Dorota Kurowicka

Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis

The increasing penetration of renewable generation in power systems necessitates the modeling of this stochastic system infeed in operation and planning studies. The system analysis leads to multivariate uncertainty analysis problems, involving non-Normal correlated random variables. In this context, the modeling of stochastic dependence is paramount for obtaining accurate results; it corresponds to the concurrent behavior of the random variables, having a major impact to the aggregate uncertainty (in problems where the random variables correspond to spatially spread stochastic infeeds) or their evolution in time (in problems where the random variables correspond to infeeds over specific time-periods). In order to investigate, measure and model stochastic dependence, one should transform all different random variables to a common domain, the rank/uniform domain, by applying the cumulative distribution function transformation. In this domain, special functions, copulae, can be used for modeling dependence. In this contribution the basic theory concerning the use of these functions for dependence modeling is presented and focus is given on a basic function, the Normal copula. The case study shows the application of the technique for the study of the large-scale integration of wind power in the Netherlands.

Selecting and estimating regular vine copulae and application to financial returns

Regular vine distributions which constitute a flexible class of multivariate dependence models are discussed. Since multivariate copulae constructed through pair-copula decompositions were introduced to the statistical community, interest in these models has been growing steadily and they are finding successful applications in various fields. Research so far has however been concentrating on so-called canonical and D-vine copulae, which are more restrictive cases of regular vine copulae. It is shown how to evaluate the density of arbitrary regular vine specifications. This opens the vine copula methodology to the flexible modeling of complex dependencies even in larger dimensions. In this regard, a new automated model selection and estimation technique based on graph theoretical considerations is presented. This comprehensive search strategy is evaluated in a large simulation study and applied to a 16-dimensional financial data set of international equity, fixed income and commodity indices which were observed over the last decade, in particular during the recent financial crisis. The analysis provides economically well interpretable results and interesting insights into the dependence structure among these indices.

DEPENDENCE MODELING:Vine Copula Handbook

This book is a collaborative effort from three workshops held over the last three years, all involving principal contributors to the vine-copula methodology. Research and applications in vines have been growing rapidly and there is now a growing need to collate basic results, and standardize terminology and methods. Specifically, this handbook will (1) trace historical developments, standardizing notation and terminology, (2) summarize results on bivariate copulae, (3) summarize results for regular vines, and (4) give an overview of its applications. In addition, many of these results are new and not readily available in any existing journals. New research directions are also discussed.

Generating random correlation matrices based on vines and extended onion method

We extend and improve two existing methods of generating random correlation matrices, the onion method of Ghosh and Henderson [S. Ghosh, S.G. Henderson, Behavior of the norta method for correlated random vector generation as the dimension increases, ACM Transactions on Modeling and Computer Simulation (TOMACS) 13 (3) (2003) 276-294] and the recently proposed method of Joe [H. Joe, Generating random correlation matrices based on partial correlations, Journal of Multivariate Analysis 97 (2006) 2177-2189] based on partial correlations. The latter is based on the so-called D-vine. We extend the methodology to any regular vine and study the relationship between the multiple correlation and partial correlations on a regular vine. We explain the onion method in terms of elliptical distributions and extend it to allow generating random correlation matrices from the same joint distribution as the vine method. The methods are compared in terms of time necessary to generate 5000 random correlation matrices of given dimensions.

Stochastic Modeling of Power Demand Due to EVs Using Copula

The driving patterns characterizing electric vehicles (EVs) are stochastic and, as a consequence, the electrical load due to EVs inherits their randomness. This paper presents a Monte Carlo procedure for the derivation of load due to EVs based on a fully stochastic method for modeling transportation patterns. Under the uncontrolled domestic charging scenario three variables are found to be crucial: the time a vehicle leaves home, the time a vehicle arrives home, and the distance traveled in between. A detailed transportation dataset is used to derive marginal cumulative distribution functions of the variables of interest. Since the variables are statistically dependent, a joint distribution function is built using a copula function. Subsequently, simulated EV trips are combined with a typical charging profile so that the energy contribution to the system is computed. The procedure is applied to analyze the effect of the EV load on the national power demand of The Netherlands under different market penetration levels and day/night electricity tariff scenarios.

Prioritizing emerging zoonoses in The Netherlands.

