Table 4 List of copula functions.
From: A copula-based multivariate flood frequency analysis under climate change effects
Copula function | \(\:{\text{C}}_\theta\left(\text{u,v}\right)\) |
|---|---|
Ali-Mikhail-Haq | \(\:\frac{\text{uv}}{1-\theta\left(\text{1-u}\right)\left(\text{1-v}\right)}\) |
Clayton | \(\:{\left[\text{max}\left({\text{u}}^{\theta}\text{+}{\text{v}}^{\theta}\text{-1,0}\right)\right]}^{\text{-}{1/\theta}}\) |
Frank | \(\:\text{-}\frac{\text{1}}{\theta}\text{ln}\left(\text{1+}\frac{{\text{(}\text{e}}^{\text{-}\theta\text{u}}\text{-}\text{1)}{\text{(}\text{e}}^{\text{-}\theta\text{v}}\text{-}\text{1)}}{{\text{e}}^{\text{-}\theta}\text{-}\text{1}}\right)\) |
Gumbel | \(\:\text{exp}\left({\text{-}\left[{\text{(-}\text{ln}\text{u}\text{)}}^{\theta}\text{+}{\text{(-}\text{ln}\text{v}\text{)}}^{\theta}\right]}^{\text{1/}\theta}\right)\) |
Joe | 1-\(\:{\left[{\text{(1-}\text{u}\text{)}}^\theta\text{+}{\text{(1-}\text{v}\text{)}}^{\theta\text{}}\text{\-}{\text{(1-}\text{u}\text{)}}^\theta{\text{(1-}\text{v}\text{)}}^\theta\right]}^{\text{1/}\theta}\) |
12 | \(\:{\text{(1+}{\left[{\text{(}{\text{u}}^{\text{-1}}\text{-1)}}^\theta\text{+}{\text{(}{\text{v}}^{\text{-1}}\text{-1)}}^\theta\right]}^{\text{1/}\theta}\text{)}}^{\text{-1}}\) |
14 | \(\:{\left[\text{1+}{\left[{\left({\text{u}}^{{-1/\theta}}\text{-1}\right)}^{\theta}\text{+}{\left({\text{v}}^{1/\theta}\text{-1}\right)}^{\theta}\right]}^{1/\theta}\right]}^{{-\theta}}\) |
19 | \(\theta/\text{ln(}{\text{e}}^{\theta\text{/}\text{u}}\text{+}{\text{e}}^{\theta\text{/}\text{v}}-{\text{e}}^{\theta}\text{)}\) |
20 | \(\lceil ln(exp(u)^{-\theta}+exp(v^{-\theta})-e\rceil^{-1/\theta}\) |
