Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods

7.4 Monotony, Regularity and Robust Risk Measure Computation

Regularity of the system model is an important property in order to improve the risk measure computation: earlier sections have shown the simplifications brought by linearity. Monotony will prove to help greatly for robust accelerated computation of both probabilistic and mixed deterministic-probabilistic risk measures.

Beyond, monotony, it is expected that other forms of regularity such as boundedness of the system model or its derivatives could also help: a few perspectives will be outlined at the end of the section.

7.4.1 Simple Examples of Monotonous Behaviours

Monotony was defined in Chapter 4 through a simple property relating to the monotonous behaviour of component-wise input-output functions constituting the system model. As commented upon in Annex Section 10.5.4, such a definition can easily be generalised to:

• models involving discrete event inputs, not only real-valued;
• local behaviour, whereby monotony is valid for known subsets of the domain of definition (or range of uncertainty) of the inputs.

An intuitive and quite useful property (also described in Annex Section 10.5.4) is the persistence of monotony through the chaining of sub-system models as well as through the iso-probabilistic transformation of input spaces, as was encountered with the FORM method (Section 7.3.5), or the copula functions (Chapter 5, Section 5.3.3).

Taking advantage of such model chaining considerations, consider the traditional ...