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