Replica-mean-field limits of fragmentation-interaction-aggregation processes

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.

No association of genetic variants in TLR4, TNF-α, IL10, IFN-γ, and IL37 in cytomegalovirus-positive renal allograft recipients with active CMV infection—Subanalysis of the prospective randomised VIPP study

Background Cytomegalovirus (CMV) infection is amongst the most important factors complicating solid organ transplantation. In a large prospective randomized clinical trial, valganciclovir prophylaxis reduced the occurrence of CMV infection and disease compared with preemptive therapy in CMV-positive renal allograft recipients (VIPP study; NCT00372229). Here, we present a subanalysis of the VIPP study, investigating single nucleotide polymorphisms (SNPs) in immune-response-related genes and their association with active CMV infection, CMV disease, graft loss or death, rejection, infections, and leukopenia. Methods Based on literature research ten SNPs were analyzed for TLR4, three for IFN-γ, six for IL10, nine for IL37, and two for TNF-α. An asymptotic independence test (Cochran-Armitage trend test) was used to examine associations between SNPs and the occurrence of CMV infection or other negative outcomes. Statistical significance was defined as p<0.05 and Bonferroni correction for multiple testing was performed. Results SNPs were analyzed on 116 blood samples. No associations were found between the analyzed SNPs and the occurrence of CMV infection, rejection and leukopenia in all patients. For IL37 rs2723186, an association with CMV disease (p = 0.0499), for IL10 rs1800872, with graft loss or death (p = 0.0207) and for IL10 rs3024496, with infections (p = 0.0258) was observed in all patients, however did not hold true after correction for multiple testing. Conclusion The study did not reveal significant associations between the analyzed SNPs and the occurrence of negative outcomes in CMV-positive renal transplant recipients after correction for multiple testing. The results of this association analysis may be of use in guiding future research efforts.

Statistical Modeling of Non-Stationary Heatwave Hazard

&lt;p&gt;The modeling of spatio-temporal trends in temperature extremes can help better understand the structure and frequency of heatwaves in a changing climate, as well as their environmental, societal, economic and global health-related risks. Here, we study annual temperature maxima over Southern Europe using a century-spanning dataset observed at 44 monitoring stations. Extending the spectral representation of max-stable processes, our modeling framework relies on a novel construction of max-infinitely divisible processes, which include covariates to capture spatio-temporal non-stationarities. Our new model keeps a popular max-stable process on the boundary of the parameter space, while flexibly capturing weakening extremal dependence at increasing quantile levels and asymptotic independence. It clearly outperforms natural alternative models. Results show that the spatial extent of heatwaves is smaller for more severe events at higher altitudes and that recent heatwaves are moderately wider. Our probabilistic assessment of the 2019 annual maxima confirms the severity of the 2019 heatwaves both spatially and at individual sites, especially when compared to climatic conditions prevailing in 1950-1975. Our applied results may be exploited in practice to understand the spatio-temporal dynamics, severity, and frequency of extreme heatwaves, and design suitable regional mitigation measures.&lt;/p&gt;

Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves

To explicitly account for asymptotic dependence between rainfall intensity maxima of different accumulation duration, a recent development for estimating Intensity-Duration-Frequency (IDF) curves involves the use of a max-stable process. In our study, we aimed to estimate the impact on the performance of the return levels resulting from an IDF model that accounts for such asymptotical dependence. To investigate this impact, we compared the performance of the return level estimates of two IDF models using the quantile skill index (QSI). One IDF model is based on a max-stable process assuming asymptotic dependence; the other is a simplified (or reduced) duration-dependent GEV model assuming asymptotic independence. The resulting QSI shows that the overall performance of the two models is very similar, with the max-stable model slightly outperforming the other model for short durations (d≤10h). From a simulation study, we conclude that max-stable processes are worth considering for IDF curve estimation when focusing on short durations if the model’s asymptotic dependence can be assumed to be properly captured.

Basin-wide spatial conditional extremes for severe ocean storms

Physical considerations and previous studies suggest that extremal dependence between ocean storm severity at two locations exhibits near asymptotic dependence at short inter-location distances, leading to asymptotic independence and perfect independence with increasing distance. We present a spatial conditional extremes (SCE) model for storm severity, characterising extremal spatial dependence of severe storms by distance and direction. The model is an extension of Shooter et al. 2019 (Environmetrics 30, e2562, 2019) and Wadsworth and Tawn (2019), incorporating piecewise linear representations for SCE model parameters with distance and direction; model variants including parametric representations of some SCE model parameters are also considered. The SCE residual process is assumed to follow the delta-Laplace form marginally, with distance-dependent parameter. Residual dependence of remote locations given conditioning location is characterised by a conditional Gaussian covariance dependent on the distances between remote locations, and distances of remote locations to the conditioning location. We apply the model using Bayesian inference to estimates extremal spatial dependence of storm peak significant wave height on a neighbourhood of 150 locations covering over 200,000 km2 in the North Sea.

