Learning of Interval and General Type-2 Fuzzy Logic Systems using Simulated Annealing

35 Pages·2016·0.34 MB·English

Learning of Interval and General Type-2 Fuzzy Logic
Systems using Simulated Annealing: Theory and Practice
M.Almaraashia,∗,R.Johnb,A. Hopgoodc,S. Ahmadid
aTheUniversityCollegeinAljamoum,UmmAl-QuraUniversity,Makkah,SaudiArabia.
bAutomatedSchedulingOptimizationandPlanningGroup(ASAP),UniversityofNottingham,NG81BB,
UK.
cHECManagementSchool,UniversityofLiege,4000Liege,Belgium.
dCenterforComputationalIntelligence,SchoolofComputerScienceandInformatics,DeMontfort
University,Leicester,LE19BH,UK.
Abstract
Thispaperreportstheuseofsimulatedannealingtodesignmoreefficientfuzzylogic
systemstomodelproblemswithassociateduncertainties. Simulatedannealingisused
withinthisworkasamethodforlearningthebestconfigurationsofintervalandgen-
eraltype-2fuzzylogicsystemstomaximizetheirmodelingability. Thecombination
ofsimulatedannealingwiththesemodelsispresentedinthemodelingoffourbench-
markproblemsincludingreal-worldproblems. Thetype-2fuzzylogicsystemmodels
are compared in their ability to model uncertainties associated with these problems.
Issues related to this combination between simulated annealing and fuzzy logic sys-
tems,includingtype-2fuzzylogicsystems,arediscussed.Theresultsdemonstratethat
learning the third dimension in type-2 fuzzy sets with a deterministic defuzzifier can
addmorecapabilitytomodelingthanintervaltype-2fuzzylogicsystems. Thisfinding
canbeseenasanimportantadvanceintype-2fuzzylogicsystemsresearchandshould
increasethelevelofinterestinthemodelingapplicationsofgeneraltype-2fuzzylogic
systems,despitetheirgreatercomputationalload.
Keywords: simulatedannealing,intervaltype-2fuzzylogicsystems,generaltype-2
fuzzylogicsystems,learning
1. Introduction
Fuzzylogicsystemshavebeenappliedsuccessfullytoabroadrangeofproblems
indifferentapplicationdomains. Onesuchtypeofapplicationisconcernedwithusing
fuzzylogicforsystemmodelingandapproximationwhereafuzzyinferencesystemis
usedtomodelhumanknowledgeortoapproximatenon-linearanddynamicsystems.
However, the existence of uncertainties and lack of information in many real-world
(cid:3)Correspondingauthor.
Emailaddress:[email protected](M.Almaraashi)
PreprintsubmittedtoInformationSciences February9,2016
problemsmakesitdifficulttomodelsuchproblemsusingexpertknowledgeonly. Ex-
amples of such problems include identifying systems with no known rule-base and
systems with only historic data observation. It becomes clear that when designing a
simplefuzzylogicsystemwithfewinputs,theexpertsmaybeabletoprovideefficient
rules but, as the complexity of the system grows, suitable rule-base and membership
functionsbecomedifficulttoacquire. Therefore,someautomatedtuningandlearning
methods are often used to cope with such situations. The objective of these methods
is to get parameterized functions that best model these problems according to cho-
sencriteria. Theuseofautomatedmethodstodesignfuzzylogicsystemshashelped
to model many real-world problems that are difficult to understand by experts and it
isnowawell-establishedmethodologyformodelingandapproximationapplications.
Themotivationforthisresearchistwo-fold:
(cid:15) Type-2 fuzzy logic systems have numerous parameters that need to be deter-
mined in the design of any system. The determination of these parameters is
anopenresearchquestionandmotivatesourapproachtolearningtype-2fuzzy
systems.
(cid:15) Thegrowthininterestintype-2fuzzylogichasnotfullymanifesteditselfinreal-
world applications using general type-2 fuzzy sets. The emphasis has been on
intervaltype-2fuzzysets, thusnottakingadvantageofthemoregeneralrepre-
sentation.Byallowingforthelearningandoptimizationoftype-2fuzzysystems
weexpecttheuseofgeneraltype-2fuzzysetstogrow.
Sothemotivationisclear,andwenowelaborateonthesepoints.
