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package org.apache.commons.rng.sampling.distribution;
import org.apache.commons.rng.UniformRandomProvider;
/**
* 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.
*
*
* @see Stable distribution (Wikipedia)
* @see Nolan (2020) Univariate Stable Distributions
* @see Chambers et al (1976) JOASA 71: 340-344
* @see Weron (1996).
* Statistics & Probability Letters. 28 (2): 165–171.
* @since 1.4
*/
public abstract class StableSampler implements SharedStateContinuousSampler {
/** pi / 2. */
private static final double PI_2 = Math.PI / 2;
/** The alpha value for the Gaussian case. */
private static final double ALPHA_GAUSSIAN = 2;
/** The alpha value for the Cauchy case. */
private static final double ALPHA_CAUCHY = 1;
/** The alpha value for the Levy case. */
private static final double ALPHA_LEVY = 0.5;
/** The alpha value for the {@code alpha -> 0} to switch to using the Weron formula.
* Note that small alpha requires robust correction of infinite samples. */
private static final double ALPHA_SMALL = 0.02;
/** The beta value for the Levy case. */
private static final double BETA_LEVY = 1.0;
/** The gamma value for the normalized case. */
private static final double GAMMA_1 = 1.0;
/** The delta value for the normalized case. */
private static final double DELTA_0 = 0.0;
/** The tau value for zero. When tau is zero, this is effectively {@code beta = 0}. */
private static final double TAU_ZERO = 0.0;
/**
* The lower support for the distribution.
* This is the lower bound of {@code (-inf, +inf)}
* If the sample is not within this bound ({@code lower < x}) then it is either
* infinite or NaN and the result should be checked.
*/
private static final double LOWER = Double.NEGATIVE_INFINITY;
/**
* The upper support for the distribution.
* This is the upper bound of {@code (-inf, +inf)}.
* If the sample is not within this bound ({@code x < upper}) then it is either
* infinite or NaN and the result should be checked.
*/
private static final double UPPER = Double.POSITIVE_INFINITY;
/** Underlying source of randomness. */
private final UniformRandomProvider rng;
// Implementation notes
//
// The Chambers-Mallows-Stuck (CMS) method uses a uniform deviate u in (0, 1) and an
// exponential deviate w to compute a stable deviate. Chambers et al (1976) published
// a formula for alpha = 1 and alpha != 1. The function is discontinuous at alpha = 1
// and to address this a trigonmoic rearrangement was provided using half angles that
// is continuous with respect to alpha. The original discontinuous formulas were proven
// in Weron (1996). The CMS rearrangement creates a deviate in the 0-parameterization
// defined by Nolan (2020); the original discontinuous functions create a deviate in the
// 1-parameterization defined by Nolan. A shift can be used to convert one parameterisation
// to the other. The shift is the magnitude of the zeta term from the 1-parameterisation.
// The following table shows how the zeta term -> inf when alpha -> 1 for
// different beta (hence the discontinuity in the function):
//
// Zeta
// Beta
// Alpha 1.0 0.5 0.25 0.1 0.0
// 0.001 0.001571 0.0007854 0.0003927 0.0001571 0.0
// 0.01 0.01571 0.007855 0.003927 0.001571 0.0
// 0.05 0.07870 0.03935 0.01968 0.007870 0.0
// 0.01 0.01571 0.007855 0.003927 0.001571 0.0
// 0.1 0.1584 0.07919 0.03960 0.01584 0.0
// 0.5 1.000 0.5000 0.2500 0.1000 0.0
// 0.9 6.314 3.157 1.578 0.6314 0.0
// 0.95 12.71 6.353 3.177 1.271 0.0
// 0.99 63.66 31.83 15.91 6.366 0.0
// 0.995 127.3 63.66 31.83 12.73 0.0
// 0.999 636.6 318.3 159.2 63.66 0.0
// 0.9995 1273 636.6 318.3 127.3 0.0
// 0.9999 6366 3183 1592 636.6 0.0
// 1.0 1.633E+16 8.166E+15 4.083E+15 1.633E+15 0.0
//
// For numerical simulation the 0-parameterization is favoured as it is continuous
// with respect to all the parameters. When approaching alpha = 1 the large magnitude
// of the zeta term used to shift the 1-parameterization results in cancellation and the
// number of bits of the output sample is effected. This sampler uses the CMS method with
// the continuous function as the base for the implementation. However it is not suitable
// for all values of alpha and beta.
//
// The method computes a value log(z) with z in the interval (0, inf). When z is 0 or infinite
// the computation can return invalid results. The open bound for the deviate u avoids
// generating an extreme value that results in cancellation, z=0 and an invalid expression.
// However due to floating point error this can occur
// when u is close to 0 or 1, and beta is -1 or 1. Thus it is not enough to create
// u by avoiding 0 or 1 and further checks are required.
// The division by the deviate w also results in an invalid expression as the term z becomes
// infinite as w -> 0. It should be noted that such events are extremely rare
// (frequency in the 1 in 10^15), or will not occur at all depending on the parameters alpha
// and beta.
//
// When alpha -> 0 then the distribution is extremely long tailed and the expression
// using log(z) often computes infinity. Certain parameters can create NaN due to
// 0 / 0, 0 * inf, or inf - inf. Thus the implementation must check the final result
// and perform a correction if required, or generate another sample.
// Correcting the original CMS formula has many edge cases depending on parameters. The
// alternative formula provided by Weron is easier to correct when infinite values are
// created. This correction is made easier by knowing that u is not 0 or 1 as certain
// conditions on the intermediate terms can be eliminated. The implementation
// thus generates u in the open interval (0,1) but leaves w unchecked and potentially 0.
// The sample is generated and the result tested against the expected support. This detects
// any NaN and infinite values. Incorrect samples due to the inability to compute log(z)
// (extremely rare) and samples where alpha -> 0 has resulted in an infinite expression
// for the value d are corrected using the Weron formula and returned within the support.
//
// The CMS algorithm is continuous for the parameters. However when alpha=1 or beta=0
// many terms cancel and these cases are handled with specialised implementations.
// The beta=0 case implements the same CMS algorithm with certain terms eliminated.
// Correction uses the alternative Weron formula. When alpha=1 the CMS algorithm can
// be corrected from infinite cases due to assumptions on the intermediate terms.
//
// The following table show the failure frequency (result not finite or, when beta=+/-1,
// within the support) for the CMS algorithm computed using 2^30 random deviates.
