Now let’s step back and look at what we just did in the lesson Some examples of functional modeling, and see if we can draw some conclusions.

The following video provides a synopsis and ties things together while underscoring a few important points.

And the following table summarizes what we found through the four examples we’ve studied so far, and through the various forms of functional model we ended up with. In the rightmost column I’ve also labeled the type of model / equations corresponding to each example.

Example? Row of \mathbf{F}(\mathbf{x},\mathbf{l}_{true})? Linear? General form? Example of?
Example 2 l_{true} - \sqrt{(E_B-E_A)^2 + (N_B-N_A)^2} = 0 no \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) =\mathbf{0} observation equations (parametric model)
Example 3 \alpha_{AB} - tan^{-1}(\dfrac{E_B-E_A}{N_B-N_A}) = 0 no \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) =\mathbf{0} observation equations (parametric model)
Example 4 N_i - mE_i - b = 0 no \mathbf{F}(\mathbf{x}, \mathbf{l}_{true}) =\mathbf{0} combined equations (combined or implicit model)
Example 5 \alpha_A + \alpha_B + \alpha_C - 180^o = 0 yes \mathbf{F}(\mathbf{l}_{true}) =\mathbf{0} condition equations (condition model)

In the geodetic sciences, we use functional models to relate the true or actual values of the observed quantities, \mathbf{l}_{true}, to the unknown parameters or observables, \mathbf{x}.

The general form can be classified into three types:

  • A combined model of form \mathbf{F}(\mathbf{x}, \mathbf{l}_{true}) =\mathbf{0} that relates observations and parameters through an implicit function (also called implicit model)
  • A parametric model of form \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) =\mathbf{0} that expresses the observations as functions of the parameters using so-called observation equations
  • A condition model of form \mathbf{F}(\mathbf{l}_{true}) =\mathbf{0} that formulates conditions between the observed quantities

It is worth noting that:

  • The choice of model is made based on the nature of the problem at hand, with considerations about how easily the equations can be formulated and solved
  • The parametric and condition models are special cases of the combined model
  • The parametric model tends to be favored in geomatics networks because the observation equations are more easily formed and solved in practice
  • A combined model can be converted into a parametric model using a set of pseudo-observations, and the solution will be the same either way
  • You can have combinations of the above models, depending on the problem at hand
  • A model is:
    • Indirect (in \mathbf{x}) when of the form \mathbf{l}_{true} = \mathbf{F}(\mathbf{x})
    • Direct(in \mathbf{x}) when of the form \mathbf{x} = \mathbf{F}(\mathbf{l}_{true})

In the next mini course we will look at the topic of linearizing functional models because, as you will see, most of those we deal with are non-linear and therefore not useful in a practical estimation process.