Functional Cluster Analysis

Summary

Cluster analysis is an exploratory technique. Functional data methods offer the advantage of allowing a greater variety of clustering matrixes to choose from. The examples involving the clustering of Canadian weather stations are meant to be illustrative, since the known locations of weather stations can be used to infer which ones should exhibit similar weather patterns. The objective is not so much to find “real” clusters of stations, but rather to learn how the weather patterns at the different stations are related. Some of the clusters obtained consist of stations that are located in the same region, which we would expect similar to have weather patterns. Other aspects of the clustering are harder to interpret (e.g., assignment of Prince Rupert and Halifax to the same cluster), although they may also indicate relationships in weather patterns for stations at some distance from each other. A cluster analysis that accounted for both precipitation and temperature (and other weather related variables such as humidity) might be preferable, provided a suitable clustering metric could be found.

Methods for determining the number of clusters in functional cluster analysis are identical to those in the classical case, and thus are not discussed further here.

If groupings for some of the data are known in advance, it may be preferable to use a discriminant function analysis to find the variables and matrix that best classify the remaining observations. In the chapter on functional generalized linear models, we use a form of discriminant function analysis, functional logistic models, to classify the weather stations.