Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm

Abstract

This study aims to model the joint probability distribution of drought duration, severity and inter-arrival time using a trivariate Plackett copula. The drought duration and inter-arrival time each follow the Weibull distribution and the drought severity follows the gamma distribution. Parameters of these univariate distributions are estimated using the method of moments (MOM), maximum likelihood method (MLM), probability weighted moments (PWM), and a genetic algorithm (GA); whereas parameters of the bivariate and trivariate Plackett copulas are estimated using the log-pseudolikelihood function method (LPLF) and GA. Streamflow data from three gaging stations, Zhuangtou, Taian and Tianyang, located in the Wei River basin, China, are employed to test the trivariate Plackett copula. The results show that the Plackett copula is capable of yielding bivariate and trivariate probability distributions of correlated drought variables.

Appendices

Appendix 1

The method for estimating the trivariate cross-product ratio entails the following steps:

• (1) Compute C _UV , C _VW , and C _UW using Eq. 3.

• (2) For computing C _UVW , Eq. 9 can be rewritten as

  $$ \begin{aligned} \left( {\psi_{UVW} - 1} \right)z^{4} & + \left[ { - \psi_{UVW} \left( {a_{1} + a_{2} + a_{3} + a_{4} } \right) + \left( {b_{1} + b_{2} + b_{3} } \right)} \right]z^{3} \\ & + \left\{ {\psi_{UVW} \left[ {a_{1} a_{2} + \left( {a_{1} + a_{2} } \right)\left( {a_{3} + a_{4} } \right) + a_{3} a_{4} } \right] - \left[ {b_{1} b_{2} + b_{3} \left( {b_{1} + b_{2} } \right)} \right]} \right\}z^{2} \\ & + \left\{ { - \psi_{UVW} \left[ {a_{1} a_{2} \left( {a_{3} + a_{4} } \right) + a_{3} a_{4} \left( {a_{1} + a_{2} } \right)} \right] + b_{1} b_{2} b_{3} } \right\}z + \psi_{UVW} a_{1} a_{2} a_{3} a_{4} = 0 \\ \end{aligned} $$

  (37)

  Let f(z) represent the left side of Eq. 37. Newton’s iterative estimating method of z can be expressed as:

  $$ z_{n + 1} = z_{n} - {\frac{{f(z_{n} )}}{{f^{\prime}(z_{n} )}}} $$

  (38)

  where f ^′(z) is the first derivative of f(z) with respect to z; z _n and z _n+1 are the nth and (n + 1)th iteratively computed values of z. Then, C _UVW can be obtained by Newton’s iteration method.

• (3) Compute P ₀₀₀, P ₀₁₀, P ₁₀₀, P ₀₁₁, P ₁₁₀, P ₁₀₁, P ₀₁₁, P ₁₁₁ using Eq. 8.

• (4) Compute ${\frac{{\partial C_{UV} }}{\partial u}},\;{\frac{{\partial C_{UV} }}{\partial v}},\;{\frac{{\partial C_{UW} }}{\partial u}},\;{\frac{{\partial C_{UW} }}{\partial w}},\;{\frac{{\partial C_{VW} }}{\partial v}},\;{\text{and}}\;{\frac{{\partial C_{VW} }}{\partial w}}$ using Eqs. 5 and 6, respectively.

• (5) The first derivative of P ₀₀₀, P ₀₁₀, P ₁₀₀, P ₀₁₁, P ₁₁₀, P ₁₀₁, P ₀₁₁, P ₁₁₁ with respect to u, v, w is expressed, respectively, as:

