arXiv subjects

Boris Beranger

Publications and source records attributed to Boris Beranger.

14 recordsLinked to original sources

arXiv2024

Modeling extremal dependence in multivariate and spatial problems: a practical perspective

From environmental sciences to finance, there is a growing demand for methods that can assess the risks of extreme events beyond those observed in available data. Extrapolating extreme events beyond the range of the data is not obvious. Risk assessments are often further complicated by the need to account for multiple variables simultaneously. Extreme value theory provides important tools for the analysis of multivariate or spatial extreme events, but these are not easily accessible to professionals without appropriate expertise. This article provides a minimal background on multivariate and spatial extremes and gives simple yet thorough instructions on how to analyse them using the R package ExtremalDep. After briefly introducing the statistical methodologies, we focus on road testing the package's toolbox through several real-world applications.

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arXiv2024

Analysing symbolic data by pseudo-marginal methods

Symbolic data analysis (SDA) aggregates large individual-level datasets into a small number of distributional summaries, such as random rectangles or random histograms. The inference is carried out using these summaries in place of the original dataset, resulting in computational gains at the loss of some information. In likelihood-based SDA, the likelihood function is characterised by an integral with a large exponent, which limits the method's utility as for typical models the integral is unavailable in closed form. In addition, the likelihood function is known to produce biased parameter estimates in some circumstances. Our article develops a Bayesian framework for SDA methods in these settings that resolves the issues resulting from integral intractability and biased parameter estimation using pseudo-marginal Markov chain Monte Carlo methods. We develop an exact but computationally expensive method based on path sampling and the Poisson estimator, and a much faster, but approximate, method based on a Taylor expansion. Through simulation and real-data examples we demonstrate the performance of the developed methods, showing large reductions in computation time compared to the full-data analysis, with only a small loss of information.

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arXiv2024

Fast and flexible inference for spatial extremes

Statistical modelling of spatial extreme events has gained increasing attention over the last few decades with max-stable processes, and more recently $r$-Pareto processes, becoming the reference tools for the statistical analysis of asymptotically dependent data. Although inference for r-Pareto processes is easier than for max-stable processes, there remain major hurdles for their application to high dimensional datasets within a reasonable timeframe. In addition, both approaches have almost exclusively focused on the Brown-Resnick model, for its Gaussian foundations, and for the continuity of its exponent measure. In this paper, we derive a class of models for which this continuity property holds and present the skewed Brown-Resnick model, an extension of the Brown-Resnick that allows for non-stationarity in the dependence structure, and the truncated extremal-t model, a refinement of the well-known extremal-$t$ model. We use an inference methodology based on the intensity function of the process which is derived from the exponent measure, and demonstrate the statistical and computational efficiency of this approach. Applications to two real-world problems illustrate valuable gains in modelling flexibility as well as appealing computational gains over reference methodologies.

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arXiv2020

Estimation and uncertainty quantification for extreme quantile regions

Estimation of extreme quantile regions, spaces in which future extreme events can occur with a given low probability, even beyond the range of the observed data, is an important task in the analysis of extremes. Existing methods to estimate such regions are available, but do not provide any measures of estimation uncertainty. We develop univariate and bivariate schemes for estimating extreme quantile regions under the Bayesian paradigm that outperforms existing approaches and provides natural measures of quantile region estimate uncertainty. We examine the method's performance in controlled simulation studies. We illustrate the applicability of the proposed method by analysing high bivariate quantiles for pairs of pollutants, conditionally on different temperature gradations, recorded in Milan, Italy.

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arXiv2020

Likelihood-based inference for modelling packet transit from thinned flow summaries

The substantial growth of network traffic speed and volume presents practical challenges to network data analysis. Packet thinning and flow aggregation protocols such as NetFlow reduce the size of datasets by providing structured data summaries, but conversely this impedes statistical inference. Methods which aim to model patterns of traffic propagation typically do not account for the packet thinning and summarisation process into the analysis, and are often simplistic, e.g.~method-of-moments. As a result, they can be of limited practical use. We introduce a likelihood-based analysis which fully incorporates packet thinning and NetFlow summarisation into the analysis. As a result, inferences can be made for models on the level of individual packets while only observing thinned flow summary information. We establish consistency of the resulting maximum likelihood estimator, derive bounds on the volume of traffic which should be observed to achieve required levels of estimator accuracy, and identify an ideal family of models. The robust performance of the estimator is examined through simulated analyses and an application on a publicly available trace dataset containing over 36m packets over a 1 minute period.

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arXiv2020

Logistic regression models for aggregated data

Logistic regression models are a popular and effective method to predict the probability of categorical response data. However inference for these models can become computationally prohibitive for large datasets. Here we adapt ideas from symbolic data analysis to summarise the collection of predictor variables into histogram form, and perform inference on this summary dataset. We develop ideas based on composite likelihoods to derive an efficient one-versus-rest approximate composite likelihood model for histogram-based random variables, constructed from low-dimensional marginal histograms obtained from the full histogram. We demonstrate that this procedure can achieve comparable classification rates compared to the standard full data multinomial analysis and against state-of-the-art subsampling algorithms for logistic regression, but at a substantially lower computational cost. Performance is explored through simulated examples, and analyses of large supersymmetry and satellite crop classification datasets.

