Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

Bacro, Jean-Noel; Gaetan, Carlo; Opitz, Thomas; Toulemonde, Gwladys

doi:10.1080/01621459.2019.1617152

Cited by 29 publications

(25 citation statements)

References 54 publications

(66 reference statements)

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“…For the family

{\overline{W}}_{θ^{'}}

, it might correspondingly be appropriate to assume

ν

does not vary over some region or, if that did not suffice to produce stable parameter estimates, to assume

ν, κ_{0}

, and

κ_{1}

do not vary. More recent work on environmental extremes often uses hierarchical models to force the scale and shape parameters of the distribution to vary smoothly in space (Bacro, Gaetan, Opitz, & Toulemonde, 2020; Castro‐Camilo, Huser, & Rue, 2019; Sharkey & Winter, 2019; Yadav et al, 2019) and this approach could also be applied to the parametric families proposed here. Such analyses are usually Bayesian and it is not clear what kinds of priors should be put on the parameters as they vary in some spatial index x .…”

Section: Discussionmentioning

confidence: 99%

A parametric model for distributions with flexible behavior in both tails

Stein

2020

Environmetrics

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Summary For many problems of inference about a marginal distribution function, while the entire distribution is important, extreme quantiles are of particular interest because rare outcomes may have large consequences. In some applications, only the extreme upper quantiles require extra attention, but in, for example, climatological applications, extremes in both tails of the distribution can be impactful. A possible approach in this setting is to use parametric families of distributions that have flexible behavior in both tails. One way to quantify this property is to require that, for any two generalized Pareto distributions, there is a member of the parametric family that behaves like one of the generalized Pareto distributions in the upper tail and like the negative of the other generalized Pareto distribution in the lower tail. This work proposes some specific quantifications of this notion and describes parametric families of distributions that satisfy these specifications. The proposed families all have closed form expressions for their densities and, hence, are convenient for use in practice. A simulation study shows how one of the proposed families can work well for estimating all quantiles when both tails of a distribution are heavy tailed. An application to climate model output shows this family can also work well when applied to daily average January temperature near Calgary, for which the evolving distribution over time due to climate change is difficult to model accurately by any standard parametric family.

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“…For the family

{\overline{W}}_{θ^{'}}

, it might correspondingly be appropriate to assume

ν

does not vary over some region or, if that did not suffice to produce stable parameter estimates, to assume

ν, κ_{0}

, and

κ_{1}

Section: Discussionmentioning

confidence: 99%

A parametric model for distributions with flexible behavior in both tails

Stein

2020

Environmetrics

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“…Therefore, we always need to choose a finite, and often relatively small, block size, which casts doubts on the validity of the max‐stability assumption in practice. A fast growing body of empirical studies on environmental extremes in the literature has indeed revealed that the max‐stability assumption arising asymptotically is often violated at finite levels (Bopp, Shaby, & Huser, 2020), and that the spatial dependence strength is often weakening as events become more extreme (see, e.g., Bacro, Gaetan, Opitz, & Toulemonde, 2020; Castro‐Camilo & Huser, 2020; Castro‐Camilo, Mhalla, & Opitz, 2020; Davison, Huser, & Thibaud, 2013; Huser, Opitz, & Thibaud, 2017; Huser & Wadsworth, 2020; Tawn, Shooter, Towe, & Lamb, 2018). In particular, under asymptotic independence , maxima become ultimately independent at the highest levels, requiring specialized models capturing the decay rate towards independence.…”

Section: Introductionmentioning

confidence: 99%

Max‐infinitely divisible models and inference for spatial extremes

Huser

Opitz

Thibaud

2020

Scandinavian J Statistics

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For many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.

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“…Let

Exp (λ)

denote the exponential distribution with rate

λ > 0

Gamma (α, β)

denote the gamma distribution with rate

α > 0

and shape

β > 0

, that is, with density

g (y) = {Γ (β)}^{- 1} α^{β} y^{β - 1} \exp (- α y)

, y > 0, and

GP (τ, ξ)

denote the GP distribution with scale

τ > 0

and shape

ξ

as defined in (1). Then we have

\begin{aligned} Y false| normalΛ & \sim Exp false(normalΛ false) \\ normalΛ & \sim Gamma false(α, β false) \end{aligned}\} \Rightarrow Y \sim GP (α / β, 1 / β);

see Reiss and Thomas (2007), Bortot and Gaetan (2014, 2016), Bopp and Shaby (2017), and Bacro et al (2020). In other words, exponentially decaying tails become heavier by making their rate parameter

Λ

random.…”

Section: Introductionmentioning

confidence: 99%

Spatial hierarchical modeling of threshold exceedances using rate mixtures

2020

Self Cite

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We develop new flexible univariate models for light‐tailed and heavy‐tailed data, which extend a hierarchical representation of the generalized Pareto (GP) limit for threshold exceedances. These models can accommodate departure from asymptotic threshold stability in finite samples while keeping the asymptotic GP distribution as a special (or boundary) case and can capture the tails and the bulk jointly without losing much flexibility. Spatial dependence is modeled through a latent process, while the data are assumed to be conditionally independent. Focusing on a gamma–gamma model construction, we design penalized complexity priors for crucial model parameters, shrinking our proposed spatial Bayesian hierarchical model toward a simpler reference whose marginal distributions are GP with moderately heavy tails. Our model can be fitted in fairly high dimensions using Markov chain Monte Carlo by exploiting the Metropolis‐adjusted Langevin algorithm (MALA), which guarantees fast convergence of Markov chains with efficient block proposals for the latent variables. We also develop an adaptive scheme to calibrate the MALA tuning parameters. Moreover, our model avoids the expensive numerical evaluations of multifold integrals in censored likelihood expressions. We demonstrate our new methodology by simulation and application to a dataset of extreme rainfall events that occurred in Germany. Our fitted gamma–gamma model provides a satisfactory performance and can be successfully used to predict rainfall extremes at unobserved locations.

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Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

Cited by 29 publications

References 54 publications

A parametric model for distributions with flexible behavior in both tails

A parametric model for distributions with flexible behavior in both tails

Max‐infinitely divisible models and inference for spatial extremes

Spatial hierarchical modeling of threshold exceedances using rate mixtures

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