A Censored Bayesian Hierarchical Model For Precipitation

Modelling of precipitation, including extremes, is important for hydrological and agricultural applications. Traditionally, because of large sample properties for data over a large threshold value, generalised Pareto (GP) distributions are often used for modelling extreme rainfall. It can be shown that under certain conditions the generalised hyperbolic (GH) distributions can approximate the power law decay of the GP distribution in the tails. Given their flexible form, this raises the possibility that distributions from the GH family serve as a model for the entire rainfall distribution thus avoiding the need to select a threshold. In this paper, we use a flexible censored hierarchical model that leverages the GH distribution to accommodate data subject to heavy tails and an excessive number of zeros. The fitted model allows estimation of probabilities and return periods of the rainfall extremes, and it produces narrower credible intervals in the tails than the traditional GP method. The model not only fits the tails of the rainfall distribution, but fits the whole distribution very well. It also efficiently represents short-term dependencies in the data so it is suitable for evaluating duration over and below thresholds as well as duration of zero rainfall.Comment: Under review at Environmentric

Revisiting and modeling power-law distributions in empirical outage data of power systems

The size distribution of planned and forced outages and following restoration times in power systems have been studied for almost two decades and has drawn great interest as they display heavy tails. Understanding of this phenomenon has been done by various threshold models, which are self-tuned at their critical points, but as many papers pointed out, explanations are intuitive, and more empirical data is needed to support hypotheses. In this paper, the authors analyze outage data collected from various public sources to calculate the outage energy and outage duration exponents of possible power-law fits. Temporal thresholds are applied to identify crossovers from initial short-time behavior to power-law tails. We revisit and add to the possible explanations of the uniformness of these exponents. By performing power spectral analyses on the outage event time series and the outage duration time series, it is found that, on the one hand, while being overwhelmed by white noise, outage events show traits of self-organized criticality (SOC), which may be modeled by a crossover from random percolation to directed percolation branching process with dissipation, coupled to a conserved density. On the other hand, in responses to outages, the heavy tails in outage duration distributions could be a consequence of the highly optimized tolerance (HOT) mechanism, based on the optimized allocation of maintenance resources.Comment: 16 pages, 8 figure

Execution time distributions in embedded safety-critical systems using extreme value theory

Several techniques have been proposed to upper-bound the worst-case execution time behaviour of programs in the domain of critical real-time embedded systems. These computing systems have strong requirements regarding the guarantees that the longest execution time a program can take is bounded. Some of those techniques use extreme value theory (EVT) as their main prediction method. In this paper, EVT is used to estimate a high quantile for different types of execution time distributions observed for a set of representative programs for the analysis of automotive applications. A major challenge appears when the dataset seems to be heavy tailed, because this contradicts the previous assumption of embedded safety-critical systems. A methodology based on the coefficient of variation is introduced for a threshold selection algorithm to determine the point above which the distribution can be considered generalised Pareto distribution. This methodology also provides an estimation of the extreme value index and high quantile estimates. We have applied these methods to execution time observations collected from the execution of 16 representative automotive benchmarks to predict an upper-bound to the maximum execution time of this program. Several comparisons with alternative approaches are discussed.The research leading to these results has received funding from the European Community’s Seventh Framework Programme [FP7/2007-2013] under the PROXIMA Project (grant agreement 611085). This study was also partially supported by the Spanish Ministry of Science and Innovation under grants MTM2012-31118 (2013-2015) and TIN2015-65316-P. Jaume Abella is partially supported by the Ministry of Economy and Competitiveness under Ramon y Cajal postdoctoral fellowship number RYC-2013- 14717.Peer ReviewedPostprint (author's final draft

Bayesian Extreme Value Mixture Modelling for Estimating VaR

A new extreme value mixture modelling approach for estimating Value-at-Risk (VaR) is proposed, overcoming the key issues of determining the threshold which defines the distribution tail and accounts for uncertainty due to threshold choice. A two-stage approach is adopted: volatility estimation followed by conditional extremal modelling of the independent innovations. Bayesian inference is used to account for all uncertainties and enables inclusion of expert prior information, potentially overcoming the inherent sparsity of extremal data. Simulations show the reliability and flexibility of the proposed mixture model, followed by VaR forecasting for capturing returns during the current financial crisis.Extreme values; Bayesian; Threshold estimation; Value-at-Risk