Policy and Complex Systems
Simplex Projection and S-Mapping
As can be observed in Figures 1 and
2 , these out-of-home care time series are highly variable from week to week and may appear to be noisy or chaotic . As a first step in EDM , we use simplex projection ( Sugihara & May , 1990 ) to isolate whether the variation in the time series is indeed chaotic , or whether , when the points in the time series are projected into a higher embedding dimension patterns of relationships between near neighbor points emerge as the attractor is folded in phase space . As the time series is projected into higher embedding dimensions , higher coefficients ( representing more predictable near neighbors ) indicate that better predictions can be made about the shape of the attractor manifold . Thus , simplex projection assists in assessing not only whether the time series displays nonlinearities , but also whether the variability of a time series is chaotic or predictable .
Further refining the analysis of whether an observed time series displays predictable nonlinear behavior , S-mapping ( Sugihara , 1994 ) uses a nonlinear tuning parameter to evaluate the nonlinearity of a time series in a given embedding dimension . The test predicts nonlinearity where values of a weighted tuning parameter , theta , in nonlinear space exceed those values in linear space . Specifically , theta amplifies the difference between observed distance and average distance between nearest neighbors . In a linear system , then , a theta of zero gives the best predictions . In a nonlinear system , however , increasing the weight ( driven by an increase of theta ) increases the locality of the prediction in phase space , indicating that nearer neighbors are more predictive than global average distances . Thus , an S-map indicates nonlinear dynamics in a time
series for a given embedding dimension determined by simplex projection .
Convergent Cross Mapping
Sugihara et al . ( 2012 ) describes CCM
as a method to detect casual relationships between weakly coupled variables in dynamic ecological populations . Theoretical ecology provides an opportunity for the study of populations because of the rich theoretical traditions coupled with mathematical analysis designed to leverage often-incomplete highly variable data sets .
CCM is rooted in Generalized Takens ’ Theorem , which states that an attractor manifold can be reconstructed from lags of different time series so long as those time series are part of the same system . Leveraging this , CCM predicts whether variables share a causal relationship governed by an attractor manifold by cross-predicting variables that are observed from the same dynamic system ( Ye , Deyle , Gilarranz , & Sugihara , 2015 ). In this case , utilizing Tennessee out-of-home care entries and lagged entries provide some assurance that the variables are coming from the same dynamic system . CCM , then , provides an analytic window into whether a predictable nonlinear attractor manifold exists within that system . Grounded in a latent , unobserved manifold that could be produced form a set of coupled difference equations representing a population growth mode similar to Equation 1l 1 :
1
Comparing Equation ( 2 ) to Equation ( 1 ), we see that Equation ( 2 ) elaborates the definition of the x ×( t ) term in Equation ( 1 ), which includes not only x from earlier in time but also the influence of a y term . In Equation ( 2 ), the entry and exit populations are coupled within the expansion of the x ×( t ) term from Equation ( 1 ). Though these models are not directly interchangeable , they represent the same general class of population growth model .
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