/* Copyright 2005 Robert Kern (robert.kern@gmail.com)
*
* Permission is hereby granted, free of charge, to any person obtaining a
* copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be included
* in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
* OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/
/* The implementations of rk_hypergeometric_hyp(), rk_hypergeometric_hrua(),
* and rk_triangular() were adapted from Ivan Frohne's rv.py which has this
* license:
*
* Copyright 1998 by Ivan Frohne; Wasilla, Alaska, U.S.A.
* All Rights Reserved
*
* Permission to use, copy, modify and distribute this software and its
* documentation for any purpose, free of charge, is granted subject to the
* following conditions:
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the software.
*
* THE SOFTWARE AND DOCUMENTATION IS PROVIDED WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO MERCHANTABILITY, FITNESS
* FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHOR
* OR COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM OR DAMAGES IN A CONTRACT
* ACTION, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
* SOFTWARE OR ITS DOCUMENTATION.
*/
#include "distributions.h"
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <limits.h>
#ifndef min
#define min(x,y) ((x<y)?x:y)
#define max(x,y) ((x>y)?x:y)
#endif
#ifndef M_PI
#define M_PI 3.14159265358979323846264338328
#endif
/*
* log-gamma function to support some of these distributions. The
* algorithm comes from SPECFUN by Shanjie Zhang and Jianming Jin and their
* book "Computation of Special Functions", 1996, John Wiley & Sons, Inc.
*/
static double loggam(double x)
{
double x0, x2, xp, gl, gl0;
long k, n;
static double a[10] = {8.333333333333333e-02,-2.777777777777778e-03,
7.936507936507937e-04,-5.952380952380952e-04,
8.417508417508418e-04,-1.917526917526918e-03,
6.410256410256410e-03,-2.955065359477124e-02,
1.796443723688307e-01,-1.39243221690590e+00};
x0 = x;
n = 0;
if ((x == 1.0) || (x == 2.0))
{
return 0.0;
}
else if (x <= 7.0)
{
n = (long)(7 - x);
x0 = x + n;
}
x2 = 1.0/(x0*x0);
xp = 2*M_PI;
gl0 = a[9];
for (k=8; k>=0; k--)
{
gl0 *= x2;
gl0 += a[k];
}
gl = gl0/x0 + 0.5*log(xp) + (x0-0.5)*log(x0) - x0;
if (x <= 7.0)
{
for (k=1; k<=n; k++)
{
gl -= log(x0-1.0);
x0 -= 1.0;
}
}
return gl;
}
double rk_normal(rk_state *state, double loc, double scale)
{
return loc + scale*rk_gauss(state);
}
double rk_standard_exponential(rk_state *state)
{
/* We use -log(1-U) since U is [0, 1) */
return -log(1.0 - rk_double(state));
}
double rk_exponential(rk_state *state, double scale)
{
return scale * rk_standard_exponential(state);
}
double rk_uniform(rk_state *state, double loc, double scale)
{
return loc + scale*rk_double(state);
}
double rk_standard_gamma(rk_state *state, double shape)
{
double b, c;
double U, V, X, Y;
if (shape == 1.0)
{
return rk_standard_exponential(state);
}
else if (shape < 1.0)
{
for (;;)
{
U = rk_double(state);
V = rk_standard_exponential(state);
if (U <= 1.0 - shape)
{
X = pow(U, 1./shape);
if (X <= V)
{
return X;
}
}
else
{
Y = -log((1-U)/shape);
X = pow(1.0 - shape + shape*Y, 1./shape);
if (X <= (V + Y))
{
return X;
}
}
}
}
else
{
b = shape - 1./3.;
c = 1./sqrt(9*b);
for (;;)
{
do
{
X = rk_gauss(state);
V = 1.0 + c*X;
} while (V <= 0.0);
V = V*V*V;
U = rk_double(state);
if (U < 1.0 - 0.0331*(X*X)*(X*X)) return (b*V);
if (log(U) < 0.5*X*X + b*(1. - V + log(V))) return (b*V);
}
}
}
double rk_gamma(rk_state *state, double shape, double scale)
{
return scale * rk_standard_gamma(state, shape);
}
double rk_beta(rk_state *state, double a, double b)
