Bayesian spatial clustering of extremal behaviour for hydrological variables

To address the need for efficient inference for a range of hydrological extreme value problems, spatial pooling of information is the standard approach for marginal tail estimation. We propose the first extreme value spatial clustering methods which account for both the similarity of the marginal tails and the spatial dependence structure of the data to determine the appropriate level of pooling. Spatial dependence is incorporated in two ways: to determine the cluster selection and to account for dependence of the data over sites within a cluster when making the marginal inference. We introduce a statistical model for the pairwise extremal dependence which incorporates distance between sites, and accommodates our belief that sites within the same cluster tend to exhibit a higher degree of dependence than sites in different clusters. By combining the models for the marginal tails and the dependence structure, we obtain a composite likelihood for the joint spatial distribution. We use a Bayesian framework which learns about both the number of clusters and their spatial structure, and that enables the inference of site-specific marginal distributions of extremes to incorporate uncertainty in the clustering allocation. The approach is illustrated using simulations, the analysis of daily precipitation levels in Norway and daily river flow levels in the UK

A joint estimation approach for monotonic regression functions in general dimensions

Regression analysis under the assumption of monotonicity is a well-studied statistical problem and has been used in a wide range of applications. However, there remains a lack of a broadly applicable methodology that permits information borrowing, for efficiency gains, when jointly estimating multiple monotonic regression functions. We introduce such a methodology by extending the isotonic regression problem presented in the article "The isotonic regression problem and its dual" (Barlow and Brunk, 1972). The presented approach can be applied to both fixed and random designs and any number of explanatory variables (regressors). Our framework penalizes pairwise differences in the values (levels) of the monotonic function estimates, with the weight of penalty being determined based on a statistical test, which results in information being shared across data sets if similarities in the regression functions exist. Function estimates are subsequently derived using an iterative optimization routine that uses existing solution algorithms for the isotonic regression problem. Simulation studies for normally and binomially distributed response data illustrate that function estimates are consistently improved if similarities between functions exist, and are not oversmoothed otherwise. We further apply our methodology to analyse two public health data sets: neonatal mortality data for Porto Alegre, Brazil, and stroke patient data for North West England

A spatio-temporal model for Red Sea surface temperature anomalies

This paper details the approach of team Lancaster to the 2019 EVA data challenge, dealing with spatio-temporal modelling of Red Sea surface temperature anomalies. We model the marginal distributions and dependence features separately; for the former, we use a combination of Gaussian and generalised Pareto distributions, while the dependence is captured using a localised Gaussian process approach. We also propose a space-time moving estimate of the cumulative distribution function that takes into account spatial variation and temporal trend in the anomalies, to be used in those regions with limited available data. The team's predictions are compared to results obtained via an empirical benchmark. Our approach performs well in terms of the threshold-weighted continuous ranked probability score criterion, chosen by the challenge organiser

Bayesian spatial monotonic multiple regression

We consider monotonic, multiple regression for contiguous regions. The regression functions vary regionally and may exhibit spatial structure. We develop Bayesian nonparametric methodology that permits estimation of both continuous and discontinuous functional shapes using marked point process and reversible jump Markov chain Monte Carlo techniques. Spatial dependence is incorporated by a flexible prior distribution which is tuned using crossvalidation and Bayesian optimization. We derive the mean and variance of the prior induced by the marked point process approach. Asymptotic results show consistency of the estimated functions. Posterior realizations enable variable selection, the detection of discontinuities and prediction. In simulations and in an application to a Norwegian insurance dataset, our method shows better performance than existing approaches

Bayesian non-parametric ordinal regression under a monotonicity constraint

Compared to the nominal scale, the ordinal scale for a categorical outcome variable has the property of making a monotonicity assumption for the covariate effects meaningful. This assumption is encoded in the commonly used proportional odds model, but there it is combined with other parametric assumptions such as linearity and additivity. Herein, the considered models are non-parametric and the only condition imposed is that the effects of the covariates on the outcome categories are stochastically monotone according to the ordinal scale. We are not aware of the existence of other comparable multivariable models that would be suitable for inference purposes. We generalize our previously proposed Bayesian monotonic multivariable regression model to ordinal outcomes, and propose an estimation procedure based on reversible jump Markov chain Monte Carlo. The model is based on a marked point process construction, which allows it to approximate arbitrary monotonic regression function shapes, and has a built-in covariate selection property. We study the performance of the proposed approach through extensive simulation studies, and demonstrate its practical application in two real data examples

