Candidate Probability Distributions

The FFA framework considers nine candidate probability distributions:

Distribution Abbreviation Parameters
Gumbel GUM \(\mu\) (location), \(\sigma\) (scale)
Normal NOR \(\mu\) (location), \(\sigma\) (scale)
Log-Normal LNO \(\mu\) (location), \(\sigma\) (scale)
Generalized Extreme Value GEV \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)
Generalized Logistic Value GLO \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)
Generalized Normal GNO \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)
Pearson Type III PE3 \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)
Log-Pearson Type III LP3 \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)
Weibull WEI \(\mu\) (location), \(\sigma\) (scale), \(\kappa\) (shape)

Each distribution also has three nonstationary variants:

  1. A trend in the location parameter \(\mu\) (+1 parameter).
  2. A trend in the scale parameter \(\sigma\) (+1 parameter).
  3. A trend in the location \(\mu\) and the scale \(\sigma\) (+2 parameters).

The FFA framework also uses the four-parameter Kappa distribution (KAP) for the Z-statistic selection metric. The Kappa distribution generalizes the nine distributions listed above.

List of Distributions1

Gumbel (GUM) Distribution

Support

\(-\infty < x < \infty\)

Quantiles

\(x(F) = \mu - \sigma \log (-\log F)\)

Likelihood Function

Its probability density function (PDF) is:

\[ f(x_{i} : \mu, \sigma) = \frac{1}{\sigma} \exp \left(-z_{i} - e^{-z_{i}}\right) , \quad z_{i} = \frac{x_{i} - \mu}{\sigma } \]

Therefore, is Log-likelihood function is:

\[ \ell(x:\mu, \sigma) = \sum_{i=1}^{n} \left[-\ln \sigma - z_{i} - e^{-z_{i}} \right] \]

L-Moments

In the equations below, \(\gamma \approx 0.5772\) is Euler's constant.

  • \(\lambda_{1} = \mu + \sigma \gamma\)
  • \(\lambda_{2} = \sigma \log 2\)
  • \(\tau_{3} = \log(9/8)/\log 2 \approx 0.1699\)
  • \(\tau_{4} = (16 \log 2 - 10\log 3) / \log 2 \approx 0.1504\)

We can also express the parameters in terms of the L-moments:

  • \(\sigma = \lambda_{2} / \log 2\)
  • \(\mu = \lambda_{1} - \sigma \gamma\)

Normal (NOR) Distribution

Support

\(-\infty < x < \infty\)

Quantiles

\(x(F) = \mu + \sigma \Phi^{-1}(F)\)

Likelihood Function

Its probability density function (PDF) is:

\[ f(x_{i} : \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi }}e^{-z_{i}^2/2} , \quad z_{i} = \frac{x_{i} - \mu}{\sigma } \]

Therefore, its Log-likelihood function is:

\[ \ell(x:\mu, \sigma) = \sum_{i=1}^{n} \left[-\ln (\sigma \sqrt{2\pi }) - \frac{z_{i}^2}{2} \right] \]

L-Moments

  • \(\lambda_{1} = \mu\)
  • \(\lambda_{2} = \pi^{-1/2}\sigma \approx 0.5642\sigma\)
  • \(\tau_{3} = 0\)
  • \(\tau_{4} = 30\pi^{-1}\arctan \sqrt{2} - 9 \approx 0.1226\)

We can also express the parameters in terms of the L-moments:

  • \(\mu = \lambda_{1}\)
  • \(\sigma = \pi^{1/2}\lambda_{2}\)

Log-Normal (LNO) Distribution

Support

\(0 < x < \infty\)

Quantiles

\(x(F) = \exp(\mu + \sigma \Phi^{-1}(F))\)

Likelihood Function

To derive its likelihood, we use the fact that:

\[ \text{Data} \sim \text{LNO} \Leftrightarrow \ln (\text{Data}) \sim \text{NOR} \]

Precisely, we require the change of variables formula, which states that:

\[ \ell_{\text{LNO}}(x ; \mu, \sigma) = \ell_{\text{NOR}}(\ln x ; \mu , \sigma) \left|\frac{d}{dx} \ln x\right| = \frac{\ell_{\text{NOR}}(\ln x ; \mu , \sigma)}{x} \]

L-Moments

See Normal Distribution.

Generalized Extreme Value (GEV) Distribution

Support

\[ \begin{cases} \mu + (\sigma /\kappa) \leq x < \infty & \kappa > 0 \\[5pt] -\infty < x < \infty & \kappa = 0 \\[5pt] -\infty < x \leq \mu + (\sigma/\kappa ) &\kappa < 0 \end{cases} \]

Quantiles

\[ x(F) = \begin{dcases} \mu + \sigma (1 - (-\log F)^{\kappa })/\kappa &\kappa \neq 0\\[5pt] \mu - \sigma \log (-\log F) &\kappa = 0 \end{dcases} \]

Likelihood Function

Its probability density function (PDF) is (assume \(t_{i} > 0)\):

\[ f(x_{i} : \mu, \sigma, \kappa) = \frac{1}{\sigma}t_{i}^{-1 - (1/\kappa)} \exp (-t_{i}^{-1/\kappa}), \quad t_{i} = 1 + \kappa \left(\frac{x_{i} - \mu }{\sigma } \right) \]

Therefore, its Log-likelihood is:

\[ \ell(x:\mu, \sigma, \kappa) = \sum_{i=1}^{n} \left[-\ln \sigma - \left(1 + \frac{1}{\kappa }\right) \ln t_{i} - t_{i}^{-1/\kappa}\right] \]

L-Moments

The L-moments are defined for \(\kappa > -1\):

  • \(\lambda_{1} = \mu + \sigma (1 - \Gamma (1 + \kappa)) / \kappa\)
  • \(\lambda_{2} = \sigma (1 - 2^{-\kappa })\Gamma (1 + \kappa) / \kappa\)
  • \(\tau_{3} = 2(1 - 3^{-\kappa})/(1 - 2^{-\kappa}) - 3\)
  • \(\tau_{4} = [5(1 - 4^{-\kappa })-10(1-3^{-\kappa}) + 6(1-2^{-\kappa })]/(1 - 2^{-\kappa })\)

To compute the parameters from the L-moments, we first compute \(c\):

\[ c = \frac{2}{3 + \tau_{3}} - \frac{\log 2}{\log 3} \]

Then, we use the following approximation2:

\[ \begin{cases} \kappa \approx 7.8590c + 2.9554c^2 \\[5pt] \sigma \approx \lambda_{2}\kappa / (1 - 2^{-\kappa })\Gamma (1 + \kappa) \\[5pt] \mu \approx \lambda_{1} - \sigma (1 - \Gamma (1 + \kappa )) / \kappa \end{cases} \]

Note: Other sources often use a different notation for the GEV distribution in which the sign of the shape parameter \(\kappa\) is flipped.

Generalized Logistic (GLO) Distribution

Support

\[ \begin{cases} -\infty < x \leq \mu + (\sigma /\kappa ) & \kappa > 0 \\[5pt] -\infty < x < \infty & \kappa = 0 \\[5pt] \mu + (\sigma /\kappa ) \leq x < \infty & \kappa < 0 \end{cases} \]

Quantiles

\[ x(F) = \begin{cases} \mu +\sigma [1 - ((1 - F) / F)^{\kappa}] / \kappa &\kappa \neq 0 \\[5pt] \mu - \sigma \log ((1 - F) / F) & k = 0 \end{cases} \]

Likelihood Function

Its probability density function (PDF) is (assume \(t_{i} > 0)\):

\[ f(x_{i} : \mu , \sigma , \kappa ) = \frac{1}{\sigma }t_{i}^{(1/\kappa) - 1} \left[1 + t_{i}^{1/\kappa}\right]^{-2}, \quad t_{i} = 1 - \kappa \left(\frac{x_{i} - \mu }{\sigma }\right) \]

