/// Calculate the eigensystem of a symmetric matrix
/// @param eigenValues :: Output variable that receives the eigenvalues of this
/// matrix.
/// @param eigenVectors :: Output variable that receives the eigenvectors of
/// this matrix.
void GSLMatrix::eigenSystem(GSLVector &eigenValues, GSLMatrix &eigenVectors) {
size_t n = size1();
if (n != size2()) {
throw std::runtime_error("Matrix eigenSystem: the matrix must be square.");
}
eigenValues.resize(n);
eigenVectors.resize(n, n);
auto workspace = gsl_eigen_symmv_alloc(n);
gsl_eigen_symmv(gsl(), eigenValues.gsl(), eigenVectors.gsl(), workspace);
gsl_eigen_symmv_free(workspace);
}
/**
* Calculates covariance matrix for fitting function's active parameters.
*/
void CostFuncFitting::calActiveCovarianceMatrix(GSLMatrix &covar,
double epsrel) {
// construct the jacobian
GSLJacobian J(m_function, m_values->size());
size_t na = this->nParams(); // number of active parameters
assert(J.getJ()->size2 == na);
covar.resize(na, na);
// calculate the derivatives
m_function->functionDeriv(*m_domain, J);
// let the GSL to compute the covariance matrix
gsl_multifit_covar(J.getJ(), epsrel, covar.gsl());
}
/**
* Calculate the transformation matrix T by numeric differentiation
* @param tm :: The output transformation matrix.
*/
void CostFuncFitting::calTransformationMatrixNumerically(GSLMatrix &tm) {
const double epsilon = std::numeric_limits<double>::epsilon() * 100;
size_t np = m_function->nParams();
size_t na = nParams();
tm.resize(na, na);
size_t ia = 0;
for (size_t i = 0; i < np; ++i) {
if (m_function->isFixed(i))
continue;
double p0 = m_function->getParameter(i);
for (size_t j = 0; j < na; ++j) {
double ap = getParameter(j);
double step = ap == 0.0 ? epsilon : ap * epsilon;
setParameter(j, ap + step);
tm.set(ia, j, (m_function->getParameter(i) - p0) / step);
setParameter(j, ap);
}
++ia;
}
}
//----------------------------------------------------------------------------------------------
/// Examine the chi squared as a function of fitting parameters and estimate
/// errors for each parameter.
void CalculateChiSquared::estimateErrors() {
// Number of fiting parameters
auto nParams = m_function->nParams();
// Create an output table for displaying slices of the chi squared and
// the probabilitydensity function
auto pdfTable = API::WorkspaceFactory::Instance().createTable();
std::string baseName = getProperty("Output");
if (baseName.empty()) {
baseName = "CalculateChiSquared";
}
declareProperty(new API::WorkspaceProperty<API::ITableWorkspace>(
"PDFs", "", Kernel::Direction::Output),
"The name of the TableWorkspace in which to store the "
"pdfs of fit parameters");
setPropertyValue("PDFs", baseName + "_pdf");
setProperty("PDFs", pdfTable);
// Create an output table for displaying the parameter errors.
auto errorsTable = API::WorkspaceFactory::Instance().createTable();
auto nameColumn = errorsTable->addColumn("str", "Parameter");
auto valueColumn = errorsTable->addColumn("double", "Value");
auto minValueColumn = errorsTable->addColumn("double", "Value at Min");
auto leftErrColumn = errorsTable->addColumn("double", "Left Error");
auto rightErrColumn = errorsTable->addColumn("double", "Right Error");
auto quadraticErrColumn = errorsTable->addColumn("double", "Quadratic Error");
auto chiMinColumn = errorsTable->addColumn("double", "Chi2 Min");
errorsTable->setRowCount(nParams);
declareProperty(new API::WorkspaceProperty<API::ITableWorkspace>(
"Errors", "", Kernel::Direction::Output),
"The name of the TableWorkspace in which to store the "
"values and errors of fit parameters");
setPropertyValue("Errors", baseName + "_errors");
setProperty("Errors", errorsTable);
// Calculate initial values
double chiSquared = 0.0;
double chiSquaredWeighted = 0.0;
double dof = 0;
API::FunctionDomain_sptr domain;
API::FunctionValues_sptr values;
m_domainCreator->createDomain(domain, values);
calcChiSquared(*m_function, nParams, *domain, *values, chiSquared,
chiSquaredWeighted, dof);
// Value of chi squared for current parameters in m_function
double chi0 = chiSquared;
// Fit data variance
double sigma2 = chiSquared / dof;
bool useWeighted = getProperty("Weighted");
if (useWeighted) {
chi0 = chiSquaredWeighted;
sigma2 = 0.0;
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "chi0=" << chi0 << std::endl;
g_log.debug() << "sigma2=" << sigma2 << std::endl;
g_log.debug() << "dof=" << dof << std::endl;
}
// Parameter bounds that define a volume in the parameter
// space within which the chi squared is being examined.