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arXiv2020

New models for symbolic data analysis

Symbolic data analysis (SDA) is an emerging area of statistics concerned with understanding and modelling data that takes distributional form (i.e. symbols), such as random lists, intervals and histograms. It was developed under the premise that the statistical unit of interest is the symbol, and that inference is required at this level. Here we consider a different perspective, which opens a new research direction in the field of SDA. We assume that, as with a standard statistical analysis, inference is required at the level of individual-level data. However, the individual-level data are aggregated into symbols - group-based distributional-valued summaries - prior to the analysis. In this way, large and complex datasets can be reduced to a smaller number of distributional summaries, that may be analysed more efficiently than the original dataset. As such, we develop SDA techniques as a new approach for the analysis of big data. In particular we introduce a new general method for constructing likelihood functions for symbolic data based on a desired probability model for the underlying measurement-level data, while only observing the distributional summaries. This approach opens the door for new classes of symbol design and construction, in addition to developing SDA as a viable tool to enable and improve upon classical data analyses, particularly for very large and complex datasets. We illustrate this new direction for SDA research through several real and simulated data analyses.

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arXiv2020

Composite likelihood methods for histogram-valued random variables

Symbolic data analysis has been proposed as a technique for summarising large and complex datasets into a much smaller and tractable number of distributions -- such as random rectangles or histograms -- each describing a portion of the larger dataset. Recent work has developed likelihood-based methods that permit fitting models for the underlying data while only observing the distributional summaries. However, while powerful, when working with random histograms this approach rapidly becomes computationally intractable as the dimension of the underlying data increases. We introduce a composite-likelihood variation of this likelihood-based approach for the analysis of random histograms in $K$ dimensions, through the construction of lower-dimensional marginal histograms. The performance of this approach is examined through simulated and real data analysis of max-stable models for spatial extremes using millions of observed datapoints in more than $K=100$ dimensions. Large computational savings are available compared to existing model fitting approaches.

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arXiv2018

Extremal properties of the univariate extended skew-normal distribution

We consider the extremal properties of the highly flexible univariate extended skew-normal distribution. We derive the well-known Mills' inequalities and Mills' ratio for the extended skew-normal distribution and establish the asymptotic extreme-value distribution for the maximum of samples drawn from this distribution.

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arXiv2018

Extremal properties of the multivariate extended skew-normal distribution

The skew-normal and related families are flexible and asymmetric parametric models suitable for modelling a diverse range of systems. We show that the multivariate maximum of a high-dimensional extended skew-normal random sample has asymptotically independent components and derive the speed of convergence of the joint tail. To describe the possible dependence among the components of the multivariate maximum, we show that under appropriate conditions an approximate multivariate extreme-value distribution that leads to a rich dependence structure can be derived.

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arXiv2016

Constructing Likelihood Functions for Interval-valued Random Variables

There is a growing need for the ability to analyse interval-valued data. However, existing descriptive frameworks to achieve this ignore the process by which interval-valued data are typically constructed; namely by the aggregation of real-valued data generated from some underlying process. In this article we develop the foundations of likelihood based statistical inference for random intervals that directly incorporates the underlying generative procedure into the analysis. That is, it permits the direct fitting of models for the underlying real-valued data given only the random interval-valued summaries. This generative approach overcomes several problems associated with existing methods, including the rarely satisfied assumption of within-interval uniformity. The new methods are illustrated by simulated and real data analyses.

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arXiv2016

Models for extremal dependence derived from skew-symmetric families

Skew-symmetric families of distributions such as the skew-normal and skew-$t$ represent supersets of the normal and $t$ distributions, and they exhibit richer classes of extremal behaviour. By defining a non-stationary skew-normal process, which allows the easy handling of positive definite, non-stationary covariance functions, we derive a new family of max-stable processes - the extremal-skew-$t$ process. This process is a superset of non-stationary processes that include the stationary extremal-$t$ processes. We provide the spectral representation and the resulting angular densities of the extremal-skew-$t$ process, and illustrate its practical implementation (Includes Supporting Information).

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arXiv2016

Exploratory data analysis for moderate extreme values using non-parametric kernel methods

In many settings it is critical to accurately model the extreme tail behaviour of a random process. Non-parametric density estimation methods are commonly implemented as exploratory data analysis techniques for this purpose as they possess excellent visualisation properties, and can naturally avoid the model specification biases implied by using parametric estimators. In particular, kernel-based estimators place minimal assumptions on the data, and provide improved visualisation over scatterplots and histograms. However kernel density estimators are known to perform poorly when estimating extreme tail behaviour, which is important when interest is in process behaviour above some large threshold, and they can over-emphasise bumps in the density for heavy tailed data. In this article we develop a transformation kernel density estimator, and demonstrate that its mean integrated squared error (MISE) efficiency is equivalent to that of standard, non-tail focused kernel density estimators. Estimator performance is illustrated in numerical studies, and in an expanded analysis of the ability of well known global climate models to reproduce observed temperature extremes in Sydney, Australia.

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arXiv2015

Extreme Dependence Models

Extreme values of real phenomena are events that occur with low frequency, but can have a large impact on real life. These are, in many practical problems, high-dimensional by nature (e.g. Tawn, 1990; Coles and Tawn, 1991). To study these events is of fundamental importance. For this purpose, probabilistic models and statistical methods are in high demand. There are several approaches to modelling multivariate extremes as described in Falk et al. (2011), linked to some extent. We describe an approach for deriving multivariate extreme value models and we illustrate the main features of some flexible extremal dependence models. We compare them by showing their utility with a real data application, in particular analyzing the extremal dependence among several pollutants recorded in the city of Leeds, UK.

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