Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data

Abstract

This study aims to model the joint probability distribution of periodic hydrologic data using meta-elliptical copulas. Monthly precipitation data from a gauging station (410120) in Texas, US, was used to illustrate parameter estimation and goodness-of-fit for univariate drought distributions using chi-square test, Kolmogorov–Smirnov test, Cramer-von Mises statistic, Anderson-Darling statistic, modified weighted Watson statistic, and Liao and Shimokawa statistic. Pearson’s classical correlation coefficient r _n , Spearman’s ρ _n, Kendall’s τ, Chi-Plots, and K-Plots were employed to assess the dependence of drought variables. Several meta-elliptical copulas and Gumbel-Hougaard, Ali-Mikhail-Haq, Frank and Clayton copulas were tested to determine the best-fit copula. Based on the root mean square error and the Akaike information criterion, meta-Gaussian and t copulas gave a better fit. A bootstrap version based on Rosenblatt’s transformation was employed to test the goodness-of-fit for meta-Gaussian and t copulas. It was found that none of meta-Gaussian and t copulas considered could be rejected at the given significance level. The meta-Gaussian copula was employed to model the dependence, and these results were found satisfactory.

Appendices

Appendix 1: p-dimensional symmetric elliptical type distributions

A p-dimensional random vector Z is said to have an elliptically contoured distribution (or simply called elliptical distribution or ECD) with parameters μ(p × 1) and ∑(p × p) if it has the stochastic representation (Fang and Fang 2002):

$$ {\mathbf{Z}}\mathop = \limits^{d} {\varvec{\upmu}} + r{\mathbf{Au}} $$

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where r ≥ 0 is a random variable, u is uniformly distributed on the unit sphere in R ^p and is independent of r, A is a p × p constant matrix such that AA ^T = ∑ and the sign $\mathop = \limits^{d}$ means that both sides of the equality have the same distribution. In particular, if r has a density, then the density of Z is of the form

$$ \left| \sum \right|^{{ - \frac{ 1}{ 2}}} {\text{g[(}}{\mathbf{Z}} - {\varvec{\upmu}} )^{\text{T}} \sum^{ - 1} ({\mathbf{Z}} - {\varvec{\upmu}}) ] $$

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where g(·) is a scale function uniquely determined by the distribution of r and is referred to as the density generator. Common p-dimensional symmetric elliptical type distributions are given in Table 1.We use the notation Z ~ EC _p (μ, ∑, g). Without loss of generality, we shall consider only the case Z ~ EC _p (0, R, g), where

$$ {\mathbf{R}} = \{ \rho_{ij} :\begin{array}{*{20}c} {\rho_{ii} = 1,\begin{array}{*{20}c} {} \\ \end{array} - 1 < \rho_{ij} < 1\begin{array}{*{20}c} {} \\ \end{array} {\text{for}}\begin{array}{*{20}c} {} \\ \end{array} } \\ \end{array} i \ne j,\begin{array}{*{20}c} {} \\ \end{array} \rho_{ij} = \rho_{ji} ;\begin{array}{*{20}c} {} \\ \end{array} i,j = 1,2, \ldots ,p\} $$

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All the marginal distributions of Z have an identical pdf:

$$ q_{g} ({\text{z}}) = \frac{{\pi^{{\frac{p - 1}{2}}} }}{{\Upgamma \left( {\frac{p - 1}{2}} \right)}}\int\limits_{{{\text{z}}^{2} }}^{\infty } {(y - z^{2} )^{{\frac{p - 1}{2} - 1}} g(y){\text{d}}y} $$

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and cdf

$$ Q_{g} ({\text{z}}) = \frac{ 1}{ 2} + \frac{{\pi^{{\frac{p - 1}{ 2}}} }}{{\Upgamma \left( {\frac{p - 1}{ 2}} \right)}}\int\limits_{0}^{z} {\int\limits_{{u^{2} }}^{\infty } {\left( {y - u^{2} } \right)^{{\frac{p - 1}{ 2} - 1}} g(y){\text{d}}y{\text{d}}u} } ;\quad z > 0 $$

