void base_reg::base_score(const mematrix<double>& resid,
const int model,
const int interaction,
const int ngpreds,
const masked_matrix& invvarmatrix,
const int nullmodel) {
mematrix<double> oX = reg_data.extract_genotypes();
mematrix<double> X = apply_model(oX, model, interaction, ngpreds,
reg_data.is_interaction_excluded, false, nullmodel);
beta.reinit(X.ncol, 1);
sebeta.reinit(X.ncol, 1);
int length_beta = X.ncol;
double N = static_cast<double>(resid.nrow);
mematrix<double> tX = transpose(X);
if (invvarmatrix.length_of_mask != 0){
tX = tX * invvarmatrix.masked_data;
}
mematrix<double> u = tX * resid;
mematrix<double> v = tX * X;
mematrix<double> csum = column_sum(X);
csum = transpose(csum) * csum;
csum = csum * (1. / N);
v = v - csum;
// use cholesky to invert
LDLT <MatrixXd> Ch = LDLT < MatrixXd > (v.data.selfadjointView<Lower>());
// before was
// mematrix<double> v_i = invert(v);
beta.data = Ch.solve(v.data.adjoint() * u.data);
//TODO(maartenk): set size of v_i directly or remove mematrix class
mematrix<double> v_i = v;
v_i.data = Ch.solve(MatrixXd(length_beta, length_beta).
Identity(length_beta, length_beta));
double sr = 0.;
double srr = 0.;
for (int i = 0; i < resid.nrow; i++)
{
sr += resid[i];
srr += resid[i] * resid[i];
}
double mean_r = sr / N;
double sigma2_internal = (srr - N * mean_r * mean_r) / (N - beta.nrow);
for (int i = 0; i < beta.nrow; i++)
sebeta[i] = sqrt(v_i.get(i, i) * sigma2_internal);
mematrix<double> chi2 = transpose(u) * v_i * u;
chi2 = chi2 * (1. / sigma2_internal);
chi2_score = chi2[0];
}
/**
* \brief Perform least squares regression
*
* Basically we solve the following linear system here:
* \f[
* \mathbf{y} = \mathbf{X} \mathbf{\beta}
* \f]
*
* This function also estimates \f$\sigma^2\f$, the variance of the
* error term. An estimator of \f$\sigma^2\f$ is given by:
* \f[
* \hat\sigma^2 = \frac{1}{n-p}||\mathbf{y} - \mathbf{X} \mathbf{\beta} ||^2,
* \f]
* with \f$n\f$ the number of rows of \f$\mathbf{X}\f$ and \f$p\f$
* the number of columns of \f$\mathbf{X}\f$.
*
* @param X The design matrix
* @param Ch
*/
void linear_reg::LeastSquaredRegression(const mematrix<double>& X,
LDLT<MatrixXd>& Ch) {
int m = X.ncol;
MatrixXd txx = MatrixXd(m, m).setZero().selfadjointView<Lower>().rankUpdate(
X.data.adjoint());
Ch = LDLT < MatrixXd > (txx.selfadjointView<Lower>());
beta.data = Ch.solve(X.data.adjoint() * reg_data.Y.data);
sigma2 = (reg_data.Y.data - (X.data * beta.data)).squaredNorm();
}
bool pdsolve(const SQMATTP& M,const VCTTP& b,VCTTP& res,double* logDet)
{
/* static */ LDLT<SQMATTP> MSr;
MSr = M.ldlt(); // Maybe replace this by a better rank-reaveling criteron
if (MSr.info()!=Success) return false;
res = MSr.solve(b);
if (logDet) {
Diagonal<const SQMATTP> MSrdiag(MSr.vectorD());
*logDet = log(MSrdiag(0));
for (int i=1;i<M.rows();i++) *logDet += log(MSrdiag(i));
}
return true;
}
bool pdsolve(const SQMATTP& M,SQMATTP& MInv,double* logDet)
{
/* static */ LDLT<SQMATTP> MSr;
/* static */ SQMATTP Ip;
/* static */ int Ipdim(0);
int p(M.rows());
if (Ipdim!=p) SetIdentity(Ip,p,&Ipdim);
MSr = M.ldlt(); // Maybe replace this by a better rank-reaveling criteron
if (MSr.info()!=Success) return false;
MInv = MSr.solve(Ip);
if (logDet) {
Diagonal<const SQMATTP> MSrdiag(MSr.vectorD());