  $$ \begin{gathered} {\frac{{\partial P_{000} }}{\partial u}} = - {\frac{{\partial P_{100} }}{\partial u}} = {\frac{{\partial C_{UVW} }}{\partial u}} \hfill \\ {\frac{{\partial P_{010} }}{\partial u}} = - {\frac{{\partial P_{110} }}{\partial u}} = - {\frac{{\partial C_{UVW} }}{\partial u}} + {\frac{{\partial C_{UW} }}{\partial u}} \hfill \\ {\frac{{\partial P_{001} }}{\partial u}} = - {\frac{{\partial P_{101} }}{\partial u}} = - {\frac{{\partial C_{UVW} }}{\partial u}} + {\frac{{\partial C_{UV} }}{\partial u}} \hfill \\ {\frac{{\partial P_{011} }}{\partial u}} = - {\frac{{\partial P_{111} }}{\partial u}} = {\frac{{\partial C_{UVW} }}{\partial u}} - {\frac{{\partial C_{UV} }}{\partial u}} - {\frac{{\partial C_{UW} }}{\partial u}} + 1 \hfill \\ \end{gathered} $$

  (39)
  $$ \begin{gathered} {\frac{{\partial P_{000} }}{\partial v}} = - {\frac{{\partial P_{010} }}{\partial v}} = {\frac{{\partial C_{UVW} }}{\partial v}} \hfill \\ {\frac{{\partial P_{100} }}{\partial v}} = - {\frac{{\partial P_{110} }}{\partial v}} = - {\frac{{\partial C_{UVW} }}{\partial v}} + {\frac{{\partial C_{VW} }}{\partial v}} \hfill \\ {\frac{{\partial P_{001} }}{\partial v}} = - {\frac{{\partial P_{011} }}{\partial v}} = - {\frac{{\partial C_{UVW} }}{\partial v}} + {\frac{{\partial C_{UV} }}{\partial v}} \hfill \\ {\frac{{\partial P_{101} }}{\partial v}} = - {\frac{{\partial P_{111} }}{\partial v}} = {\frac{{\partial C_{UVW} }}{\partial v}} - {\frac{{\partial C_{UV} }}{\partial v}} - {\frac{{\partial C_{VW} }}{\partial v}} + 1 \hfill \\ \end{gathered} $$

  (40)
  $$ \begin{gathered} {\frac{{\partial P_{000} }}{\partial w}} = - {\frac{{\partial P_{001} }}{\partial w}} = {\frac{{\partial C_{UVW} }}{\partial w}} \hfill \\ {\frac{{\partial P_{100} }}{\partial w}} = - {\frac{{\partial P_{101} }}{\partial w}} = - {\frac{{\partial C_{UVW} }}{\partial w}} + {\frac{{\partial C_{VW} }}{\partial w}} \hfill \\ {\frac{{\partial P_{010} }}{\partial w}} = - {\frac{{\partial P_{011} }}{\partial w}} = - {\frac{{\partial C_{UVW} }}{\partial w}} + {\frac{{\partial C_{UW} }}{\partial w}} \hfill \\ {\frac{{\partial P_{110} }}{\partial w}} = - {\frac{{\partial P_{111} }}{\partial w}} = {\frac{{\partial C_{UVW} }}{\partial u}} - {\frac{{\partial C_{UW} }}{\partial w}} - {\frac{{\partial C_{VW} }}{\partial w}} + 1 \hfill \\ \end{gathered} $$

  (41)

• (6) Compute ${\frac{{\partial C_{UVW} }}{\partial u}},\;{\frac{{\partial C_{UVW} }}{\partial v}}$ and ${\frac{{\partial C_{UVW} }}{\partial w}}$ as:

  $$ \begin{aligned} & {\frac{{\partial P_{000} }}{\partial u}}P_{011} P_{101} P_{110} + P_{000}\, {\frac{{\partial P_{011} }}{\partial u}}P_{101} P_{110} + P_{000} P_{011}\, {\frac{{\partial P_{101} }}{\partial u}}P_{110} + P_{000} P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial u}} \\ & - \psi_{UVW} \left( {{\frac{{\partial P_{111} }}{\partial u}}P_{100} P_{010} P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial u}}P_{010} P_{001} + P_{111} P_{100}\, {\frac{{\partial P_{010} }}{\partial u}}P_{001} + P_{111} P_{100} P_{010}\, {\frac{{\partial P_{001} }}{\partial u}}} \right) = 0 \\ \end{aligned} $$

