SCIENCES

Mechanics, Field Director – Gilles Pijaudier-Cabot

Numerical Methods in Mechanics, Subject Head – Sylvain Drapier

Coordinated by

Christian Gogu

Maurice LEMAIRE

SIGMA Clermont, Clermont-Ferrand, and Phimeca Engineering, Cournon, France

For centuries, mechanics has been a rational science motivated by astronomers observing the solar system with remarkable predictions of ephemeris, until Henri Poincaré brought out the chaos. For their part, the builders knew that their forecasts could be thwarted by dubious or unforeseen events, of which they were protected from by the so-called safety coefficients introduced by Claude-Henri Navier.

The 20th Century was marked by advances in materials and mechanical system behavior models, and by the considerable development of digital solutions. However, there will always be an uncertain gap (not an error) between the model and the physical reality.

If Andrey Kolmogorov made the theory of probabilities quasi-perfect, the statistical model necessary for its implementation remains, and will remain, uncertain in mechanics, even if the big data of the 21st Century will bring much to the exploration of the Mediocristan (the life around the median), but little to that of the Extremistan (the life at borders), according to the mapping of Nassim-Nicholas Taleb.

Immersed in the uncertainties of its models and its data, the domain of mechanics was initially based on expertise and feedback to justify its new works. Only recently did it realize that probability theory could provide an additional precision, by eliciting the content of its rules and codes. Engineers gradually accept to go from quantities of interest justification by a deterministic calculation, to their probability distributions acceptance stressing the failure.

This paradigm shift is cultural, scientific and technological all at once. It is cultural because it involves a new way of thinking that both scientific and educational communities were unprepared for. Mechanics of materials and structures, statistics (often limited to process control) and probabilities fall under disciplinary fields independent from one another.

It is scientific because it involves new algorithm development, recognizing the uncertain nature of models and parameters by the probability theory or other approaches. It is technological because it involves a new approach to rules, now based on probabilistic content, like the EN Eurocodes did. Finally, it involves thought on the decision: to entrust it to the probabilities in Mediocristan and to the consequences of the failures in Extremistan.

Some visionaries became aware of the importance of variability from the beginning of the 20th Century. A reliability index concept introduced in 1949, by Rzhanitzyn in steel construction, was quickly popularized because of the need to provide answers to new situations, for example in off-shore platforms engineering, and by the development of computer tools, which is now motivating methodological works to be implemented numerically.

Since the early 1980s, several scientific communities in the world have gradually seized the subject, bringing more arguments to the probabilistic approach and giving it a scientific consistency and operational tools exiting academic applications. Allow me to have the pleasure of pointing out that the Clermont-Ferrand team has done its part.

Christian Gogu joined talented authors from university centers and research organizations to contribute to the Mechanics and Uncertainty French School, and I want to congratulate them.

In this book, the reader will find a clarification of the concepts of epistemic and random, because the first requirement is to ensure that the vocabulary is not random. This book’s contents provide details about historical methods, as well as the introduction of methods that overcome their shortcomings. It addresses the issue of sensitivity factors, essential for both reliability and optimization. It sheds light on the fundamental uncertainty of the statistical model, which should be considered as fractal. It takes into account the uncertainty of the models and not only that of the parameters. It gives meaning to the notion of optimization with uncertain variables.

However, it leaves open questions on the curse of dimensionality and on the time-variant problem; it will be the topic of the next edition in the coming years. It illustrates the approaches by various technological fields in mechanical engineering for aeronautical and space engineering, and in civil engineering for nuclear engineering. In this book the reader will find both matter to initiation, as well as an introduction to advanced tools.

I hope that this book will be a motivation for researchers, and especially young researchers, to find a scientific field in which to express their skills, where it is necessary to innovate for the scientific and technological progress of humanity.

November 2020

Christian GOGU

Clément Ader Institute, Paul Sabatier University, Toulouse, France

This preface is intended to thank all those who contributed to this book and to establish the context of each contribution.

First of all, I would like to thank Sylvain Drapier, professor at the École des Mines de Saint-Étienne, whom I had as a professor during my studies and whom I subsequently worked with at the Tribology and Systems Dynamics Laboratory, where I completed my PhD thesis concerning the identification of mechanical properties under uncertainty. In charge of the topic “Numerical modeling in mechanics” for the new Sciences encyclopedia launched by ISTE Editions, Sylvain Drapier invited me to be the editor of this work and I thank him for the trust he has placed in me.

I would then like to give my warm thanks to Maurice Lemaire, Professor Emeritus at IFMA/SIGMA Clermont and Scientific Advisor to Phimeca, for agreeing to write the foreword of this book. I met Maurice a long time ago, and came to know him personally first and then professionally. He has pioneered the research on considering uncertainties in mechanical engineering and it is an honor to have his contribution to this book.

Thereafter I would like to thank all the authors of the different chapters in this book, while establishing the context of each of their contributions in order to highlight the structuring and articulation of the book.

Without being exhaustive, this book aims to present a number of fundamental concepts and classical approaches, as well as several recent developments and advanced methods. It is divided into two parts, each with four chapters. The first part is dedicated to the modeling, quantification and propagation of uncertainties. The second part is dedicated to taking uncertainties into account through reliability analyses and optimization under uncertainty.

Chapter 1, which I have written, aims to review a number of conventional uncertainty modeling methods. It also seeks to introduce a number of notations and terminologies. A distinction is specifically made between random and epistemic uncertainties, and the different modeling methods associated with each are presented, illustrated and discussed.

Chapter 2, written by François Willot, illustrates the use of probabilistic tools to characterize environments exhibiting a microheterogeneous structure, a typical example of which is concrete. Such a characterization of “random” microstructures makes it possible to later return to a statistical characterization of macroscopic properties, necessary to characterize the reliability or robustness of the structures.

Chapter 3, written by David Bouhjiti, Julien Baroth and Frédéric Dufour, illustrates methods of uncertainty propagation on large civil engineering structures, in this case containment facilities for nuclear power plants. It discusses the essential notions of variability of material properties and their characterization by random fields. The stochastic finite-element technique, as well as several improvements, is then introduced in order to achieve the propagation of random fields to the quantities of interest.

Chapter 4, co-authored with Sylvain Dubreuil, Nathalie Bartoli and Thierry Lefebvre, presents a new method for characterizing and then reducing uncertainties, due to the use of substitution models based on Gaussian processes within multidisciplinary analysis. This method is illustrated with a classical example of multidisciplinary coupling, in this case, a fluid-structure coupling on an aeronautical wing.

Chapter 5, written by Jean-Marc Bourinet, reviews a large number of methods for calculating the probabilities of rare events, typically failure probabilities. The reader will find an introduction to the fundamental concepts and to these different methods for calculating failure probabilities.

Chapter 6, written by Cécile Mattrand, Pierre Beaurepaire and Nicolas Gayton, reviews a number of relatively recent developments associated with a certain type of method, known as the adaptive method, for calculating the probabilities of failure. These methods are based on the use of kriging surrogate models and their adaptive enrichment, in order to calculate failure probability with the most moderate numerical cost possible.

Chapter 7, written by Vincent Chabridon, Mathieu Balesdent, Guillaume Perrin, Jérôme Morio, Jean-Marc Bourinet and Nicolas Gayton, proposes a new method for the calculation of the (global) sensitivity of the probability of failure, taking into account the uncertainty regarding the parameters of the distributions, characterizing the variabilities of the problem. This method is illustrated by an example in the field of space launchers.

