Monte Carlo Estimation for Imprecise Probabilities: Basic Properties

We describe Monte Carlo methods for estimating lower envelopes of expectations of real random variables. We prove that the estimation bias is negative and that its absolute value shrinks with increasing sample size. We discuss fairly practical techniques for proving strong consistency of the estimators and use these to prove the consistency of an example in the literature. We also provide an example where there is no consistency

A STOCHASTIC SIMULATION-BASED HYBRID INTERVAL FUZZY PROGRAMMING APPROACH FOR OPTIMIZING THE TREATMENT OF RECOVERED OILY WATER

In this paper, a stochastic simulation-based hybrid interval fuzzy programming (SHIFP) approach is developed to aid the decision-making process by solving fuzzy linear optimization problems. Fuzzy set theory, probability theory, and interval analysis are integrated to take into account the effect of imprecise information, subjective judgment, and variable environmental conditions. A case study related to oily water treatment during offshore oil spill clean-up operations is conducted to demonstrate the applicability of the proposed approach. The results suggest that producing a random sequence of triangular fuzzy numbers in a given interval is equivalent to a normal distribution when using the centroid defuzzification method. It also shows that the defuzzified optimal solutions follow the normal distribution and range from 3,000-3,700 tons, given the budget constraint (CAD 110,000-150,000). The normality seems to be able to propagate throughout the optimization process, yet this interesting finding deserves more in-depth study and needs more rigorous mathematical proof to validate its applicability and feasibility. In addition, the optimal decision variables can be categorized into several groups with different probability such that decision makers can wisely allocate limited resources with higher confidence in a short period of time. This study is expected to advise the industries and authorities on how to distribute resources and maximize the treatment efficiency of oily water in a short period of time, particularly in the context of harsh environments

Uncertainty Analysis of the Adequacy Assessment Model of a Distributed Generation System

Due to the inherent aleatory uncertainties in renewable generators, the reliability/adequacy assessments of distributed generation (DG) systems have been particularly focused on the probabilistic modeling of random behaviors, given sufficient informative data. However, another type of uncertainty (epistemic uncertainty) must be accounted for in the modeling, due to incomplete knowledge of the phenomena and imprecise evaluation of the related characteristic parameters. In circumstances of few informative data, this type of uncertainty calls for alternative methods of representation, propagation, analysis and interpretation. In this study, we make a first attempt to identify, model, and jointly propagate aleatory and epistemic uncertainties in the context of DG systems modeling for adequacy assessment. Probability and possibility distributions are used to model the aleatory and epistemic uncertainties, respectively. Evidence theory is used to incorporate the two uncertainties under a single framework. Based on the plausibility and belief functions of evidence theory, the hybrid propagation approach is introduced. A demonstration is given on a DG system adapted from the IEEE 34 nodes distribution test feeder. Compared to the pure probabilistic approach, it is shown that the hybrid propagation is capable of explicitly expressing the imprecision in the knowledge on the DG parameters into the final adequacy values assessed. It also effectively captures the growth of uncertainties with higher DG penetration levels

Integrating and Ranking Uncertain Scientific Data

Mediator-based data integration systems resolve exploratory queries by joining data elements across sources. In the presence of uncertainties, such multiple expansions can quickly lead to spurious connections and incorrect results. The BioRank project investigates formalisms for modeling uncertainty during scientific data integration and for ranking uncertain query results. Our motivating application is protein function prediction. In this paper we show that: (i) explicit modeling of uncertainties as probabilities increases our ability to predict less-known or previously unknown functions (though it does not improve predicting the well-known). This suggests that probabilistic uncertainty models offer utility for scientific knowledge discovery; (ii) small perturbations in the input probabilities tend to produce only minor changes in the quality of our result rankings. This suggests that our methods are robust against slight variations in the way uncertainties are transformed into probabilities; and (iii) several techniques allow us to evaluate our probabilistic rankings efficiently. This suggests that probabilistic query evaluation is not as hard for real-world problems as theory indicates

Robust estimation of risks from small samples

Data-driven risk analysis involves the inference of probability distributions from measured or simulated data. In the case of a highly reliable system, such as the electricity grid, the amount of relevant data is often exceedingly limited, but the impact of estimation errors may be very large. This paper presents a robust nonparametric Bayesian method to infer possible underlying distributions. The method obtains rigorous error bounds even for small samples taken from ill-behaved distributions. The approach taken has a natural interpretation in terms of the intervals between ordered observations, where allocation of probability mass across intervals is well-specified, but the location of that mass within each interval is unconstrained. This formulation gives rise to a straightforward computational resampling method: Bayesian Interval Sampling. In a comparison with common alternative approaches, it is shown to satisfy strict error bounds even for ill-behaved distributions.Comment: 13 pages, 3 figures; supplementary information provided. A revised version of this manuscript has been accepted for publication in Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Science