1975 The concept of a linguistic variable and its application to approximate reasoning,
Part 1. Information Sciences 8 199–249
[2] Mamdani E H 1977 Applications of fuzzy logic to approximate reasoning using linguistic
synthesis IEEE Transactions on Computers 26 no 12 1182–91
[3] Terano T, Asai K and Sugeno M 1992 Fuzzy systems theory and its applications (San Diego:
Academic Press)
[4] Kudinov Y I, Kudinov I Y, Pashchenko F F and Pashchenko A F 2013 Program complex for
identification of fuzzy models Proc. of 25th Chinese Control and Decision Conference
(CCDC) 2492–4
[5] Kudinov Y I, Pashchenko F F, Pashchenko A F, Kelina A Y and Kolesnikov V A 2017
Optimization settings in the fuzzy combined Mamdani PID controller IOP Conf. Series:
Journal of Physics: Conf. Series 891 012294
[6] Get Started with Fuzzy Logic Toolbox, available at:
https://www.mathworks.com/help/fuzzy/getting-started-with-fuzzy-logic-toolbox.html
[7] Kozhomberdieva G I, Garina M I and Burakov D P 2013 Using decision making theory for
appraisement problems according to the CMMI® model Programmnye produkty i sistemy
[Software & Systems] 4 117–24 (in Russian)
[8] Kozhomberdieva G I, Burakov D P, and Garina M I 2017 Using Bayes' theorem to estimate
CMMI® practices implementation Programmnye produkty i sistemy [Software & Systems] 1
67–74 (in Russian)
[9] Kozhomberdieva G I, Burakov D P 2018 Bayesian approach to solving the problems of quality
appraisement Myagkie izmereniya i vychisleniya 5 15–26 (in Russian)
[10] Kozhomberdieva G I, Burakov D P 2019 Bayesian logical-probabilistic model of fuzzy inference:
stages of conclusions obtaining and defuzzification Nechetkie sistemy i myagkie vychisleniya
[Fuzzy Systems and Soft Computing] 14 no 2 92−110 (in Russian)
[11] Morris W 1968 Management science: a Bayesian introduction (New York: Prentice-Hall Publ.,
Enqlewood Cliffs)
[12] Jøsang A 2016 Subjective logic: a formalism for reasoning under uncertainty (Springer Verlag)
[13] Kosko B 1990 Fuzziness vs. probability Int. J. General Systems 17 211–240
[14] Merekin Y V 1963 Solving problems of probabilistic calculation of single-cycle circuits by the
orthogonalization method. Vychislitel'nye sistemy 5 10–21 (in Russian)
[15] Feller W 1970 An introduction to probability theory and its applications vol 1, 3rd edition (New
York: JohnWiley&Sons)