Class StableSampler

java.lang.Object
org.apache.commons.rng.sampling.distribution.StableSampler
All Implemented Interfaces:
ContinuousSampler, SharedStateContinuousSampler, SharedStateSampler<SharedStateContinuousSampler>
Direct Known Subclasses:
StableSampler.BaseStableSampler, StableSampler.GaussianStableSampler, StableSampler.LevyStableSampler, StableSampler.TransformedStableSampler

public abstract class StableSampler extends Object implements SharedStateContinuousSampler
Samples from a stable distribution.

Several different parameterizations exist for the stable distribution. This sampler uses the 0-parameterization distribution described in Nolan (2020) "Univariate Stable Distributions: Models for Heavy Tailed Data". Springer Series in Operations Research and Financial Engineering. Springer. Sections 1.7 and 3.3.3.

The random variable \( X \) has the stable distribution \( S(\alpha, \beta, \gamma, \delta; 0) \) if its characteristic function is given by:

\[ E(e^{iuX}) = \begin{cases} \exp \left (- \gamma^\alpha |u|^\alpha \left [1 - i \beta (\tan \frac{\pi \alpha}{2})(\text{sgn}(u)) \right ] + i \delta u \right ) & \alpha \neq 1 \\ \exp \left (- \gamma |u| \left [1 + i \beta \frac{2}{\pi} (\text{sgn}(u)) \log |u| \right ] + i \delta u \right ) & \alpha = 1 \end{cases} \]

The function is continuous with respect to all the parameters; the parameters \( \alpha \) and \( \beta \) determine the shape and the parameters \( \gamma \) and \( \delta \) determine the scale and location. The support of the distribution is:

\[ \text{support} f(x|\alpha,\beta,\gamma,\delta; 0) = \begin{cases} [\delta - \gamma \tan \frac{\pi \alpha}{2}, \infty) & \alpha \lt 1\ and\ \beta = 1 \\ (-\infty, \delta + \gamma \tan \frac{\pi \alpha}{2}] & \alpha \lt 1\ and\ \beta = -1 \\ (-\infty, \infty) & otherwise \end{cases} \]

The implementation uses the Chambers-Mallows-Stuck (CMS) method as described in:

  • Chambers, Mallows & Stuck (1976) "A Method for Simulating Stable Random Variables". Journal of the American Statistical Association. 71 (354): 340–344.
  • Weron (1996) "On the Chambers-Mallows-Stuck method for simulating skewed stable random variables". Statistics & Probability Letters. 28 (2): 165–171.
Since:
1.4
See Also:
  • Field Details

    • PI_2

      private static final double PI_2
      pi / 2.
      See Also:
    • ALPHA_GAUSSIAN

      private static final double ALPHA_GAUSSIAN
      The alpha value for the Gaussian case.
      See Also:
    • ALPHA_CAUCHY

      private static final double ALPHA_CAUCHY
      The alpha value for the Cauchy case.
      See Also:
    • ALPHA_LEVY

      private static final double ALPHA_LEVY
      The alpha value for the Levy case.
      See Also:
    • ALPHA_SMALL

      private static final double ALPHA_SMALL
      The alpha value for the alpha -> 0 to switch to using the Weron formula. Note that small alpha requires robust correction of infinite samples.
      See Also:
    • BETA_LEVY

      private static final double BETA_LEVY
      The beta value for the Levy case.
      See Also:
    • GAMMA_1

      private static final double GAMMA_1
      The gamma value for the normalized case.
      See Also:
    • DELTA_0

      private static final double DELTA_0
      The delta value for the normalized case.
      See Also:
    • TAU_ZERO

      private static final double TAU_ZERO
      The tau value for zero. When tau is zero, this is effectively beta = 0.
      See Also:
    • LOWER

      private static final double LOWER
      The lower support for the distribution. This is the lower bound of (-inf, +inf) If the sample is not within this bound (lower < x) then it is either infinite or NaN and the result should be checked.
      See Also:
    • UPPER

      private static final double UPPER
      The upper support for the distribution. This is the upper bound of (-inf, +inf). If the sample is not within this bound (x < upper) then it is either infinite or NaN and the result should be checked.
      See Also:
    • rng

      private final UniformRandomProvider rng
      Underlying source of randomness.
  • Constructor Details

    • StableSampler

      StableSampler(UniformRandomProvider rng)
      Parameters:
      rng - Generator of uniformly distributed random numbers.
  • Method Details

    • sample

      public abstract double sample()
      Generate a sample from a stable distribution.

