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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2003, 2004, 2007 Ferdinando Ametrano
Copyright (C) 2006 Yiping Chen
Copyright (C) 2007 Neil Firth
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include <ql/math/comparison.hpp>
#include <ql/math/matrixutilities/choleskydecomposition.hpp>
#include <ql/math/matrixutilities/pseudosqrt.hpp>
#include <ql/math/matrixutilities/symmetricschurdecomposition.hpp>
#include <ql/math/optimization/conjugategradient.hpp>
#include <ql/math/optimization/constraint.hpp>
#include <ql/math/optimization/problem.hpp>
#include <utility>
namespace QuantLib {
namespace {
#if defined(QL_EXTRA_SAFETY_CHECKS)
void checkSymmetry(const Matrix& matrix) {
Size size = matrix.rows();
QL_REQUIRE(size == matrix.columns(),
"non square matrix: " << size << " rows, " <<
matrix.columns() << " columns");
for (Size i=0; i<size; ++i)
for (Size j=0; j<i; ++j)
QL_REQUIRE(close(matrix[i][j], matrix[j][i]),
"non symmetric matrix: " <<
"[" << i << "][" << j << "]=" << matrix[i][j] <<
", [" << j << "][" << i << "]=" << matrix[j][i]);
}
#endif
void normalizePseudoRoot(const Matrix& matrix,
Matrix& pseudo) {
Size size = matrix.rows();
QL_REQUIRE(size == pseudo.rows(),
"matrix/pseudo mismatch: matrix rows are " << size <<
" while pseudo rows are " << pseudo.columns());
Size pseudoCols = pseudo.columns();
// row normalization
for (Size i=0; i<size; ++i) {
Real norm = 0.0;
for (Size j=0; j<pseudoCols; ++j)
norm += pseudo[i][j]*pseudo[i][j];
if (norm>0.0) {
Real normAdj = std::sqrt(matrix[i][i]/norm);
for (Size j=0; j<pseudoCols; ++j)
pseudo[i][j] *= normAdj;
}
}
}
//cost function for hypersphere and lower-diagonal algorithm
class HypersphereCostFunction : public CostFunction {
private:
Size size_;
bool lowerDiagonal_;
Matrix targetMatrix_;
Array targetVariance_;
mutable Matrix currentRoot_, tempMatrix_, currentMatrix_;
public:
HypersphereCostFunction(const Matrix& targetMatrix,
Array targetVariance,
bool lowerDiagonal)
: size_(targetMatrix.rows()), lowerDiagonal_(lowerDiagonal),
targetMatrix_(targetMatrix), targetVariance_(std::move(targetVariance)),
currentRoot_(size_, size_), tempMatrix_(size_, size_), currentMatrix_(size_, size_) {}
Array values(const Array&) const override {
QL_FAIL("values method not implemented");
}
Real value(const Array& x) const override {
Size i,j,k;
std::fill(currentRoot_.begin(), currentRoot_.end(), 1.0);
if (lowerDiagonal_) {
for (i=0; i<size_; i++) {
for (k=0; k<size_; k++) {
if (k>i) {
currentRoot_[i][k]=0;
} else {
for (j=0; j<=k; j++) {
if (j == k && k!=i)
currentRoot_[i][k] *=
std::cos(x[i*(i-1)/2+j]);
else if (j!=i)
currentRoot_[i][k] *=
std::sin(x[i*(i-1)/2+j]);
}
}
}
}
} else {
for (i=0; i<size_; i++) {
for (k=0; k<size_; k++) {
for (j=0; j<=k; j++) {
if (j == k && k!=size_-1)
currentRoot_[i][k] *=
std::cos(x[j*size_+i]);
else if (j!=size_-1)
currentRoot_[i][k] *=
std::sin(x[j*size_+i]);
}
}
}
}
Real temp, error=0;
tempMatrix_ = transpose(currentRoot_);
currentMatrix_ = currentRoot_ * tempMatrix_;
for (i=0;i<size_;i++) {
for (j=0;j<size_;j++) {
temp = currentMatrix_[i][j]*targetVariance_[i]
*targetVariance_[j]-targetMatrix_[i][j];
error += temp*temp;
}
}
return error;
}
};