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Note q _g (z) = q _g (−z) and Q _g (z) = 1 − Q _g (−z) for z > 0 (Kotz and Nadarajah 2001; Nadarajah and Kotz 2005; Nadarajah 2006). Let X = (X ₁, X ₁,…,X _p )^T be a random vector with each component X _i having a given continuous pdf f _i (x _i ) and cdf F _i (x _i ). Let the random vector Z = (Z ₁, Z ₂,…, Z _p )^T ~ EC _p (0, R, g). Suppose

$$ Z_{i} = Q_{g}^{ - 1} [F_{i} (x_{i} )];\quad i = 1, 2, \ldots ,p $$

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where Q ⁻¹ _g is the inverse of Q _g . Then the density function of X is given by

$$ h(x_{ 1} ,x_{ 2} , \ldots ,x_{p} ) = \phi \{ Q_{g}^{ - 1} [F_{ 1} (x_{ 1} )],Q_{g}^{ - 1} [F_{ 2} (x_{ 2} )] \ldots ,Q_{g}^{ - 1} [F_{p} (x_{p} )]\} \prod\limits_{i = 1}^{p} {f_{i} (x_{i} )} $$

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where ϕ is the p-variant density weight function:

$$ \phi ({\text{z}}_{ 1} ,{\text{z}}_{ 2} , \ldots ,{\text{z}}_{p} ) = \frac{{\left| {\mathbf{R}} \right|^{{ - \frac{ 1}{ 2}}} g({\mathbf{z}}^{\text{T}} \sum^{ - 1} {\mathbf{z}})}}{{\prod\limits_{i = 1}^{p} {q_{g} (z_{i} )} }} $$

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The n-dimensional random vector X is said to have a meta-elliptical distribution, if its density function is given by Eq. 12. Denote X ~ ME _p (0, R, g; F ₁, F ₂,…,F _p ). The function $\phi \{ {\text{Q}}_{g}^{ - 1} [F_{1} (x_{1} ),{\text{Q}}_{g}^{ - 1} ]F_{2} (x_{2} ), \ldots ,Q_{g}^{ - 1} F_{p} (x_{p} )\}$ is referred to as the density weighting function (Fang et al. 2002).

Appendix 2: Bivariate meta-Gaussian distribution

A meta-Gaussian distribution with marginal distribution F ₁(z ₁) and F ₂(z₂)

$$ C(u,v) = \int\limits_{ - \infty }^{{\Upphi^{ - 1} (u)}} {\int\limits_{ - \infty }^{{\Upphi^{ - 1} (v)}} {\frac{1}{{2\pi \sqrt {1 - \rho^{2} } }}{ \exp }\left[ { - \frac{{s^{2} - 2\rho st + t^{2} }}{{2(1 - \rho^{2} )}}} \right]dsdt} } $$

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where u = F ₁(z₁), v = F ₂(z₂); Φ⁻¹(·) is inverse of the normal distribution. Its pdf is

$$ c(u,v) = \frac{1}{{\sqrt {1 - \rho^{2} } }}\frac{{{ \exp }\left\{ { - \frac{{[\Upphi^{ - 1} (u)]^{2} - 2\rho \Upphi^{ - 1} (u)\Upphi^{ - 1} (v) + [\Upphi^{ - 1} (v)]^{2} }}{{2(1 - \rho^{2} )}}} \right\}}}{{{\text{exp}}\left\{ { - \frac{1}{2}\left[ {\Upphi^{ - 1} \left( u \right)} \right]^{2} } \right\}{ \exp }\left\{ { - \frac{1}{2}[\Upphi^{ - 1} (v)]^{2} } \right\}}} $$

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The marginal pdf and cdf of Z are

$$ q_{g} (z) = { \exp }\left( { - \frac{{z^{2} }}{2}} \right) $$

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$$ Q_{g} (z) = \int\limits_{ - \infty }^{z} {{ \exp }\left( { - \frac{{y^{2} }}{2}} \right)dy} $$

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2.1 Bivariate meta-t _v distribution

A meta-t _v distribution with marginal distribution F ₁(z ₁) and F ₂(z₂)

$$ C(u,v) = \int\limits_{ - \infty }^{{t_{\upsilon }^{ - 1} (u)}} {\int\limits_{ - \infty }^{{t_{\upsilon }^{ - 1} (v)}} {\frac{1}{{2\pi \sqrt {1 - \rho^{2} } }}} } \left[ {1 + \frac{{s^{2} - 2\rho st + t^{2} }}{{(1 - \rho^{2} )\upsilon }}} \right]^{{ - \frac{\upsilon + 2}{2}}} dsdt $$

