Copula-based evaluations of drought variations in Guangdong, South China

Abstract

Changing characteristics of hazardous weather-related events have been arousing considerable public interest in recent years. Guangdong is an economically developed province in China and is prone to natural hazards. Using monthly precipitation data covering a period of 1956–2008 from 127 rain gauge stations, the probabilistic behaviors of SPI-based droughts were investigated with copulas functions. Results indicated a higher risk of droughts along the coastal regions and the western Guangdong, particularly the Pearl River Delta (PRD) region. Joint probabilities of droughts with higher intensity and longer duration were found to have relatively even geographical distribution across Guangdong. The northern parts of Guangdong are higher in altitude and have a lower risk of droughts. Identification of regions characterized by droughts of different severity and durations across Guangdong is important for scientific management of water resource and agricultural activities and also the development of social resilience under the influence of climate changes.

Appendix

The Copula family is described as the follows:

• (1) Gumbel–Hougaard (GH) family

The GH family can be formulated as:

$$ C(u,v) = \exp \{ - [( - \ln u)^{\theta } + ( - \ln v)^{\theta } ]^{1/\theta } \} , \quad \theta \in [1,\infty ) $$

(13)

where u and v are the marginal distribution functions of two hydrological variables; θ is the parameter of GH family; C is the copula having the function of capturing the essential features of the dependence among random variables. Parameters u, v, and θ in the following equations have the same meaning as those in Eq. 13; so, no further explanations are provided thereafter. The copula generator of GH family (Zhang and Singh 2007) is

$$ \varphi (t) = ( - \ln t)^{\theta } $$

(14)

t is the u ₁ or v ₁, being specific values of u and v in Eq. 13 varying from 0 to 1. The relationship between the parameter, θ, and the Kendall’s coefficient of correlation, τ, between X and Y is τ = 1 − θ ⁻¹. The GH copula performs well in the description of correlation structure of two variables characterized by positive correlation.

• (2) Clayton family

The Clayton family can be formulated as:

$$ C(u,v) = (u^{ - \theta } + v^{ - \theta } - 1)^{ - 1/\theta } ,\quad \theta \in (0,\infty ) $$

(15)

The copula generator of the Clayton family is

$$ \varphi (t) = (t^{ - \theta } - 1)/\theta ,\quad \theta \in (0,\infty ) $$

(16)

The relationship between the parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is

$$ \tau = \frac{\theta }{2 + \theta },\quad \theta \in (0,\infty ) $$

(17)

The Clayton family is good at the description of two variables having positive correlation.

• (3) Frank family

The Frank family can be formulated as:

$$ C(u,v) = - \frac{1}{\theta }\ln \left[1 + \frac{{(e^{ - \theta u} - 1)(e^{ - \theta v} - 1)}}{{(e^{ - \theta } - 1)}}\right],\quad \theta \in R $$

(18)

The copula generator of the Frank family is

$$ \varphi (t) = - \ln \frac{{e^{ - \theta t} - 1}}{{e^{ - \theta } - 1}},\quad \theta \in R $$

(19)

The relationship between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is

$$ \tau = 1 + \frac{4}{\theta }\left[ {\frac{1}{\theta }\int\limits_{0}^{\theta } {\frac{t}{{e^{t} - 1}}dt - 1} } \right],\quad \theta \in R $$

(20)

The Frank family describes two variables in positive correlation well.

• (1) t copula family

The t family can be formulated as:

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

(21)

$t_{v,R}^{n}$ is the R-dimensional standard t distribution with degree of freedom of v. $t_{v}^{ - 1}$ is the inverse function of $t_{v,R}^{n}$.

The relationship between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is

$$ \theta = \sin \left( {\frac{\pi \tau }{2}} \right) $$

(22)

1.1 Conditional copula

Let X and Y be random variables with marginal distributions as u = F _X (x) and v = F _Y (y). The conditional distribution function of X given Y = y can be expressed by the copula method as

$$ H(X < x|Y = y) = C_{\theta } (u|v = v_{1} ) = \mathop {\lim }\limits_{\Updelta v \to 0} \frac{{C_{\theta } (u,v + \Updelta v) - C_{\theta } (u,v)}}{\Updelta v} = \frac{\partial }{\partial v}C_{\theta } (u,v)|v = v_{1} $$

(23)

An equivalent formula for the conditional distribution function for Y given X = x can be obtained. The conditional distribution function of X given Y ≥ y can be expressed by the copula method as

$$ C_{\theta } (u|v < v_{1} ) = \frac{{C_{\theta } (u,v)}}{v} $$

(24)

Similarly, an equivalent formula for the conditional distribution function for Y given X < x can be obtained.