Learningandoptimization. Thisresearchisconcernedwiththelearningoftype-
2fuzzylogicsystems,bothgeneralandinterval. Type-2fuzzylogicsystemsarenow
well established as both a research topic and an application tool. The motivation for
the use of type-2 fuzzy sets is that type-1 fuzzy logic has problems when faced with
environmentsthatcontainuncertaintiesthataretypicalinalargenumberofreal-world
applications. Theseuncertaintiesintheenvironmenttranslateintouncertaintiesabout
membership functions [38]. Type-1 fuzzy logic cannot fully handle these uncertain-
ties because it is precise in nature and for many applications it is unable to model
knowledge adequately, while type-2 fuzzy logic offers a higher level of imprecision
modeling[26]. Theextradimensionandparametersintype-2fuzzysetsaresupposed
to provide more design freedom and flexibility than type-1 fuzzy sets. The use of
automatedlearningmethodsbecomesimportantascomplexitygrowswhendesigning
type-2fuzzylogicsystems.
Many approaches have been proposed to learn and tune type-1 and type-2 fuzzy
logicsystemsincludingsearchalgorithmssuchasgeneticalgorithmsandparticleswarm
algorithms, as well as local search algorithms and classical learning methods. Com-
pared to genetic algorithms, few researchers have studied use of simulated annealing
to learn type-1 fuzzy logic systems such as [16, 12, 49]. So far as we are aware, the
onlyresearchreportedontheuseofsimulatedannealingtodesigntype-2fuzzylogic
systemsistheauthors’previousworkin[3,4,5,6].
Helping develop real-world applications. Another motivation for this research
comesfromthelackofapplicationsusinggeneraltype-2fuzzylogicsystems. Type-2
2
fuzzylogicisagrowingresearchtopicwithmuchevidenceofsuccessfulapplications.
However, almost all developments of type-2 fuzzy logic systems have been based on
intervaltype-2fuzzylogic[45][27]. Theheavycomputationalloadassociatedwiththe
generalizedformoftype-2setsisthemaindriverforthelackofapplicationsofgeneral
type-2 fuzzy sets compared with the interval model. This prior work has reinforced
thecommonconceptthatintervaltype-2fuzzylogicsystemscanaddmoremodeling
capabilitiesthantype-1fuzzylogicsystemsbutwithextracomputationalcost. Learn-
ing and optimization of general type-2 fuzzy logic systems are open areas for more
research,aswellastheongoingresearchonhowtoreducethecomplexityofgeneral
type-2fuzzylogicsystems,especiallyinthetype-reductionphaseofthesystem. The
largenumberofmethodsusedtodesigntype-1andintervaltype-2fuzzylogicsystems
canbeseenaspotentialcandidatesforgeneraltype-2fuzzylogicsystemsandsomeof
themmightuncoverfurtherpossibilitiesformodelinguncertainty.However,recentad-
vancesingeneraltype-2fuzzylogicsystemsresearch,includingnewrepresentations,
optimizedoperationsandfastertype-reductionmethods,indicateanexpectedgrowthin
applications. Despitethelargernumberofcomputationsassociatedwithgeneraltype-
2 fuzzy sets, there may well be benefits compared to interval type-2 fuzzy sets. This
ability can be unveiled using automated designing methods rather than being chosen
bythedesignermanually. Automatedmethodscanfine-tuneinitialfuzzylogicsystem
designs due to the lack of a rational basis for choosing secondary membership func-
tionsforgeneraltype-2fuzzysets[36,p.302]. Thisissueenforcestheneedforusing
automated methods in such problem. The other factor affecting the usage of general
type-2fuzzylogicsystemsisthelackofpracticalparameterizationmethodstohandle
the third dimension in general type-2 fuzzy sets. In general, a general type-2 fuzzy
logicsystemhasthepotentialtomodelmoreuncertaintiesdespitethelargeamountof
computationsassociatedwithitespeciallywhenappliedtononreal-timeapplications.
Inconsequence,thequestionofhowmuchgeneraltype-2fuzzylogicsystemscanadd
tomodelingperformanceoverintervaltype-2fuzzylogicsystemsisanotherissuethat
warrantsinvestigation.
