The method for generating random d x d correlation matrices with a partial correlation C-vine is extended so that each correlation can have a distribution that is asymmetric on (-1, 1) or on (0, 1). With the recursion formulas from the partial correlation C-vine to the correlation matrix, first and second moments can be derived, in the case of the same distribution for each partial correlation in tree l of the vine (1 <= l < d). Algorithms and conditions are given so that, after a permutation step, all random correlations have a common mean and second moment. The algorithms can be useful for simulation experiments to generate random correlation matrices that cover the whole space or with the restriction that each correlation is positive.

The work presented here marks a further advance in expert uncertainty quantification. In a recent probabilistic evaluation of ice sheet process contributions to sea level rise, tail dependence was elicited and propagated through an uncertainty analysis for the first time. The elicited correlations and tail dependencies concerned pairings of three processes: Accumulation, Discharge and Run-off, which operate on major ice sheets in the West and East Antarctic and in Greenland. The elicitation enumerated dependencies between these processes under selected global temperature change scenarios over different future time horizons. These expert judgments allowed us to populate a Paired Copula Bayesian network model to obtain the estimated contributions of these ice sheets for future sea level rise. Including positive central tendency dependence and tail dependence increases the fatness of the upper tails of projected sea level rise distributions, an amplification important for designing and evaluating possible mitigation strategies. Detailing and jointly computing distributional dependencies and tail dependencies can be crucial components of good practice for assessing the influence of uncertainties on extreme values when modelling stochastic multifactorial processes.

Consider a random vector $\mathbf{y}=\mathbf{\Sigma}^{1/2}\mathbf{x}$, where the $p$ elements of the vector $\mathbf{x}$ are i.i.d. real-valued random variables with zero mean and finite fourth moment, and $\mathbf{\Sigma}^{1/2}$ is a deterministic $p\times p$ matrix such that the spectral norm of the population correlation matrix $\mathbf{R}$ of $\mathbf{y}$ is uniformly bounded. In this paper, we find that the log determinant of the sample correlation matrix $\hat{\mathbf{R}}$ based on a sample of size $n$ from the distribution of $\mathbf{y}$ satisfies a CLT (central limit theorem) for $p/n\to \gamma\in (0, 1]$ and $p\leq n$. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of $\mathbf{y}$ is unknown, we show that after recentering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. At last, the obtained findings are applied for testing of uncorrelatedness of $p$ random variables. Surprisingly, in the null case $\mathbf{R}=\mathbf{I}$, the test statistic becomes completely pivotal and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

Multivariate statistical models can be simplified by assuming that a pattern of conditional independence is presented in the given data. A popular way of capturing the (conditional) independence is to use probabilistic graphical models. The relationship between strongly chordal graphs and m-saturated vines is proved. Moreover, an algorithm to construct an m-saturated vine structure corresponding to strongly chordal graph is provided. This allows the reduction of regular vine copula models complexity. When the underlying data is sparse our approach leads to model estimation improvement when compared with current heuristic methods. Furthermore, due to reduction of model complexity it is possible to evaluate all vine structures as well as to fit non-simplified vines. These advantages have been shown in the simulated and real data examples.1

An extension of the D-vine based forward regression procedure to a R-vine forward regression is proposed. In this extension any R-vine structure can be taken into account. Moreover, a new heuristic is proposed to determine which R-vine structure is the most appropriate to model the conditional distribution of the response variable given the covariates. It is shown in the simulation that the performance of the heuristic is comparable to the D-vine based approach. Furthermore, it is explained how to extend the heuristic into a situation when more than one response variable are of interest. Finally, the proposed R-vine regression is applied to perform a stress analysis on the manufacturing sector which shows its impact on the whole economy.