{
double Ga, Gb;
if ((a <= 1.0) && (b <= 1.0))
{
double U, V, X, Y;
/* Use Johnk's algorithm */
while (1)
{
U = rk_double(state);
V = rk_double(state);
X = pow(U, 1.0/a);
Y = pow(V, 1.0/b);
if ((X + Y) <= 1.0)
{
if (X +Y > 0)
{
return X / (X + Y);
}
else
{
double logX = log(U) / a;
double logY = log(V) / b;
double logM = logX > logY ? logX : logY;
logX -= logM;
logY -= logM;
return exp(logX - log(exp(logX) + exp(logY)));
}
}
}
}
else
{
Ga = rk_standard_gamma(state, a);
Gb = rk_standard_gamma(state, b);
return Ga/(Ga + Gb);
}
}
double rk_chisquare(rk_state *state, double df)
{
return 2.0*rk_standard_gamma(state, df/2.0);
}
double rk_noncentral_chisquare(rk_state *state, double df, double nonc)
{
if (nonc == 0){
return rk_chisquare(state, df);
}
if(1 < df)
{
const double Chi2 = rk_chisquare(state, df - 1);
const double N = rk_gauss(state) + sqrt(nonc);
return Chi2 + N*N;
}
else
{
const long i = rk_poisson(state, nonc / 2.0);
return rk_chisquare(state, df + 2 * i);
}
}
double rk_f(rk_state *state, double dfnum, double dfden)
{
return ((rk_chisquare(state, dfnum) * dfden) /
(rk_chisquare(state, dfden) * dfnum));
}
double rk_noncentral_f(rk_state *state, double dfnum, double dfden, double nonc)
{
double t = rk_noncentral_chisquare(state, dfnum, nonc) * dfden;
return t / (rk_chisquare(state, dfden) * dfnum);
}
long rk_binomial_btpe(rk_state *state, long n, double p)
{
double r,q,fm,p1,xm,xl,xr,c,laml,lamr,p2,p3,p4;
double a,u,v,s,F,rho,t,A,nrq,x1,x2,f1,f2,z,z2,w,w2,x;
long m,y,k,i;
if (!(state->has_binomial) ||
(state->nsave != n) ||
(state->psave != p))
{
/* initialize */
state->nsave = n;
state->psave = p;
state->has_binomial = 1;
state->r = r = min(p, 1.0-p);
state->q = q = 1.0 - r;
state->fm = fm = n*r+r;
state->m = m = (long)floor(state->fm);
state->p1 = p1 = floor(2.195*sqrt(n*r*q)-4.6*q) + 0.5;
state->xm = xm = m + 0.5;
state->xl = xl = xm - p1;
state->xr = xr = xm + p1;
state->c = c = 0.134 + 20.5/(15.3 + m);
a = (fm - xl)/(fm-xl*r);
state->laml = laml = a*(1.0 + a/2.0);
a = (xr - fm)/(xr*q);
state->lamr = lamr = a*(1.0 + a/2.0);
state->p2 = p2 = p1*(1.0 + 2.0*c);
state->p3 = p3 = p2 + c/laml;
state->p4 = p4 = p3 + c/lamr;
}
else
{
r = state->r;
q = state->q;
fm = state->fm;
m = state->m;
p1 = state->p1;
xm = state->xm;
xl = state->xl;
xr = state->xr;
c = state->c;
laml = state->laml;
lamr = state->lamr;
p2 = state->p2;
p3 = state->p3;
p4 = state->p4;
}
/* sigh ... */
Step10:
nrq = n*r*q;
u = rk_double(state)*p4;
v = rk_double(state);
if (u > p1) goto Step20;
y = (long)floor(xm - p1*v + u);
goto Step60;
Step20:
if (u > p2) goto Step30;
x = xl + (u - p1)/c;
v = v*c + 1.0 - fabs(m - x + 0.5)/p1;
if (v > 1.0) goto Step10;
y = (long)floor(x);
goto Step50;
Step30:
if (u > p3) goto Step40;
y = (long)floor(xl + log(v)/laml);
if (y < 0) goto Step10;
v = v*(u-p2)*laml;
goto Step50;
Step40:
y = (long)floor(xr - log(v)/lamr);
if (y > n) goto Step10;
v = v*(u-p3)*lamr;
Step50:
k = labs(y - m);
if ((k > 20) && (k < ((nrq)/2.0 - 1))) goto Step52;
s = r/q;
a = s*(n+1);
F = 1.0;
if (m < y)
{
for (i=m+1; i<=y; i++)
{
F *= (a/i - s);
}
}
else if (m > y)
{
for (i=y+1; i<=m; i++)
{
F /= (a/i - s);
}
}
if (v > F) goto Step10;
goto Step60;
Step52:
rho = (k/(nrq))*((k*(k/3.0 + 0.625) + 0.16666666666666666)/nrq + 0.5);
t = -k*k/(2*nrq);
A = log(v);
if (A < (t - rho)) goto Step60;
if (A > (t + rho)) goto Step10;
x1 = y+1;
f1 = m+1;
z = n+1-m;
w = n-y+1;
x2 = x1*x1;
f2 = f1*f1;
z2 = z*z;
w2 = w*w;
if (A > (xm*log(f1/x1)
+ (n-m+0.5)*log(z/w)
+ (y-m)*log(w*r/(x1*q))
+ (13680.-(462.-(132.-(99.-140./f2)/f2)/f2)/f2)/f1/166320.