Chapter 8, written by Quentin Mercier and Fabrice Poirion, presents a new method for solving stochastic multiobjective optimization problems and, more particularly, those where the expectation of the different objectives is to be optimized. It is also shown how this formulation can be used to efficiently solve optimization problems under reliability constraints, by reformulating them as multiobjective optimization problems.

Christian GOGU

Clément Ader Institute, Paul Sabatier University, Toulouse, France

Several decades ago, the scientific and engineering community began to recognize the value of considering uncertainties in the design, optimization and risk analysis of complex systems, such as aircraft, space vehicles or nuclear power plants (Wagner 2003; Lemaire 2014). These uncertainties can manifest themselves in many forms, originating from many different sources. In the field of mechanics, sources of uncertainty can typically be due to:

With respect to these different sources of uncertainty, a distinction is often made between aleatory and epistemic uncertainties (Vose 2008; National Research Council 2009), although this distinction is debatable, as will be discussed in more detail in section 1.2.

Aleatory uncertainty is also referred to as irreducible uncertainty, stochastic uncertainty, inherent uncertainty or type I uncertainty. This uncertainty typically arises from environmental stochasticity, fluctuations in time, variations in space, heterogeneities and other intrinsic differences in a system. It is often referred to as irreducible uncertainty because it cannot be reduced further except by modification of the problem under consideration. On the other hand, it can be better characterized when it is empirically estimated. For example, the characterization of the variability of a material property can be improved by increasing the number of samples used, allowing a better estimate of statistical properties such as the mean and standard deviation.

An example of random uncertainty can be seen in an (unbiased) coin toss (that is, “heads or tails”). The intrinsic characteristics of the toss create uncertainty about the outcome of the toss, with a probability of 0.5 of obtaining tails. Assuming that tails is an undesirable outcome, it would then be desirable to reduce the probability of obtaining tails in order to reduce the uncertainty of the desired outcome (obtaining heads). However, without breaking the rules of the game, that is, without modifying the problem under consideration, it is not possible to reduce this uncertainty, hence the term “irreducible uncertainty”. On the other hand, if this uncertainty is characterized empirically on the basis of several tosses, it is obvious that this uncertainty can be better characterized by increasing the number of tosses.

In terms of an engineering example, the amplitude of the gusts that an aircraft will be likely to encounter during its lifetime can be seen as a random or irreducible uncertainty. Indeed, when designing a new model of aircraft, it would be desirable for the engineer to reduce this uncertainty as much as possible in order to reduce the weight of the aircraft structure. Unfortunately, since this uncertainty is essentially related to environmental stochasticity, the engineer has no way to reduce it without changing the design being considered, therefore, the uncertainty is seen as irreducible. However, it could perhaps be reduced by changing the problem under study. For example, the engineer might want to consider installing sensors that allow the aircraft to detect the amplitude of turbulence in its trajectory in advance, thus allowing pilots to undertake avoidance maneuvers. Nevertheless, such a choice would have numerous and serious consequences. (Would such an aircraft be certifiable? Is it more economical to avoid turbulence rather than designing the aircraft to withstand it? Would passenger comfort be satisfactory? etc.) Usually, the problem is thus considered fixed and the irreducible or non-irreducible nature of the uncertainties is estimated on a given problem.

Epistemic uncertainty, also called reducible uncertainty or type II uncertainty, is the result of a lack of knowledge. This type of uncertainty is usually associated with uncertainty in measurements, a small number of experimental data, censorship or the unobservability of phenomena or scientific ignorance, which, in general terms, amounts to a lack of knowledge of one kind or another. It is often referred to as reducible uncertainty, as it can potentially be reduced by additional actions (for example, additional studies), leading to an improvement in knowledge. It should be noted that once epistemic uncertainty has been reduced, random uncertainty may remain, which could then become the predominant uncertainty and would therefore be irreducible.

An example of epistemic uncertainty would be that associated with the estimation of the age (in years) of an individual. Let us assume that we have just heard on the radio that a Nobel Prize winning scientist is going to give an acceptance speech in our town and we would like to know the age of the scientist. There is no mention of that on the radio. At this point, let us say that we can only estimate the age of this person between 30 and 90 years old. Therefore, there is a lot of uncertainty about their age at that point. However, as mentioned earlier, epistemic uncertainty can be reduced through improved knowledge. If we attend the acceptance speech, we will have the opportunity to see this person, which could allow us to reduce the uncertainty about their age to between, let us say, 50 and 60 years of age. If we even go to talk to them after the speech, we may even be able to get additional information that will allow us to further reduce the uncertainty about their age. In this case, the uncertainty could even be reduced to zero if we can find out their date of birth.

A typical example of epistemic uncertainty in engineering problems is measurement uncertainty. Similar to the scientist’s age, the quantity that must be measured has a true value that is fixed (considering measurements at the macroscopic, not the quantum scale). Nevertheless, measurement instrumentation usually only allows this quantity to be determined with uncertainty. This uncertainty can be reduced by developing better instruments, using a better knowledge of the measurement phenomena involved, hence the term reducible uncertainty, although, in general, this does not mean that the uncertainty can be reduced to zero.

Probability theory has historically provided the first framework for modeling and quantifying uncertainties. Today, it is generally accepted that aleatory uncertainties can be adequately handled by probability theory. While the probabilistic approach can also be used to model epistemic uncertainties, other alternative representations have been proposed for such uncertainties, such as interval analysis, fuzzy set theory, possibility theory or evidence theory. These alternative approaches address, in particular, the need to quantify uncertainty when little or no data (either numerical or experimental) are available. Attempts to unify all of these approaches under a generalized theory of inaccurate probabilities have also been undertaken (Walley 2000; Klir 2004), without, however, leading to fully satisfactory approaches.

The purpose of this chapter is therefore to provide an overview of some of the different approaches used to represent and quantify uncertainties, both aleatory and epistemic. It is organized as follows. In section 1.2, a discussion about the need to distinguish between epistemic and aleatory uncertainty is presented. In section 1.3, an illustration of the probabilistic modeling approach is provided, including illustrations of some cases where its use may be problematic for representing epistemic uncertainty. In section 1.4, p-box theory is presented, which is an extension of probability theory that is designed to better address problematic cases in modeling epistemic uncertainties. In section 1.5, interval analysis is briefly discussed and in section 1.6 fuzzy set theory is addressed. In section 1.7, possibility theory is introduced, while evidence theory (or Dempster–Shafer theory) is presented in section 1.8. Some concluding remarks and discussions are provided in section 1.9.

The need to classify uncertainty into epistemic and aleatory has been the subject of much debate and discussion (Hoffman and Hammonds 1994; Apostolakis 1999; Der Kiureghian and Ditlevsen 2009; Lemaire 2014). It therefore seemed useful to create a section dedicated to the challenges and the usefulness of this distinction. This debate arises from the controversy between Niels Bohr and Albert Einstein concerning the nature of randomness observed at the quantum scale. For the former, it was a fundamental randomness, while for the latter, it was linked to an incomplete knowledge of phenomena at these scales. To illustrate his point of view, Einstein stated that “God does not play dice”. Subsequent theoretical and experimental work by physicists seems to have proved Niels Bohr right: randomness observed at these scales is intrinsic to quantum nature. Now that the debate at the microscopic scale has been closed, one may raise the same question at the macroscopic level of mechanics, which is the subject of this book. Are there uncertainties at the macroscopic level of an intrinsically aleatory nature that can affect mechanical systems? Or are all uncertainties at the macroscopic level related only to a lack of knowledge?