//
// CMS failure rate
// Beta
// Alpha 1.0 0.5 0.25 0.1 0.0
// 1.999 0.0 0.0 0.0 0.0 0.0
// 1.99 0.0 0.0 0.0 0.0 0.0
// 1.9 0.0 0.0 0.0 0.0 0.0
// 1.5 0.0 0.0 0.0 0.0 0.0
// 1.1 0.0 0.0 0.0 0.0 0.0
// 1.0 0.0 0.0 0.0 0.0 0.0
// 0.9 0.0 0.0 0.0 0.0 0.0
// 0.5 0.0 0.0 0.0 0.0 0.0
// 0.25 0.0 0.0 0.0 0.0 0.0
// 0.1 0.0 0.0 0.0 0.0 0.0
// 0.05 0.0003458 0.0 0.0 0.0 0.0
// 0.02 0.009028 6.938E-7 7.180E-7 7.320E-7 6.873E-7
// 0.01 0.004878 0.0008555 0.0008553 0.0008554 0.0008570
// 0.005 0.1519 0.02896 0.02897 0.02897 0.02897
// 0.001 0.6038 0.3903 0.3903 0.3903 0.3903
//
// The sampler switches to using the error checked Weron implementation when alpha < 0.02.
// Unit tests demonstrate the two samplers (CMS or Weron) product the same result within
// a tolerance. The switch point is based on a consistent failure rate above 1 in a million.
// At this point zeta is small and cancellation leading to loss of bits in the sample is
// minimal.
//
// In common use the sampler will not have a measurable failure rate. The output will
// be continuous as alpha -> 1 and beta -> 0. The evaluated function produces symmetric
// samples when u and beta are mirrored around 0.5 and 0 respectively. To achieve this
// the computation of certain parameters has been changed from the original implementation
// to avoid evaluating Math.tan outside the interval (-pi/2, pi/2).
//
// Note: Chambers et al (1976) use an approximation to tan(x) / x in the RSTAB routine.
// A JMH performance test is available in the RNG examples module comparing Math.tan
// with various approximations. The functions are faster than Math.tan(x) / x.
// This implementation uses a higher accuracy approximation than the original RSTAB
// implementation; it has a mean ULP difference to Math.tan of less than 1 and has
// a noticeable performance gain.
/**
* Base class for implementations of a stable distribution that requires an exponential
* random deviate.
*/
private abstract static class BaseStableSampler extends StableSampler {
/** pi/2 scaled by 2^-53. */
private static final double PI_2_SCALED = 0x1.0p-54 * Math.PI;
/** pi/4 scaled by 2^-53. */
private static final double PI_4_SCALED = 0x1.0p-55 * Math.PI;
/** -pi / 2. */
private static final double NEG_PI_2 = -Math.PI / 2;
/** -pi / 4. */
private static final double NEG_PI_4 = -Math.PI / 4;
/** The exponential sampler. */
private final ContinuousSampler expSampler;
/**
* @param rng Underlying source of randomness
*/
BaseStableSampler(UniformRandomProvider rng) {
super(rng);
expSampler = ZigguratSampler.Exponential.of(rng);
}
/**
* Gets a random value for the omega parameter ({@code w}).
* This is an exponential random variable with mean 1.
*
* Warning: For simplicity this does not check the variate is not 0.
* The calling CMS algorithm should detect and handle incorrect samples as a result
* of this unlikely edge case.
*
* @return omega
*/
double getOmega() {
// Note: Ideally this should not have a value of 0 as the CMS algorithm divides
// by w and it creates infinity. This can result in NaN output.
// Under certain parameterizations non-zero small w also creates NaN output.
// Thus output should be checked regardless.
return expSampler.sample();
}
/**
* Gets a random value for the phi parameter.
* This is a uniform random variable in {@code (-pi/2, pi/2)}.
*
* @return phi
*/
double getPhi() {
// See getPhiBy2 for method details.
final double x = (nextLong() >> 10) * PI_2_SCALED;
// Deliberate floating-point equality check
if (x == NEG_PI_2) {
return getPhi();
}
return x;
}
/**
* Gets a random value for the phi parameter divided by 2.
* This is a uniform random variable in {@code (-pi/4, pi/4)}.
*
*
Note: Ideally this should not have a value of -pi/4 or pi/4 as the CMS algorithm
* can generate infinite values when the phi/2 uniform deviate is +/-pi/4. This
* can result in NaN output. Under certain parameterizations phi/2 close to the limits
* also create NaN output. Thus output should be checked regardless. Avoiding
* the extreme values simplifies the number of checks that are required.
*
* @return phi / 2
*/
double getPhiBy2() {
// As per o.a.c.rng.core.utils.NumberFactory.makeDouble(long) but using a
// signed shift of 10 in place of an unsigned shift of 11. With a factor of 2^-53
// this would produce a double in [-1, 1).
// Here the multiplication factor incorporates pi/4 to avoid a separate
// multiplication.
final double x = (nextLong() >> 10) * PI_4_SCALED;
// Deliberate floating-point equality check
if (x == NEG_PI_4) {
// Sample again using recursion.
// A stack overflow due to a broken RNG will eventually occur
// rather than the alternative which is an infinite loop
// while x == -pi/4.
return getPhiBy2();
}
return x;
}
}
/**
* Class for implementations of a stable distribution transformed by scale and location.
*/
private static class TransformedStableSampler extends StableSampler {
/** Underlying normalized stable sampler. */
private final StableSampler sampler;
/** The scale parameter. */
private final double gamma;
/** The location parameter. */
private final double delta;
/**
* @param sampler Normalized stable sampler.
* @param gamma Scale parameter. Must be strictly positive.
* @param delta Location parameter.
*/
TransformedStableSampler(StableSampler sampler, double gamma, double delta) {
// No RNG required
super(null);
this.sampler = sampler;
this.gamma = gamma;
this.delta = delta;
}
@Override
public double sample() {
return gamma * sampler.sample() + delta;
}
@Override
public StableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new TransformedStableSampler(sampler.withUniformRandomProvider(rng),
gamma, delta);
}
@Override
public String toString() {
// Avoid a null pointer from the unset RNG instance in the parent class
return sampler.toString();
}
}
/**
* Implement the {@code alpha = 2} stable distribution case (Gaussian distribution).
*/
private static class GaussianStableSampler extends StableSampler {
/** sqrt(2). */
private static final double ROOT_2 = Math.sqrt(2);
/** Underlying normalized Gaussian sampler. */
private final NormalizedGaussianSampler sampler;
/** The standard deviation. */
private final double stdDev;
/** The mean. */
private final double mean;
/**
* @param rng Underlying source of randomness
* @param gamma Scale parameter. Must be strictly positive.
* @param delta Location parameter.
*/
GaussianStableSampler(UniformRandomProvider rng, double gamma, double delta) {
super(rng);
this.sampler = ZigguratSampler.NormalizedGaussian.of(rng);
// A standardized stable sampler with alpha=2 has variance 2.
// Set the standard deviation as sqrt(2) * scale.
// Avoid this being infinity to avoid inf * 0 in the sample
this.stdDev = Math.min(Double.MAX_VALUE, ROOT_2 * gamma);
this.mean = delta;
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
GaussianStableSampler(UniformRandomProvider rng, GaussianStableSampler source) {
super(rng);
this.sampler = ZigguratSampler.NormalizedGaussian.of(rng);
this.stdDev = source.stdDev;
this.mean = source.mean;
}
@Override
public double sample() {
return stdDev * sampler.sample() + mean;
}
@Override
public GaussianStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new GaussianStableSampler(rng, this);
}
}
/**
* Implement the {@code alpha = 1} and {@code beta = 0} stable distribution case
* (Cauchy distribution).
*/
private static class CauchyStableSampler extends BaseStableSampler {
/** The scale parameter. */
private final double gamma;
/** The location parameter. */
private final double delta;
/**
* @param rng Underlying source of randomness
* @param gamma Scale parameter. Must be strictly positive.
* @param delta Location parameter.
*/
CauchyStableSampler(UniformRandomProvider rng, double gamma, double delta) {
super(rng);
this.gamma = gamma;
this.delta = delta;
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
CauchyStableSampler(UniformRandomProvider rng, CauchyStableSampler source) {
super(rng);
this.gamma = source.gamma;
this.delta = source.delta;
}
@Override
public double sample() {
// Note:
// The CMS beta=0 with alpha=1 sampler reduces to:
// S = 2 * a / a2, with a = tan(x), a2 = 1 - a^2, x = phi/2
// This is a double angle identity for tan:
// 2 * tan(x) / (1 - tan^2(x)) = tan(2x)
// Here we use the double angle identity for consistency with the other samplers.
final double phiby2 = getPhiBy2();
final double a = phiby2 * SpecialMath.tan2(phiby2);
final double a2 = 1 - a * a;
final double x = 2 * a / a2;
return gamma * x + delta;
}
@Override
public CauchyStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new CauchyStableSampler(rng, this);
}
}
/**
* Implement the {@code alpha = 0.5} and {@code beta = 1} stable distribution case
* (Levy distribution).
*
* Note: This sampler can be used to output the symmetric case when
* {@code beta = -1} by negating {@code gamma}.
*/
private static class LevyStableSampler extends StableSampler {
/** Underlying normalized Gaussian sampler. */
private final NormalizedGaussianSampler sampler;
/** The scale parameter. */
private final double gamma;
/** The location parameter. */
private final double delta;
/**
* @param rng Underlying source of randomness
* @param gamma Scale parameter. Must be strictly positive.
* @param delta Location parameter.
*/
LevyStableSampler(UniformRandomProvider rng, double gamma, double delta) {
super(rng);
this.sampler = ZigguratSampler.NormalizedGaussian.of(rng);
this.gamma = gamma;
this.delta = delta;
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
LevyStableSampler(UniformRandomProvider rng, LevyStableSampler source) {
super(rng);
this.sampler = ZigguratSampler.NormalizedGaussian.of(rng);
this.gamma = source.gamma;
this.delta = source.delta;
}
@Override
public double sample() {
// Levy(Z) = 1 / N(0,1)^2, where N(0,1) is a standard normalized variate
final double norm = sampler.sample();
// Here we must transform from the 1-parameterization to the 0-parameterization.
// This is a shift of -beta * tan(pi * alpha / 2) = -1 when alpha=0.5, beta=1.
final double z = (1.0 / (norm * norm)) - 1.0;
// In the 0-parameterization the scale and location are a linear transform.
return gamma * z + delta;
}
@Override
public LevyStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new LevyStableSampler(rng, this);
}
}
/**
* Implement the generic stable distribution case: {@code alpha < 2} and
* {@code beta != 0}. This routine assumes {@code alpha != 1}.
*
*
Implements the Chambers-Mallows-Stuck (CMS) method using the
* formula provided in Weron (1996) "On the Chambers-Mallows-Stuck method for
* simulating skewed stable random variables" Statistics & Probability
* Letters. 28 (2): 165–171. This method is easier to correct from infinite and
* NaN results by boxing intermediate infinite values.
*
*
The formula produces a stable deviate from the 1-parameterization that is
* discontinuous at {@code alpha=1}. A shift is used to create the 0-parameterization.
* This shift is very large as {@code alpha -> 1} and the output loses bits of precision
* in the deviate due to cancellation. It is not recommended to use this sampler when
* {@code alpha -> 1} except for edge case correction.
*
*
This produces non-NaN output for all parameters alpha, beta, u and w with
* the correct orientation for extremes of the distribution support.
* The formulas used are symmetric with regard to beta and u.
*
* @see Weron, R
* (1996). Statistics & Probability Letters. 28 (2): 165–171.
*/
static class WeronStableSampler extends BaseStableSampler {
/** Epsilon (1 - alpha). */
protected final double eps;
/** Alpha (1 - eps). */
protected final double meps1;
/** Cache of expression value used in generation. */
protected final double zeta;
/** Cache of expression value used in generation. */
protected final double atanZeta;
/** Cache of expression value used in generation. */
protected final double scale;
/** 1 / alpha = 1 / (1 - eps). */
protected final double inv1mEps;
/** (1 / alpha) - 1 = eps / (1 - eps). */
protected final double epsDiv1mEps;
/** The inclusive lower support for the distribution. */
protected final double lower;
/** The inclusive upper support for the distribution. */
protected final double upper;
/**
* @param rng Underlying source of randomness
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
*/
WeronStableSampler(UniformRandomProvider rng, double alpha, double beta) {
super(rng);
eps = 1 - alpha;
// When alpha < 0.5, 1 - eps == alpha is not always true as the reverse is not exact.
// Here we store 1 - eps in place of alpha. Thus eps + (1 - eps) = 1.
meps1 = 1 - eps;
// Compute pre-factors for the Weron formula used during error correction.
if (meps1 > 1) {
// Avoid calling tan outside the domain limit [-pi/2, pi/2].
zeta = beta * Math.tan((2 - meps1) * PI_2);
} else {
zeta = -beta * Math.tan(meps1 * PI_2);
}
// Do not store xi = Math.atan(-zeta) / meps1 due to floating-point division errors.