Technology forecasting is anessential starting point for conceptual design of anycomplex engineering system. Infact, many research projects are focused on developing a small set of promising technologies to a suitable readiness level. However, selecting a set of technologies from a larger pool is a nontrivial task, opposed by uncertainty and subjective tradeoffs. This paper proposes a probabilistic method to represent technologies and quantify their effects, while accounting for uncertainty. Using probabilistic inversion, technologies can be selected froma larger set tomeet a certain combination of requirements. Several test cases illustrate the method and how it may be used in conceptual design projects. It is concluded that probabilistic inversion enables answering technology development and selection queries, which would be challenging to answer with traditional deterministic approaches, or purely forward uncertainty propagation approaches.

The selection of vine structure to represent dependencies in a data set with a regular vine copula model is still an open question. Up to date, the most popular heuristic to choose the vine structure is to construct consecutive trees by capturing largest correlations in lower trees. However, this might not lead to the optimal vine structure. A new heuristic based on sampling orders implied by regular vines is investigated. The idea is to start with an initial vine structure, that can be chosen with any existing procedure and search for a regular vine copula representing the data better within vines having 2 common sampling orders with this structure. Several algorithms are proposed to support the new heuristic. Both in the simulation study and real data analysis, the potential of the new heuristic to find a structure fitting the data better than the initial vine copula model, is shown.

The availability of resources is crucial for the socio-economic stability of our society. For more than two decades, there was a debate on how to structure this issue within the context of life-Cycle assessment (LCA). The classical approach with LCA is to describe “scarcity” for future generations (100–1000 years) in terms of absolute depletion. The problem, however, is that the long-term availability is simply not known (within a factor of 100–1000). Outside the LCA community, the short-term supply risks (10–30 years) were predicted, resulting in the list of critical raw materials (CRM) of the European Union (EU), and the British risk list. The methodology used, however, cannot easily be transposed and applied into LCA calculations. This paper presents a new approach to the issue of short-term material supply shortages, based on subsequent sudden price jumps, which can lead to socio-economic instability. The basic approach is that each resource is characterized by its own specific supply chain with its specific price volatility. The eco-costs of material scarcity are derived from the so-called value at risk (VAR), a well-known statistical risk indicator in the financial world. This paper provides a list of indicators for 42 metals. An advantage of the system is that it is directly related to business risks, and is relatively easy to understand. A disadvantage is that “statistics of the past” might not be replicated in the future (e.g., when changing from structural oversupply to overdemand, or vice versa, which appeared an issue for two companion metals over the last 30 years). Further research is recommended to improve the statistics.

Disruptions such as rolling stock breakdown, signal failures, and accidents are recurrent events during daily railway operation. Such events disrupt the deployment of resources and cause delay to passengers. Obtaining a reliable disruption length estimation can potentially reduce the negative impact caused by the disruption. Different factors such as the location, cause of disruption, traffic density, etc. can determine the disruption length. The uncertainty inherent to the variability of each factor and the unavailability of sufficient data results in a wide distribution of disruption lengths from which a certain value should be selected as the length prediction. The rescheduling measure considered in this research is short-turning the trains that are heading to the disrupted area. To investigate the impact of the disruption length estimates on the rescheduling strategy and the resulting passengers delays, this research presents a framework consisting of three models: a disruption length model, short-turning model and passenger assignment model. The framework is applied to a part of the Dutch railway network. The results show the effects of short (optimistic) and long (pessimistic) estimates on the number of affected passengers, generalized travel time and number of passengers rerouting and transferring.

This paper describes a novel sensitivity analysis method, able to handle dependency relationships between model parameters. The starting point is the popular Morris (1991) algorithm, which was initially devised under the assumption of parameter independence. This important limitation is tackled by allowing the user to incorporate dependency information through a copula. The set of model runs obtained using latin hypercube sampling, are then used for deriving appropriate sensitivity measures. Delft3D-WAQ (Deltares, 2010) is a sediment transport model with strong correlations between input parameters. Despite this, the parameter ranking obtained with the newly proposed method is in accordance with the knowledge obtained from expert judgment. However, under the same conditions, the classic Morris method elicits its results from model runs which break the assumptions of the underlying physical processes. This leads to the conclusion that the proposed extension is superior to the classic Morris algorithm and can accommodate a wide range of use cases.

