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Using time-delayed mutual information to discover and interpret temporal correlation structure in complex populations
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Figures

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FIG. 1.

(Color) Graphically comparing (average PDF) and (PDF of the aggregate) for a collection of three collections of Gaussian random numbers whose distributions have means 0, 2, and 4 respectively.

Image of FIG. 2.

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FIG. 2.

The graphical schematic for the TDMI analysis of a population; note that by TDMI Present, we mean that the relevant TDMI measure (e.g., ) is greater than bias.

Image of FIG. 3.

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FIG. 3.

(Color) The graphs of the quadratic map (Eq. (50) ) and the Gauss map (Eq. (51) )—note the significant difference between the graphs of the mappings, and invariant density (PDF of the orbit) for the quadratic map, Gauss map, and the sum of the quadratic and Gauss maps—note the significant differences between the relative p’s.

Image of FIG. 4.

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FIG. 4.

(Color) PDFs of glucose measurements for individuals within a population and for a population for two data sets, the 100 patients with the largest records and 20 000 random patients.

Image of FIG. 5.

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FIG. 5.

(Color) Comparisons of the supports and PDF graph variations for two data sets, the 100 patients with the largest records and 5000 random patients.

Image of FIG. 6.

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FIG. 6.

(Color online) The TDMI for both and with δt bins of 6 h for a period of a few days for D 7 and D 8; note that the bias estimates can be found in Tables VIII and VII . With respect to (a), note the following: for δt ≤ 6 h, δI > 0 and for δt > 6 h, δI ≈ 0; the KDE and histogram estimates are extremely similar; the diurnal (daily) periodic variation in correlation of glucose is clearly evident in both and . With respect to (b), note the following: for all δt δI is consistent and likely zero within bias; the KDE and histogram estimates differ greatly, implying the presence of small sample size effects in the average TDMI calculation; the diurnal (daily) periodic variation in correlation of glucose is clearly evident in both and in all but the KDE estimated TDMI average.

Tables

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Algorithm 1:

How to interpret the TDMI for a population of time series

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Table I.

Summary of all the non-TDMI based metrics used to assess homogeneity in a population (both among the graphs and the supports) used to verify the TDMI-type analysis.

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Table II.

Summary of all the TDMI-based metrics used to interpret the TDMI and determine the population composition.

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Table III.

Complete list of the simulated data sets.

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Table IV.

TDMI results and homogeneity metrics for the simulated data sets one through six.

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Table V.

Heuristic homogeneity metrics for the simulated data sets one through six.

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Table VI.

Complete list of the real patient data sets.

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Table VII.

TDMI results and homogeneity metrics for the real patient data sets seven and eight; note all δt times are in hours.

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Table VIII.

TDMI results and homogeneity metrics for the real patient data sets seven and eight; note all δt times are in hours.

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Table IX.

Time independent TDMI results for the real patient data sets seven and eight.

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Table X.

Heuristic homogeneity metrics for the real patient data sets seven and eight.

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/content/aip/journal/chaos/22/1/10.1063/1.3675621
2012-01-24
2014-04-16

Abstract

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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Scitation: Using time-delayed mutual information to discover and interpret temporal correlation structure in complex populations
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/1/10.1063/1.3675621
10.1063/1.3675621
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