  (42)
  $$ \begin{aligned} & {\frac{{\partial P_{000} }}{\partial v}}P_{011} P_{101} P_{110} + P_{000}\, {\frac{{\partial P_{011} }}{\partial v}}P_{101} P_{110} + P_{000} P_{011} \,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + P_{000} P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial v}} \\ & - \psi_{UVW} \left( {{\frac{{\partial P_{111} }}{\partial v}}P_{100} P_{010} P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}P_{010} P_{001} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + P_{111} P_{100} P_{010} \,{\frac{{\partial P_{001} }}{\partial v}}} \right) = 0 \\ \end{aligned} $$

  (43)
  $$ \begin{aligned} & {\frac{{\partial P_{000} }}{\partial w}}P_{011} P_{101} P_{110} + P_{000}\, {\frac{{\partial P_{011} }}{\partial w}}P_{101} P_{110} + P_{000} P_{011} \,{\frac{{\partial P_{101} }}{\partial w}}P_{110} + P_{000} P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial w}} \\ & - \psi_{UVW} \left( {{\frac{{\partial P_{111} }}{\partial w}}P_{100} P_{010} P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial w}}P_{010} P_{001} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial w}}P_{001} + P_{111} P_{100} P_{010} \,{\frac{{\partial P_{001} }}{\partial w}}} \right) = 0 \\ \end{aligned} $$

  (44)

• (7) Compute the first derivative of P ₀₀₀, P ₀₁₀, P ₁₀₀, P ₀₁₁, P ₁₁₀ P ₁₀₁, P ₀₁₁, P ₁₁₁ with respect to u, v, w using Eqs. 39, 40 and 41, respectively.

• (8) Compute the probability density of a bivariate Plackett copula ${\frac{{\partial^{2} C_{UV} }}{\partial u\partial v}},\;{\frac{{\partial^{2} C_{VW} }}{\partial v\partial w}}$ and ${\frac{{\partial^{2} C_{UW} }}{\partial u\partial w}}$ using Eq. 4, respectively.

• (9) Compute second-order derivatives of P ₀₀₀, P ₀₁₀, P ₁₀₀, P ₀₁₁, P ₁₁₀ P ₁₀₁, P ₀₁₁, P ₁₁₁ with respect to u, v, w as:

  $$ \begin{gathered} {\frac{{\partial^{2} P_{000} }}{\partial u\partial v}} = - {\frac{{\partial^{2} P_{100} }}{\partial u\partial v}} = - {\frac{{\partial^{2} P_{010} }}{\partial u\partial v}} = {\frac{{\partial^{2} P_{110} }}{\partial u\partial v}} = {\frac{{\partial^{2} C_{UVW} }}{\partial u\partial v}} \hfill \\ {\frac{{\partial^{2} P_{001} }}{\partial u\partial v}} = - {\frac{{\partial^{2} P_{101} }}{\partial u\partial v}} = - {\frac{{\partial^{2} P_{011} }}{\partial u\partial v}} = {\frac{{\partial^{2} P_{111} }}{\partial u\partial v}} = - {\frac{{\partial^{2} C_{UVW} }}{\partial u\partial v}} + {\frac{{\partial^{2} C_{UV} }}{\partial u\partial v}} \hfill \\ \end{gathered} $$

  (45)
  $$ \begin{gathered} {\frac{{\partial^{2} P_{000} }}{\partial u\partial w}} = - {\frac{{\partial^{2} P_{100} }}{\partial u\partial w}} = - {\frac{{\partial^{2} P_{001} }}{\partial u\partial w}} = {\frac{{\partial^{2} P_{101} }}{\partial u\partial w}} = {\frac{{\partial^{2} C_{UVW} }}{\partial u\partial w}} \hfill \\ {\frac{{\partial^{2} P_{010} }}{\partial u\partial w}} = - {\frac{{\partial^{2} P_{110} }}{\partial u\partial w}} = - {\frac{{\partial^{2} P_{011} }}{\partial u\partial w}} = {\frac{{\partial^{2} P_{111} }}{\partial u\partial w}} = - {\frac{{\partial^{2} C_{UVW} }}{\partial u\partial w}} + {\frac{{\partial^{2} C_{UV} }}{\partial u\partial w}} \hfill \\ \end{gathered} $$

