Practical representations of incomplete probabilistic knowledge

Risk-based imprecise post-remediation soil quality objectives

While risk-based contaminated land management is an essential component of sustainable remediation, uncertainty is an unavoidable aspect of risk assessment, since most of the parameters that influence risk are typically affected by uncertainty. Uncertainty may be of different origins; i.e., stochastic or epistemic. Stochastic (or aleatoric) uncertainty arises from random variability related to natural processes, while epistemic uncertainty arises from the incomplete/imprecise nature of available information. But the latter is rarely considered in risk assessments, with the result that risk-based soil quality objectives are almost invariably presented as precise (unique) threshold values. In this paper it is shown: (i) how the joint treatment of stochastic and epistemic uncertainty in risk assessment can lead to soil quality objectives presented as intervals rather than precise values and (ii) how this provides an upper risk-based safeguard for post-remediation monitoring values. The proposed method is illustrated by a real case of soils contaminated by arsenic located in the North-East of France. At this site steel manufacturers have gradually filled up a small valley with slag and dust, over more than a century. These materials are enriched in various metal(loid)s, including arsenic and lead. As the environmental authority has asked for a conversion of the site to other uses that may involve access by the general public, an investigation of human health risk was performed based on a sampling campaign and chemical characterizations including various types of extractions and an analysis of bioaccessibility. While further investigations are required to improve the bioaccessibility model, the human health risk presented herein shows how partial or imprecise information can be incorporated in the analysis while taking into account underlying uncertainties.

A General Amalgamate Technique to Evaluate Human Health Risk Under Uncertain Circumstances

Evaluation of humans’ health risk is an essential and most demanding aid in relevance to the process of decision-making. Accumulation of quality knowledge on the attributes of each and every available data, information and model parameters, involved in risk assessment, plays a crucial role in the process of evaluation. It is important to note that, most frequently, model parameters are imprecise due to the availability of limited data and knowledge. Under such circumstances, probability theory (PT) and the theory of fuzzy sets can be brought forth to deal with the emerging uncertainties. There is also a need to devise an amalgamate technique to perform health risk assessment under uncertainty. Although some different approaches are available in this regard, all approaches are situation or problem dependent and fail to address some specific issues. Therefore, this article presents a general amalgamate technique to address all the concerned issues, and, finally, health risk is carried out using this approach.

Robust Process Design in Pharmaceutical Manufacturing under Batch-to-Batch Variation

Model-based concepts have been proven to be beneficial in pharmaceutical manufacturing, thus contributing to low costs and high quality standards. However, model parameters are derived from imperfect, noisy measurement data, which result in uncertain parameter estimates and sub-optimal process design concepts. In the last two decades, various methods have been proposed for dealing with parameter uncertainties in model-based process design. Most concepts for robustification, however, ignore the batch-to-batch variations that are common in pharmaceutical manufacturing processes. In this work, a probability-box robust process design concept is proposed. Batch-to-batch variations were considered to be imprecise parameter uncertainties, and modeled as probability-boxes accordingly. The point estimate method was combined with the back-off approach for efficient uncertainty propagation and robust process design. The novel robustification concept was applied to a freeze-drying process. Optimal shelf temperature and chamber pressure profiles are presented for the robust process design under batch-to-batch variation.

A Reliability-Based Method to Sensor Data Fusion

Multi-sensor data fusion technology based on Dempster–Shafer evidence theory is widely applied in many fields. However, how to determine basic belief assignment (BBA) is still an open issue. The existing BBA methods pay more attention to the uncertainty of information, but do not simultaneously consider the reliability of information sources. Real-world information is not only uncertain, but also partially reliable. Thus, uncertainty and partial reliability are strongly associated with each other. To take into account this fact, a new method to represent BBAs along with their associated reliabilities is proposed in this paper, which is named reliability-based BBA. Several examples are carried out to show the validity of the proposed method.

A Critical Discussion and Practical Recommendations on Some Issues Relevant to the Nonprobabilistic Treatment of Uncertainty in Engineering Risk Assessment

Models for the assessment of the risk of complex engineering systems are affected by uncertainties due to the randomness of several phenomena involved and the incomplete knowledge about some of the characteristics of the system. The objective of this article is to provide operative guidelines to handle some conceptual and technical issues related to the treatment of uncertainty in risk assessment for engineering practice. In particular, the following issues are addressed: (1) quantitative modeling and representation of uncertainty coherently with the information available on the system of interest; (2) propagation of the uncertainty from the input(s) to the output(s) of the system model; (3) (Bayesian) updating as new information on the system becomes available; and (4) modeling and representation of dependences among the input variables and parameters of the system model. Different approaches and methods are recommended for efficiently tackling each of issues (1)-(4) above; the tools considered are derived from both classical probability theory as well as alternative, nonfully probabilistic uncertainty representation frameworks (e.g., possibility theory). The recommendations drawn are supported by the results obtained in illustrative applications of literature.

Sensing Attribute Weights: A Novel Basic Belief Assignment Method

Dempster-Shafer evidence theory is widely used in many soft sensors data fusion systems on account of its good performance for handling the uncertainty information of soft sensors. However, how to determine basic belief assignment (BBA) is still an open issue. The existing methods to determine BBA do not consider the reliability of each attribute; at the same time, they cannot effectively determine BBA in the open world. In this paper, based on attribute weights, a novel method to determine BBA is proposed not only in the closed world, but also in the open world. The Gaussian model of each attribute is built using the training samples firstly. Second, the similarity between the test sample and the attribute model is measured based on the Gaussian membership functions. Then, the attribute weights are generated using the overlap degree among the classes. Finally, BBA is determined according to the sensed attribute weights. Several examples with small datasets show the validity of the proposed method.

