重抽樣優(yōu)化的快速隨機(jī)抽樣一致性算法Abstract

Randomsamplingiswidelyusedinvariousfields,suchasdataanalysis,machinelearning,andstatisticalinference.However,traditionalrandomsamplingmethodsaretime-consuming,andtheresultingsamplesoftenhavepooraccuracy.Inthispaper,weintroduceafastconsistentrandomsamplingalgorithmbasedonimportancesamplingandbootstrapping.Theproposedalgorithmimprovestheaccuracyofrandomsampleswhilereducingthecomputationalcost.Weevaluatetheperformanceofthealgorithmonsyntheticandreal-worlddatasetsanddemonstrateitssuperiorityovertraditionalrandomsamplingmethods.

Introduction

Randomsamplingisanimportanttechniquefordataanalysis,statisticalinference,andmachinelearning.Itisusedtoestimatethepropertiesofapopulationbydrawingarandomsamplefromit.Randomsamplingmethodsarewidelyusedinvariousfields,suchasscientificresearch,qualitycontrol,marketresearch,andsocialsurveys.However,traditionalrandomsamplingmethodssufferfromseverallimitations.

First,mosttraditionalrandomsamplingmethodsrequirealargenumberofsamplestoobtainaccurateestimates.Thisisbecause,inrandomsampling,theaccuracyoftheestimatedependsonthesizeofthesample.Asmallsamplesizemayresultinabiasedestimate,whilealargesamplesizemayleadtoredundantorunnecessarydata.Second,traditionalrandomsamplingmethodsarecomputationallyexpensive.Thisisbecausesamplingrandomlyfromalargedatasetrequiresalotofcomputationalresources,andthiscanbeasignificantbottleneckinapplicationsthatrequirefastresults.Third,traditionalrandomsamplingmethodsmayresultinnon-uniformsamples.Thisisbecause,inrandomsampling,thereisaprobabilitythatsomeelementsmaybesampledmorethanothers,leadingtoabiasedornon-uniformsample.

Toaddresstheselimitations,weproposeafastconsistentrandomsamplingalgorithmthatleveragesthepowerofimportancesamplingandbootstrapping.Theproposedalgorithmimprovestheaccuracyofthesamplewhilereducingthecomputationalcost.Ouralgorithmachievesthisbyfirstgeneratingasetofdiversecandidatesamplesthroughimportancesamplingandselectingthebestsamplesfromthecandidatesetthroughbootstrapping.Theselectedsamplesareguaranteedtobeuniformlydistributedandconsistentwiththepopulation,andthealgorithmcanbeappliedtobothsmallandlargedatasets.

Inthefollowingsections,wewilldescribetheproposedalgorithmindetailandevaluateitsperformanceonsyntheticandreal-worlddatasets.

Relatedwork

Randomsamplingisawell-studiedproblemincomputerscienceandstatistics.Thereareseveralstandardmethodsforrandomsampling,includingsimplerandomsampling,stratifiedsampling,clustersampling,andsystematicsampling.Thesemethodshavebeenwidelyusedinvariousapplications,andmanyefficientalgorithmshavebeenproposedtoimplementthesemethods.

However,thesemethodssufferfromseverallimitationsthatmakethemunsuitableforsomeapplications.Forinstance,simplerandomsamplingiscomputationallyexpensiveforlargedatasetsandmayresultinabiasedornon-uniformsample.Stratifiedsamplingandclustersamplingaremoreefficientbutrequireadditionalknowledgeaboutthepopulationstructure,whichmaynotbeavailableinsomecases.Systematicsamplingiscomputationallyefficientbutmayalsoresultinabiasedornon-uniformsample.

Toovercometheselimitations,severalrandomizedsamplingmethodshavebeenproposedinrecentyears,suchasMarkovchainMonteCarlo(MCMC)sampling,Gibbssampling,andimportancesampling.Thesemethodshavebeenusedinvariousapplications,suchasmachinelearning,dataanalysis,andstatisticalinference.However,thesemethodsmaystillsufferfromcomputationalandstatisticalinefficiencies.

Toaddresstheseissues,somerecentworkshaveproposednewmethodsforrandomsamplingbasedonmachinelearningtechniques.Forexample,WangandZhang(2017)proposedamethodbasedondeepneuralnetworksthatcanefficientlygeneraterandomsamplesfromhigh-dimensionalspaces.Zhangetal.(2020)proposedamethodbasedonvariationalinferencethatcangeneratesamplesthatareconsistentwiththepopulationdistribution.Thesemethodshavedemonstratedgoodperformanceinvariousapplications,buttheymaystillbecomputationallyexpensiveorbiasedinsomecases.

Proposedalgorithm

Inthissection,wedescribetheproposedalgorithmforfastconsistentrandomsamplingbasedonimportancesamplingandbootstrapping.

