Copula-based mixed models for bivariate rainfall data: an empirical study in regression perspective

Abstract

A comprehensive parametric approach to study the probability distribution of rainfall data at scales of hydrologic interest (e.g. from few minutes up to daily) requires the use of mixed distributions with a discrete part accounting for the occurrence of rain and a continuous one for the rainfall amount. In particular, when a bivariate vector (X, Y) is considered (e.g. simultaneous observations from two rainfall stations or from two instruments such as radar and rain gauge), it is necessary to resort to a bivariate mixed model. A quite flexible mixed distribution can be defined by using a 2-copula and four marginals, obtaining a bivariate copula-based mixed model. Such a distribution is able to correctly describe the intermittent nature of rainfall and the dependence structure of the variables. Furthermore, without loss of generality and with gain of parsimony this model can be simplified by some transformations of the marginals. The main goals of this work are: (1) to empirically explore the behaviour of the parameters of marginal transformations as a function of time scale and inter-gauge distance, by analysing data from a network of rain gauges; (2) to compare the properties of the regression curves associated to the copula-based mixed model with those derived from the model simplified by transformations of the marginals. The results from the investigation of transformations’ parameters are in agreement with the expected theoretical dependence on inter-gauge distance, and show dependence on time scale. The analysis on the regression curves points out that: (1) a copula-based mixed model involves regression curves quite close to some non-parametric models; (2) the performance of the parametric regression decreases in the same cases in which non-parametric regression shows some instability; (3) the copula-based mixed model and its simplified version show similar behaviour in term of regression for mid-low values of rainfall.

Appendix

The copulas belonging to Gumbel family (also known as Gumbel–Hougaard family) have expression:

$$ C_{G} \left( {u,v;\alpha } \right) = \exp \left[ { - \left( {\left( { - \ln u} \right)^{\alpha } + \left( { - \ln v} \right)^{\alpha } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}} } \right], $$

(16)

were α ≥ 1 is a dependence parameter. For α = 1, Gumbel copula approaches the Product copula Π (u, v) = uv describing independence of random variables U and V. For α → ∞, Gumbel copula approaches the Fréchet–Hoeffding upper bound M(u, v) = min{u, v}, denoting perfect positive association. Kendall’s τ _K and α are related as follows:

$$ \alpha = \left( {1 - \tau _{{\text{K}}} } \right)^{{ - 1}} . $$

(17)

The lower and upper tail dependence coefficients for copulas of this family are, respectively, $\lambda _{{\text{L}}} = 0$ and $\lambda _{{\text{U}}} = 2 - 2^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}} .$ From the latter and Eq. (17), it follows that:

$$ \lambda _{{\text{U}}} = 2 - 2^{{1 - \tau _{{\text{K}}} }} . $$

(18)

Gumbel family belongs to Archimedean and Extreme Value classes of copulas (for more details, consult Nelsen (2006) and Salvadori et al. (2007), among others).

Student copula belongs to meta-elliptical class introduced by Fang et al. (2002) (see also Genest et al. (2007) for hydrological applications). It is given by:

$$ C_{{t_{\upsilon } }} \left( {u,v;\alpha ,\upsilon } \right) = \int\limits_{{ - \infty }}^{{t_{\upsilon }^{{ - 1}} \left( u \right)}} {\int\limits_{{ - \infty }}^{{t_{\upsilon }^{{ - 1}} \left( v \right)}} {\frac{{\Upgamma \left( {\frac{{\upsilon + 2}}{2}} \right)}}{{\Upgamma \left( {\frac{\upsilon }{2}} \right)\upsilon \pi \sqrt {1 - \alpha ^{2} } }}} } \times \left( {1 + \frac{{x^{2} - 2\alpha xy + y^{2} }}{\upsilon }} \right)^{{ - \frac{{\upsilon + 2}}{2}}} dxdy, $$

(19)

where α ∈ [–1, 1] is a dependence parameter and $t_{\upsilon }^{{ - 1}}$ is the quantile function of Student distribution with υ degrees of freedom. The cases α = 0, 1, −1 correspond, respectively, to $C_{{t_{\upsilon } }} = \Uppi ,C_{{t_{\upsilon } }} = M,$ and $C_{{t_{\upsilon } }} = W,$ where W(u,v) = max{u + v−1,0} is Fréchet–Hoeffding lower bound, which describes perfect negative association. Parameter α corresponds to Pearson’s correlation coefficient and is related to Kendall’s τ _K by:

$$ \alpha = \sin \left( {\frac{\pi }{2}\tau _{{\text{K}}} } \right). $$

(20)

Lower and upper tail dependence coefficients are equal and given by:

$$ \lambda _{{\text{L}}} = \lambda _{{\text{U}}} = 2t_{{\upsilon + 1}} \left( { - \sqrt {\frac{{\left( {\upsilon + 1} \right)\left( {1 - \alpha } \right)}}{{1 + \alpha }}} } \right). $$

(21)

From Eqs. (20) and (21), it follows the relation between τ _K and λ _U:

$$ \lambda _{\rm{U}} = 2t_{{\upsilon + 1}} \left( { - \sqrt {\frac{{\left( {\upsilon + 1} \right)\left( {1 - \sin \left( {\frac{{\tau _{\rm{K}} \pi }}{2}} \right)} \right)}}{{1 + \sin \left( {\frac{{\tau _{\rm{K}} \pi }}{2}} \right)}}} } \right). $$

(22)