Building your practice in functional modeling

In previous examples and mini course (such as our introduction to math models for geomatics networks), we kept things simpler than they are in reality.

For example, recall our simple width of the room example that helped us express an error and a residual as follows for the i^{th} measurement in a survey:

    \begin{equation*} e_i = l_i - x \end{equation*}

and

    \begin{equation*} r_i = l_i - \hat{x} \end{equation*}

These made sense as the difference between the measurements and the actual and estimated parameters, respectively, but toward the end of the same example we went further and clarified that it’s a rare case where we would be able to directly measure the parameter of interest.

To deal with that we represented the mapping from parameter space to measurement space by some nominal function f() and used that write the above equations as follows:

    \begin{equation*} e_i = l_i - f(x) \end{equation*}

and

    \begin{equation*} r_i = l_i - f(\hat{x}) \end{equation*}

Conceptually this kind of works. But it’s not enough for our work in geomatics networks.

We need to be able to more rigorously represent the relationship between the parameters of interest and the measurements accessible to us in a survey. This is called functional modeling.

We’re going to represent the relationship between our parameters and our measurements as follows:

    \begin{equation*} \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0} \end{equation*}

I want you to deeply understand what functional modeling is all about, and I want you to build a practice in it. That is the subject of this mini course.