It is the author’s view that macroscopic mechanical systems can be affected by uncertainty of an inherently aleatory nature. This is notably related to two aspects: free will and the chaotic nature of some systems. Free will refers to the ability of human beings to freely self-determine and thus to be responsible for their actions. An opposite school of thought, articulated by Pierre Simon Laplace through “Laplace’s demon”, would establish that perfect knowledge of the present and past would make it possible to perfectly predict the future, including with regard to human actions. In such a vision, all uncertainties would obviously be of an epistemic nature, and an improvement in knowledge would make it possible to reduce them. Since all our societies are based, particularly in terms of justice systems, on the responsibility of each individual for their actions, we shall admit here the existence of the faculty of free will. Free will thus leads to a degree of aleatory uncertainty, which cannot be reduced by improvement in knowledge.

An example from the mechanical field is now provided to illustrate the uncertainty associated with free will. Anyone who has seen a construction excavator at work has quickly realized that it is an illusion to try to predict the load that its articulated arms will be subjected to during its lifetime, as the conditions under which the machine is used can be extremely unexpected. The ingenuity (if we look on the positive side) of the operators of the machine means that they will always find new uses for the machine, which the engineer designing the machine would never have thought of. For example, using the bucket to push a truck that the excavator has just loaded and that skids up the slope to get out of the hole, and also using it for different kinds of demolitions. Consequently, it is not possible to consider the uncertainty about the loads being applied on the arms of the machine as an uncertainty of a purely epistemic nature, solely linked to lack of knowledge and therefore reducible by improving knowledge. As a matter of fact, this would mean that an improvement in knowledge would ultimately be able to predict the ways in which operators will decide to use the machine, which is obviously contrary to free will.

Another uncertainty of an intrinsically aleatory nature at the macroscopic level is that related to chaotic systems. Chaos theory has shown that such systems cannot be predicted beyond a more or less long-term horizon. A classic example of a chaotic system is the atmospheric system, which is reflected by the fact that the state of the system (namely what is commonly known as weather) cannot be predicted beyond a certain period of time; this is the famous butterfly effect. Note that it is the system itself that is chaotic and not just our modeling of the system. An improvement of our models will therefore not change anything in the system itself, which will remain chaotic. Because of their chaotic nature, and therefore unpredictability in the more or less long term, chaotic systems are a source of aleatory uncertainty, which cannot be reduced by improvement in knowledge.

In order to illustrate this with a mechanical case, let us again consider the case of the gusts that an aircraft will experience. When designing a new model of aircraft, it is useful for the engineer to know the amplitude of the gusts that the aircraft will experience during its lifetime in order to be able to size the aircraft structure as efficiently as possible. The amplitude of these gusts can then be considered as aleatory, irreducible uncertainty. Indeed, due to the chaotic system that is the atmosphere, it is not possible to predict air movements on a scale of a few meters over a time scale of several decades (that is, the lifetime of the aircraft). Moreover, it should be noted that the chaotic nature of the atmospheric system is further amplified by interaction with human free will. Uncertainty about human response to climate change makes it difficult to predict the state of the atmospheric system on the time and space scales under consideration. The engineer therefore has no means of reducing this uncertainty about the amplitude of gusts by improving knowledge.

After having seen two sources of uncertainty that are inherently aleatory in nature at the macroscopic scale, one may ask whether other uncertainties can be classified as aleatory. While the previous two sources are intrinsically aleatory, the classification of other sources of uncertainty may depend on the situation. It should not be forgotten that the classification between epistemic and aleatory uncertainty introduced in the previous section is specific to the problem under consideration. When we considered the favorable outcome of the “coin toss” as aleatory uncertainty, the rules of the game are fixed (in particular, considering that the toss cannot be biased). If it is considered that biasing the toss is part of the rules of the game, then the uncertainty about the outcome is obviously reducible.

In mechanical problems, the engineer is responsible for articulating the problem, and, in particular, for setting “the rules of the game”, namely what can and cannot be changed. The classification is then specific to the problem under consideration and an uncertainty on the same physical quantity can then be classified as being epistemic for certain problems and aleatory for others.

To illustrate this, let us take the example of the uncertainty on the limit stress at failure of a spar of an aircraft wing. Without further details on the problem under consideration, it is not possible to say whether this uncertainty should be classified as epistemic or aleatory. Indeed, in the first case, assume that the problem considered is that of designing a new wing, keeping the same raw material supplier and the existing quality control process, and that we are focusing on the uncertainty in the limit stress at failure of the spar of one of the future aircraft that will be manufactured. Since the raw material batches obtained from the supplier have some variability, which cannot be reduced by staying within the scope of the problem under consideration (that is, without changing supplier and the quality control process), the uncertainty about the limit stress at failure in question can be classified as aleatory, irreducible uncertainty. In the second case, assume that we are studying the limit stress at failure of the spar of a particular aircraft that has already been manufactured. The uncertainty about this limit stress at failure is of interest to ensure that this aircraft will be able to withstand a higher load associated with a particular mission being considered for this aircraft. In this case, the failure limit stress has a single “true” value, but which is not known. The uncertainty associated with the failure limit stress is this time classified as epistemic uncertainty. It can be reduced by improving knowledge, for example, here by testing a sample kept by the manufacturer from the batch of material used for the spar.

In light of this ambivalence, the interest of classifying uncertainties into epistemic and aleatory for a given problem can be questioned. From the author’s perspective, the usefulness of this classification is most apparent from an engineering point of view in terms of distinguishing between reducible and irreducible uncertainties. This distinction is important because the solutions to be provided by the engineer to these two types of uncertainties are very different. For the former, the solutions essentially consist of seeking to reduce uncertainties, while for the latter, the solutions essentially consist of designing the systems in their presence (for example, by integrating partial safety factors, redundancies or by explicitly calculating the reliability of the system).

The interest of alternative approaches to modeling epistemic uncertainties is also questionable compared to the probabilistic approach typically used for aleatory uncertainties. Clearly, since the distinction between epistemic and aleatory uncertainty is based on the engineer’s decisions regarding the problem to be considered, this, in itself, cannot justify different modeling methods for these two types of uncertainties. Taking the example of the limit stress at failure of a spar, illustrating the dependence of the classification on the problem under consideration, we can elaborate by noting that in the first case probabilistic modeling is typically adopted to model the variability of the stress at failure. It is obvious that when we shift to case 2, there is no reason not to maintain the probabilistic approach to model the epistemic uncertainty before trying to reduce it by way of testing (using Bayesian approaches, for example). There are thus many cases where probabilistic modeling is well suited to modeling epistemic uncertainties.

On the other hand, there are also situations where probabilistic modeling may not be the most appropriate way to model uncertainties. These are typically cases where tests or simulations on the uncertain quantity of interest are difficult, if not impossible, to obtain. In such cases, the available data are not sufficient to construct a suitable probabilistic model. Alternative approaches that are less data-intensive and based on expert opinions have been proposed and will be reviewed in the following sections. Finally, it should be noted that, in practice, for the same quantity, there is often an epistemic (reducible) and an aleatory (irreducible) part of uncertainty, making the analysis even more complex.