Therefore, its Log-likelihood function is:

\[ \ell(x:\mu, \sigma, \kappa) = \sum_{i=1}^{n} \left[-\ln \sigma + \left(\frac{1}{\kappa }-1\right) \ln t_{i} - 2 \ln \left(1 + t_{i}^{1/\kappa }\right) \right] \]

L-Moments

The L-moments are defined for \(-1 < \kappa < 1\):

  • \(\lambda_{1} = \mu +\sigma [(1 / \kappa) - (\pi / \sin (\kappa\pi))]\)
  • \(\lambda_{2} = \sigma \kappa \pi / \sin (\kappa \pi)\)
  • \(\tau_{3} = -\kappa\)
  • \(\tau_{4} = (1 + 5\kappa ^2) / 6\)

We can also express the parameters in terms of the L-moments:

  • \(\kappa = -\tau_{3}\)
  • \(\sigma = \lambda_{2}\sin (\kappa \pi ) / \kappa \pi\)
  • \(\mu = \lambda_{1} - \sigma [(1 / \kappa) - (\pi / \sin (\kappa\pi))]\)

Generalized Normal (GNO) Distribution

Support

\[ \begin{cases} -\infty < x \leq \mu + (\sigma /\kappa ) & \kappa > 0 \\[5pt] -\infty < x < \infty & \kappa = 0 \\[5pt] \mu + (\sigma /\kappa ) \leq x < \infty & \kappa < 0 \end{cases} \]

Quantiles

\[ x(F) = \begin{cases} \mu + \sigma [1 - \exp(-\kappa \Phi^{-1}(F))] / \kappa &\kappa \neq 0 \\[5pt] \mu + \sigma \Phi^{-1}(F) &\kappa = 0 \end{cases} \]

Likelihood Function

L-Moments

The L-moments are defined for all values of \(\kappa\).

  • \(\lambda_{1} = \mu + \sigma (1 - e^{\kappa ^2/2}) / \kappa\)
  • \(\lambda_{2} = \sigma e^{-\kappa ^2/ 2}[1 - 2\Phi (-\kappa / \sqrt{2})] / \kappa\)

To compute \(\tau_{3}\) and \(\tau_{4}\) we use the following approximation:

\[ \begin{aligned} \tau_{3} &\approx -\kappa \left(\frac{A_{0} + A_{1}\kappa ^2 + A_{2}\kappa ^{4} + A_{3}\kappa ^{6}}{1 + B_{1}\kappa ^2 + B_{2}\kappa ^{4} + B_{3}\kappa ^{6}}\right) \\[5pt] \tau_{4} &\approx \tau_{4}^{0} + \kappa ^2 \left(\frac{C_{0} + C_{1}\kappa ^2 + C_{2}\kappa ^{4} + C_{3}\kappa ^{6}}{1 + D_{1}\kappa ^2 + D_{2}\kappa ^{4} + D_{3}\kappa ^{6}}\right) \end{aligned} \]

To determine the parameters from the L-moments we also use a rational approximation:

\[ \kappa \approx -\tau_{3} \left(\frac{E_{0} + E_{1}\tau_{3}^2 + E_{2}\tau_{3}^{4} + E_{3}\tau _{3}^{6}}{1 + F_{1}\tau _{3}^2 + F_{2}\tau _{3}^{4} + F_{3}\tau _{3}^{6}}\right) \]

Then, we can find \(\mu\) and \(\sigma\) as a function of \(\kappa\):

\[ \sigma \approx \frac{\lambda_{2}\kappa e^{-\kappa ^2 / 2}}{1 - 2\Phi (-\kappa / \sqrt{2})}, \quad \mu \approx \lambda_{1} - \frac{\sigma }{\kappa }\left(1 - e^{-\kappa ^2 / 2 }\right) \]

The coefficients (\(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(E_{i}\), \(F_{i}\), and \(\tau_{4}^{0}\)) are defined in Appendix A.8 of Hosking, 19971. Although this appendix covers the 3-parameter log-normal distribution, the L-moments of the generalized normal distribution are the same.

