Class LevyDistribution

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

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

\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} \]

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

See Also:

  • Field Details

    • HALF_OVER_ERFCINV_HALF_SQUARED

      private static final double HALF_OVER_ERFCINV_HALF_SQUARED
      1 / 2(erfc^-1 (0.5))^2. Computed using Matlab's VPA to 30 digits.
      See Also:

    • mu

      private final double mu
      Location parameter.

    • c

      private final double c
      Scale parameter.

    • halfC

      private final double halfC
      Half of c (for calculations).

  • Constructor Details

    • LevyDistribution

      private LevyDistribution(double mu, double c)
      Parameters:
      mu - Location parameter.
      c - Scale parameter.

  • Method Details

    • of

      public static LevyDistribution of(double mu, double c)
      Creates a Levy distribution.
      Parameters:
      mu - Location parameter.
      c - Scale parameter.
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if c <= 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.

      If x is less than the location parameter then 0 is returned, as in these cases the distribution is not defined.

      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.

      The mean is equal to positive infinity.

      Returns:
      positive infinity.

    • getVariance

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

      The variance is equal to positive infinity.

      Returns:
      positive infinity.

    • 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 the location.

      Returns:
      location.

    • 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

    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.