Chapter II
Extreme Events
23
To determine which distribution is most appropriate for our data, we fit both distributions
(Gumbel or Weibull) to our data and compare the goodness of fit. The goodness of fit can be
evaluated using graphical methods such as the probability plot.
Figure II-7 Representation of the possible distributions of the GEV method.
In order to graphically represent the real distribution of our wave height data, we use a script in
the python language of the seaborn library (Figure II-8)
Figure II-8 Graphical representation of the wave height distribution.
To proceed to the second method (POT), we will opt for the results obtained with the Gumbel
distribution. On the other hand, we neglect the results obtained with the Weibull distribution
because our data graph Figure II-8 is similar to the one of Gumbel in the Figure II-7.
II.2.2. Peaks over Threshold Method (POT)
Let í µí± 1 , í µí± 2 ,⋯,í µí± í µí± be a sequence of random variables that are independent and identically
distributed with some unknown distribution function í µí°¹. Then, the POT method can be
summarized as that we want to find the distribution of all excess observations that are greater
than some chosen threshold í µí±¢. Denote this distribution by í µí°¹ í µí±¢ (í µí±¥). Thus, we have:
Extreme Events
23
To determine which distribution is most appropriate for our data, we fit both distributions
(Gumbel or Weibull) to our data and compare the goodness of fit. The goodness of fit can be
evaluated using graphical methods such as the probability plot.
Figure II-7 Representation of the possible distributions of the GEV method.
In order to graphically represent the real distribution of our wave height data, we use a script in
the python language of the seaborn library (Figure II-8)
Figure II-8 Graphical representation of the wave height distribution.
To proceed to the second method (POT), we will opt for the results obtained with the Gumbel
distribution. On the other hand, we neglect the results obtained with the Weibull distribution
because our data graph Figure II-8 is similar to the one of Gumbel in the Figure II-7.
II.2.2. Peaks over Threshold Method (POT)
Let í µí± 1 , í µí± 2 ,⋯,í µí± í µí± be a sequence of random variables that are independent and identically
distributed with some unknown distribution function í µí°¹. Then, the POT method can be
summarized as that we want to find the distribution of all excess observations that are greater
than some chosen threshold í µí±¢. Denote this distribution by í µí°¹ í µí±¢ (í µí±¥). Thus, we have: