public void HypergeometricIncompleteBeta()
{
foreach (double a in abcs)
{
if (a <= 0.0)
{
continue;
}
foreach (double b in abcs)
{
if (b <= 0.0)
{
continue;
}
foreach (double x in xs)
{
if ((x < 0.0) || (x > 1.0))
{
continue;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.Pow(x, a) * Math.Pow(1.0 - x, b) * AdvancedMath.Hypergeometric2F1(a + b, 1.0, a + 1.0, x) / a,
AdvancedMath.Beta(a, b, x)
));
}
}
}
}
public void HypergeometricIntegral()
{
// A&S 15.3.1
// F(a, b, c, x) = \frac{\Gamma(c)}{\Gamma(b) \Gamma(c - b)}
// \int_{0}^{1} \! dt \, t^{b-1} (1 - t)^{c - b - 1} (1 - x t)^{-a}
// Choose limits on a, b, c so that singularities of integrand are numerically integrable.
foreach (double a in TestUtilities.GenerateRealValues(1.0, 10.0, 2))
{
foreach (double b in TestUtilities.GenerateRealValues(0.6, 10.0, 2))
{
foreach (double c in TestUtilities.GenerateRealValues(b + 0.6, 10.0, 2))
{
foreach (double x in xs)
{
double I = FunctionMath.Integrate(
t => Math.Pow(t, b - 1.0) * Math.Pow(1.0 - t, c - b - 1.0) * Math.Pow(1.0 - x * t, -a),
Interval.FromEndpoints(0.0, 1.0)
);
double B = AdvancedMath.Beta(b, c - b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.Hypergeometric2F1(a, b, c, x), I / B));
}
}
}
}
}
/// <summary>
/// Initializes a new β distribution.
/// </summary>
/// <param name="alpha">The left shape parameter, which controls the form of the distribution near x=0.</param>
/// <param name="beta">The right shape parameter, which controls the form of the distribution near x=1.</param>
/// <remarks>
/// <para>The <paramref name="alpha"/> shape parameter controls the form of the distribution near x=0. The
/// <paramref name="beta"/> shape parameter controls the form of the distribution near z=1. If a shape parameter
/// is less than one, the PDF diverges on the side of the distribution it controls. If a shape parameter
/// is greater than one, the PDF goes to zero on the side of the distribution it controls. If the left and right
/// shape parameters are equal, the distribution is symmetric about x=1/2.</para>
/// </remarks>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="alpha"/> or <paramref name="beta"/> is non-positive.</exception>
public BetaDistribution(double alpha, double beta)
{
if (alpha <= 0.0)
{
throw new ArgumentOutOfRangeException(nameof(alpha));
}
if (beta <= 0.0)
{
throw new ArgumentOutOfRangeException(nameof(beta));
}
this.a = alpha;
this.b = beta;
// cache value of B(alpha, beta) to avoid having to re-calculate it whenever needed
this.bigB = AdvancedMath.Beta(alpha, beta);
// get a beta generator
if (alpha < 0.75 && beta < 0.75)
{
this.betaRng = new JoehnkBetaGenerator(alpha, beta);
}
else if (alpha > 1.0 && beta > 1.0)
{
this.betaRng = new ChengBetaGenerator(alpha, beta);
}
else
{
this.betaRng = DeviateGeneratorFactory.GetBetaGenerator(alpha, beta);
}
// get a beta inverter
this.betaInverter = new BetaInverter(alpha, beta);
}
public void EllipticFBetaRelationship()
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(phi, 1.0 / Math.Sqrt(2.0)),
AdvancedMath.Beta(1.0 / 2.0, 1.0 / 4.0, 1.0 - MoreMath.Pow(Math.Cos(phi), 4)) / Math.Sqrt(8.0)
));
