Smoothing Techniques – Comparing Smoothing Splines vs LOESS for Smoothing

loessregressionsplines

I wish to better understand the pros/cons for using either loess or a smoothing splines for smoothing some curve.

Another variation of my question is if there is a way to construct a smoothing spline in a way that will yield the same results as using loess.

Any reference or insight are welcomed.

Best Answer

Here is some R code/example that will let you compare the fits for a loess fit and a spline fit:

library(TeachingDemos)
library(splines)

tmpfun <- function(x,y,span=.75,df=3) {
    plot(x,y)
    fit1 <- lm(y ~ ns(x,df))
    xx <- seq( min(x), max(x), length.out=250 )
    yy <- predict(fit1, data.frame(x=xx))
    lines(xx,yy, col='blue')
    fit2 <- loess(y~x, span=span)
    yy <- predict(fit2, data.frame(x=xx))
    lines(xx,yy, col='green')
    invisible(NULL)
}

tmplst <- list( 
    span=list('slider', from=0.1, to=1.5, resolution=0.05, init=0.75),
    df=list('slider', from=3, to=25, resolution=1, init=3))

tkexamp( tmpfun(ethanol$E, ethanol$NOx), tmplst )

You can try it with your data and change the code to try other types or options. You may also want to look at the loess.demo function in the TeachingDemos package for a better understanding of what the loess algorythm does. Note that what you see from loess is often a combination of loess with a second interpolation smoothing (sometimes itself a spline), the loess.demo function actually shows both the smoothed and the raw loess fit.

Theoretically you can always find a spline that approximates another continuous function as close as you want, but it is unlikely that there will be a simple choice of knots that will reliably give a close approximation to a loess fit for any data set.

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