Statistical methods for weather-related insurance claims

Severe weather events, for instance, heavy rainfall, snow-melt or droughts, cause large losses of lives and money every year. Insurance companies offer some form of protection against such undesirable outcomes, and decision makers want to take precautions to prevent future catastrophes. Both, decision makers and insurance companies, are hence interested to understand which weather events induce a high risk. This information then allows the insurance companies to set premiums for their policies by predicting future losses. Further, the relationship between damages and weather is also important to assess the impact of climate change. Several aspects have to be considered in the statistical modelling of this relationship. For instance, some regions in the world are more used to severe rainfall events than others and, hence, presumably less vulnerable to small amounts of rainfall than others. Spatial statistics provides a statistical framework which allows for a spatially varying relationship while accounting for certain similarities for areas which are geographically close. Further, damages, especially large losses, are rather rare and the statistical analysis is hence usually based on a low number of observations. Methods from extreme value theory consider the modelling of such events and may hence be beneficial. This thesis aims to develop statistical models for the relationship between damages, in particular property insurance claims, and weather events, based on daily Norwegian insurance and weather data. To improve existing models, new methodology is introduced which allows for substantial flexibility of the statistical model. The risk induced by certain weather events is assumed to be spatially varying across Norway but with neighbouring regions exhibiting similar vulnerability. To account for certain non-linear effects, the class of monotonic regression functions is considered. Specifically, this work is the first to de- fine flexible dependence structures for such functions. In particular, the first approach considers a Bayesian framework and estimates are obtained by Markov chain Monte Carlo algorithms while the second approach is optimization-based. The last part of the thesis derives extreme value models for discrete data and estimates them in a Bayesian framework. In particular, a mixture model which allows for a flexible tail behaviour is motivated by an exploratory analysis of the highest claims in the data. Additionally, the data are restructured based on spatial and temporal patterns and then combined with the proposed extreme value mixture model. All these approaches, monotonic regression and extreme value analysis, lead to an improved model fit and a better understanding of the relationship between insurance claims and weather events

Extreme value modelling of water-related insurance claims

This paper considers the dependence between weather events, for example, rainfall or snow-melt, and the number of water-related property insurance claims. Weather events which cause severe damages are of general interest; decision makers want to take efficient actions against them while the insurance companies want to set adequate premiums. The modelling is challenging since the underlying dynamics vary across geographical regions due to differences in topology, construction designs and climate. We develop new methodology to improve the existing models which fail to model high numbers of claims. The statistical framework is based on both mixture and extremalmixture modelling, with the latter being based on a discretized generalized Pareto distribution. Furthermore, we propose a temporal clustering algorithm and derive new covariates which lead to a better understanding of the association between claims and weather events. The modelling of the claims, conditional on the locally observed weather events, both fit the marginal distributions well and capture the spatial dependence between locations. Our methodology is applied to three cities across Norway to demonstrate its benefits

A Bayesian spatio-temporal model for precipitation extremes - STOR team contribution to the EVA2017 challenge

This paper concerns our approach to the EVA2017 challenge, the aim of which was to predict extreme precipitation quantiles across several sites in the Netherlands. Our approach uses a Bayesian hierarchical structure, which combines Gamma and generalised Pareto distributions. We impose a spatio-temporal structure in the model parameters via an autoregressive prior. Estimates are obtained using Markov chain Monte Carlo techniques and spatial interpolation. This approach has been successful in the context of the challenge, providing reasonable improvements over the benchmark

Bayesian spatial monotonic multiple regression

We consider monotonic, multiple regression for contiguous regions. The regression functions vary regionally and may exhibit spatial structure. We develop Bayesian nonparametric methodology that permits estimation of both continuous and discontinuous functional shapes using marked point process and reversible jump Markov chain Monte Carlo techniques. Spatial dependence is incorporated by a flexible prior distribution which is tuned using cross-validation and Bayesian optimization. We derive the mean and variance of the prior induced by the marked point process approach. Asymptotic results show consistency of the estimated functions. Posterior realizations enable variable selection, the detection of discontinuities and prediction. In simulations and in an application to a Norwegian insurance data set, our methodology shows better performance than existing approaches