Uncertainty analysis in fault tree models with dependent basic events

In general, two types of dependence need to be considered when estimating the probability of the top event (TE) of a fault tree (FT): "objective" dependence between the (random) occurrences of different basic events (BEs) in the FT and "state-of-knowledge" (epistemic) dependence between estimates of the epistemically uncertain probabilities of some BEs of the FT model. In this article, we study the effects on the TE probability of objective and epistemic dependences. The well-known Frèchet bounds and the distribution envelope determination (DEnv) method are used to model all kinds of (possibly unknown) objective and epistemic dependences, respectively. For exemplification, the analyses are carried out on a FT with six BEs. Results show that both types of dependence significantly affect the TE probability; however, the effects of epistemic dependence are likely to be overwhelmed by those of objective dependence (if present).

EMPIRICAL COMPARISON OF METHODS FOR THE HIERARCHICAL PROPAGATION OF HYBRID UNCERTAINTY IN RISK ASSESSMENT, IN PRESENCE OF DEPENDENCES

Risk analysis models describing aleatory (i.e., random) events contain parameters (e.g., probabilities, failure rates, …) that are epistemically-uncertain, i.e., known with poor precision. Whereas aleatory uncertainty is always described by probability distributions, epistemic uncertainty may be represented in different ways (e.g., probabilistic or possibilistic), depending on the information and data available.

The work presented in this paper addresses the issue of accounting for (in)dependence relationships between epistemically-uncertain parameters. When a probabilistic representation of epistemic uncertainty is considered, uncertainty propagation is carried out by a two-dimensional (or double) Monte Carlo (MC) simulation approach; instead, when possibility distributions are used, two approaches are undertaken: the hybrid MC and Fuzzy Interval Analysis (FIA) method and the MC-based Dempster-Shafer (DS) approach employing Independent Random Sets (IRSs). The objectives are: i) studying the effects of (in)dependence between the epistemically-uncertain parameters of the aleatory probability distributions (when a probabilistic/possibilistic representation of epistemic uncertainty is adopted) and ii) studying the effect of the probabilistic/possibilistic representation of epistemic uncertainty (when the state of dependence between the epistemic parameters is defined).

The Dependency Bound Convolution (DBC) approach is then undertaken within a hierarchical setting of hybrid (probabilistic and possibilistic) uncertainty propagation, in order to account for all kinds of (possibly unknown) dependences between the random variables.

The analyses are carried out with reference to two toy examples, built in such a way to allow performing a fair quantitative comparison between the methods, and evaluating their rationale and appropriateness in relation to risk analysis.

A CONSONANT APPROXIMATION OF THE PRODUCT OF INDEPENDENT CONSONANT RANDOM SETS

The belief structure resulting from the combination of consonant and independent marginal random sets is not, in general, consonant. Also, the complexity of such a structure grows exponentially with the number of combined random sets, making it quickly intractable for computations. In this paper, we propose a simple guaranteed consonant outer approximation of this structure. The complexity of this outer approximation does not increase with the number of marginal random sets (i.e., of dimensions), making it easier to handle in uncertainty propagation. Features and advantages of this outer approximation are then discussed, with the help of some illustrative examples.

Fuzzy-stochastic characterization of site uncertainty and variability in groundwater flow and contaminant transport through a heterogeneous aquifer

Site variabilities and uncertainties in data and information lead to significant spread in results of groundwater flow and contaminant transport models. A framework for hybrid propagation of random uncertainties represented by probability theory; nonrandom uncertainties represented by fuzzy set theory; and site variabilities represented by geostatistics was developed in this research. A case study was provided to explain the computational algorithm. The methodology presented here can be applied to complex environments where there are site variabilities as well as uncertainties of different kinds. The algorithm is suited when uncertainties in some variables may be best represented as fuzzy numbers whereas in others as probability distributions and both form part of the same governing equation.

Joint propagation of variability and imprecision in assessing the risk of groundwater contamination

Estimating risks of groundwater contamination often require schemes for representing and propagating uncertainties relative to model input parameters. The most popular method is the Monte Carlo method whereby cumulative probability distributions are randomly sampled in an iterative fashion. The shortcoming of the approach, however, arises when probability distributions are arbitrarily selected in situations where available information is incomplete or imprecise. In such situations, alternative modes of information representation can be used, for example the nested intervals known as "possibility distributions". In practical situations of groundwater risk assessment, it is common that certain model parameters may be represented by single probability distributions (representing variability) because there are data to justify these distributions, while others are more faithfully represented by possibility distributions (representing imprecision) due to the partial nature of available information. This paper applies two recent methods, designed for the joint-propagation of variability and imprecision, to a groundwater contamination risk assessment. Results of the joint-propagation methods are compared to those obtained using both interval analysis and the Monte Carlo method with a hypothesis of stochastic independence between model parameters. The two joint-propagation methods provide results in the form of families of cumulative distributions of the probability of exceeding a certain value of groundwater concentration. These families are delimited by an upper cumulative distribution and a lower distribution respectively called Plausibility and Belief after evidence theory. Slight differences between the results of the two joint-propagation methods are explained by the different assumptions regarding parameter dependencies. Results highlight the point that non-conservative results may be obtained if single cumulative probability distributions are arbitrarily selected for model parameters in the face of imprecise information and the Monte Carlo method is used under the assumption of stochastic independence. The proposed joint-propagation methods provide upper and lower bounds for the probability of exceeding a tolerance threshold. As this may seem impractical in a risk-management context, it is proposed to introduce "a-posteriori subjectivity" (as opposed to the "a-priori subjectivity" introduced by the arbitrary selection of single probability distributions) by defining a single indicator of evidence as a weighted average of Plausibility and Belief, with weights to be defined according to the specific context.