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

6.2 One-Dimensional Introduction of the Gradual Inverse Algorithms

To clarify the distinctive features and mathematical properties of inverse algorithms, this section will introduce them within simplified one-dimensional formulations.

6.2.1 Direct Least Square Calibration with two Alternative Interpretations

Returning to the example provided at the beginning of the chapter:

(6.14)

(6.15)

where y, y_m and x are scalars, but d can be a vector and the observable system model H is not necessarily linear; u represents the model-measurement residual, encompassing potential measurement error and/or model imprecision. Suppose we have acquired a data sample of n measurements for given environmental conditions:

(6.16)

The easiest way to proceed is with a straightforward least-square calibration of the uncertain x, which means that calibration is assumed to equate to the selection of a proper point value for x:

(6.17)

In fact, if it is reasonable to assume that the uncertainty (or variability) ...