Package org.apache.commons.rng.sampling.distribution

Class StableSampler

  • All Implemented Interfaces:
    ContinuousSampler, SharedStateContinuousSampler, SharedStateSampler<SharedStateContinuousSampler>
    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:
    Stable distribution (Wikipedia), Nolan (2020) Univariate Stable Distributions, Chambers et al (1976) JOASA 71: 340-344, Weron (1996). Statistics & Probability Letters. 28 (2): 165–171.

    • Method Detail

      • 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.

      • 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:
        of(UniformRandomProvider, double, double)