Probability theory is the most widely used approach for the representation of uncertainties. Its theoretical framework is recalled here from an angle that makes it easy to compare it subsequently with alternative uncertainty representation approaches. We shall also present some situations illustrating the limits of probability theory for representing some types of epistemic uncertainties.

Formally, probabilities are defined on a probability space EP defined as follows.

DEFINITION 1.1.– Let Ω be a set. A σ-algebra or σ-field on Ω refers to a set T of subsets of Ω verifying:

DEFINITION 1.2.– Let Ω be a set and T a σ-algebra of Ω. A probability measure on (Ω, T) is a function ℙ: T → ℝ satisfying the following three axioms, called Kolmogorov axioms:

DEFINITION 1.3.– Let Ω be a set, T a σ-algebra of Ω and ℙ a probability measure. The triplet (Ω, T, ℙ) is called a probability space EP.

Concretely, Ω is called a universe, comprising the set of the possible events (or outcomes) under consideration. T is a subset of the universe, that is, a subset of possible events, having a σ-algebra structure. ℙ is a probability measure that allows a certain probability to be attributed to any event in the universe.

PROPERTY 1.1.– Let (Ω, T, ℙ) denote a probability space. Then ∀A ∈ T, ℙ (A) + ℙ (AC) = 1.

This property is trivial within the context of probability theory, but we shall see that in alternative theories of representation of uncertainties it appears differently.

DEFINITION 1.4.– Let (Ω, T, ℙ) be a probability space. Any measurable function from Ω to ℝ is called a real-valued random variable X. x = X(ω) is called a realization of the random variable X:

[1.1]

DEFINITION 1.5.– The probability distribution of the real-valued random variable X is the probability measure, denoted ℙ X, resulting from the transport of the probability measure ℙ on Ω into a probability measure ℙX on ℝ.

The probability distribution ℙX is then a function that allows any subset of ℝ to be associated with an associated probability. Note that there are different types of probability distributions, some examples of which are the binomial, geometric or Bernoulli distributions for discrete random variables and normal (or Gaussian), uniform or gamma distributions for continuous random variables. In the following, we will be interested, essentially, in continuous random variables, which are those most often used in engineering problems.

DEFINITION 1.6.– The cumulative distribution function (CDF), sometimes referred to as the distribution function (DF), of the probability distribution ℙX is the function FX defined by:

[1.2]

The CDF thus associates to any real value x the value of the probability that the random variable of distribution ℙX is less than or equal to the value x.

DEFINITION 1.7.– The complementary cumulative distribution function (CCDF), sometimes called the complementary cumulative probability function (CCPF), of the probability distribution ℙX is the function defined by:

[1.3]

The CCDF is introduced here because it is sometimes used in risk analysis, where it provides a means to know what is the probability that a quantity will exceed a threshold.

An example of CDF and CCDF is provided in Figure 1.1 for a Gaussian distribution of mean 4 and standard deviation 1.

Uncertainty of an aleatory nature can typically be modeled by a random variable with a certain probability distribution. In an engineering context, the probability distributions of the random variable quantities of interest can rarely be described in a fully analytical manner. In this case, these distributions are instead constructed from samples of the quantities of interest, which can be obtained either experimentally or by simulation. These samples can then be used to construct an empirical estimate of the distribution function of each random variable.

DEFINITION 1.12.– Let (x1, …, xn) denote n samples of a random variable of distribution function FX. The empirical cumulative distribution function (ECDF) is then defined by:

[1.8]

where 1_A is the indicator function of event A. It can be shown that the ECDF is an unbiased estimator of the CDF. An illustration of the ECDF obtained from 100 samples from a reduced centered normal distribution is provided in Figure 1.2. The CDF of this normal distribution is also shown.

There are many situations where epistemic uncertainties can be represented strictly within the same probabilistic framework as described in the previous sections.

To illustrate such situations, let us take an example addressed earlier. Consider the limit stress at failure of a wing spar. Let us assume that the variability characterization provided by the raw material supplier allows us to model the variability on the limit stress at failure of the spars of all the aircraft that will be produced by a given probability distribution ℙX. Now we assume that we are interested in a specific aircraft among those that have already been produced. The failure limit stress of the spar of this aircraft has a “true” value that is fixed, but which we do not know. The associated uncertainty is therefore epistemic in nature. How can we model this epistemic uncertainty?

The spar in question represents one realization among all the realizations (that is, all the spars that will be produced); the failure limit stress of this spar thus represents a realization of the random variable X characterized by the distribution ℙX. It is then absolutely reasonable to consider at first that the epistemic uncertainty that we want to represent can be modeled by a random variable of the same distribution ℙX. Since this uncertainty is epistemic, it is however possible to reduce it, for example, by carrying out characterization tests, which could provide a new probability distribution for this epistemic uncertainty, a distribution which will typically be “narrower” than ℙX. In any case, the epistemic uncertainty on the failure limit stress of a specific aircraft spar is modeled here all along within a probabilistic framework, a framework that seems to be well suited to this problem.

On the other hand, there are other situations where retaining this probabilistic framework to represent epistemic uncertainties may be more problematic. These are typically cases where the amount of data is limited. This happens quite frequently, since having large sample sizes (experimental or by simulation) is often expensive or simply not feasible in practice. What can be said, for example, about the inherent variability of systems that have been manufactured only a few times, such as the space shuttle, which has been manufactured only five times?

In such cases, the key questions to ask before adopting a probabilistic modeling for these epistemic uncertainties are as follows:

In the following, we present three examples of situations where the probabilistic modeling of epistemic uncertainties can be problematic or lead to counterintuitive results.

Problems can arise when epistemic uncertainties are represented by uniform probability distributions, which are supposed to account for a lack of knowledge about what happens between the upper and lower bounds of the uniform distribution. In the event, an expert can only provide two bounds to characterize the uncertainty on a quantity of interest and the principle of indifference could lead us to believe that modeling this uncertainty using a uniform distribution between these two bounds is a reasonable choice.

However, this can be problematic, as a uniform distribution has relatively strong specific implications in probability theory. In particular, making the assumption of a uniform distribution between two bounds is a much stronger assumption than the assumption that the quantity can be indifferently (that is, without preferential concentration) located anywhere between the two bounds. As a matter of fact, a uniform distribution implies a particular uncertainty structure between the two bounds, which is not without consequence. To illustrate this, consider the following example.

Let us consider a bar with a square cross-section subjected to a specific tensile force. In order to calculate the stress in the bar, we need the cross-sectional area of the bar. However, the dimensions of the bar are not precisely known. They thus comprise an epistemic uncertainty. Assume that the only thing that is known is that the value of the side of the square is between a and b. How should we model these uncertainties? The principle of indifference could lead us to model the side X of the square using a uniform distribution between a and b. However, the same principle of indifference could lead us to model these epistemic uncertainties by considering that it is the area of section X² that is uniformly distributed between a² and b², since ultimately it is the section that we need for the calculation of the stress. Which of these two models should we choose?