*logDet = log(MSrdiag(0));
for (int i=1;i<M.rows();i++) *logDet += log(MSrdiag(i));
}
return true;
}
Image smooth_detector(const Image& source, Interpolation level, int r) {
Image output(source.rows(), source.columns(), 1, numeric_limits<float>::max());
const MatrixXf reg_matrix = ComputeRegMatrix(level, r);
const LDLT<MatrixXf> solver = (reg_matrix.transpose() * reg_matrix).ldlt();
for (int pr = 0; pr <= source.rows() - r; ++pr) {
for (int pc = 0; pc <= source.columns() - r; ++pc) {
VectorXf dist = VectorXf::Zero(r * r);
for (int ch = 0; ch < source.channels(); ++ch) {
EigenImage y = ExtractPatch(source, r, pr, pc, ch);
VectorXf reg_surf = solver.solve(reg_matrix.transpose() * y.asvector());
dist += (reg_matrix * reg_surf - y.asvector()).cwiseAbs2();
}
dist = dist.cwiseSqrt();
for (int row = pr; row < min(output.rows(), pr + r); ++row) {
for (int col = pc; col < min(output.columns(), pc + r); ++col) {
output.val(col, row) = min(output.val(col, row), dist((row - pr) * r + col - pc));
}
}
}
}
return output;
}
/**
* \brief Solve the linear system in case the --mmscore option was
* specified.
*
* Specifying the --mmscore command line option requires a file name
* argument as well. This file should contain the inverse
* variance-covariance matrix file. This function is run when Linear
* regression is done in combination with the mmscore option. It
* solves the 'mmscore' equation as specified in Eq. (5) in section
* 8.2.1 of the ProbABEL manual:
* \f[
* \hat{\beta}_g = (\mathbf{X}^T_g
* \mathbf{V}^{-1}_{\hat{\sigma}^2,\hat{h}^2}
* \mathbf{X}_g)^{-1}
* \mathbf{X}^T_g \mathbf{V}^{-1}_{\hat{\sigma}^2,\hat{h}^2}
* \mathbf{R}_{\hat{\beta}_x},
* \f]
* where \f$\mathbf{V}^{-1}_{\hat{\sigma}^2,\hat{h}^2}\f$ is the
* inverse variance-covariance matrix, and
* \f$\mathbf{R}_{\hat{\beta}_x}\f$ is the vector containing the
* residuals obtained from the base regression model i.e. the
* phenotype. In this function, the phenotype is stored in the
* variable \c Y.
*
* @param X The design matrix \f$X_g\f$. \c X should only contain the
* parts involving genotype data (including any interactions involving
* a genetic term), all other covariates should have been regressed out
* before running ProbABEL.
* @param W_masked The inverse variance-covariance matrix
* \f$\mathbf{V}^{-1}_{\hat{\sigma}^2,\hat{h}^2}\f$.
* @param Ch Reference to the LDLT Cholesky decomposition of the
* matrix to be inverted to get \f$\hat\beta_g\f$:
* \f[
* \mathbf{X}^T_g
* \mathbf{V}^{-1}_{\hat{\sigma}^2,\hat{h}^2}
* \mathbf{X}_g.
* \f]
* On return this variable contains said matrix.
*/
void linear_reg::mmscore_regression(const mematrix<double>& X,
const masked_matrix& W_masked,
LDLT<MatrixXd>& Ch) {
VectorXd Y = reg_data.Y.data.col(0);
/*
in ProbABEL <0.50 this calculation was performed like t(X)*W
This changed to W*X since this is better vectorized since the left hand
side has more rows: this introduces an additional transpose, but can be
neglected compared to the speedup this brings(about a factor 2 for the
palinear with 1 predictor)
This function solves the system
(X^T W X) beta = X^T W Y.
Since W is symmetric (WX)^T = X^T W, so this can be rewritten as
(WX)^T X beta = (WX)^T Y,
which is solved using LDLT Cholesky decomposition.
