/// <summary>
/// Compute the regularized upper incomplete Gamma function by a series expansion
/// </summary>
/// <param name="a">The shape parameter, > 0</param>
/// <param name="x">The lower bound of the integral, >= 0</param>
/// <returns></returns>
private static double GammaUpperSeries(double a, double x)
{
// this series should only be applied when x is small
// the series is: 1 - x^a sum_{k=0}^inf (-x)^k /(k! Gamma(a+k+1))
// = (1 - 1/Gamma(a+1)) + (1 - x^a)/Gamma(a+1) - x^a sum_{k=1}^inf (-x)^k/(k! Gamma(a+k+1))
double xaMinus1 = MMath.ExpMinus1(a * Math.Log(x));
double aReciprocalFactorial, aReciprocalFactorialMinus1;
if (a > 0.3)
{
aReciprocalFactorial = 1 / MMath.Gamma(a + 1);
aReciprocalFactorialMinus1 = aReciprocalFactorial - 1;
}
else
{
aReciprocalFactorialMinus1 = ReciprocalFactorialMinus1(a);
aReciprocalFactorial = 1 + aReciprocalFactorialMinus1;
}
// offset = 1 - x^a/Gamma(a+1)
double offset = -xaMinus1 * aReciprocalFactorial - aReciprocalFactorialMinus1;
double scale = 1 - offset;
double term = x / (a + 1) * a;
double sum = term;
for (int i = 1; i < 1000; i++)
{
term *= -(a + i) * x / ((a + i + 1) * (i + 1));
double sumOld = sum;
sum += term;
//Console.WriteLine("{0}: {1}", i, sum);
if (AreEqual(sum, sumOld))
{
return(scale * sum + offset);
}
}
throw new Exception(string.Format("GammaUpperSeries not converging for a={0} x={1}", a, x));
}
/// <summary>
/// Sample from a Gaussian(0,1) truncated at the given upper and lower bounds
/// </summary>
/// <param name="lowerBound">Can be -Infinity.</param>
/// <param name="upperBound">Must be >= <paramref name="lowerBound"/>. Can be Infinity.</param>
/// <returns>A real number >= <paramref name="lowerBound"/> and < <paramref name="upperBound"/></returns>
public static double NormalBetween(double lowerBound, double upperBound, IPolyrand random = null)
{
if (double.IsNaN(lowerBound))
{
throw new ArgumentException("lowerBound is NaN");
}
if (double.IsNaN(upperBound))
{
throw new ArgumentException("upperBound is NaN");
}
double delta = upperBound - lowerBound;
if (delta == 0)
{
return(lowerBound);
}
if (delta < 0)
{
throw new ArgumentException("upperBound (" + upperBound + ") < lowerBound (" + lowerBound + ")");
}
// Switch between the following 3 options:
// 1. Gaussian rejection, with acceptance rate Z = NormalCdf(upperBound) - NormalCdf(lowerBound)
// 2. Uniform rejection, with acceptance rate sqrt(2*pi)*Z/delta if the interval contains 0
// 3. Truncated exponential rejection, with acceptance rate
// = sqrt(2*pi)*Z*lambda*exp(-lambda^2/2)/(exp(-lambda*lowerBound)-exp(-lambda*upperBound))
// = sqrt(2*pi)*Z*lowerBound*exp(lowerBound^2/2)/(1-exp(-lowerBound*(upperBound-lowerBound)))
// (3) has the highest acceptance rate under the following conditions:
// lowerBound > 0.5 or (lowerBound > 0 and delta < 2.5)
// (2) has the highest acceptance rate if the interval contains 0 and delta < sqrt(2*pi)
// (1) has the highest acceptance rate otherwise
if (lowerBound > 0.5 || (lowerBound > 0 && delta < 2.5))
{
// Rejection sampler using truncated exponential proposal
double lambda = lowerBound;
double s = MMath.ExpMinus1(-lambda * delta);
double c = 2 * lambda * lambda;
while (true)
{
double x = -MMath.Log1Plus(s * NextDouble(random));
double u = -System.Math.Log(NextDouble(random));
if (c * u > x * x)
{
return(x / lambda + lowerBound);
}
}
throw new Exception("failed to sample");
}
else if (upperBound < -0.5 || (upperBound < 0 && delta < 2.5))
{
return(-NormalBetween(-upperBound, -lowerBound));
}
else if (lowerBound <= 0 && upperBound >= 0 && delta < MMath.Sqrt2PI)
{
// Uniform rejection
while (true)
{
double x = NextDouble(random) * delta + lowerBound;
double u = -System.Math.Log(NextDouble(random));
if (2 * u > x * x)
{
return(x);
}
}
}
else
{
// Gaussian rejection
while (true)
{
double x = Rand.Normal(random);
if (x >= lowerBound && x < upperBound)
{
return(x);
}
}
}
}