  (46)
  $$ \begin{gathered} {\frac{{\partial^{2} P_{000} }}{\partial v\partial w}} = - {\frac{{\partial^{2} P_{010} }}{\partial v\partial w}} = - {\frac{{\partial^{2} P_{001} }}{\partial v\partial w}} = {\frac{{\partial^{2} P_{011} }}{\partial v\partial w}} = {\frac{{\partial^{2} C_{UVW} }}{\partial v\partial w}} \hfill \\ {\frac{{\partial^{2} P_{100} }}{\partial v\partial w}} = - {\frac{{\partial^{2} P_{110} }}{\partial v\partial w}} = - {\frac{{\partial^{2} P_{101} }}{\partial v\partial w}} = {\frac{{\partial^{2} P_{111} }}{\partial v\partial w}} = - {\frac{{\partial^{2} C_{UVW} }}{\partial v\partial w}} + {\frac{{\partial^{2} C_{UV} }}{\partial v\partial w}} \hfill \\ \end{gathered} $$

  (47)

• (8) Compute ${\frac{{\partial^{2} C_{UVW} }}{\partial u\partial v}},\;{\frac{{\partial^{2} C_{UVW} }}{\partial v\partial w}}$ and ${\frac{{\partial^{2} C_{UVW} }}{\partial u\partial w}}.$

  ${\frac{{\partial^{2} C_{UVW} }}{\partial u\partial v}}$ can be obtained as:

  $$ \begin{aligned} {\frac{{\partial^{2} P_{000} }}{\partial u\partial v}}P_{011} P_{101} P_{110} & + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial P_{011} }}{\partial v}}P_{101} P_{110} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} \,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} P_{101}\, {\frac{{\partial P_{110} }}{\partial v}} \\ & + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial P_{011} }}{\partial u}}P_{101} P_{110} + P_{000} \,{\frac{{\partial^{2} P_{011} }}{\partial u\partial v}}P_{101} P_{110} + P_{000}\, {\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + P_{000} \,{\frac{{\partial P_{011} }}{\partial u}}P_{101}\, {\frac{{\partial P_{110} }}{\partial v}} \\ & + {\frac{{\partial P_{000} }}{\partial v}}P_{011} \,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + P_{000} \,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + P_{000} P_{011} \,{\frac{{\partial^{2} P_{101} }}{\partial u\partial v}}P_{110} + P_{000} P_{011} \,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial P_{110} }}{\partial v}} \\ & + {\frac{{\partial P_{000} }}{\partial v}}P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial u}} + P_{000} \,{\frac{{\partial P_{011} }}{\partial v}}P_{101} \,{\frac{{\partial P_{110} }}{\partial u}} + P_{000} P_{011} \,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000} P_{011} P_{101} \,{\frac{{\partial^{2} P_{110} }}{\partial u\partial v}} \\ & - \psi_{UVW} \left( {{\frac{{\partial^{2} P_{111} }}{\partial u\partial v}}P_{100} P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial P_{100} }}{\partial v}}P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial u}}P_{100} \,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + {\frac{{\partial P_{111} }}{\partial u}}P_{100} P_{010} \,{\frac{{\partial P_{001} }}{\partial v}}} \right. \\ & + {\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial P_{100} }}{\partial u}}P_{010} P_{001} + P_{111} \,{\frac{{\partial^{2} P_{100} }}{\partial u\partial v}}P_{010} P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial u}}P_{010} \,{\frac{{\partial P_{001} }}{\partial v}} \\ & + {\frac{{\partial P_{111} }}{\partial v}}P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + P_{111} P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial u\partial v}}P_{001} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial P_{001} }}{\partial v}} \\ & \left. { + {\frac{{\partial P_{111} }}{\partial v}}P_{100} P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} P_{100} P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial u\partial v}}} \right) = 0 \\ \end{aligned} $$

  (48)

  Similarly, applying the ${\frac{\partial }{\partial w}}$ operation to Eqs. 42 and 43, respectively, we can obtain ${\frac{{\partial^{2} C_{UVW} }}{\partial v\partial w}}$ and ${\frac{{\partial^{2} C_{UVW} }}{\partial u\partial w}}$.

