/*!
\brief Partial pivoted LU to construct low-rank.
*/
void partial_Piv_LU(const int start_Row, const int start_Col, const int n_Rows, const int n_Cols, const double tolerance, int& computed_Rank, MatrixXcd& U, MatrixXcd& V) {
/********************************/
/* PURPOSE OF EXISTENCE */
/********************************/
/*!
Obtains the low-rank decomposition of the matrix to a desired tolerance using the partial pivoting LU algorithm, i.e., given a sub-matrix 'A' and tolerance 'epsilon', computes matrices 'U' and 'V' such that ||A-UV||_F < epsilon. The norm is Frobenius norm.
*/
/************************/
/* INPUTS */
/************************/
/// start_Row - Starting row of the sub-matrix.
/// start_Col - Starting column of the sub-matrix.
/// n_Rows - Number of rows of the sub-matrix.
/// n_Cols - Number of columns of the sub-matrix.
/// tolerance - Tolerance of low-rank approximation.
/************************/
/* OUTPUTS */
/************************/
/// computed_Rank - Rank obtained for the given tolerance.
/// U - Matrix forming the column basis.
/// V - Matrix forming the row basis.
/// If the matrix is small enough, do not do anything
int tolerable_Rank = 5;
if (n_Cols <= tolerable_Rank){
kernel->get_Matrix(start_Row, start_Col, n_Rows, n_Cols, U);
V = MatrixXcd::Identity(n_Cols, n_Cols);
computed_Rank = n_Cols;
return;
}
else if (n_Rows <= tolerable_Rank){
U = MatrixXcd::Identity(n_Rows, n_Rows);
kernel->get_Matrix(start_Row, start_Col, n_Rows, n_Cols, V);
computed_Rank = n_Rows;
return;
}
vector<int> rowIndex; /// This stores the row indices, which have already been used.
vector<int> colIndex; /// This stores the column indices, which have already been used.
vector<VectorXcd> u; /// Stores the column basis.
vector<VectorXcd> v; /// Stores the row basis.
srand (time(NULL));
complex<double> max, unused_max, Gamma;
/* INITIALIZATION */
/// Initialize the matrix norm and the the first row index
double matrix_Norm = 0;
rowIndex.push_back(0);
int pivot;
computed_Rank = 0;
VectorXcd a, row, col;
double row_Squared_Norm, row_Norm, col_Squared_Norm, col_Norm;
/// Repeat till the desired tolerance is obtained
do {
/// Generation of the row
kernel->get_Matrix_Row(start_Col, n_Cols, start_Row+rowIndex.back(), a);
/// Row of the residuum and the pivot column
row = a;
for (int l=0; l<computed_Rank; ++l) {
row = row-u[l](rowIndex.back())*v[l];
}
pivot = kernel->max_Abs_Vector(row, colIndex, max);
int max_tries = 100;
int count = 0;
int count1 = 0;
/// This randomization is needed if in the middle of the algorithm the row happens to be exactly the linear combination of the previous rows.
while (abs(max)<tolerance && count < max_tries) {
int new_rowIndex;
rowIndex.pop_back();
do {
new_rowIndex = rand()%n_Rows;
++count1;
} while (find(rowIndex.begin(),rowIndex.end(),new_rowIndex)!=rowIndex.end() && count1 < max_tries);
count1 = 0;
rowIndex.push_back(new_rowIndex);
/// Generation of the row
kernel->get_Matrix_Row(start_Col, n_Cols, start_Row+rowIndex.back(), a);
/// Row of the residuum and the pivot column
row = a;
for (int l=0; l<computed_Rank; ++l) {
row = row-u[l](rowIndex.back())*v[l];
}
pivot = kernel->max_Abs_Vector(row, colIndex, max);
++count;
}
if (count == max_tries) break;
count = 0;
colIndex.push_back(pivot);
/// Normalizing constant
Gamma = 1.0/(max);
/// Generation of the column
kernel->get_Matrix_Col(start_Row, n_Rows, start_Col+colIndex.back(), a);
/// Column of the residuum and the pivot row
col = a;
for (int l=0; l<computed_Rank; ++l) {
col = col-v[l](colIndex.back())*u[l];
}
pivot = kernel->max_Abs_Vector(col, rowIndex, unused_max);
/// This randomization is needed if in the middle of the algorithm the columns happens to be exactly the linear combination of the previous columns.