// Directly store Math.atan(-zeta).
atanZeta = Math.atan(-zeta);
scale = Math.pow(1 + zeta * zeta, 0.5 / meps1);
// Note: These terms are used interchangeably in formulas
// 1 1
// ------- = -----
// (1-eps) alpha
inv1mEps = 1.0 / meps1;
// 1 eps (1-alpha) 1
// ------- - 1 = ------- = --------- = ----- - 1
// (1-eps) (1-eps) alpha alpha
epsDiv1mEps = inv1mEps - 1;
// Compute the support. This applies when alpha < 1 and beta = +/-1
if (alpha < 1 && Math.abs(beta) == 1) {
if (beta == 1) {
// alpha < 0, beta = 1
lower = zeta;
upper = UPPER;
} else {
// alpha < 0, beta = -1
lower = LOWER;
upper = zeta;
}
} else {
lower = LOWER;
upper = UPPER;
}
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
WeronStableSampler(UniformRandomProvider rng, WeronStableSampler source) {
super(rng);
this.eps = source.eps;
this.meps1 = source.meps1;
this.zeta = source.zeta;
this.atanZeta = source.atanZeta;
this.scale = source.scale;
this.inv1mEps = source.inv1mEps;
this.epsDiv1mEps = source.epsDiv1mEps;
this.lower = source.lower;
this.upper = source.upper;
}
@Override
public double sample() {
final double phi = getPhi();
final double w = getOmega();
return createSample(phi, w);
}
/**
* Create the sample. This routine is robust to edge cases and returns a deviate
* at the extremes of the support. It correctly handles {@code alpha -> 0} when
* the sample is increasingly likely to be +/- infinity.
*
* @param phi Uniform deviate in {@code (-pi/2, pi/2)}
* @param w Exponential deviate
* @return x
*/
protected double createSample(double phi, double w) {
// Here we use the formula provided by Weron for the 1-parameterization.
// Note: Adding back zeta creates the 0-parameterization defined in Nolan (1998):
// X ~ S0_alpha(s,beta,u0) with s=1, u0=0 for a standard random variable.
// As alpha -> 1 the translation zeta to create the stable deviate
// in the 0-parameterization is increasingly large as tan(pi/2) -> infinity.
// The max translation is approximately 1e16.
// Without this translation the stable deviate is in the 1-parameterization
// and the function is not continuous with respect to alpha.
// Due to the large zeta when alpha -> 1 the number of bits of the output variable
// are very low due to cancellation.
// As alpha -> 0 or 2 then zeta -> 0 and cancellation is not relevant.
// The formula can be modified for infinite terms to compute a result for extreme
// deviates u and w when the CMS formula fails.
// Note the following term is subject to floating point error:
// xi = atan(-zeta) / alpha
// alphaPhiXi = alpha * (phi + xi)
// This is required: cos(phi - alphaPhiXi) > 0 => phi - alphaPhiXi in (-pi/2, pi/2).
// Thus we compute atan(-zeta) and use it to compute two terms:
// [1] alpha * (phi + xi) = alpha * (phi + atan(-zeta) / alpha) = alpha * phi + atan(-zeta)
// [2] phi - alpha * (phi + xi) = phi - alpha * phi - atan(-zeta) = (1-alpha) * phi - atan(-zeta)
// Compute terms
// Either term can be infinite or 0. Certain parameters compute 0 * inf.
// t1=inf occurs alpha -> 0.
// t1=0 occurs when beta = tan(-alpha * phi) / tan(alpha * pi / 2).
// t2=inf occurs when w -> 0 and alpha -> 0.
// t2=0 occurs when alpha -> 0 and phi -> pi/2.
// Detect zeros and return as zeta.
// Note sin(alpha * phi + atanZeta) is zero when:
// alpha * phi = -atan(-zeta)
// tan(-alpha * phi) = -zeta
// = beta * tan(alpha * pi / 2)
// Since |phi| < pi/2 this requires beta to have an opposite sign to phi
// and a magnitude < 1. This is possible and in this case avoid a possible
// 0 / 0 by setting the result as if term t1=0 and the result is zeta.
double t1 = Math.sin(meps1 * phi + atanZeta);
if (t1 == 0) {
return zeta;
}
// Since cos(phi) is in (0, 1] this term will not create a
// large magnitude to create t1 = 0.
t1 /= Math.pow(Math.cos(phi), inv1mEps);
// Iff Math.cos(eps * phi - atanZeta) is zero then 0 / 0 can occur if w=0.
// Iff Math.cos(eps * phi - atanZeta) is below zero then NaN will occur
// in the power function. These cases are avoided by phi=(-pi/2, pi/2) and direct
// use of arctan(-zeta).
final double t2 = Math.pow(Math.cos(eps * phi - atanZeta) / w, epsDiv1mEps);
if (t2 == 0) {
return zeta;
}
final double x = t1 * t2 * scale + zeta;
// Check the bounds. Applies when alpha < 1 and beta = +/-1.
if (x <= lower) {
return lower;
}
return x < upper ? x : upper;
}
@Override
public WeronStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new WeronStableSampler(rng, this);
}
}
/**
* Implement the generic stable distribution case: {@code alpha < 2} and
* {@code beta != 0}. This routine assumes {@code alpha != 1}.
*
*
Implements the Chambers-Mallows-Stuck (CMS) method from Chambers, et al
* (1976) A Method for Simulating Stable Random Variables. Journal of the
* American Statistical Association Vol. 71, No. 354, pp. 340-344.
*
*
The formula produces a stable deviate from the 0-parameterization that is
* continuous at {@code alpha=1}.
*
*
This is an implementation of the Fortran routine RSTAB. In the event the
* computation fails then an alternative computation is performed using the
* formula provided in Weron (1996) "On the Chambers-Mallows-Stuck method for
* simulating skewed stable random variables" Statistics & Probability
* Letters. 28 (2): 165–171. This method is easier to correct from infinite and
* NaN results. The error correction path is extremely unlikely to occur during
* use unless {@code alpha -> 0}. In general use it requires the random deviates
* w or u are extreme. See the unit tests for conditions that create them.
*
*
This produces non-NaN output for all parameters alpha, beta, u and w with
* the correct orientation for extremes of the distribution support.
* The formulas used are symmetric with regard to beta and u.
*/
static class CMSStableSampler extends WeronStableSampler {
/** 1/2. */
private static final double HALF = 0.5;
/** Cache of expression value used in generation. */
private final double tau;
/**
* @param rng Underlying source of randomness
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
*/
CMSStableSampler(UniformRandomProvider rng, double alpha, double beta) {
super(rng, alpha, beta);
// Compute the RSTAB pre-factor.
tau = getTau(alpha, beta);
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
CMSStableSampler(UniformRandomProvider rng, CMSStableSampler source) {
super(rng, source);
this.tau = source.tau;
}
/**
* Gets tau. This is a factor used in the CMS algorithm. If this is zero then
* a special case of {@code beta -> 0} has occurred.
