While working on a investment planning model to combat damages resulting from flooding, I received the results from a rainfall model that calculates damages as a result of excessive rain and flooding. The picture the engineers produced looks like:
This picture has the scenario on the x-axis (sorted by damage) and the damages on the y-axis. This picture is very much like a load-duration curve in power generation.
For a more “statistical” picture we can use standard histogram (after binning the data):
Gamma distribution
We can use standard techniques to fit a distribution. When considering a Gamma distribution (1), a simple approach is the method of moments. The mean and the variance of the Gamma distribution with parameters \(k\) and \(\theta\) are given by:
| \[\begin{align} &E[X] = k \cdot \theta\\ & Var[X] = k \cdot \theta^2 \end{align} \] |
Using sample mean \(\mu\) and standard deviation \(\sigma\), we can solve:
| \[\begin{align} & k \cdot \theta = \mu\\ & \sqrt{k} \cdot \theta = \sigma \end{align} \] |
This can be easily solved numerically and it actually seems to work:
Weibull distribution
An alternative distribution that is sometimes suggested is the Weibull distribution (2). The method of moments estimator for the Weibull distribution with parameters \(\lambda\) and \(k\) can be found by solving the system:
| \[\begin{align} & \lambda \Gamma(1+1/k) = \mu\\ &\lambda \sqrt{\Gamma(1+2/k)+\left(\Gamma(1+1/k)\right)^2} = \sigma \end{align}\] |
An alternative approach would be to use an MLE (Maximum Likelihood Estimation) technique. This yields a system of equations:
| \[\begin{align} & \lambda^k = \frac{1}{n}\sum_{i=1}^n x_i^k\\ &k^{-1} = \frac{\sum_{i=1}^n x_i^k \ln x_i}{\sum_{i=1}^n x_i^k} – \frac{1}{n}\sum_{i=1}^n \ln x_i \end{align}\] |
This gives is a very similar picture:
References
- Gamma distribution, https://en.wikipedia.org/wiki/Gamma_distribution
- Weibull distribution, https://en.wikipedia.org/wiki/Weibull_distribution
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