广东农垦经济和社会发展“十二五”规.ppt

ComparisonMethodology MeaningofasampleConfidenceintervalsMakingdecisionsandcomparingalternativesSpecialconsiderationsinconfidenceintervalsSamplesizes EstimatingConfidenceIntervals TwoformulasforconfidenceintervalsOver30samplesfromanydistribution z distributionSmallsamplefromnormallydistributedpopulation t distributionCommonerror usingt distributionfornon normalpopulationCentralLimitTheoremoftensavesus ThezDistribution Intervaloneithersideofmean Significancelevel issmallforlargeconfidencelevelsTablesofzaretricky becareful ThetDistribution Formulaisalmostthesame Usableonlyfornormallydistributedpopulations Butworkswithsmallsamples MakingDecisions Whydoweuseconfidenceintervals SummarizeserrorinsamplemeanGiveswaytodecideifmeasurementismeaningfulAllowscomparisonsinfaceoferrorButremember at90 confidence 10 ofsamplemeansdonotincludepopulationmean TestingforZeroMean Ispopulationmeansignificantlynonzero Ifconfidenceintervalincludes0 answerisnoCantestforanyvalue meanofsumsissumofmeans Example ourheightsamplesareconsistentwithaverageheightof170cmAlsoconsistentwith160and180 ComparingAlternatives OftenneedtofindbettersystemChoosefastestcomputertobuyProveouralgorithmrunsfasterDifferentmethodsforpaired unpairedobservationsPairedifithtestoneachsystemwassameUnpairedotherwise ComparingPairedObservations Treatproblemas1sampleofnpairsForeachtestcalculateperformancedifferenceCalculateconfidenceintervalfordifferencesIfintervalincludeszero systemsaren tdifferentIfnot signindicateswhichisbetter Example ComparingPairedObservations Dohomebaseballteamsoutscorevisitors Samplefrom9 4 96 Example ComparingPairedObservations H V2 2 756 1 767321 16Mean1 4 90 interval 0 75 3 6 Can trejectthehypothesisthatdifferenceis0 70 intervalis 0 10 2 76 ComparingUnpairedObservations AsampleofsizenaandnbforeachalternativeAandBStartwithconfidenceintervalsIfnooverlap Systemsaredifferentandhighermeanisbetter forHBmetrics IfoverlapandeachCIcontainsothermean SystemsarenotdifferentatthislevelIfclosecall couldlowerconfidencelevelIfoverlapandonemeanisn tinotherCIMustdot test Thet test 1 1 Computesamplemeansand2 Computesamplestandarddeviationssaandsb3 Computemeandifference 4 Computestandarddeviationofdifference Thet test 2 5 Computeeffectivedegreesoffreedom 6 Computetheconfidenceinterval 7 Ifintervalincludeszero nodifference ComparingProportions Ifkofntrialsgiveacertainresult thenconfidenceintervalisIfintervalincludes0 5 can tsaywhichoutcomeisstatisticallymeaningfulMusthavek 10togetvalidresults SpecialConsiderations SelectingaconfidencelevelHypothesistestingOne sidedconfidenceintervals SelectingaConfidenceLevel Dependsoncostofbeingwrong90 95 arecommonvaluesforscientificpapersGenerally usehighestvaluethatletsyoumakeafirmstatementButit sbettertobeconsistentthroughoutagivenpaper HypothesisTesting Thenullhypothesis H0 iscommoninstatisticsConfusingduetodoublenegativeGiveslessinformationthanconfidenceintervalOftenhardertocomputeShouldunderstandthatrejectingnullhypothesisimpliesresultismeaningful One SidedConfidenceIntervals Two sidedintervalstestformeanbeingoutsideacertainrange see errorbands inpreviousgraphs One sidedtestsusefulifonlyinterestedinonelimitUsez1 ort1 ninsteadofz1 2 ort1 2 ninformulas SampleSizes BiggersamplesizesgivenarrowerintervalsSmallervaluesoft vasnincreasesinformulasButsamplecollectionisoftenexpensiveWhatistheminimumwecangetawaywith Startwithasmallnumberofpreliminarymeasurementstoestimatevariance ChoosingaSampleSize Togetagivenpercentageerror r Here zrepresentseitherzortasappropriateForaproportionp k n ExampleofChoosingSampleSize Fiverunsofacompilationtook22 5 19 8 21 1 26 7 20 2secondsHowmanyrunstoget 5 confidenceintervalat90 confidencelevel 22 1 s 2 8 t0 95 4 2 132 LinearRegressionModels Whatisa good model EstimatingmodelparametersAllocatingvariationConfidenceintervalsforregressionsVerifyingassumptionsvisually WhatIsa Good Model Forcorrelateddata modelpredictsresponsegivenaninputModelshouldbeequationthatfitsdataStandarddefinitionof