Appendix 2
Regressions

Very often, we have a set of n measurements (x_i, y_i) of a parameter, y, in different x points. However, describing the parameter using n measurement points is generally not sufficient, especially if we wish to make estimates at points where no measurements are available.

To make it possible to have an estimate of y at every possible point defined by x, a mathematical model can be developed from the knowledge of the measurements available. This model is noted y = f(x) and is called an estimator. It allows us to calculate the y value of the considered parameter, whatever point x retains.

A2.1. Determining a linear estimator

By adopting a linear regression, the sought model is of the form:

where:

• – y^M is the y value estimated by the model for x; and
• – a and b are the model parameters. They are inferred from the n data points (x_i, y_i) so that the estimations y_i^M are as close as possible to the measurement. y_i; that is, the difference between the actual measurements and the estimations is as small as possible.

Indeed, for the available n measurements (x_i, y_i), the model error at point i is defined by the difference, at this point, between the measured value of y and the model’s value, y^M:

Coefficients a and b are determined by minimizing the sum, S, of squared errors:

or, substituting for ε(i):

and replacing y^M(X_i) by its expression:

By developing the squared expression, we obtain:

S is minimal for and :

Likewise: .

Hence: .

We can then infer the expression of a:

A2.2. Performance of the estimator

One way to give an idea about the performance of an estimator is to compute the sum of the squared errors obtained.

Recall that in the case of a linear regression, the model is:

with:

and:

Once the parameters, a and b, of the model are calculated, we can determine, at every measurement point defined by (x_i, y_i), the difference between the value of y estimated by the model and that obtained through measurements. This difference is given by:

The performance of the model is then measured by the sum of squared differences: