get_MPM

This function takes as input the identified maxima, and output the Most Probable Maximum.

The algorithm is as follow, according to DNV-OS-E301 Guidance Note (Dec 2024), Ch.2 Sec.2:

Fit a Weibull distribution to your dataset of maxima:
- Weibull cumulative distribution function:
  
  \[F(x; k, \lambda) = 1 - e^{-(x / \lambda)^k}\]
- Fit data to obtain: - shape parameter: ( k ) - scale parameter: ( lambda ) - location parameter (if using 3-parameter): ( theta ) [optional]
For maxima analysis according to extreme value theory, the logarithm of a Weibull random variable is Gumbel distributed:
- If ( X sim mathrm{Weibull}(lambda, k) ), then ( G = log(X) ) is Gumbel (minimum) distributed with: - Location parameter: ( mu = log(lambda) ) - Scale parameter: ( beta = 1/k ) - So ( G sim mathrm{Gumbel}_{min}(log(lambda), 1/k) ) [web:24][web:27]
For the transformation used in engineering standards (maxima conversion):
- Number of expected independent maxima:
  
  \[n = \frac{\text{total_duration}}{\text{average_period}}\]
- Use 3600 seconds as the reference duration (e.g., for 1 hour maxima count):
  
  \[n = \frac{3600}{T_\text{avg}}\]
- The Most Probable Maximum (MPM) for the Gumbel distribution: .. math:
```
\text{MPM} = \mu + \beta \cdot \ln(n)
```
  where:
  - ( mu ) is the location parameter (from Weibull fit)
  - ( beta ) is the scale parameter (from Weibull fit or conversion)
  - ( n ) is the expected number of maxima
- If Weibull fit gives shape ( k ), scale ( lambda ): .. math:
```
\mu = \log(\lambda)
\quad
\beta = 1/k
\quad
\text{MPM} = \log(\lambda) + \frac{1}{k} \cdot \ln(n)
```

References: - DNV-OS-E301 Guidance Note (Dec 2024), Ch.2 Sec.2