Extreme Value Theory and Floods

Question

Let's start with an example problem case. Say we measure two variables that are non-normally distributed and correlated. For example, we look at various rivers and for every river we look at the maximum level of that river over a certain time-period. In addition, we also count how many months each river caused flooding. For the probability distribution of the maximum level of the river we can look to Extreme Value Theory which tells us that maximums are Gumbel distributed. How many times flooding occurred will be modeled according to a Beta distribution which just tells us the probability of flooding to occur as a function of how many times flooding vs non-flooding occurred.

Source: https://twiecki.io/blog/2018/05/03/copulas/

Does anyone know why the maximums are Gumbel distributed? Shouldn't their distribution depend on the data itself?

Does anyone know why the number of floods are modeled by a beta distribution? Again, shouldn't their distribution depend on the data itself (e.g. maybe normally distributed)?

Answer 1

You have two questions so I'll answer them in turn

(1) Why are the maxima Gumbel distributed?

Essentially, this comes from the extremal types theorem which is a limiting distribution for block maxima. The most common limits distribution is the CLT which states

$$ \frac{\bar{X} - \mu}{\sigma} \to N(0,1) \text{ as } n \to \infty$$.

Now suppose we have data index by say time, and want to analyse the largest possible values the data could take. This could be amount of flooding, measured by volume of water spilling out of a river bank every day. In the $k$th year, having $N$ observations, we denote the block maxima as $M_k = \max \{y_1, y_2, \ldots, y_N \}$. You can think of these as e.g. annual maxima. Then the extremal types theorem says, for suitable sequences of constants, $a_k$, $b_k$ that

$$P\left(\frac{M_k - a_k}{b_k} \leq x\right) \to G(x)$$ where $G$ has one of three forms. The three forms are

Gumbel: $G(x) = \exp\{ -\exp(-x)\}$

Frechet: $G(x) = \exp(-x^{-\alpha}) \text{ with } (x, \alpha > 0)$

Weibull: $G(x) = \exp\{ - (-x)^{\alpha} \}\text{ with } (x<0, \alpha > 0)$

and for $x>0$ the Weibull form takes the value $1$.

Note that, like the CLT, this result hold regardless of the distribution of the $y$ values. Although, the particular form might depend on the $y$ values. For instance, if the data are positive (daily rainfall counts) the Frechet would be a sensible form, whereas the Weibull would not support this type of observation.

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(2) Why are the number of floods modelled by a Beta distribution?

The article mentions that they are going to model this by modelling the probability of a flood. That is $P(\text{flood}) = \frac{\text{num. floods}}{\text{num. obs}} \in [0,1]$. The Beta distribution has support on $(0,1)$ which is well suited to probabilities as they lie in $[0,1]$. If we modelled the probability by a Normal distribution, we might end up making statement like $P(\text{flood}) = -0.3$ which is not a valid statement. The Beta distribution will always give a value in $(0,1)$ as the answer. Thus the statements are always, in some sense, valid.