Explain current validation tests.

pelicun/tests/validation/0/readme.md

@@ -0,0 +1,4 @@
+# Loss function validation test
+
+In this example, a single loss function is defined as a 1:1 mapping of the input EDP.
+This means that the resulting loss distribution will be the same as the EDP distribution, allowing us to test and confirm that this is what happens.

pelicun/tests/validation/0/test_loss_function.py

@@ -41,9 +41,10 @@
 """
 Validation test on loss functions.
 
-In this example, the loss function is a 1:1 mapping of the input EDP.
-This means that the resulting loss distribution will be the same as
-the EDP input distribution.
+In this example, a single loss function is defined as a 1:1 mapping of
+the input EDP.  This means that the resulting loss distribution will
+be the same as the EDP distribution, allowing us to test and confirm
+that this is what happens.
 
 """
 

pelicun/tests/validation/1/readme.md

@@ -0,0 +1,20 @@
+# Damage state probability validation test
+
+Here we test whether we get the correct damage state probabilities for a single component with two damage states.
+For such a component, assuming the EDP demand and the fragility curve capacities are all lognormal, there is a closed-form solution for the probability of each damage state.
+We utilize those equations to ensure that the probabilities obtained from our Monte-Carlo sample are in line with our expectations.
+
+## Equations
+
+If $\mathrm{Y} \sim \textrm{LogNormal}(\delta, \beta)$, then  $\mathrm{X} = \log(\mathrm{Y}) \sim \textrm{Normal}(\mu, \sigma)$ with $\mu = \log(\delta)$ and $\sigma = \beta$.
+
+```math
+\begin{align*}
+\mathrm{P}(\mathrm{DS}=0) &= 1 - \Phi\left(\frac{\log(\delta_D) - \log(\delta_{C1})}{\sqrt{\beta_{D}^2 + \beta_{C1}^2}}\right), \\
+\mathrm{P}(\mathrm{DS}=1) &= \Phi\left(\frac{\log(\delta_D) - \log(\delta_{C1})}{\sqrt{\beta_D^2 + \beta_{C1}^2}}\right) - \Phi\left(\frac{\log(\delta_{D}) - \log(\delta_{C2})}{\sqrt{\beta_D^2 + \beta_{C2}^2}}\right), \\
+\mathrm{P}(\mathrm{DS}=2) &= \Phi\left(\frac{\log(\delta_D) - \log(\delta_{C2})}{\sqrt{\beta_D^2 + \beta_{C2}^2}}\right), \\
+\end{align*}
+```
+where $\Phi$ is the cumulative distribution function of the standard normal distribution, $\delta_{C1}$, $\delta_{C2}$, $\beta_{C1}$, $\beta_{C2}$ are the medians and dispersions of the fragility curve capacities, and $\delta_{D}$, $\beta_{D}$ is the median and dispersion of the EDP demand.
+
+The equations inherently asume that the capacity RVs for the damage states are perfectly correlated, which is the case for sequential damage states.