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where u = F ₁(z₁), v = F ₂(z₂); ${\text{t}}_{\upsilon }^{ - 1} \left( \cdot \right)$ is the inverse function for t distribution with υ degrees of freedom. Its pdf is

$$ c(u,v) = \frac{{\Upgamma \left( {\frac{\upsilon + 2}{2}} \right)\Upgamma \left( {\frac{\upsilon }{2}} \right)}}{{\sqrt {1 - \rho^{2} } \left[ {\Upgamma \left( {\frac{\upsilon + 1}{2}} \right)} \right]^{2} }}\frac{{\left\{ {1 + \frac{{\left[ {t_{\upsilon }^{ - 1} (u)} \right]^{2} }}{\upsilon }} \right\}^{{\frac{\upsilon + 1}{2}}} \left\{ {1 + \frac{{\left[ {t_{\upsilon }^{ - 1} (v)} \right]^{2} }}{\upsilon }} \right\}^{{\frac{\upsilon + 1}{2}}} }}{{\left\{ {1 + \frac{{\left[ {t_{\upsilon }^{ - 1} (u)} \right]^{2} - 2\rho t_{\upsilon }^{ - 1} (u)t_{\upsilon }^{ - 1} (v) + \left[ {t_{\upsilon }^{ - 1} (v)} \right]^{2} }}{{(1 - \rho^{2} )\upsilon }}} \right\}^{{\frac{\upsilon + 2}{2}}} }} $$

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The marginal pdf and cdf of Z are:

$$ q_{t} (z) = \frac{{\Upgamma \left( {\frac{\upsilon + 1}{2}} \right)}}{{\sqrt {\pi \upsilon } \Upgamma \left( {\frac{\upsilon }{2}} \right)}}\left( {1 + \frac{{z^{2} }}{\upsilon }} \right)^{{ - \left( {\frac{\upsilon + 1}{2}} \right)}} $$

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$$ Q_{t} (z) = \frac{{\Upgamma \left( {\frac{\upsilon + 1}{2}} \right)}}{{\sqrt {\pi \upsilon } \Upgamma \left( {\frac{\upsilon }{2}} \right)}}\int\limits_{ - \infty }^{z} {\left( {1 + \frac{{y^{2} }}{\upsilon }} \right)^{{ - \left( {\frac{\upsilon + 1}{2}} \right)}} dy} $$

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2.2 Bivariate meta-Cauchy distribution

A meta-Cauchy distribution with marginal distribution F ₁(z ₁) and F ₂(z₂)

$$ C(u,v) = \int\limits_{ - \infty }^{{\omega^{ - 1} (u)}} {\int\limits_{ - \infty }^{{\omega^{ - 1} \left( {\text{v}} \right)}} {\frac{1}{{2\pi \sqrt {1 - \rho^{2} } }}\left[ {1 + \frac{{s^{2} - 2\rho st + {\text{t}}^{2} }}{{(1 - \rho^{2} )}}} \right]^{{ - \frac{3}{2}}} dsdt} } $$

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where u = F ₁(z₁), v = F ₂(z₂); $\omega^{ - 1} \left( \cdot \right)$ is inverse function for Cauchy distribution.

$$ c(u,v) = \Upgamma \left( {\frac{1}{2}} \right)\Upgamma \left( {\frac{3}{2}} \right)\frac{{\{ 1 + [\psi^{ - 1} (u)]^{2} \} \{ 1 + [\psi^{ - 1} (v)]^{2} \} }}{{\sqrt {1 - \rho^{2} } \left\{ {1 + \frac{{[\psi^{ - 1} (u)]^{2} + [\psi^{ - 1} (v)]^{2} - 2\rho \psi^{ - 1} (u)\psi^{ - 1} (v)}}{{(1 - \rho^{2} )}}} \right\}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} }} $$

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The marginal pdf and cdf of Z are

$$ q_{c} (z) = \frac{1}{{\pi (1 + z^{2} )}} $$

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$$ Q_{c} (z) = \int\limits_{ - \infty }^{z} {\frac{1}{{\pi \left( {1 + y^{2} } \right)}}dy} = \frac{1}{\pi }{\text{arctg(}}z )+ \frac{1}{2} $$

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