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Table 1 Three ordinary families of Archimedean Copulas (Clayton, Frank, and Gumbel Copula) and their generator, parameter space, and their formula

From: Polar motion prediction using the combination of SSA and Copula-based analysis

Family

Generator

Parameter

Formula

Clayton

\(\phi ^{Cl}(x)=\frac{1}{\theta }(t^{-\theta }-1)\)

\(-\,1 \le \theta\)

\(C_\theta (u,v)= \max [(u^{-\theta }+v^{-\theta }-1),0]^{-\frac{1}{\theta }}\)

Frank

\(\phi ^{Fr}(t)=-\ln \left\{ \frac{\mathrm{e}^{-\theta t}-1}{\mathrm{e}^{-\theta }-1}\right\}\)

\(-\,\infty< \theta <\infty\)

\(C_\theta (u,v)= \frac{1}{\theta }\ln (1+ \frac{(\mathrm{e}^{-\theta u}-1)(\mathrm{e}^{-\theta v})}{\mathrm{e}^{-\theta }-1})\)

Gumbel

\(\phi (t)=(-\ln t)^\theta\)

\(1 \le \theta\)

\(C_\theta (u,v)= \mathrm{e}^{-((-\ln (u)^\theta )+(-\ln (v)^\theta ))^{\frac{1}{\theta }}}\)

  1. \(\theta\) is the parameter of the Copula called the dependence parameter, which measures the dependence between the marginal