Self-assessment for: Parametric least squares estimation equations

The following are some recommended self-assessment questions for the lesson called Parametric Least Squares Estimation (the solution).

Question 1

If you stand pretty still on the street corner and open the mapping application on your smart phone, what it does can be thought of pretty much as a parametric least squares adjustment. When it first begins to estimate your location, how / from where do you think it gets reasonable values of the following?

a) the vector x_0

b) the variance-covariance matrix \mathbf{C}_{\mathbf{l}}

For this question I’m not assuming you know much about how the algorithms on such a phone work. Rather, I want you to use your common sense – based in part on your experience in the course you took on least squares estimation – to demonstrate that you know what x_0 and \mathbf{C}_{\mathbf{l}} are and that you understand what they mean.

Question 2

In the simple example given above, your phone is going to be making some measurements, as we discussed in class.

a) Between what and what would be the measurements be made?

b) Which of the functional models we saw in this topic would apply?

Question 3

Assume there are four GPS satellites in the sky at the time of the measurements in the above example, and that you only need to solve for the three components of your unknown position, e.g. x, y, and z in a 3D earth-centered cartesian frame. What will be the design matrix \mathbf{A}?

Question 4

When you turn on the mapping application in your phone, it will show a blue circle as in the image below (which I borrowed from here).

Does this correspond more closely to the information in \mathbf{C}_{\hat{\mathbf{x}}}, \mathbf{C}_{\hat{\mathbf{l}}}, \mathbf{C}_{\hat{\mathbf{r}}}, or \mathbf{C}_{\mathbf{w}}? Justify your answer.

Question 5

Again in the same example:

a) What form will the variance-covariance matrix \mathbf{C}_\mathbf{l} take if the measurements are assumed to be uncorrelated? You do not need to write out the matrix here. Just tell me what form it will have.

b) Will the variance-covariance matrix \mathbf{C}_{\hat{\mathbf{x}}} take the same form?

Question 6

Assume that a least squares adjustment is carried out in the above example to find your position, using the equations provided in the topic. And assume that everything looks great until you compute \hat{\sigma}_0^2 and find it to be different at statistically significantly level from {\sigma}_0^2.

a) What does this mean?

b) What could you do about it?

The following schedules are for Alex’s in-class students:

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