32lowess - Smoothing-part3 f x when y = f x (Loess smoother...

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Unformatted text preview: Smoothing-part3 Penalizedsplinesisnottheonlywaytoestimate f ( x ) when y = f ( x )+ TwoothersarekernelsmoothingandtheLowess(Loess) smoother I’llonlytalkaboutLowess PenalizedsplinesandLowesshavesamegoal. Lowessismoread-hoc.OnlypracticaldifferenceI’vefoundisthat Lowessincludesanoutlierdetection/downweightingstep LowessisLOcallyWEightedpolynomialregreSSion OriginalpaperisCleveland,W.S.(1979)Robustlocallyweighted regressionandsmoothingscatterplots.JASA74:829-836 c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 1/15 Lowessconcepts anyfunction, f ( x ) ,iscloselyapproximatedby β + β 1 x overasmall enoughrangeof x sopickapoint, x ,mayormaynotbean x inthedata fitalinearregressiontothesubsetofthedatawith x valuesnear x predict f ( x ) from ˆ β + ˆ β 1 x repeatforasmany x ’sasneededtoestimate f ( x ) Acoupleofrefinements: Useweightedregressionwithweightsdependentondistancefrom x “Robustify”thefitbydownweightingtheinfluenceofregression outliers(farverticallyfrom ˆ f ( x )) c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 2/15 Asimplealgorithmthatdoesn’tworkwell: Considerallpointswithinspecifieddistance d of x FitOLSregressiontothesepoints Imaginefittingthecurveto5pointsaround x Thenshiftingto x + . 01 thatnowincludes6points that6thpointiswithin d of x + . 01 butnotwithin d of x Resultisavery“jumpy”fittedcurve becausethat6’thpointhasaverylargeinfluenceonthefitted value Lowessincorporatesweightingsopoints ≈ d from x havesome influence,butnotmuch. Influenceincreasesas x getsclosetoadatapoint Alsohardtospecifyanappropriate d :dependsonrangeof x values c 2011Dept.Statistics(IowaStateUniversity) Stat511section31 3/15 Lowessdetails Whatis“closeto x ”? Define“close”intermsofafractionofthedatapoints f =smoothingparameter,...
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32lowess - Smoothing-part3 f x when y = f x (Loess smoother...

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