Spatial conditional extremes via the Gibbs sampler.

&lt;div&gt; &lt;div&gt; &lt;div&gt; &lt;p&gt;Spatial conditional extremes via the Gibbs sampler.&lt;/p&gt; &lt;p&gt;Adrian Casey, University of Edinburgh&lt;/p&gt; &lt;p&gt;January 14, 2020&lt;/p&gt; &lt;p&gt;Conditional extreme value theory has been successfully applied to spatial extremes problems. In this statistical method, data from observation sites are modelled as appropriate asymptotic characterisations of random vectors &lt;strong&gt;X,&lt;/strong&gt; conditioned on one of their components being extreme. The method is generic and applies to a broad range of dependence structures including asymptotic dependence and asymptotic independence. However, one issue that affects the conditional extremes method is the necessity to model and fit a multi-dimensional residual distribution; this can be challenging in spatial problems with a large number of sites.&lt;/p&gt; &lt;p&gt;We describe early-stage work that takes a local approach to spatial extremes; this approach explores lower dimensional structures that are based on asymptotic representations of Markov random fields. The main element of this new method is a model for the behaviour of a random component X&lt;sub&gt;i &lt;/sub&gt; given that its nearest neighbours exceed a sufficiently large threshold. When combined with a model for the case where the neighbours are below this threshold, a Gibbs sampling scheme serves to induce a model for the full conditional extremes distribution by taking repeated samples from these local (univariate) distributions.&lt;/p&gt; &lt;p&gt;The new method is demonstrated on a data set of significant wave heights from the North Sea basin. Markov chain Monte-Carlo diagnostics and goodness-of-fit tests illustrate the performance of the method. The potential for extrapolation into the outer reaches of the conditional extreme tails is then examined.&lt;/p&gt; &lt;p&gt;Joint work with Ioannis Papastathopoulos.&lt;/p&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt;

Examining the extremal dependence structure of precipitation in Norway

&lt;p&gt;Extreme precipitation can lead to great floods and landslides and cause severe damage and economical losses. It is therefore of great importance that we manage to assess the risk of future extremes. Furthermore, natural hazards are spatiotemporal phenomena that require extensive modelling in both space and time. Extreme value theory (EVT) can be used for statistical modelling of spatial extremes, such as extreme precipitation over a catchment. An important concept when modelling a natural hazard is the degree of extremal dependence for the given phenomenon. Extremal dependence describes the possibility of multiple extremes occurring at the same time. For the stochastic variables X and Y, with distribution functions F&lt;sub&gt;X &lt;/sub&gt;and F&lt;sub&gt;Y&lt;/sub&gt;, the measure&lt;/p&gt;&lt;p&gt;&amp;#967; = lim&lt;sub&gt;u&lt;/sub&gt;&lt;sub&gt;&amp;#8594;1 &lt;/sub&gt;P(F&lt;sub&gt;X&lt;/sub&gt;(X) &gt; u &amp;#921; F&lt;sub&gt;Y&lt;/sub&gt;(Y) &gt; u)&lt;/p&gt;&lt;p&gt;describes the pairwise extremal dependence between X and Y. If &amp;#967; = 0, then the variables are &lt;em&gt;asymptotically independent&lt;/em&gt;. If &amp;#967; &gt; 0, they are&lt;br&gt;&lt;em&gt;asymptotically dependent&lt;/em&gt;. Thus, extremes tend to occur simultaneously in space for processes that are asymptotically dependent, while this seldom occurs for asymptotically independent processes. It is a general belief that extreme precipitation tends to be asymptotically independent. However, to our knowledge, not much work has been put into analysing the extremal dependence structure of precipitation. Different statistical models have been developed, which can be applied for modelling spatial extremes. The most popular model is the &lt;em&gt;max-stable process&lt;/em&gt;. Unfortunately, this model does not provide a good fit to asymptotically independent processes. Other models have been developed for better incorporating asymptotic independence, but most have not been extensively applied yet. We aim to examine the extremal dependence structure of precipitation in Norway, with the ultimate goal of modelling and simulating extreme precipitation. This is achieved by examining multiple popular statistics for extremal dependence, as well as comparing different spatial EVT models. This analysis is performed on hourly, gridded precipitation data from the MetCoOp Ensemble Prediction System (MEPS), which is publicly available from the internet: http://thredds.met.no/thredds/catalog/meps25epsarchive/catalog.html.&lt;/p&gt;