+ (13680.-(462.-(132.-(99.-140./z2)/z2)/z2)/z2)/z/166320.
+ (13680.-(462.-(132.-(99.-140./x2)/x2)/x2)/x2)/x1/166320.
+ (13680.-(462.-(132.-(99.-140./w2)/w2)/w2)/w2)/w/166320.))
{
goto Step10;
}
Step60:
if (p > 0.5)
{
y = n - y;
}
return y;
}
long rk_binomial_inversion(rk_state *state, long n, double p)
{
double q, qn, np, px, U;
long X, bound;
if (!(state->has_binomial) ||
(state->nsave != n) ||
(state->psave != p))
{
state->nsave = n;
state->psave = p;
state->has_binomial = 1;
state->q = q = 1.0 - p;
state->r = qn = exp(n * log(q));
state->c = np = n*p;
state->m = bound = min(n, np + 10.0*sqrt(np*q + 1));
} else
{
q = state->q;
qn = state->r;
np = state->c;
bound = state->m;
}
X = 0;
px = qn;
U = rk_double(state);
while (U > px)
{
X++;
if (X > bound)
{
X = 0;
px = qn;
U = rk_double(state);
} else
{
U -= px;
px = ((n-X+1) * p * px)/(X*q);
}
}
return X;
}
long rk_binomial(rk_state *state, long n, double p)
{
double q;
if (p <= 0.5)
{
if (p*n <= 30.0)
{
return rk_binomial_inversion(state, n, p);
}
else
{
return rk_binomial_btpe(state, n, p);
}
}
else
{
q = 1.0-p;
if (q*n <= 30.0)
{
return n - rk_binomial_inversion(state, n, q);
}
else
{
return n - rk_binomial_btpe(state, n, q);
}
}
}
long rk_negative_binomial(rk_state *state, double n, double p)
{
double Y;
Y = rk_gamma(state, n, (1-p)/p);
return rk_poisson(state, Y);
}
long rk_poisson_mult(rk_state *state, double lam)
{
long X;
double prod, U, enlam;
enlam = exp(-lam);
X = 0;
prod = 1.0;
while (1)
{
U = rk_double(state);
prod *= U;
if (prod > enlam)
{
X += 1;
}
else
{
return X;
}
}
}
/*
* The transformed rejection method for generating Poisson random variables
* W. Hoermann
* Insurance: Mathematics and Economics 12, 39-45 (1993)
*/
#define LS2PI 0.91893853320467267
#define TWELFTH 0.083333333333333333333333
long rk_poisson_ptrs(rk_state *state, double lam)
{
long k;
double U, V, slam, loglam, a, b, invalpha, vr, us;
slam = sqrt(lam);
loglam = log(lam);
b = 0.931 + 2.53*slam;
a = -0.059 + 0.02483*b;
invalpha = 1.1239 + 1.1328/(b-3.4);
vr = 0.9277 - 3.6224/(b-2);
while (1)
{
U = rk_double(state) - 0.5;
V = rk_double(state);
us = 0.5 - fabs(U);
k = (long)floor((2*a/us + b)*U + lam + 0.43);
if ((us >= 0.07) && (V <= vr))
{
return k;
}
if ((k < 0) ||
((us < 0.013) && (V > us)))
{
continue;
}
if ((log(V) + log(invalpha) - log(a/(us*us)+b)) <=
(-lam + k*loglam - loggam(k+1)))
{
return k;
}
}
}
long rk_poisson(rk_state *state, double lam)
{
if (lam >= 10)
{
return rk_poisson_ptrs(state, lam);
}
else if (lam == 0)
{
return 0;
}
else
{
return rk_poisson_mult(state, lam);
}
}
double rk_standard_cauchy(rk_state *state)
{
return rk_gauss(state) / rk_gauss(state);
}
double rk_standard_t(rk_state *state, double df)
{
double N, G, X;
N = rk_gauss(state);
G = rk_standard_gamma(state, df/2);
X = sqrt(df/2)*N/sqrt(G);
return X;
}
/* Uses the rejection algorithm compared against the wrapped Cauchy
distribution suggested by Best and Fisher and documented in
Chapter 9 of Luc's Non-Uniform Random Variate Generation.