*/
MatrixXd WX = W_masked.masked_data->data * X.data;
MatrixXd XWT = WX.transpose();
Ch = LDLT<MatrixXd>(XWT * X.data);
VectorXd beta_vec = Ch.solve(XWT * Y);
sigma2 = (Y - WX * beta_vec).squaredNorm();
beta.data = beta_vec;
}
/**
* \brief Estimate the parameters for linear regression.
*
* @param verbose Turns verbose printing of various matrices on if
* non-zero.
* @param model The number of the genetic model (e.g. additive,
* recessive, ...) that is to be applied by the apply_model() function.
* @param interaction
* @param ngpreds Number of genomic predictors (1 for dosages, 2 for
* probabilities).
* @param invvarmatrixin
* @param robust If non-zero calculate robust standard errors.
* @param nullmodel If non-zero calculate the null model (excluding
* SNP information).
*/
void linear_reg::estimate(const int verbose,
const int model,
const int interaction,
const int ngpreds,
masked_matrix& invvarmatrixin,
const int robust,
const int nullmodel) {
// suda interaction parameter
// model should come here
//regdata rdata = rdatain.get_unmasked_data();
if (verbose)
{
cout << reg_data.is_interaction_excluded
<< " <-rdata.is_interaction_excluded\n";
// std::cout << "invvarmatrix:\n";
// invvarmatrixin.masked_data->print();
std::cout << "rdata.X:\n";
reg_data.X.print();
}
mematrix<double> X = apply_model(reg_data.X, model, interaction, ngpreds,
reg_data.is_interaction_excluded, false, nullmodel);
if (verbose)
{
std::cout << "X:\n";
X.print();
std::cout << "Y:\n";
reg_data.Y.print();
}
int length_beta = X.ncol;
beta.reinit(length_beta, 1);
sebeta.reinit(length_beta, 1);
//Han Chen
if (length_beta > 1)
{
if (model == 0 && interaction != 0 && ngpreds == 2 && length_beta > 2)
{
covariance.reinit(length_beta - 2, 1);
}
else
{
covariance.reinit(length_beta - 1, 1);
}
}
double sigma2_internal;
LDLT <MatrixXd> Ch;
if (invvarmatrixin.length_of_mask != 0)
{
//retrieve masked data W
invvarmatrixin.update_mask(reg_data.masked_data);
mmscore_regression(X, invvarmatrixin, Ch);
double N = X.nrow;
//sigma2_internal = sigma2 / (N - static_cast<double>(length_beta));
// Ugly fix to the fact that if we do mmscore, sigma2 is already
// in the matrix...
// YSA, 2009.07.20
sigma2_internal = 1.0;
sigma2 /= N;
}
else // NO mm-score regression : normal least square regression
{
LeastSquaredRegression(X, Ch);
double N = static_cast<double>(X.nrow);
double P = static_cast<double>(length_beta);
sigma2_internal = sigma2 / (N - P);
sigma2 /= N;
}
/*
loglik = 0.;
double ss=0;
for (int i=0;i<rdata.nids;i++) {
double resid = rdata.Y[i] - beta.get(0,0); // intercept
for (int j=1;j<beta.nrow;j++) resid -= beta.get(j,0)*X.get(i,j);
// residuals[i] = resid;
ss += resid*resid;
}
sigma2 = ss/N;
*/
//cout << "estimate " << rdata.nids << "\n";
//(rdata.X).print();
//for (int i=0;i<rdata.nids;i++) cout << rdata.masked_data[i] << " ";
//cout << endl;
logLikelihood(X);
MatrixXd tXX_inv = Ch.solve(MatrixXd(length_beta, length_beta).
Identity(length_beta, length_beta));
mematrix<double> robust_sigma2(X.ncol, X.ncol);
int offset = X.ncol- 1;
//if additive and interaction and 2 predictors and more than 2 betas
if (model == 0 && interaction != 0 && ngpreds == 2 && length_beta > 2){
offset = X.ncol - 2;
}
if (robust)
{
RobustSEandCovariance(X, robust_sigma2, tXX_inv, offset);
}
else
{
PlainSEandCovariance(sigma2_internal, tXX_inv, offset);
}
}