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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
* Γ (Gamma) 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 GammaFun {
//@formatter:off
private static final double STIR[] = { 7.87311395793093628397E-4,
-2.29549961613378126380E-4, -2.68132617805781232825E-3,
3.47222221605458667310E-3, 8.33333333333482257126E-2, };
private static final double MAX_STIR = 143.01608;
private static final double P[] = { 1.60119522476751861407E-4,
1.19135147006586384913E-3, 1.04213797561761569935E-2,
4.76367800457137231464E-2, 2.07448227648435975150E-1,
4.94214826801497100753E-1, 9.99999999999999996796E-1 };
private static final double Q[] = { -2.31581873324120129819E-5,
5.39605580493303397842E-4, -4.45641913851797240494E-3,
1.18139785222060435552E-2, 3.58236398605498653373E-2,
-2.34591795718243348568E-1, 7.14304917030273074085E-2,
1.00000000000000000320E0 };
private static final double A[] = { 8.11614167470508450300E-4,
-5.95061904284301438324E-4, 7.93650340457716943945E-4,
-2.77777777730099687205E-3, 8.33333333333331927722E-2 };
private static final double B[] = { -1.37825152569120859100E3,
-3.88016315134637840924E4, -3.31612992738871184744E5,
-1.16237097492762307383E6, -1.72173700820839662146E6,
-8.53555664245765465627E5 };
private static final double C[] = { -3.51815701436523470549E2,
-1.70642106651881159223E4, -2.20528590553854454839E5,
-1.13933444367982507207E6, -2.53252307177582951285E6,
-2.01889141433532773231E6 };
private static final double DIGAMMA_C7[][] = {
{1.3524999667726346383e4, 4.5285601699547289655e4, 4.5135168469736662555e4,
1.8529011818582610168e4, 3.3291525149406935532e3, 2.4068032474357201831e2,
5.1577892000139084710, 6.2283506918984745826e-3},
{6.9389111753763444376e-7, 1.9768574263046736421e4, 4.1255160835353832333e4,
2.9390287119932681918e4, 9.0819666074855170271e3,
1.2447477785670856039e3, 6.7429129516378593773e1, 1.0}
};
private static final double DIGAMMA_C4[][] = {
{-2.728175751315296783e-15, -6.481571237661965099e-1, -4.486165439180193579,
-7.016772277667586642, -2.129404451310105168},
{7.777885485229616042, 5.461177381032150702e1,
8.929207004818613702e1, 3.227034937911433614e1, 1.0}
};
/* Chebyshev coefficients for trigamma (x + 3), 0 <= x <= 1. In Yudell
Luke: The special functions and their approximations, Vol. II,
Academic Press, p. 301, 1969. */
private static final int TRIGAMMA_N = 15;
private static final double TRIGAMMA_A[] = { 2.0 * 0.33483869791094938576,
-0.05518748204873009463, 0.00451019073601150186, -0.00036570588830372083,
2.943462746822336e-5, -2.35277681515061e-6, 1.8685317663281e-7,
-1.475072018379e-8, 1.15799333714e-9, -9.043917904e-11,
7.029627e-12, -5.4398873e-13, 0.4192525e-13, -3.21903e-15, 0.2463e-15,
-1.878e-17, 0.0, 0.0 };
//@formatter:on
/**
* Returns the Gamma function of the argument.