The research reported here introduces a new method for learning general type-
2 fuzzy systems with a unique combination of learning the footprint of uncertainty
(FOU)followedbylearningthesecondarymembershipfunctions(SMF).Inaddition,
we show that when using the vertical slice type reducer we have improvement over
otherapproachesimplementedhere. Furthermore,intervaltype-2fuzzylogicsystems
wereappliedtoanswerthequestionoftowhatextentgeneraltype-2fuzzysetscanadd
moreabilitiesandflexibilitiestomodelingthanintervaltype-2fuzzysets. Adetailed
analysisiscarriedoutofthelearningofgeneraltype-2fuzzysystemsonasetofreal-
world data with and without added noise and, as such, provides significant insight
intohowthefutureoflearninggeneraltype-2fuzzysystemscanbecarriedout. These
methodsareappliedtofourbenchmarkproblems:noise-freeMackey-Glasstimeseries
forecasting[34],noisyMackey-Glasstimeseriesforecasting[34],andtworeal-world
problems,namelytheestimationofthelow-voltageelectricallinelengthinruraltowns
andtheestimationofthemedium-voltageelectricallinemaintenancecost[11].
Therestofthispaperstartswithareviewofthemethodsandconceptsusedinthis
workinsection2andissuesrelatedtothedesignofgeneraltype-2fuzzylogicsystems
insections3and3.2. Themethodologyandtheresultsaredetailedinsections4and5
3
andsomeconclusionsaredrawninsection6.
2. Background
2.1. Type-2fuzzysetsandsystems
Atype-2fuzzyset[36,p.83][38],denotedA~,ischaracterizedbyatype-2member-
shipfunction(cid:22) (x;u)wherex2X andu2J (cid:18)[0;1]:Forexample:
A~ x
A~=f((x;u);(cid:22) (x;u))j8x2X;8u2J (cid:18)[0;1]g (1)
A~ x
where 0 (cid:20) (cid:22) (x;u) (cid:20) 1. When all the secondary grades (cid:22) (x;u) equal 1 then A~
A~ A~
is an interval type-2 fuzzy set. Interval type-2 fuzzy sets are easier to compute than
general type-2 fuzzy sets. The footprint of uncertainty (FOU) is a 2D representation
ofanintervaltype-2setandrepresentstheunionofallprimarymembershipsandcan
be described by a lower and upper membership functions. The ease of computation
andrepresentationofintervaltype-2fuzzysetsisthemainreasonfortheirwideusage
inreal-worldapplications. Theprincipalmembershipfunction[36,p.86]occurswhen
thereisonlyonesecondarygradeequalto1ateachsecondarymembershipfunctionof
type-2set. Therefore,theprincipalmembershipfunctionistheunionofallsuchpoints
atwhichtheunityoccursasfollows:
∫
(cid:22) (x)= u=xwheref (u)=1 (2)
principal x
x∈X
Using the Zadeh extension principle, union and intersection of type-2 fuzzy sets are
defined(knownasjoinandmeetrespectively)[29]. KarnikandMendel[29]haspro-
posedamethodtocalculatemeetandjoinoperationswhenallsecondarymembership
functionsarenormalandconvex. CouplandandJohn[15]haspresentedanextension
tothisformulatoallowtheuseofnon-normalsecondarymembershipfunctions.