http://cg.scs.carleton.ca/~luc/rnbookindex.html
(but corrected to match the algorithm in R and Python)
*/
double rk_vonmises(rk_state *state, double mu, double kappa)
{
double s;
double U, V, W, Y, Z;
double result, mod;
int neg;
if (kappa < 1e-8)
{
return M_PI * (2*rk_double(state)-1);
}
else
{
/* with double precision rho is zero until 1.4e-8 */
if (kappa < 1e-5) {
/*
* second order taylor expansion around kappa = 0
* precise until relatively large kappas as second order is 0
*/
s = (1./kappa + kappa);
}
else {
double r = 1 + sqrt(1 + 4*kappa*kappa);
double rho = (r - sqrt(2*r)) / (2*kappa);
s = (1 + rho*rho)/(2*rho);
}
while (1)
{
U = rk_double(state);
Z = cos(M_PI*U);
W = (1 + s*Z)/(s + Z);
Y = kappa * (s - W);
V = rk_double(state);
if ((Y*(2-Y) - V >= 0) || (log(Y/V)+1 - Y >= 0))
{
break;
}
}
U = rk_double(state);
result = acos(W);
if (U < 0.5)
{
result = -result;
}
result += mu;
neg = (result < 0);
mod = fabs(result);
mod = (fmod(mod+M_PI, 2*M_PI)-M_PI);
if (neg)
{
mod *= -1;
}
return mod;
}
}
double rk_pareto(rk_state *state, double a)
{
return exp(rk_standard_exponential(state)/a) - 1;
}
double rk_weibull(rk_state *state, double a)
{
return pow(rk_standard_exponential(state), 1./a);
}
double rk_power(rk_state *state, double a)
{
return pow(1 - exp(-rk_standard_exponential(state)), 1./a);
}
double rk_laplace(rk_state *state, double loc, double scale)
{
double U;
U = rk_double(state);
if (U < 0.5)
{
U = loc + scale * log(U + U);
} else
{
U = loc - scale * log(2.0 - U - U);
}
return U;
}
double rk_gumbel(rk_state *state, double loc, double scale)
{
double U;
U = 1.0 - rk_double(state);
return loc - scale * log(-log(U));
}
double rk_logistic(rk_state *state, double loc, double scale)
{
double U;
U = rk_double(state);
return loc + scale * log(U/(1.0 - U));
}
double rk_lognormal(rk_state *state, double mean, double sigma)
{
return exp(rk_normal(state, mean, sigma));
}
double rk_rayleigh(rk_state *state, double mode)
{
return mode*sqrt(-2.0 * log(1.0 - rk_double(state)));
}
double rk_wald(rk_state *state, double mean, double scale)
{
double U, X, Y;
double mu_2l;
mu_2l = mean / (2*scale);
Y = rk_gauss(state);
Y = mean*Y*Y;
X = mean + mu_2l*(Y - sqrt(4*scale*Y + Y*Y));
U = rk_double(state);
if (U <= mean/(mean+X))
{
return X;
} else
{
return mean*mean/X;
}
}
long rk_zipf(rk_state *state, double a)
{
double am1, b;
am1 = a - 1.0;
b = pow(2.0, am1);
while (1) {
double T, U, V, X;
U = 1.0 - rk_double(state);
V = rk_double(state);
X = floor(pow(U, -1.0/am1));
/*
* The real result may be above what can be represented in a signed
* long. Since this is a straightforward rejection algorithm, we can
* just reject this value. This function then models a Zipf
* distribution truncated to sys.maxint.
