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Statistical distributions library (in statu nascendi)
/*
Copyright ? 1999 CERN - European Organization for Nuclear Research.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
is hereby granted without fee, provided that the above copyright notice appear in all copies and
that both that copyright notice and this permission notice appear in supporting documentation.
CERN makes no representations about the suitability of this software for any purpose.
It is provided "as is" without expressed or implied warranty.
*/
package math;
import static math.MathConsts.*;
/**
* This is a utility class that provides computation methods related to the Beta
* family of functions.
*
* Implementation:
*
* Some code taken and adapted from the Java 2D Graph
* Package 2.4, which in turn is a port from the Cephes 2.2
* Math Library (C). Most Cephes code (missing from the 2D Graph Package)
* directly ported.
*
* @author [email protected]
* @version 0.9, 22-Jun-99
*/
public final class Beta {
private static final double THRESHOLD = 3.0 * MACH_EPS;
/**
* Returns the Beta function of the arguments.
*
*
* - -
* | (alpha) | (beta)
* Beta( alpha, beta ) = ------------------.
* -
* | (alpha+beta)
*
*/
public static double beta(final double alpha, final double beta) {
double y = alpha + beta;
y = Gamma.gamma(y);
if (y == 0.0) {
return 1.0;
}
if (alpha > beta) {
y = Gamma.gamma(alpha) / y;
y *= Gamma.gamma(beta);
} else {
y = Gamma.gamma(beta) / y;
y *= Gamma.gamma(alpha);
}
return y;
}
/**
* Returns the natural logarithm of the beta function.
*/
public static double lnBeta(final double alpha, final double beta) {
return Gamma.lnGamma(alpha) + Gamma.lnGamma(beta)
- Gamma.lnGamma(alpha + beta);
}
/**
* Returns the Incomplete Beta Function evaluated from zero to x.
*
* @param alpha
* the alpha parameter of the beta distribution.
* @param beta
* the beta parameter of the beta distribution.
* @param x
* the integration end point.
*/
public static double incompleteBeta(final double alpha, final double beta,
final double x) {
if (alpha <= 0.0 || beta <= 0.0) {
throw new BetaException("Domain error",
"alpha <= 0.0 || beta <= 0.0", 1);
}
if ((x <= 0.0) || (x >= 1.0)) {
if (x == 0.0) {
return 0.0;
}
if (x == 1.0) {
return 1.0;
}
throw new BetaException("Domain error", x,
"(x < 0.0) || (x > 1.0)", 2);
}
if ((beta * x) <= 1.0 && x <= 0.95) {
return powerSeries(alpha, beta, x);
}
double w = 1.0 - x;
double a;
double b;
double xc;
double x_;
boolean flag = false;
/* Reverse a and b if x is greater than the mean. */
if (x > (alpha / (alpha + beta))) {
flag = true;
a = beta;
b = alpha;
xc = x;
x_ = w;
} else {
a = alpha;
b = beta;
xc = w;
x_ = x;
}
double t;
if (flag && (b * x_) <= 1.0 && x_ <= 0.95) {
t = powerSeries(a, b, x_);
if (t <= MACH_EPS) {
t = 1.0 - MACH_EPS;
} else {
t = 1.0 - t;
}
return t;
}
/* Choose expansion for better convergence. */
double y = x_ * (a + b - 2.0) - (a - 1.0);
if (y < 0.0) {
w = incompleteBetaFraction1(a, b, x_);
} else {
w = incompleteBetaFraction2(a, b, x_) / xc;
}
// @formatter:off
/*
Multiply w by the factor
a b _ _ _
x (1-x) | (a+b) / ( a | (a) | (b) )
*/
// @formatter:on
y = a * Math.log(x_);
t = b * Math.log(xc);
if ((a + b) < MAX_GAMMA && Math.abs(y) < MAX_LOG
&& Math.abs(t) < MAX_LOG) {
t = FastMath.pow(xc, b);
t *= FastMath.pow(x_, a);
t /= a;
t *= w;
t *= Gamma.gamma(a + b) / (Gamma.gamma(a) * Gamma.gamma(b));
if (flag) {
if (t <= MACH_EPS) {
t = 1.0 - MACH_EPS;
} else {
t = 1.0 - t;
}
}
return t;
}
/* Resort to logarithms */
y += t + Gamma.lnGamma(a + b) - Gamma.lnGamma(a) - Gamma.lnGamma(b);
y += Math.log(w / a);
if (y < MIN_LOG) {
t = 0.0;
} else {
t = FastMath.exp(y);
}
if (flag) {
if (t <= MACH_EPS) {
t = 1.0 - MACH_EPS;
} else {
t = 1.0 - t;
}
}
return t;
}
/**
* Power series for incomplete beta integral.
