int FitNullModel(Matrix& mat_Xnull, Matrix& mat_y,
const EigenMatrix& kinshipU, const EigenMatrix& kinshipS){
// type conversion
Eigen::MatrixXf x;
Eigen::MatrixXf y;
G_to_Eigen(mat_Xnull, &x);
G_to_Eigen(mat_y, &y);
this->lambda = kinshipS.mat;
const Eigen::MatrixXf& U = kinshipU.mat;
// rotate
this->ux = U.transpose() * x;
this->uy = U.transpose() * y;
// get beta, sigma2_g and delta
// where delta = sigma2_e / sigma2_g
double loglik[101];
int maxIndex = -1;
double maxLogLik = 0;
for (int i = 0; i <= 100; ++i ){
double d = exp(-10 + i * 0.2);
getBetaSigma2(d);
loglik[i] = getLogLikelihood(d);
// fprintf(stderr, "%d\tdelta=%g\tll=%lf\n", i, delta, loglik[i]);
if (std::isnan(loglik[i])) {
continue;
}
if (maxIndex < 0 || loglik[i] > maxLogLik) {
maxIndex = i;
maxLogLik = loglik[i];
}
}
if (maxIndex < -1) {
fprintf(stderr, "Cannot optimize\n");
return -1;
}
if (maxIndex == 0 || maxIndex == 100) {
// on the boundary
// do not try maximize it.
} else {
gsl_function F;
F.function = goalFunction;
F.params = this;
Minimizer minimizer;
double lb = exp(-10 + (maxIndex-1) * 0.2);
double ub = exp(-10 + (maxIndex+1) * 0.2);
double start = exp(-10 + maxIndex * 0.2);
if (minimizer.minimize(F, start, lb, ub)) {
// fprintf(stderr, "Minimization failed, fall back to initial guess.\n");
this->delta = start;
} else {
this->delta = minimizer.getX();
// fprintf(stderr, "minimization succeed when delta = %g, sigma2_g = %g\n", this->delta, this->sigma2_g);
}
}
// store some intermediate results
// fprintf(stderr, "maxIndex = %d, delta = %g, Try brent\n", maxIndex, delta);
// fprintf(stderr, "beta[%d][%d] = %g\n", (int)beta.rows(), (int)beta.cols(), beta(0,0));
this->h2 = 1.0 /(1.0 + this->delta);
this->sigma2 = this->sigma2_g * this->h2;
// we derive different formular to replace original eqn (7)
this->gamma = (this->lambda.array() / (this->lambda.array() + this->delta)).sum() / this->sigma2_g / (this->ux.rows() - 1 ) ;
// fprintf(stderr, "gamma = %g\n", this->gamma);
// transformedY = \Sigma^{-1} * (y_tilda) and y_tilda = y - X * \beta
// since \Sigma = (\sigma^2_g * h^2 ) * (U * (\lambda + delta) * U')
// transformedY = 1 / (\sigma^2_g * h^2 ) * (U * (\lambda+delta)^{-1} * U' * (y_tilda))
// = 1 / (\sigma^2_g * h^2 ) * (U * \lambda^{-1} * (uResid))
// since h^2 = 1 / (1+delta)
// transformedY = (1 + delta/ (\sigma^2_g ) * (U * \lambda^{-1} * (uResid))
Eigen::MatrixXf resid = y - x * (x.transpose() * x).eval().ldlt().solve(x.transpose() * y); // this is y_tilda
this->transformedY.noalias() = U.transpose() * resid;
this->transformedY = (this->lambda.array() + this->delta).inverse().matrix().asDiagonal() * this->transformedY;
this->transformedY = U * this->transformedY;
this->transformedY /= this->sigma2_g;
// fprintf(stderr, "transformedY(0,0) = %g\n", transformedY(0,0));
this->ySigmaY= (resid.array() * transformedY.array()).sum();
return 0;
}
double surface_tension(Minimizer min, Functional f0, LiquidProperties prop,
bool verbose, const char *plotname) {
int numptspersize = 100;
int size = 64;
const int gas_size = 10;
Lattice lat(Cartesian(1,0,0), Cartesian(0,1,0), Cartesian(0,0,size*prop.lengthscale));
GridDescription gd(lat, 1, 1, numptspersize*size);
Grid potential(gd);
// Set the density to range from vapor to liquid
const double Veff_liquid = -prop.kT*log(prop.liquid_density);
const double Veff_gas = -prop.kT*log(prop.vapor_density);
for (int i=0; i<gd.NxNyNz*gas_size/size; i++) potential[i] = Veff_gas;
for (int i=gd.NxNyNz*gas_size/size; i<gd.NxNyNz; i++) potential[i] = Veff_liquid;
// Enable the following line for debugging...