There are some representations for type-2 fuzzy sets that have been proposed in
theliteraturesuchasvertical-slicerepresentation[36,p.83][38],wavy-slicerepresen-
tation [38], geometric representation [15], alpha-planes [41], alpha cuts [22] and Z-
slices [45]. The most well-known representations among them are the vertical-slice
and wavy-slice representations. The vertical-slice representation [36, p.83] represent
fuzzysetsbyusingsecondarysetsinavertical-slicemannerwhere:
A~=f(x;(cid:22) (x))j8x2Xg (3)
A~
∫
(cid:22) (x)=(cid:22) (ujx)= f (u)=u (4)
A~ A~ x
∀u∈Jx⊆[0;1]
Thisrepresentationisveryusefulforcomputation. Inwavy-slicerepresentation[38],
atype-2fuzzysetisrepresentedasaunionofembeddedtype-2fuzzysetswhereeach
embeddedtype-2fuzzysetA~ hasthesamedomainoftype-2fuzzysetA~. Thetype-
e
2 embedded set A~ has been defined for discrete universes of discourse X and U,
e
an embedded type-2 set A~ has N elements, where A~ contains exactly one element
e e
4
fromJ ;J ;:::;J , namelyu ;u ;:::;u , eachwithassociatedsecondarygrade,
x1 x2 xN 1 2 N
namelyf (u );f (u );:::;f (u )[36,p.83][38]. Forexample:
x1 1 x2 2 xN N
∑N
A~ = [f (u )=u ]=x ;u 2J (cid:18)U =[0;1]: (5)
e xi i i i i xi
i=1
In this definition, the embedded set contains N elements represented using the
primary memberships u 2 J that is linked to its secondary membership grades
i xi
f (u )inorderedpairs. Sotype-2fuzzysetAcanbeshownasaunionofembedded
xi i
type-2fuzzysetsasfollows:
∑n
A~= A~j (6)
e
j=1
where the total number of type-2 embedded sets in type-2 fuzzy set A is calculated
usingthenumberofdiscretisedpointsintheprimarydomainN andtheprimarymem-
bershipfunctionsM (knownasthesecondarydomain)asfollows:
∏N
n= M (7)
i
i=1
whereA~jdenotesthejthetype-2embeddedfuzzysetintype-2fuzzysetA~.Thewavy-
e
slicerepresentationknownastheMendel-JohnRepresentationTheorem(RT)hasbeen
proposedby[38]. Itisusefulfortheoreticalderivationsbutnotusefulforpracticaluse
becauseoftheastronomicalnumberintheunionofembeddedsets. However,itisvery
useful when dealing with interval type-2 fuzzy sets due to the ability of using type-1
fuzzymathematicswhichiseasytodealwith[39].
Type-2fuzzylogicsystemsarerule-basedsystemsthataresimilartotype-1fuzzy
logic systems in terms of the structure and components but type-2 FLS has an extra
outputprocesscomponentwhichiscalledthetype-reducerbeforedefuzzification. The
components of a type-2 Mamdani fuzzy system are fuzzifier, rules, inference engine,
type-reducer and defuzzifier. The type-reducer reduces output type-2 fuzzy sets to
type-1fuzzysetsthenthedefuzzifierreducesittoacrispoutput. Thetype-reduction
stageisthemostcomputationallyexpensivestageinatype-2fuzzylogicsystem.
2.2. Type-2fuzzysetsanduncertaintymodelling
Type-1 fuzzy logic has been used successfully in a wide range of problems such
ascontrolsystemdesign,decisionmaking,classification,systemmodelingandinfor-
mation retrieval. However, the type-1 approach is not able to directly model all un-
certaintiesandminimizetheireffects[38]. Theseuncertaintiesexistinalargenumber
of real-world applications. They can be a result of uncertainty in inputs, uncertainty
in outputs, uncertainty that is related to the linguistic differences, uncertainty caused
bythechangeofconditionsintheoperation,anduncertaintyassociatedwiththenoisy
datawhentrainingthefuzzylogicsystem[36,p.68]. Alltheseuncertaintiestranslate
into uncertainties about the membership functions of the fuzzy sets [38]. Therefore,
theexistenceofuncertaintiesinthemajorityofreal-worldapplicationsmakestheuse
5
oftype-1fuzzylogicinappropriateinmanycasesespeciallywithproblemsrelatedto
inefficiency of performance in fuzzy logic control [21]. Problems related to model-
inguncertaintyusingmembershipfunctionsoftype-1fuzzysetshavebeenrecognized
earlyand[50]introducedhighertypesoffuzzysetscalledtype-nfuzzysetsincluding
type-2 fuzzy sets [37]. Type-2 fuzzy logic systems have many advantages compared
withtype-1fuzzylogicsystems,includingtheabilitytohandledifferenttypesofuncer-
taintiesandtheabilitytomodelproblemswithfewerrules[21]. Twofactorsshouldbe
considered regarding the the widespread perception that a general type-2 fuzzy logic
system should outperform the interval form which also should outperform a type-1
fuzzylogicsystem[46]. Thesetwofactorsarethedependenceofperformanceonthe
choiceofthemodelparametersaswellasonthevariabilityofuncertaintywithinthe
application[46]. Therefore,agoodchoiceofthemodel’sparametersusingautomated
