Lateral Loaded Pile Reliability Analysis Using the Random Set Method

The method of reliability estimation with hybrid uncertainties including random and fuzzy variables is proposed. By means of random set theory, random variables and fuzzy variables are transformed into random set, and then a unified model of reliability is proposed in the unified framework of random set theory. The limit-state function of a structure with hybrid uncertainties is mapped into a random set by using the extension principle of function based on random set, and then the limit cumulative distribution functions (CDFs) of reliability are determined. According to the upper and the lower CDF, the interval estimation of reliability can be obtained. The traditional reliability theory based on probability can be viewed as one of the sub-cases of the method proposed. The proposed method is demonstrated with an example.

Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory

A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type

Conditional event algebra (CEA) was developed in order to represent conditional probabilities with differing antecedents by the probability evaluation of well-defined individual “conditional” events in a single larger space extending the original unconditional one. These conditional events can then be combined logically before being evaluated. A major application of CEA is to data fusion problems, especially the testing of hypotheses concerning the similarity or redundancy among inference rules through use of probabilistic distance functions which critically require probabilistic conjunctions of conditional events. Relational event algebra (REA) is a further extension of CEA, whereby functions of probabilities formally representing single event probabilities — not just divisions as in the case of CEA — are shown to represent actual “relational” events relative to appropriately determined larger probability spaces. Analogously, utilizing the logical combinations of such relational events allows for testing of hypotheses of similarity between data fusion models represented by functions of probabilities. Independent of, and prior to this work, it was proven that a major portion of fuzzy logic — a basic tool for treating natural language descriptions — can be directly related to probability theory via the use of one point random set coverage functions. In this paper, it is demonstrated that a natural extension of the one point coverage link between fuzzy logic and random set theory can be used in conjunction with CEA and REA to test for similarity of natural language descriptions.

This paper aims at pointing out a variety of statistical structural models in which parameters of interest are subsets rather than points in parameter spaces, especially in partially identified econometric models. For conducting inferences in such situations, we emphasize the need to use random set theory and elaborate upon statistics of random sets.

The design of high-rise buildings often necessitates ground excavation, where buildings are in close proximity to the construction, thus there is a potential for damage to these structures. This paper studies an efficient user-friendly framework for dealing with uncertainties in a deep excavation in layers of cemented coarse grained soil located in Tehran, Iran by non-deterministic Random Set (RS) method. In order to enhance the acceptability of the method among engineers, a pertinent code was written in FISH language of FLAC2D software which enables the designers to run all simulations simultaneously, without cumbersome procedure of changing input variables in every individual analysis. This could drastically decrease the computational effort and cost imposed to the project, which is of great importance especially to the owners. The results are presented in terms of probability of occurrence and most likely values of the horizontal displacement at top of the wall at every stage of construction. Moreover, a methodology for assessing the credibility of the uncertainty model is presented using a quality indicator. It was concluded that performing RS analysis before the beginning of every stage could cause great economical savings, while improving the safety of the project.

Reasoning with fuzzy and uncertain evidence using epistemic random fuzzy sets: General framework and practical models

Parametric families of continuous belief functions based on generalized Gaussian random fuzzy numbers

This chapter reviews the microeconometrics literature on partial identification, focusing on the developments of the last thirty years. The topics presented illustrate that the available data combined with credible maintained assumptions may yield much information about a parameter of interest, even if they do not reveal it exactly. Special attention is devoted to discussing the challenges associated with, and some of the solutions put forward to, (1) obtain a tractable characterization of the values for the parameters of interest which are observationally equivalent, given the available data and maintained assumptions; (2) estimate this set of values; (3) conduct test of hypotheses and make confidence statements. The chapter reviews advances in partial identification analysis both as applied to learning (functionals of) probability distributions that are well-defined in the absence of models, as well as to learning parameters that are well-defined only in the context of particular models. A simple organizing principle is highlighted: the source of the identification problem can often be traced to a collection of random variables that are consistent with the available data and maintained assumptions. This collection may be part of the observed data or be a model implication. In either case, it can be formalized as a random set. Random set theory is then used as a mathematical framework to unify a number of special results and produce a general methodology to carry out partial identification analysis.

