Modeling Complexity in Human Built Systems
Table 2 . Comparison of convergent cross mapping coefficients
Simplex Projection and S-Mapping
Simplex projection suggested embedding dimensions of E = 7 for the foster series , and E = 6 for the congregate care . Figure 3 contains a representative Simplex plot for congregate care entries . Each time series displayed some degree of non-linearity , though each series also contained some linear component . This suggests that both linear and nonlinear analysis techniques have information to contribute to the analysis of out-of-home care time series . Figure 4 contains a representative S-map plot for congregate care entries .
Convergent Cross Mapping
The cross map for each time series indicated some convergence of couple correlations accompanying increases in library size . This is indicative of causal coupled relationships between the two variables represented in each line . The strongest relationship was shown , as expected , in the congregate care time series ( see Figure 5 ). The foster care time series showed the weakest relationship ( see Figure 6 ). However , the foster care time series did show the largest marginal increase in predictive value between the linear models and the CCM model in the lagged foster care exits causing foster care entries series . These values are listed in Table 2 .
The ad hoc significance test showed p-values exceeding p < 0.001 for all crossmapped data sets in congregate and foster care time series .
Analysis , Implications , and Limitations
Out-of-home care time series display predictable nonlinear causal dynamics at higher embedding dimensions . The complex shapes of the linear time series for congregate care and foster care are not generated through random linear processes , but through the contribution of nonlinear coupling . Moreover , these dynamics can be explored using only one lagged predictive variable . The short lag time ( tau ) of 1 week employed in this study suggests that outof-home care time series are proximately sensitive to fluctuations in population sizes . This insight has implications for policy conversations that are frequently carried out on time scales that are quite lengthy in comparison . Understanding that there are nonlinear relationships continued within these time series suggests that some variability
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