The two choices might seem indifferent, but unfortunately, they are not. Indeed, by modeling uncertainty using a uniform distribution on X, when it is propagated to X², it will not lead to a uniform distribution on X². This is illustrated in Figure 1.3 when X follows a uniform distribution between 0 and 1. The distribution of X² is then far from being a uniform distribution, with low values of X² much more likely than high values. This dilemma illustrates one of the difficulties that can arise when trying to model certain knowledge gaps by imposing a priori choices of probability distributions.

Figure 1.3. Distribution function of a uniform distribution on X between [0,1] (magenta line) and the corresponding distribution function of X2 (blue line). For a color version of this figure, see www.iste.co.uk/gogu/uncertainties.zip

A second example illustrating counterintuitive results that could emerge from inadequate modeling of epistemic uncertainties is based on Nikolaidis et al. (2004). Let us consider a structure subjected to vibrational stresses. Due to the resonance phenomenon, some excitation frequencies must be avoided in order to prevent failure of the structure. If we impose that the structure’s response (for example, its maximum deflection) must be below a threshold, this delimits a zone of failure, illustrated in Figure 1.4 for a generic frequency response function.

Let us now consider that there is a large epistemic uncertainty on the values of the frequencies at which the structure in service will be excited and that only the bounds a and b on these frequencies are known. In the absence of additional information, the principle of indifference is applied and a uniform distribution in [a,b] is initially considered for the excitation frequencies (see Figure 1.4).

This defines the probability of failure by the area below the distribution within the failure zone. Now let us suppose that the engineer is looking to be more conservative and considers the uncertainty about the excitation frequencies to be higher. Therefore, they consider a wider uniform distribution in [ã, ] (see Figure 1.4) to account for the higher epistemic uncertainty. This results in a decrease in the area under the distribution that lies within the failure zone and thus decreases the probability of failure. This example thus illustrates a situation where increasing epistemic uncertainty (that is, increasing the lack of knowledge) decreases the probability of failure. This is somewhat counterintuitive, as one would normally expect that high epistemic uncertainty (that is, a significant lack of knowledge) would lead to unreliable products (namely, having a high probability of failure) and that products can be made more reliable by increasing knowledge (namely, decreasing epistemic uncertainty).

Probability box (or p-box) theory has been developed to deal with cases where the probability distribution ℙX of a random variable X cannot be accurately determined, thus combining an aleatory component and an epistemic component of uncertainty. Within the framework of probability box theory, the distribution function FX associated with the unknown distribution ℙX can only be bounded (exactly or at a given level of confidence): the left bound is a distribution function that will be denoted and the right bound is a distribution function denoted F. This is illustrated in Figure 1.5.

Any distribution function comprised between and F can thus represent the probability distribution of X. The spacing between and F represents the magnitude of the epistemic uncertainty associated with the lack of knowledge of the distribution of X. If and F are superimposed, there is no epistemic uncertainty and the aleatory uncertainty is represented by the distribution function FX = = F.

Figure 1.5. Illustration of a probability box

A probability box can be interpreted in two different ways: as bounds on the cumulative probability for a given value of x or as bounds on the values of x for a given confidence level. In the example in Figure 1.5, for example, one can read that the probability that x is less than 3 is comprised between 0.05 and 0.3. We can also read that the 95% quantile of X is between x = 8 and 17.

From a formal point of view, a probability box can be defined, adopting the formalism of Ferson et al. (2015), as follows:

DEFINITION 1.15.– Let and F be non-decreasing functions from ℝ to [0,1] satisfying the condition , ∀x ∈ ℝ. Let [F, ] denote the set of all non-decreasing functions F from ℝ to [0,1] satisfying F (x)≤ F(x)≤ (x),∀ x ∈ ℝ. [F, ] is then called a “probability box”.

Note that this approach can be seen as an extension of inaccurate probabilities (Walley 1991) to distribution functions and, consequently, to probability distributions. There are different variants existing derived from this concept of a probability box. For example, in addition to bounding the distribution function F with and F one can impose additional constraints such as:

There are also different types of probability boxes:

Probability box-based approaches have been applied in many areas of engineering. In the context of mechanics, we can cite the application to the analysis of the buckling load of the Ariane 5 fairing (Oberguggenberger et al. 2009). Roy and Balch (2012) applied probability boxes for predicting the thrust of a supersonic nozzle, while Zhang et al. (2011) applied them to the analysis of a finite element lattice.

Probability boxes-based approaches are generally well suited for analyses where aleatory and epistemic uncertainties are simultaneously present. Its major disadvantage lies in the difficulty of obtaining bounding functions and F in practice, especially in the presence of small amounts of data (a small number of samples or even no samples at all and only the availability of expert opinions, etc.). Indeed, when few samples are available, the bounding that can usually be obtained on the distribution function is too large to be useful in practice. Sometimes there is even a total absence of samples (that is, there have been no formalized experiments that can be traced and exploited) and only expert opinions can provide information on existing uncertainties. How can the bounds and F be constructed based on these expert opinions? Various approaches, which we shall review later, and, in particular, the Dempster–Shafer theory, which is conceptually quite close to probability boxes, have sought to overcome these difficulties.

As we have seen in previous sections, one of the major problems in modeling epistemic uncertainties resides in the determination of the probability distribution (or its bounding) that a certain physical quantity follows. However, in a large majority of cases, the uncertainty can at least be bounded. A range with very wide bounds can often be obtained by considering constraints of physical nature on the quantities of interest. For example, physical considerations make it possible to limit the Poisson ratio of a homogeneous isotropic material between –1 and 0.5. Even if relatively narrow, this range is not very informative. The opinion of an expert, on the other hand, could allow this range to be further significantly reduced. In our example, an expert could, for example, state that, for the material that we are considering, the Poisson ratio will be between 0.2 and 0.4. Note that, based on this information, we cannot characterize how the uncertainty varies within this range (are the boundary values as plausible as the center values?). Nonetheless, there are situations where a range is the only information available.

Interval analysis can be used to model such cases where the uncertainty is provided by bounds on the quantity of interest. The uncertainty on the quantities x₁, …, xn is then described by their lower bound and their upper bound . A lot of research work has then focused on the propagation of uncertainties from x₁, …, xn to a quantity y = g(x₁, …, xn), where g is any function modeling the relation between the input and output variables. The problem of determining the lower and upper bound on y is equivalent to solving a global optimization problem. Consequently, many global optimization algorithms can be applied to solve this problem (Hansen and Walster 2003). Note that in cases where the function g is monotonic over the range of variation of the input variables (namely, on the range domain), then the lower and upper bounds on y can be determined very efficiently: due to the monotonicity, y has merely to be evaluated at the vertices of the hypercube constituted by the bounds of the input variables. The lower and upper bounds are then necessarily the minimum and maximum among these points, respectively. This method is often known as the vertex method. Other techniques use Taylor expansions to approximate the bounds on the output variable. Note that sampling techniques, such as Monte-Carlo simulation, can also be used to solve this problem, but this quickly becomes prohibitively time-consuming and methods using global optimization algorithms are usually more efficient. For a review of the different algorithms for efficient interval analysis, the reader may refer to Kreinovich and Xiang (2008).

A major disadvantage of interval approaches is the lack of a measure of uncertainty, similar to probability within the context of probability theory. In interval analysis, uncertainty is characterized by the boundaries of the interval only, but no information is available on the likelihood of different values within that interval. In probabilistic approaches, the likelihood of different values is characterized by the PDF, which defines the probability that the quantity of interest lies within a certain range (for example, an interval). Such a measure is not available in the interval approach.

