/// <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); } } } }