*
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @return tau
*/
static double getTau(double alpha, double beta) {
final double eps = 1 - alpha;
final double meps1 = 1 - eps;
// Compute RSTAB prefactor
double tau;
// tau is symmetric around alpha=1
// tau -> beta / pi/2 as alpha -> 1
// tau -> 0 as alpha -> 2 or 0
// Avoid calling tan as the value approaches the domain limit [-pi/2, pi/2].
if (Math.abs(eps) < HALF) {
// 0.5 < alpha < 1.5. Note: This works when eps=0 as tan(0) / 0 == 1.
tau = beta / (SpecialMath.tan2(eps * PI_2) * PI_2);
} else {
// alpha >= 1.5 or alpha <= 0.5.
// Do not call tan with alpha > 1 as it wraps in the domain [-pi/2, pi/2].
// Since pi is approximate the symmetry is lost by wrapping.
// Keep within the domain using (2-alpha).
if (meps1 > 1) {
tau = beta * PI_2 * eps * (2 - meps1) * -SpecialMath.tan2((2 - meps1) * PI_2);
} else {
tau = beta * PI_2 * eps * meps1 * SpecialMath.tan2(meps1 * PI_2);
}
}
return tau;
}
@Override
public double sample() {
final double phiby2 = getPhiBy2();
final double w = getOmega();
// Compute as per the RSTAB routine.
// Generic stable distribution that is continuous as alpha -> 1.
// This is a trigonomic rearrangement of equation 4.1 from Chambers et al (1976)
// as implemented in the Fortran program RSTAB.
// Uses the special functions:
// tan2 = tan(x) / x
// d2 = (exp(x) - 1) / x
// The method is implemented as per the RSTAB routine with the exceptions:
// 1. The function tan2(x) is implemented with a higher precision approximation.
// 2. The sample is tested against the expected distribution support.
// Infinite intermediate terms that create infinite or NaN are corrected by
// switching the formula and handling infinite terms.
// Compute some tangents
// a in (-1, 1)
// bb in [1, 4/pi)
// b in (-1, 1)
final double a = phiby2 * SpecialMath.tan2(phiby2);
final double bb = SpecialMath.tan2(eps * phiby2);
final double b = eps * phiby2 * bb;
// Compute some necessary subexpressions
final double da = a * a;
final double db = b * b;
// a2 in (0, 1]
final double a2 = 1 - da;
// a2p in [1, 2)
final double a2p = 1 + da;
// b2 in (0, 1]
final double b2 = 1 - db;
// b2p in [1, 2)
final double b2p = 1 + db;
// Compute coefficient.
// numerator=0 is not possible *in theory* when the uniform deviate generating phi
// is in the open interval (0, 1). In practice it is possible to obtain <=0 due
// to round-off error, typically when beta -> +/-1 and phiby2 -> -/+pi/4.
// This can happen for any alpha.
final double z = a2p * (b2 + 2 * phiby2 * bb * tau) / (w * a2 * b2p);
// Compute the exponential-type expression
// Note: z may be infinite, typically when w->0 and a2->0.
// This can produce NaN under certain parameterizations due to multiplication by 0.
final double alogz = Math.log(z);
final double d = SpecialMath.d2(epsDiv1mEps * alogz) * (alogz * inv1mEps);
// Pre-compute the multiplication factor.
// The numerator may be zero. The denominator is not zero as a2 is bounded to
// above zero when the uniform deviate that generates phiby2 is not 0 or 1.
// The min value of a2 is 2^-52. Assume f cannot be infinite as the numerator
// is computed with a in (-1, 1); b in (-1, 1); phiby2 in (-pi/4, pi/4); tau in
// [-2/pi, 2/pi]; bb in [1, 4/pi); a2 in (0, 1] limiting the numerator magnitude.
final double f = (2 * ((a - b) * (1 + a * b) - phiby2 * tau * bb * (b * a2 - 2 * a))) /
(a2 * b2p);
// Compute the stable deviate:
final double x = (1 + eps * d) * f + tau * d;
// Test the support
if (lower < x && x < upper) {
return x;
}
// Error correction path:
// x is at the bounds, infinite or NaN (created by 0 / 0, 0 * inf, or inf - inf).
// This is caused by extreme parameterizations of alpha or beta, or extreme values
// from the random deviates.
// Alternatively alpha < 1 and beta = +/-1 and the sample x is at the edge or
// outside the support due to floating point error.
// Here we use the formula provided by Weron which is easier to correct
// when deviates are extreme or alpha -> 0. The formula is not continuous
// as alpha -> 1 without a shift which reduces the precision of the sample;
// for rare edge case correction this has minimal effect on sampler output.
return createSample(phiby2 * 2, w);
}
@Override
public CMSStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new CMSStableSampler(rng, this);
}
}
/**
* Implement the stable distribution case: {@code alpha == 1} and {@code beta != 0}.
*
*
Implements the same algorithm as the {@link CMSStableSampler} with
* the {@code alpha} assumed to be 1.
*
*
This sampler specifically requires that {@code beta / (pi/2) != 0}; otherwise
* the parameters equal {@code alpha=1, beta=0} as the Cauchy distribution case.
*/
static class Alpha1CMSStableSampler extends BaseStableSampler {
/** Cache of expression value used in generation. */
private final double tau;
/**
* @param rng Underlying source of randomness
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
*/
Alpha1CMSStableSampler(UniformRandomProvider rng, double beta) {
super(rng);
tau = beta / PI_2;
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
Alpha1CMSStableSampler(UniformRandomProvider rng, Alpha1CMSStableSampler source) {
super(rng);
this.tau = source.tau;
}
@Override
public double sample() {
final double phiby2 = getPhiBy2();
final double w = getOmega();
// Compute some tangents
final double a = phiby2 * SpecialMath.tan2(phiby2);
// Compute some necessary subexpressions
final double da = a * a;
final double a2 = 1 - da;
final double a2p = 1 + da;
// Compute coefficient.
// numerator=0 is not possible when the uniform deviate generating phi
// is in the open interval (0, 1) and alpha=1.
final double z = a2p * (1 + 2 * phiby2 * tau) / (w * a2);
// Compute the exponential-type expression
// Note: z may be infinite, typically when w->0 and a2->0.
// This can produce NaN under certain parameterizations due to multiplication by 0.