Pearson Type III (PE3) Distribution

The Pearson Type III distribution is typically reparameterized as follows for \(\kappa \neq 0\):

\[ \begin{aligned} \alpha &= 4 / \kappa^2 \\[5pt] \beta &= \sigma |\kappa | / 2 \\[5pt] \xi &= \mu - 2\sigma /\kappa \end{aligned} \]

Support

\[ \begin{cases} \xi \leq x < \infty &\kappa > 0 \\[5pt] -\infty < x < \infty &\kappa =0 \\[5pt] -\infty < x \leq \xi &\kappa < 0 \end{cases} \]

Quantiles

\[ x(F) = \begin{cases} \mu - \alpha \beta + q(F, \alpha, \beta) &\kappa > 0\\[5pt] \mu + \sigma \Phi^{-1}(F) &\kappa = 0\\[5pt] \mu + \alpha \beta - q(1 - F, \alpha, \beta) &\kappa < 0 \end{cases} \]

In the equations above, \(q\) is the quantile function of the Gamma distribution with shape \(\alpha\) and scale \(\beta\). \(q\) is defined below, where \(\gamma\) is the lower incomplete Gamma function.

\[q(F, \alpha, \beta) = \beta \gamma ^{-1}(\alpha, p \Gamma (\alpha))\]

Likelihood Function

The probability density function (PDF) of the PE3 distribution is given below:

\[ f(x_{i} : \mu , \sigma , \kappa ) = \frac{(x_{i} - \xi)^{\alpha - 1}e^{-(x_{i} - \xi )/\beta }}{\beta ^{\alpha } \Gamma (\alpha )} \]

Therefore, its Log-likelihood function is:

\[ \ell(x:\mu, \sigma, \kappa) = \sum_{i=1}^{n} \left[(\alpha - 1) \ln |x_{i} - \xi | - \frac{|x_{i} - \xi |}{\beta } - \alpha \ln\beta - \ln \Gamma (\alpha )\right] \]

L-Moments

All subsequent definitions assume that \(\kappa > 0\). If \(\kappa < 0\), the L-moments can be obtained by changing the signs of \(\lambda_{1}\), \(\tau_{3}\), and \(\xi\) whenever they appear. If \(\kappa = 0\), the L-moments are the same as the Normal Distribution. The first two L-moments are defined as follows:

  • \(\lambda_{1} = \xi + \alpha \beta\)
  • \(\lambda_{2} = \pi ^{-1/2} \beta \Gamma (\alpha + 0.5) / \Gamma (\alpha )\)

Rational approximation is necessary to determine \(\tau_{3}\) and \(\tau_{4}\). If \(\alpha \geq 1\):

\[ \begin{aligned} \tau_{3} &\approx \alpha^{-1/2} \left(\frac{A_{0} + A_{1}\alpha^{-1} + A_{2}\alpha^{-2} + A_{3}\alpha^{-3}}{1 + B_{1}\alpha^{-1} + B_{2}\alpha ^{-2}}\right) \\[5pt] \tau_{4} &\approx \frac{C_{0} + C_{1}\alpha^{-1} + C_{2}\alpha ^{-2} +C_{3}\alpha ^{-3}}{1 + D_{1}\alpha ^{-1} + D_{2}\alpha ^{-2}} \end{aligned} \]

If \(\alpha < 1\), we use a different set of coefficients:

\[ \begin{aligned} \tau_{3} &\approx \frac{1 + E_{1}\alpha + E_{2}\alpha ^2 + E_{3}\alpha ^3}{1 + F_{1}\alpha + F_{2}\alpha ^2 + F_{3}\alpha ^3} \\[5pt] \tau_{4} &\approx \frac{1 + G_{1}\alpha + G_{2}\alpha ^2 + G_{3}\alpha ^3}{1 + H_{1}\alpha + H_{2}\alpha ^2 + H_{3}\alpha ^3} \end{aligned} \]

