/*
* Compute the natural log of Gamma(x), accurate to 10 decimal places.
*
* This implementation is based on:
*
* Pike, M.C., I.D. Hill (1966) Algorithm 291: Logarithm of Gamma function
* [S14]. Communications of the ACM 9(9):684.
*/
static inline double
ln_gamma(double x) {
double f, z;
assert(x > 0.0);
if (x < 7.0) {
f = 1.0;
z = x;
while (z < 7.0) {
f *= z;
z += 1.0;
}
x = z;
f = -log(f);
} else {
f = 0.0;
}
z = 1.0 / (x * x);
return f + (x-0.5) * log(x) - x + 0.918938533204673 +
(((-0.000595238095238 * z + 0.000793650793651) * z -
0.002777777777778) * z + 0.083333333333333) / x;
}
/*
* Compute the incomplete Gamma ratio for [0..x], where p is the shape
* parameter, and ln_gamma_p is ln_gamma(p).
*
* This implementation is based on:
*
* Bhattacharjee, G.P. (1970) Algorithm AS 32: The incomplete Gamma integral.
* Applied Statistics 19:285-287.
*/
static inline double
i_gamma(double x, double p, double ln_gamma_p) {
double acu, factor, oflo, gin, term, rn, a, b, an, dif;
double pn[6];
unsigned i;
assert(p > 0.0);
assert(x >= 0.0);
if (x == 0.0) {
return 0.0;
}
acu = 1.0e-10;
oflo = 1.0e30;
gin = 0.0;
factor = exp(p * log(x) - x - ln_gamma_p);
if (x <= 1.0 || x < p) {
/* Calculation by series expansion. */
gin = 1.0;
term = 1.0;
rn = p;
while (true) {
rn += 1.0;
term *= x / rn;
gin += term;
if (term <= acu) {
gin *= factor / p;
return gin;
}
}
} else {
/* Calculation by continued fraction. */
a = 1.0 - p;
b = a + x + 1.0;
term = 0.0;
pn[0] = 1.0;
pn[1] = x;
pn[2] = x + 1.0;
pn[3] = x * b;
gin = pn[2] / pn[3];
while (true) {
a += 1.0;
b += 2.0;
term += 1.0;
an = a * term;
for (i = 0; i < 2; i++) {
pn[i+4] = b * pn[i+2] - an * pn[i];
}
if (pn[5] != 0.0) {
rn = pn[4] / pn[5];
dif = fabs(gin - rn);
if (dif <= acu && dif <= acu * rn) {
gin = 1.0 - factor * gin;
return gin;
}
gin = rn;
}
for (i = 0; i < 4; i++) {
pn[i] = pn[i+2];
}
if (fabs(pn[4]) >= oflo) {
for (i = 0; i < 4; i++) {
pn[i] /= oflo;
}
}
}
}
}
/*
* Given a value p in [0..1] of the lower tail area of the normal distribution,
* compute the limit on the definite integral from [-inf..z] that satisfies p,
* accurate to 16 decimal places.
*
* This implementation is based on:
*
* Wichura, M.J. (1988) Algorithm AS 241: The percentage points of the normal
* distribution. Applied Statistics 37(3):477-484.
*/
static inline double
pt_norm(double p) {
double q, r, ret;
assert(p > 0.0 && p < 1.0);
q = p - 0.5;
if (fabs(q) <= 0.425) {
/* p close to 1/2. */
r = 0.180625 - q * q;
return q * (((((((2.5090809287301226727e3 * r +
3.3430575583588128105e4) * r + 6.7265770927008700853e4) * r
+ 4.5921953931549871457e4) * r + 1.3731693765509461125e4) *
r + 1.9715909503065514427e3) * r + 1.3314166789178437745e2)
* r + 3.3871328727963666080e0) /
(((((((5.2264952788528545610e3 * r +
2.8729085735721942674e4) * r + 3.9307895800092710610e4) * r
+ 2.1213794301586595867e4) * r + 5.3941960214247511077e3) *
r + 6.8718700749205790830e2) * r + 4.2313330701600911252e1)
* r + 1.0);
} else {
if (q < 0.0) {
r = p;
} else {
r = 1.0 - p;
}
assert(r > 0.0);
r = sqrt(-log(r));
if (r <= 5.0) {
/* p neither close to 1/2 nor 0 or 1. */
r -= 1.6;
ret = ((((((((7.74545014278341407640e-4 * r +
2.27238449892691845833e-2) * r +
2.41780725177450611770e-1) * r +
1.27045825245236838258e0) * r +
3.64784832476320460504e0) * r +
5.76949722146069140550e0) * r +
4.63033784615654529590e0) * r +
1.42343711074968357734e0) /
(((((((1.05075007164441684324e-9 * r +
5.47593808499534494600e-4) * r +
1.51986665636164571966e-2)
* r + 1.48103976427480074590e-1) * r +
6.89767334985100004550e-1) * r +
1.67638483018380384940e0) * r +
2.05319162663775882187e0) * r + 1.0));
} else {
/* p near 0 or 1. */
r -= 5.0;
ret = ((((((((2.01033439929228813265e-7 * r +
2.71155556874348757815e-5) * r +
1.24266094738807843860e-3) * r +
2.65321895265761230930e-2) * r +
2.96560571828504891230e-1) * r +
1.78482653991729133580e0) * r +
5.46378491116411436990e0) * r +
6.65790464350110377720e0) /
(((((((2.04426310338993978564e-15 * r +
1.42151175831644588870e-7) * r +
1.84631831751005468180e-5) * r +
7.86869131145613259100e-4) * r +
1.48753612908506148525e-2) * r +
1.36929880922735805310e-1) * r +
5.99832206555887937690e-1)
* r + 1.0));
}
if (q < 0.0) {
ret = -ret;
}
return ret;
}
}
/*
* Given a value p in [0..1] of the lower tail area of the Chi^2 distribution
* with df degrees of freedom, where ln_gamma_df_2 is ln_gamma(df/2.0), compute
* the upper limit on the definite integral from [0..z] that satisfies p,
* accurate to 12 decimal places.