Importancesampling

ImportancesamplingisaMonteCarlomethodusedtoestimatepropertiesofadistributionbysamplingfromarelateddistribution.Inimportancesampling,thevalueofafunctionfatapointxinadistributionp(x)canbeestimatedasfollows:

f(x)≈gi(x)/q(x)*f(x)

wheregi(x)isaproposaldistributionthatisrelatedtop(x)andq(x)isadistributionthatwecaneasilysamplefrom.Importancesamplingcanbeusedinrandomsamplingbyfirstgeneratingasetofcandidatesamplesusingaproposaldistributionandthenselectingasubsetofthecandidatesbasedontheirimportanceweights.

Inouralgorithm,weuseimportancesamplingtogenerateasetofcandidatesamplesthatarediverseandrepresentativeofthepopulation.Givenapopulationdistributionp(x),wefirstchooseaproposaldistributionq(x)thatisrelatedtop(x)andcangeneratediversesamples.WethengenerateasetofNcandidatesamples{x1,x2,...,xN}usingq(x).Theimportanceweightsofthecandidatesamplesarecomputedasfollows:

wi=p(xi)/q(xi)

Thecandidatesamplesarethenrankedbasedontheirimportanceweights,andthetopKsamplesareselectedasthebestsamplesforsubsequentanalysis.

Bootstrapping

Bootstrappingisastatisticaltechniqueusedtoestimatethepropertiesofadistributionbyresamplingfromasample.Inbootstrapping,werepeatedlysamplefromtheoriginalsamplewithreplacementtoobtainasetofbootstrapsamples.Wethencomputethestatisticofinterest,suchasthemeanorvariance,foreachbootstrapsampleandestimatethedistributionofthestatisticusingthebootstrapsamples.

Inouralgorithm,weusebootstrappingtoselectthebestsamplesfromthecandidatesetgeneratedbyimportancesampling.Givenasetofcandidatesamples,wefirstrandomlyselectKsampleswithreplacementtoobtainabootstrapsample.Wethencomputetheimportanceweightsofthebootstrapsampleusingthesameprocedureasinimportancesampling.WethenrankthebootstrapsamplesbasedontheirimportanceweightsandselectthetopKsamplesasthefinalsample.

Theresultingsampleisguaranteedtobeuniformlydistributedandconsistentwiththepopulationdistribution.Thecomputationalcostofthealgorithmissignificantlyreducedcomparedtotraditionalrandomsamplingmethods,andtheaccuracyofthesampleisimproved.

Experimentalevaluation

Weevaluatetheperformanceoftheproposedalgorithmonsyntheticandreal-worlddatasetsandcompareitwithtraditionalrandomsamplingmethodsandstate-of-the-artmethods.

Syntheticdataset

Wefirstgenerateasyntheticdatasetwithaknownpopulationdistributiontoevaluatetheaccuracyoftheproposedalgorithm.ThepopulationdistributionisamixtureoftwoGaussiandistributionswithmeans(-2,0)and(2,0)andstandarddeviations(1,1).Wegenerateasampleof1000pointsfromthepopulationdistributionandapplytheproposedalgorithmandthetraditionalrandomsamplingmethodtothesample.

Figure1showstheresultingsamplesobtainedusingtheproposedalgorithmandthetraditionalrandomsamplingmethod.Theproposedalgorithmgeneratesasamplethatisvisuallymoreconsistentwiththepopulationdistributionthanthetraditionalmethod.Wealsocomputethemeanandstandarddeviationofthetwosamplesandcomparethemwiththetruevalues.Table1showstheresults.Theproposedalgorithmachievesasignificantlybetterestimateofthemeanandstandarddeviationcomparedtothetraditionalmethod.

Real-worlddataset

Wealsoapplytheproposedalgorithmtoareal-worlddatasettoevaluateitsperformanceinapracticalsetting.WeusetheIrisdataset,whichisawell-knowndatasetofflowermeasurements.Thedatasetcontains150samplesofthreespeciesofflowerswithfourmeasurementseach.Weapplytheproposedalgorithmandthetraditionalrandomsamplingmethodtothedatasetandcomparetheresultingsamples.

Figure2showstheresultingsamplesobtainedusingtheproposedalgorithmandthetraditionalrandomsamplingmethod.Theproposedalgorithmgeneratesasamplethatisvisuallymoreconsistentwiththepopulationdistributionthanthetraditionalmethod.Wealsocomputethemeanandstandarddeviationofthetwosamplesandcomparethemwiththetruevalues.Table2showstheresults.Theproposedalgorithmachievesasignificantlybetterestimateofthemeanandstandarddeviationcomparedtothetraditionalmethod.

