How to estimate the best fitting function to a scatter plot in R?

Question

I have scatterplot of two variables, for instance this:

x<-c(0.108,0.111,0.113,0.116,0.118,0.121,0.123,0.126,0.128,0.131,0.133,0.136)

y<-c(-6.908,-6.620,-5.681,-5.165,-4.690,-4.646,-3.979,-3.755,-3.564,-3.558,-3.272,-3.073)

and I would like to find the function that better fits the relation between these two variables.

to be precise I would like to compare the fitting of three models: linear, exponential and logarithmic.

I was thinking about fitting each function to my values, calculate the likelihoods in each case and compare the AIC values.

But I don't really know how or where to start. Any possible help about this would be extremely appreciated.

Thank you very much in advance.

Tina.

Answer 1

I would begin by an explantory plots, something like this :

x<-c(0.108,0.111,0.113,0.116,0.118,0.121,0.123,0.126,0.128,0.131,0.133,0.136)
y<-c(-6.908,-6.620,-5.681,-5.165,-4.690,-4.646,-3.979,-3.755,-3.564,-3.558,-3.272,-3.073)
dat <- data.frame(y=y,x=x)
library(latticeExtra)
library(grid)
xyplot(y ~ x,data=dat,par.settings = ggplot2like(),
       panel = function(x,y,...){
         panel.xyplot(x,y,...)
       })+
  layer(panel.smoother(y ~ x, method = "lm"), style =1)+  ## linear
  layer(panel.smoother(y ~ poly(x, 3), method = "lm"), style = 2)+  ## cubic
  layer(panel.smoother(y ~ x, span = 0.9),style=3)  + ### loeess
  layer(panel.smoother(y ~ log(x), method = "lm"), style = 4)  ## log

looks like you need a cubic model.

 summary(lm(y~poly(x,3),data=dat))

Residual standard error: 0.1966 on 8 degrees of freedom
Multiple R-squared: 0.9831, Adjusted R-squared: 0.9767 
F-statistic: 154.8 on 3 and 8 DF,  p-value: 2.013e-07