Mathematical model

The mathematical model in observation theory is the combination of the functional and stochastic model. The functional model describes the “functional” relation between observables \(\underline{y}\) and unknown parameters \(x\). This may be a linear or non-linear relation:

Linear model: \(\underline{y}=Ax +\underline{e}\)   or   \(E\{\underline{y}\}= Ax\)

Non-linear model: \(\underline{y}=A(x) +\underline{e}\)   or   \(E\{\underline{y}\}= A(x)\)

We will mainly focus on linear models, and therefore in the following only the linear functional model will be considered.

The stochastic model describes the uncertainty or precision of the observables, and is therefore given by the covariance matrix of the observables:

\[D\{\underline{y}\} = Q_{yy}\]

Note that \(Q_{yy}=Q_{ee}\), since \(\underline{y}=Ax +\underline{e}\) and \(Ax\) is deterministic.

In summary, the mathematical model is given by:

Functional model        : \(E\{\underline{y}\}= Ax\)

Stochastic model        : \(D\{\underline{y}\} = Q_{yy}\)

The functional model is the first moment of the observables, i.e. the mean or expectation. The stochastic model is the second central moment of the observables, i.e. the dipersion being equal to the covariance matrix.