Modeling Complexity in Human Built Systems
of biological populations ( Wulczyn , 1996 ). In both ecology and the study of human-built systems , limited available data have influenced the ability to test dynamic models . The advantage of using the foster care population to study human built systems has to do with the quality of the data used to track the population . States are required to record when a child enters foster care and when a child leaves foster care . The time stamps used to track these comings and goings are recorded at the daily level , which means the time scale of the time series data can be adjusted easily and with very little measurement error . Importantly , a large number of state systems have been tracking the whereabouts of foster children for decades . Consequently , we have sufficient time points to study the dynamical properties of the time series data over extended periods . Due to the record keeping requirements of the outof-home care system , the volume and granularity of the data provides a unique opportunity for theoretical and empirical advances in systems science .
The subset of FCDA data used in these analyses represents 81,142 child entries into out-of-home placements in the State of Tennessee from 2000 through 2014 . These data are census data , representing all out-of-home care entries in the population during that period . These entry data are provided at the daily level , representing 5,479 days worth of entries during that time . For analysis , that data was summed at the weekly level , and provided 783 weeks of aggregated entries . The anonymized , aggregate state-level entry data were reported in two different care-type categories : congregate care ( Entries , Figure 1a ; Exits , Figure 1b ); and traditional foster care ( Entries , Figure 2a ; Exits , Figure 2b ).
Theory and Method
The analysis in this paper is rooted
in empirical dynamical modeling
( EDM ), a method for time-series analysis developed by ecologist George Sugihara ( Sugihara et al ., 2012 ). EDM uses coupled population growth models from the same family as the coupled form of partial adjustment equations proposed in Wulczyn ( 1996 ). For non-chaotic systems that can be described as a point in high-dimensional space , EDM provides information on appropriate embedding dimensions through simplex projection , insights on nonlinearity in the system though S-mapping , and analysis of causal relationships between variables through convergent cross mapping ( CCM ).
Lag Determination
A
threshold issue is selecting the appropriate lag to describe the temporal relationship between variables . The lag structure represents the flow of feedback information through the system from stimulus to response . Though out-of-home care time series do display seasonality and periodicity , there is no theoretical ground for the selection of lag length for a finegrained aggregate out-of-home care time series . Thus , lag structure must be determined empirically . The lag structure of the data was produced using the general-to-specific methodology as described by Enders ( 2010 ), where a stepwise longitudinal linear regression is used to narrow down potential lag lengths and then the multivariate model is paired down using criterion testing ( in this case , the Akaike Information Criterion ). In the present analysis , we were seeking a univariate lag model for application in later analysis .
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