Solved – Are LOESS and GAM with one covariate the same

generalized-additive-modelloessmgcv

I am learning about Generalized Additive Models (GAMS) and have found in multiple locations a claim that a GAM with one independent variable is identical to a bivariate LOESS regression. Is this claim correct?

I have been unable to obtain the same answer using the loess function in base R and the gam function in the R package mgcv. Here is a link to a similar question from 2009 on a different forum: http://r.789695.n4.nabble.com/Difference-between-gam-and-loess-td881045.html

Below is my R code, which I modified from code provided at the link above. The gam function in R has numerous options. If a GAM with one independent variable is identical to a bivariate LOESS regression could someone provide the option settings needed? I have tried a wide range of values for span in the loess function and am only showing a few.

Thank you for any advice or assistance.

library(mgcv)

set.seed(1234) 

# generate data
x <- sort(runif(100)) 
y <- sin(2*pi*x) + rnorm(10, sd=0.1) 

mgcv.1 <- gam(y ~ s(x), family=gaussian(), weights=NULL, subset=NULL,
          offset=NULL, method = "GCV.Cp",
          optimizer=c("outer", "newton"), control=list(), scale=0,
          select=FALSE, knots=NULL, sp=NULL, min.sp=NULL, H=NULL, gamma=1,
          fit=TRUE, paraPen=NULL, G=NULL, drop.unused.levels=TRUE,
          bs="cr")

base.r <- loess(y ~ x, degree=1, span=0.50) ; summary(base.r$fitted - mgcv.1$fitted)
base.r <- loess(y ~ x, degree=1, span=0.75) ; summary(base.r$fitted - mgcv.1$fitted)
base.r <- loess(y ~ x, degree=2, span=0.50) ; summary(base.r$fitted - mgcv.1$fitted)
base.r <- loess(y ~ x, degree=2, span=0.75) ; summary(base.r$fitted - mgcv.1$fitted)

EDIT

According to lecture notes at the link here:

http://polisci.msu.edu/jacoby/icpsr/regress3/lectures/week4/16.GAM.pdf

the gam function in mgcv only fits smoothing splines, but local polynomial regression could be done in S-PLUS. Perhaps that is why I cannot obtain the same answer with mgcv as I get with the loess function? Perhaps another GAM package in R does allow local polynomial regression?

Best Answer

Not really a full answer, but too long for a comment: s sets up a spline, whereas loess does a local regression.

In the gam package (maybe mgcv too, not too familiar with that one) you can also feed a local regression, as in 

library(gam)

set.seed(1234) 

# generate data
x <- sort(runif(100)) 
y <- sin(2*pi*x) + rnorm(10, sd=0.1) 

gam.1 <- gam(y ~ lo(x))
base.r <- loess(y ~ x) 
summary(base.r$fitted - gam.1$fitted)
plot(base.r$fitted,gam.1$fitted)

That does not produce the same fitted values either, but maybe you can further play around with the settings of lo and loess.

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