Consider a particular input quantity X. In order to characterize (represent) the input uncertainty, data on that particular quantity needs to be available. Convention- ally, probability distributions have been constructed using only point data (e.g., [27]).
Sometimes sufficient point data may not be available to construct such probability distributions. This problem is further complicated if there are interval data. There are several sources of interval data in engineering applications [57–59]. Sometimes, the only information available might come from physical and theoretical constraints that impose bounds on the quantities of interest. Data collected based on temporally spaced inspections may lead to intervals. Uncertainty and errors associated with calibrated instruments may result in experimental observations that are best described using intervals. Sometimes, subject matter experts may describe uncertain quantities using a range of values. Interval data needs to be treated carefully, especially when there are multiple intervals from different sources (say, from multiple experts) and the width of each interval is comparable to the magnitude of the quantity.
The presence of interval data complicates uncertainty representation and propaga- tion because non-probabilistic interval analysis methods have commonly been used to quantify the uncertainty due to interval data. Sometimes, intervals have been approx- imated with uniform distributions, based on the principle of maximum entropy [60].
This approach may be suitable if a quantity is represented with only a single interval;
if multiple intervals are available for a particular quantity, then this method is not suitable since there may be multiple possible uniform distributions.
Sandia National Laboratories conducted an epistemic uncertainty workshop [57]
that invited various views on the quantification and propagation of epistemic uncer- tainty [58], mostly in the form of interval data. Several researchers published different approaches to tackle interval uncertainty in a special issue of the journal Reliability Engineering and Systems Safety [57]. The following paragraphs briefly discuss the different approaches for the treatment of epistemic uncertainty and develop the mo- tivation for the likelihood-based approach proposed in this chapter.
Most probabilistic techniques rely on the existence of sufficient point values for the stochastic quantity of interest. An empirical distribution function (EDF) can be constructed and popular inference techniques such as least squares, moment matching and maximum likelihood can be used to fit parametric probability distributions. The concept of empirical distribution can be extended to interval data sets to arrive at the so-called empirical p-box [59], which is the collection of all possible EDFs for the given set of intervals. Zaman et al. [61] have used the Johnson family of distributions to represent interval uncertainty using a family of distributions which are bounded by the aforementioned p-box. Similar to frequentist p-boxes, Bayesian p-boxes have also been used to represent uncertainty [62].
Researchers have also investigated the use of non-probabilistic approaches for the treatment of epistemic uncertainty due to interval data. Evidence theory [63, 64]
has been proposed to handle interval data. This theory is based on the assumption that the sources of interval data are independent. However, data obtained from different sources needs to be properly aggregated. Dempster’s rule [65, 66] is a popular scheme of aggregation used for this purpose; several improved rules have also been proposed that acknowledge the conflicts that can potentially exist among evidences from different sources. Convex models of uncertainty [67, 68] use a family of convex functions to represent realizations of uncertain quantities and this approach has also
been used to handle interval data. Zadeh’s Extension Principle [69] can be used to construct the possibility distribution of an interval variable which can then be used for uncertainty representation and propagation. Rao and Annamdas [70] presented the idea of weighted fuzzy theory for intervals, where fuzzy set-based representations of interval variables from evidences of different credibilities are combined to estimate the system margin of failure.
The above mentioned methods for uncertainty quantification can be computation- ally expensive in the context of uncertainty propagation. The application of proba- bilistic techniques to interval data is computationally expensive and too cumbersome to apply without severe restrictions because there is an infinite number of possible empirical distributions bounded by the p-box. On the hand, non-probabilistic tech- niques are interval analysis-based approaches [64] and are computationally expensive wherein the cost increases exponentially with the number of uncertain variables, and with the increase in non-linearity of the response function that depends on these uncertain variables. Suppose that some variables are described using intervals and some other physically variable (aleatory) quantities are described using probability distributions. For every combination of interval values, the probabilistic analysis for the aleatory variables has to be repeated, resulting in a computationally expensive double-loop (second-order) sampling analysis.
The primary motivation of this chapter is to accurately perform uncertainty prop- agation in the presence of sparse point and/or interval data and simultaneously reduce the computational effort involved in uncertainty propagation. This is facilitated by the inherent use of the Bayesian philosophy, which allows probabilistic representation of epistemic variables which are described using intervals. (Recall that a frequen- tist approach will not permit the assignment of probability distributions to epistemic variables.) Therefore, the computational expense can be reduced by including both
aleatory and epistemic variables in a single analysis loop, and well-known probabilistic methods are alone sufficient for uncertainty propagation.
However, it is not straightforward to construct probability distributions in the presence of sparse point and/or interval data. If there are sufficient point values, then both parametric (where the parameters are estimated using the method of mo- ments [27]) and non-parametric (using kernel density estimation [29]) probability distributions can be constructed based on the point values. In the presence of sparse point data, there is large uncertainty in the parameter estimates which need to be accounted for; this issue is explained in detail in the next subsection. In the presence of interval data, it is not easy to calculate moments or construct kernels; hence the construction of a probability density function (PDF) is not straightforward. In order to overcome these challenges, this chapter proposes a new likelihood-based methodol- ogy to facilitate probabilistic representation of quantities described using sparse point and/or interval data.