// use REML equation to estimate variance component
// input P, calculate PA, H, R and varcmp
void gcta::ai_reml(eigenMatrix &P, eigenMatrix &Hi, eigenVector &Py, eigenVector &prev_varcmp, eigenVector &varcmp, double dlogL)
{
int i=0, j=0;
Py=P*_y;
eigenVector cvec(_n);
eigenMatrix APy(_n, _r_indx.size());
for(i=0; i<_r_indx.size(); i++) (APy.col(i))=(_A.block(0,_r_indx[i]*_n,_n,_n))*Py;
// Calculate Hi
eigenVector R(_r_indx.size()+1);
for(i=0; i<_r_indx.size(); i++){
R(i)=(Py.transpose()*(APy.col(i)))(0,0);
cvec=P*(APy.col(i));
Hi(i,i)=((APy.col(i)).transpose()*cvec)(0,0);
for(j=0; j<i; j++) Hi(j,i)=Hi(i,j)=((APy.col(j)).transpose()*cvec)(0,0);
}
cvec=P*Py;
for(j=0; j<_r_indx.size(); j++) Hi(j,_r_indx.size())=Hi(_r_indx.size(),j)=((APy.col(j)).transpose()*cvec)(0,0);
R(_r_indx.size())=(Py.transpose()*Py)(0,0);
Hi(_r_indx.size(),_r_indx.size())=(Py.transpose()*cvec)(0,0);
Hi=0.5*Hi;
// Calcualte tr(PA) and dL
eigenVector tr_PA;
calcu_tr_PA(P, tr_PA);
R=-0.5*(tr_PA-R);
// Calculate variance component
comput_inverse_logdet_LU(Hi, "Error: AI matrix is not invertible.");
eigenVector delta(_r_indx.size()+1);
delta=Hi*R;
if(dlogL>1.0) varcmp=prev_varcmp+0.316*delta;
else varcmp=prev_varcmp+delta;
}
// use REML equation to estimate variance component
// input P, calculate PA, H, R and varcmp
void gcta::reml_equation(eigenMatrix &P, eigenMatrix &Hi, eigenVector &Py, eigenVector &varcmp)
{
// Calculate Hi
calcu_Hi(P, Hi);
// Calculate R
Py=P*_y;
eigenVector R(_r_indx.size()+1);
for(int i=0; i<_r_indx.size(); i++) R(i)=(Py.transpose()*(_A.block(0,_r_indx[i]*_n,_n,_n))*Py)(0,0);
R(_r_indx.size())=Py.squaredNorm();
// Calculate variance component
varcmp=Hi*R;
Hi=2*Hi; // for calculation of SE
}
// input P, calculate varcmp
void gcta::em_reml(eigenMatrix &P, eigenVector &Py, eigenVector &prev_varcmp, eigenVector &varcmp)
{
int i=0;
// Calculate trace(PA)
eigenVector tr_PA;
calcu_tr_PA(P, tr_PA);
// Calculate R
Py=P*_y;
eigenVector R(_r_indx.size()+1);
for(i=0; i<_r_indx.size(); i++) R(i)=(Py.transpose()*(_A.block(0,_r_indx[i]*_n,_n,_n))*Py)(0,0);
R(_r_indx.size())=Py.squaredNorm();
// Calculate variance component
for(i=0; i<_r_indx.size(); i++) varcmp(i)=(prev_varcmp(i)*_n-prev_varcmp(i)*prev_varcmp(i)*tr_PA(i)+prev_varcmp(i)*prev_varcmp(i)*R(i))/_n;
int j=_r_indx.size();
varcmp(j)=(prev_varcmp(j)*_n-prev_varcmp(j)*prev_varcmp(j)*tr_PA(j)+prev_varcmp(j)*prev_varcmp(j)*R(j))/_n;
}
bool gcta::bending_eigenval(eigenVector &eval)
{
int j=0;
double eval_m=eval.mean();
if(eval.minCoeff()>0.0) return false;
double S=0.0, P=0.0;
for(j=0; j<eval.size(); j++){
if(eval[j]>=0) continue;
S+=eval[j];
P=-eval[j];
}
double W=S*S*100.0+1;
for(j=0; j<eval.size(); j++){
if(eval[j]>=0) continue;
eval[j]=P*(S-eval[j])*(S-eval[j])/W;
}
eval*=eval_m/eval.mean();
return true;
}
void gcta::makex_eigenVector(int j, eigenVector &x, bool resize, bool minus_2p)
{
int i=0;
if(resize) x.resize(_keep.size());
for(i=0; i<_keep.size(); i++){
if(!_snp_1[_include[j]][_keep[i]] || _snp_2[_include[j]][_keep[i]]){
if(_allele1[_include[j]]==_ref_A[_include[j]]) x[i]=(_snp_1[_include[j]][_keep[i]]+_snp_2[_include[j]][_keep[i]]);
else x[i]=2.0-(_snp_1[_include[j]][_keep[i]]+_snp_2[_include[j]][_keep[i]]);
}
else x[i]=_mu[_include[j]];
if(minus_2p) x[i]-=_mu[_include[j]];
}
}
bool gcta::comput_inverse_logdet_LDLT(eigenMatrix &Vi, eigenVector &prev_varcmp, double &logdet)
{
int i=0, n=Vi.cols();
LDLT<eigenMatrix> ldlt(Vi);
