  This paper addresses how to calculate and interpret the time-delayed mutual
information for a complex, diversely and sparsely measured, possibly
non-stationary population of time-series of unknown composition and origin. The
primary vehicle used for this analysis is a comparison between the time-delayed
mutual information averaged over the population and the time-delayed mutual
information of an aggregated population (here aggregation implies the
population is conjoined before any statistical estimates are implemented).
Through the use of information theoretic tools, a sequence of practically
implementable calculations are detailed that allow for the average and
aggregate time-delayed mutual information to be interpreted. Moreover, these
calculations can be also be used to understand the degree of homo- or
heterogeneity present in the population. To demonstrate that the proposed
methods can be used in nearly any situation, the methods are applied and
demonstrated on the time series of glucose measurements from two different
subpopulations of individuals from the Columbia University Medical Center
electronic health record repository, revealing a picture of the composition of
the population as well as physiological features.
