31Spline2 - Smoothing-part2

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Unformatted text preview: Smoothing-part2 Nextpage:fittedpenalizedregressionsplinesfor3smoothing parameters: 0,100,and5.7 5.7istheoptimalchoice,tobediscussedshortly optimalcurveisasequenceofstraightlines continuous,but1stderivativeisnotcontinuous Smoothedfitslooksmootherifcontinuousin1stderivativeand in2ndderivative Suggestsjoiningtogethercubicpieceswithappropriate constraintsonthepiecessothatthe1stand2ndderivativesare continuous Manyveryslightlydifferentapproaches cubicregressionsplines(cubicsmoothingsplines) thinplatesplines c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 1/26 5 1 1 5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 Age of diagnosis lo g C- p e p t id e c o n c e n t r a t io n ~0 100 5.7 c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 2/26 Welltalkaboutthinplatesplinesbecausetheyprovideaneasyto implementwaytofitmultiple X s E y = f ( x 1 , x 2 ) aswellasE y = f ( x 1 )+ f ( x 2 ) Thedegree3thinplatesplinewithknotsat ( k 1 , k 2 ,..., k K ) f ( x )= + 1 x + 2 x 2 + K i = 1 u k | x k i | 5 0.0 0.2 0.4 0.6 0.8 1.0 . . 2 .4 .6 .8 1 .0 c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 3/26 Howmuchtosmooth? i.e.what 2 ?orwhat u k s reminder:0 nosmoothing(linearorquadraticintps) large closefittodatapoints Welltalkaboutthreeapproaches: 1. Crossvalidation 2. Generalizedcrossvalidation 3. Mixedmodels c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 4/26 Crossvalidation Generalmethodtoestimateoutofsamplepredictionerror Concept:Developamodel,wanttoassesshowwellitpredicts MightuserMSEP ( y i y i ) 2 asacriterion. Problem:datausedtwice,oncetodevelopmodelandagainto assesspredictionaccuracy rMSEPsystematicallyunderestimates ( y i y i ) 2 ,where y arenewobservations,notusedinmodeldevelopment Training/testsetapproach:splitdataintwoparts Trainingdata:usedtodevelopmodel,usually50%,80%or90%of dataset Testset:usedtoassesspredictionaccuracy Wantalargetrainingdataset(togetagoodmodel)andalarge testset(togetapreciseestimateofrMSEP) c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 5/26 Crossvalidationgetsthebestofboth. leave-one-outcv:fitmodelwithoutobs i ,usethatmodelto compute y i 10-foldcv:sameidea,blocksof N / 10 observations Canbeusedtochooseasmoothingparameter Find 2 thatminimizescvpredictionerror CV ( 2 )= n i = 1 y i f i ( x i ; 2 ) 2 , where f i ( x i ; 2 ) isthepredictedvalueof y i usingapenalized linearsplinefunctionestimatedwithsmoothingparameter 2 from thedatasetthatexcludesthe i th observation. Find 2 valuethatminimizes CV ( 2 ) .Perhapscompute CV ( 2 ) for agridof 2 values Requiresa LOT ofcomputing(eachobs,many 2 ) c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 6/26 Approximationto CV ( 2...
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31Spline2 - Smoothing-part2

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