/* The t-distribution has the form
p(x) dx = (Gamma((nu + 1)/2)/(sqrt(pi nu) Gamma(nu/2))
* (1 + (x^2)/nu)^-((nu + 1)/2) dx
The method used here is the one described in Knuth
*/
public static double Sampler(PZRandomUnivariate random, double nu)
{
if (nu <= 2)
{
double Y1 = NormalDistribution.SamplerPolar(random, 0.0, 1.0);
double Y2 = ChiSquareDistribution.Sampler(random, nu);
double t = Y1 / System.Math.Sqrt(Y2 / nu);
return t;
}
else
{
double Y1, Y2, Z, t;
do
{
Y1 = NormalDistribution.SamplerPolar(random, 0.0, 1.0);
Y2 = ExponentialDistribution.Sampler(random, 1 / (nu / 2 - 1));
Z = Y1 * Y1 / (nu - 2);
}
while (1 - Z < 0 || System.Math.Log(-Y2 - Z) > (1 - Z));
/* Note that there is a typo in Knuth's formula, the line below
is taken from the original paper of Marsaglia, Mathematics of
Computation, 34 (1980), p 234-256 */
t = Y1 / System.Math.Sqrt((1 - 2 / nu) * (1 - Z));
return t;
}
}
/// <summary>
/// </summary>
/// <returns></returns>
public static double Sampler(PZRandomUnivariate random, double k)
{
double chisq = 2 * GammaDistribution.SampleMarsagliaAndTsang(random, k / 2, 1.0);
return chisq;
}
public static double Sampler(PZRandomUnivariate random)
{
double r = random.Sample();
return r;
}
/// <summary>
/// U(0, range], return int value
/// </summary>
/// <param name="random"></param>
/// <param name="range"></param>
/// <returns></returns>
public static int SamplerInt(PZRandomUnivariate random, int range)
{
return 0;
}
/// <summary>
/// New version based on Marsaglia and Tsang, "A Simple Method for
/// generating gamma variables", ACM Transactions on Mathematical
/// Software, Vol 26, No 3 (2000), p363-372.
/// Implemented by [email protected], minor modifications for GSL
/// by Brian Gough
/// </summary>
/// <returns></returns>
public static double SampleMarsagliaAndTsang(PZRandomUnivariate random, double alpha, double beta)
{
if (alpha < 1)
{
double u = UniformDistribution.Sampler(random);
return SampleMarsagliaAndTsang(random, 1.0 + alpha, beta) * System.Math.Pow(u, 1.0 / alpha);
}
{
double x, v, u;
double d = alpha - 1.0 / 3.0;
double c = (1.0 / 3.0) / System.Math.Sqrt(d);
while (true)
{
do
{
x = NormalDistribution.SamplerPolar(random, 0.0, 1.0);
v = 1.0 + c * x;
}
while (v <= 0);
v = v * v * v;
u = random.Sample();
if (u < 1 - 0.0331 * x * x * x * x)
break;
if (System.Math.Log(u) < 0.5 * x * x + d * (1 - v + System.Math.Log(v)))
break;
}
return beta * d * v;
}
}
/// <summary>
/// http://en.wikipedia.org/wiki/Gamma_distribution
/// Knuth
/// </summary>
/// <returns></returns>
public static double SampleKnuth(PZRandomUnivariate random, double k, double theta)
{
double integralk = System.Math.Floor(k);
double fractionalk = k - integralk;
double xi = 0.0;
if (fractionalk > 0)
{
// Gamma(delta, 1), delta = fractional of k
// step 1
int m = 1;
double delta = fractionalk;
double v0 = System.Math.E / (System.Math.E + delta);
double xim = 0.0;
double etam = 0.0;
while (true)
{
// step 2
double v2mM1 = random.Sample();
double v2m = random.Sample();
// step 3
if (v2mM1 <= v0)
{
// step 4
xim = System.Math.Pow(v2mM1, 1 / delta);
etam = v2m * System.Math.Pow(xim, delta - 1.0);
}
else
{
// ste 5
xim = 1 - System.Math.Log(v2mM1);
etam = v2m * System.Math.Exp(-1.0 * xim);
}
// step 6
if (etam > System.Math.Pow(xim, delta - 1) * System.Math.Exp(-1.0 * xim))
m++;
// return step 2
else
{
// step 7
xi = xim; // xi ~ Gamma(delta, 1)
break;
}
}
}
// Gamma(n, 1), n = integral of k
double sum = 0.0;
double ui;
for (int i = 0; i < integralk; i++)
{
ui = random.Sample();
sum += System.Math.Log(ui);
}
return theta * (xi - sum);
}
/* Ratio method (Kinderman-Monahan); see Knuth v2, 3rd ed, p130.
