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

curve-fitting, model-fitting, r

Solution

Here is an example of comparing five models. Due to the form of the first two models we are able to use `lm` to get good starting values. (Note that models using different transforms of `y` should not be compared so we should not use `lm1` and `lm2` as comparison models but only for starting values.) Now run an `nls` for each of the first two. After these two models we try polynomials of various degrees in `x`. Fortunately `lm` and `nls` use consistent `AIC` definitions (although its not necessarily true that other R model fitting functions have consistent `AIC` definitions) so we can just use `lm` for the polynomials. Finally we plot the data and fits of the first two models.

The lower the AIC the better so `nls1` is best followed by `lm3.2` following by `nls2` .

lm1 <- lm(1/y ~ x)
nls1 <- nls(y ~ 1/(a + b*x), start = setNames(coef(lm1), c("a", "b")))
AIC(nls1) # -2.390924

lm2 <- lm(1/y ~ log(x))
nls2 <- nls(y ~ 1/(a + b*log(x)), start = setNames(coef(lm2), c("a", "b")))
AIC(nls2) # -1.29101

lm3.1 <- lm(y ~ x) 
AIC(lm3.1) # 13.43161

lm3.2 <- lm(y ~ poly(x, 2))
AIC(lm3.2) # -1.525982

lm3.3 <- lm(y ~ poly(x, 3))
AIC(lm3.3) # 0.1498972

plot(y ~ x)

lines(fitted(nls1) ~ x, lty = 1) # solid line
lines(fitted(nls2) ~ x, lty = 2) # dashed line

ADDED a few more models and subsequently fixed them up and changed notation. Also to follow up on Ben Bolker's comment we can replace `AIC` everywhere above with `AICc` from the AICcmodavg package.

Problem

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.

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