In this paper, we develop the conditional evidence theory and apply it to knowledge discovery in database. In this theory, we assume that a priori knowledge about generic situation and evidence about situation at hand can be modelled by two independent random sets. Dempster’s rule of combination is a popular method used in evidence theory, we think that this rule can be applied to knowledge revision, but isn’t appropriate for knowledge updating. Based on random set theory, we develop a new bayesian updating rule in evidence theory. More importantly, we show that bayesian updating rule can be performed incrementally by using Möbius transforms.KeywordsKnowledge DiscoveryFuzzy MeasureEvidence TheoryPossibility TheoryConditional BeliefThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The development of data fusion systems capable of incorporating soft human-generated data into the fusion process is an emerging trend in the fusion community, motivated mainly by asymmetric warfare situations where the observational opportunities for traditional hard sensors are restricted. This paper describes an extension of our prototype soft/hard data fusion system, based on the random set theory, from centralized into a fully distributed computational framework. A fully distributed data fusion algorithm relies only on information exchange between local sensor nodes and hence promises enhanced scalability, reliability, and robustness in contrast to the conventional centralized fusion approach. We propose a novel approach for distributed estimation of average soft data using the consensus propagation algorithm. The distributed estimation of aggregated hard data is accomplished through an average consensus filter. Based on the proposed approach, we describe a single-target tracking system capable of processing soft and hard data. The preliminary experiments demonstrate the efficiency of the consensus propagation based approach for distributed aggregation of soft data, as well as the advantages of incorporating soft data into the distributed data fusion process.

Rock slope reliability analysis based on distinct element method and random set theory

This thesis contains three essays on inference in econometric models. Chapter 1 considers the question of bootstrap inference for Propensity Score Matching. Propensity Score Matching, where the propensity scores are estimated in a first step, is widely used for estimating treatment effects. In this context, the naive bootstrap is invalid (Abadie and Imbens, 2008). This chapter proposes a novel bootstrap procedure for this context, and demonstrates its consistency. Simulations and real data examples demonstrate the superior performance of the proposed method relative to using the asymptotic distribution for inference, especially when the degree of overlap in propensity scores is poor. General versions of the procedure can also be applied to other causal effect estimators such as inverse probability weighting and propensity score subclassification, potentially leading to higher order refinements for inference in such contexts. Chapter 2 tackles the question of inference in incomplete econometric models. In many economic and statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, nonparametric bounds on treatment effects under imperfect compliance can be expressed by means of random sets. This chapter develops a concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Chapter 3 considers inference on the cumulative distribution function (CDF) in the classical measurement error model. It proposes both asymptotic and bootstrap based uniform confidence bands for the estimator of the CDF under measurement error. The proposed techniques can also be used to obtain confidence bands for quantiles, and perform various CDF-based tests such as goodness-offit tests for parametric models of densities, two sample homogeneity tests, and tests for stochastic dominance; all for the first time under measurement error.

The slopes of open-pit mines are often at risk of failure. To identify this hazard, stability analyses are performed. An important element of these stability analyses is the reliable selection of input parameter values for the calculations. This selection is difficult because the slopes of the open pit are disturbed by mining activities. In such conditions, rheological processes, intensified by weathering, develop in open-pit slopes. This study is aimed at setting the strength parameters for the stability analysis of open-pit slopes with a developed slide process, using the random set method. The study was performed on the example of the open pit of the Bełchatów lignite mine, central Poland. A four-stage methodology, according to the random set method, was proposed. The methodology covered the following: site investigation, sensitivity analyses, shear strength reduction (SSR) analyses using numerical calculations, and probability analyses of the factor of safety (FoS) calculation results. The setting of the input parameters took into account the peak and residual strength parameters for each lithological unit in the physical model of the open-pit slope. Samples for laboratory tests were taken from the cores of nine test boreholes. The sensitivity analysis included all peak and residual strength parameters for each lithological unit in the body. As a result of the sensitivity analysis, specific strength parameters were adopted that would have a great impact upon the results of the calculations. Selected sets of parameter values were then used for the FoS calculations. The resultant FoS values revealed the probable slide planes. The positions of the slide planes were consistent with the interpreted slide surfaces based on the control boreholes and terrain observations. Knowledge of the slide planes positions and the values of the strength parameters enabled the designing of a securing approach for this landslide, and the taking of preventive measures to reduce this risk.

Quantifying coal rib stability and support requirements using bonded block modeling and reliability analysis based on the random set theory