text: #include <stdlib.h>
  #include <string.h>
  #include <math.h>
  #include "bcf.h"
  #include "kmin.h"
  
  static double g_q2p[256];
  
  #define ITER_MAX 50
  #define ITER_TRY 10
  #define EPS 1e-5
  
  extern double kf_gammaq(double, double);
  
  /*
  	Generic routines
   */
  // get the 3 genotype likelihoods
  static double *get_pdg3(const bcf1_t *b)
  {
  	double *pdg;
  	const uint8_t *PL = 0;
  	int i, PL_len = 0;
  	// initialize g_q2p if necessary
  	if (g_q2p[0] == 0.)
  		for (i = 0; i < 256; ++i)
  			g_q2p[i] = pow(10., -i / 10.);
  	// set PL and PL_len
  	for (i = 0; i < b->n_gi; ++i) {
  		if (b->gi[i].fmt == bcf_str2int("PL", 2)) {
  			PL = (const uint8_t*)b->gi[i].data;
  			PL_len = b->gi[i].len;
  			break;
  		}
  	}
  	if (i == b->n_gi) return 0; // no PL
  	// fill pdg
  	pdg = malloc(3 * b->n_smpl * sizeof(double));
  	for (i = 0; i < b->n_smpl; ++i) {
  		const uint8_t *pi = PL + i * PL_len;
  		double *p = pdg + i * 3;
  		p[0] = g_q2p[pi[2]]; p[1] = g_q2p[pi[1]]; p[2] = g_q2p[pi[0]];
  	}
  	return pdg;
  }
  
  // estimate site allele frequency in a very naive and inaccurate way
  static double est_freq(int n, const double *pdg)
  {
  	int i, gcnt[3], tmp1;
  	// get a rough estimate of the genotype frequency
  	gcnt[0] = gcnt[1] = gcnt[2] = 0;
  	for (i = 0; i < n; ++i) {
  		const double *p = pdg + i * 3;
  		if (p[0] != 1. || p[1] != 1. || p[2] != 1.) {
  			int which = p[0] > p[1]? 0 : 1;
  			which = p[which] > p[2]? which : 2;
  			++gcnt[which];
  		}
  	}
  	tmp1 = gcnt[0] + gcnt[1] + gcnt[2];
  	return (tmp1 == 0)? -1.0 : (.5 * gcnt[1] + gcnt[2]) / tmp1;
  }
  
  /*
  	Single-locus EM
   */
  
  typedef struct {
  	int beg, end;
  	const double *pdg;
  } minaux1_t;
  
  static double prob1(double f, void *data)
  {
  	minaux1_t *a = (minaux1_t*)data;
  	double p = 1., l = 0., f3[3];
  	int i;
  //	printf("brent %lg\n", f);
  	if (f < 0 || f > 1) return 1e300;
  	f3[0] = (1.-f)*(1.-f); f3[1] = 2.*f*(1.-f); f3[2] = f*f;
  	for (i = a->beg; i < a->end; ++i) {
  		const double *pdg = a->pdg + i * 3;
  		p *= pdg[0] * f3[0] + pdg[1] * f3[1] + pdg[2] * f3[2];
  		if (p < 1e-200) l -= log(p), p = 1.;
  	}
  	return l - log(p);
  }
  
  // one EM iteration for allele frequency estimate
  static double freq_iter(double *f, const double *_pdg, int beg, int end)
  {
  	double f0 = *f, f3[3], err;
  	int i;
  //	printf("em %lg\n", *f);
  	f3[0] = (1.-f0)*(1.-f0); f3[1] = 2.*f0*(1.-f0); f3[2] = f0*f0;
  	for (i = beg, f0 = 0.; i < end; ++i) {
  		const double *pdg = _pdg + i * 3;
  		f0 += (pdg[1] * f3[1] + 2. * pdg[2] * f3[2])
  			/ (pdg[0] * f3[0] + pdg[1] * f3[1] + pdg[2] * f3[2]);
  	}
  	f0 /= (end - beg) * 2;
  	err = fabs(f0 - *f);
  	*f = f0;
  	return err;
  }
  
  /* The following function combines EM and Brent's method. When the signal from
   * the data is strong, EM is faster but sometimes, EM may converge very slowly.
   * When this happens, we switch to Brent's method. The idea is learned from
   * Rasmus Nielsen.
   */
  static double freqml(double f0, int beg, int end, const double *pdg)
  {
  	int i;
  	double f;
  	for (i = 0, f = f0; i < ITER_TRY; ++i)
  		if (freq_iter(&f, pdg, beg, end) < EPS) break;
  	if (i == ITER_TRY) { // haven't converged yet; try Brent's method
  		minaux1_t a;
  		a.beg = beg; a.end = end; a.pdg = pdg;
  		kmin_brent(prob1, f0 == f? .5*f0 : f0, f, (void*)&a, EPS, &f);
  	}
  	return f;
  }
  
