Journal on Policy & Complex Systems Volume 3, Issue 2 | Page 116

Modeling Complexity in Human Built Systems
X ( t + 1 ) = X ( t )[ r x
– rxX ( t ) – β x , y
( t )] ( 2 ) X ( t + 1 ) = Y ( t )[ ry – ryX ( t ) – β y , x
( t )]
CCM dispenses with the need to resolve that model by assuming that if the predictor variables are coupled and share an attractor manifold , that variable values should be predictably nearby on that manifold structure .
CCM tests for the degree to which entries are functionally coupled with lagged entries . This is accomplished by building an increasingly large library of randomly drawn lagged pair series — starting with a handful of pairs — a short library length — and building to a large number of pairs — a large library length . In the present analysis , library lengths increased from 10 pairs to 400 pairs , at 10-pair steps . Mean correlations are taken across these library lengths , across a number of bootstrapped random draws . In the present analysis , each library length was drawn 100 times . If the variables are coupled around an attractor manifold then the cross-mapped estimates are predicted to improve as the library length increases .
An ad hoc hypothesis test is performed to determine the significance of the CCM findings . This test uses random distribution of the time series data in two forms : first , overall random shuffles of the data set , and second , a semi-annual seasonal shuffle of the data set . These random distributions are analyzed using the CCM method described above , and then compared to the observed distributions to see the percentage of matched library pairs where the CCM coefficient of the random data set exceeds that of the observed data set . This value approximates a p-value .
Results and Analysis
Lag Determination
A pattern emerged using the general-to-specific lag determination methodology that a lag value of one was either the most explanatory , or among the two or three most explanatory , lag values both in terms of overall effect size , r-squared value , and information loss from multivariate lag models to a univariate lag model . Thus , a lag of one was selected for all analyses in this data set . Table 1 describes the coefficient , confidence interval , and r-squared for the series of univariate regression analyses in this study .
Table 1 . Lag determination coefficients ( Lag = 1 )
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