*/
public static double gamma(double x) {
double q = Math.abs(x);
double p;
double z;
if (q > 33.0) {
if (x < 0.0) {
p = Math.floor(q);
if (p == q) {
throw new GammaException("Overflow", x, "p == q", 1);
}
z = q - p;
if (z > 0.5) {
p += 1.0;
z = q - p;
}
z = q * FastMath.sin(Math.PI * z);
if (z == 0.0) {
throw new GammaException("Overflow", x, "z == 0.0", 2);
}
z = Math.abs(z);
z = Math.PI / (z * stirlingFormula(q));
return -z;
} else {
return stirlingFormula(x);
}
}
z = 1.0;
while (x >= 3.0) {
x -= 1.0;
z *= x;
}
while (x < 0.0) {
if (x == 0.0) {
throw new GammaException("Singular", "x == 0.0", 3);
} else if (x > -1.E-9) {
return (z / ((1.0 + 0.5772156649015329 * x) * x));
}
z /= x;
x += 1.0;
}
while (x < 2.0) {
if (x == 0.0) {
throw new GammaException("Singular", "x == 0.0", 4);
} else if (x < 1.e-9) {
return (z / ((1.0 + 0.5772156649015329 * x) * x));
}
z /= x;
x += 1.0;
}
if ((x == 2.0) || (x == 3.0)) {
return z;
}
x -= 2.0;
p = Polynomial.polevl(x, P, 6);
q = Polynomial.polevl(x, Q, 7);
return z * p / q;
}
/**
* Returns the natural logarithm of the gamma function.
*/
public static double lnGamma(double x) {
double p;
double q;
if (x < -34.0) {
q = -x;
final double w = lnGamma(q);
p = Math.floor(q);
if (p == q) {
throw new GammaException("Overflow", x, "p == q", 1);
}
double z = q - p;
if (z > 0.5) {
p += 1.0;
z = p - q;
}
z = q * FastMath.sin(Math.PI * z);
if (z == 0.0) {
throw new GammaException("Overflow", x, "z == 0.0", 2);
}
z = LN_PI - Math.log(z) - w;
return z;
}
if (x < 13.0) {
double z = 1.0;
while (x >= 3.0) {
x -= 1.0;
z *= x;
}
while (x < 2.0) {
if (x == 0.0) {
throw new GammaException("Overflow", "x == 0.0", 3);
}
z /= x;
x += 1.0;
}
if (z < 0.0) {
z = -z;
}
if (x == 2.0) {
return Math.log(z);
}
x -= 2.0;
p = x * Polynomial.polevl(x, B, 5) / Polynomial.p1evl(x, C, 6);
return (Math.log(z) + p);
}
if (x > 2.556348e305) {
throw new GammaException("Overflow", "x > 2.556348e305", 4);
}
q = (x - 0.5) * Math.log(x) - x + 0.91893853320467274178;
if (x > 1.0e8) {
return q;
}
p = 1.0 / (x * x);
if (x >= 1000.0) {
q += ((7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3)
* p + 0.0833333333333333333333)
/ x;
} else {
q += Polynomial.polevl(p, A, 4) / x;
}
return q;
}
/**
* Returns the Incomplete Gamma function.
*
* @param a
* the parameter of the gamma distribution.
* @param x
* the integration end point.
*/
public static double incompleteGamma(final double a, final double x) {
if (x <= 0 || a <= 0) {
return 0.0;
}
if (x > 1.0 && x > a) {
return 1.0 - incompleteGammaComplement(a, x);
}
/* Compute x**a * exp(-x) / gamma(a) */
final double ax = a * Math.log(x) - x - lnGamma(a);
if (ax < -MAX_LOG) {
return 0.0;
}
/* power series */
double r = a;
double c = 1.0;
double ans = 1.0;
do {
r += 1.0;
c *= x / r;
ans += c;
} while (c / ans > MACH_EPS);
return (ans * FastMath.exp(ax) / a);
}
/**
* Returns the Complemented Incomplete Gamma function.
*
* @param a
* the parameter of the gamma distribution.
* @param x
* the integration start point.