GSLVector lBounds(nParams);
GSLVector rBounds(nParams);
// Number of points in lines for plotting
size_t n = 100;
pdfTable->setRowCount(n);
const double fac = 1e-4;
// Loop over each parameter
for (size_t ip = 0; ip < nParams; ++ip) {
// Add columns for the parameter to the pdf table.
auto parName = m_function->parameterName(ip);
nameColumn->read(ip, parName);
// Parameter values
auto col1 = pdfTable->addColumn("double", parName);
col1->setPlotType(1);
// Chi squared values
auto col2 = pdfTable->addColumn("double", parName + "_chi2");
col2->setPlotType(2);
// PDF values
auto col3 = pdfTable->addColumn("double", parName + "_pdf");
col3->setPlotType(2);
double par0 = m_function->getParameter(ip);
double shift = fabs(par0 * fac);
if (shift == 0.0) {
shift = fac;
}
// Make a slice along this parameter
GSLVector dir(nParams);
dir.zero();
dir[ip] = 1.0;
ChiSlice slice(*m_function, dir, *domain, *values, chi0, sigma2);
// Find the bounds withn which the PDF is significantly above zero.
// The bounds are defined relative to par0:
// par0 + lBound is the lowest value of the parameter (lBound <= 0)
// par0 + rBound is the highest value of the parameter (rBound >= 0)
double lBound = slice.findBound(-shift);
double rBound = slice.findBound(shift);
lBounds[ip] = lBound;
rBounds[ip] = rBound;
// Approximate the slice with a polynomial.
// P is a vector of values of the polynomial at special points.
// A is a vector of Chebyshev expansion coefficients.
// The polynomial is defined on interval [lBound, rBound]
// The value of the polynomial at 0 == chi squared at par0
std::vector<double> P, A;
bool ok = true;
auto base = slice.makeApprox(lBound, rBound, P, A, ok);
if (!ok) {
g_log.warning() << "Approximation failed for parameter " << ip << std::endl;
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "Parameter " << ip << std::endl;
g_log.debug() << "Slice approximated by polynomial of order "
<< base->size() - 1;
g_log.debug() << " between " << lBound << " and " << rBound << std::endl;
}
// Write n slice points into the output table.
double dp = (rBound - lBound) / static_cast<double>(n);
for (size_t i = 0; i < n; ++i) {
double par = lBound + dp * static_cast<double>(i);
double chi = base->eval(par, P);
col1->fromDouble(i, par0 + par);
col2->fromDouble(i, chi);
}
// Check if par0 is a minimum point of the chi squared
std::vector<double> AD;
// Calculate the derivative polynomial.
// AD are the Chebyshev expansion of the derivative.
base->derivative(A, AD);
// Find the roots of the derivative polynomial
std::vector<double> minima = base->roots(AD);
if (minima.empty()) {
minima.push_back(par0);
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "Minima: ";
}
// If only 1 extremum is found assume (without checking) that it's a
// minimum.
// If there are more than 1, find the one with the smallest chi^2.
double chiMin = std::numeric_limits<double>::max();
double parMin = par0;
for (size_t i = 0; i < minima.size(); ++i) {
double value = base->eval(minima[i], P);
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << minima[i] << " (" << value << ") ";
}
if (value < chiMin) {
chiMin = value;
parMin = minima[i];
}
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << std::endl;
g_log.debug() << "Smallest minimum at " << parMin << " is " << chiMin
<< std::endl;
}
// Points of intersections with line chi^2 = 1/2 give an estimate of
// the standard deviation of this parameter if it's uncorrelated with the
// others.
A[0] -= 0.5; // Now A are the coefficients of the original polynomial
// shifted down by 1/2.
std::vector<double> roots = base->roots(A);
std::sort(roots.begin(), roots.end());
if (roots.empty()) {
// Something went wrong; use the whole interval.
roots.resize(2);
roots[0] = lBound;
roots[1] = rBound;
} else if (roots.size() == 1) {
// Only one root found; use a bound for the other root.
if (roots.front() < 0) {
roots.push_back(rBound);
} else {
roots.insert(roots.begin(), lBound);
}
} else if (roots.size() > 2) {
// More than 2 roots; use the smallest and the biggest
auto smallest = roots.front();
auto biggest = roots.back();
roots.resize(2);
roots[0] = smallest;
roots[1] = biggest;
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "Roots: ";
for (size_t i = 0; i < roots.size(); ++i) {
g_log.debug() << roots[i] << ' ';
}
g_log.debug() << std::endl;
}
// Output parameter info to the table.
valueColumn->fromDouble(ip, par0);
minValueColumn->fromDouble(ip, par0 + parMin);
leftErrColumn->fromDouble(ip, roots[0] - parMin);
rightErrColumn->fromDouble(ip, roots[1] - parMin);
chiMinColumn->fromDouble(ip, chiMin);
// Output the PDF
for (size_t i = 0; i < n; ++i) {
double chi = col2->toDouble(i);
col3->fromDouble(i, exp(-chi + chiMin));
}
// make sure function parameters don't change.
m_function->setParameter(ip, par0);
}
// Improve estimates for standard deviations.
