Q4) I have a time series on the maximum daily rainfall in (mm) per year, over 30 years from a weather station in a forest. What distribution does this rainfall maxima likely follow?
Gumbel/Log Weibull
The distribution of maximum values in datasets often aligns with the Extreme Value Distribution. For block maxima, like maximum daily rainfall in a year, the Gumbel distribution is typically used. This is due to the Fisher-Tippett-Gnedenko theorem, which suggests that block maxima, under certain conditions, will converge to one of three extreme value distributions, with Gumbel being the one for maxima (known as the ‘Type I EV’)
To give some context: if you recorded the highest daily rainfall each year and plotted these values, the resulting shape would likely resemble the Gumbel distribution. This is because the Gumbel captures the tail behavior of underlying distributions.
However, in practice, I would validate the distribution fit against actual data using methods like goodness-of-fit tests or QQ plots.*
More about the Gumbel
A beautiful cumulative probability distribution function: In Machine Learning: The Gumbel-Softmax trick is used for sampling from a categorical distribution. This method utilizes the Gumbel distribution to produce differentiable approximations to the categorical distribution, which is useful in the context of gradient-based optimization.
* Don’t worry, we cover these.
Extras ###### Q3) **What distribution models data that has a peak at the center and heavy tails, especially in physics and finance?** `Cauchy`
The Cauchy distribution is known for its heavy tails. It does not have a defined mean or variance.