* K+M, ACM Trans Math Software 3 (1977) 257-260.
*
* [Added by Charles Karney] This is an implementation of Leva's
* modifications to the original K+M method; see:
* J. L. Leva, ACM Trans Math Software 18 (1992) 449-453 and 454-455. */
public static double SamplerRatio(PZRandomUnivariate random, double mu, double sigma)
{
double u, v, x, y, Q;
double s = 0.449871; /* Constants from Leva */
double t = -0.386595;
double a = 0.19600;
double b = 0.25472;
double r1 = 0.27597;
double r2 = 0.27846;
do /* This loop is executed 1.369 times on average */
{
/* Generate a point P = (u, v) uniform in a rectangle enclosing
the K+M region v^2 <= - 4 u^2 log(u). */
/* u in (0, 1] to avoid singularity at u = 0 */
u = 1 - random.Sample();
/* v is in the asymmetric interval [-0.5, 0.5). However v = -0.5
is rejected in the last part of the while clause. The
resulting normal deviate is strictly symmetric about 0
(provided that v is symmetric once v = -0.5 is excluded). */
v = random.Sample() - 0.5;
/* Constant 1.7156 > sqrt(8/e) (for accuracy); but not by too
much (for efficiency). */
v *= 1.7156;
/* Compute Leva's quadratic form Q */
x = u - s;
y = System.Math.Abs(v) - t;
Q = x * x + y * (a * y - b * x);
/* Accept P if Q < r1 (Leva) */
/* Reject P if Q > r2 (Leva) */
/* Accept if v^2 <= -4 u^2 log(u) (K+M) */
/* This final test is executed 0.012 times on average. */
}
while (Q >= r1 && (Q > r2 || v * v > -4 * u * u * System.Math.Log(u)));
double r = mu + sigma * (v / u);
return r; /* Return slope */
}
/* Of the two methods provided below, I think the Polar method is more
* efficient, but only when you are actually producing two random
* deviates. We don't produce two, because then we'd have to save one
* in a static variable for the next call, and that would screws up
* re-entrant or threaded code, so we only produce one. This makes
* the Ratio method suddenly more appealing.
*
* [Added by Charles Karney] We use Leva's implementation of the Ratio
* method which avoids calling log() nearly all the time and makes the
* Ratio method faster than the Polar method (when it produces just one
* result per call). Timing per call (gcc -O2 on 866MHz Pentium,
* average over 10^8 calls)
*
* Polar method: 660 ns
* Ratio method: 368 ns
*
*/
/* Polar (Box-Mueller) method; See Knuth v2, 3rd ed, p122 */
public static double SamplerPolar(PZRandomUnivariate random, double mu, double sigma)
{
double x, y, r2;
do
{
/* choose x,y in uniform square (-1,-1) to (+1,+1) */
x = -1 + 2 * random.Sample();
y = -1 + 2 * random.Sample();
/* see if it is in the unit circle */
r2 = x * x + y * y;
}
while (r2 > 1.0 || r2 == 0);
/* Box-Muller transform */
double r = mu + sigma * y * System.Math.Sqrt(-2.0 * System.Math.Log(r2) / r2);
return r;
}
/* The exponential distribution has the form
p(x) dx = exp(-x/mu) dx/mu
for x = 0 ... +infty */
public static double Sampler(PZRandomUnivariate random, double mu)
{
double e = random.Sample();
double r = -mu * System.Math.Log(e);
return r;
}