// When alpha=1 the expression
// d = d2((eps / (1-eps)) * alogz) * (alogz / (1-eps)) is eliminated to 1 * log(z)
final double d = Math.log(z);
// Pre-compute the multiplication factor.
final double f = (2 * (a - phiby2 * tau * (-2 * a))) / a2;
// Compute the stable deviate:
// This does not require correction as f is finite (as per the alpha != 1 case),
// tau is non-zero and only d can be infinite due to an extreme w -> 0.
return f + tau * d;
}
@Override
public Alpha1CMSStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new Alpha1CMSStableSampler(rng, this);
}
}
/**
* Implement the generic stable distribution case: {@code alpha < 2} and {@code beta == 0}.
*
*
Implements the same algorithm as the {@link WeronStableSampler} with
* the {@code beta} assumed to be 0.
*
*
This routine assumes {@code alpha != 1}; {@code alpha=1, beta=0} is the Cauchy
* distribution case.
*/
static class Beta0WeronStableSampler extends BaseStableSampler {
/** Epsilon (1 - alpha). */
protected final double eps;
/** Epsilon (1 - alpha). */
protected final double meps1;
/** 1 / alpha = 1 / (1 - eps). */
protected final double inv1mEps;
/** (1 / alpha) - 1 = eps / (1 - eps). */
protected final double epsDiv1mEps;
/**
* @param rng Underlying source of randomness
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
*/
Beta0WeronStableSampler(UniformRandomProvider rng, double alpha) {
super(rng);
eps = 1 - alpha;
meps1 = 1 - eps;
inv1mEps = 1.0 / meps1;
epsDiv1mEps = inv1mEps - 1;
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
Beta0WeronStableSampler(UniformRandomProvider rng, Beta0WeronStableSampler source) {
super(rng);
this.eps = source.eps;
this.meps1 = source.meps1;
this.inv1mEps = source.inv1mEps;
this.epsDiv1mEps = source.epsDiv1mEps;
}
@Override
public double sample() {
final double phi = getPhi();
final double w = getOmega();
return createSample(phi, w);
}
/**
* Create the sample. This routine is robust to edge cases and returns a deviate
* at the extremes of the support. It correctly handles {@code alpha -> 0} when
* the sample is increasingly likely to be +/- infinity.
*
* @param phi Uniform deviate in {@code (-pi/2, pi/2)}
* @param w Exponential deviate
* @return x
*/
protected double createSample(double phi, double w) {
// As per the Weron sampler with beta=0 and terms eliminated.
// Note that if alpha=1 this reduces to sin(phi) / cos(phi) => Cauchy case.
// Compute terms.
// Either term can be infinite or 0. Certain parameters compute 0 * inf.
// Detect zeros and return as 0.
// Note sin(alpha * phi) is only ever zero when phi=0. No value of alpha
// multiplied by small phi can create zero due to the limited
// precision of alpha imposed by alpha = 1 - (1-alpha). At this point cos(phi) = 1.
// Thus 0/0 cannot occur.
final double t1 = Math.sin(meps1 * phi) / Math.pow(Math.cos(phi), inv1mEps);
if (t1 == 0) {
return 0;
}
final double t2 = Math.pow(Math.cos(eps * phi) / w, epsDiv1mEps);
if (t2 == 0) {
return 0;
}
return t1 * t2;
}
@Override
public Beta0WeronStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new Beta0WeronStableSampler(rng, this);
}
}
/**
* Implement the generic stable distribution case: {@code alpha < 2} and {@code beta == 0}.
*
*
Implements the same algorithm as the {@link CMSStableSampler} with
* the {@code beta} assumed to be 0.
*
*
This routine assumes {@code alpha != 1}; {@code alpha=1, beta=0} is the Cauchy
* distribution case.
*/
static class Beta0CMSStableSampler extends Beta0WeronStableSampler {
/**
* @param rng Underlying source of randomness
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
*/
Beta0CMSStableSampler(UniformRandomProvider rng, double alpha) {
super(rng, alpha);
}
/**
* @param rng Underlying source of randomness
* @param source Source to copy.
*/
Beta0CMSStableSampler(UniformRandomProvider rng, Beta0CMSStableSampler source) {
super(rng, source);
}
@Override
public double sample() {
final double phiby2 = getPhiBy2();
final double w = getOmega();
// Compute some tangents
final double a = phiby2 * SpecialMath.tan2(phiby2);
final double b = eps * phiby2 * SpecialMath.tan2(eps * phiby2);
// Compute some necessary subexpressions
final double da = a * a;
final double db = b * b;
final double a2 = 1 - da;
final double a2p = 1 + da;
final double b2 = 1 - db;
final double b2p = 1 + db;
// Compute coefficient.
final double z = a2p * b2 / (w * a2 * b2p);
// Compute the exponential-type expression
final double alogz = Math.log(z);
final double d = SpecialMath.d2(epsDiv1mEps * alogz) * (alogz * inv1mEps);
// Pre-compute the multiplication factor.
// The numerator may be zero. The denominator is not zero as a2 is bounded to
// above zero when the uniform deviate that generates phiby2 is not 0 or 1.
final double f = (2 * ((a - b) * (1 + a * b))) / (a2 * b2p);
// Compute the stable deviate:
final double x = (1 + eps * d) * f;
// Test the support
if (LOWER < x && x < UPPER) {
return x;
}
// Error correction path.
// Here we use the formula provided by Weron which is easier to correct
// when deviates are extreme or alpha -> 0.
return createSample(phiby2 * 2, w);
}
@Override
public Beta0CMSStableSampler withUniformRandomProvider(UniformRandomProvider rng) {
return new Beta0CMSStableSampler(rng, this);
}
}
/**
* Implement special math functions required by the CMS algorithm.
