// [TestCase(0.99999999999999999, 8.4937932241095980744447188132289548161213991737094E+0)]
public void GetInverseCdfValue_TestCase_BenchmarkResult(double probability, double expected)
{
double actual = StandardNormalDistribution.GetInverseCdfValue(probability);
Assert.That(actual, Is.EqualTo(expected).Within(1E-7)); // low tolerance only!
}
public void GetCdfValue_TestCase_BenchmarkResult(double x, double expected)
{
double actual = StandardNormalDistribution.GetCdfValue(x);
Assert.That(actual, Is.EqualTo(expected).Within(1E-14));
}
/// <summary>Gets a specific value of the cummulative distribution function.
/// </summary>
/// <param name="x">The value where to evaluate.</param>
/// <returns>The specified value of the cummulative distribution function.</returns>
public double GetCdfValue(double x)
{
return(StandardNormalDistribution.GetCdfValue((x - Mu) / Sigma));
}
/// <summary>Gets a specific value of the inverse of the cumulative distribution function.
/// </summary>
/// <param name="probability">The probability where to evaluate.</param>
/// <returns>The specified value of the inverse of the cumulative distribution function.</returns>
public double GetInverseCdfValue(double probability)
{
return(Math.Exp(StandardNormalDistribution.GetInverseCdfValue(probability) * Sigma + Mu) + Shift);
}
/// <summary>Evaluate the cummulative distribution function of a bivariate normal random variable
/// at a given point.
/// </summary>
/// <param name="a">The first part of the evaluation point.</param>
/// <param name="b">The second part of the evaluation point.</param>
/// <param name="rho">The correlation coefficient.</param>
/// <returns>Returns the value of the double-integral
/// <para>
/// N_2(a,b,\rho) = \frac{1}{ 2\pi \sqrt{1-\rho^2}} * \int_{-\infty}^a \int_{-\infty}^b exp\bigl[ - \frac{ x^2 - 2 \rho xy +y^2}{2 (1-\rho^2)} \bigr] dy dx.
/// </para></returns>
/// <remarks>This function is based on the method described by
/// Drezner, Z and G.O. Wesolowsky, (1989), On the computation of the bivariate normal integral,
/// Journal of Statist. Comput. Simul. 35, pp. 101-107,
/// with major modifications for double precision, and for |R| close to 1.
/// This implementation is based on the Fortran function BVND of A. Genz,
/// http://www.sci.wsu.edu/math/faculty/genz/homepage.</remarks>
public static double StandardCDFValue(double a, double b, double rho)
{
double cdfValue = 0.0;
double absOfRho = Math.Abs(rho);
int gaussLegendreRule;
if (absOfRho < 0.3)
{
gaussLegendreRule = (int)eGaussLegendreRule.GaussLegendre6;
}
else if (absOfRho < 0.75)
{
gaussLegendreRule = (int)eGaussLegendreRule.GaussLegendre12;
}
else
{
gaussLegendreRule = (int)eGaussLegendreRule.GaussLegendre20;
}
int numberOfEvaluations = sm_GaussWeights[gaussLegendreRule].Length;
double[] gaussLegendreWeights = sm_GaussWeights[gaussLegendreRule];
double[] gaussLegendreEvaluationPoints = sm_GaussEvaluationPoints[gaussLegendreRule];
double aTimesb = a * b;
if (absOfRho < 0.925)
{
if (absOfRho > 0)
{
double halfOfaSquarePlusbSquare = (a * a + b * b) / 2;
double rhoAsin = Math.Asin(rho);
for (int i = 0; i < numberOfEvaluations; i++)
{
for (int j = -1; j <= 2; j += 2)
{
double tempValue = Math.Sin(rhoAsin * (j * gaussLegendreEvaluationPoints[i] + 1) / 2);
cdfValue = cdfValue + gaussLegendreWeights[i] * Math.Exp((tempValue * aTimesb - halfOfaSquarePlusbSquare) / (1 - tempValue * tempValue));
}
}
cdfValue *= rhoAsin / (2 * MathConsts.TwoPi);
}
return(cdfValue + StandardNormalDistribution.GetCdfValue(a) * StandardNormalDistribution.GetCdfValue(b));
}
else
{
if (rho < 0)
{
aTimesb = -aTimesb;
}
if (absOfRho < 1)
{
double AS = (1 - rho) * (1 + rho);
double A = Math.Sqrt(AS);
double BS = (a - b) * (a - b);
double C = (4 - aTimesb) / 8;
double D = (12 - aTimesb) / 16;
double ASR = -(BS / AS + aTimesb) / 2;
if (ASR > -100)
{
cdfValue = A * Math.Exp(ASR) * (1 - C * (BS - AS) * (1 - D * BS / 5) / 3 + C * D * AS * AS / 5);
}
if (-aTimesb < 100)
{
double B = Math.Sqrt(BS);
cdfValue = cdfValue - Math.Exp(-aTimesb / 2) * MathConsts.SqrtTwoPi * StandardNormalDistribution.GetCdfValue(-B / A) * B * (1 - C * BS * (1 - D * BS / 5) / 3);
}
A = A / 2;
for (int i = 0; i < numberOfEvaluations; i++)
{
for (int j = -1; j <= 2; j += 2)
{
double XS = (A * (j * gaussLegendreEvaluationPoints[i] + 1)) * (A * (j * gaussLegendreEvaluationPoints[i] + 1));
double RS = Math.Sqrt(1 - XS);
ASR = -(BS / XS + aTimesb) / 2;
if (ASR > -100)
{
cdfValue = cdfValue + A * gaussLegendreWeights[i] * Math.Exp(ASR) * (Math.Exp(-aTimesb * (1 - RS) / (2 * (1 + RS))) / RS - (1 + C * XS * (1 + D * XS)));
}
}
}
cdfValue = -cdfValue / MathConsts.TwoPi;
}
if (rho > 0)
{
cdfValue = cdfValue + StandardNormalDistribution.GetCdfValue(Math.Min(a, b));
}
else
{
cdfValue = -cdfValue;
if (b < a)
{
cdfValue = cdfValue + StandardNormalDistribution.GetCdfValue(-b) - StandardNormalDistribution.GetCdfValue(-a);
}
}
}
return(cdfValue);
}