*/
if (X > LONG_MAX || X < 1.0) {
continue;
}
T = pow(1.0 + 1.0/X, am1);
if (V*X*(T - 1.0)/(b - 1.0) <= T/b) {
return (long)X;
}
}
}
long rk_geometric_search(rk_state *state, double p)
{
double U;
long X;
double sum, prod, q;
X = 1;
sum = prod = p;
q = 1.0 - p;
U = rk_double(state);
while (U > sum)
{
prod *= q;
sum += prod;
X++;
}
return X;
}
long rk_geometric_inversion(rk_state *state, double p)
{
return (long)ceil(log(1.0-rk_double(state))/log(1.0-p));
}
long rk_geometric(rk_state *state, double p)
{
if (p >= 0.333333333333333333333333)
{
return rk_geometric_search(state, p);
} else
{
return rk_geometric_inversion(state, p);
}
}
long rk_hypergeometric_hyp(rk_state *state, long good, long bad, long sample)
{
long d1, K, Z;
double d2, U, Y;
d1 = bad + good - sample;
d2 = (double)min(bad, good);
Y = d2;
K = sample;
while (Y > 0.0)
{
U = rk_double(state);
Y -= (long)floor(U + Y/(d1 + K));
K--;
if (K == 0) break;
}
Z = (long)(d2 - Y);
if (good > bad) Z = sample - Z;
return Z;
}
/* D1 = 2*sqrt(2/e) */
/* D2 = 3 - 2*sqrt(3/e) */
#define D1 1.7155277699214135
#define D2 0.8989161620588988
long rk_hypergeometric_hrua(rk_state *state, long good, long bad, long sample)
{
long mingoodbad, maxgoodbad, popsize, m, d9;
double d4, d5, d6, d7, d8, d10, d11;
long Z;
double T, W, X, Y;
mingoodbad = min(good, bad);
popsize = good + bad;
maxgoodbad = max(good, bad);
m = min(sample, popsize - sample);
d4 = ((double)mingoodbad) / popsize;
d5 = 1.0 - d4;
d6 = m*d4 + 0.5;
d7 = sqrt((double)(popsize - m) * sample * d4 * d5 / (popsize - 1) + 0.5);
d8 = D1*d7 + D2;
d9 = (long)floor((double)(m + 1) * (mingoodbad + 1) / (popsize + 2));
d10 = (loggam(d9+1) + loggam(mingoodbad-d9+1) + loggam(m-d9+1) +
loggam(maxgoodbad-m+d9+1));
d11 = min(min(m, mingoodbad)+1.0, floor(d6+16*d7));
/* 16 for 16-decimal-digit precision in D1 and D2 */
while (1)
{
X = rk_double(state);
Y = rk_double(state);
W = d6 + d8*(Y- 0.5)/X;
/* fast rejection: */
if ((W < 0.0) || (W >= d11)) continue;
Z = (long)floor(W);
T = d10 - (loggam(Z+1) + loggam(mingoodbad-Z+1) + loggam(m-Z+1) +
loggam(maxgoodbad-m+Z+1));
/* fast acceptance: */
if ((X*(4.0-X)-3.0) <= T) break;
/* fast rejection: */
if (X*(X-T) >= 1) continue;
if (2.0*log(X) <= T) break; /* acceptance */
}
/* this is a correction to HRUA* by Ivan Frohne in rv.py */
if (good > bad) Z = m - Z;
/* another fix from rv.py to allow sample to exceed popsize/2 */
if (m < sample) Z = good - Z;
return Z;
}
#undef D1
#undef D2
long rk_hypergeometric(rk_state *state, long good, long bad, long sample)
{
if (sample > 10)
{
return rk_hypergeometric_hrua(state, good, bad, sample);
} else
{
return rk_hypergeometric_hyp(state, good, bad, sample);
}
}
double rk_triangular(rk_state *state, double left, double mode, double right)
{
double base, leftbase, ratio, leftprod, rightprod;
double U;
base = right - left;
leftbase = mode - left;
ratio = leftbase / base;
leftprod = leftbase*base;
rightprod = (right - mode)*base;
U = rk_double(state);
if (U <= ratio)
{
return left + sqrt(U*leftprod);
} else
{
return right - sqrt((1.0 - U) * rightprod);
}
}
long rk_logseries(rk_state *state, double p)
{
double q, r, U, V;
long result;
r = log(1.0 - p);
while (1) {
V = rk_double(state);
if (V >= p) {
return 1;
}
U = rk_double(state);
q = 1.0 - exp(r*U);
if (V <= q*q) {
result = (long)floor(1 + log(V)/log(q));
if (result < 1) {
continue;
}
else {
return result;
}
}
if (V >= q) {
return 1;
}
return 2;
}
}