kfunc.c

#include <math.h>
#include <stdlib.h>
#include "htslib/kfunc.h"

/* Log gamma function
 * \log{\Gamma(z)}
 * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
 */
double kf_lgamma(double z)
{
	double x = 0;
	x += 0.1659470187408462e-06 / (z+7);
	x += 0.9934937113930748e-05 / (z+6);
	x -= 0.1385710331296526     / (z+5);
	x += 12.50734324009056      / (z+4);
	x -= 176.6150291498386      / (z+3);
	x += 771.3234287757674      / (z+2);
	x -= 1259.139216722289      / (z+1);
	x += 676.5203681218835      / z;
	x += 0.9999999999995183;
	return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
}

/* complementary error function
 * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
 * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
 */
double kf_erfc(double x)
{
	const double p0 = 220.2068679123761;
	const double p1 = 221.2135961699311;
	const double p2 = 112.0792914978709;
	const double p3 = 33.912866078383;
	const double p4 = 6.37396220353165;
	const double p5 = .7003830644436881;
	const double p6 = .03526249659989109;
	const double q0 = 440.4137358247522;
	const double q1 = 793.8265125199484;
	const double q2 = 637.3336333788311;
	const double q3 = 296.5642487796737;
	const double q4 = 86.78073220294608;
	const double q5 = 16.06417757920695;
	const double q6 = 1.755667163182642;
	const double q7 = .08838834764831844;
	double expntl, z, p;
	z = fabs(x) * M_SQRT2;
	if (z > 37.) return x > 0.? 0. : 2.;
	expntl = exp(z * z * - .5);
	if (z < 10. / M_SQRT2) // for small z
	    p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
			/ (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
	else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
	return x > 0.? 2. * p : 2. * (1. - p);
}

/* The following computes regularized incomplete gamma functions.
 * Formulas are taken from Wiki, with additional input from Numerical
 * Recipes in C (for modified Lentz's algorithm) and AS245
 * (http://lib.stat.cmu.edu/apstat/245).
 *
 * A good online calculator is available at:
 *
 *   http://www.danielsoper.com/statcalc/calc23.aspx
 *
 * It calculates upper incomplete gamma function, which equals
 * kf_gammaq(s,z)*tgamma(s).
 */

#define KF_GAMMA_EPS 1e-14
#define KF_TINY 1e-290

// regularized lower incomplete gamma function, by series expansion
static double _kf_gammap(double s, double z)
{
	double sum, x;
	int k;
	for (k = 1, sum = x = 1.; k < 100; ++k) {
		sum += (x *= z / (s + k));
		if (x / sum < KF_GAMMA_EPS) break;
	}
	return exp(s * log(z) - z - kf_lgamma(s + 1.) + log(sum));
}
// regularized upper incomplete gamma function, by continued fraction
static double _kf_gammaq(double s, double z)
{
	int j;
	double C, D, f;
	f = 1. + z - s; C = f; D = 0.;
	// Modified Lentz's algorithm for computing continued fraction
	// See Numerical Recipes in C, 2nd edition, section 5.2
	for (j = 1; j < 100; ++j) {
		double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
		D = b + a * D;
		if (D < KF_TINY) D = KF_TINY;
		C = b + a / C;
		if (C < KF_TINY) C = KF_TINY;
		D = 1. / D;
		d = C * D;
		f *= d;
		if (fabs(d - 1.) < KF_GAMMA_EPS) break;
	}
	return exp(s * log(z) - z - kf_lgamma(s) - log(f));
}

double kf_gammap(double s, double z)
{
	return z <= 1. || z < s? _kf_gammap(s, z) : 1. - _kf_gammaq(s, z);
}

double kf_gammaq(double s, double z)
{
	return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
}

/* Regularized incomplete beta function. The method is taken from
 * Numerical Recipe in C, 2nd edition, section 6.4. The following web
 * page calculates the incomplete beta function, which equals
 * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
 *
 *   http://www.danielsoper.com/statcalc/calc36.aspx
 */
static double kf_betai_aux(double a, double b, double x)
{
	double C, D, f;
	int j;
	if (x == 0.) return 0.;
	if (x == 1.) return 1.;
	f = 1.; C = f; D = 0.;
	// Modified Lentz's algorithm for computing continued fraction
	for (j = 1; j < 200; ++j) {
		double aa, d;
		int m = j>>1;
		aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
			: m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
		D = 1. + aa * D;
		if (D < KF_TINY) D = KF_TINY;
		C = 1. + aa / C;
		if (C < KF_TINY) C = KF_TINY;
		D = 1. / D;
		d = C * D;
		f *= d;
		if (fabs(d - 1.) < KF_GAMMA_EPS) break;
	}
	return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
}
double kf_betai(double a, double b, double x)
{
	return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
}