*
* This implementation is based on:
*
* Best, D.J., D.E. Roberts (1975) Algorithm AS 91: The percentage points of
* the Chi^2 distribution. Applied Statistics 24(3):385-388.
*
* Shea, B.L. (1991) Algorithm AS R85: A remark on AS 91: The percentage
* points of the Chi^2 distribution. Applied Statistics 40(1):233-235.
*/
static inline double
pt_chi2(double p, double df, double ln_gamma_df_2) {
double e, aa, xx, c, ch, a, q, p1, p2, t, x, b, s1, s2, s3, s4, s5, s6;
unsigned i;
assert(p >= 0.0 && p < 1.0);
assert(df > 0.0);
e = 5.0e-7;
aa = 0.6931471805;
xx = 0.5 * df;
c = xx - 1.0;
if (df < -1.24 * log(p)) {
/* Starting approximation for small Chi^2. */
ch = pow(p * xx * exp(ln_gamma_df_2 + xx * aa), 1.0 / xx);
if (ch - e < 0.0) {
return ch;
}
} else {
if (df > 0.32) {
x = pt_norm(p);
/*
* Starting approximation using Wilson and Hilferty
* estimate.
*/
p1 = 0.222222 / df;
ch = df * pow(x * sqrt(p1) + 1.0 - p1, 3.0);
/* Starting approximation for p tending to 1. */
if (ch > 2.2 * df + 6.0) {
ch = -2.0 * (log(1.0 - p) - c * log(0.5 * ch) +
ln_gamma_df_2);
}
} else {
ch = 0.4;
a = log(1.0 - p);
while (true) {
q = ch;
p1 = 1.0 + ch * (4.67 + ch);
p2 = ch * (6.73 + ch * (6.66 + ch));
t = -0.5 + (4.67 + 2.0 * ch) / p1 - (6.73 + ch
* (13.32 + 3.0 * ch)) / p2;
ch -= (1.0 - exp(a + ln_gamma_df_2 + 0.5 * ch +
c * aa) * p2 / p1) / t;
if (fabs(q / ch - 1.0) - 0.01 <= 0.0) {
break;
}
}
}
}
for (i = 0; i < 20; i++) {
/* Calculation of seven-term Taylor series. */
q = ch;
p1 = 0.5 * ch;
if (p1 < 0.0) {
return -1.0;
}
p2 = p - i_gamma(p1, xx, ln_gamma_df_2);
t = p2 * exp(xx * aa + ln_gamma_df_2 + p1 - c * log(ch));
b = t / ch;
a = 0.5 * t - b * c;
s1 = (210.0 + a * (140.0 + a * (105.0 + a * (84.0 + a * (70.0 +
60.0 * a))))) / 420.0;
s2 = (420.0 + a * (735.0 + a * (966.0 + a * (1141.0 + 1278.0 *
a)))) / 2520.0;
s3 = (210.0 + a * (462.0 + a * (707.0 + 932.0 * a))) / 2520.0;
s4 = (252.0 + a * (672.0 + 1182.0 * a) + c * (294.0 + a *
(889.0 + 1740.0 * a))) / 5040.0;
s5 = (84.0 + 264.0 * a + c * (175.0 + 606.0 * a)) / 2520.0;
s6 = (120.0 + c * (346.0 + 127.0 * c)) / 5040.0;
ch += t * (1.0 + 0.5 * t * s1 - b * c * (s1 - b * (s2 - b * (s3
- b * (s4 - b * (s5 - b * s6))))));
if (fabs(q / ch - 1.0) <= e) {
break;
}
}
return ch;
}
/*
* Given a value p in [0..1] and Gamma distribution shape and scale parameters,
* compute the upper limit on the definite integral from [0..z] that satisfies
* p.
*/
static inline double
pt_gamma(double p, double shape, double scale, double ln_gamma_shape) {
return pt_chi2(p, shape * 2.0, ln_gamma_shape) * 0.5 * scale;
}