• (9) Compute the probability density ${\frac{{\partial^{3} C_{UVW} }}{\partial u\partial v\partial w}}$ of the trivariate Plackett copula.

  Applying the ${\frac{\partial }{\partial w}}$ operation to Eq. 45, we can obtain the third-order derivatives as:

  $$ \begin{gathered} {\frac{{\partial^{3} P_{000} }}{\partial u\partial v\partial w}} = - {\frac{{\partial^{3} P_{100} }}{\partial u\partial v\partial w}} = - {\frac{{\partial^{3} P_{010} }}{\partial u\partial v\partial w}} = {\frac{{\partial^{3} P_{110} }}{\partial u\partial v\partial w}} = {\frac{{\partial^{3} C_{UVW} }}{\partial u\partial v\partial w}} \hfill \\ {\frac{{\partial^{3} P_{001} }}{\partial u\partial v\partial w}} = - {\frac{{\partial^{3} P_{101} }}{\partial u\partial v\partial w}} = - {\frac{{\partial^{3} P_{011} }}{\partial u\partial v\partial w}} = {\frac{{\partial^{3} P_{111} }}{\partial u\partial v\partial w}} = - {\frac{{\partial^{3} C_{UVW} }}{\partial u\partial v\partial w}} + {\frac{{\partial^{3} C_{UV} }}{\partial u\partial v\partial w}} \hfill \\ \end{gathered} $$

  (49)

  Applying ${\frac{\partial }{\partial w}}$ operation to Eq. 48, a new equation is obtained. Because the new equation is too long which contains 128 terms (see Appendix 2), it is omitted here. Substituting Eq. 49 into the new equation, ${\frac{{\partial^{3} C_{UVW} }}{\partial u\partial v\partial w}}$ is the only unknown and one can then obtain the final formula for computation.

Appendix 2: Third order derivative

The final formula of ${\frac{{\partial^{3} C_{UVW} }}{\partial u\partial v\partial w}}$ is expressed as