methods is desirable to get clearer conclusions regarding this comparison. Despite
thesepromisingindicatorsofthegeneraltype-2fuzzylogicsystems,almostalldevel-
opmentsoftype-2fuzzylogicsystemshavebeenbasedonintervaltype-2fuzzylogic
systems. However,newrepresentationsallowustoconsidergeneraltype-2fuzzylogic
systems. These representations include geometric T2FLS [15], alpha-planes [41], al-
phacuts[22]andZ-slices[45,10][47]. Therehavebeenanumberofdevelopmentsin
reducingthecomputationsforgeneraltype-2fuzzylogicsystems. Fortype-reduction,
the geometric defuzzifier [15], the sampling defuzzifier [19] followed by importance
samplingdefuzzifier[31]andacentroiddefuzzifierbasedonthealpharepresentation
[32]havebeenproposed. Oneattempttodesigngeneraltype-2setsbasedonzSlices
representationwasproposedin[10]wheresurveydataanddevicecharacteristicswere
used to build zSlices automatically. Other work using an alpha-planes representation
has been applied, e.g. as a method for edge-detection [35] and a learning method to
forecast Mackey-Glass time-series [41]. The latter showed a better performance of
general type-2 fuzzy logic systems using a simpler model known as “triangle quasi-
type-2fuzzylogicsystem”firstpresentedin[40]. Someotherresearchersusedsome
neuralnetworkconceptsorclassificationalgorithmssuchas:type2AdaptiveNetwork
Based Fuzzy Inference System (ANFIS) [28], general type-2 fuzzy neural network
(GT2FNN) [24] and fuzzy C-means algorithm with a model known as “efficient tri-
angular type-2 fuzzy logic system” [43]. To the best of the authors’ knowledge, no
attempt to employ a learning method to general type-2 fuzzy logic systems using the
vertical-slicesrepresentationhasbeenreported. Toachievethisobjective, apartfrom
using a practical type-reducer, some kinds of parametrization are needed for general
type-2 sets to allow learning or optimization techniques to deal with these parame-
terseasilyratherthanhavingallthesecondarygradesormembershipfunctionschosen
manually.Theparametrizationmethodshouldpreservemostofthefreedomassociated
withGT2FLS.
Ourproposedpracticaldesignmethodologyaimstoreducethecomputationsneeded
to get the best footprint of uncertainty (FOU). The proposed parametrization method
wasfirstpresentedin[6]and[7]. Inaddition,thispaperpresentsanovelapproachfor
learningallparametersofgeneraltype-2fuzzylogicsystemsusingsimulatedannealing
underthevertical-slicesrepresentation.
6
2.3. Simulatedannealingandtype-2fuzzylogicsystems
Thesimulatedannealingalgorithmisasimpleandgeneraloptimizationalgorithm
for finding global minima [30]. It has been used widely to search for optimal or
nearly optimal solutions in a wide range of optimization problems. In this work, it
acts as a learning algorithm to automatically design fuzzy logic systems by search-
ing for the best configurations of these systems. One of the motivations for using
simulated annealing with fuzzy systems is that it does not require the existence of
mathematicalpropertiessuchasdifferentiabilityintheproblem,whichallowsthepos-
sibility of using all fuzzy structure components including non-differentiable t-norms
and non-differentiable membership functions. Although the combination might have
morecomplexityandlongersearchtimethanlocalsearchalgorithms,itismorelikely
to find the global or near global optima of the configuration of fuzzy logic systems
thanlocalsearchapproaches. Thisisduetotheabilityofsimulatedannealingtoavoid
localoptimabyacceptinghigher-coststateswithsomeprobabilityinordertoexplore
theproblemspace. Inaddition,simulatedannealingcansuithighdimensionalityprob-
lems as it scales well with the increase of variable numbers, which makes it a good
candidate for the optimization of fuzzy logic systems [16]. Also, it is able to handle
costfunctionswithdifferentdegreesofnon-linearities,discontinuities,andstochastic-
ity[23].Theproblemofoptimizingmembershipfunctionsofthefuzzylogicsystemin
ordertominimizetheobjectivefunctionisacomplexproblemduetothelargenumber
ofparametersusedaswellasthethenon-differentiableandnon-continuousobjective
functions[20]. Thesimulatedannealingconvergencenormallyrequiresanexponential
timewhichcausesthealgorithmtobeimpracticalinsomecases[1,p.14]. Oneofthe
criticismsofsimulatedannealingisthedifficultyinfine-tuningitsparameters,soitcan
betime-consumingfordeveloperstofindanoptimalfit[23]. Theformalizationsand
configurations for simulated annealing to design fuzzy logic systems can be chosen
fromalargenumberofchoicesproposedintheliterature.
