32lowess(1) - Smoothing - part 3 I Penalized splines is not...

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Unformatted text preview: Smoothing - part 3 I Penalized splines is not the only way to estimate f ( x ) when y = f ( x ) + I Two others are kernel smoothing and the Lowess (Loess) smoother I Ill only talk about Lowess I Penalized splines and Lowess have same goal. I Lowess is more ad-hoc. Only practical difference Ive found is that Lowess includes an outlier detection/downweighting step I Lowess is LOcally WEighted polynomial regreSSion I Original paper is Cleveland, W.S. (1979) Robust locally weighted regression and smoothing scatterplots. JASA 74:829-836 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 1 / 15 Lowess concepts I any function, f ( x ) , is closely approximated by + 1 x over a small enough range of x I so pick a point, x , may or may not be an x in the data I fit a linear regression to the subset of the data with x values near x I predict f ( x ) from + 1 x I repeat for as many x s as needed to estimate f ( x ) I A couple of refinements: I Use weighted regression with weights dependent on distance from x I Robustify the fit by downweighting the influence of regression outliers (far vertically from f ( x )) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 2 / 15 I A simple algorithm that doesnt work well: I Consider all points within specified distance d of x I Fit OLS regression to these points I Imagine fitting the curve to 5 points around x I Then shifting to x + . 01 that now includes 6 points I that 6th point is within d of x + . 01 but not within d of x I Result is a very jumpy fitted curve because that 6th point has a very large influence on the fitted value I Lowess incorporates weighting so points d from x have some influence, but not much. I Influence increases as x gets close to a data point I Also hard to specify an appropriate d : depends on range of x values c 2011 Dept. Statistics (Iowa State University) Stat 511 section 31 3 / 15 Lowess details I What is close to...
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32lowess(1) - Smoothing - part 3 I Penalized splines is not...

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