Assessment Progress:

The course navigation elements this site work best when you’re signed in »

Welcome (back)!

All of the Creative Commons tools on this site are free. We just ask that you sign in / sign up using one of your existing social accounts:

This will take you to their secure sign in tools - we'll never see your password and you don't have to create and remember yet another one.

That said, we will receive your name and email address from them as well as any other profile information you approve for them to share.

And, by signing in, you are agreeing to our full terms of use (including privacy policy and license agreement) found here.


The following are some recommended practical self-assessment questions for the lesson called A Quick Practical Introduction to Parametric Least Squares Estimation. They’re intended for you to work through to test your own knowledge of the key concepts we covered there.

Question 1

Reproduce for yourself the full solution that I showed in the video found here, i.e. solve the drone location problem for yourself in Excel.

With the measurements and key equations already in hand, i.e. \mathbf{A} and \mathbf{w}, you should be able to implement something like this for yourself in Excel in 5 to 8 minutes. This means building the variance covariance matrix for the observations, \mathbf{C}_{\mathbf{l}}, and then doing the matrix calculations as shown in the video to implement the following.

1. Calculating the variance-covariance matrix of the estimated unknown parameters:

    \begin{equation*} \underset{u\times u}{\mathbf{C}_{\hat{\mathbf{x}}}} =\mathbf{C}_{\hat{\boldsymbol{\delta}}} =(\mathbf{A}^\mathsf{T}\mathbf{C}_{\mathbf{l}}^{-1}\mathbf{A})^{-1} \end{equation*}

2. And then computing the estimated parameters themselves:

    \begin{equation*} \underset{u\times 1}{\hat{\mathbf{x}}}=\mathbf{x}_0+\hat{\boldsymbol{\delta}} \end{equation*}


    \begin{equation*} \underset{u\times 1}{\hat{\boldsymbol{\delta}}} = -(\mathbf{A}^\mathsf{T}\mathbf{C}_{\mathbf{l}}^{-1}\mathbf{A})^{-1}\mathbf{A}^\mathsf{T}\mathbf{C}_{\mathbf{l}}^{-1}\mathbf{w} \end{equation*}

3. And then the vector containing the estimated residuals:

    \begin{align*} \underset{n\times 1}{\hat{\mathbf{r}}} &= \mathbf{A}\hat{\boldsymbol{\delta}} + \mathbf{w} \end{align*}

If you’re in my class then I want to see your solution on one tab of the spreadsheet you hand in. Show the final estimates of the coordinates and the associated uncertainty, as well as the work you did to get them.

If you’re one of my students, then you’re expected to answer this on your own and submit according to the directions provided in class, i.e. you don’t  need to submit them through this website. Don’t forget that our TA and I are both here to help you in the associated lab (and/or tutorial) sessions.

Aim to provide succinct answers to these applications questions in the same document you created for the conceptual self-assessment questions.

I’d also like you to submit an Excel spreadsheet along with that document. This will likely require a bit of thoughtful organization on your part. For example, It would work well to put the answers to both of the above application problems onto their own sheet / tab within your spreadsheet, and name it accordingly. And then you can refer to that specific tab from your written document.

When you answer other applied question sets in future topics, you should do the calculations on separate sheets / tabs, also appropriately named. And refer to them where required to show your work. This way, you’ll have all of your Excel work in one nicely indexed place and will only need to hand in a single spreadsheet as an appendix to each set of self-assessments.

You can click through to other self-assessments or lessons (if any) using the button below, and return here whenever you wish.