A meta-Gaussian distribution for sub-hourly rainfall

Abstract

Meta-Gaussian models are ubiquitous in the statistical literature. They provide a flexible building block to represent non-Gaussian distributions which inherit modeling and inference methods available in the Gaussian framework. In particular they have been widely used for modeling rainfall distributions. The first step when working with meta-Gaussian models consists in choosing an appropriate transformation which allows to map the Gaussian distribution to the target rainfall distribution. Many transfer functions have been proposed in the literature but most of them are not appropriate to describe heavy-tailed distributions, which is known to be a usual feature for rainfall at sub-daily scales. In this context, we propose and study a new meta-Gaussian model that can handle heavy-tailed observations. It leads to a four parameter model for which each parameter is linked to a different part of the distribution: a first one describes the probability of rainfall occurrence, two of them are related to the lower and upper tailed features of the distribution, and the last one is just a scaling parameter. Theoretical arguments are given to justify the proposed model. A statistical analysis of seven French rain gauges indicates the flexibility of our approach under different climatological regions and different aggregation times, here from 6 min to 24 h. Our distribution outperforms other meta-Gaussian models that have been proposed in the literature and, in particular, it captures well heavier tail behaviours below the hourly scale.

Appendices

Appendix A: Some theoretical properties of the GP meta-Gaussian distribution

The density, cdf and quantile function of a meta-Gaussian model as defined in (1) are

$$\begin{aligned}{} & {} f(y) = c \times \left\{ \begin{array}{ll} {\phi _{\mu }\left( \psi ^{-1}(y)\right) }/{\psi '\left( \psi ^{-1}(y)\right) } &{} \text {if } y>0 \\ \Phi _{\mu }(0) &{}\text {if }y=0 \end{array} \right. ,\\{} & {} F(y) = c \times \left\{ \begin{array}{ll} \Phi _\mu (\psi ^{-1}(y)) &{} \text {if } y>0 \\ \Phi _\mu (0) &{}\text {if } y=0 \end{array} \right. , \\{} & {} F^{-1}(u) = \left\{ \begin{array}{ll} \psi (\Phi ^{-1}_\mu (u/c)) &{} \text {if } u>\Phi _\mu (0) \\ 0 &{}\text {if } u=\Phi _\mu (0) \end{array} \right. , \end{aligned}$$

with $\phi _\mu$ and $\Phi _\mu$ denoting respectively the pdf and cdf of a normal distribution with mean $\mu$. c is the normalisation constant that deals with the probability of truncation when $\xi <0$ with the GP meta-Gaussian transform. Hence $c=1$ for the classical transform (4), and for the GP meta-Gaussian transform (9) $c = 1/\Phi _\mu (x_{sup})$, with $x_{sup}$ the upper bound in the Gaussian domain as defined in (10).

An explicit expression of the moments was found for the GP meta-Gaussian distribution when $\xi \ge 0$. Let us write $Y_+$ the wet measurements.

$$\begin{aligned} E(Y_+^p)&= \frac{1}{\sqrt{2\pi }(1-\Phi (-\mu ))} \int _{0}^{+\infty } \psi (x)^p \exp \left\{ -\frac{1}{2}(x-\mu )^2\right\} dx\\&=\frac{\sigma ^p}{\sqrt{2\pi }(1-\Phi (-\mu ))}\exp \left( -\frac{\mu ^2}{2}\right) \\&\quad \int _{0}^{+\infty } x^{p/\alpha } \exp \left\{ -\frac{1-\xi p}{2}x^2+\mu x\right\} dx \end{aligned}$$

By identification in Gradshteyn and Ryzhik (2007) (eq. 3.462.1, page 365), with $\gamma =-\mu$, $\nu -1=p/\alpha$ and $\beta =(1-\xi p)/2$,

$$\begin{aligned} E(Y_+^p)&= \frac{\sigma ^p(1-\xi p)^{-\frac{1}{2}(\frac{p}{\alpha }+1)}}{\sqrt{2\pi }(1-\Phi (-\mu ))} \exp \left\{ \frac{\mu ^2}{2}\left( \frac{1}{2(1-\xi p)}-1\right) \right\} \Gamma \\&\quad \left( \frac{p}{\alpha }+1\right) D_{-(\frac{p}{\alpha }+1)}\left( -\frac{\mu }{\sqrt{1-\xi p}} \right) \end{aligned}$$

$\Gamma$ is the Gamma function and $D_\nu$ can be expressed with Kummer’s confluent hypergeometric function of first kind (Gradshteyn and Ryzhik 2007), Eq. 9.240, page 1028. This expression is valid if $-\alpha<p<1/\xi$.