Background To support the development of early warning and surveillance systems of emerging zoonoses, we present a general method to prioritize pathogens using a quantitative, stochastic multi-criteria model, parameterized for the Netherlands. Methodology/Principal Findings A risk score was based on seven criteria, reflecting assessments of the epidemiology and impact of these pathogens on society. Criteria were weighed, based on the preferences of a panel of judges with a background in infectious disease control. Conclusions/Significance Pathogens with the highest risk for the Netherlands included pathogens in the livestock reservoir with a high actual human disease burden (e.g. Campylobacter spp., Toxoplasma gondii, Coxiella burnetii) or a low current but higher historic burden (e.g. Mycobacterium bovis), rare zoonotic pathogens in domestic animals with severe disease manifestations in humans (e.g. BSE prion, Capnocytophaga canimorsus) as well as arthropod-borne and wildlife associated pathogens which may pose a severe risk in future (e.g. Japanese encephalitis virus and West-Nile virus). These agents are key targets for development of early warning and surveillance.

Hybrid Method for Quantifying and Analyzing Bayesian Belief Nets

Bayesian belief nets (BBNs) have become a popular tool for specifying high-dimensional probabilistic models. Commercial tools with an advanced graphical user interface that support BBNs construction and inference are available. Thus, building and working with BBNs is very efficient as long as one is not forced to quantify complex BBNs. A high assessment burden of discrete BBNs is often caused by the discretization of continuous variables. Until recently, continuous BBNs were restricted to the joint normal distribution. We present the ‘copula–vine’ approach to continuous BBNs. This approach is quite general and allows traceable and defendable quantification methods, but it comes at a price: these BBNs must be evaluated by Monte Carlo simulation. Updating such a BBN requires re-sampling the whole structure. The advantages of fast updating algorithms for discrete BBNs are decisive. A hybrid method advanced here samples the continuous BBN once, and then discretizes this so as to enable fast updating. This combines the reduced assessment burden and modelling flexibility of the continuous BBNs with the fast updating algorithms of discrete BBNs. Sampling large complex structures only once can still involve time consuming numerical calculations. Therefore a new sampling protocol based on normal vines is developed. Normal vines are used to realize the dependence structure specified via (conditional) rank correlations on the continuous BBN. We will emphasize the advantages of this method by means of examples. Copyright

Further development of a Causal model for Air Transport Safety (CATS): Building the mathematical heart

The development of the Netherlands international airport Schiphol has been the subject of fierce political debate for several decades. One of the considerations has been the safety of the population living around the airport, the density of which has been and still is growing. In the debate about the acceptability of the risks associated with the air traffic above, The Netherlands extensive use has been made of statistical models relating the movement of airplanes to the risks on the ground. Although these models are adequate for the debate and for physical planning around the airport, the need has arisen to gain a more thorough understanding of the accident genesis in air traffic, with the ultimate aim of improving the safety situation in air traffic in general and around Schiphol in particular. To this aim, a research effort has started to develop causal models for air traffic risks in the expectation that these will ultimately give the insight needed. The concept was described in an earlier paper. In this paper, the backbone of the model and the way event sequence diagrams, fault-trees and Bayesian belief nets are linked to form a homogeneous mathematical model suitable as a tool to analyse causal chains and quantify risks is described.

Eliciting conditional and unconditional rank correlations from conditional probabilities

Abstract Causes of uncertainties may be interrelated and may introduce dependencies. Ignoring these dependencies may lead to large errors. A number of graphical models in probability theory such as dependence trees, vines and (continuous) Bayesian belief nets [Cooke RM. Markov and entropy properties of tree and vine-dependent variables. In: Proceedings of the ASA section on Bayesian statistical science, 1997; Kurowicka D, Cooke RM. Distribution-free continuous Bayesian belief nets. In: Proceedings of mathematical methods in reliability conference, 2004; Bedford TJ, Cooke RM. Vines—a new graphical model for dependent random variables. Ann Stat 2002; 30(4):1031–68; Kurowicka D, Cooke RM. Uncertainty analysis with high dimensional dependence modelling. New York: Wiley; 2006; Hanea AM, et al. Hybrid methods for quantifying and analyzing Bayesian belief nets. In: Proceedings of the 2005 ENBIS5 conference, 2005; Shachter RD, Kenley CR. Gaussian influence diagrams. Manage Sci 1998; 35(5) [15] .] have been developed to capture dependencies between random variables. The input for these models are various marginal distributions and dependence information, usually in the form of conditional rank correlations. Often expert elicitation is required. This paper focuses on dependence representation, and dependence elicitation. The techniques presented are illustrated with an application from aviation safety.

Techniques for generic probabilistic inversion

Probabilistic inversion problems are defined, existing algorithms for solving such problems are discussed, and new algorithms based on iterative re-weighting of a sample are introduced. One of these is the well-known iterative proportional fitting whose properties were already studied (Csiszar, Ann. Probab. (3) (1975) 146). A variant on this is shown to have fixed points minimizing an information functional, even if the problem is not feasible, and is shown to have only feasible fixed points if the problem is feasible. The algorithm is not shown to converge, but the relative information of successive iterates is shown to converge to zero. Applications to atmospheric dispersion and environmental transport are discussed.