// Optimization function for hypersphere and lower-diagonal algorithm
Matrix hypersphereOptimize(const Matrix& targetMatrix,
const Matrix& currentRoot,
const bool lowerDiagonal) {
Size i,j,k,size = targetMatrix.rows();
Matrix result(currentRoot);
Array variance(size, 0);
for (i=0; i<size; i++){
variance[i]=std::sqrt(targetMatrix[i][i]);
}
if (lowerDiagonal) {
Matrix approxMatrix(result*transpose(result));
result = CholeskyDecomposition(approxMatrix, true);
for (i=0; i<size; i++) {
for (j=0; j<size; j++) {
result[i][j]/=std::sqrt(approxMatrix[i][i]);
}
}
} else {
for (i=0; i<size; i++) {
for (j=0; j<size; j++) {
result[i][j]/=variance[i];
}
}
}
ConjugateGradient optimize;
EndCriteria endCriteria(100, 10, 1e-8, 1e-8, 1e-8);
HypersphereCostFunction costFunction(targetMatrix, variance,
lowerDiagonal);
NoConstraint constraint;
// hypersphere vector optimization
if (lowerDiagonal) {
Array theta(size * (size-1)/2);
const Real eps=1e-16;
for (i=1; i<size; i++) {
for (j=0; j<i; j++) {
theta[i*(i-1)/2+j]=result[i][j];
if (theta[i*(i-1)/2+j]>1-eps)
theta[i*(i-1)/2+j]=1-eps;
if (theta[i*(i-1)/2+j]<-1+eps)
theta[i*(i-1)/2+j]=-1+eps;
for (k=0; k<j; k++) {
theta[i*(i-1)/2+j] /= std::sin(theta[i*(i-1)/2+k]);
if (theta[i*(i-1)/2+j]>1-eps)
theta[i*(i-1)/2+j]=1-eps;
if (theta[i*(i-1)/2+j]<-1+eps)
theta[i*(i-1)/2+j]=-1+eps;
}
theta[i*(i-1)/2+j] = std::acos(theta[i*(i-1)/2+j]);
if (j==i-1) {
if (result[i][i]<0)
theta[i*(i-1)/2+j]=-theta[i*(i-1)/2+j];
}
}
}
Problem p(costFunction, constraint, theta);
optimize.minimize(p, endCriteria);
theta = p.currentValue();
std::fill(result.begin(),result.end(),1.0);
for (i=0; i<size; i++) {
for (k=0; k<size; k++) {
if (k>i) {
result[i][k]=0;
} else {
for (j=0; j<=k; j++) {
if (j == k && k!=i)
result[i][k] *=
std::cos(theta[i*(i-1)/2+j]);
else if (j!=i)
result[i][k] *=
std::sin(theta[i*(i-1)/2+j]);
}
}
}
}
} else {
Array theta(size * (size-1));
const Real eps=1e-16;
for (i=0; i<size; i++) {
for (j=0; j<size-1; j++) {
theta[j*size+i]=result[i][j];
if (theta[j*size+i]>1-eps)
theta[j*size+i]=1-eps;
if (theta[j*size+i]<-1+eps)
theta[j*size+i]=-1+eps;
for (k=0;k<j;k++) {
theta[j*size+i] /= std::sin(theta[k*size+i]);
if (theta[j*size+i]>1-eps)
theta[j*size+i]=1-eps;
if (theta[j*size+i]<-1+eps)
theta[j*size+i]=-1+eps;
}
theta[j*size+i] = std::acos(theta[j*size+i]);
if (j==size-2) {
if (result[i][j+1]<0)
theta[j*size+i]=-theta[j*size+i];
}
}
}
Problem p(costFunction, constraint, theta);
optimize.minimize(p, endCriteria);
theta=p.currentValue();
std::fill(result.begin(),result.end(),1.0);
for (i=0; i<size; i++) {
for (k=0; k<size; k++) {
for (j=0; j<=k; j++) {
if (j == k && k!=size-1)
result[i][k] *= std::cos(theta[j*size+i]);
else if (j!=size-1)
result[i][k] *= std::sin(theta[j*size+i]);
}
}
}
}
for (i=0; i<size; i++) {
for (j=0; j<size; j++) {
result[i][j]*=variance[i];
}
}
return result;
}
// Matrix infinity norm. See Golub and van Loan (2.3.10) or
// <http://en.wikipedia.org/wiki/Matrix_norm>
Real normInf(const Matrix& M) {
Size rows = M.rows();
Size cols = M.columns();
Real norm = 0.0;
for (Size i=0; i<rows; ++i) {
Real colSum = 0.0;
for (Size j=0; j<cols; ++j)
colSum += std::fabs(M[i][j]);
norm = std::max(norm, colSum);
}
return norm;
}
// Take a matrix and make all the diagonal entries 1.