*
* Use when b*x is small and x not too close to 1.
*/
private static double powerSeries(final double a, final double b,
final double x) {
double n = 2.0;
double s = 0.0;
double u = (1.0 - b) * x;
double v = u / (a + 1.0);
double t1 = v;
double t = u;
final double ai = 1.0 / a;
final double z = MACH_EPS * ai;
while (Math.abs(v) > z) {
u = (n - b) * x / n;
t *= u;
v = t / (a + n);
s += v;
n += 1.0;
}
s += t1;
s += ai;
u = a * Math.log(x);
if ((a + b) < MAX_GAMMA && Math.abs(u) < MAX_LOG) {
t = Gamma.gamma(a + b) / (Gamma.gamma(a) * Gamma.gamma(b));
s = s * t * FastMath.pow(x, a);
} else {
t = Gamma.lnGamma(a + b) - Gamma.lnGamma(a) - Gamma.lnGamma(b) + u
+ Math.log(s);
if (t < MIN_LOG) {
s = 0.0;
} else {
s = FastMath.exp(t);
}
}
return s;
}
/**
* Continued fraction expansion #1 for incomplete beta integral.
*/
private static double incompleteBetaFraction1(final double a,
final double b, final double x) {
double k1 = a;
double k2 = a + b;
double k3 = a;
double k4 = a + 1.0;
double k5 = 1.0;
double k6 = b - 1.0;
double k7 = k4;
double k8 = a + 2.0;
double pkm1 = 1.0;
double qkm1 = 1.0;
double pkm2 = 0.0;
double qkm2 = 1.0;
double r = 1.0;
double ans = 1.0;
int n = 0;
do {
double xk = -(x * k1 * k2) / (k3 * k4);
double pk = pkm1 + pkm2 * xk;
double qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = (x * k5 * k6) / (k7 * k8);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (qk != 0) {
r = pk / qk;
}
double t;
if (r != 0) {
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
if (t < THRESHOLD) {
return ans;
}
k1 += 1.0;
k2 += 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 -= 1.0;
k7 += 2.0;
k8 += 2.0;
if ((Math.abs(qk) + Math.abs(pk)) > BIG) {
pkm2 *= BIG_INV;
pkm1 *= BIG_INV;
qkm2 *= BIG_INV;
qkm1 *= BIG_INV;
}
if ((Math.abs(qk) < BIG_INV) || (Math.abs(pk) < BIG_INV)) {
pkm2 *= BIG;
pkm1 *= BIG;
qkm2 *= BIG;
qkm1 *= BIG;
}
} while (++n < 300);
return ans;
}
/**
* Continued fraction expansion #2 for incomplete beta integral.
*/
private static double incompleteBetaFraction2(final double a,
final double b, final double x) {
double k1 = a;
double k2 = b - 1.0;
double k3 = a;
double k4 = a + 1.0;
double k5 = 1.0;
double k6 = a + b;
double k7 = a + 1.0;
double k8 = a + 2.0;
double pkm1 = 1.0;
double qkm1 = 1.0;
double pkm2 = 0.0;
double qkm2 = 1.0;
final double z = x / (1.0 - x);
double r = 1.0;
double ans = 1.0;
int n = 0;
do {
double xk = -(z * k1 * k2) / (k3 * k4);
double pk = pkm1 + pkm2 * xk;
double qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = (z * k5 * k6) / (k7 * k8);
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (qk != 0) {
r = pk / qk;
}
double t;
if (r != 0) {
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
if (t < THRESHOLD) {
return ans;
}
k1 += 1.0;
k2 -= 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 += 1.0;
k7 += 2.0;
k8 += 2.0;
if ((Math.abs(qk) + Math.abs(pk)) > BIG) {
pkm2 *= BIG_INV;
pkm1 *= BIG_INV;
qkm2 *= BIG_INV;
qkm1 *= BIG_INV;
}
if ((Math.abs(qk) < BIG_INV) || (Math.abs(pk) < BIG_INV)) {
pkm2 *= BIG;
pkm1 *= BIG;
qkm2 *= BIG;
qkm1 *= BIG;
}
} while (++n < 300);
return ans;
}
private Beta() {
}
}