Advances in the Theory of Probabilistic and Fuzzy Data Scientific Methods with Applications

201 Pages·2021·5.095 MB·English

by  József Dombi, Tamás Jónás

Studies in Computational Intelligence 814
József Dombi
Tamás Jónás
Advances in
the Theory
of Probabilistic
and Fuzzy Data
Scientific Methods
with Applications
Studies in Computational Intelligence
Volume 814
Series Editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
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ó á ó á
J zsef Dombi Tam s J n s
(cid:129)
Advances in the Theory
of Probabilistic and Fuzzy
fi
Data Scienti c Methods
with Applications
123
József Dombi TamásJónás
Institute of Informatics Institute of Business Economics
University of Szeged EötvösLorándUniversity
Szeged,Hungary Budapest, Hungary
ISSN 1860-949X ISSN 1860-9503 (electronic)
Studies in Computational Intelligence
ISBN978-3-030-51948-3 ISBN978-3-030-51949-0 (eBook)
https://doi.org/10.1007/978-3-030-51949-0
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Preface
This monograph focuses on advances in the areas of soft computational and
probabilisticmethodsthatwepublishedoverthepastfewyears.Thetextconsistsof
five separate chapters, in which theoretical results and application possibilities are
discussed as well. The proposed methods and techniques are connected with var-
iousareasofscience,engineeringandthebusinesseconomics.Itispointedouthow
theplausibilityandbeliefmeasuresandthe‚-additiveand”-additivemeasurescan
be used to model uncertainty. A chapter is dedicated to a probability distribution
family which may be viewed as an alternative to four notable probability distri-
butions. This distribution family can be applied in reliability engineering and in
economicsaswell.Afuzzyarithmetic-basedtimeseriesmodelispresentedwiththe
aimofprovidingtheuserswithaneasy-to-usemodelingandforecastingtechnique.
FlexiblefuzzynumbersarethenproposedforLikertscale-basedevaluations.These
fuzzy numbers can be successfully applied in performance evaluations where the
evaluators need to incorporate vagueness, which may originate from their uncer-
tainty or from the variability of the perceived performance values.
We are confident that the new methods and techniques can be successfully
applied to solving a wide range of problems in many areas including engineering,
economics, biology, chemistry and the medical sciences.
Szeged, Hungary József Dombi
Budapest, Hungary Tamás Jónás
v
Introduction
Nowadays, application of computer science is common in many areas like engi-
neering, economics, biology, chemistry and the medical sciences. Most of the
computational techniques are hybrids as they utilize various branches of mathe-
matics.Thismonographsummarizesournovelresultsthatwerepublishedoverthe
pastfewyears.Therearefiveseparatechapterswhichdescribesoftcomputational,
probabilistic and hybrid methods. The latter methods may be viewed as synergies
betweenthesoftcomputationalandtheprobabilisticapproaches.Themonograph—
in line with our motivations—besides the theoretical results, also provides the
reader with potential applications in the areas of engineering and business
economics.
The Chap. 1 may be regarded as a brief review of some important monotone
measures; namely, thebelief-, probability- andplausibilitymeasures. The practical
importanceofthesemeasuresliesinthefactthathumandecisionsunderuncertainty
arecommonlybaseduponbeliefsandplausibilitiesassociatedwithpossibleevents.
Here, the most important properties of these measures are summarized and the
uncertainty modeling capabilities of the belief- and plausibility measures are
demonstrated.Inthischapter,itisshownhowthewell-knownEllsbergparadoxcan
be resolved by using belief- and plausibility measures, which may be viewed as
lower and upper probability measures, respectively.
Chapter 2 discusses one of the most widely applied classes of monotone mea-
sures, the class of ‚-additive measures (Sugeno ‚-measures). Although many
theoretical and practical articles have been published concerning ‚-additive mea-
sure, its properties and its applicability, there have been no studies that dealt with
thegeneralformof‚-additivemeasureoftheunionofnsets.Withtheaimoffilling
this gap, here, the general Poincaré formula for ‚-additive measures is presented.
Thisformula,whichmaybeviewedasageneralizationofthewell-knownPoincaré
formulainprobabilitytheory,tellsushow‚-additivemeasureoftheunionofnsets
can be computed.
In this chapter, it is also shown how ‚-additive measure is connected with the
belief-, probability- and plausibility measures. Furthermore, new characteristic
inequalities for belief- and plausibility measures using ‚-additive measure are
vii
viii Introduction
presented. It is a well-known fact that ‚-additive measurecorresponds to a belief-,
probability- and plausibility measure, if ‚[0, ‚¼0 and ‚2ð(cid:2)1;0Þ, respec-
tively.However,sincethezerovalueof‚doesnotdividethedomainð(cid:2)1;1Þinto
two symmetric sub-domains, it is difficult to judge the ‘plausibilitiness’ or ‘be-
liefness’ of a ‚-additive measure based on the value of parameter ‚. Here, the
so-called ”-additive measure, which may be treated as an alternatively parameter-
ized ‚-additive measure, is introduced and it is demonstrated that ” parameter has
an important semantic meaning. Namely, ” is the fix point of ”-additive comple-
ment operation. Also, it is shown that ”-additive measure corresponds to a belief-,
probability- and plausibility measure, if ” 2ð0;1=2Þ, ” ¼1=2 and ” 2ð1=2;1Þ,
respectively. Therefore, ”-additive measure can be utilized to characterize the
‘plausibilitiness’or‘beliefness’onanormalizedscale.Moreover,inthischapter,it