*/
static final class SpecialMath {
/** pi/4. */
private static final double PI_4 = Math.PI / 4;
/** 4/pi. */
private static final double FOUR_PI = 4 / Math.PI;
/** tan2 product constant. */
private static final double P0 = -0.5712939549476836914932149599e10;
/** tan2 product constant. */
private static final double P1 = 0.4946855977542506692946040594e9;
/** tan2 product constant. */
private static final double P2 = -0.9429037070546336747758930844e7;
/** tan2 product constant. */
private static final double P3 = 0.5282725819868891894772108334e5;
/** tan2 product constant. */
private static final double P4 = -0.6983913274721550913090621370e2;
/** tan2 quotient constant. */
private static final double Q0 = -0.7273940551075393257142652672e10;
/** tan2 quotient constant. */
private static final double Q1 = 0.2125497341858248436051062591e10;
/** tan2 quotient constant. */
private static final double Q2 = -0.8000791217568674135274814656e8;
/** tan2 quotient constant. */
private static final double Q3 = 0.8232855955751828560307269007e6;
/** tan2 quotient constant. */
private static final double Q4 = -0.2396576810261093558391373322e4;
/**
* The threshold to switch to using {@link Math#expm1(double)}. The following
* table shows the mean (max) ULP difference between using expm1 and exp using
* random +/-x with different exponents (n=2^30):
*
*
* x exp positive x negative x
* 64.0 6 0.10004021506756544 (2) 0.0 (0)
* 32.0 5 0.11177831795066595 (2) 0.0 (0)
* 16.0 4 0.0986650362610817 (2) 9.313225746154785E-10 (1)
* 8.0 3 0.09863092936575413 (2) 4.9658119678497314E-6 (1)
* 4.0 2 0.10015273280441761 (2) 4.547201097011566E-4 (1)
* 2.0 1 0.14359260816127062 (2) 0.005623611621558666 (2)
* 1.0 0 0.20160607434809208 (2) 0.03312791418284178 (2)
* 0.5 -1 0.3993037799373269 (2) 0.28186883218586445 (2)
* 0.25 -2 0.6307008266448975 (2) 0.5192863345146179 (2)
* 0.125 -3 1.3862918205559254 (4) 1.386285437270999 (4)
* 0.0625 -4 2.772640804760158 (8) 2.772612397558987 (8)
*
*
* The threshold of 0.5 has a mean ULP below 0.5 and max ULP of 2. The
* transition is monotonic. Neither is true for the next threshold of 0.25.
*/
private static final double SWITCH_TO_EXPM1 = 0.5;
/** No instances. */
private SpecialMath() {}
/**
* Evaluate {@code (exp(x) - 1) / x}. For {@code x} in the range {@code [-inf, inf]} returns
* a result in {@code [0, inf]}.
*
*
* - For {@code x=-inf} this returns {@code 0}.
*
- For {@code x=0} this returns {@code 1}.
*
- For {@code x=inf} this returns {@code inf}.
*
- For {@code x=nan} this returns {@code nan}.
*
*
* This corrects {@code 0 / 0} and {@code inf / inf} division from
* {@code NaN} to either {@code 1} or the upper bound respectively.
*
* @param x value to evaluate
* @return {@code (exp(x) - 1) / x}.
*/
static double d2(double x) {
// Here expm1 is only used when use of expm1 and exp consistently
// compute different results by more than 0.5 ULP.
if (Math.abs(x) < SWITCH_TO_EXPM1) {
// Deliberate comparison to floating-point zero
if (x == 0) {
// Avoid 0 / 0 error
return 1.0;
}
return Math.expm1(x) / x;
}
// No use of expm1. Accuracy as x moves away from 0 is not required as the result
// is divided by x and the accuracy of the final result is within a few ULP.
if (x < Double.POSITIVE_INFINITY) {
return (Math.exp(x) - 1) / x;
}
// Upper bound (or NaN)
return x;
}
/**
* Evaluate {@code tan(x) / x}.
*
*
For {@code x} in the range {@code [0, pi/4]} this returns
* a value in the range {@code [1, 4 / pi]}.
*
*
The following properties are desirable for the CMS algorithm:
*
*
* - For {@code x=0} this returns {@code 1}.
*
- For {@code x=pi/4} this returns {@code 4/pi}.
*
- For {@code x=pi/4} this multiplied by {@code x} returns {@code 1}.
*
*
* This method is called by the CMS algorithm when {@code x < pi/4}.
* In this case the method is almost as accurate as {@code Math.tan(x) / x}, does
* not require checking for {@code x=0} and is faster.
*
* @param x the x
* @return {@code tan(x) / x}.
*/
static double tan2(double x) {
if (Math.abs(x) > PI_4) {
// Reduction is not supported. Delegate to the JDK.
return Math.tan(x) / x;
}
// Testing with approximation 4283 from Hart et al, as used in the RSTAB
// routine, showed the method was not accurate enough for use with
// double computation. Hart et al state it has max relative error = 1e-10.66.
// For tan(x) / x with x in [0, pi/4] values outside [1, 4/pi] were computed.
// When testing verses Math.tan(x) the mean ULP difference is 93436.3.
// Approximation 4288 from Hart et al (1968, P. 252).
// Max relative error = 1e-26.68 (for tan(x)).
// When testing verses Math.tan(x) the mean ULP difference is 0.590597.
// The approximation is defined as:
// tan(x*pi/4) = x * P(x^2) / Q(x^2)
// with P and Q polynomials of x squared.
//
// To create tan(x):
// tan(x) = xi * P(xi^2) / Q(xi^2), xi = x * 4/pi
// tan(x) / x = xi * P(xi^2) / Q(xi^2) / x
// tan(x) / x = 4/pi * (P(xi^2) / Q(xi^2))
// = P(xi^2) / (pi/4 * Q(xi^2))
// The later has a smaller mean ULP difference to Math.tan(x) / x.
final double xi = x * FOUR_PI;
// Use the power form with a reverse summation order to have smaller
// magnitude terms first. Note: x < 1 so greater powers are smaller.
// This has essentially the same accuracy as the nested form of the polynomials
// for a marginal performance increase. See JMH examples for performance tests.
final double x2 = xi * xi;
final double x4 = x2 * x2;
final double x6 = x4 * x2;
final double x8 = x4 * x4;
return (x8 * P4 + x6 * P3 + x4 * P2 + x2 * P1 + P0) /
(PI_4 * (x8 * x2 + x8 * Q4 + x6 * Q3 + x4 * Q2 + x2 * Q1 + Q0));
}
}
/**
* @param rng Generator of uniformly distributed random numbers.
*/
StableSampler(UniformRandomProvider rng) {
this.rng = rng;
}
/**
* Generate a sample from a stable distribution.
*
*
The distribution uses the 0-parameterization: S(alpha, beta, gamma, delta; 0).
*/
@Override
public abstract double sample();
/** {@inheritDoc} */
// Redeclare the signature to return a StableSampler not a SharedStateContinuousSampler
@Override
public abstract StableSampler withUniformRandomProvider(UniformRandomProvider rng);
/**
* Generates a {@code long} value.
* Used by algorithm implementations without exposing access to the RNG.
*
* @return the next random value
*/
long nextLong() {
return rng.nextLong();
}
/** {@inheritDoc} */
@Override
public String toString() {
// All variations use the same string representation, i.e. no changes
// for the Gaussian, Levy or Cauchy case.
return "Stable deviate [" + rng.toString() + "]";
}
/**
* Creates a standardized sampler of a stable distribution with zero location and unit scale.