$$ \begin{aligned} {\frac{{\partial^{3} P_{000} }}{\partial u\partial v\partial w}}P_{011} P_{101} P_{110} & + {\frac{{\partial^{2} P_{000} }}{\partial u\partial v}}\,{\frac{{\partial P_{011} }}{\partial w}}P_{101} P_{110} + {\frac{{\partial^{2} P_{000} }}{\partial u\partial v}}P_{011} \,{\frac{{\partial P_{101} }}{\partial w}}P_{110} + {\frac{{\partial^{2} P_{000} }}{\partial u\partial v}}P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial u\partial w}}\,{\frac{{\partial P_{011} }}{\partial v}}P_{101} P_{110} + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial^{2} P_{011} }}{\partial v\partial w}}P_{101} P_{110} + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial P_{101} }}{\partial w}}P_{110} + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial P_{011} }}{\partial v}}P_{101} \,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial u\partial w}}P_{011} \,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial P_{011} }}{\partial w}}\,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} \,{\frac{{\partial^{2} P_{101} }}{\partial v\partial w}}P_{110} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} \,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial u\partial w}}P_{011} P_{101} \,{\frac{{\partial P_{110} }}{\partial v}} + {\frac{{\partial P_{000} }}{\partial u}}\,{\frac{{\partial P_{011} }}{\partial w}}P_{101} \,{\frac{{\partial P_{110} }}{\partial v}} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} \,{\frac{{\partial P_{101} }}{\partial w}}\,{\frac{{\partial P_{110} }}{\partial v}} + {\frac{{\partial P_{000} }}{\partial u}}P_{011} P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial v\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial v\partial w}}\,{\frac{{\partial P_{011} }}{\partial u}}P_{101} P_{110} + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial w}}P_{101} P_{110} + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial P_{101} }}{\partial w}}P_{110} + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial P_{011} }}{\partial u}}P_{101}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial v}}P_{101} P_{110} + P_{000}\,{\frac{{\partial^{3} P_{011} }}{\partial u\partial v\partial w}}P_{101} P_{110} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial v}}\,{\frac{{\partial P_{101} }}{\partial w}}P_{110} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial v}}P_{101}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}\,{\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial w}}\,{\frac{{\partial P_{101} }}{\partial v}}P_{110} + P_{000}\,{\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial^{2} P_{101} }}{\partial v\partial w}}P_{110} + P_{000}\,{\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}\,{\frac{{\partial P_{011} }}{\partial u}}P_{101}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial u\partial w}}P_{101}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial u}}\,{\frac{{\partial P_{101} }}{\partial w}}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial u}}P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial v\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial v\partial w}}P_{011}\,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial P_{011} }}{\partial w}}\,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + {\frac{{\partial P_{000} }}{\partial v}}P_{011}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial w}}P_{110} + {\frac{{\partial P_{000} }}{\partial v}}P_{011}\,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}\,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial v\partial w}}\,{\frac{{\partial P_{101} }}{\partial u}}P_{110} + P_{000}\,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial v}}P_{110} + P_{000}\,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}P_{011}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial v}}P_{110} + P_{000}\,{\frac{{\partial P_{011} }}{\partial w}}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial w}}P_{110} + P_{000} P_{011}\,{\frac{{\partial^{3} P_{101} }}{\partial u\partial v\partial w}}P_{110} + P_{000} P_{011}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial w}}\,{\frac{{\partial P_{110} }}{\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}P_{011}\,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial w}}\,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000} P_{011}\,{\frac{{\partial^{2} P_{101} }}{\partial u\partial w}}\,{\frac{{\partial P_{110} }}{\partial v}} + P_{000} P_{011}\,{\frac{{\partial P_{101} }}{\partial u}}\,{\frac{{\partial^{2} P_{110} }}{\partial v\partial w}} \\ & + {\frac{{\partial^{2} P_{000} }}{\partial v\partial w}}P_{011} P_{101}\,{\frac{{\partial P_{110} }}{\partial