Modeling jointly low, moderate, and heavy rainfall intensities without a threshold selection

Abstract

In statistics, extreme events are often defined as excesses above a given large threshold. This definition allows hydrologists and flood planners to apply Extreme-Value Theory (EVT) to their time series of interest. Even in the stationary univariate context, this approach has at least two main drawbacks. First, working with excesses implies that a lot of observations (those below the chosen threshold) are completely disregarded. The range of precipitation is artificially shopped down into two pieces, namely large intensities and the rest, which necessarily imposes different statistical models for each piece. Second, this strategy raises a nontrivial and very practical difficultly: how to choose the optimal threshold which correctly discriminates between low and heavy rainfall intensities. To address these issues, we propose a statistical model in which EVT results apply not only to heavy, but also to low precipitation amounts (zeros excluded). Our model is in compliance with EVT on both ends of the spectrum and allows a smooth transition between the two tails, while keeping a low number of parameters. In terms of inference, we have implemented and tested two classical methods of estimation: likelihood maximization and probability weighed moments. Last but not least, there is no need to choose a threshold to define low and high excesses. The performance and flexibility of this approach are illustrated on simulated and hourly precipitation recorded in Lyon, France.

An objective cross-validation framework for mapping rainfall hazard based on rain gauge data

“…Noting that these distributions tend to underestimate extreme rainfall amounts (Katz et al, 2002), a recent flurry of research developed hybrid models based on mixtures of distributions for low and heavy amounts (Vrac and Naveau, 2007;Furrer and Katz, 2008;Li et al, 2012). 20 More recently Naveau et al (2016) proposed a family of distributions that is able to model the full spectrum of rainfall, while avoiding the use of mixtures of distributions. Several studies compared marginal models for rainfall (e.g.…”

“…Although less common, another family of models for nonzero rainfall relies on univariate extreme value theory, which tells that probabilities of the form pr(R ≤ r|r > q), with q large, can be approximated by either an exponential or a Generalized Pareto tail (Coles, 2001, chapter 4). This led Naveau et al (2016) to propose the extended exponential and extended Generalized Pareto distributions, whose CDF are given in Table 5 1. Note that less parsimonious models are given in Naveau et al (2016) but they are not be considered in the present study.…”

“…This led Naveau et al (2016) to propose the extended exponential and extended Generalized Pareto distributions, whose CDF are given in Table 5 1. Note that less parsimonious models are given in Naveau et al (2016) but they are not be considered in the present study. The extended exponential and extended Generalized Pareto distributions of Table 1 insure that the occurrence probability of small (but nonzero) rainfall amounts is driven by κ while the upper tail of nonzero rainfall is equivalent to a Generalized Pareto tail.…”

“…For rainfall analysis, two options are popular: maximum likelihood (ML) estimation and a method of moments based on probability weighted moments (PWM). However, as noted in Naveau et al (2016), ML estimation may fail 15 for rainfall because the discretization due to instrumental precision strongly affects low values, which biaises ML estimation if not accounted for. One way to circumvent this issue is to resort to censored likelihood but choice of the censoring threshold is in itself an issue.…”

“…Results on our data (not shown) reveal that the threshold has to be no smaller than 5mm. PWM, on the other side, is much more robust against discretization since it is based on summary statistics, rather than on the exact values of observations (Naveau et al, 2016). For this reason, we estimate in this study the distributions of precipitation amounts Table 2.…”