*
*
Special cases:
*
*
* - {@code alpha=2} returns a Gaussian distribution sampler with
* {@code mean=0} and {@code variance=2} (Note: {@code beta} has no effect on the distribution).
*
- {@code alpha=1} and {@code beta=0} returns a Cauchy distribution sampler with
* {@code location=0} and {@code scale=1}.
*
- {@code alpha=0.5} and {@code beta=1} returns a Levy distribution sampler with
* {@code location=-1} and {@code 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 {@code 1 - alpha} in the interval {@code [-1, 1)}.
*
* @param rng Generator of uniformly distributed random numbers.
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @return the sampler
* @throws IllegalArgumentException if {@code 1 - alpha < -1}; or {@code 1 - alpha >= 1};
* or {@code beta < -1}; or {@code beta > 1}.
*/
public static StableSampler of(UniformRandomProvider rng,
double alpha,
double beta) {
validateParameters(alpha, beta);
return create(rng, alpha, beta);
}
/**
* Creates a sampler of a stable distribution. This applies a transformation to the
* standardized sampler.
*
*
The random variable \( X \) has
* the stable distribution \( S(\alpha, \beta, \gamma, \sigma; 0) \) 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 {@code 1 - alpha} in the interval {@code [-1, 1)}.
*
* @param rng Generator of uniformly distributed random numbers.
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @param gamma Scale parameter. Must be strictly positive and finite.
* @param delta Location parameter. Must be finite.
* @return the sampler
* @throws IllegalArgumentException if {@code 1 - alpha < -1}; or {@code 1 - alpha >= 1};
* or {@code beta < -1}; or {@code beta > 1}; or {@code gamma <= 0}; or
* {@code gamma} or {@code delta} are not finite.
* @see #of(UniformRandomProvider, double, double)
*/
public static StableSampler of(UniformRandomProvider rng,
double alpha,
double beta,
double gamma,
double delta) {
validateParameters(alpha, beta, gamma, delta);
// Choose the algorithm.
// Reuse the special cases as they have transformation support.
if (alpha == ALPHA_GAUSSIAN) {
// Note: beta has no effect and is ignored.
return new GaussianStableSampler(rng, gamma, delta);
}
// Note: As beta -> 0 the result cannot be computed differently to beta = 0.
if (alpha == ALPHA_CAUCHY && CMSStableSampler.getTau(ALPHA_CAUCHY, beta) == TAU_ZERO) {
return new CauchyStableSampler(rng, gamma, delta);
}
if (alpha == ALPHA_LEVY && Math.abs(beta) == BETA_LEVY) {
// Support mirroring for negative beta by inverting the beta=1 Levy sample
// using a negative gamma. Note: The delta is not mirrored as it is a shift
// applied to the scaled and mirrored distribution.
return new LevyStableSampler(rng, beta * gamma, delta);
}
// Standardized sampler
final StableSampler sampler = create(rng, alpha, beta);
// Transform
return new TransformedStableSampler(sampler, gamma, delta);
}
/**
* Creates a standardized sampler of a stable distribution with zero location and unit scale.
*
* @param rng Generator of uniformly distributed random numbers.
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @return the sampler
*/
private static StableSampler create(UniformRandomProvider rng,
double alpha,
double beta) {
// Choose the algorithm.
// The special case samplers have transformation support and use gamma=1.0, delta=0.0.
// As alpha -> 0 the computation increasingly requires correction
// of infinity to the distribution support.
if (alpha == ALPHA_GAUSSIAN) {
// Note: beta has no effect and is ignored.
return new GaussianStableSampler(rng, GAMMA_1, DELTA_0);
}
// Note: As beta -> 0 the result cannot be computed differently to beta = 0.
// This is based on the computation factor tau:
final double tau = CMSStableSampler.getTau(alpha, beta);
if (tau == TAU_ZERO) {
// Symmetric case (beta skew parameter is effectively zero)
if (alpha == ALPHA_CAUCHY) {
return new CauchyStableSampler(rng, GAMMA_1, DELTA_0);
}
if (alpha <= ALPHA_SMALL) {
// alpha -> 0 requires robust error correction
return new Beta0WeronStableSampler(rng, alpha);
}
return new Beta0CMSStableSampler(rng, alpha);
}
// Here beta is significant.
if (alpha == 1) {
return new Alpha1CMSStableSampler(rng, beta);
}
if (alpha == ALPHA_LEVY && Math.abs(beta) == BETA_LEVY) {
// Support mirroring for negative beta by inverting the beta=1 Levy sample
// using a negative gamma. Note: The delta is not mirrored as it is a shift
// applied to the scaled and mirrored distribution.
return new LevyStableSampler(rng, beta, DELTA_0);
}
if (alpha <= ALPHA_SMALL) {
// alpha -> 0 requires robust error correction
return new WeronStableSampler(rng, alpha, beta);
}
return new CMSStableSampler(rng, alpha, beta);
}
/**
* Validate the parameters are in the correct range.
*
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @throws IllegalArgumentException if {@code 1 - alpha < -1}; or {@code 1 - alpha >= 1};
* or {@code beta < -1}; or {@code beta > 1}.
*/
private static void validateParameters(double alpha, double beta) {
// The epsilon (1-alpha) value must be in the interval [-1, 1).
// Logic inversion will identify NaN
final double eps = 1 - alpha;
if (!(-1 <= eps && eps < 1)) {
throw new IllegalArgumentException("alpha is not in the interval (0, 2]: " + alpha);
}
if (!(-1 <= beta && beta <= 1)) {
throw new IllegalArgumentException("beta is not in the interval [-1, 1]: " + beta);
}
}
/**
* Validate the parameters are in the correct range.
*
* @param alpha Stability parameter. Must be in the interval {@code (0, 2]}.
* @param beta Skewness parameter. Must be in the interval {@code [-1, 1]}.
* @param gamma Scale parameter. Must be strictly positive and finite.
* @param delta Location parameter. Must be finite.
* @throws IllegalArgumentException if {@code 1 - alpha < -1}; or {@code 1 - alpha >= 1};
* or {@code beta < -1}; or {@code beta > 1}; or {@code gamma <= 0}; or
* {@code gamma} or {@code delta} are not finite.
*/
private static void validateParameters(double alpha, double beta,
double gamma, double delta) {
validateParameters(alpha, beta);
// Logic inversion will identify NaN
if (!(0 < gamma && gamma <= Double.MAX_VALUE)) {
throw new IllegalArgumentException("gamma is not strictly positive and finite: " + gamma);
}
if (!Double.isFinite(delta)) {
throw new IllegalArgumentException("delta is not finite: " + delta);
}
}
}