Modelling seasonal data with GAMs

In previous posts I have looked at how generalized additive models (GAMs) can be used to model non-linear trends in time series data. At the time a number of readers commented that they were interested in modelling data that had more than just a trend component; how do you model data collected throughout the year over many years with a GAM? In this post I will show one way that I have found particularly useful in my research.