3. Designingandlearningofgeneraltype-2fuzzylogicsystems
3.1. Apracticalchoiceforgeneraltype-2fuzzyset
Inordertogetaneffectiveandpracticalformofgeneraltype-2set,thechosenform
should:
(cid:15) havealowcomputationalburden;
(cid:15) preservemostofthefreedomassociatedwithgeneraltype-2sets.
These two objectives are normally in conflict as more freedom (through parameters)
requires more computations. Therefore, some trade-offs are needed using some pa-
rameterization mechanisms. One way to do this is to have parameterized secondary
membershipfunctionsthatareasymmetricandconvex. Forexample,consideratrian-
gularsecondarymembershipfunctionwithanapexintheareabetweenthelowestand
thehighestFOUpoints(FOU andFOU )andprimarymembershipsforeach
lower upper
xinthedomain. Theasymmetryispreferredtoallowoptimizingtheapexlocationof
theSMFwhentheirprimarymembershipsarefixed. Theotherpreferredpropertyisto
7
haveaconvexSMFtoallowquickmeetandjoinoperationswhenusingthesesetsin
GT2FLS.
Thesecondarymembershipfunctionofageneraltype-2fuzzysetisitselfatype-1
fuzzyset. Ourapproachistolearnthe‘apex’ofthesecondarymembershipfunctions
using a location indicator of the apex point called the “apex factor” (AF). The apex
factor for each SMF takes the value in [0;1] where if it is zero it takes a value at the
SMFandatunitytheUMF.ThisworksforanyasymmetricandconvexshapeofSMF
includingnon-normal. Inotherwords,toallowlearningthebestlocationfortheapex
foreachSMF,afunctiontodeterminetheSMF’sapexlocationsinFOUforeachxin
the primary domain is needed. The values of the apex locations must be bounded by
thehighestandthelowestFOU points(FOU andFOU )foreachxinthe
upper lower
domain.Anexampleofthisapproachistohaveapiecewiselinearfunctionorasmooth
piecewise-polynomial function and to use some interpolation methods. However, it
couldbepossibletoensurethisconditionofboundarieswhendesigningthefirstmodel
ofthegeneraltype-2setbutthisisverydifficulttotraceandensureforeachxinthe
continuousdomainwhenlearningSMF (x)astheinterpolationmightdefinesome
apex
apexesdomainsoutsidetheFOUboundaries. Therefore,anewparametricformulais
proposedherethatnormalizestheFOUapexlocationstobewithinFOU(x)foreach
xintheprimarydomain. ThisisdonebydefiningtheSMF (x)asfollowing[6]:
apex
SMF (x)=1=(FOU (x)+g(x)(cid:2)(FOU (x)(cid:0)FOU (x))): 06g(x)61
apex low up low
(8)
whereg(x)isaparametercalled“apexfactor”thatisusedasanapexlocationindicator
foreachx. Thisparametercanbeusedtochangetheapexlocationwithouttheneed
tocheckforboundaryconditions. Forexamplewheng(x) = 0:5, thelocationofthe
apexisinthemiddlebetweenFOU (x)andFOU (x)andtheresultingSMF
upper lower
issymmetrical. Therefore,thisparameterisactingasavariablerepresentingtheapex
locationswhendoingsomeoptimizationorlearningforthegeneraltype-2set. Anex-
ampleoftheuseofthisparameteristouseapiecewiselinearfunctiontodeterminethis
parameterforallxintheprimarydomain. Forexample,supposethatk ;k ;::::::;k
1 2 n
areorderedpointsinthexdomainandg(k1);g(k );::::::;g(k )aretheirapexfactors
2 n
whichbothdefinethepiecewiselinearfunction,then:

0:5; x<k1
g(x)= g(ki)+ kix+−1k−ix (cid:2)(g(ki+1)(cid:0)g(ki)); ki 6x6ki+1 (9)
0:5; x>k
n
Therefore,eachpointintheprimarydomainislinkedwithoneapexfactorg(x). An-
other similar function to determine the height of the apexes when non-normal SMFs
areusedcanbedesignedthesameway. Forexample:

 1; x<k
1
h(x)= h(ki)+ kix+−1k−ix (cid:2)(h(ki+1)(cid:0)h(ki)); ki 6x6ki+1 (10)
1; x>k
n
Thisformisnotidenticaltotheprincipalfunctiondescribedin[36,p.86]orthefuzzy
truthnumbersproposedin[43]becauseforeachxvalue,theSMF canbenon-normal.
