/// <summary>
/// Computes <c>GammaLn(x) - (x-0.5)*log(x) + x - 0.5*log(2*pi)</c> for x >= 10
/// </summary>
/// <param name="x">A real number >= 10</param>
/// <returns></returns>
private static double GammaLnSeries(double x)
{
// GammaLnSeries(10) = 0.008330563433362871
if (x < 10)
{
return(MMath.GammaLn(x) - (x - 0.5) * Math.Log(x) + x - LnSqrt2PI);
}
else
{
// the series is: sum_{i=1}^inf B_{2i} / (2i*(2i-1)*x^(2i-1))
double sum = 0;
double term = 1.0 / x;
double delta = term * term;
for (int i = 0; i < c_gammaln_series.Length; i++)
{
sum += c_gammaln_series[i] * term;
term *= delta;
}
return(sum);
}
}
public static int Poisson(double mean, IPolyrand random = null)
{
// TODO: There are more efficient samplers
if (mean < 0)
{
throw new ArgumentException("mean < 0");
}
else if (mean == 0.0)
{
return(0);
}
else if (mean < 10)
{
double L = System.Math.Exp(-mean);
double p = 1.0;
int k = 0;
do
{
k++;
p *= NextDouble(random);
} while (p > L);
return(k - 1);
}
else
{
// mean >= 10
// Devroye ch10.3, with corrections
// Reference: "Non-Uniform Random Variate Generation" by Luc Devroye (1986)
double mu = System.Math.Floor(mean);
double muLogFact = MMath.GammaLn(mu + 1);
double logMeanMu = System.Math.Log(mean / mu);
double delta = System.Math.Max(6, System.Math.Min(mu, System.Math.Sqrt(2 * mu * System.Math.Log(128 * mu / System.Math.PI))));
double c1 = System.Math.Sqrt(System.Math.PI * mu / 2);
double c2 = c1 + System.Math.Sqrt(System.Math.PI * (mu + delta / 2) / 2) * System.Math.Exp(1 / (2 * mu + delta));
double c3 = c2 + 2;
double c4 = c3 + System.Math.Exp(1.0 / 78);
double c = c4 + 2 / delta * (2 * mu + delta) * System.Math.Exp(-delta / (2 * mu + delta) * (1 + delta / 2));
while (true)
{
double u = NextDouble(random) * c;
double x, w;
if (u <= c1)
{
double n = Rand.Normal(random);
double y = -System.Math.Abs(n) * System.Math.Sqrt(mu) - 1;
x = System.Math.Floor(y);
if (x < -mu)
{
continue;
}
w = -n * n / 2;
}
else if (u <= c2)
{
double n = Rand.Normal(random);
double y = 1 + System.Math.Abs(n) * System.Math.Sqrt(mu + delta / 2);
x = System.Math.Ceiling(y);
if (x > delta)
{
continue;
}
w = (2 - y) * y / (2 * mu + delta);
}
else if (u <= c3)
{
x = 0;
w = 0;
}
else if (u <= c4)
{
x = 1;
w = 0;
}
else
{
double v = -System.Math.Log(NextDouble(random));
double y = delta + v * 2 / delta * (2 * mu + delta);
x = System.Math.Ceiling(y);
w = -delta / (2 * mu + delta) * (1 + y / 2);
}
double e = -System.Math.Log(NextDouble(random));
w -= e + x * logMeanMu;
double qx = x * System.Math.Log(mu) - MMath.GammaLn(mu + x + 1) + muLogFact;
if (w <= qx)
{
return((int)System.Math.Round(x + mu));
}
}
}
}
