@Michael: A statistician directly involved with climate data said that I would be best sticking with Spearman's rank correlation, as the data is non-Gaussian. He also mentioned that I could look at other mechanisms to see how well the curve fits, but that that information is relatively meaningless. Thanks for your help!

@Michael: I think I can use ANOVA and Chi-squared. This should give me the probability that the curve fits the data. Thank you.

@Michael: The calculation uses R as follows: data.frame( x, fitted( gam( y ~ s(x) ) ) ), where x & y are years and measurements, respectively. The documentation notes that the method is iteratively reweighted least squares. There are no parameters.

@Michael: I do not understand your question; I thought the fit was the model. Essentially, I have two data sets and I want to see how closely they relate. In the chart, the green dots are measured temperature values and the fitted orange line is the Generalized Additive Model. I am trying to find a suitable estimator for the goodness-of-fit of the model to the measurements.

I switched from MySQL to PostgreSQL to use autoregressive models in PL/R. The project ends July 15th, so I don't have enough time to remodel 273 million data points. I will be speaking with a climate researcher. The data is exposed through various inputs, including starting month-day to ending month-day (e.g., March 22 - June 22). It does not make sense to divide the data categorically into seasonal constraints. The users have the ability to probe the data like that, and in many other ways. I wanted to know if it made sense statistically to toss insufficient annual data.