#ifdef KF_MAIN
#include <stdio.h>
int main(int argc, char *argv[])
{
	double x = 5.5, y = 3;
	double a, b;
	printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
	printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
	a = 2; b = 2; x = 0.5;
	printf("incomplete-beta(%lg,%lg,%lg): %lg\n", a, b, x, kf_betai(a, b, x) / exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b)));
	return 0;
}
#endif


// log\binom{n}{k}
static double lbinom(int n, int k)
{
    if (k == 0 || n == k) return 0;
    return lgamma(n+1) - lgamma(k+1) - lgamma(n-k+1);
}

// n11  n12  | n1_
// n21  n22  | n2_
//-----------+----
// n_1  n_2  | n

// hypergeometric distribution
static double hypergeo(int n11, int n1_, int n_1, int n)
{
    return exp(lbinom(n1_, n11) + lbinom(n-n1_, n_1-n11) - lbinom(n, n_1));
}

typedef struct {
    int n11, n1_, n_1, n;
    double p;
} hgacc_t;

// incremental version of hypergenometric distribution
static double hypergeo_acc(int n11, int n1_, int n_1, int n, hgacc_t *aux)
{
    if (n1_ || n_1 || n) {
        aux->n11 = n11; aux->n1_ = n1_; aux->n_1 = n_1; aux->n = n;
    } else { // then only n11 changed; the rest fixed
        if (n11%11 && n11 + aux->n - aux->n1_ - aux->n_1) {
            if (n11 == aux->n11 + 1) { // incremental
                aux->p *= (double)(aux->n1_ - aux->n11) / n11
                    * (aux->n_1 - aux->n11) / (n11 + aux->n - aux->n1_ - aux->n_1);
                aux->n11 = n11;
                return aux->p;
            }
            if (n11 == aux->n11 - 1) { // incremental
                aux->p *= (double)aux->n11 / (aux->n1_ - n11)
                    * (aux->n11 + aux->n - aux->n1_ - aux->n_1) / (aux->n_1 - n11);
                aux->n11 = n11;
                return aux->p;
            }
        }
        aux->n11 = n11;
    }
    aux->p = hypergeo(aux->n11, aux->n1_, aux->n_1, aux->n);
    return aux->p;
}

double kt_fisher_exact(int n11, int n12, int n21, int n22, double *_left, double *_right, double *two)
{
    int i, j, max, min;
    double p, q, left, right;
    hgacc_t aux;
    int n1_, n_1, n;

    n1_ = n11 + n12; n_1 = n11 + n21; n = n11 + n12 + n21 + n22; // calculate n1_, n_1 and n
    max = (n_1 < n1_) ? n_1 : n1_; // max n11, for right tail
    min = n1_ + n_1 - n;    // not sure why n11-n22 is used instead of min(n_1,n1_)
    if (min < 0) min = 0; // min n11, for left tail
    *two = *_left = *_right = 1.;
    if (min == max) return 1.; // no need to do test
    q = hypergeo_acc(n11, n1_, n_1, n, &aux); // the probability of the current table
    // left tail
    p = hypergeo_acc(min, 0, 0, 0, &aux);
    for (left = 0., i = min + 1; p < 0.99999999 * q && i<=max; ++i) // loop until underflow
        left += p, p = hypergeo_acc(i, 0, 0, 0, &aux);
    --i;
    if (p < 1.00000001 * q) left += p;
    else --i;
    // right tail
    p = hypergeo_acc(max, 0, 0, 0, &aux);
    for (right = 0., j = max - 1; p < 0.99999999 * q && j>=0; --j) // loop until underflow
        right += p, p = hypergeo_acc(j, 0, 0, 0, &aux);
    ++j;
    if (p < 1.00000001 * q) right += p;
    else ++j;
    // two-tail
    *two = left + right;
    if (*two > 1.) *two = 1.;
    // adjust left and right
    if (abs(i - n11) < abs(j - n11)) right = 1. - left + q;
    else left = 1.0 - right + q;
    *_left = left; *_right = right;
    return q;
}