while (abs(max)<tolerance && count < max_tries) {
colIndex.pop_back();
int new_colIndex;
do {
new_colIndex = rand()%n_Cols;
} while (find(colIndex.begin(),colIndex.end(),new_colIndex)!=colIndex.end() && count1 < max_tries);
count1 = 0;
colIndex.push_back(new_colIndex);
/// Generation of the column
kernel->get_Matrix_Col(start_Row, n_Rows, start_Col+colIndex.back(), a);
/// Column of the residuum and the pivot row
col = a;
for (int l=0; l<computed_Rank; ++l) {
col = col-u[l](colIndex.back())*v[l];
}
pivot = kernel->max_Abs_Vector(col, rowIndex, unused_max);
++count;
}
if (count == max_tries) break;
count = 0;
rowIndex.push_back(pivot);
/// New vectors
u.push_back(Gamma*col);
v.push_back(row);
/// New approximation of matrix norm
row_Squared_Norm = row.squaredNorm();
row_Norm = sqrt(row_Squared_Norm);
col_Squared_Norm = col.squaredNorm();
col_Norm = sqrt(col_Squared_Norm);
matrix_Norm = matrix_Norm + abs(Gamma*Gamma*row_Squared_Norm*col_Squared_Norm);
for (int j=0; j<computed_Rank; ++j) {
matrix_Norm = matrix_Norm + 2.0*abs(u[j].dot(u.back()))*abs(v[j].dot(v.back()));
}
++computed_Rank;
} while (row_Norm*col_Norm > abs(max)*tolerance*matrix_Norm && computed_Rank <= fmin(n_Rows, n_Cols));
/// If the computed_Rank is close to full-rank then return the trivial full-rank decomposition
if (computed_Rank>=fmin(n_Rows, n_Cols)) {
if (n_Rows < n_Cols) {
U = MatrixXcd::Identity(n_Rows,n_Rows);
kernel->get_Matrix(start_Row, start_Col, n_Rows, n_Cols, V);
computed_Rank = n_Rows;
return;
}
else {
kernel->get_Matrix(start_Row, start_Col, n_Rows, n_Cols, U);
V = MatrixXcd::Identity(n_Cols,n_Cols);
computed_Rank = n_Cols;
return;
}
}
U = MatrixXcd(n_Rows,computed_Rank);
V = MatrixXcd(computed_Rank,n_Cols);
for (int j=0; j<computed_Rank; ++j) {
U.col(j) = u[j];
V.row(j) = v[j];
}
};
void UnbiasedSquaredPhaseLagIndex::compute(ConnectivitySettings::IntermediateTrialData& inputData,
QVector<QPair<int,MatrixXcd> >& vecPairCsdSum,
QVector<QPair<int,MatrixXd> >& vecPairCsdImagSignSum,
QMutex& mutex,
int iNRows,
int iNFreqs,
int iNfft,
const QPair<MatrixXd, VectorXd>& tapers)
{
if(inputData.vecPairCsdImagSign.size() == iNRows) {
//qDebug() << "UnbiasedSquaredPhaseLagIndex::compute - vecPairCsdImagSign was already computed for this trial.";
return;
}
inputData.vecPairCsdImagSign.clear();
int i,j;
// Calculate tapered spectra if not available already
// This code was copied and changed modified Utils/Spectra since we do not want to call the function due to time loss.
if(inputData.vecTapSpectra.size() != iNRows) {
inputData.vecTapSpectra.clear();
RowVectorXd vecInputFFT, rowData;
RowVectorXcd vecTmpFreq;
MatrixXcd matTapSpectrum(tapers.first.rows(), iNFreqs);
QVector<Eigen::MatrixXcd> vecTapSpectra;
FFT<double> fft;
fft.SetFlag(fft.HalfSpectrum);
for (i = 0; i < iNRows; ++i) {
// Substract mean
rowData.array() = inputData.matData.row(i).array() - inputData.matData.row(i).mean();
// Calculate tapered spectra if not available already
for(j = 0; j < tapers.first.rows(); j++) {
vecInputFFT = rowData.cwiseProduct(tapers.first.row(j));
// FFT for freq domain returning the half spectrum and multiply taper weights
fft.fwd(vecTmpFreq, vecInputFFT, iNfft);
matTapSpectrum.row(j) = vecTmpFreq * tapers.second(j);
}
inputData.vecTapSpectra.append(matTapSpectrum);
}
}
// Compute CSD
if(inputData.vecPairCsd.isEmpty()) {
double denomCSD = sqrt(tapers.second.cwiseAbs2().sum()) * sqrt(tapers.second.cwiseAbs2().sum()) / 2.0;
bool bNfftEven = false;
if (iNfft % 2 == 0){
bNfftEven = true;
}
MatrixXcd matCsd = MatrixXcd(iNRows, iNFreqs);
for (i = 0; i < iNRows; ++i) {
for (j = i; j < iNRows; ++j) {
// Compute CSD (average over tapers if necessary)
matCsd.row(j) = inputData.vecTapSpectra.at(i).cwiseProduct(inputData.vecTapSpectra.at(j).conjugate()).colwise().sum() / denomCSD;
// Divide first and last element by 2 due to half spectrum
matCsd.row(j)(0) /= 2.0;
if(bNfftEven) {
matCsd.row(j).tail(1) /= 2.0;
}
}
inputData.vecPairCsd.append(QPair<int,MatrixXcd>(i,matCsd));
inputData.vecPairCsdImagSign.append(QPair<int,MatrixXd>(i,matCsd.imag().cwiseSign()));
}
mutex.lock();
if(vecPairCsdSum.isEmpty()) {
vecPairCsdSum = inputData.vecPairCsd;
vecPairCsdImagSignSum = inputData.vecPairCsdImagSign;
} else {
for (int j = 0; j < vecPairCsdSum.size(); ++j) {
vecPairCsdSum[j].second += inputData.vecPairCsd.at(j).second;
vecPairCsdImagSignSum[j].second += inputData.vecPairCsdImagSign.at(j).second;
}
}
mutex.unlock();
} else {
if(inputData.vecPairCsdImagSign.isEmpty()) {
for (i = 0; i < inputData.vecPairCsd.size(); ++i) {
inputData.vecPairCsdImagSign.append(QPair<int,MatrixXd>(i,inputData.vecPairCsd.at(i).second.imag().cwiseSign()));
}
mutex.lock();
if(vecPairCsdImagSignSum.isEmpty()) {
vecPairCsdImagSignSum = inputData.vecPairCsdImagSign;
} else {
for (int j = 0; j < vecPairCsdImagSignSum.size(); ++j) {
vecPairCsdImagSignSum[j].second += inputData.vecPairCsdImagSign.at(j).second;
}
}
mutex.unlock();
}
}
}