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SensitivityAnalysisofEnumeratedTreesofIncreasingBooleanExpressions
SaketAnand,DavidMadigan,RichardMammone,FredRobertsEnumerationandSelectionofOptimumDecisionTreeABC0101AsetofdecisiontreesisconstructedforeachcompleteandmonotonicbooleanfunctionwhereinputsrepresenttestsperformedbyeachsensorABCY00000010010001101000101111011111Y=f(A,B,C)wherefiscompleteandmonotonicThecostofeachtreeisevaluatedandtheoptimumtreeselected.CA0A101B1EnumerationandSelectionofOptimumDecisionTreeThedecisiontreesareconstructedusing4sensorsForthreesensors,thereare114monotonicandcompletebooleanexpressions.Thesecanbeimplementedusing11808distincttrees.Thetreesareevaluatedandrankedusingthecostfunction1.Thetreewiththelowestcostisselectedastheoptimumdecisiontree.1Stroud,P.D.andSaegerKJ.,“EnumerationofIncreasingBooleanExpressionsandAlternativeDigraphImplementationsforDiagnosticApplications〞,ProceedingsVol.IV,Computer,CommunicationandControlTechnologiesCostFunctionusedforevaluatingthedecisiontrees.CTot=
CFalsePositive*PFalsePositive+CFalseNegative*PFalseNegative
+CfixedTheErrorProbabilityoftheentiretreeiscomputedfromtheerrorprobabilitiesoftheindividualsensors.where,
CFalsePositiveisthecostoffalsepositive(TypeIerror)
CFalseNegativeisthecostoffalsenegative(TypeIIerror)
PFalsePositiveistheprobabilityofafalsepositiveoccurring
PFalseNegative
istheprobabilityofafalsenegativeoccurring
Cfixedisthefixedcostofutilizationofthetree.ProbabilityofErrorforIndividualSensorsForithsensor,thetype1(P(Yi=1|X=0))andtype2(P(Yi=0|X=1))errorsaremodeledusingGaussiandistributions.StateofnatureX=0representsabsenceofabomb.StateofnatureX=1representspresenceofabomb.Yirepresentstheoutcomeofsensori.Itischaracterizedby:Ki,discriminationcoefficientTi,decisionthresholdΣi,varianceofthedistributionsKiP(Yi|X=1)P(Yi|X=0)TiCharacteristicsofatypicalsensorReceiverOperatingCharacteristic(ROC)CurveTheROCcurveistheplotoftheProbabilityofcorrectdetection(PD)vs.theProbabilityoffalsepositive(PF).TheROCcurveisusedtoselectanoperatingpoint,whichprovidesthetradeoffbetweenthePDandPFEachsensorhasaROCcurveandthecombinationofthesensorsintoadecisiontreehasacompositeROCcurve.TheparameterwhichisvariedtogetdifferentoperatingpointsontheROCcurveisthesensorThresholdandacombinationofThresholdsforthedecisiontree.EqualErrorRate(EER)istheoperatingpointontheROCcurvewhere,
PF
=
1-PDP(Yi|X=1)P(Yi|X=0)TiKiPDPFOperatingPoint101EERStroud-SaegerExperimentsStroud-SaegerrankedalltreesformedfromfourgivensensorsA,B,CandDaccordingtoincreasingtreecosts.Thecostfunctionusedwasasshowninearlierslides.Valuesusedintheirexperiment:CA=.25;KA=4.37;ΣA=1;CB=.25;KB=1.53;ΣB=1;CC=10;KC=2.9;ΣC=1;CD=30;KD=4.6;ΣD=1;whereCiistheindividualcostofutilizationofsensori,KiisthesensordiscriminationpowerandΣiistherelativespreadfactorforsensori.Valuesofothervariablesarenotknown.CostSensitivitytoGlobalParametersValuesusedintheexperiment:CA=.25;P(YA=1|X=1)=.9856;P(YA=1|X=0)=.0144;CB=1;P(YB=1|X=1)=.7779;P(YB=1|X=0)=.2221;CC=10;P(YC=1|X=1)=.9265;P(YC=1|X=0)=.0735;CD=30;P(YC=1|X=1)=.9893;P(YC=1|X=0)=.0107; whereCiistheindividualcostofutilizationofsensori.Theprobabilitieshavebeencomputedforathresholdcorrespondingtotheequalerrorrate.CFalseNegativetobevariedbetween25millionand500billiondollarsLowandhighestimatesofdirectandindirectcostsincurredduetoafalsenegative.
CFalsePositive
tobevariedbetween180and720dollarsCostincurredduetofalsepositive(4men*(3-6hrs)*(15–30$/hr)P(X=1)tobevariedbetween3/109and1/100,000StructureoftreeswhichcamefirstRankwith3sensors(A,CandD)Treenumber49BooleanExpr:01010111Treenumber37BooleanExpr:00011111acbc10101ab1c001abc0111Treenumber55BooleanExpr:01111111Frequencyofoptimaltreeswith3sensors(A,CandD)whenoneparameterwasvariedConstantParameter(s)VariableParameter(s)TreeNumbersFrequency(outof10,000)EquivalentBooleanExpressionP(X=1)=1.281x10-6,CFalsePositive=492.61CFalseNegative375680001111155943201111111P(X=1)=0.8373x10-5,CFalseNegative=4.2681x1011CFalsePositive55994601111111CFalseNegative=4.4747x1011,CFalsePositive=351.9526P(X=1)37540001111155994601111111RandomlyselectedfixedparametervaluesVariationofCTotvs.CFalseNegative
P(X=1)andCFalsePositive
werekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalseNegative
inthespecifiedrange.RandomlyselectedfixedparametervaluesP(X=1)andCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalsePositiveinthespecifiedrange.RandomlyselectedfixedparametervaluesVariationofCTotvs.CFalsePositiveCFalsePositiveandCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofP(X=1)inthespecifiedrange.RandomlyselectedfixedparametervaluesVariationofCTotvs.P(X=1)Frequencyofoptimaltreeswith3sensors(A,CandD)whenoneparameterwasvariedFixedparametervaluesselectedatStroudandSaegervaluesConstantParameter(s)VariableParameter(s)TreeNumbersFrequency(outof10,000)EquivalentBooleanExpressionP(X=1)=3x10-8,CFalsePositive=600CFalseNegative371000000011111P(X=1)=3x10-8,CFalseNegative=5x1010CFalsePositive371000000011111CFalseNegative=5x1010,CFalsePositive=600P(X=1)4910801010111376940001111155919801111111VariationofCTotvs.CFalseNegative
P(X=1)andCFalsePositive
werekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalseNegative
inthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesP(X=1)andCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalsePositiveinthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotvs.CFalsePositiveCFalsePositiveandCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofP(X=1)inthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotvs.P(X=1)VariationofCTotwrtCFalseNegativeandCFalsePositive
CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesVariationofCTotwrtCFalseNegativeandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesVariationofCTotwrtCFalsePositiveandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesVariationofCTotwrtCFalseNegativeandCFalsePositive
CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotwrtCFalseNegativeandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotwrtCFalsePositiveandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesTreeStructureandcorrespondingBooleanExpressionsTreenumber11785BooleanExpr:Treenumber11605BooleanExpr:a1cd011b1a1bc1d01d01TreeStructureandcorrespondingBooleanExpressionsTreenumber9133BooleanExpr:Treenumber8965BooleanExpr:d01acd011b0d01acd01b01cd01b1TreeStructureandcorrespondingBooleanExpressionsTreenumber6797BooleanExpr:Treenumber2473BooleanExpr:0000000101111111acd0101cd01b10abc0d1cd01b101TreeStructureandcorrespondingBooleanExpressionsTreenumber11305BooleanExpr:ad101cd01b1VariationofCTotvs.CFalseNegative
P(X=1)andCFalsePositive
werekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalseNegative
inthespecifiedrange.RandomlyselectedfixedparametervaluesP(X=1)andCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalsePositiveinthespecifiedrange.RandomlyselectedfixedparametervaluesVariationofCTotvs.CFalsePositiveCFalsePositiveandCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofP(X=1)inthespecifiedrange.RandomlyselectedfixedparametervaluesVariationofCTotvs.P(X=1)VariationofCTotvs.CFalseNegative
P(X=1)andCFalsePositive
werekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalseNegative
inthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesP(X=1)andCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofCFalsePositiveinthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotvs.CFalsePositiveCFalsePositiveandCFalseNegativewerekeptconstantatthespecifiedvalueandCTotwascomputedfor10,000randomlyselectedvaluesofP(X=1)inthespecifiedrange.FixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotvs.P(X=1)Frequencyofoptimaltreeswith4sensorswhentwoparameterswerevaried.Thefixedparameterswererandomlyselected.
RandomlyselectedfixedparametervaluesConstantParameter(s)VariableParameter(s)TreeNumbersFrequency(outof10,000)EquivalentBooleanExpressionCFalsePositive=453.6849CFalseNegativeP(X=1)50510000000001111111679718000100010111111189655000010101011111119001700010101011111119017600010101011111119133235000101011111111111605862101010111111111111178510620111111111111111CFalseNegative=4.7485x1010P(X=1),CFalsePositive261710000000111111111679716000100010111111189651210001010101111111900170001010101111111901713000101010111111191333920001010111111111113059901010101011111111160593510101011111111111P(X=1)=0.6344x10-5CFalseNegative,CFalsePositive679720001000101111111896513000101010111111191336500010101111111111130513010101010111111111605792801010111111111111178519790111111111111111VariationofCTotwrtCFalseNegativeandCFalsePositive
CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesVariationofCTotwrtCFalseNegativeandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesVariationofCTotwrtCFalsePositiveandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedRandomlyselectedfixedparametervaluesFrequencyofoptimaltreeswith4sensorswhentwoparameterswerevaried.ThefixedparameterswereselectedattheStroudandSaegervalues.ConstantParameter(s)VariableParameter(s)TreeNumbersFrequency(outof10,000)EquivalentBooleanExpressionCFalsePositive=600CFalseNegativeP(X=1)50510000000001111111247320000000101111111250910000000101111111679718000100010111111189651380001010101111111900119000101010111111190177000101010111111191331840001010111111111113056501010101011111111160592320101011111111111117853330111111111111111CFalseNegative=5x1010P(X=1),CFalsePositive67971400010001011111118965117000101010111111190019000101010111111190179000101010111111191333740001010111111111113059601010101011111111160593810101011111111111P(X=1)=3x10-8,CFalseNegative,CFalsePositive505110000000001111111775500000001000011112473420000000101111111261740000000011111111167975580001000101111111896538330001010101111111913354060001010111111111116051050101011111111111VariationofCTotwrtCFalseNegativeandCFalsePositive
CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotwrtCFalseNegativeandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesVariationofCTotwrtCFalsePositiveandP(X=1)CTot=
CFalsePositive*P(X=0)*P(Y=1|X=0)+CFalseNegative*P(X=1)*P(Y=0|X=1)
+CfixedFixedparametervaluesselectedatStroudandSaegervaluesSensitivitytoSensorPerformanceCA=.25;KA=4.37;ΣA=1CB=.25;KB=1.53;ΣB=1CC=15;KC=2.9;ΣC=1CD=30;KD=4.6;ΣD=1Theprobabilityoffalsepositivefortheithsensoriscomputedas:P(Yi=1|X=0)=0.5erfc[Ti/√2]Theprobabilityofdetectionfortheithsensoriscomputedas:P(Yi=1|X=1)=0.5erfc[(Ti-Ki)/(Σ√2)]whereCiistheindividualcostofutilizationofsensori,KiisthediscriminationpowerofthesensorandΣiisthespreadfactorforthesensorFollowingexperimentshavebeendoneusingsensorsA,B,CandDasdescribedbelowbyvaryingtheindividualsensorthresholdsTA,TBandTCfrom-4.0to+4.0instepsof0.4.ThesevalueswerechosensincetheygaveusaROCcurvefortheindividualsensorsoveracompleterangeP(Yi=1|X=0)andP(Yi=1|X=1)Frequencyofoptimaltreeswith3sensorswhentheThresholdswerevaried.Thefixedparameters(CFalsePositive,CFalseNegative,P(X=1))wereselectedrandomly.Fifteentreesattainedrankone,outofwhichtreenumber37wasthemostfrequent.ConstantsTreeNumbersFrequencyBooleanExpressionCFalseNegative==5.0125x109P(X=1)=5.05x10-6andCFalsePositive=450271140001011129146000101112183000000014926401010111513220101011125957000101112314750001010115243700010011384572000111111952560001010115828000000014558730011011155105870111111171339200000111371751500011111Performance(ROC)ofBestDecisionTreeforTreenumber37Performance(ROC)ofBestDecisionTreeforTreenumber37Frequencyofoptimaltreeswith4sensorswhentheThresholdswerevaried.Thefixedparameters(CFalsePositive,CFalseNegative,P(X=1))wereselectedrandomly.244treesattainedrankone,outofwhichtreenumber445wasthemostfrequent.Only15mostfrequentlyoccurringoptimaltreesoutofthe241aretabulatedbelow.ConstantsTreeNumbersFrequencyBooleanExpressionCFalseNegative==4.8668x1011P(X=1)=7.5361x10-6andCFalsePositive=499.754451
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