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

Notation

Mathematical quantities included in the text are always written in italic.

x = (xⁱ)_{i=1. . .p}: Vector (of dimension p) of uncertain model inputs representing sources of uncertainties, uncertain variables or events

e = (eⁱ)_{i=1. . .p′}: Vector (of dimension p′) of uncertain model inputs representing uncertain events

d: Vector (of dimension n_d) of fixed model inputs representing design or decision variables or conditional scenarios

d°: Vector of fixed model inputs taken at their reference value in a comparative study

z = (z^l)_{l=1. . .r}: Vector (of dimension r) of the model output variables or events of interest

y = (y^k)_{k=1. . .q}: Vector (of dimension q) of model observable output variables

u, u_mod, u_mes: Vectors (of dimension q) of model error or more generally measurement-model deviations: u represents the total deviation, potentially broken down into u_mes a metrological error component and u_mod, a model error component: u = u_mod + u_mes

x_t, e_t, y_t, z_t,_{. . .}: Vector of time-dependent variables that is vector time series. The time series nature is left implicit in cases where it does not matter much, thus omitting the subscript t in simplified notation x, e, y, z,_{. . .}

y_m = (y_m^k)_{k=1. . .q}: Vector (of dimension q) of measurements on observable outputs, so that y_m = y + u

G(.) or G(.): Deterministic function (generally very complex) representing the system model linking uncertain or fixed input vectors to the vector of output variables of interest: z = G(x, d). Notation will ...