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30Spline(1) - Nonparametric regression using smoothing...

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Nonparametric regression using smoothing splines I Smoothing is fitting a smooth curve to data in a scatterplot I Will focus on two variables: Y and one X I Our model: y i = f ( x i ) + ε i , where ε 1 , ε 1 , . . . ε n are independent with mean 0 I f is some unknown smooth function I Up to now f has a specified form with unknown parameters I f could be linear or nonlinear in the parameters, I functional form always specified I If f not determined by the subject matter, we may prefer to let the data suggest a function form c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 1 / 20
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I Why estimate f ? I can see features of the relationship between X and Y that are obscured by error variation I summarizes the relationship between X and Y I provide a diagnostic for a presumed parametric form I Example: Diabetes data set in Hastie and Tibshirani’s book Generalized Additive Models I Examine relationship between age of diagnosis of diabetes and log of the serum C-peptide concentration I Here’s what happens if we fit increasing orders of polynomial, then fit an estimated f c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 2 / 20
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5 10 15 3.0 4.0 5.0 6.0 Age at Diagnosis log C-peptide concentration c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 3 / 20
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5 10 15 3.0 4.0 5.0 6.0 Age at Diagnosis log C-peptide concentration linear fit c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 4 / 20
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