Report

Report#

Complete this report by hand, using only the handwritten reference notes you prepared during the setup notebook. No laptop, internet, AI, or other resources may be used during the written hour.

Questions#

Question 1 — Method of moments#

1.1#

Briefly explain which distributions you fit to \(q\) and \(h\), and why.

1.2#

Consider a Gumbel distribution for the discharge \(q\), with probability density function

\[f_Q(q)=\frac{1}{\beta}\exp\left[-\frac{q-\mu}{\beta}-\exp\left(-\frac{q-\mu}{\beta}\right)\right],\qquad \beta>0.\]

For this distribution,

\[ E[Q] = \mu + \gamma\beta \]

and

\[ \text{Var}(Q) = \frac{\pi^2}{6}\beta^2, \]

where \(\gamma\approx0.5772\) is the Euler–Mascheroni constant.

Using the mean and standard deviation of \(q\) from your handwritten reference notes:

  1. Apply the method of moments by equating the theoretical mean and variance above to the sample mean and variance from your notes. Show your derivation.

  2. Compare your result with the fitted parameter values from the notebook.

  3. Briefly explain why the two estimation methods do not have to give exactly the same parameter values.

Question 2 — Goodness of fit#

During Section 4 of the setup notebook you assessed the fitted distributions for \(h\) and \(q\) graphically with the help of exceedance probability plots. Sketch these distributions qualitatively.

For each variable, qualitatively describe how well your function fits the data. How is the fit in the tails, and what might be the consequences of differences in the tail behaviour?

Support your answer using specific evidence from the graphical checks recorded in your handwritten notes.

Question 3 — Contingency probabilities#

Choose one of the two raw contingency tables in your handwritten reference notes, copy the raw data table into this report, and use the same table throughout this question. Clearly state which variable pair you selected.

3.1#

Convert the raw counts in your table into joint probabilities. Show your calculation for one cell.

3.2#

Determine the marginal probability distribution of each variable. Show your calculation for one marginal value.

3.3#

Choose one category of one of the variables. Among the observations in that category, determine the probability distribution of the other variable. Show the calculations of how you obtained the probabilities and explain in words what this conditional distribution represents.

3.4#

Select one cell from the conditional distribution in part (c). Now reverse the conditioning direction and calculate the corresponding conditional probability for that cell. Compare the two conditional probabilities and explain why they are generally different. Under what conditions would they be identical?

Question 4 — Cumulative probability#

Consider one of the two marginal distributions from the contingency table you used in Question 3.

Would constructing a cumulative distribution function (CDF) for this variable give a meaningful probability description?

Explain what property of a variable determines whether cumulative probabilities such as

\[ P(X\leq x) \]

have a useful interpretation.

Question 5 — Modelling assumptions#

Compare the two correlation coefficients in your handwritten reference notes, \(\rho_{obs}\) and \(\rho_{sim}\).

5.1#

What do these results tell you about the assumption made when \(h\) and \(q\) were sampled in the Monte Carlo simulation?

5.2#

Does your Monte Carlo simulation generate realistic samples for the simulation of \(u=\frac{q}{wh}\)? Justify your choice.

Last Question: How did things go? (Optional)#

Use this space to let us know if you ran into any challenges while working on this GA, and if you have any feedback to report.

End of file.

By Max Ramgraber, Patricia Mares Nasarre and Robert Lanzafame, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.