Report#
Answer the following questions based on the analyses performed in the notebook. Explain your reasoning clearly. No Python code is required.
Background and key results#
Soil properties vary naturally because of differences in composition, depositional history, stress history, and water content. In this exercise, three properties are analysed jointly: unit weight \(\gamma\) (\(\mathrm{kN/m^3}\)), water content \(w\) (%), and plasticity index \(PI\) (%).
You may use the following results obtained in the notebook (with \(\mu\) for mean, \(\sigma\) for standard deviation and \(\rho\) for the correlation coefficient):
\(\mu_{\gamma} = 15.243\ \mathrm{kN/m^3}\), \(\sigma_{\gamma} = 1.603\ \mathrm{kN/m^3}\)
\(\mu_w = 78.946\%\), \(\sigma_w = 42.304\%\)
\(\mu_{PI} = 38.050\%\), \(\sigma_{PI} = 21.733\%\)
\(\rho_{\gamma,w}=-0.8495\)
\(\rho_{\gamma,PI}=-0.6420\)
\(\rho_{w,PI}=0.6837\)
Question 1#
The variables \(\gamma\), \(w\), and \(PI\) are not independent.
a) Which pair has the strongest linear dependence? State whether the relationship is positive or negative, and explain how you can tell from the correlation coefficients.
b) What information does a covariance matrix contain? Sketch its structure for \(\gamma\), \(w\), and \(PI\) (write out the matrix with the variables, no need for numerical values). Indicate where variances and covariances appear and why the matrix is symmetric.
Question 2#
The notebook compares the observations with samples generated from the fitted multivariate Gaussian distribution.
a) Examine the marginal distributions along the diagonal. Describe one important difference between an observed marginal distribution and the corresponding Gaussian model.
b) Examine the scatter plots. Does the Gaussian model reproduce the general direction of dependence? Identify one feature of an observed joint distribution that is not well represented by the Gaussian model.
Question 3#
Consider the following event:
a) Using samples generated from the fitted multivariate Gaussian distribution, explain how you would estimate the probability that all three conditions are satisfied simultaneously.
b) The estimated probability depends on the number of generated samples. How could you check whether the number of samples used is sufficient to obtain a stable probability estimate?
c) Suppose \(\gamma\), \(w\), and \(PI\) were incorrectly treated as independent. Would you expect the estimated probability of satisfying all three conditions to be different? Explain qualitatively.
Question 4#
In the notebook, you investigated the conditional distribution given \(w=120\%\), which is above the mean water content. Now consider a soil sample for which
which is below the mean value \(\mu_w=78.946\%\).
Without repeating the full conditional Gaussian calculation, answer the following questions.
a) Compared with the unconditional mean of \(\gamma\), would you expect the conditional mean of \(\gamma\) given \(w=40\%\) to be higher or lower? Explain using \(\rho_{\gamma,w}\).
b) Would the uncertainty in \(\gamma\) after conditioning on \(w=40\%\) be larger or smaller than its unconditional uncertainty? Explain briefly.
c) Sketch the unconditional marginal PDF of \(\gamma\) and the conditional marginal PDF of \(\gamma\) given \(w=40\%\) on the same axes. Show the relative positions of their means and their relative spreads. Exact values are not required.
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.
By Zheng Guan, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.