fits isleast squaresMinimizesquarederrorWhilekeepingmeanerrorzeroMinimizesvarianceoferrors Least SquaredError IfthenerrorinestimateforxiisMinimizeSumofSquaredErrors SSE Subjecttotheconstraint EstimatingModelParameters BestregressionparametersarewhereNoteerrorinbook ParameterEstimationExample Executiontimeofascriptforvariousloopcounts 6 8 2 32 xy 88 54 x2 264b0 2 32 0 29 6 8 0 35 GraphofParameterEstimationExample VariantsofLinearRegression Somenon linearrelationshipscanbehandledbytransformationsFory aebxtakelogarithmofy doregressiononlog y b0 b1x letb b1 Fory a blog x takelogofxbeforefittingparameters letb b1 a b0Fory axb takelogofbothxandy letb b1 AllocatingVariation Ifnoregression bestguessofyisObservedvaluesofydifferfrom givingrisetoerrors variance Regressiongivesbetterguess buttherearestillerrorsWecanevaluatequalityofregressionbyallocatingsourcesoferrors TheTotalSumofSquares Withoutregression squarederroris TheSumofSquaresfromRegression RecallthatregressionerrorisErrorwithoutregressionisSSTSoregressionexplainsSSR SST SSERegressionqualitymeasuredbycoefficientofdetermination EvaluatingCoefficientofDetermination ComputeComputeCompute ExampleofCoefficientofDetermination Forpreviousregressionexample y 11 60 y2 29 79 xy 88 54 SSE 29 79 0 35 11 60 0 29 88 54 0 05SST 29 79 26 9 2 89SSR 2 89 05 2 84R2 2 89 0 05 2 89 0 98 StandardDeviationofErrors VarianceoferrorsisSSEdividedbydegreesoffreedomDOFisn 2becausewe vecalculated2regressionparametersfromthedataSovariance meansquarederror MSE isSSE n 2 Standarddeviationoferrorsissquareroot CheckingDegreesofFreedom Degreesoffreedomalwaysequate SS0has1 computedfrom SSThasn 1 computedfromdataand whichusesup1 SSEhasn 2 needs2regressionparameters So ExampleofStandardDeviationofErrors Forourregressionexample SSEwas0 05 soMSEis0 05 3 0 017andse 0 13Notehighqualityofourregression R2 0 98se 0 13Whysuchanicestraight linefit ConfidenceIntervalsforRegressions Regressionisdonefromasinglepopulationsample sizen DifferentsamplemightgivedifferentresultsTruemodelisy 0 1xParametersb0andb1arereallymeanstakenfromapopulationsample CalculatingIntervalsforRegressionParameters Standarddeviationsofparameters Confidenceintervalsarebitsbiwherethasn 2degreesoffreedom ExampleofRegressionConfidenceIntervals Recallse 0 13 n 5 x2 264 6 8SoUsinga90 confidencelevel t0 95 3 2 353 RegressionConfidenceExample cont d Thus b0intervalisNotsignificantat90 Andb1isSignificantat90 andwouldsurviveeven99 9 test ConfidenceIntervalsforNonlinearRegressions Fornonlinearfitsusingexponentialtransformations ConfidenceintervalsapplytotransformedparametersNotvalidtoperforminversetransformationonintervals ConfidenceIntervalsforPredictions PreviousconfidenceintervalsareforparametersHowcertaincanwebethattheparametersarecorrect PurposeofregressionispredictionHowaccuratearethepredictions Regressiongivesmeanofpredictedresponse basedonsamplewetook PredictingmSamples StandarddeviationformeanoffuturesampleofmobservationsatxpisNotedeviationdropsasm Varianceminimalatx Uset quantileswithn 2DOFforinterval ExampleofConfidenceofPredictions Usingpreviousequation whatispredictedtimeforasinglerunof8loops Time 0 35 0 29 8 2 67Standarddeviationoferrorsse 0 1390 intervalisthen VerifyingAssumptionsVisually Regressionsarebasedonassumptions LinearrelationshipbetweenresponseyandpredictorxOrnonlinearrelationshipusedinfittingPredictorxnonstochasticanderror freeModelerrorsstatisticallyindependentWithdistributionN 0 c forconstantcIfassumptionsviolated modelmisleadingorinvalid TestingLinearity Scatterplotxvs ytoseebasiccurvetype Linear PiecewiseLinear Outlier Nonlinear Power TestingIndependenceofErrors Scatter plot iversusShouldbenovisibletrendExamplefromourcurvefit MoreonTestingIndependence MaybeusefultoploterrorresidualsversusexperimentnumberInpreviousexample thisgivessameplotexceptforxscalingNofoolprooftests TestingforNormalErrors Preparequantile quantileplotExampleforourregression TestingforConstantStandardDeviation Tongue twiste