1. Distribution

1.1. Purpose

Fit Weibull, Rayleigh or Gumbel distributions to data and estimate responses based on the fitted probability distributions. See here for description of how to set this up in the post processor or workflow.

1.2. Description

1.2.1. Fitting of Rayleigh distribution

The cumulative distribution function for a Rayleigh distribution is

\[F(x)=1−\exp(−\frac{(x-\lambda)^2}{2\sigma^2}), \quad x \geq \lambda,\]

where \(\lambda\) is the location parameter and \(\sigma\) is the scale parameter. A maximum likelihood estimator is used to find the parameters.

1.2.2. Fitting of Weibull distribution

The cumulative distribution function for a 3-parameter Weibull distribution is

\[F(x)=1−\exp(−(\frac{x-\alpha}{\beta})^\gamma ), \quad x \geq \alpha,\]

where \(\alpha\) is location, \(\beta\) is scale and \(\gamma\) is shape. The Weibull parameters are using the method of moments. If the user specifies a threshold value, then the parameters are fitted using an iteration over the method of least squares:

  1. The location parameter is calculated using the method of moments.

  2. For location parameters from 0.5 to 1.5 times the location parameter found using method of moments, estimate the scale and shape using method of least squares.

  3. The combination of location, scale and shape which gives the smallest residual of the least square fit is chosen.

1.2.3. Fitting of Gumbel distribution

The cumulative distribution function for a Gumbel distribution is

\[F(x)=\exp(−\exp(−\frac{x−\mu}{\beta} )),\]

where \(\mu\) is location and \(\beta\) is scale. The Gumbel parameters can be fitted using the method of moments.

1.2.4. Estimation of responses

The user may choose to compute the response for a given probability level. For Rayleigh and Weibull distributions the user has the option Use return period, which enables the user to give a return period (for instance 3 hours) and estimate a corresponding characteristic extreme value. The characteristic value \(u_N\) of \(\{x_i\}^N_{i=1}\) is defined as \(P(u_N) = 1 − 1/(N+1)\). If the simulation length is \(T_{sim}\) and the user wants an estimate of the peak value corresponding to a length of \(T_{return period}\), then this can be found as \(P(u_{return period}) = 1 − \frac{T_{sim}}{(N+1) T_{return period}}\).

1.3. Input

Only input from Peaks is allowed for the Rayleigh distribution. Any equidistant signal (see Signal Types ) is allowed in Weibull and Gumbel. Note that the option Use return period requires input from Peaks.

1.4. Output

There is one output slot:

  • Distribution: Shows the fitted distribution, the sample points, sample extreme, estimated extreme value and the parameters of the fitted distribution.