public void BetaInversion()
{
// test that beta distribution is accurately inverted over a wide range of a, b, P
foreach (double a in TestUtilities.GenerateRealValues(0.01, 100.0, 8)) {
foreach (double b in cs) {
BetaDistribution B = new BetaDistribution(a, b);
foreach (double P in probabilities) {
Console.WriteLine("a={0} b={1} P={2}", a, b, P);
double x = B.InverseLeftProbability(P);
Console.WriteLine(" x={0} P(x)={1}", x, B.LeftProbability(x));
// we would like to test that P(x) = P, but floating point limitations prevent us from meeting this standard
// P(x) changes so fast at extremes that sometimes even the minimal change in x causes a change
// in P(x) larger than our target precision; so instead we test that our routine gets us
// as close as it can, by checking that P(x-e) < P < P(x+e)
double Px = B.LeftProbability(x);
double Pxm = B.LeftProbability(Math.Min(0.0, x * (1.0 - TestUtilities.TargetPrecision)));
double Pxp = B.LeftProbability(Math.Max(x * (1.0 + TestUtilities.TargetPrecision), 1.0));
Assert.IsTrue((Pxm <= P) && (P <= Pxp));
}
}
}
}
/// <inheritdoc />
public override double InverseLeftProbability(double P)
{
if ((P < 0.0) || (P > 1.0))
{
throw new ArgumentOutOfRangeException(nameof(P));
}
BetaDistribution b = new BetaDistribution(nu / 2.0, 1.0 / 2.0);
if (P <= 0.5)
{
double x = b.InverseLeftProbability(2.0 * P);
return(-Math.Sqrt(nu * (1.0 - x) / x));
}
else
{
double Q = 1.0 - P;
double x = b.InverseLeftProbability(2.0 * Q);
return(Math.Sqrt(nu * (1.0 - x) / x));
}
}
public void TransformedBetaMoments()
{
// For x ~ B(a,b), kth cumulant of \ln\left(\frac{x}{1-x}\right) = \psi^{(k-1)}(a) \pm \psi^{(k-1)}(b)
BetaDistribution D = new BetaDistribution(2.0, 3.0);
double m = D.ExpectationValue(x => Math.Log(x / (1.0 - x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m, AdvancedMath.Psi(D.Alpha) - AdvancedMath.Psi(D.Beta)));
double c2 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 2));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c2, AdvancedMath.Psi(1, D.Alpha) + AdvancedMath.Psi(1, D.Beta)));
double c3 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 3));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c3, AdvancedMath.Psi(2, D.Alpha) - AdvancedMath.Psi(2, D.Beta)));
double c4 = D.ExpectationValue(x => MoreMath.Pow(Math.Log(x / (1.0 - x)) - m, 4));
Assert.IsTrue(TestUtilities.IsNearlyEqual(c4 - 3.0 * c2 * c2, AdvancedMath.Psi(3, D.Alpha) + AdvancedMath.Psi(3, D.Beta)));
}
public void Repro()
{
double I = 0.9998063099306;
double a = 0.00034509911609819255;
double b = 6.8453983996634218;
var bd = new BetaDistribution(a, b);
double x = bd.InverseLeftProbability(I);
Console.WriteLine(x);
Console.WriteLine(bd.LeftProbability(x));
Console.WriteLine(I);
}
public void BetaExercise()
{
double[] c = new double[] { 1.0 / 32768.0, 1.0 / 1024.0, 1.0 / 64.0, 1.0 / 8.0, 1.0 / 2.0, 1.0, 2.0, 8.0, 64.0, 1024.0, 32768.0 };
double[] p = new double[] { 1.0 / 65536.0, 1.0 / 256.0, 1.0 / 16.0, 1.0 / 4.0, 1.0 - 1.0 / 4.0, 1.0 - 1.0 / 16.0, 1.0 - 1.0 / 256.0 };
Stopwatch s = Stopwatch.StartNew();
double sum = 0.0;
foreach (double a in c) {
foreach (double b in c) {
BetaDistribution BD = new BetaDistribution(a, b);
foreach (double x in p) {
//Console.WriteLine("a={0} b={1} P={2}", a, b, x);
//Console.WriteLine(BD.InverseLeftProbability(x));
sum += BD.InverseLeftProbability(x);
}
}
}
s.Stop();
Console.WriteLine(sum);
Console.WriteLine(s.ElapsedMilliseconds);