Appendix B: Pareto tail for meta-Gaussian models

Proposition 1

Let Z be any positive absolutely continuous random variable with pdf $f_Z$ and with a Pareto survival function ${\overline{F}}_Z$. Let X be any standardized normal distributed random variable, and let us define the positive random variable

$$\begin{aligned} Y \overset{d}{=}\ \psi ( X), \end{aligned}$$

where $\overset{d}{=}$ means equality in distribution and $\psi (.)$ represents a continuous and increasing function from the real line to $[0, \infty )$. The two random variables Z and Y are tail-equivalent if and only if

$$\begin{aligned} \lim _{x\rightarrow \infty } \frac{x \psi (x)}{\psi '(x)}= \frac{1}{\xi }, \end{aligned}$$

(B1)

where $\xi$ corresponds the common positive GP shape parameter of Z.

Proof of Proposition 1:

Let $\phi$ and ${\overline{\Phi }}$ denote respectively the pdf and survival function of a standard normal distribution X.

Recall that Z and Y are tail-equivalent, if and only

$$\begin{aligned} \lim _{y\rightarrow \infty } \frac{{\overline{F}}_Z(y)}{{\mathbb {P}}[Y> y]} = c \in (0, \infty ), \end{aligned}$$

This condition is satisfied if they have the same tail index. Assuming a Pareto tail with positive shape parameter $\xi$ for Z implies that Z is regularly varying with index $1/\xi$. Proposition A.3.8(b) from Embrechts et al. (2013) recalled that this regular variation type is equivalent to

$$\begin{aligned} \lim _{z\rightarrow \infty } \frac{z \times f_Z(z)}{{\overline{F}}_Z(z)} = \frac{1}{\xi }. \end{aligned}$$

Hence, to show that Y and Z are tail equivalent, one needs to determine under which condition it can be written that

$$\begin{aligned} \lim _{z\rightarrow \infty } \frac{z \times F(z)}{{\overline{F}}_Y(z)} = \frac{1}{\xi }. \end{aligned}$$

where f and ${\overline{F}}$ denote the pdf and survival function of Y, respectively.

By construction, the survival function of Y equals to

$$\begin{aligned} {\overline{F}}_Y(z) = {\mathbb {P}}[X> \psi ^{-1}(z)]= {\overline{\Phi }} \left[ \psi ^{-1}(z) \right] , \end{aligned}$$

The density of Y is

$$\begin{aligned} f(z) = \left( \psi ^{-1}(z) \right) ' \phi \left[ \psi ^{-1}(z) \right] . \end{aligned}$$

Then one can write

$$\begin{aligned} \frac{z \times f(z)}{{\overline{F}}_Y(z)} = \left( z \times \psi ^{-1}(z) \times \left( \psi ^{-1}(z) \right) ' \right) \times \left( \frac{ \phi \left[ \psi ^{-1}(z) \right] }{\psi ^{-1}(z) {\overline{\Phi }} \left[ \psi ^{-1}(z) \right] } \right) . \end{aligned}$$

Mill’s ratio (Embrechts et al. 2013) tells us that the ratio in the last bracket goes to one as $\psi ^{-1}(z)$ goes to $\infty$ (i.e. as z grows). Hence, the condition

$$\begin{aligned} \lim _{z\rightarrow \infty } \left( z \times \psi ^{-1}(z) \times \left( \psi ^{-1}(z) \right) ' \right) = \frac{1}{\xi }, \end{aligned}$$

(B2)

is equivalent to

$$\begin{aligned} \lim _{z\rightarrow \infty } \frac{z \times f(z)}{{\overline{F}}_Y(z)} = \frac{1}{\xi }. \end{aligned}$$

This is equivalent to have tail equivalence between Z and Y.

Changing variables with $z=\psi (x)$, $x=\psi ^{-1}(z)$ and $\left( \psi ^{-1}(z) \right) ' =dx/dz$, condition (B2) is equivalent to condition (B1).

This is the necessary and sufficient condition on $\psi (.)$ to build a Pareto random variable of tail index $\xi$ from a standardized normal random variable X. $\square$