}
}
/// <inheritdoc />
public override double ProbabilityDensity(double x)
{
if (Math.Abs(x) > 1.0)
{
return(0.0);
}
else
{
return(Math.Pow((1.0 - x) * (1.0 + x), (n - 4) / 2.0) / AdvancedMath.Beta(0.5, (n - 2) / 2.0));
}
}
/// <inheritdoc />
public override double LeftProbability(double x)
{
double t2 = x * x;
if (x < 0.0)
{
return(0.5 * AdvancedMath.Beta(0.5 * nu, 0.5, nu / (nu + t2)) / AdvancedMath.Beta(0.5, 0.5 * nu));
}
else
{
return(0.5 * (1.0 + AdvancedMath.Beta(0.5, 0.5 * nu, t2 / (nu + t2)) / AdvancedMath.Beta(0.5, 0.5 * nu)));
}
}
/// <inheritdoc />
public override double RightExclusiveProbability(int k)
{
if (k < 0)
{
return(1.0);
}
else if (k >= n)
{
return(0.0);
}
else
{
// use direct summation in tails
return(AdvancedMath.Beta(k + 1, n - k, p) / AdvancedMath.Beta(k + 1, n - k));
}
}
/// <inheritdoc />
public override double ProbabilityMass(int k)
{
if (k < 0)
{
return(0.0);
}
else
{
return(
//AdvancedIntegerMath.BinomialCoefficient(k + r - 1, k) *
MoreMath.Pow(p, k) * Math.Pow(q, r) /
AdvancedMath.Beta(r, k + 1) / (k + r)
);
// note: n \choose k = \frac{1}{(n+1) B(n - k + 1, k + 1)}
}
}
/// <inheritdoc />
public override double LeftProbability(double x)
{
if (x <= -1.0)
{
return(0.0);
}
else if (x < 0.0)
{
return(AdvancedMath.Beta((n - 2) / 2.0, 0.5, (1.0 - x) * (1.0 + x)) / AdvancedMath.Beta((n - 2) / 2.0, 0.5) / 2.0);
}
else if (x < 1.0)
{
return((1.0 + AdvancedMath.Beta(0.5, (n - 2) / 2.0, x * x) / AdvancedMath.Beta(0.5, (n - 2) / 2.0)) / 2.0);
}
else
{
return(1.0);
}
}
/// <summary>
/// Initializes a new β distribution.
/// </summary>
/// <param name="alpha">The left shape parameter, which controls the form of the distribution near x=0.</param>
/// <param name="beta">The right shape parameter, which controls the form of the distribution near x=1.</param>
/// <remarks>
/// <para>The <paramref name="alpha"/> shape parameter controls the form of the distribution near x=0. The
/// <paramref name="beta"/> shape parameter controls the form of the distribution near z=1. If a shape parameter
/// is less than one, the PDF diverges on the side of the distribution it controls. If a shape parameter
/// is greater than one, the PDF goes to zero on the side of the distribution it controls. If the left and right
/// shapre parameters are equal, the distribution is symmetric about x=1/2.</para>
/// </remarks>
/// <seealso href="http://en.wikipedia.org/wiki/Beta_distribution" />
public BetaDistribution(double alpha, double beta)
{
if (alpha <= 0.0)
{
throw new ArgumentOutOfRangeException("alpha");
}
if (beta <= 0.0)
{
throw new ArgumentOutOfRangeException("beta");
}
this.alpha = alpha;
this.beta = beta;
// cache value of B(alpha, beta) to avoid having to re-calculate it whenever needed
this.bigB = AdvancedMath.Beta(alpha, beta);
// get a beta generator
this.betaRng = DeviateGeneratorFactory.GetBetaGenerator(alpha, beta);
// get a beta inverter
this.betaInverter = new BetaInverter(alpha, beta);
}
/// <inheritdoc />
public override double ProbabilityDensity(double x)
{
return(Math.Pow(1.0 + x * x / nu, -0.5 * (nu + 1.0)) / AdvancedMath.Beta(0.5, 0.5 * nu) / Math.Sqrt(nu));
}