isdescribedhow‚-additiveand”-additivemeasuresareconnectedwithroughsets,
multi-attribute utility functions and with certain operators offuzzy logic.
Anewfour-parameter probabilitydistributionfunctionfamilyisintroduced and
someofitsapplicationsarediscussedinChap.3.Thisnewdistributionfunctionis
called the pliant probability distribution function, which may be viewed as an
alternative to some notable distribution functions. The cumulative distribution
function of the novel probability distribution is based on the so-called omega
function. It is demonstrated here that both the omega and the exponential function
fðxÞ¼eaxb ðx;a;b2R;b[0Þ may be deduced from a common differential
equation that is called the generalized exponential differential equation.
Furthermore,itisshownthattheomegafunction,whichhasparametersa,bandd
ða;b;d 2R;b;d[0Þ, is asymptotically identical with the exponential function
fðxÞ¼eaxb.
Also,itisshownindetailhowthepliantprobabilitydistributionfunctioncanbe
utilized to approximate some well-known probability distribution functions, the
formulas of which include exponential terms. Namely, it is demonstrated that the
asymptotic pliantdistributionfunctionmay coincidewith theWeibull, exponential
and logistic probability distribution functions. Then, it is shown that with appro-
priate parameter settings, the pliant probability distribution function can approxi-
mate the standard normal probability distribution function quite well, while the
approximating formula is very simple and contains only one parameter. Here, it is
also noted that this approximation formula may be viewed as a special case of a
modifier operator in the continuous-valued logic. The flexibility of this new
probability distribution function lays the foundations for its applications in many
fieldsofscienceandinwiderangeofmodelingproblems.Itispointedoutthatthe
pliant probability distribution function, as an alternative to the Weibull- and
exponentialdistributionfunctions,canbeutilizedformodelingconstant,monotonic
and bathtub-shaped hazard functions in reliability theory. Furthermore, it is
demonstrated that a function transformed from the new probability distribution
function can be applied in the so-called kappa regression analysis method, which
may be viewed as an alternative to logistic regression.
Introduction ix
In Chap. 4, a new fuzzy time series modeling technique is proposed. This
technique is based on fuzzy inference methods in which the fuzzy output is
obtained by fuzzy arithmetic operations; namely, via the weighted summation of
pliantnumbersthataretheconsequentsoffuzzyrules.Thisfuzzyinferencemethod
is called the Pliant Arithmetic-Based Fuzzy Inference System (PAFIS). The
defuzzification method of the system has two notable advantages. Firstly, it does
not require any numerical integration to generate the crisp output; and secondly, it
runsinaconstanttime.Afterwards,itisexplainedhowthePliantArithmetic-Based
Fuzzy Time Series (PAFTS) model can be derived by utilizing the
above-mentioned inference methods.
ThegreatestadvantagesofourPAFTSmodelliesinitssimplicityandeasy-to-use
characteristics. Once the fuzzy rule consequents are obtained, the time series mod-
eling and forecasting become very simple. In order to evaluate the forecasting
performance of the PAFTS method, 17 time series were analyzed including the
Australian Beer Consumption (ABC) from 1956 to 1994, the Istanbul Stock
ExchangeMarket(BIST100)Indexforthetimeperiodbetween2009and2013,and
theTaiwan Stock ExchangeCapitalization Weighted Stock Index (TAIEX) for the
timeperiodbetween1999and2004.Theforecastingresultsoftheproposedmethods
were compared with the results of various well-known methods. Based on our
empirical results, the PAFTS method may be viewed as novel viable time series
modelingtechnique.
In Chap. 5, the so-called flexible fuzzy number, the membership function of
which may have various shapes depending on its shape parameter ‚ is introduced.
Namely,thismembershipfunctioncanbebell-shapedð‚[1Þ,triangularð‚¼1Þ,
or‘reverse’bell-shapedð0\‚\1Þ.Thisfeatureoftheflexiblefuzzynumberallows
us toexpressthe soft equation‘x isapproximately equal to x ’ invariousways.
0
Here, it is demonstrated that if the shape parameter ‚ is fixed, then the set of
flexible fuzzy numbers is closed under the multiplication by scalar, fuzzy addition
and weighted average operations. The pliancy of flexible fuzzy numbers and the
above-mentioned properties of the operations over them make these numbers
suitableforLikertscale-basedfuzzyevaluations.Ontheonehand,theycanbeused
todealwiththevaguenessthatmayoriginatefromtheuncertaintyoftheevaluators
orfromthevariabilityoftheperceivedperformancevalues.Ontheotherhand,ina
multi-dimensional Likert scale-based evaluation, having the rating result in each
evaluation dimension represented by a flexible fuzzy number, these results can be
readily aggregated into one flexible fuzzy number using the above-mentioned
arithmetic operations.
Next,asageneralizationoftheflexiblefuzzynumberstheextendedflexiblefuzzy
numbers, which may have different left hand side and right hand side shape
parameters, are introduced. Also, it is demonstrated that the asymptotic flexible
fuzzynumberisjustaquasifuzzynumbercomposedofanincreasinglefthandside
and a decreasing right hand side sigmoid function. Exploiting this property of the
extendedflexiblefuzzynumbersallowsustoperformapproximatefuzzyarithmetic
x Introduction
operationsovertheminasimpleway.Owingtotheiradvantageouspropertiesboth
oftheflexiblefuzzynumbersandtheextendedflexiblefuzzynumbers,theproposed
fuzzy evaluation methods can bereadilyimplemented and utilized in practice.