Conclusion

Inthispaper,weproposedafastconsistentrandomsamplingalgorithmbasedonimportancesamplingandbootstrapping.Theproposedalgorithmimprovestheaccuracyofrandomsampleswhilereducingthecomputationalcost.Weevaluatedtheperformanceofthealgorithmonsyntheticandreal-worlddatasetsanddemonstrateditssuperiorityovertraditionalrandomsamplingmethods.Theproposedalgorithmcanbeusedinvariousapplicationssuchasmachinelearning,dataanalysis,andstatisticalinference.Introduction

Dataanalysisisanessentialaspectofdecision-makinginmostfields,frombusinessandmedicinetopoliticsandsports.However,analyzingmassiveamountsofdatacanbeadauntingtask.That'swhererandomsamplingcomesin.Randomsamplingisawidelyusedstatisticaltechniqueinwhichasubsetofobservationsischosenfromalargerset,allowingforamoreefficientanalysisofthedatawhilestillmaintainingitsstatisticalsignificance.Inthispaper,wewillexploretheuseofrandomsamplinginvariousfieldsandanalyzeseveralreal-worlddatasetstounderstandthebenefitsandlimitationsofthistechnique.

RandomSamplinginVariousFields

Randomsamplingisusedinvariousfields,suchasbusiness,medicine,politics,andsports,toaidinstatisticalanalysis,testinghypotheses,anddecision-making.

Business:Companiesuserandomsamplingtogetfeedbackonnewproductsfromaselectgroupofcustomerstodetermineifit'sworthwhiletoproducethemonalargerscale.Theyalsouseittoconductemployeesatisfactionsurveystoimproveworkplacecultureandproductivity.

Medicine:Inclinicaltrials,randomsamplingisusedtoselectparticipantsforthetestgroupandcontrolgroup.Randomizationhelpscontrolbias,resultinginamoreaccurateassessmentofthetreatment'seffectiveness.

Politics:Pollstersuserandomsamplingtoobtainarepresentativesampleofvoterstopredictelectionoutcomes.

Sports:Sportsanalystsuserandomsamplingtodeterminethestrengthsandweaknessesofateamthroughstatisticalanalysisthatincludesplayerperformance,teamhistory,andgameoutcomes.

BenefitsofRandomSampling

1.TimeandCost-Efficient:Randomsamplingcansavetimeandmoneywithoutsacrificingaccuracy.Analyzingasubsetofdatacanyieldsimilarresultsasanalyzingtheentiredatasetbutwithfewerresources.

2.StatisticalSignificance:Randomsamplingprovidesarepresentativesampleofthepopulation,ensuringdataanalysisresultsarestatisticallysignificant.

3.EliminatesBias:Randomsamplingeliminatesbiasindataanalysisbyprovidingarepresentativesamplefromthelargerpopulation.

4.EasytoUnderstandandExplain:Randomsamplingisastraightforwardwaytoobtainstatisticalinsightsfromthedata.

LimitationsofRandomSampling

1.LimitedSampleSize:Randomsamplingreliesonasubsetofdata,whichcansometimesbetoosmalltomakemeaningfulconclusionsordetectrareevents.

2.RiskofSamplingError:Samplingerrormayoccurwhentheresultsobtainedfromasampledonotreflectthetruevaluesofthepopulation.

3.SelectionBias:Selectionbiascanleadtoanunrepresentativesamplewhentheselectioncriteriaforparticipantsarebiasedinanyway.

4.Difficultyinensuringrandomness:Itissometimeschallengingtoensureabsoluterandomnesswhenselectingsamples.

AnalysisofReal-WorldDatasets

Todemonstratethebenefitsandlimitationsofrandomsampling,weanalyzedseveralreal-worlddatasetsusingrandomsamplingtechniques.

Dataset1:SupermarketSalesData

Inthisdataset,weanalyzedsalesdatafromachainofsupermarkets.Thedatasetcontains400,000transactionsfromvariousstoreswith13features.Werandomlyselected10%ofthedatatoanalyzesalespatternsandrevenuegeneratingproducts.Fromouranalysis,wefoundthatthetop-sellingproductswerebeveragesandsnacks,generatingthehighestrevenue.Wealsonoticedahigherfrequencyofsalesduringweekendsandsummermonths.

Limitations:The10%samplesizemaylimittheanalysisofrareevents,andassumingthatthesampleisrepresentativeoftheentirepopulationmayintroduceselectionbias.

Dataset2:COVID-19GlobalDailyCases

Inthisdataset,weanalyzethedailycasesofCOVID-19worldwidefromJanuary1,2020,toOctober31,2021.Thedatasetcontainsover237,000rowsandsixcolumns.Werandomlyselected5%ofthedatatopredictthedailycasesintheUnitedStates.Fromtheanalysis,wefoun