Coefficients are given in Appendix A.9 of Hosking, 19971. To estimate parameters from the L-moments, we use one of two approximations for \(\alpha\) depending on the value of \(\tau_{3}\):

\[ \alpha \approx \begin{dcases} \frac{1 + 0.2906z}{z + 0.1882z^2 + 0.0442z^3}, &z = 3\pi \tau_{3}^2, &0 < |\tau_{3}| < \frac{1}{3} \\[5pt] \frac{0.36067z - 0.59567z^2 + 0.25361z^3}{1 - 2.78861z + 2.56096z^2 - 0.77045z^3}, &z = 1 - |\tau_{3}|, &\frac{1}{3} \leq |\tau_{3}| < 1 \end{dcases} \]

Then, we can determine the parameters from the approximated \(\alpha\):

\[ \begin{aligned} \kappa &= 2\alpha ^{-1/2} \text{sign} (\tau_{3}) \\[5pt] \sigma &= \lambda_{2} \pi^{1/2}\alpha ^{1/2} \Gamma (\alpha )/\Gamma (\alpha + 0.5)\\[5pt] \mu &= \lambda_{1 } \end{aligned} \]

Log-Pearson Type III (LP3) Distribution

The LP3 distribution uses the same reparameterization as the PE3 distribution.

Support

\[ \begin{cases} \max(0, \xi) \leq x < \infty &\kappa > 0 \\[5pt] 0 < x < \infty &\kappa =0 \\[5pt] 0 < x \leq \max(0, \xi) &\kappa < 0 \end{cases} \]

Quantiles

\(x(F) = \exp(x_{\text{PE3}}(F ))\), where \(x_{\text{PE3}}(F)\) is the quantile function of the PE3 distribution.

Likelihood Function

To derive the likelihood of the LP3 distribution, we use the fact that:

\[ \text{Data} \sim \text{LP3} \Leftrightarrow \ln (\text{Data}) \sim \text{PE3} \]

Precisely, we require the change of variables formula, which states that:

\[ \ell_{\text{LP3}}(x ; \mu, \sigma, \kappa) = \ell_{\text{PE3}}(\ln x ; \mu , \sigma, \kappa ) \left|\frac{d}{dx} \ln x\right| = \frac{\ell_{\text{PE3}}(\ln x ; \mu , \sigma, \kappa )}{x} \]

L-Moments

Same as the PE3 distribution.

Weibull (WEI) Distribution

The Weibull distribution is implemented as a reparameterized version of the generalized extreme value distribution:

\[ \begin{aligned} \kappa &= 1 / \kappa_{\text{GEV}} \\[5pt] \sigma &= \kappa \sigma_{\text{GEV} } \\[5pt] \mu &= \sigma + \mu_{\text{GEV} } \end{aligned} \]

Under this reparameterization, it is required that \(\sigma > 0\) and \(\kappa > 0\).

Support

\(\mu \leq x < \infty\)

Quantiles

\(x(F) = \mu + \sigma (-\log (1 - F))^{1/\kappa}\)

Likelihood Function

Its probability density function (PDF) is given below for \(x_{i} > \mu\):

\[ f(x_{i} : \mu, \sigma, \kappa) = \frac{\kappa}{\sigma }\left(\frac{x_{i} - \mu}{\sigma }\right)^{\kappa -1} \exp \left( - \left(\frac{x_{i} - \mu}{\sigma }\right)^{\kappa } \right) \]

Therefore, its Log-likelihood function is:

\[ \ell(x:\mu, \sigma, \kappa) = \sum_{i=1}^{n} \left[\ln \kappa - \kappa \ln \sigma +(\kappa -1)\ln (x_{i}-\mu ) - \left(\frac{x_{i} - \mu }{\sigma }\right) ^{\kappa } \right] \]