8
The lowest and highest FOU points (FOU and FOU ) for each x can be
lower upper
defined by another function such as trapezoidal, Gaussian or triangular functions or
any other functions used to define interval type-2 sets. An example of the proposed
method is shown in figure 1 and an example of learning the secondary membership
functions is shown in figure 2. The chosen form is based on the latest general type-
2 literature using the vertical-slice representation and the novel method we proposed
heretodeterminetheapexlocationsandheightsofSMFs. Although,thisisnotanew
representationandcannotbegeneralizedforallformsofgeneraltype-2sets,theaim
ofthismethodistohavegeneraltype-2fuzzysetssimplifiedforpracticalusage.
Figure1:Generaltype-2fuzzysetdefinedbyitsFOUandSMF.TheFOUisdefinedbytwoGaussianfunc-
tionswhiletheSMFisatriangularshapeddefinedbylinearinterpolationoftwopiecewiselinearfunctions.
Thegreendottedlineistheapexfactorwhichhasdifferentvaluesbetween0and1.
3.2. Thechoiceforthedefuzzificationmethod
Thebottleneckpartofthegeneraltype-2fuzzylogicsystemisthedefuzzification
phase. Thisisduetothehighcomputationalburdenassociatedwiththetype-reduction
process. Therefore, special attention should be given to the choice of such methods.
Theaimofthissubsectionistohighlightthisissueanditseffectsonthelearningpro-
cess. Whenusingrepresentationmethodsotherthanthevertical-slicesrepresentation,
thereare fewproposedmethods that havebeenused for this purpose. An exampleis
a type-reducer proposed by [43] using triangular type-2 fuzzy sets which uses fuzzy
9
Figure2:AnexampleoflearningatriangularSMFbyadaptingtheapexlocation.

1
0.8
e
d
Gra0.6
y
ar
d
on0.4
ec
S
0.2
0
0 0.2 0.4 0.6 0.8 1
Secondary Domain
truth numbers where all the secondary membership functions are normal and convex
for a unique entity in the secondary domain (primary membership functions). This
type-reducerusestheiterativeKMalgorithmandsomeinterpolationoperationstoget
anapproximatecentroidfortriangulartype-2fuzzysets. Methodsthatuseotherrep-
resentations include the one proposed in [32] which uses the iterative KM algorithm
under the alpha-plane representation and the one based on z-slices in [44]. Based on
ourchoicefortherepresentationofgeneraltype-2fuzzysetsusingvertical-slices,the
availabledefuzzificationsoptionsare:
1. Theexhaustivebrute-forcehighlyexpensivetype-reductionmethodpresentedin
[36,p.248-254]whichcomputestheunionofallthecentroidsofalltheembed-
ded type-2 fuzzy sets involved in the general type-2 fuzzy set. This method is
impracticaltouseforourpurpose. Inpractice,thenumberofembeddedsetsis
normallyastronomicalandabovethecurrentdatastructure. Forinstance, fora
generaltype-2fuzzysetsdiscretizedintoourchoiceof101x-domainpointsand
each vertical slice into 9 points, the number of embedded sets is 2:39(cid:2)1096
whichisfarabovethecurrentdatastructure. Inourchosenlanguage(C++),the
longest data structure size can be allocated is unsigned integer = 2;14(cid:2)109.
Thisnumberofembeddedsetsareunionedtogetonesampleoutputinonefuzzy
logic system evaluation in one iteration of the optimization process. Table 1
shows how type-reduction complexity evolves in our problem with some rea-
sonablechoicesoffuzzylogicsysteminputsamplesandreasonablenumberof
simulated annealing iterations. Note that table 1 is for the type-reduction op-
erations only (i.e. does not include fuzzification and other fuzzy logic system
operations). Therefore,thischoiceisimpracticalforourwork.
2. The recursive algorithm introduced by [17], which includes some interesting
ideastoreducethesecomputationsbutwhosecomplexityisstillveryhigh.
3. Theverticalslicecentroidtype-reducer(VSCTR)whichwasinitiallyproposed
10