u}} + {\frac{{\partial P_{000} }}{\partial v}}\,{\frac{{\partial P_{011} }}{\partial w}}P_{101}\,{\frac{{\partial P_{110} }}{\partial u}} + {\frac{{\partial P_{000} }}{\partial v}}P_{011}\,{\frac{{\partial P_{101} }}{\partial w}}\,{\frac{{\partial P_{110} }}{\partial u}} + {\frac{{\partial P_{000} }}{\partial v}}P_{011} P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}\,{\frac{{\partial P_{011} }}{\partial v}}P_{101}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000}\,{\frac{{\partial^{2} P_{011} }}{\partial v\partial w}}P_{101}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial v}}\,{\frac{{\partial P_{101} }}{\partial w}}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial v}}P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}P_{011}\,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial w}}\,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000} P_{011}\,{\frac{{\partial^{2} P_{101} }}{\partial v\partial w}}\,{\frac{{\partial P_{110} }}{\partial u}} + P_{000} P_{011}\,{\frac{{\partial P_{101} }}{\partial v}}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial w}} \\ & + {\frac{{\partial P_{000} }}{\partial w}}P_{011} P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial v}} + P_{000}\,{\frac{{\partial P_{011} }}{\partial w}}P_{101}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial v}} + P_{000} P_{011}\,{\frac{{\partial P_{101} }}{\partial w}}\,{\frac{{\partial^{2} P_{110} }}{\partial u\partial v}} + P_{000} P_{011} P_{101}\,{\frac{{\partial^{3} P_{110} }}{\partial u\partial v\partial w}} \\ & - \psi_{UVW} \left( {{\frac{{\partial^{3} P_{111} }}{\partial u\partial v\partial w}}P_{100} P_{010} P_{001} + {\frac{{\partial^{2} P_{111} }}{\partial u\partial v}}\,{\frac{{\partial P_{100} }}{\partial w}}P_{010} P_{001} + {\frac{{\partial^{2} P_{111} }}{\partial u\partial v}}P_{100}\,{\frac{{\partial P_{010} }}{\partial w}}P_{001} + {\frac{{\partial^{2} P_{111} }}{\partial u\partial v}}P_{100} P_{010}\,{\frac{{\partial P_{001} }}{\partial w}}} \right. \\ & + {\frac{{\partial^{2} P_{111} }}{\partial u\partial w}}\,{\frac{{\partial P_{100} }}{\partial v}}P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial^{2} P_{100} }}{\partial v\partial w}}P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial P_{010} }}{\partial w}}P_{001} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial P_{100} }}{\partial v}}P_{010}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + {\frac{{\partial^{2} P_{111} }}{\partial u\partial w}}P_{100}\,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial P_{100} }}{\partial w}}\,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + {\frac{{\partial P_{111} }}{\partial u}}P_{100}\,{\frac{{\partial^{2} P_{010} }}{\partial v\partial w}}P_{001} + {\frac{{\partial P_{111} }}{\partial u}}P_{100}\,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + {\frac{{\partial^{2} P_{111} }}{\partial u\partial w}}P_{100} P_{010}\,{\frac{{\partial P_{001} }}{\partial v}} + {\frac{{\partial P_{111} }}{\partial u}}\,{\frac{{\partial P_{100} }}{\partial w}}P_{010}\,{\frac{{\partial P_{001} }}{\partial v}} + {\frac{{\partial P_{111} }}{\partial u}}P_{100}\,{\frac{{\partial P_{010} }}{\partial w}}\,{\frac{{\partial P_{001} }}{\partial v}} + {\frac{{\partial P_{111} }}{\partial u}}P_{100} P_{010}\,{\frac{{\partial^{2} P_{001} }}{\partial v\partial w}} \\ & + {\frac{{\partial^{2} P_{111} }}{\partial v\partial w}}\,{\frac{{\partial P_{100} }}{\partial u}}P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial^{2} P_{100} }}{\partial u\partial w}}P_{010} P_{001} + {\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial P_{010} }}{\partial w}}P_{001} + {\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial P_{100} }}{\partial u}}P_{010}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + {\frac{{\partial P_{111} }}{\partial w}}\,{\frac{{\partial^{2} P_{100} }}{\partial u\partial v}}P_{010} P_{001} + P_{111} \,{\frac{{\partial^{3} P_{100} }}{\partial u\partial v\partial w}}P_{010} P_{001} + P_{111}\,{\frac{{\partial^{2} P_{100} }}{\partial u\partial v}}\,{\frac{{\partial P_{010} }}{\partial w}}P_{001} + P_{111}\,{\frac{{\partial^{2} P_{100} }}{\partial u\partial v}}P_{010}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + {\frac{{\partial P_{111} }}{\partial w}}\,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + P_{111}\,{\frac{{\partial^{2} P_{100} }}{\partial u\partial w}}\,{\frac{{\partial P_{010} }}{\partial v}}P_{001} + P_{111}\,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial^{2} P_{010} }}{\partial v\partial w}}P_{001} + P_{111}\,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}\,{\frac{{\partial P_{100} }}{\partial