// If parameters are correlated the found deviations
// most likely underestimate the true values.
unfixParameters();
GSLJacobian J(m_function, values->size());
m_function->functionDeriv(*domain, J);
refixParameters();
// Calculate the hessian at the current point.
GSLMatrix H;
if (useWeighted) {
H.resize(nParams, nParams);
for (size_t i = 0; i < nParams; ++i) {
for (size_t j = i; j < nParams; ++j) {
double h = 0.0;
for (size_t k = 0; k < values->size(); ++k) {
double w = values->getFitWeight(k);
h += J.get(k, i) * J.get(k, j) * w * w;
}
H.set(i, j, h);
if (i != j) {
H.set(j, i, h);
}
}
}
} else {
H = Tr(J.matrix()) * J.matrix();
}
// Square roots of the diagonals of the covariance matrix give
// the standard deviations in the quadratic approximation of the chi^2.
GSLMatrix V(H);
if (!useWeighted) {
V *= 1. / sigma2;
}
V.invert();
// In a non-quadratic asymmetric case the following procedure can give a
// better result:
// Find the direction in which the chi^2 changes slowest and the positive and
// negative deviations in that direction. The change in a parameter at those
// points can be a better estimate for the standard deviation.
GSLVector v(nParams);
GSLMatrix Q(nParams, nParams);
// One of the eigenvectors of the hessian is the direction of the slowest
// change.
H.eigenSystem(v, Q);
// Loop over the eigenvectors
for (size_t i = 0; i < nParams; ++i) {
auto dir = Q.copyColumn(i);
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "Direction " << i << std::endl;
g_log.debug() << dir << std::endl;
}
// Make a slice in that direction
ChiSlice slice(*m_function, dir, *domain, *values, chi0, sigma2);
double rBound0 = dir.dot(rBounds);
double lBound0 = dir.dot(lBounds);
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "lBound " << lBound0 << std::endl;
g_log.debug() << "rBound " << rBound0 << std::endl;
}
double lBound = slice.findBound(lBound0);
double rBound = slice.findBound(rBound0);
std::vector<double> P, A;
// Use a polynomial approximation
bool ok = true;
auto base = slice.makeApprox(lBound, rBound, P, A, ok);
if (!ok) {
g_log.warning() << "Approximation failed in direction " << i << std::endl;
}
// Find the deviation points where the chi^2 = 1/2
A[0] -= 0.5;
std::vector<double> roots = base->roots(A);
std::sort(roots.begin(), roots.end());
// Sort out the roots
auto nRoots = roots.size();
if (nRoots == 0) {
roots.resize(2, 0.0);
} else if (nRoots == 1) {
if (roots.front() > 0.0) {
roots.insert(roots.begin(), 0.0);
} else {
roots.push_back(0.0);
}
} else if (nRoots > 2) {
roots[1] = roots.back();
roots.resize(2);
}
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << "Roots " << roots[0] << " (" << slice(roots[0]) << ") " << roots[1] << " (" << slice(roots[1]) << ") " << std::endl;
}
// Loop over the parameters and see if there deviations along
// this direction is greater than any previous value.
for (size_t ip = 0; ip < nParams; ++ip) {
auto lError = roots.front() * dir[ip];
auto rError = roots.back() * dir[ip];
if (lError > rError) {
std::swap(lError, rError);
}
if (lError < leftErrColumn->toDouble(ip)) {
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << " left for " << ip << ' ' << lError << ' ' << leftErrColumn->toDouble(ip) << std::endl;
}
leftErrColumn->fromDouble(ip, lError);
}
if (rError > rightErrColumn->toDouble(ip)) {
if (g_log.is(Kernel::Logger::Priority::PRIO_DEBUG)) {
g_log.debug() << " right for " << ip << ' ' << rError << ' ' << rightErrColumn->toDouble(ip) << std::endl;
}
rightErrColumn->fromDouble(ip, rError);
}
}
// Output the quadratic estimate for comparrison.
quadraticErrColumn->fromDouble(i, sqrt(V.get(i, i)));
}
}