Many challenges are associated with the modelling of a complex system. One of the most prominent issues is probably the necessity to cope with the unavoidable uncertainty affecting both the model of the system and the collected data. Uncertainty has to be properly characterised and quantified to improve confidence in the model and its results. In general, uncertainty can affect the data and the model in several ways. For instance:
• noises can affect signals;
• privacy issues can lead to a partial censoring of the data;
• lack of data, small sample sizes, e.g. due to time-economic constraints; • low image/signal resolutions;
• tolerances imprecision and measurement tools limitations; • gaps in the data or missing points in the data;
• qualitative or subjective information; • limited numerical/computational precision; • linguistic vagueness;
• inherent variability;
• low-fidelity models (surrogates, reduced, simplified models); • conflicting evidence (e.g from sensors or experts).
In some situation, when data is scarce or at all available, expert elicitation (i.e. expert guessing) may be the only viable way of carrying on with the analysis.
In the last decades, generalised uncertainty quantification methods have been de- veloped to better cope with the combination of epistemic and aleatory uncertainty in a unified mathematical framework. In particular, characterise lack of data and im- precision (i.e. epistemic uncertainty) using only classical probability is particularly challenging as, for its treatment, strong artificial assumptions are often required (and are generally hardly justifiable). Generalised probabilistic methods have been recently developed to better cope with the uncertain quantification tasks when problems affected by a combination of epistemic and aleatory uncertainties are challenging the analyst. Generalised methods are generally non-intrusive and applicable to any computational model. These methods do not require (or require less) unwarranted, hard-to-justify as- sumptions compared to their classical probabilistic counterpart, thus better preserving the real information content in the data, and can quantify the effect of both aleatory and epistemic uncertainty without mixing them. Their capability to differentiate be- tween aleatory and epistemic uncertainty offer several advantages from a modeller and analyst perspective. First, this allows modellers to clearly defined which among the input factors uncertainties can be, theoretically, reduced (i.e. the one for which can be economically viable to collect more data). Secondly, the output of a generalised prob- abilistic framework points out if the uncertainty in the system performance is mainly due to lack of knowledge or inherent variability.
This is valuable information for decision makers which will have quantitative evidence to discuss:
1) A necessity to achieve higher precision in the results;
2) The inability/ability to take a decision based on the assessed uncertainty; 3) The need to invest in further data gathering;
4) On which uncertain factors focus the data collection; 5) Uncertainty reduction (precision-cost) trade-off;
Although those methods are powerful and flexible, thse find a limited application in real world industrial environments. This is possibly due to lack of clear guidance for practitioner, relatively new and not fully developed theory and to the few applications needed to display capability of the methods. Moreover, generalised probabilistic approaches are very intensive computationally speaking and, in many cases, this restricts their applicability.
Figure 1.1 presents the generalised framework for uncertainty quantification adopted in this work. Figure 1.1 presents some of the key components of the framework which can be summarised as follows:
1: High-Fidelity Computational Model: The high-fidelity model is a realistic representation of a system or a process. For the goals of uncertainty quantification it can be conveniently regarded as a black-box model. This can include a digital twin of the real physical system, but it can also be a combination of models (e.g. system-environment models coupled) or an entire work-flow, e.g. different pieces of software, physical experiments, framework for the reliability, vulnerability, re- silience assessment.
2: Uncertainty Characterisation (Epistemic): The epistemic uncertainty char- acterization goal is to model the uncertainty associated to features for which the information is limited, imprecise, vague, qualitative or affected by any mixture of lack of data and variability. Mathematical tools deriving from imprecise proba- bility theory (such as P-boxes, Possibility distributions, Credal Sets, Fuzzy Vari- ables or Dempster-Shafer structures) can be used for the modelling as described in Chapter 2.
3: Uncertainty Characterisation (Aleatory): Aleatory uncertainty can be effec- tively characterised by e.g. cumulative distribution function, probability masses, probability distributions etc. Classical probability theory is commonly used to characterise aleatory uncertainty.
4: Uncertainty Propagation (Aleatory): The propagation of aleatory uncer- tainty deals with the effect of inherent variability of model inputs to the quan- tities of interest in the model outputs. Aleatory uncertainty propagation can be performed using classical sampling methods, e.g. Monte Carlo (MC), Lathing Hy- percube Sampling (LHS), or if appropriate, more efficient methods such as subset simulation, line sampling, importance sampling, etc.
5: Uncertainty Propagation (Generalised): Generalised propagation of the un- certainties consists of a combined analysis of the effect of epistemic and aleatory uncertainty (without mixing them) on the model output quantities of interest. This is generally performed using time costly double loop methods and optimiza- tion methods.
6: Emulators/Surrogates: In case of time-consuming codes, a low-fidelity, compu- tationally cheap substitute of the high-fidelity model can be adopted. This model will act as an approximated model (also known as emulator or surrogate model) on top of which uncertainty quantification can be performed efficiently. Some
representative examples are the Response Surfaces models, Polynomial Chaos Ex- pansions, Gaussian Process Emulator (GPE), Artificial Neural Networks (ANN), Extreme Learning Machines.
7: Aleatory feedback loops: When new information is made available to the analyst (e.g. new information which comes from the propagation/quantification of aleatory uncertainty), it can be used to update and refine the model and produce better inferences on output quantities of interest. For instance, sensitivity analysis and dimensionality reduction can be performed on the aleatory space (e.g. by applying screening methods or variance-based sensitivity analysis) or Bayesian inference used to refine the model.
8: Epistemic feedback loop: Analogously to the aleatory feedback loop, the epis- temic feedback loop adopts the new information to improve the model but also to reduce the extent of the epistemic uncertainty by a better characterisation of the imprecise inputs. The uncertainty model can be update and refine in this phase, imprecisely defined parameters can be better specified and also sensitivity analysis can be performed (e.g. epistemic space pinching, Bayesian updating of the epistemic space, etc.).