*/
public static double incompleteGammaComplement(final double a,
final double x) {
if (x <= 0 || a <= 0) {
return 1.0;
}
if (x < 1.0 || x < a) {
return 1.0 - incompleteGamma(a, x);
}
final double ax = a * Math.log(x) - x - lnGamma(a);
if (ax < -MAX_LOG) {
return 0.0;
}
/* continued fraction */
double y = 1.0 - a;
double z = x + y + 1.0;
double c = 0.0;
double pkm2 = 1.0;
double qkm2 = x;
double pkm1 = x + 1.0;
double qkm1 = z * x;
double ans = pkm1 / qkm1;
double t;
do {
c += 1.0;
y += 1.0;
z += 2.0;
final double yc = y * c;
final double pk = pkm1 * z - pkm2 * yc;
final double qk = qkm1 * z - qkm2 * yc;
if (qk != 0) {
final double r = pk / qk;
t = Math.abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if (Math.abs(pk) > BIG) {
pkm2 *= BIG_INV;
pkm1 *= BIG_INV;
qkm2 *= BIG_INV;
qkm1 *= BIG_INV;
}
} while (t > MACH_EPS);
return ans * FastMath.exp(ax);
}
/**
* Returns the Gamma function computed by Stirling's formula. The polynomial
* STIR is valid for 33 <= x <= 172.
*/
private static double stirlingFormula(final double x) {
double w = 1.0 / x;
w = 1.0 + w * Polynomial.polevl(w, STIR, 4);
double y = FastMath.exp(x);
if (x > MAX_STIR) {
/* Avoid overflow in Math.pow() */
final double v = FastMath.pow(x, 0.5 * x - 0.25);
y = v * (v / y);
} else {
y = FastMath.pow(x, x - 0.5) / y;
}
y = SQRT_TWO_PI * y * w;
return y;
}
/**
* Returns the value of the logarithmic derivative of the Gamma
* function {@code psi(x) = Gamma?(x) / Gamma(x)}.
*/
public static double digamma(double x) {
if (Double.isNaN(x)) {
return Double.NaN;
}
double digX = 0.0;
if (x >= 3.0) {
double prodPj = 0.0;
double prodQj = 0.0;
double x2 = 1.0 / (x * x);
for (int j = 4; j >= 0; j--) {
prodPj = prodPj * x2 + DIGAMMA_C4[0][j];
prodQj = prodQj * x2 + DIGAMMA_C4[1][j];
}
digX = Math.log(x) - (0.5 / x) + (prodPj / prodQj);
} else if (x >= 0.5) {
double prodPj = 0.0;
double prodQj = 0.0;
for (int j = 7; j >= 0; j--) {
prodPj = x * prodPj + DIGAMMA_C7[0][j];
prodQj = x * prodQj + DIGAMMA_C7[1][j];
}
digX = (x - 1.46163214496836234126) * (prodPj / prodQj);
} else {
double f = (1.0 - x) - Math.floor(1.0 - x);
digX = digamma(1.0 - x) + (Math.PI / FastMath.tan(Math.PI * f));
}
return digX;
}
/**
* Returns the value of the trigamma function {@code d(psi(x))/dx}, the
* derivative of the {@link #digamma(double)} function, evaluated at
* {@code x}.
*/
public static double trigamma(double x) {
if (Double.isNaN(x)) {
return Double.NaN;
}
double y;
double sum;
if (x < 0.5) {
y = (1.0 - x) - Math.floor(1.0 - x);
sum = Math.PI / FastMath.sin(Math.PI * y);
return (sum * sum) - trigamma(1.0 - x);
}
if (x >= 40.0) {
// asymptotic series
y = 1.0 / (x * x);
sum = 1.0 + y * (1.0 / 6.0 - y * (1.0 / 30.0 - y * (1.0 / 42.0 - 1.0 / 30.0 * y)));
sum += 0.5 / x;
return sum / x;
}
int p = (int) x;
y = x - p;
sum = 0.0;
if (p > 3) {
for (int i = 3; i < p; i++) {
sum -= 1.0 / ((y + i) * (y + i));
}
} else if (p < 3) {
for (int i = 2; i >= p; i--) {
sum += 1.0 / ((y + i) * (y + i));
}
}
return sum + Polynomial.evalChebyStar(TRIGAMMA_A, TRIGAMMA_N, y);
}
private GammaFun() {
}
}