Interval arithmetic and associated uncertainty propagation methods will not be discussed in more detail because of the lack of an uncertainty measure in interval analysis. This reduces the usefulness of the method for reliability- or robustness-based applications where quantification of the risk associated with various decisions is required. Nevertheless, the interval approach is still useful in situations where only the worst case needs to be considered.

While bounds on an interval may be sufficient to quantify epistemic uncertainty within the context of interval analysis, there are many situations where additional information is available, making it possible to refine uncertainty quantification.

Let us consider the case where experts believe that the Poisson ratio of a material is bounded by the interval [0.2;0.4] for a given material. It is very likely that, with their expertise, they will be able to provide a more refined analysis. For example, they can state that most of the time the Poisson ratio of the material will be between 0.3 and 0.35 and only in rare cases it will fall outside this narrower range. This information on the epistemic uncertainty provided by experts can be modeled within the context of fuzzy set theory by a trapezoidal membership function, as illustrated in Figure 1.6. The values in the interval [0.3;0.35] are the most likely values and their corresponding value in terms of the membership function is 1. The membership function then linearly decreases down to the bounds 0.2 and 0.4 provided by the experts. For any value α of the membership function, we can obtain an interval on the quantity of interest, called α-cut. In the example in Figure 1.6, we illustrate the interval for a level α = 0.6, denoted 0.6-cut.

Triangular membership functions are also frequently used in fuzzy set theory. They allow a single value to be modeled as the most probable, as well as bounds outside which the quantity of interest cannot be located. More generally, any kind of membership functions can be used, but this is very rare, in practice, because of the difficulty of specifying them for concrete problems.

Figure 1.6. Illustration of a trapezoidal membership function in fuzzy set theory

As with previous approaches, a fundamental question concerns the propagation of uncertainties through a function g. As for any value α of the membership function, we can associate an interval; this is tantamount to ultimately performing interval analysis propagation at each level α. The methods discussed in the previous section apply in this way. Purely algebraic propagation is also possible for simple operations (additions, multiplications, powers). For a more detailed view of fuzzy set theory, the reader may refer to Zimmermann (2011).

One of the major drawbacks of fuzzy set theory lies, as for interval analysis, in the absence of a measure of uncertainty, which is equivalent to probability in probability theory. Uncertainty propagation is carried out at a level α, but the link with a measure allowing the quantification of the risk associated with this level α is still missing. In order to remedy this, a measure of uncertainty called possibility has been introduced. This has led to possibility theory, which will be presented in the next section.

Formally, possibility theory is defined on a possibility space Epos, defined as follows:

DEFINITION 1.16.– Let Ω be a set and E the set of the subsets of Ω. A function Π : E → ℝ is called a possibility measure (or possibility distribution function [PoDF]) if it satisfies the following axioms:

DEFINITION 1.17.– Let Ω be a set, called universe, E the set of the subsets of Ω and Π a measure of possibility. The triplet (Ω, E, Π) is called the possibility space EPos.

Note that the possibility measure (or PoDF) Π provides a measure of the likelihood of each element of E. The membership functions introduced for fuzzy sets (for example, triangular, trapezoidal functions) are typically used as PoDFs.

DEFINITION 1.18.– Let ξ be an uncertain variable to which the possibility distribution Π is associated. Let and be the lower and upper bounds of the α-cut of ξ. Let f be a non-negative weighting function f : [0,1] → ℝ , monotonically decreasing and verifying the condition of normalization in the integral sense. The f-weighted possibilistic mean operator is:

[1.11]

This operator is the equivalent of expectation in the context of probability theory. By introducing various weighting functions, different levels of importance can be assigned to the different α-cuts of the possibility distribution. There are also similar expressions existing for higher order moments in a possibility context.

In order to be able to compare the likelihood of different events, two quantities, the possibility and necessity of an event, are introduced.

DEFINITION 1.19.– Let (Ω, E, Π) be a space of possibility. We call the possibility and necessity of an event e ∈ E, respectively:

[1.12]

[1.13]

PROPERTY 1.2.– Let (Ω, E, Π) be a space of possibility and an event e ∈ E. We have the following properties:

[1.14]

[1.15]

[1.16]

[1.17]

[1.18]

More intuitively, we can say that possibility measures the degree to which the facts do not contradict the assumption that an event can happen. If an event has a possibility of 1, it means that there is no reason to believe that the event cannot happen. It does not mean, however, that the event will certainly happen. In order for an event to be certain, both its possibility and its necessity must be equal to 1. On the other hand, if we consider that an event cannot happen, then it must be assigned a possibility of 0.

It should be noted that, similarly to what has been done in probability theory, a cumulative possibility function (CPoF), a cumulative necessity function (CNeF), a cumulative complementary possibility function (CCPoF) and a cumulative complementary necessity function (CCNeF) can be associated with a PoDF. Figure 1.7 illustrates these different functions with an example.

Figure 1.7. Illustrations of (a) a possibility distribution function (PoDF); (b) the cumulative possibility function (CPoF) and the corresponding cumulative necessity function (CNeF)

Note that for the example shown in Figure 1.7, the most likely values are those in the interval [5,7], which have a possibility of 1. The larger intervals encompassing this latter then have progressively smaller possibility values, until they reach 0 for the interval [1,10]. Values below 1 or above 10 are thus considered impossible. All the properties of equations [1.13]–[1.17] can be verified on these curves. Note also that the stepwise nature of the cumulative functions is typical of the modeling of epistemic uncertainties using alternative approaches. Indeed, the quantification of uncertainty is, most of the time, based on experts who would assign likelihood values to the different intervals, thus allowing the construction of the PoDF. The more finely the expert can elicit the uncertainty (by assigning likelihood values to many successively larger nested intervals), the smaller the steps will be. Finally, in terms of uncertainty propagation, given that the PoDFs are, most of the time, based on fuzzy set theory membership functions, they also make use of the same propagation techniques discussed in section 1.6. Propagation is thus typically done by interval propagation for different α-cuts.

This section aims to highlight the commonalities and differences between probability and possibility theories. First of all, in terms of axioms, the main difference lies in terms of σ-additivity for probability theory and subadditivity for possibility theory (see the definitions given in sections 1.3.1 and 1.7.1). Let us recall that the probability of the union of disjoint events is equal to the sum of the probabilities of the events. Thus, if {A₁, …, An} is a partition of the universe, then the sum of the probabilities of the A_i must be equal to 1. There is no similar constraint in terms of possibilities of events Ai. For example, the sum of the probabilities of the events “tomorrow it will snow” and “tomorrow it will not snow” must be equal to 1. On the other hand, if we assign a possibility of 0.7 to “it will snow tomorrow”, we must assign a possibility of 1 to the event “it will not snow tomorrow”. This is because the maximum possibility in the universe must be equal to 1. Thus, since the possibility of the universe Ω is the maximum possibility of the events in the universe, this implies that the possibility of at least one event in the partition must be equal to 1.

Both probability theory and possibility theory work with distribution functions, called PDF in the probabilistic framework and PoDF within the possibility framework. However, despite visual similarities, these two functions account for significantly different modeling. Figure 1.8 illustrates some of these differences.