  // one EM iteration for genotype frequency estimate
  static double g3_iter(double g[3], const double *_pdg, int beg, int end)
  {
  	double err, gg[3];
  	int i;
  	gg[0] = gg[1] = gg[2] = 0.;
  //	printf("%lg,%lg,%lg\n", g[0], g[1], g[2]);
  	for (i = beg; i < end; ++i) {
  		double sum, tmp[3];
  		const double *pdg = _pdg + i * 3;
  		tmp[0] = pdg[0] * g[0]; tmp[1] = pdg[1] * g[1]; tmp[2] = pdg[2] * g[2];
  		sum = (tmp[0] + tmp[1] + tmp[2]) * (end - beg);
  		gg[0] += tmp[0] / sum; gg[1] += tmp[1] / sum; gg[2] += tmp[2] / sum;
  	}
  	err = fabs(gg[0] - g[0]) > fabs(gg[1] - g[1])? fabs(gg[0] - g[0]) : fabs(gg[1] - g[1]);
  	err = err > fabs(gg[2] - g[2])? err : fabs(gg[2] - g[2]);
  	g[0] = gg[0]; g[1] = gg[1]; g[2] = gg[2];
  	return err;
  }
  
  // perform likelihood ratio test
  static double lk_ratio_test(int n, int n1, const double *pdg, double f3[3][3])
  {
  	double r;
  	int i;
  	for (i = 0, r = 1.; i < n1; ++i) {
  		const double *p = pdg + i * 3;
  		r *= (p[0] * f3[1][0] + p[1] * f3[1][1] + p[2] * f3[1][2])
  			/ (p[0] * f3[0][0] + p[1] * f3[0][1] + p[2] * f3[0][2]);
  	}
  	for (; i < n; ++i) {
  		const double *p = pdg + i * 3;
  		r *= (p[0] * f3[2][0] + p[1] * f3[2][1] + p[2] * f3[2][2])
  			/ (p[0] * f3[0][0] + p[1] * f3[0][1] + p[2] * f3[0][2]);
  	}
  	return r;
  }
  
  // x[0]: ref frequency
  // x[1..3]: alt-alt, alt-ref, ref-ref frequenc
  // x[4]: HWE P-value
  // x[5..6]: group1 freq, group2 freq
  // x[7]: 1-degree P-value
  // x[8]: 2-degree P-value
  int bcf_em1(const bcf1_t *b, int n1, int flag, double x[10])
  {
  	double *pdg;
  	int i, n, n2;
  	if (b->n_alleles < 2) return -1; // one allele only
  	// initialization
  	if (n1 < 0 || n1 > b->n_smpl) n1 = 0;
  	if (flag & 1<<7) flag |= 7<<5; // compute group freq if LRT is required
  	if (flag & 0xf<<1) flag |= 0xf<<1;
  	n = b->n_smpl; n2 = n - n1;
  	pdg = get_pdg3(b);
  	if (pdg == 0) return -1;
  	for (i = 0; i < 10; ++i) x[i] = -1.; // set to negative
  	{
  		if ((x[0] = est_freq(n, pdg)) < 0.) {
  			free(pdg);
  			return -1; // no data
  		}
  		x[0] = freqml(x[0], 0, n, pdg);
  	}
  	if (flag & (0xf<<1|3<<8)) { // estimate the genotype frequency and test HWE
  		double *g = x + 1, f3[3], r;
  		f3[0] = g[0] = (1 - x[0]) * (1 - x[0]);
  		f3[1] = g[1] = 2 * x[0] * (1 - x[0]);
  		f3[2] = g[2] = x[0] * x[0];
  		for (i = 0; i < ITER_MAX; ++i)
  			if (g3_iter(g, pdg, 0, n) < EPS) break;
  		// Hardy-Weinberg equilibrium (HWE)
  		for (i = 0, r = 1.; i < n; ++i) {
  			double *p = pdg + i * 3;
  			r *= (p[0] * g[0] + p[1] * g[1] + p[2] * g[2]) / (p[0] * f3[0] + p[1] * f3[1] + p[2] * f3[2]);
  		}
  		x[4] = kf_gammaq(.5, log(r));
  	}
  	if ((flag & 7<<5) && n1 > 0 && n1 < n) { // group frequency
  		x[5] = freqml(x[0], 0, n1, pdg);
  		x[6] = freqml(x[0], n1, n, pdg);
  	}
  	if ((flag & 1<<7) && n1 > 0 && n1 < n) { // 1-degree P-value
  		double f[3], f3[3][3], tmp;
  		f[0] = x[0]; f[1] = x[5]; f[2] = x[6];
  		for (i = 0; i < 3; ++i)
  			f3[i][0] = (1-f[i])*(1-f[i]), f3[i][1] = 2*f[i]*(1-f[i]), f3[i][2] = f[i]*f[i];
  		tmp = log(lk_ratio_test(n, n1, pdg, f3));
  		if (tmp < 0) tmp = 0;
  		x[7] = kf_gammaq(.5, tmp);
  	}
  	if ((flag & 3<<8) && n1 > 0 && n1 < n) { // 2-degree P-value
  		double g[3][3], tmp;
  		for (i = 0; i < 3; ++i) memcpy(g[i], x + 1, 3 * sizeof(double));
  		for (i = 0; i < ITER_MAX; ++i)
  			if (g3_iter(g[1], pdg, 0, n1) < EPS) break;
  		for (i = 0; i < ITER_MAX; ++i)
  			if (g3_iter(g[2], pdg, n1, n) < EPS) break;
  		tmp = log(lk_ratio_test(n, n1, pdg, g));
  		if (tmp < 0) tmp = 0;
  		x[8] = kf_gammaq(1., tmp);
  	}
  	// free
  	free(pdg);
  	return 0;
  }
  
