Class GumbelDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.GumbelDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class GumbelDistribution extends AbstractContinuousDistribution
Implementation of the Gumbel distribution.

The probability density function of \( X \) is:

\[ f(x; \mu, \beta) = \frac{1}{\beta} e^{-(z+e^{-z})} \]

where \[ z = \frac{x - \mu}{\beta} \]

for \( \mu \) the location, \( \beta > 0 \) the scale, and \( x \in (-\infty, \infty) \).

See Also:

  • Field Details

    • SUPPORT_LO

      private static final double SUPPORT_LO
      Support lower bound.
      See Also:

    • SUPPORT_HI

      private static final double SUPPORT_HI
      Support upper bound.
      See Also:

    • PI_SQUARED_OVER_SIX

      private static final double PI_SQUARED_OVER_SIX
      π2/6. https://oeis.org/A013661.
      See Also:

    • EULER

      private static final double EULER
      Approximation of Euler's constant. https://oeis.org/A001620.
      See Also:

    • LN_LN_2

      private static final double LN_LN_2
      ln(ln(2)). https://oeis.org/A074785.
      See Also:

    • mu

      private final double mu
      Location parameter.

    • beta

      private final double beta
      Scale parameter.

  • Constructor Details

    • GumbelDistribution

      private GumbelDistribution(double mu, double beta)
      Parameters:
      mu - Location parameter.
      beta - Scale parameter (must be positive).

  • Method Details

    • of

      public static GumbelDistribution of(double mu, double beta)
      Creates a Gumbel distribution.
      Parameters:
      mu - Location parameter.
      beta - Scale parameter (must be positive).
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if beta <= 0

    • getLocation

      public double getLocation()
      Gets the location parameter of this distribution.
      Returns:
      the location parameter.

    • getScale

      public double getScale()
      Gets the scale parameter of this distribution.
      Returns:
      the scale parameter.

    • density

      public double density(double x)
      Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.
      Parameters:
      x - Point at which the PDF is evaluated.
      Returns:
      the value of the probability density function at x.

    • logDensity

      public double logDensity(double x)
      Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
      Parameters:
      x - Point at which the PDF is evaluated.
      Returns:
      the logarithm of the value of the probability density function at x.

    • cumulativeProbability

      public double cumulativeProbability(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.
      Parameters:
      x - Point at which the CDF is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value less than or equal to x.

    • survivalProbability

      public double survivalProbability(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.

      By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

      Parameters:
      x - Point at which the survival function is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value greater than x.

    • inverseCumulativeProbability

      public double inverseCumulativeProbability(double p)
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

      The default implementation returns:

      Specified by:
      inverseCumulativeProbability in interface ContinuousDistribution
      Overrides:
      inverseCumulativeProbability in class AbstractContinuousDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

    • inverseSurvivalProbability

      public double inverseSurvivalProbability(double p)
      Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

      By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

      The default implementation returns:

      Specified by:
      inverseSurvivalProbability in interface ContinuousDistribution
      Overrides:
      inverseSurvivalProbability in class AbstractContinuousDistribution
      Parameters:
      p - Survival probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

    • getMean

      public double getMean()
      Gets the mean of this distribution.

      For location parameter \( \mu \) and scale parameter \( \beta \), the mean is:

      \[ \mu + \beta \gamma \]

      where \( \gamma \) is the Euler-Mascheroni constant.

      Returns:
      the mean.

    • getVariance

      public double getVariance()
      Gets the variance of this distribution.

      For scale parameter \( \beta \), the variance is:

      \[ \frac{\pi^2}{6} \beta^2 \]

      Returns:
      the variance.

    • getSupportLowerBound

      public double getSupportLowerBound()
      Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

      The lower bound of the support is always negative infinity.

      Returns:
      negative infinity.

    • getSupportUpperBound

      public double getSupportUpperBound()
      Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).

      The upper bound of the support is always positive infinity.

      Returns:
      positive infinity.

    • getMedian

      double getMedian()
      Gets the median. This is used to determine if the arguments to the AbstractContinuousDistribution.probability(double, double) function are in the upper or lower domain.

      The default implementation calls AbstractContinuousDistribution.inverseCumulativeProbability(double) with a value of 0.5.

      Overrides:
      getMedian in class AbstractContinuousDistribution
      Returns:
      the median