Matrix projectToUnitDiagonalMatrix(const Matrix& M) {
Size size = M.rows();
QL_REQUIRE(size == M.columns(),
"matrix not square");
Matrix result(M);
for (Size i=0; i<size; ++i)
result[i][i] = 1.0;
return result;
}
// Take a matrix and make all the eigenvalues non-negative
Matrix projectToPositiveSemidefiniteMatrix(Matrix& M) {
Size size = M.rows();
QL_REQUIRE(size == M.columns(),
"matrix not square");
Matrix diagonal(size, size, 0.0);
SymmetricSchurDecomposition jd(M);
for (Size i=0; i<size; ++i)
diagonal[i][i] = std::max<Real>(jd.eigenvalues()[i], 0.0);
Matrix result =
jd.eigenvectors()*diagonal*transpose(jd.eigenvectors());
return result;
}
// implementation of the Higham algorithm to find the nearest
// correlation matrix.
Matrix highamImplementation(const Matrix& A, const Size maxIterations, const Real& tolerance) {
Size size = A.rows();
Matrix R, Y(A), X(A), deltaS(size, size, 0.0);
Matrix lastX(X);
Matrix lastY(Y);
for (Size i=0; i<maxIterations; ++i) {
R = Y - deltaS;
X = projectToPositiveSemidefiniteMatrix(R);
deltaS = X - R;
Y = projectToUnitDiagonalMatrix(X);
// convergence test
if (std::max(normInf(X-lastX)/normInf(X),
std::max(normInf(Y-lastY)/normInf(Y),
normInf(Y-X)/normInf(Y)))
<= tolerance)
{
break;
}
lastX = X;
lastY = Y;
}
// ensure we return a symmetric matrix
for (Size i=0; i<size; ++i)
for (Size j=0; j<i; ++j)
Y[i][j] = Y[j][i];
return Y;
}
}
Matrix pseudoSqrt(const Matrix& matrix, SalvagingAlgorithm::Type sa) {
Size size = matrix.rows();
#if defined(QL_EXTRA_SAFETY_CHECKS)
checkSymmetry(matrix);
#else
QL_REQUIRE(size == matrix.columns(),
"non square matrix: " << size << " rows, " <<
matrix.columns() << " columns");
#endif
// spectral (a.k.a Principal Component) analysis
SymmetricSchurDecomposition jd(matrix);
Matrix diagonal(size, size, 0.0);
// salvaging algorithm
Matrix result(size, size);
bool negative;
switch (sa) {
case SalvagingAlgorithm::None:
// eigenvalues are sorted in decreasing order
QL_REQUIRE(jd.eigenvalues()[size-1]>=-1e-16,
"negative eigenvalue(s) ("
<< std::scientific << jd.eigenvalues()[size-1]
<< ")");
result = CholeskyDecomposition(matrix, true);
break;
case SalvagingAlgorithm::Spectral:
// negative eigenvalues set to zero
for (Size i=0; i<size; i++)
diagonal[i][i] =
std::sqrt(std::max<Real>(jd.eigenvalues()[i], 0.0));
result = jd.eigenvectors() * diagonal;
normalizePseudoRoot(matrix, result);
break;
case SalvagingAlgorithm::Hypersphere:
// negative eigenvalues set to zero
negative=false;
for (Size i=0; i<size; ++i){
diagonal[i][i] =
std::sqrt(std::max<Real>(jd.eigenvalues()[i], 0.0));
if (jd.eigenvalues()[i]<0.0) negative=true;
}
result = jd.eigenvectors() * diagonal;
normalizePseudoRoot(matrix, result);
if (negative)
result = hypersphereOptimize(matrix, result, false);
break;
case SalvagingAlgorithm::LowerDiagonal:
// negative eigenvalues set to zero
negative=false;
for (Size i=0; i<size; ++i){
diagonal[i][i] =
std::sqrt(std::max<Real>(jd.eigenvalues()[i], 0.0));
if (jd.eigenvalues()[i]<0.0) negative=true;
}
result = jd.eigenvectors() * diagonal;
normalizePseudoRoot(matrix, result);
if (negative)
result = hypersphereOptimize(matrix, result, true);
break;
case SalvagingAlgorithm::Higham: {
int maxIterations = 40;
Real tol = 1e-6;
result = highamImplementation(matrix, maxIterations, tol);
result = CholeskyDecomposition(result, true);
}
break;
default:
QL_FAIL("unknown salvaging algorithm");
}
return result;
}
Matrix rankReducedSqrt(const Matrix& matrix,
Size maxRank,
Real componentRetainedPercentage,
SalvagingAlgorithm::Type sa) {
Size size = matrix.rows();
#if defined(QL_EXTRA_SAFETY_CHECKS)
checkSymmetry(matrix);
#else
QL_REQUIRE(size == matrix.columns(),
"non square matrix: " << size << " rows, " <<
matrix.columns() << " columns");
#endif
QL_REQUIRE(componentRetainedPercentage>0.0,
"no eigenvalues retained");
QL_REQUIRE(componentRetainedPercentage<=1.0,
"percentage to be retained > 100%");
QL_REQUIRE(maxRank>=1,
"max rank required < 1");
// spectral (a.k.a Principal Component) analysis
SymmetricSchurDecomposition jd(matrix);
Array eigenValues = jd.eigenvalues();
// salvaging algorithm
switch (sa) {
case SalvagingAlgorithm::None:
// eigenvalues are sorted in decreasing order
QL_REQUIRE(eigenValues[size-1]>=-1e-16,
"negative eigenvalue(s) ("
<< std::scientific << eigenValues[size-1]
<< ")");
break;
case SalvagingAlgorithm::Spectral:
// negative eigenvalues set to zero
for (Size i=0; i<size; ++i)
eigenValues[i] = std::max<Real>(eigenValues[i], 0.0);
break;
case SalvagingAlgorithm::Higham:
{
int maxIterations = 40;
Real tolerance = 1e-6;
Matrix adjustedMatrix = highamImplementation(matrix, maxIterations, tolerance);
jd = SymmetricSchurDecomposition(adjustedMatrix);
eigenValues = jd.eigenvalues();
}
break;
default:
QL_FAIL("unknown or invalid salvaging algorithm");
}
// factor reduction
Real enough = componentRetainedPercentage *
std::accumulate(eigenValues.begin(),
eigenValues.end(), Real(0.0));
if (componentRetainedPercentage == 1.0) {
// numerical glitches might cause some factors to be discarded
enough *= 1.1;
}
// retain at least one factor
Real components = eigenValues[0];
Size retainedFactors = 1;
for (Size i=1; components<enough && i<size; ++i) {
components += eigenValues[i];
retainedFactors++;
}
// output is granted to have a rank<=maxRank
retainedFactors=std::min(retainedFactors, maxRank);
Matrix diagonal(size, retainedFactors, 0.0);
for (Size i=0; i<retainedFactors; ++i)
diagonal[i][i] = std::sqrt(eigenValues[i]);
Matrix result = jd.eigenvectors() * diagonal;
normalizePseudoRoot(matrix, result);
return result;
}
}
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