/* em.c -- mathematical functions.

Copyright (C) 2010, 2011 Broad Institute.

Portions copyright (C) 2013-2014 Genome Research Ltd.

Author: Heng Li <lh3@live.co.uk>

Permission is hereby granted, free of charge, to any person obtaining a copy

of this software and associated documentation files (the "Software"), to deal

in the Software without restriction, including without limitation the rights

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

copies of the Software, and to permit persons to whom the Software is

furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in

all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN

THE SOFTWARE. */

#include <stdlib.h>

#include <string.h>

#include <math.h>

#include <htslib/vcf.h>

#include "kmin.h"

#include "call.h"

#define ITER_MAX 50

#define ITER_TRY 10

#define EPS 1e-5

extern double kf_gammaq(double, double);

/*

Generic routines

*/

// 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(call_t *call, const bcf1_t *rec, int n1, int flag, double x[10])

{

double *pdg;

int i, n; //, n2;

if (rec->n_allele < 2) return -1; // one allele only

// initialization

if (n1 < 0 || n1 > rec->n_sample) n1 = 0;

if (flag & 1<<7) flag |= 7<<5; // compute group freq if LRT is required

if (flag & 0xf<<1) flag |= 0xf<<1;

n = rec->n_sample; //n2 = n - n1;

pdg = call->pdg;

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.) 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);

}

return 0;

}

/*

Two-locus EM (LD)

*/

#define _G1(h, k) ((h>>1&1) + (k>>1&1))

#define _G2(h, k) ((h&1) + (k&1))

#if 0

// 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;

}

#endif