/**
* Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10. Based on the <em>NSWC Library
* of Mathematics Subroutines</em> double precision implementation, {@code DLGDIV}. In {@code
* BetaTest.testLogGammaMinusLogGammaSum()}, this private method is accessed through reflection.
*
* @param a First argument.
* @param b Second argument.
* @return the value of {@code log(Gamma(b) / Gamma(a + b))}.
* @throws NumberIsTooSmallException if {@code a < 0.0} or {@code b < 10.0}.
*/
private static double logGammaMinusLogGammaSum(final double a, final double b)
throws NumberIsTooSmallException {
if (a < 0.0) {
throw new NumberIsTooSmallException(a, 0.0, true);
}
if (b < 10.0) {
throw new NumberIsTooSmallException(b, 10.0, true);
}
/*
* d = a + b - 0.5
*/
final double d;
final double w;
if (a <= b) {
d = b + (a - 0.5);
w = deltaMinusDeltaSum(a, b);
} else {
d = a + (b - 0.5);
w = deltaMinusDeltaSum(b, a);
}
final double u = d * FastMath.log1p(a / b);
final double v = a * (FastMath.log(b) - 1.0);
return u <= v ? (w - u) - v : (w - v) - u;
}
/**
* Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2. Based on the <em>NSWC Library of
* Mathematics Subroutines</em> double precision implementation, {@code DGSMLN}. In {@code
* BetaTest.testLogGammaSum()}, this private method is accessed through reflection.
*
* @param a First argument.
* @param b Second argument.
* @return the value of {@code log(Gamma(a + b))}.
* @throws OutOfRangeException if {@code a} or {@code b} is lower than {@code 1.0} or greater than
* {@code 2.0}.
*/
private static double logGammaSum(final double a, final double b) throws OutOfRangeException {
if ((a < 1.0) || (a > 2.0)) {
throw new OutOfRangeException(a, 1.0, 2.0);
}
if ((b < 1.0) || (b > 2.0)) {
throw new OutOfRangeException(b, 1.0, 2.0);
}
final double x = (a - 1.0) + (b - 1.0);
if (x <= 0.5) {
return Gamma.logGamma1p(1.0 + x);
} else if (x <= 1.5) {
return Gamma.logGamma1p(x) + FastMath.log1p(x);
} else {
return Gamma.logGamma1p(x - 1.0) + FastMath.log(x * (1.0 + x));
}
}
/** {@inheritDoc} * */
@Override
public double logDensity(double x) {
recomputeZ();
if (x < 0 || x > 1) {
return Double.NEGATIVE_INFINITY;
} else if (x == 0) {
if (alpha < 1) {
throw new NumberIsTooSmallException(
LocalizedFormats.CANNOT_COMPUTE_BETA_DENSITY_AT_0_FOR_SOME_ALPHA, alpha, 1, false);
}
return Double.NEGATIVE_INFINITY;
} else if (x == 1) {
if (beta < 1) {
throw new NumberIsTooSmallException(
LocalizedFormats.CANNOT_COMPUTE_BETA_DENSITY_AT_1_FOR_SOME_BETA, beta, 1, false);
}
return Double.NEGATIVE_INFINITY;
} else {
double logX = FastMath.log(x);
double log1mX = FastMath.log1p(-x);
return (alpha - 1) * logX + (beta - 1) * log1mX - z;
}
}
@Test
public void testSamplerHelper1() {
final double tol = 1e-12;
final double[] testValues = {
FastMath.nextUp(-1.),
-1e-1,
-1e-2,
-1e-3,
-1e-4,
-1e-5,
-1e-6,
-1e-7,
-1e-8,
-1e-9,
-1e-10,
-1e-11,
0.,
1e-11,
1e-10,
1e-9,
1e-8,
1e-7,
1e-6,
1e-5,
1e-4,
1e-3,
1e-2,
1e-1,
1e0
};
for (final double testValue : testValues) {
final double expected = FastMath.log1p(testValue);
TestUtils.assertRelativelyEquals(
expected, ZipfRejectionInversionSampler.helper1(testValue) * testValue, tol);
}
}
/** {@inheritDoc} */
public double value(double x) {
return FastMath.log1p(x);
}
/**
* Returns the value of log B(p, q) for 0 ≤ x ≤ 1 and p, q > 0. Based on the <em>NSWC Library of
* Mathematics Subroutines</em> implementation, {@code DBETLN}.
*
* @param p First argument.
* @param q Second argument.
* @return the value of {@code log(Beta(p, q))}, {@code NaN} if {@code p <= 0} or {@code q <= 0}.
*/
public static double logBeta(final double p, final double q) {
if (Double.isNaN(p) || Double.isNaN(q) || (p <= 0.0) || (q <= 0.0)) {
return Double.NaN;
}
final double a = FastMath.min(p, q);
final double b = FastMath.max(p, q);
if (a >= 10.0) {
final double w = sumDeltaMinusDeltaSum(a, b);
final double h = a / b;
final double c = h / (1.0 + h);
final double u = -(a - 0.5) * FastMath.log(c);
final double v = b * FastMath.log1p(h);
if (u <= v) {
return (((-0.5 * FastMath.log(b) + HALF_LOG_TWO_PI) + w) - u) - v;
} else {
return (((-0.5 * FastMath.log(b) + HALF_LOG_TWO_PI) + w) - v) - u;
}
} else if (a > 2.0) {
if (b > 1000.0) {
final int n = (int) FastMath.floor(a - 1.0);
double prod = 1.0;
double ared = a;
for (int i = 0; i < n; i++) {
ared -= 1.0;
prod *= ared / (1.0 + ared / b);
}
return (FastMath.log(prod) - n * FastMath.log(b))
+ (Gamma.logGamma(ared) + logGammaMinusLogGammaSum(ared, b));
} else {
double prod1 = 1.0;
double ared = a;
while (ared > 2.0) {
ared -= 1.0;
final double h = ared / b;
prod1 *= h / (1.0 + h);
}
if (b < 10.0) {
double prod2 = 1.0;
double bred = b;
while (bred > 2.0) {
bred -= 1.0;
prod2 *= bred / (ared + bred);
}
return FastMath.log(prod1)
+ FastMath.log(prod2)
+ (Gamma.logGamma(ared) + (Gamma.logGamma(bred) - logGammaSum(ared, bred)));
} else {
return FastMath.log(prod1) + Gamma.logGamma(ared) + logGammaMinusLogGammaSum(ared, b);
}
}
} else if (a >= 1.0) {
if (b > 2.0) {
if (b < 10.0) {
double prod = 1.0;
double bred = b;
while (bred > 2.0) {
bred -= 1.0;
prod *= bred / (a + bred);
}
return FastMath.log(prod)
+ (Gamma.logGamma(a) + (Gamma.logGamma(bred) - logGammaSum(a, bred)));
} else {
return Gamma.logGamma(a) + logGammaMinusLogGammaSum(a, b);
}
} else {
return Gamma.logGamma(a) + Gamma.logGamma(b) - logGammaSum(a, b);
}
} else {
if (b >= 10.0) {
return Gamma.logGamma(a) + logGammaMinusLogGammaSum(a, b);
} else {
// The following command is the original NSWC implementation.
// return Gamma.logGamma(a) +
// (Gamma.logGamma(b) - Gamma.logGamma(a + b));
// The following command turns out to be more accurate.
return FastMath.log(Gamma.gamma(a) * Gamma.gamma(b) / Gamma.gamma(a + b));
}
}
}