}
public void BetaFitUncertainty()
{
// check that the uncertainty in reported fit parameters is actually meaningful
// it should be the standard deviation of fit parameter values in a sample of many fits
// define a population distribution
Distribution distribution = new BetaDistribution(1.0 / 3.0, 2.0);
// draw a lot of samples from it; fit each sample and
// record the reported parameter value and error of each
BivariateSample values = new BivariateSample();
BivariateSample uncertainties = new BivariateSample();
for (int i = 0; i < 50; i++) {
Sample sample = CreateSample(distribution, 10, i);
FitResult fit = BetaDistribution.FitToSample(sample);
UncertainValue a = fit.Parameter(0);
UncertainValue b = fit.Parameter(1);
values.Add(a.Value, b.Value);
uncertainties.Add(a.Uncertainty, b.Uncertainty);
}
// the reported errors should agree with the standard deviation of the reported parameters
Assert.IsTrue(values.X.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(uncertainties.X.Mean));
Assert.IsTrue(values.Y.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(uncertainties.Y.Mean));
}
public void BetaFit()
{
// Create a very non-normal Beta distribution
BetaDistribution B = new BetaDistribution(0.25, 3.0);
// Sample from it
Sample S = CreateSample(B, 128);
// Fit the sample to a beta distribution
// The fit should agree with the population and the fit should be good
FitResult RB = BetaDistribution.FitToSample(S);
Assert.IsTrue(RB.Parameter(0).ConfidenceInterval(0.99).ClosedContains(B.Alpha));
Assert.IsTrue(RB.Parameter(1).ConfidenceInterval(0.99).ClosedContains(B.Beta));
Assert.IsTrue(RB.GoodnessOfFit.LeftProbability < 0.99);
// Fit to a normal distribution should be bad
FitResult RN = NormalDistribution.FitToSample(S);
Assert.IsTrue(RN.GoodnessOfFit.LeftProbability > 0.99);
}
public void Bug7788()
{
// Beta with high parameters used to have incorrect inverse probabilty,
// which in this case resulted in zero median and consistently zero random values.
Distribution beta = new BetaDistribution(1713.0, 58743.0);
Console.WriteLine(beta.Mean);
Console.WriteLine(beta.StandardDeviation);
Assert.IsTrue(beta.Median != 0.0);
Assert.IsTrue(Math.Abs(beta.Median - beta.Median) <= beta.StandardDeviation);
Random rng = new Random(1);
for (int i = 0; i < 10; i++) {
Assert.IsTrue(beta.GetRandomValue(rng) != 0.0);
}
}
public void Bug7684()
{
// These values caused incomplete Beta to be called with argument outside the interval [0,1].
double I1 = 0.9999902;
double a1 = 0.0000434313636267175;
double b1 = 18474.36071078790000;
BetaDistribution D1 = new BetaDistribution(a1, b1);
double x1 = D1.InverseLeftProbability(I1);
Console.WriteLine("{0} {1} {2}", x1, D1.LeftProbability(x1), I1);
double I2 = 0.9998063099306;
double a2 = 0.00034509911609819255;
double b2 = 6.8453983996634218;
BetaDistribution D2 = new BetaDistribution(a2, b2);
double x2 = D2.InverseLeftProbability(I2);
Console.WriteLine("{0} {1} {2}", x2, D2.LeftProbability(x2), I2);
}
/// <inheritdoc />
public override double InverseLeftProbability(double P)
{
if ((P < 0.0) || (P > 1.0)) throw new ArgumentOutOfRangeException("P");
BetaDistribution b = new BetaDistribution(nu / 2.0, 1.0 / 2.0);
if (P <= 0.5) {
double x = b.InverseLeftProbability(2.0 * P);
return (-Math.Sqrt(nu * (1.0 - x) / x));
} else {
double Q = 1.0 - P;
double x = b.InverseLeftProbability(2.0 * Q);
return (Math.Sqrt(nu * (1.0 - x) / x));
}
}