We can see the following differences in Figure 1.8:

Figure 1.8. Comparison between a probability density function f(x) and a possibility distribution function Π(x)

In order to ensure consistency between variables modeled in a probabilistic manner and variables modeled in a possibilistic manner, the following condition is typically imposed: the possibility of an event must be greater than or equal to its probability. This is consistent with the usual meaning of the words probable and possible: when we say “it is probable that it will snow tomorrow”, this shows a stronger conviction than when we say “it is possible that it will snow tomorrow”. This condition of consistency implies that reasoning in terms of possibilities is more conservative than reasoning in terms of probabilities.

The differences in the probabilistic and possibilistic axioms, as well as the consistency requirement specified above, imply a very different interpretation of distribution functions (CDF or CPoF) in these two frameworks. Let us examine how the two theories model the information X in the interval [8,12]. If we want to model this in a purely probabilistic framework (and not in a context of probability boxes), we need to make an additional assumption, such as the principle of indifference between values in this interval, which leads to modeling by a uniform distribution between 8 and 12. The corresponding probability density is shown in Figure 1.9. The figure also represents a PoDF modeling the same information (X is in the interval [8,12]). Note that multiple possibility distributions could have been plotted. In particular, a uniform distribution with a possibility value of 1 might have been suitable. Such a distribution would have been very conservative, being the most uninformative distribution that could have been constructed. On the other hand, the triangular distribution plotted in Figure 1.9 is the most informative distribution that can be plotted and at the same time meets the consistency condition, which requires that the possibility of any event X be greater than or equal to its probability.

Figure 1.9. Probability density function and possibility distribution function satisfying the consistency condition

To conclude this comparison, it should be noted that possibility and necessity can be considered as upper and lower bounds of true probability in the presence of epistemic uncertainty. This implies that the cumulative functions of possibility (CPoF) and necessity (CNeF) will bound the cumulative probability distribution function (CDF).

This is illustrated in Figure 1.10 for the previous example of the variable X within the range [8,12]. The CPF of the uniform probability distribution is effectively bounded by the CPoF and the CNeF. This bounding models the presence of epistemic uncertainty. Indeed, the statement X in the interval [8,12] is very vague and many probability distributions could have been suitable. We had to introduce the principle of indifference, which may be debatable, to obtain the uniform distribution and its associated CDF. In the absence of this assumption, any PDF whose CDF would have been between the CPoF and CNeF curves could have been suitable. The CPoF and CNeF bounding is thus similar in spirit to that obtained in the context of probability boxes (p-boxes), but obtained within the context of possibility theory, based on axioms that are quite different from those of probability theory.

Figure 1.10. Distribution function (DF), cumulative possibility function (CPoF) and cumulative necessity function (CNeF). For a color version of this figure, see www.iste.co.uk/gogu/uncertainties.zip

Possibility theory addresses the problem of quantifying uncertainties when solely based on expert opinion, which will assign likelihood levels to different values of the quantity of interest via possibility distributions Π(x) (typically via triangular or trapezoidal distributions). One of the fundamental questions that then arises is how to deal with divergent expert opinions. For this purpose, different rules for combining possibility distributions have been established.

Let Π₁ and Π₂ be two possibility distributions. These distributions can be aggregated according to the following rules:

[1.19]

These three modes of combination usually make it possible to combine distributions of possibility from different sources (experts, for example). If there is good agreement between the sources (for example, trapezoidal distributions with overlapping cores), then either the connective or disjunctive modes are both well suited. The choice between the two depends on whether one wishes to consider only consensus or whether one wishes to integrate divergent views as well.

In the absence of good agreement between sources (for example, trapezoidal distributions with non-overlapping kernels), the conjunctive and disjunctive modes are less well suited. The disjunctive mode introduces a multimodal distribution (two peaks), which is generally not desirable in terms of quantifying and propagating uncertainties. The conjunctive mode by itself is too limiting, as it only considers consensus, which can be very limited or even null. The intermediate mode is a proposed solution to resolve these two problems associated with the conjunctive and disjunctive modes. It should be noted that variations on the intermediate mode have also been proposed in order to give more confidence to one distribution (in other words, an expert) than to another. The reader may refer to Dubois and Prade (1992) for more information.

The theory of belief functions (or evidence theory) is another theory for modeling uncertainties of an epistemic nature. It was developed by Dempster (1967) and Shafer (1976) and is consequently sometimes known as the Demspter–Shafer theory. This approach is similar in spirit to the probability box theory and possibility theory, in that it seeks to obtain a bounding for a CDF.

Formally, evidence theory is defined on a belief space EE defined as follows:

DEFINITION 1.20.– Let Ω be a set and E the set of the subsets of Ω. A function m : E → ℝ is called belief mass function if it satisfies the following axioms:

DEFINITION 1.21.– Let Ω be a set, called universe, E the set of subsets of Ω and m a belief mass function. The triplet (Ω, E, m) is called the belief space EE.

Note that the belief mass function m provides, as was the case for the possibility distribution, a measure of the relative likelihood of each element of E. In the terminology of the theory of belief functions, an element of E that has a non-zero mass is called a focal element, denoted Ω_i. Note that these focal elements are, most of the time, defined such as to form a set of disjoint elements. The quantity mi = m(Ω_i) is then called mass (or sometimes basic probability assignment [BPA]) associated with this focal element.

In order to be able to compare the likelihood of different events, two quantities, plausibility and belief of the event, are introduced.

DEFINITION 1.22.– Let (Ω, E, m) be a belief space. We call belief (noted Bel) and plausibility (denoted Pl) of an event e ∈ E, respectively:

[1.20]

[1.21]

From a conceptual point of view, the mass of a focal element m (Ω_i) provides the true likelihood that is attributed to that focal element, but without specifying how this likelihood is distributed among the subsets of Ωi. Therefore, the belief Bel(e) of an event e represents the minimal likelihood that can be associated with e, while the plausibility Pl(e) represents the maximum likelihood that can be associated with e.

In the vast majority of applications of evidence theory, focal elements are defined such as to form a set of adjacent polytopes (or even orthotopes). To illustrate this, consider a universe with nine focal elements Ω1 to Ω9 of mass m1 to m9 (see Figure 1.11). Let us now consider the belief and plausibility of the event e, shaded in the figure. According to the definition, the plausibility Pl(e) is obtained by adding the masses of the focal elements that have a non-zero intersection with e (in our case Ω2, Ω3, Ω5, Ω6, Ω8 and Ω9). This means that each focal point included even partially in e contributes to the plausibility of e. The belief Bel(e) is calculated by summing only the masses of the focal elements entirely included in e (in our case, Ω3 and Ω6). Belief and plausibility can thus be seen as a bounding for the likelihood of an event, reflecting the fact that the distribution of likelihood within a focal element is not known, due to epistemic uncertainty. By considering Ω8 we can, for example, consider that the likelihood of the small gray triangle included in Ω8 is very low and that the mass m8 thus contributes almost nothing to the likelihood of e. This hypothesis comes closer to the calculation of belief. On the other hand, we can also assume that all the likelihood of Ω8 is concentrated in this small gray triangle and all the rest of Ω8 is not at all likely. In this case, m8 must actually contribute in full to the likelihood of e. This assumption is thus close to the plausibility calculation. Both cases are thus extreme cases, bounding the true likelihood.

Figure 1.11. Example of an uncertainty structure (focal elements Ωi of mass mi) in the theory of belief functions. An event e is grayed out here

Note that, in a similar way to what has been done in probability and possibility theory, a cumulative plausibility function (CPlF), a cumulative belief function (CBF), a complementary cumulative plausibility function (CCPlF) and a complementary cumulative belief function (CCBF) can be associated with a BPA function. Figure 1.12 illustrates these different functions in an example where a BPA function assigns a mass of 0.1 to the following intervals: [1,3], [1,4], [1,10], [2,4], [2,6], [5,8], [5,10], [7,8], [7,10], [9,10].

Note that, while the curves in Figures 1.7 and 1.12 are similar in terms of their stepwise nature, there are some notable differences. The major difference is that in possibility theory, the necessity must be zero for the possibility to be different from 1 and vice versa. Such a constraint does not exist in evidence theory, which allows plausibility and belief to be simultaneously different from 0 and 1. This thus allows for a much narrower bounding of an unknown cumulative probability function than could be achieved in possibility theory.

In terms of uncertainty propagation, it is typically achieved for an orthorectangular uncertainty structure using interval propagation for each focal element, using the methods mentioned in section 1.5 for interval analysis.

Figure 1.12. Illustrations of (a) a BPA function m(x); (b) the corresponding cumulative plausibility function (CPlF) and cumulative belief function (CBF)

Evidence theory is based on the definition of a structure of uncertainties in the form of masses attributed to different focal elements. The definition of this uncertainty structure comes from experts. Analogously, as in the framework of possibility theory, rules have been defined in order to be able to aggregate different and sometimes divergent sources of uncertainty, for example, from different experts.

Let m1 and m2 denote two BPA functions with the respective sets Ci and Cj as focal elements. These functions can be aggregated according to the following rules:

[1.27]

In this rule, the conflict is simply averaged with a weighting based on the confidence attributed to each expert. Other rules, as well as variations around the three presented, have been proposed and the reader can refer to Sentz and Ferson (2002) for more information.

The quantification of epistemic uncertainties is a major challenge for achieving an integral framework for the quantification and propagation of uncertainties in industrial problems. Depending on the nature of the lack of knowledge, these uncertainties can be modeled in different ways. In some situations, epistemic uncertainties may be due to insufficient sampling, in which case the probabilistic framework, through the use of probability boxes (p-boxes), is probably the most relevant framework.

In other situations, no samples may be available and obtaining them would be too time consuming or costly, so only expert opinion can provide information on the uncertainties involved. Some consider that such advice cannot be objective and does not need to be mixed with rigorous, sample-based uncertainty quantification within a probabilistic framework. Others consider that, although imperfect, because it is subjective, expert opinions can contain useful information for uncertainty quantification and, rather than ignoring them, have focused on developing theories in order to be able to model the uncertainties expressed in this way. This is particularly the case with the theories of possibility and evidence that we have reviewed. Basically, both theories have the same objective, which is to bound an unknown probability distribution function by the construction of an uncertainty structure based on expert opinion. It is worth noting, however, that possibility theory is based on axioms which are quite different from those of probability theory, which can sometimes lead to counterintuitive results. These axioms also imply a relatively wide bounding for the unknown probability distribution function. Evidence theory, on the other hand, allows a narrower bounding for the distribution function, provided that it is possible to construct BPA functions with a very fine granularity. It also does not involve the more constraining axioms of possibility theory. To conclude, let us also note that in most real problems, epistemic and aleatory uncertainty modeling must be simultaneously considered. This combination of different approaches remains a challenge, especially from the point of view of the associated numerical costs.

Apostolakis, G. (1999). The distinction between aleatory and epistemic uncertainties is important: An example from the inclusion of aging effects into PSA. Proceedings of PSA ‘99, International Topical Meeting on Probabilistic Safety Assessment, 135–142.

Dempster, A.P. (1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat., 38, 325–339.

Der Kiureghian, A. and Ditlevsen, O. (2009). Aleatory or epistemic? Does it matter? Structural Safety, 31(2), 105–112.

Dubois, D. and Prade, H. (1992). Combination of fuzzy information in the framework of possibility theory. In Data Fusion in Robotics and Machine Intelligence, Della Riccia, G., Lenz, H.J. and Fruse, R. (eds). Academic Press, New York.

Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. The Annals of Mathematical Statistics, 27(3), 642–669.

Ferson, S., Joslyn, C.A., Helton, J.C., Oberkampf, W.L. and Sentz, K. (2004). Summary from the epistemic uncertainty workshop: Consensus amid diversity. Reliability Engineering and System Safety, 85, 355–369.

Ferson, S., Kreinovich, V., Grinzburg, L., Myers, D. and Sentz, K. (2015). Constructing probability boxes and Dempster–Shafer structures. Technical report, SAND-2015-4166J, Sandia National Lab, Albuquerque.

Hansen, E. and Walster, G.W. (2003). Global Optimization Using Interval Analysis: Revised and Expanded. CRC Press, Boca Raton.

Hoffman, F.O. and Hammonds, J.S. (1994). Propagation of uncertainty in risk assessments: The need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Analysis, 14(5), 707–712.

Klir, G.J. (2004). Generalized information theory: Aims, results, and open problems. Reliability Engineering and System Safety, 85, 355–369.

Kreinovich, V. and Xiang, G. (2008). Fast algorithms for computing statistics under interval uncertainty: An overview. In Interval/Probabilistic Uncertainty and Non-classical Logics, Huynh, V.N., Nakamori, Y., Ono, H., Lawry, J., Kreinovich, V., Nguyen, H.T. (eds). Springer, Berlin.

Lemaire, M. (2014). Mechanics and Uncertainty. ISTE Ltd, London and John Wiley & Sons, New York.

National Research Council (2009). Evaluation of Quantification of Margins and Uncertainties Methodology for Assessing and Certifying the Reliability of the Nuclear Stockpile. The National Academies Press, Washington, D.C.

Nikolaidis, E., Chen, S., Cudney, H., Hatftka, R.T. and Rosca, R. (2004). Comparison of probability and possibility for design against catastrophic failure under uncertainty. Journal of Mechanical Design, 126(3), 386–394.

Oberguggenberger, M., King, J. and Schmelzer, B. (2009). Classical and imprecise probability methods for sensitivity analysis in engineering: A case study. International Journal of Approximate Reasoning, 50(4), 680–693.

Roy, C.J. and Balch, M.S. (2012). A holistic approach to uncertainty quantification with application to supersonic nozzle thrust. International Journal for Uncertainty Quantification, 2(4), 363–381.

Sentz, K. and Ferson, S. (2002). Combination of evidence in Dempster–Shafer theory. Technical report, SAND2002-0835, Sandia National Lab, Albuquerque.

Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton University Press, Princeton. Silverman, B.W. (2018). Density Estimation for Statistics and Data Analysis. Routledge, New York.

Vose, D. (2008). Risk Analysis: A Quantitative Guide. John Wiley & Sons, New York. Wagner, R.L. (2003). Science, uncertainty and risk: The problem of complex phenomena. APS News, 1(12), 8.

Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London.

Walley, P. (2000). Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning, 24, 125–148.

Zhang, H., Mullen, R.L. and Muhanna, R.L. (2011). Structural analysis with probability-boxes. International Journal of Reliability and Safety, 6(1–3), 110–129.

Zimmermann, H.J. (2011). Fuzzy Set Theory and Its Applications. Springer, The Hague.