Imprecise probabilities in engineering analyses
TL;DR: Evidence theory, probability bounds analysis with p-boxes, and fuzzy probabilities are discussed with emphasis on their key features and on their relationships to one another.
About: This article is published in Mechanical Systems and Signal Processing.The article was published on 2013-05-01 and is currently open access. It has received 382 citations till now. The article focuses on the topics: Probabilistic logic & Probability bounds analysis.
Summary (2 min read)
Jump to: [1. Introduction] – [2. Facets of epistemic uncertainty] – [3.1. Emergence in engineering] – [3.2. Engineering application elds] – [4.1. Conceptual categories] – [4.2. Evidence theory] – [4.3. Interval probabilities] – [4.4. Probability bounds analysis with p-boxes] and [5. Conclusions]
1. Introduction
- The analysis and reliability assessment of engineered structures and systems involves uncertainty and imprecision in parameters and models of di erent types.
- Information is often not available in the form of precise models and parameter values; it rather appears as imprecise, di use, uctuating, incomplete, fragmentary, vague, ambiguous, dubious, or linguistic.
- If this balance is violated or not achieved, computational results may deviate signi cantly from reality, and the associated decisions may lead to serious consequences.
- Solutions to this problem are discussed in the literature from various perspectives using di erent mathematical concepts.
- Beyond this coarse review, the collection of papers in this Special Issue provides detailed insight into various imprecise probability approaches and high- lights their bene ts in engineering analyses.
2. Facets of epistemic uncertainty
- If subjective probabilistic statements can be formulated on rational grounds and some data of suitable quality are available, then Bayesian updating can play its important role.
- This conceptual understanding together with the classi cation into probabilistic uncertainty and imprecision provides intuitive motivation for imprecise probabilities and their terminology.
- Suppose the dependency between load and de ection is known deterministically, but a load parameter is available in the form of bounds only.
- Fuzzy numbers are a generalization and re nement of intervals for representing imprecise parameters and quantities.
3.1. Emergence in engineering
- A key feature of imprecise probabilities is the identi cation of bounds on probabilities for events of interest; the uncertainty of an event is character- ized with two measure values a lower probability and an upper probability [54].
- The distance between the probability bounds re ects the indeterminacy in model speci cations expressed as imprecision of the models.
- Imprecise probabilities include a large variety of speci c theories and mathematical models associated with an entire class of measures.
- The adoption of imprecise probabilities and related theories for the solution of engineering problems can be traced, in its early stage, with the publications [83, 84, 85, 86, 87].
3.2. Engineering application elds
- From the initial developments imprecise probabilities have emerged into several application elds in engineering with structured approaches.
- In [94] intervals are employed for the description of the imprecision in probabilistic models for a structural reliability assessment.
- The developments in this area have been extended to applicability to larger, realistic and practical problems.
- The proposed interval-valued sensitivity index measures the relative contribution of individual model input variables, in the form of intervals or sets of distribution functions, to the uncertainty in the model output.
4.1. Conceptual categories
- The ideas of imprecise probabilities may be categorized into three basic groups of concepts associated with three di erent technical approaches to construct imprecise probabilistic models.
- In general, these coarse specications may be the best information available, or they may arise from limitations in measurement feasibility.
- This imprecision may arise, for example, when con icting information regarding the distribution type is obtained from statistical tests, that is, when the test results for di erent distributions as well as for compound distributions thereof with any mixing ratio are similar.
- The authors shall see below that interval probabilities can be used to represent this group of concepts.
- But from a practical point of view, this categorization and the associated features of the concepts as elucidated in the subsequent sections can provide the engineer with a good sense for the modeling of a problem.
4.2. Evidence theory
- If the information available possesses some probabilistic or probability-related background, but does not meet the preconditions to be speci ed as a random variable, evidence theory often provides a suitable basis for an appropriate quanti cation and subsequent processing.
- On this basis various speci c uncertainty measures may be derived within a range from plausibility to belief which include traditional probability as a special case.
- The compliance with the traditional de nition of probability is, however, not complete because the focal subsets are not required to be disjoint.
- The subjects included in the discussion concern uncertainty quanti cation, computation of model predictions and system responses, simulation, optimization and design, and decision making.
- A crucial point in the practical application of evidence theory is realizing the basic probability assignment in each particular case.
