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This lesson is all about how to linearize the parametric functional models we’ve been looking at so far.

It takes you through: 1) why this matters; 2) what are the key equations I’m going to need for linearizing parametric models?; 3) deriving those key equations; and 4) an illustrative numerical example that makes things tangible.

By the time your done, you will have what you need to set up what you need for pretty much any parametric geospatial estimation problem on your own.

You can jump to different points in the video here:

  • 0:00 – Introduction
  • 3:41 – So what? Why this matters?
  • 10:57 – The punchline – The key equation for the linearized form
  • 14:11 – How do we linearize?
    • This portion of the video is about deriving and understanding the linearized form of the parametric model. You don’t need to be able to do this derivation but I find it really important in knowing what’s really happening when you use the linearized equations.
  • 35:01 – A recap of the result and how it’s almost like magic!
  • 40:40 – A numerical example to tie it all together
  • 55:04 – Taking that result back to an earlier example
  • 58:16 – You finally have enough to do some real geospatial estimation!

Please give me a sign!

In the video above and in the self-assessment questions, everything starts well (right derivation of A from the Taylor Series) and ends well (right numerical values in the sample solutions), but I managed to make a slip of the signs in between those.

You can check out the 2-page note with example linked here.

But, in short, it boils down to the fact that:

    \begin{align*} \underset{n\times u}{\mathbf{A} } & = - \left.\frac{ \partial{\mathbf{F}} }{\partial{\mathbf{x}}}\right|_0 \\ \end{align*}

for the parametric case, and not:

    \begin{align*} \underset{n\times u}{\mathbf{A} } & = + \left.\frac{ \partial{\mathbf{F}} }{\partial{\mathbf{x}}}\right|_0 \\ \end{align*}

for the parametric case.

(In case you’re interested, you can look here at how the same is not the case for the combined case, i.e. where \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) =\mathbf{0}. But that’s just shared here for fun, and not part of our course. Basing the derivation on that more general case is in part what nudged my sign error though …)

This summarizes things and provides an example.

Self-assessment problems

The following conceptual and applied self-assessment questions are meant to help you make sure you understand the material and be as efficient as possible in preparing for the quiz and related final exam questions.