Extreme Value Statistics

Abstract

When studying peaks in electricity demand, we may be interested in understanding the risk of a certain large level for demand being exceeded. For example, there is potential interest in finding the probability that the electricity demand of a business or household exceeds the contractual limit. An alternative, yet in principle equivalent way, involves assessment of maximal needs for electricity over a certain period of time, like a day, a week or a season within a year. This would stem from the potential interested in quantifying the largest electricity consumption for a substation, household or business. In either case, we are trying to infer about extreme traits in electricity loads for a certain region assumed fairly homogeneous with the ultimate aim of predicting the likelihood of an extreme event which might have never been observed before. While the exact truth may be not be possible to determine, it may be possible come up with an educated guess (an estimate) and ascertain confidence margins around it. In this chapter, not only we will list and describe mainstream statistical methodology for drawing inference on extreme and rare events, but we will also endeavour to elucidate what sets the semiparametric approach apart from the classical parametric approach and how these two eventually align with one another. For details on possible approaches and related statistical methodologies we refer to [1, 2]. We hope that, through this chapter, practitioners and users of extreme value theory will be able to see how the theoretical results in Chap. 3 translate in practice and how conditions can be checked. We will be mainly concerned with semi-parametric inference for univariate extremes. This statistical methodology builds strongly on the fundamental results expounded in the previous section, most notably the theory of extended regular variation (see e.g. [3], Appendix B). Despite the numerous approaches whereby extreme values can be statistically analysed, these are generally classed into two main frameworks: methods for maxima over fixed intervals (blocks) and methods for exceedances (peaks) over high

Extreme value modelling of SARS-CoV-2 community transmission using discrete generalized Pareto distributions

“…For the sake of simplicity, we used a straightforward threshold selection rule, which is to spot a stability region in the estimates (as a function of the threshold value) and choose an estimate whose value is representative of those reached in this region. This practice, colloquially known as ‘eyeballing’, is standard in applied extreme value analysis: see for example the discussion on p. 77 of chapter 4 in [26]. It applies reasonably well to the D-GPD sample paths, because they are overall much smoother and more stable than the standard Hill and GPD maximum likelihood sample paths, which are not designed to handle the discreteness of the data.…”

Modeling and Analysis of Residential Electricity Consumption Statistics: A Tracy-Widom Mixture Density Approximation

“…Fortunately, there exist in the literature some numerical evaluations of Tracy-Widom distributions [29], [32], [33] that one can use. Inspired by the works of K. Johansson [30] and I. Johnstone [35], by finding a proper pair of location parameter µ and scale parameter σ, we can apply a certain type of Tracy-Widom distribution, e.g., F 1 , to arbitrary data, not just eigenvalues of a random matrix. Here, we have formulated a quadratic programming (QP) problem to solve for µ and σ of type-I Tracy-Widom distribution F 1 .…”

“…Most existing research on load demands are concentrated on average values; however, extreme value analysis (e.g., peak load demands) is critical to power system operational safety, since an unexpected load demand impulse may a cause grid blackout. Classical EVT is committed to statistical analysis of the maximum (or minimum) of a set of uncorrelated random variables [35]. Furthermore, the conventional study of load demand always considers a customer as an independent individual, when in fact, in most realworld systems, e.g., power systems, the underlying random variables are typically correlated.…”