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

7.1 Classifying the Risk Measure Computational Issues

Chapter 2 listed the potential risk measures c_z as a series of functionals of the output uncertainty model, formally:

(7.1)

Such a functional may combine:

• for purely probabilistic risk measures:

  – integration involving the density functions f_X(.), π(.) as well as the predictive system model G(.) and ancillary operations such as computing an indicator function 1_{G(x, d)<zs}: it generally corresponds to computing an expected utility, that is expectation, exceedance probability and so on;

  – ancillary numerical operations may be needed for non-expectation based probabilistic measures: for example power or quotient when estimating the variance, a coefficient of variation or moments of the outputs Z; inverting an integral function when computing a quantile; smoothing a series of threshold probabilities to estimate the distribution and so on.

• for mixed deterministic-probabilistic risk measures:

  – any of the previous operations;

  – mixed with optimisation of the similar functions or intermediate integration results with respect to some penalised variables x^pn or or of the raw outputs along a time interval or space field when Z is defined as a maximum output.

Note also that a decision criterion associated with a risk measure also ...