Maximum entropy-Gumbel-Hougaard copula method for simulation of monthly streamflow in Xiangxi river, China

Abstract

A maximum entropy-Gumbel-Hougaard copula (MEGHC) method has been proposed for monthly streamflow simulation. The marginal distributions of monthly streamflows are estimated through the maximum entropy (ME) method with the first four non-central moments (i.e. mean, standard deviation, skewness and kurtosis) being the constraints. The Lagrange multipliers in ME-based marginal distributions are determined using the conjugate gradient (CG) method which is of superlinear convergence, simple recurrence formula and less calculation. Then the joint distributions of two adjacent monthly streamflows are constructed using the Gumbel-Hougaard copula (GHC) method. The developed MEGHC method has been applied for monthly streamflow simulation in Xiangxi river, China. The goodness-of-fit statistical tests, consisting of K–S test, A–D test, RMSE and Rosenblatt transformation with Cramér von Mises statistic, show that the MEGHC method can reflect dependence structure in adjacent monthly streamflows of Xiangxi river, China. Comparison between simulated streamflow generated by MEGHC and observations indicates the satisfactory performance of MEGHC with small relative errors.

Appendices

Appendix 1: A bootstrap procedure for CG method

The method of CG is presented as follows:

• (1) Starting from some initial value λ ₍₀₎;

• (2) Let the initial value of search direction d ₍₀₎:

  $$ d_{(0)} = - g_{(0)} = - \frac{{\partial f\left( {\lambda_{\left( 0 \right)} } \right)}}{{\partial \lambda_{(0)i} }}\quad (i = 1, \, 2, \, \ldots \, , \, m) $$

  where f(·) is the objective function; g ₍₀₎ is the gradient of f(λ ₍₀₎); m is the size of random variable λ _(·).

• (3) For steps k = 0, 1, …, n−1, calculate:

  $$ f(\lambda_{(k)} + \alpha_{(k)} d_{(k)} ) = \hbox{min} f(\lambda_{(k)} + \alpha d_{(k)} ) $$
  $$ \lambda_{(k + 1)} = \lambda_{(k)} + \alpha_{(k)} d_{(k)} $$
  $$ g_{\left( k \right)} = \nabla f\left( {\lambda_{\left( k \right)} } \right) $$
  $$ \beta_{(k)} = \frac{{g_{(k + 1)}^{T} (g_{(k + 1)} - g_{(k)} )}}{{\left\| {g_{(k)} } \right\|_{2}^{2} }} $$
  $$ d_{(k + 1)} = - g_{(k + 1)} + \beta_{(k)} d_{(k)} $$

  where g _(k) is gradient of f(λ _(k)); α, α _(k) and β _(k) are parameters of step k; d _(k+1) is the search direction of step k + 1.

Appendix 2: The effectiveness of empirical distribution generated by Gringorten plotting position formula

See Fig. S1.

Appendix 3: A parametric bootstrap procedure for P value of statistic S ^(B) _n

The procedure for determining P value of statistic S ^(B) _n is presented as follows:

• (1) Compute empirical distribution D _n and parameter θ;

• (2) Compute the test statistic S ^(B) _n ;

• (3) For steps k = 1, 2,…, N (N is a large integer), calculate:

  $$ {\mathbf{U}}_{i,k}^{*} = {\mathbf{R}}_{i,k}^{*} / ({n + 1}), \quad i = 1,2,\ldots,n $$
  $$ {\mathbf{E}}_{i,k}^{*} = \Re_{\theta_{i,k}^{*}} ({{\mathbf{U}}_{i,k}^{*}}), \quad i = 1,2,\ldots,n $$
  $$ D_{{n,k}}^{*} \left( {\mathbf{u}} \right) = \frac{1}{n}\sum\limits_{{i = 1}}^{n} 1 ({\mathbf{E}}_{{i,k}}^{*} \le {\mathbf{u}}),\quad {\mathbf{u}} \in [0,1] $$
  $$ S_{n}^{( B)*}=n \int_{0}^{1} {\left\{ {D_{n,k}^{*}({{\mathbf{E}}_{i,k}^{*}})-C_{\bot} ({{\mathbf{E}}_{i,k}^{*}})} \right\}^{2} d{\mathbf{E}}_{i,k}^{*}} $$
  $$ P value^{{\left[ {Cramer - vom\,\, Mise} \right]}} = \frac{1}{N}\sum\limits_{{k = 1}}^{N} {\left\{ {1\left( {S_{{n,k}}^{{\left( B \right)*}} - S_{n}^{{\left( B \right)}} } \right) \ge\, 0} \right\}} $$

  where R ^*_1,k , R ^*_2,k ,…, R ^* _n,k are ran vectors which are associated with the random sample Y ^*_1,k , Y ^*_2,k ,…,Y ^* _n,k from $C_{{\theta_{n} }}$.