4.3. Interval probabilities
- Initial applications, although few, already indicate the usefulness of interval probabilities and demonstrate their capabilities.
- Numerically, conditional probability is quite di erent from the probability of the conditional.
4.4. Probability bounds analysis with p-boxes
- Probability bounds analysis [162, 163, 164, 61] is another of the uncertainty quanti cation approaches that are considered part of the theory of imprecise probability [44].
- A p-box represents a class of probability distributions consistent with these constraints.
- There are several other techniques that originated in the eld of interval analysis that are likewise fruitfully extended to probability bounds analysis, three of which the authors mention here.
- Taylor arithmetic can also project uncertainty expressed as p-boxes whether they represent epistemic uncertainty or aleatory uncertainty or both.
- Like Monte Carlo simulation, it sidesteps the problem of repeated uncertain variables to approximate best-possible bounds.
5. Conclusions
- In solving engineering problems, it is extremely important to properly take uncertainty and imprecision into consideration.
- In engineering applications, there are two main sources of uncertainty and imprecision.
- T. Augustin, F. Coolen (Eds.), Special Issue: Imprecise probability in statistical inference and decision making, Vol. 51 of International Journal of Approximate Reasoning, 2010. [73].
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Citations
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TL;DR: In this paper, a numerical strategy for the efficient estimation of set-valued failure probabilities, coupling Monte Carlo with optimization methods, is presented in order to both speed up the reliability analysis, and provide a better estimate for the lower and upper bounds of the failure probability.
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TL;DR: In this article, the authors present a review of the state-of-the-art in probability-interval hybrid uncertainty analysis and provide an outlook for future research in this area.
Abstract: Traditional structural uncertainty analysis is mainly based on probability models and requires the establishment of accurate parametric probability distribution functions using large numbers of experimental samples. In many actual engineering problems, the probability distributions of some parameters can be established due to sufficient samples available, whereas for some parameters, due to the lack or poor quality of samples, only their variation intervals can be obtained, or their probability distribution types can be determined based on the existing data while some of the distribution parameters such as mean and standard deviation can only be given interval estimations. This thus will constitute an important type of probability-interval hybrid uncertain problem, in which the aleatory and epistemic uncertainties both exist. The probability-interval hybrid uncertainty analysis provides an important mean for reliability analysis and design of many complex structures, and has become one of the research focuses in the field of structural uncertainty analysis over the past decades. This paper reviews the four main research directions in this area, i.e., uncertainty modeling, uncertainty propagation analysis, structural reliability analysis, and reliability-based design optimization. It summarizes the main scientific problems, technical difficulties, and current research status of each direction. Based on the review, this paper also provides an outlook for future research in probability-interval hybrid uncertainty analysis.
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Abstract: Prognostics and Health Management (PHM) is a multidisciplinary field aiming at maintaining physical systems in their optimal functioning conditions. The system under study is assumed to be monitored by sensors from which are obtained measurements reflecting the system’s health state. A health indicator (HI) is estimated to feed a data-driven PHM solution developed to predict the remaining useful life (RUL). In this paper, the values taken by an HI are assumed imprecise (IHI). An IHI is interpreted as a planar figure called polygon and a case-based reasoning (CBR) approach is adapted to estimate the RUL. This adaptation makes use of computational geometry tools in order to estimate the nearest cases to a given testing instance. The proposed algorithm called RULCLIPPER is assessed and compared on datasets generated by the NASA’s turbofan simulator (C-MAPSS) including the four turbofan testing datasets and the two testing datasets of the PHM’08 data challenge. These datasets represent 1360 testing instances and cover different realistic and difficult cases considering operating conditions and fault modes with unknown characteristics. The problem of feature selection, health indicator estimation, RUL fusion and ensembles are also tackled. The proposed algorithm is shown to be efficient with few parameter tuning on all datasets.
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"Imprecise probabilities in engineer..." refers background in this paper
...Imprecise probabilities have a close relationship to the theory of random sets [56, 57] and cover, for example, the concept of upper and lower probabilities [58], sets of probability measures [59], distribution envelopes [60], probability bounds analysis using p-boxes [61], interval probabilities [62], Choquet capacities [63] of various orders, and evidence theory (or Dempster-Shafer Theory) [64, 65]...
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