Statistical modeling of extreme with Weibull Distribution

Extreme environmental conditions at sea can have significant impacts on navigation safety and/or successful search and rescue operations. Statistical techniques are crucial for accurately quantifying the likelihood of extreme events and monitoring changes in their frequency and intensity. Extreme events are by definition rare and the SEA service will use the Weibull extreme value distributions in order to evaluate the intensity and frequency of rare events that lie far in the tails of the probability distribution of ocean variables (i.e. events that occur few time in a year in a certain region).

A valid analysis of extremes in the tails of the distribution requires long time series to obtain reasonable estimates of the intensity and frequency of rare events. The service will use time series of daily values of past ten years model outputs to estimate the pdf and evaluate extreme events belonging to the tails of the distribution for significant wave height and period, temperature and intensity of the surface currents.

Weibull Distribution

The probability density function (PDF) of a random variable x having the Weibull distribution (Weibull 1951) is defined for positive values, x>0, as:

\[f(x) = \frac{\kappa}{\lambda} \left( \frac{x-\mu}{\lambda} \right)^{(k-1)} e^{-\left(\frac{x-\mu}{\lambda}\right)^k}\]

where λ is the scale parameter, κ is the shape parameter, also known as the Weibull slope and μ is the location parameter to be estimated for a given sample of data. Frequently, the location parameter is not used, and the value for this parameter can be set to zero.

The form of the density function of the Weibull distribution changes drastically with the value of k.

  • For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing. The density function has infinite negative slope at x = 0.
  • For k = 1, the density function tends to 1/λ as x approaches zero from above and is strictly decreasing. The density function has a finite negative slope at x = 0.
  • For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite positive slope at x = 0 if 1 < k < 2 and null slope at x = 0 if k > 2. F or k = 2 the density has a finite positive slope at x = 0.

As k goes to infinity, the Weibull distribution converges to a Dirac delta distribution centered at x = λ.

The first four moments (mean, standard deviation, skewness, and kurtosis) of the Weibull distribution are calculated by

Statistical properties of the Weibull distribution

  • Mean

    The mean is a measure of the central tendency of the distribution.

    \[ mean(x) = \lambda \Gamma(1+\frac{1}{k}) \]

    where Γ is the gamma function.

  • Median

    The median is the central value of an ordered distribution.

    \[ median(x) = \lambda \left( ln2 \right)^{1/k} \]

    where Γ is the gamma function.

  • Mode

    The Mode is the value that occurs most frequently in the dataset.

    \[ mode(x) = \lambda \left( \frac{k-1}{k}\right)^{1/k} \]

    where Γ is the gamma function.

  • Variance and Standard Deviation

    The standard deviation is a measure of the spread of data around the mean value.

    \[ std(x) = \lambda \left[ \Gamma \left( 1+\frac{2}{k} \right) - \Gamma^2 \left( 1+\frac{1}{k} \right)\right]^{1/2} \]

    where Γ is the gamma function.

  • Skewness

    Skewness is a measure of the asymmetry of a PDF. A positive (negative) skewness of a variable x indicates that the PDF is characterized by an elongated tail in the direction of positive (negative) fluctuations away from the mean.

    \[ skw(x) = \frac{\Gamma \left( 1+\frac{3}{k} \right) - 3 \Gamma \left( 1+\frac{1}{k} \right) \Gamma \left( 1+\frac{2}{k} \right) + 2 \Gamma^3 \left( 1+\frac{1}{k} \right)}{\left[ \Gamma \left( 1+\frac{2}{k} \right) - \Gamma^2 \left( 1+\frac{1}{k} \right)\right]^{3/2}} \]

    where Γ is the gamma function. The skewness depend only on the parameter κ.

  • Kurtosis

    Kurtosis is a measure of the flatness or peakedness of a distribution. A variable x has positive (negative) kurtosis if its PDF is more sharply peaked (broadly peaked) and has longer (shorter) tails than a Gaussian distribution with the same mean and standard deviation.

    \[ kurt(x) = \frac{\Gamma \left( 1+\frac{4}{k} \right) - 4 \Gamma \left( 1+\frac{1}{k} \right) \Gamma \left( 1+\frac{3}{k} \right) + 6 \Gamma^2 \left( 1+\frac{1}{k} \right) \Gamma \left( 1+\frac{2}{k} \right) - 3 \Gamma^4 \left( 1+\frac{1}{k} \right)}{\left[ \Gamma \left( 1+\frac{2}{k} \right) - \Gamma^2 \left( 1+\frac{1}{k} \right)\right]^{2}} \]

    where Γ is the gamma function. The kurtosis depend only on the parameter κ.

Weibull Function Graph