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

5.1 The General Issue

The issues developed in the present chapter concern the estimation of the uncertainty model in the case where there are direct observations available. This means that there is no need for the inversion of the observational system model Π_ooM_s(.) derived in the general estimation program in Chapter 2, as shown in Figure 5.1 and Equation 5.1.

(5.1)

Figure 5.1 General model estimation program.

When model inputs are directly observable, as represented in Figure 5.2, that is Y_mj = X_j where X may represent any sub-vector of the uncertain input components, variable or discrete event (X, E) in full notation), the estimation program becomes much simpler. The simplification may even be extended if direct observations of the outputs of interest for decision-making are directly available, that is Y_m = X = Z. In that most favourable case, z = G(x,d) = x is the (degenerated) system model and the estimation of the risk measure is restricted to a pure statistical issue. Putting aside this ultimate simplification, which impacts only the subsequent propagation phase, the generic statistical program for directly observable inputs is formulated as the following:

(5.2)

Figure 5.2 ...