4_Rotshtein


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[,][,]
Fig.1.Predictionmodelstructure.
FuzzyApproximator
Toapplyfuzzyknowledgebases(Tables1and2)wewillusethegeneralizedfuzzyapproximator(Fig.2)introduced
in[14,15].
Thisapproximatordescribesthedependence
fxxx
(,,...,)
TABLE1.ExpertKnowledgeMatricesfor
Relations(2)and(3)
510
BLBLBLBLBL
BWSLBLSLBWBL
SWBLSLSLSW
SLSLSLSLSL
DSLSLDDSL
SWDSLSLSW
DDDDD
SLSWSWDSLD
DDSWSWD
SWSWSWSWSW
DBWBWSWDSW
SLSWSWBWSL
BWBWBWBWBW
SLBWSWBWSLBW
BLSWBWSWBL
TABLE2.ExpertKnowledgeMatricesfor
Relation(1)
BLBLBLBW
BWDBLDd
SWBLSLSL
SWSLDSL
DSLSLDd
SWDSLSL
SLSWSWDd
SLDSWSW
SLSWSWBW
DBWBWSWd
SLSWSWBW
BWBWBWBL
SWBWSWDd
BWDBWSL
Fig.2.
Generalizedfuzzy
TABLE3.ExpertKnowledgeMatrix
RuleNo.
IF&#xinpu;&#xts00;inputs
Weightof
therule



isalinguistictermevaluatingavariable
intherow
isthenumberofconjunctionrows
correspondingtotheclass
oftheoutputvariable
isanumberfromtheinterval[0,1],characterizingthe
subjectivemeasureofexpert'sconfidenceastoastatementwiththenumber
,areformedbyquantizationoftherange
[,]
oftheoutputvariableinto
[,][,)[,)[,]
yyyyyyyy

111

Accordingto[1416],thefollowingobjectapproximationcorrespondstofuzzyknowledgebase(4):
yyyyyy
yyy
ddd


()()()
()()(


,
(5)
ywx
()maxmin()
i
i










1
1
,(7)
isthemembershipfunctionoftheoutput
intheclass
dyy
jjj
[,]
isthemembership
functionofavariable
inaterm
,and
Cccccc
BLSLDSWBW
1515151515
(,,,,)
Cccccc
BLSLDSWBW
610610610610610
(,,,,)
Cccccc
BLSLDSWBW
11121112111211121112
(,,,,)
,,,,,

XylM
becomposedfromthetournamentdata,where
,...,
),(
,...,
),(
Xxxxxxx
llllll
1256710
1112
}aretheresultsof
previousmatchesfortheteams
inthe
thexperiment,and
MubRANDOMyy
()([,])
MucRANDOMcc
()([,])
RANDOMxx
([,])
istheoperationoffindingarandomnumberuniformlydistributedovertheinterval
[,]
Selectionofparentchromosomesforthecrossoveroperationshouldnotberandom.Weusedtheselectionprocedure
givingprioritytothebestsolutions.Thegreaterthefitnessfunctionofsomechromosome,thehighertheprobabilitythatthis
chromosomewillgeneratedaughterchromosomes[15,20].Asafitnessfunction,wetakecriterion(10)withminussign,i.e.,
thegreaterthedegreeofchromosomeadaptabilitytotheoptimizationcriterion,thegreaterthefitnessfunction.Whilethe
624
Fig.4.Structureofthecrossoveroperation.
Fig.3.Structureofachromosome.
NEURALTUNINGOFTHEFUZZYMODEL
RelationshipsfortheTuning
wtwt
jpjp
()()

ctct
()()

,(12)
btbt
()()

isthecenterof



m
m
j
1
u
u
u


u
u
T
u
u
minimizingthecriterion
Eyy
aretheruleweightsandmembershipfunction
Fig.6.Membershipfunctionsaftertuning.
TABLE6.RuleWeightsin
Relation(3)
TABLE7.RuleWeightsin
Relation(1)
TABLE5.RuleWeightsin
Relation(2)
Thelearningsamplingconsistsoftheresultsof1056matchesovereightyearsfrom1994to2001.Tables58and
Fig.6showtheresultsoffuzzymodeltuning.
Formodeltesting,weusedtheresultsof350matchesfrom1991to1993.Afragmentofthetestingsampleand
predictionresultsispresentedinTable9,where
and
areteamnames;
and
arerealandexperimentalresults;
TABLE9.FragmentofPredictionResults
CONSTRAINTSOFTHEPREDICTIONMODEL
modeldidnottakeintoaccountthefollowingimportantfactors.
1.Numberofinjuredplayers.Informalizingthisfactor,itisnecessarytotakeintoaccounttheimportanceand
performanceoftheinjuredplayer,whomayinfluencethematchresult.
2.Thenumberofbookedandbenchedplayers.
TABLE10.EfficiencyIndicesofTuningAlgorithms
Thepartialderivativesappearinginrelations(11)(13)characterizetheerrorsensitivity(
)tothevariationof
123

1112
1234
1112

1112
1234
1112

12356

123568

123568

12357

610
123579
610

610
123579
610



y
,
dydy
()()









121112
jpjpjpjp
zzxx
(()()()())

121112
(()()(())
jpjpjpjp
zzxx
121112

jpjpjpjp
zzxx
()()(()
1112

121112
(()()(())
jpjpjpjp
zzxx
121112

jpjpjpjp
zzxx
()()()()


1
1
5
1











x
w
,


2
6
10
2











x
w
,
()()
125
,,...,
()()
6710
,,...,

jpjpjpjp
zzxx
()()()()
121112



jp
w
x
1
1
5
,



jp
w
x
2
6
10
,
cxb
cxb
()()


cxb
cxb
()()
()()
1212
,,...,
1.A.G.IvakhnenkoandV.G.Lapa,PredictionofRandomProcesses[inRussian],NaukovaDumka,Kiev(1971).

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