u}}P_{010} \,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} \,{\frac{{\partial^{2} P_{100} }}{\partial u\partial w}}P_{010} \,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial u}}\,{\frac{{\partial P_{010} }}{\partial w}}\,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial u}}P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial v\partial w}} \\ & + \,{\frac{{\partial^{2} P_{111} }}{\partial v\partial w}}P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + \,{\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial P_{100} }}{\partial w}}\,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + \,{\frac{{\partial P_{111} }}{\partial v}}P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial u\partial w}}P_{001} + \,{\frac{{\partial P_{111} }}{\partial v}}P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}\,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + P_{111} \,{\frac{{\partial^{2} P_{100} }}{\partial v\partial w}}\,{\frac{{\partial P_{010} }}{\partial u}}P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial^{2} P_{010} }}{\partial u\partial w}}P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial u\partial v}}P_{001} + P_{111} \,{\frac{{\partial P_{100} }}{\partial w}}\,{\frac{{\partial^{2} P_{010} }}{\partial u\partial v}}P_{001} + P_{111} P_{100} \,{\frac{{\partial^{3} P_{010} }}{\partial u\partial v\partial w}}P_{001} + P_{111} P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial u\partial v}}\,{\frac{{\partial P_{001} }}{\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial w}}\,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial u\partial w}}\,{\frac{{\partial P_{001} }}{\partial v}} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial u}}\,{\frac{{\partial^{2} P_{001} }}{\partial v\partial w}} \\ & + \,{\frac{{\partial^{2} P_{111} }}{\partial v\partial w}}P_{100} P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + \,{\frac{{\partial P_{111} }}{\partial v}}\,{\frac{{\partial P_{100} }}{\partial w}}P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + \,{\frac{{\partial P_{111} }}{\partial v}}P_{100} \,{\frac{{\partial P_{010} }}{\partial w}}\,{\frac{{\partial P_{001} }}{\partial u}} + \,{\frac{{\partial P_{111} }}{\partial v}}P_{100} P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial u\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}\,{\frac{{\partial P_{100} }}{\partial v}}P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} \,{\frac{{\partial^{2} P_{100} }}{\partial v\partial w}}P_{010} \,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}\,{\frac{{\partial P_{010} }}{\partial w}}\,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial v}}P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial u\partial w}} \\ & + \,{\frac{{\partial P_{111} }}{\partial w}}P_{100} \,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial w}}\,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} P_{100} \,{\frac{{\partial^{2} P_{010} }}{\partial v\partial w}}\,{\frac{{\partial P_{001} }}{\partial u}} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial v}}\,{\frac{{\partial^{2} P_{001} }}{\partial u\partial w}} \\ & \left. { + \,{\frac{{\partial P_{111} }}{\partial w}}P_{100} P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial u\partial v}} + P_{111} \,{\frac{{\partial P_{100} }}{\partial w}}P_{010} \,{\frac{{\partial^{2} P_{001} }}{\partial u\partial v}} + P_{111} P_{100} \,{\frac{{\partial P_{010} }}{\partial w}}\,{\frac{{\partial^{2} P_{001} }}{\partial u\partial v}} + P_{111} P_{100} P_{010} \,{\frac{{\partial^{3} P_{001} }}{\partial u\partial v\partial w}}} \right) = 0 \\ \end{aligned} $$

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Appendix 3: Estimation of trivariate cross product ratio

• (1) Compute C _UV , C _VW , and C _UW using Eq. 3.

• (2) Compute the empirical trivariate probability distribution P ₀(i), i = 1, 2,…, n.

• (3) Set the parameters of GA, including population size, probability of selection and crossover and mutation, calculation terminal condition.

• (4) Generate randomly the cross-product ratio of trivariate Plackett copula and form an initial population.

• (5) Compute the individual ψ _UVW using Eqs. 7–10 and evaluate the fitness of each individual in the population using Eq. 17.

• (6) Repeat the following calculation steps until termination condition: ① Select best-ranking individuals to reproduce. ② Breed new generation through genetic operations (crossover and mutation) and give birth to their offsprings. ③ Compute the individual ψ _UVW using Eqs. 7–10 and evaluate the individual fitness of offsprings using Eq. 17. ④ Replace the worst ranked part of population with offspring.