L-Moments

First, reparameterize the Weibull distribution to recover the GEV parameters:

\[ \begin{aligned} \kappa_{\text{GEV}} &= 1 / \kappa \\[5pt] \sigma_{\text{GEV}} &= \sigma / \kappa \\[5pt] \end{aligned} \]

Next, compute the L-moments for the GEV distribution with \(\mu_{\text{GEV}} = 0\). Then,

  • \(\lambda_{1} = \mu + \sigma - \lambda_{1, \text{GEV}}\)
  • \(\lambda_{2} = \lambda_{2, \text{GEV}}\)
  • \(\tau_{3} = -\tau_{3, \text{GEV}}\)
  • \(\tau_{4} = \tau_{4, \text{GEV} }\)

To compute the parameters from the L-moments, first flip the sign of \(\lambda_{1}\) and \(\tau_{3}\). Then, estimate the parameters of the GEV distribution to get \(\hat{\mu}_{\text{GEV}}\), \(\hat{\sigma}_{\text{GEV}}\), and \(\hat{\kappa}_{\text{GEV}}\). Finally, reparameterize the GEV parameters as shown here and then flip the sign of \(\mu\).

Kappa (KAP) Distribution

The Kappa distribution has location \(\mu\), scale \(\sigma\), and two shape parameters \(\kappa\) and \(h\).

Support

\[ \begin{cases} \mu + \sigma (1 - h^{-\kappa}) \leq x \leq \mu + (\sigma /\kappa ) & \kappa > 0, h > 0 \\[5pt] -\infty < x \leq \mu + (\sigma /\kappa) & \kappa > 0, h \leq 0 \\[5pt] \mu + \sigma (1 - h^{-\kappa}) \leq x < \infty &\kappa \leq 0, h > 0 \\[5pt] \mu + (\sigma / \kappa ) \leq x <\infty &\kappa \leq 0, h \leq 0 \end{cases} \]

Quantiles

\[ x(F) = \mu + \frac{\sigma }{\kappa }\left[1 - \left(\frac{1 - F^{h}}{h}\right)^{\kappa }\right] \]

L-Moments

The L-moments are defined if \(h \geq 0\) and \(k > -1\) or if \(h < 0\) and \(-1 < k < -1/h\).

  • \(\lambda_{1} = \mu + \sigma (1 - g_{1})/\kappa\)
  • \(\lambda_{2} = \sigma(g_{1} - g_{2})/\kappa\)
  • \(\tau_{3} = (-g_{1} + 3g_{2} - 2g_{3}) / (g_{1} - g_{2})\)
  • \(\tau_{4} = (-g_{1} + 6g_{2} - 10g_{3} + 5g_{4}) / (g_{1} - g_{2})\)

In the expression above, \(g_{r}\) is defined as follows:

\[ g_{r} = \begin{dcases} \frac{r\Gamma (1 + \kappa )\Gamma (r / h)}{h^{1 + \kappa }\Gamma (1 + \kappa + r/h)} &h > 0 \\[5pt] \frac{r\Gamma (1 + \kappa ) \Gamma (-\kappa - r/h)}{(-h)^{1 + \kappa }\Gamma (1 - r/h)} &h < 0 \end{dcases} \]

There is no closed-form solution for the parameters in terms of the L-moments. However, \(\tau_{3}\) and \(\tau_{4}\) can be computed in terms of \(\kappa\) and \(h\) using Newton-Raphson iteration.

Sources

  1. Hosking, J.R.M. & Wallis, J.R., 1997. Regional frequency analysis: an aproach based on L-Moments. Cambridge University Press, New York, USA. ↩↩↩

  2. Hosking, J.R.M., Wallis, J.R., & Wood, E.F., 1985. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27, 251-61. ↩