//f0.run_finite_difference_test("f0", prop.kT, potential);
min.minimize(f0, gd, &potential);
while (min.improve_energy(verbose))
if (verbose) {
printf("Working on liberated interface...\n");
fflush(stdout);
}
const double Einterface = f0.integral(prop.kT, potential);
double Ninterface = 0;
Grid interface_density(gd, EffectivePotentialToDensity()(prop.kT, gd, potential));
for (int i=0;i<gd.NxNyNz;i++) Ninterface += interface_density[i]*gd.dvolume;
if (verbose) printf("Got interface energy of %g.\n", Einterface);
for (int i=0; i<gd.NxNyNz; i++) potential[i] = Veff_gas;
min.minimize(f0, gd, &potential);
while (min.improve_energy(verbose))
if (verbose) {
printf("Working on gas...\n");
fflush(stdout);
}
const double Egas = f0.integral(prop.kT, potential);
double Ngas = 0;
{
Grid density(gd, EffectivePotentialToDensity()(prop.kT, gd, potential));
for (int i=0;i<gd.NxNyNz;i++) Ngas += density[i]*gd.dvolume;
}
for (int i=0; i<gd.NxNyNz; i++) potential[i] = Veff_liquid;
if (verbose) {
printf("\n\n\nWorking on liquid...\n");
fflush(stdout);
}
min.minimize(f0, gd, &potential);
while (min.improve_energy(verbose))
if (verbose) {
printf("Working on liquid...\n");
fflush(stdout);
}
const double Eliquid = f0.integral(prop.kT, potential);
double Nliquid = 0;
{
Grid density(gd, EffectivePotentialToDensity()(prop.kT, gd, potential));
for (int i=0;i<gd.NxNyNz;i++) Nliquid += density[i]*gd.dvolume;
}
const double X = Ninterface/Nliquid; // Fraction of volume effectively filled with liquid.
const double surface_tension = (Einterface - Eliquid*X - Egas*(1-X))/2;
if (verbose) {
printf("\n\n");
printf("interface energy is %.15g\n", Einterface);
printf("gas energy is %.15g\n", Egas);
printf("liquid energy is %.15g\n", Eliquid);
printf("Ninterface/liquid/gas = %g/%g/%g\n", Ninterface, Nliquid, Ngas);
printf("X is %g\n", X);
printf("surface tension is %.10g\n", surface_tension);
}
if (plotname)
interface_density.Dump1D(plotname, Cartesian(0,0,0),
Cartesian(0,0,size*prop.lengthscale));
return surface_tension;
}
int FitNullModel(Matrix& mat_Xnull, Matrix& mat_y,
const EigenMatrix& kinshipU, const EigenMatrix& kinshipS){
// sanity check
if (mat_Xnull.rows != mat_y.rows) return -1;
if (mat_Xnull.rows != kinshipU.mat.rows()) return -1;
if (mat_Xnull.rows != kinshipS.mat.rows()) return -1;
// type conversion
G_to_Eigen(mat_Xnull, &this->ux);
G_to_Eigen(mat_y, &this->uy);
this->lambda = kinshipS.mat;
const Eigen::MatrixXf& U = kinshipU.mat;
// rotate
this->ux = U.transpose() * this->ux;
this->uy = U.transpose() * this->uy;
// get beta, sigma and delta
// where delta = sigma2_e / sigma2_g
double loglik[101];
int maxIndex = -1;
double maxLogLik = 0;
for (int i = 0; i <= 100; ++i ){
delta = exp(-10 + i * 0.2);
getBetaSigma2(delta);
loglik[i] = getLogLikelihood(delta);
#ifdef DEBUG
fprintf(stderr, "%d\tdelta=%g\tll=%lf\n", i, delta, loglik[i]);
fprintf(stderr, "beta(0)=%lf\tsigma2=%lf\n",
beta(0), sigma2);
#endif
if (std::isnan(loglik[i])) {
continue;
}
if (maxIndex < 0 || loglik[i] > maxLogLik) {
maxIndex = i;
maxLogLik = loglik[i];
}
}
if (maxIndex < -1) {
fprintf(stderr, "Cannot optimize\n");
return -1;
}
#if 0
fprintf(stderr, "maxIndex = %d\tll=%lf\t\tbeta(0)=%lf\tsigma2=%lf\n",
maxIndex, maxLogLik, beta(0), sigma2);
#endif
if (maxIndex == 0 || maxIndex == 100) {
// on the boundary
// do not try maximize it.
} else {
gsl_function F;
F.function = goalFunction;
F.params = this;
Minimizer minimizer;
double lb = exp(-10 + (maxIndex-1) * 0.2);
double ub = exp(-10 + (maxIndex+1) * 0.2);
double start = exp(-10 + maxIndex * 0.2);
if (minimizer.minimize(F, start, lb, ub)) {
fprintf(stderr, "Minimization failed, fall back to initial guess.\n");
this->delta = start;
} else {
this->delta = minimizer.getX();
#ifdef DEBUG
fprintf(stderr, "minimization succeed when delta = %g, sigma2 = %g\n", this->delta, this->sigma2);
#endif
}
}
// store some intermediate results
#ifdef DEBUG
fprintf(stderr, "delta = sigma2_e/sigma2_g, and sigma2 is sigma2_g\n");
fprintf(stderr, "maxIndex = %d, delta = %g, Try brent\n", maxIndex, delta);
fprintf(stderr, "beta[0][0] = %g\t sigma2_g = %g\tsigma2_e = %g\n", beta(0,0), this->sigma2, delta * sigma2);
#endif
// if (this->test == MetaCov::LRT) {
// this->nullLikelihood = getLogLikelihood(this->delta);
// } else if (this->test == MetaCov::SCORE) {
// this->uResid = this->uy - this->ux * this->beta;
// }
return 0;
}