eigenVector d=ldlt.vectorD();
if(d.minCoeff()<0){
if(prev_varcmp.minCoeff()>0) return false;
else throw("Error: the matrix V becomes negative-definite because of one of the variance component is negative.\nPlease re-run the analysis without the --reml-no-constrain option.");
}
logdet=0.0;
for(i=0; i<n; i++) logdet+=log(d[i]);
Vi.setIdentity();
ldlt.solveInPlace(Vi);
return true;
}
// input P, calculate tr(PA)
void gcta::calcu_tr_PA(eigenMatrix &P, eigenVector &tr_PA)
{
int i=0, k=0, l=0;
double d_buf=0.0;
// Calculate trace(PA)
tr_PA.resize(_r_indx.size()+1);
for(i=0; i<_r_indx.size(); i++){
d_buf=0.0;
for(k=0; k<_n; k++){
for(l=0; l<_n; l++) d_buf+=P(k,l)*(_A.block(0,_r_indx[i]*_n,_n,_n))(k,l);
}
tr_PA(i)=d_buf;
}
tr_PA(_r_indx.size())=P.trace();
}
double gcta::reml_iteration(double y_Ssq, eigenMatrix &Vi_X, eigenMatrix &Xt_Vi_X_i, eigenMatrix &Hi, eigenVector &Py, eigenVector &varcmp, bool prior_var_flag, bool no_constrain)
{
char *mtd_str[3]={"AI-REML algorithm", "REML equation ...", "EM-REML algorithm ..."};
int i=0, constrain_num=0, iter=0, reml_mtd_tmp=_reml_mtd;
double logdet=0.0, logdet_Xt_Vi_X=0.0, prev_lgL=-1e20, lgL=-1e20, dlogL=1000.0;
eigenVector prev_varcmp(varcmp), varcomp_init(varcmp);
_Vi.resize(_n,_n);
_P(_n,_n);
for(iter=0; iter<_reml_max_iter; iter++){
if(iter==0){
prev_varcmp=varcomp_init;
if(prior_var_flag){
cout<<"User-specified prior values of variance components: "<<varcmp.transpose()<<endl;
//continue;
}
else{
_reml_mtd=2;
cout<<"Calculating prior values of variance components by EM-REML ..."<<endl;
}
}
if(iter==1){
_reml_mtd=reml_mtd_tmp;
cout<<"Running "<<mtd_str[_reml_mtd]<<" ..."<<"\nIter.\tlogL\t";
for(i=0; i<_r_indx.size(); i++) cout<<_var_name[_r_indx[i]]<<"\t";
cout<<_var_name[_var_name.size()-1]<<endl;
}
if(!calcu_Vi(_Vi, prev_varcmp, logdet, iter)) continue; // Calculate Vi
logdet_Xt_Vi_X=calcu_P(_Vi, Vi_X, Xt_Vi_X_i, _P); // Calculate P
if(_reml_mtd==0) ai_reml(_P, Hi, Py, prev_varcmp, varcmp, dlogL);
else if(_reml_mtd==1) reml_equation(_P, Hi, Py, varcmp);
else if(_reml_mtd==2) em_reml(_P, Py, prev_varcmp, varcmp);
lgL=-0.5*(logdet_Xt_Vi_X+logdet+(_y.transpose()*Py)(0,0));
// output log
if(!no_constrain) constrain_num=constrain_varcmp(varcmp, y_Ssq);
if(iter>0){
cout<<iter<<"\t"<<setiosflags(ios::fixed)<<setprecision(2)<<lgL<<"\t";
for(i=0; i<_r_indx.size()+1; i++) cout<<setprecision(5)<<varcmp[i]<<"\t";
if(constrain_num>0) cout<<"("<<constrain_num<<" component(s) constrained)"<<endl;
else cout<<endl;
}
else{
if(!prior_var_flag) cout<<"Prior values updated from EM-REML: "<<varcmp.transpose()<<endl;
cout<<"logL: "<<lgL<<endl;
}
if(constrain_num*2>_r_indx.size()+1) throw("Error: analysis stopped because more than half of the variance components are constrained. The result would be unreliable.\n Please have a try to add the option --reml-no-constrain.");
// convergence
dlogL=lgL-prev_lgL;
if((varcmp-prev_varcmp).squaredNorm()/varcmp.squaredNorm()<1e-8 && (fabs(dlogL)<1e-4 || (fabs(dlogL)<1e-2 && dlogL<0))){
if(_reml_mtd==2){ calcu_Hi(_P, Hi); Hi=2*Hi; } // for calculation of SE
break;
}
prev_varcmp=varcmp; prev_lgL=lgL;
}
if(iter==_reml_max_iter){
stringstream errmsg;
errmsg<<"Error: Log-likelihood not converged (stop after "<<_reml_max_iter<<" iteractions). \nYou can specify the option --reml-maxit to allow for more iterations."<<endl;
throw(errmsg.str());
}
else cout<<"Log-likelihood ratio converged."<<endl;
return lgL;
}