  /*
  	Two-locus EM (LD)
   */
  
  #define _G1(h, k) ((h>>1&1) + (k>>1&1))
  #define _G2(h, k) ((h&1) + (k&1))
  
  // 0: the previous site; 1: the current site
  static int pair_freq_iter(int n, double *pdg[2], double f[4])
  {
  	double ff[4];
  	int i, k, h;
  //	printf("%lf,%lf,%lf,%lf\n", f[0], f[1], f[2], f[3]);
  	memset(ff, 0, 4 * sizeof(double));
  	for (i = 0; i < n; ++i) {
  		double *p[2], sum, tmp;
  		p[0] = pdg[0] + i * 3; p[1] = pdg[1] + i * 3;
  		for (k = 0, sum = 0.; k < 4; ++k)
  			for (h = 0; h < 4; ++h)
  				sum += f[k] * f[h] * p[0][_G1(k,h)] * p[1][_G2(k,h)];
  		for (k = 0; k < 4; ++k) {
  			tmp = f[0] * (p[0][_G1(0,k)] * p[1][_G2(0,k)] + p[0][_G1(k,0)] * p[1][_G2(k,0)])
  				+ f[1] * (p[0][_G1(1,k)] * p[1][_G2(1,k)] + p[0][_G1(k,1)] * p[1][_G2(k,1)])
  				+ f[2] * (p[0][_G1(2,k)] * p[1][_G2(2,k)] + p[0][_G1(k,2)] * p[1][_G2(k,2)])
  				+ f[3] * (p[0][_G1(3,k)] * p[1][_G2(3,k)] + p[0][_G1(k,3)] * p[1][_G2(k,3)]);
  			ff[k] += f[k] * tmp / sum;
  		}
  	}
  	for (k = 0; k < 4; ++k) f[k] = ff[k] / (2 * n);
  	return 0;
  }
  
  double bcf_pair_freq(const bcf1_t *b0, const bcf1_t *b1, double f[4])
  {
  	const bcf1_t *b[2];
  	int i, j, n_smpl;
  	double *pdg[2], flast[4], r, f0[2];
  	// initialize others
  	if (b0->n_smpl != b1->n_smpl) return -1; // different number of samples
  	n_smpl = b0->n_smpl;
  	b[0] = b0; b[1] = b1;
  	f[0] = f[1] = f[2] = f[3] = -1.;
  	if (b[0]->n_alleles < 2 || b[1]->n_alleles < 2) return -1; // one allele only
  	pdg[0] = get_pdg3(b0); pdg[1] = get_pdg3(b1);
  	if (pdg[0] == 0 || pdg[1] == 0) {
  		free(pdg[0]); free(pdg[1]);
  		return -1;
  	}
  	// set the initial value
  	f0[0] = est_freq(n_smpl, pdg[0]);
  	f0[1] = est_freq(n_smpl, pdg[1]);
  	f[0] = (1 - f0[0]) * (1 - f0[1]); f[3] = f0[0] * f0[1];
  	f[1] = (1 - f0[0]) * f0[1]; f[2] = f0[0] * (1 - f0[1]);
  	// iteration
  	for (j = 0; j < ITER_MAX; ++j) {
  		double eps = 0;
  		memcpy(flast, f, 4 * sizeof(double));
  		pair_freq_iter(n_smpl, pdg, f);
  		for (i = 0; i < 4; ++i) {
  			double x = fabs(f[i] - flast[i]);
  			if (x > eps) eps = x;
  		}
  		if (eps < EPS) break;
  	}
  	// free
  	free(pdg[0]); free(pdg[1]);
  	{ // calculate r^2
  		double p[2], q[2], D;
  		p[0] = f[0] + f[1]; q[0] = 1 - p[0];
  		p[1] = f[0] + f[2]; q[1] = 1 - p[1];
  		D = f[0] * f[3] - f[1] * f[2];
  		r = sqrt(D * D / (p[0] * p[1] * q[0] * q[1]));
  //		printf("R(%lf,%lf,%lf,%lf)=%lf\n", f[0], f[1], f[2], f[3], r);
  		if (isnan(r)) r = -1.;
  	}
  	return r;
  }
  