# Pairwise correlations of time series variables

I am analyzing a group of related time-series variables, and am wondering the best way to think of in-group correlations. For example, suppose that my population is daily leaf-fall for all trees. The subgroup of all Quercus Nigra trees in Whateverton, Georgia are all very highly correlated in the sense that, on any given day, if the leaf-fall rate of one tree is high, one could expect the leaf-fall rate of other trees in that group to be high. On the other hand, the group of all trees is not (or at least less) correlated, since if the leaf-fall rate of one Quercus Nigra tree in Whateverton is high, we cannot expect the leaf-fall rate of an evergreen tree in South America (or another arbitrary tree) to be high.

Between two individual time-series I can use a cross correlation. What would be the best way to generalize this to the concept of in-group correlation?

Hierarchical Clustering sounds like what you are looking for. There are packages for R and Python that you can use. You can do it with time series by using appropriate distance metrics and appropriate ways of combining the distances. The result is, depending on the metrics you choose, a set of groups where correlations within the group are higher than correlations between groups.

Cross-correlations can be computed between any two columns of numbers. Interpretating the cross correlation is another matter. Consider this blog http://empslocal.ex.ac.uk/people/staff/dbs202/cat/stats/corr.html . Correlation can arise when the two series being observed/measure are the resultant of a third variable just think about the relationship between fire damage and the # of fireman at a fire or the relationship between the # of churches in a town and the # of bars. Rather than repeat what is already written on SE in this regard just sort on "user:3382 correlation" to get some more of my reflctions on this subject . You might also like a 1926 paper https://www.jstor.org/stable/2341482?seq=1#page_scan_tab_contents .

• Not too concerned about interpreting it. The problem I have is less "I need to cause B to happen, can I enact A, thereby causing B?", and more " I need to select a subset of my population with a difficult-to-measure quality B, can I select on A, thereby getting a very B-ish subset?", for which plain old correlation, causation or no, should suffice....
– Him
Dec 27, 2016 at 18:49
• Specifically, I am trying to select a "very independent" subset of my time series variables. It may be sufficient that they do not correlate. I was hoping to use something like distance correlation, a lack of which implies independence, but I'm not sure that I can apply it to time-series variables.
– Him
Dec 27, 2016 at 18:51

I didn't quite understand your original question, but from your comment above to the answer provided byu @IrishStat ("I am trying to select a 'very independent' subset of my time series variables") I have a feeling that you might want to look at dynamic factor analysis (see this or google "dynamic factor analysis" for many other references.

This will not give you a subset of time series, but a reduced dimension set of latent variables that you can then relate to individual time series in a more or less ad-hoc way.