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

6.4 Specificities for Parameter Identification, Calibration or Data Assimilation Algorithms

As shown in the previous sections, parameter identification or calibration algorithms identify the ‘central value’ of the input variables, X being replaced by a fixed (although unknown) parameter x_m, representing, for instance, system properties in a mechanical model or initial conditions in a meteorological model. A review of the particular features will be given for those cases, expanding to the general domain of data assimilation and parameter identification, and the main existing algorithms.

6.4.1 The BLUE Algorithm for Linear gaussian Parameter Identification

Consider the simplest formulation where the system model is linear (or linearised), the uncertainty model is Gaussian and where data only is assimilated to estimate the value x_m in absence of any prior expertise available. The program to be estimated is the following:

(6.130)

The BLUE Estimator

The log-likelihood is the following quadratic function of x_m:

(6.131)

the maximisation of which enjoys the following closed-form solution (refer for a demonstration onto the general solution of a quadratic program given in Annex Section 10.5.2):

(6.132)

It may be seen as an algebraic n-dimensional generalisation of the estimator that was ...