Decreasing the uncertainty in the lengths of railway disruptions is a major help to disruption management. To assist the Dutch Operational Control Center Rail (OCCR) during disruptions, we propose the Copula Bayesian Network method to construct a disruption length prediction model. Computational efficiency and fast inference features make the method attractive for the OCCR's real-time decision making environment. The method considers the factors influencing the length of a disruption and models the dependence between them to produce a prediction. As an illustration, a model for track circuit (TC) disruptions in the Dutch railway network is presented in this paper. Factors influencing the TC disruption length are considered and a disruption length model is constructed. We show that the resulting model's prediction power is sound and discuss its real-life use and challenges to be tackled in practice. (C) 2016 Elsevier Ltd. All rights reserved.

The use of different copula-based models to represent the joint distribution of an eight-dimensional mixed discrete and continuous problem consisting of five discrete and three continuous variables is investigated. The discussion starts with the theoretical properties of the copula-based models. Four different models are constructed for the data collected for the purpose of predicting the length of disruption caused by problems with the train detection system in the Dutch railway network and their performance is tested. The more complex models turn out to represent the data better. Nevertheless, it is shown that the simpler eight dimensional Normal copula still constitutes a statistically sound model for the data.

We present a sampling algorithm for a regular vine on n variables which starts at an arbitrary variable. A sampling order whose nested conditional probabilities can be written as products of (conditional) copulas in the vine and univariate margins is said to be implied by the regular vine. We show that there are 2n−1 implied sampling orders for any regular vine on n variables. We show that two regular vines on n and m distinct variables can be merged in 2n+m−2 ways. This greatly simplifies the proof of the number of regular vines on n variables. A notion of sampling proximity based on numbers of shared implied sampling orders is introduced, and we use this notion to define a heuristic for searching vine space that avoids proximate vines.

We study properties of a local dependence function of Wang for copulas. In this paper this dependence function is called the mixed derivative measure of interactions as it is a mixed derivative of a log of a density function. It is stressed that this measure is not margin free in the sense that the interaction function of a density and the corresponding copula are not equal. We show that there is no Archimedean copula with constant interactions. The interaction function is positive (negative) for an Archimedean copula density whose second derivative of the generator is log convex (log concave). Moreover, the only Archimedean copula with interactions proportional to its density is Frank’s copula. We obtain some preliminary results concerning the connection between the behaviour of the interaction function and the tail dependence of the distribution. Moreover, the notion of an interaction function has been extended to more than two dimensional case, and we study its properties for a canonical Archimedean copula.

The highly uncertain nature of a railway disruption complicates the tasks carried by the Dutch Operational Control Centre Rail (OCCR) in the Netherlands. A good prediction of disruption length is believed to help the decision making in dealing with the disruption. Zilko, et al. [Non-Parametric Bayesian Network to Forecast Railway Disruption Lengths, In: The Second International Conference on Railway Technology: Research, Development and Maintenance (Railways 2014), Ajaccio, France (2014)] proposes the use of the Non- Parametric Bayesian Network (NPBN) method, a graphical model based on a probabilistic approach that represents the dependence between the variables of interest, to predict the disruption length. The model offers an attractive feature for the real-time decision making environment in the OCCR, which is its ef?ciency and fast inference. This paper extends the model construction. More variables are added into the NPBN model to increase the prediction power of the model. From the data analysis, it turns out that some of the influ-encing variables are discrete variables; thus resulting in a mixed discrete and continuous model. This raises some serious questions because the NPBN method is originally designed to work with continuous variables. This paper investigates how the presence of discrete variables affects the NPBN method and how one can proceed with the mixed discrete and continuous model. As an illustration, a model about the railway disruption in the Netherlands caused by track circuit failures is presented in this paper as well.