1.4.2 Interactive Exercises

Exercise 1

For a situation like in this illustration

Figure 1.5: Example of an aircraft and 4 beacons.

you can change the positions of the beacons, their accuracy and their measurement-values. As a result you'll get the estimated airplane position and a plot of the resulting confidence region. A cross (+) in the resulting plot marks the approximated best airplane position obtained by the least-squares method. The confidence region corresponding to the first specified confidence level is depicted as red ellipse, the second specified level is shown as blue ellipse, the third as cyan (bluegreen) ellipse and the fourth as magenta (purple). If more different confidence levels are specified these colours are repeated. Be aware, that some confidence regions can be empty, depending on the quality of the estimation. (Compare exercise 3 in the online version !)

Exercise 2: Simulating measurements

Starting with a known (true) airplane position, you can now simulate measurements by pressing the "Simulate measurement"-button! To do this:

• Type in a true airplane position below.
• If you like, you can change the VOR and DME positions, and the degree of accuracy of the measurement devices (given by their standard deviations) in the table above.
• Each time, you press "Simulate measurement", you get new "measured values" with random errors according to the specified standard deviation of each device. These values are shown in the above table.

In the resulting plot of the confidence regions, the true airplane position is marked with a green diamond, the estimated airplane position with a red cross.

Exercise 3

1. Choose a confidence level, e.g. 0.5. Press "Simulate measurements" repeatedly (10 times, e.g.) and count the hits, i.e. how many times the confidence regions actually contain the green diamond, marking the true airplane position. Compare this with the expected value of hits!
2. How large is the probability of having exactly 5 hits out of 10 trials? How large is the probability of having 4 to 6 hits? If you want to use Maple and don't have it on your computer, try the Maple front-end.
3. Choose a low confidence level, e.g. 0.1, and then press "Simulate measurements" repeatedly. Count how may times no confidence region is drawn. Determine the probability of getting an empty confidence region, assuming that the distribution of S0 is a chi-square distribution with two degrees of freedom, whereas the confidence-region is defined using a chi-square distribution with four degrees of freedom.

If the interactive exercise fails to work correctly, please send us a bug report containing OS, browser version and detailed bug description. Please specify the page exactly. If possible, also send the output on the Java Console of your browser. (Ruhig auch auf Deutsch!)