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Using time-delayed mutual information to discover and interpret temporal correlation structure in complex populations
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27. It may seem odd to normalize indices, but this just keeps the domain of between zero and one.
28. To see the variation in the PDF estimates due to small sample sizes, observe the PDF estimates for different sets of uniform random numbers with small cardinality.
29. Note, the L1 difference is not technically a distance function or a metric because it does not satisfy the triangle inequality.
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This paper addresses how to calculate and interpret the time-delayed mutual information (TDMI) 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 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.
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