      The distribution uses the 0-parameterization: S(alpha, beta, gamma, delta; 0).

      Specified by:
      sample in interface ContinuousSampler
      Returns:
      a sample.
    • withUniformRandomProvider

      public abstract StableSampler withUniformRandomProvider(UniformRandomProvider rng)
      Create a new instance of the sampler with the same underlying state using the given uniform random provider as the source of randomness.
      Specified by:
      withUniformRandomProvider in interface SharedStateSampler<SharedStateContinuousSampler>
      Parameters:
      rng - Generator of uniformly distributed random numbers.
      Returns:
      the sampler
    • nextLong

      long nextLong()
      Generates a long value. Used by algorithm implementations without exposing access to the RNG.
      Returns:
      the next random value
    • toString

      public String toString()
      Overrides:
      toString in class Object
    • of

      public static StableSampler of(UniformRandomProvider rng, double alpha, double beta)
      Creates a standardized sampler of a stable distribution with zero location and unit scale.

      Special cases:

      • alpha=2 returns a Gaussian distribution sampler with mean=0 and variance=2 (Note: beta has no effect on the distribution).
      • alpha=1 and beta=0 returns a Cauchy distribution sampler with location=0 and scale=1.
      • alpha=0.5 and beta=1 returns a Levy distribution sampler with location=-1 and scale=1. This location shift is due to the 0-parameterization of the stable distribution.

      Note: To allow the computation of the stable distribution the parameter alpha is validated using 1 - alpha in the interval [-1, 1).

      Parameters:
      rng - Generator of uniformly distributed random numbers.
      alpha - Stability parameter. Must be in the interval (0, 2].
      beta - Skewness parameter. Must be in the interval [-1, 1].
      Returns:
      the sampler
      Throws:
      IllegalArgumentException - if 1 - alpha < -1; or 1 - alpha >= 1; or beta < -1; or beta > 1.
    • of

      public static StableSampler of(UniformRandomProvider rng, double alpha, double beta, double gamma, double delta)
      Creates a sampler of a stable distribution. This applies a transformation to the standardized sampler.

      The random variable \( X \) has the stable distribution \( S(\alpha, \beta, \gamma, \delta; 0) \) if:

      \[ X = \gamma Z_0 + \delta \]

      where \( Z_0 = S(\alpha, \beta; 0) \) is a standardized stable distribution.

      Note: To allow the computation of the stable distribution the parameter alpha is validated using 1 - alpha in the interval [-1, 1).

      Parameters:
      rng - Generator of uniformly distributed random numbers.
      alpha - Stability parameter. Must be in the interval (0, 2].
      beta - Skewness parameter. Must be in the interval [-1, 1].
      gamma - Scale parameter. Must be strictly positive and finite.
      delta - Location parameter. Must be finite.
      Returns:
      the sampler
      Throws:
      IllegalArgumentException - if 1 - alpha < -1; or 1 - alpha >= 1; or beta < -1; or beta > 1; or gamma <= 0; or gamma or delta are not finite.
      See Also:
    • create

      private static StableSampler create(UniformRandomProvider rng, double alpha, double beta)
      Creates a standardized sampler of a stable distribution with zero location and unit scale.
      Parameters:
      rng - Generator of uniformly distributed random numbers.
      alpha - Stability parameter. Must be in the interval (0, 2].
      beta - Skewness parameter. Must be in the interval [-1, 1].
      Returns:
      the sampler
    • validateParameters

      private static void validateParameters(double alpha, double beta)
      Validate the parameters are in the correct range.
      Parameters:
      alpha - Stability parameter. Must be in the interval (0, 2].
      beta - Skewness parameter. Must be in the interval [-1, 1].
      Throws:
      IllegalArgumentException - if 1 - alpha < -1; or 1 - alpha >= 1; or beta < -1; or beta > 1.
    • validateParameters

      private static void validateParameters(double alpha, double beta, double gamma, double delta)
      Validate the parameters are in the correct range.
      Parameters:
      alpha - Stability parameter. Must be in the interval (0, 2].
      beta - Skewness parameter. Must be in the interval [-1, 1].
      gamma - Scale parameter. Must be strictly positive and finite.
      delta - Location parameter. Must be finite.
      Throws:
      IllegalArgumentException - if 1 - alpha < -1; or 1 - alpha >= 1; or beta < -1; or beta > 1; or gamma <= 0; or gamma or delta are not finite.