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.
Solution: The strongest linear dependence is between \(\gamma\) and \(w\), because \(|\rho_{\gamma,w}|=0.8495\) is larger than \(|\rho_{\gamma,PI}|=0.6420\) and \(|\rho_{w,PI}|=0.6837\). The relationship is negative because \(\rho_{\gamma,w}<0\). Thus, larger water contents tend to be associated with smaller unit weights in this dataset.
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.
Solution:
The diagonal entries are variances and the off-diagonal entries are pairwise covariances. The matrix is symmetric because \(\mathrm{Cov}(X,Y)=\mathrm{Cov}(Y,X)\).
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.
Solution: A correct answer should identify a visible feature in the plotted soil data that a Gaussian marginal does not reproduce well, such as skewness, a long tail, a sharp bound, or another departure from a symmetric bell shape. The key point is that Gaussian marginals are symmetric and bell-shaped.
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.
Solution: Yes. It should reproduce the overall negative dependence for \((\gamma,w)\) and \((\gamma,PI)\) and positive dependence for \((w,PI)\). However, Gaussian samples form regular elliptical clouds, so observed asymmetry, clustering, bounds, outliers, or non-elliptical dependence will not be represented exactly.
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.
Solution: For each generated triplet \((\gamma,w,PI)\), check whether all three conditions are satisfied. The probability is estimated as
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?
Solution: Repeat the probability calculation using increasing numbers of samples, for example \(N=100\), \(1{,}000\), \(10{,}000\), \(100{,}000\), and \(1{,}000{,}000\). Compare the estimated probabilities. If the estimate changes very little as N is further increased, the number of samples can be considered sufficient for the required accuracy.
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.
Solution: Yes. Under independence, the joint probability would be calculated as the product of the three marginal probabilities,
However, the variables are correlated. Since \(\gamma\) is negatively correlated with both \(w\) and \(PI\), while \(w\) and \(PI\) are positively correlated, lower \(\gamma\), higher \(w\), and higher \(PI\) tend to occur together. Ignoring these dependencies will therefore lead to a lower estimate of the joint probability.
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 \(\rho_{\gamma,w}\).
Solution: Higher. Because \(w=40\%\) is below its mean and \(\gamma\) and \(w\) are strongly negatively correlated, a below-average \(w\) tends to be associated with an above-average \(\gamma\). Thus \(\mu_{\gamma\mid w=40}>\mu_\gamma\).
b) Would the uncertainty in \(\gamma\) after conditioning on \(w=40\%\) be larger or smaller than its unconditional uncertainty? Explain briefly.
Solution: Smaller. Observing \(w\) provides information about \(\gamma\) because the variables are correlated. For a multivariate Gaussian model, the conditional variance is smaller than the unconditional variance whenever the correlation is non-zero. Therefore \(\sigma_{\gamma\mid w}<\sigma_\gamma\).
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.
Solution: The conditional distribution should be shifted to the right because \(\mu_{\gamma\mid w=40}>\mu_\gamma\), and it should be narrower because conditioning on \(w\) reduces uncertainty in \(\gamma\).
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.