31Spline2(1)

31Spline2(1) - Smoothing - part 2 I Next page: fitted...

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Unformatted text preview: Smoothing - part 2 I Next page: fitted penalized regression splines for 3 smoothing parameters: 0, 100, and 5.7 I 5.7 is the optimal choice, to be discussed shortly I optimal curve is a sequence of straight lines I continuous, but 1st derivative is not continuous I Smoothed fits look smoother if continuous in 1st derivative and in 2nd derivative I Suggests joining together cubic pieces with appropriate constraints on the pieces so that the 1st and 2nd derivatives are continuous I Many very slightly different approaches I cubic regression splines (cubic smoothing splines) I thin plate splines c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 1 / 26 5 10 15 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Age of diagnosis log C-peptide concentration ~0 100 5.7 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 2 / 26 I Well talk about thin plate splines because they provide an easy to implement way to fit multiple X s E y = f ( x 1 , x 2 ) as well as E y = f ( x 1 ) + f ( x 2 ) I The degree 3 thin plate spline with knots at ( k 1 , k 2 , .. . , k K ) f ( x ) = + 1 x + 2 x 2 + K X i = 1 u k | x- k i | 5 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 3 / 26 I How much to smooth? I i.e. what 2 ? or what u k s I reminder: 0 no smoothing (linear or quadratic in tps) large close fit to data points I Well talk about three approaches: 1. Cross validation 2. Generalized cross validation 3. Mixed models c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 4 / 26 Cross validation I General method to estimate out of sample prediction error I Concept: Develop a model, want to assess how well it predicts I Might use rMSEP p ( y i- y i ) 2 as a criterion. I Problem: data used twice, once to develop model and again to assess prediction accuracy I rMSEP systematically underestimates p ( y * i- y * i ) 2 , where y * are new observations, not used in model development I Training/test set approach: split data in two parts I Training data: used to develop model, usually 50%, 80% or 90% of data set I Test set: used to assess prediction accuracy I Want a large training data set (to get a good model) and a large test set (to get a precise estimate of rMSEP) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 5 / 26 I Cross validation gets the best of both. I leave-one-out cv: fit model without obs i , use that model to compute y i I 10-fold cv: same idea, blocks of N / 10 observations I Can be used to choose a smoothing parameter I Find 2 that minimizes cv prediction error I CV ( 2 ) = n X i = 1 n y i- f- i ( x i ; 2 ) o 2 , where f- i ( x i ; 2 ) is the predicted value of y i using a penalized linear spline function estimated with smoothing parameter...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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31Spline2(1) - Smoothing - part 2 I Next page: fitted...

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