Generalized Extreme Value Distribution

This is a location-scale family distribution.

Parameters¶

Probability Density Function¶

This is denoted as the probability that our rv $Y$ will be equivalent to some specific value

We can define the probability density function

where the function $t(y;\boldsymbol{\theta})$ is defined as:

From

Cumulative Distribution Function¶

This is denoted as the probability that our rv $Y$ will be less than or equal to some specific value $y$.

We can define the cumulative density function

where the function $t(y;\boldsymbol{\theta})$ is defined in equation (4).

Survival Function¶

This is the probability that our value of interest $y$ is less than ...

We denote this as:

We can plug in the CDF function into this equation

where the function $t(y;\boldsymbol{\theta})$ is defined in equation (4).

Quantile Function¶

This is also known as the Point-Percentile-Function or the inverse CDF. This function maps an input threshold, $y_0$, to a value $y$ st the probability of $Y$ being less than or equal to $y$ is $y_p$.

We can take the inverse of this function to see that it is the inverse CDF which we denote as the quantile function.

where $y_p\in[0,1]$ is the data within the probability transform domain. These can be computed in closed form

# function for kappa > 0
def quantile(, loc, scale, shape):
    level = loc - scale / shape * (1 - (- log(1 - p)) ** (- shape))
    return level

# function for kappa = 0
def quantile(p, loc, scale):
    level = loc - scale * log(- log())
    return level

Return Period¶

We can calculate the RP using equation (8). Practically, we set this to the survival function of the GEVD (equation (6)).

To make things simpler, we can simply use the quantile function in equation (12) and set the probability to

However, if we expand this out, we get

Average Recurrence Interval¶

We can calculate the ARI using equation (14). Practically, we set this to the survival function of the GEVD (equation (6)).

To make things simpler, we can simply use the quantile function in equation (12) and set the probability to

However, if we expand this out and simplify, we get

Joint Distribution¶

We can write the likelihood that the observations, $y$, follow the GEVD distribution. So, given some observations, $\mathcal{D}=\{y_n\}_{n=1}^{N}$, which we believe follow the GEVD distribution, we can write the joint distribution decomposition as

This implies that the global prior parameters come from some distribution

and that these parameters get passed through our data likelihood term

Log Probability¶

Recall the PDF for our iid samples is

where $t(y_n;\boldsymbol{\theta})$ is defined in equation (4). We can add the log term to get

which we can expand as

which reduces to

def gev_logpdf(x, location, scale, shape):
    # calculate location scale: z=(y−μ​)/σ
    z = (x - mu) / sigma
    # calculate t(z) = 1+κz
    t = 1.0 + shape * z
    # grab max value
    t = np.max(t, 0)
    # term 1: −log σ
    t1 = - np.log(sigma)
    # term 2: − (1+κz) ** −1/κ
    t2 = - np.power(t, -1.0 / xi)
    # term 3: - (1+1/κ)log(1+κz)
    t3 = - (1.0 / xi + 1.0) * np.log(t) 
    return  t1 + t2 + t3

y: Array["T"] = ...
params: PyTree = ...
# apply vectorized operation
nll: Array["T"] = vectorize(gev_logpdf, y, params)
# take the sume
nll: Scalar = sum(nll)

Reparameterization¶

In this instance, we are assuming that there is a threshold parameter, $y_0$. We can write the reparameterization of this distribution as

Rescaling¶

where $h$ is in years and $h^*$ is in days. We can write all of the parameters with these rescaled ones

Literature Review¶

Theory. Leadbetter et al. (1983) is another very popular book.

Applications. García et al. (2023) investigated annual temperature extremes in Extremadura, Spain where they applied a Gaussian process to account for spatial dependencies for the GEVD. Räty et al. (2022) looked at sea level extremes in the Finnish coastal region where they applied a Bayesian hierarchical model to account for spatial dependencies for the GEVD.

Algorithms. Moins et al. (2023) look at a reparameterization framework to improve the convergence of the MCMC inference algorithms. Koh et al. (2021) investigate spatiotemporal extremes of daily large wildfires in the French Mediterranean where they employ a Bayesian Hierarchical model for a PP for the events and a GPD for the marks.