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/***********************************************/
/**
* @file gravityfield2EmpiricalCovariance.cpp
*
* @brief Estimate a empircal covariance function from time series.
*
* @author Torsten Mayer-Guerr
* @date 2009-03-30
*
*/
/***********************************************/
// Latex documentation
#define DOCSTRING docstring
static const char *docstring = R"(
This program estimates an spatial and temporal covariance matrix from
a time series of gravity fields.
Firstly for every time step $t_i$
a spherical harmonics vector $\M x_i$ from the time variable gravity field
is generated. The coefficients of the spherical harmonics expansion are
in the sequence given by \configClass{numbering}{sphericalHarmonicsNumberingType}.
If set the expansion is limited in the range between \config{minDegree}
and \config{maxDegree} inclusivly. The coefficients are related to the
reference radius~\config{R} and the Earth gravitational constant \config{GM}.
In the next step the empirical covariance matrix is estimated according to
\begin{equation}
\M\Sigma(\Delta i)_{full} = \frac{1}{N}\sum_{i=1}^N \M x_i \M x_{i+\Delta i}^T,
\end{equation}
where $\Delta i$ is given by \config{differenceStep}.
From the diagonal elements of $\M\Sigma(\Delta i)$ the isotropic accuracies
are computed
\begin{equation}
\sigma_n^2 = \frac{1}{2n+1}\sum_{m=0}^n \sigma_{cnm}^2+\sigma_{snm}^2,
\end{equation}
and a diagonal matrix is constructed $\Sigma_{iso} = \text{diag}(\sigma_2^2,\ldots,\sigma_N^2)$.
The result is computed:
\begin{equation}
\M\Sigma(\Delta i) = \alpha_{full}\M\Sigma(\Delta i)_{full}+\alpha_{iso}\M\Sigma(\Delta i)_{iso}.
\end{equation}
)";
/***********************************************/
#include "programs/program.h"
#include "files/fileMatrix.h"
#include "files/fileSphericalHarmonics.h"
#include "classes/timeSeries/timeSeries.h"
#include "classes/gravityfield/gravityfield.h"
#include "classes/sphericalHarmonicsNumbering/sphericalHarmonicsNumbering.h"
/***** CLASS ***********************************/
/** @brief Estimate a empircal covariance function from time series.
* @ingroup programsGroup */
class Gravityfield2EmpiricalCovariance
{
public:
void run(Config &config, Parallel::CommunicatorPtr comm);
};
GROOPS_REGISTER_PROGRAM(Gravityfield2EmpiricalCovariance, SINGLEPROCESS, "Estimate a empircal covariance function from time series", Gravityfield, Covariance)
/***********************************************/
void Gravityfield2EmpiricalCovariance::run(Config &config, Parallel::CommunicatorPtr /*comm*/)
{
try
{
FileName outputName, fileNamePotCoeff;
UInt minDegree, maxDegree = INFINITYDEGREE;
Time time;
Double GM, R;
GravityfieldPtr gravityfield;
TimeSeriesPtr timeSeriesPtr, timesIntervalPtr;
Bool removeMean;
UInt delay;
SphericalHarmonicsNumberingPtr numbering;
Double factorFullMatrix, factorIsotropic;
readConfig(config, "outputfileCovarianceMatrix", outputName, Config::MUSTSET, "", "");
readConfig(config, "outputfilePotentialCoefficients", fileNamePotCoeff, Config::OPTIONAL, "", "");
readConfig(config, "gravityfield", gravityfield, Config::MUSTSET, "", "");
readConfig(config, "minDegree", minDegree, Config::DEFAULT, "2", "");
readConfig(config, "maxDegree", maxDegree, Config::MUSTSET, "", "");
readConfig(config, "GM", GM, Config::DEFAULT, STRING_DEFAULT_GM, "Geocentric gravitational constant");
readConfig(config, "R", R, Config::DEFAULT, STRING_DEFAULT_R, "reference radius");
readConfig(config, "numbering", numbering, Config::MUSTSET, "", "numbering scheme for solution vector");
readConfig(config, "removeMean", removeMean, Config::DEFAULT, "1", "");
readConfig(config, "timeSeries", timeSeriesPtr, Config::MUSTSET, "", "sampling of the gravityfield");
readConfig(config, "differenceStep", delay, Config::DEFAULT, "0", "choose dt for: x,i(t) - x,j(t+dt)");
readConfig(config, "factorFullMatrixPart", factorFullMatrix, Config::DEFAULT, "1", "");
readConfig(config, "factorIsotropicPart", factorIsotropic, Config::DEFAULT, "0.1", "");
readConfig(config, "intervals", timesIntervalPtr, Config::DEFAULT, "", "");
if(isCreateSchema(config)) return;
// ============================
// init time intervals
// -------------------
std::vector<Time> timeSeries = timeSeriesPtr->times();
std::vector<Time> timesInterval;
if(timesIntervalPtr)
timesInterval = timesIntervalPtr->times();
if(timesInterval.size()==0)
{
timesInterval.push_back(timeSeries.at(0));
timesInterval.push_back(timeSeries.back()+seconds2time(1));
}
// ============================
// numbering of covariance matrix
// ------------------------------
std::vector< std::vector<UInt> > idxC, idxS;
numbering->numbering(maxDegree, minDegree, idxC, idxS);
const UInt dim = numbering->parameterCount(maxDegree, minDegree);
Matrix CovFull(dim, dim);
logStatus<<"estimate covariance covariance function in each interval"<<Log::endl;
UInt countTotal = 0;
Single::forEach(timesInterval.size()-1, [&](UInt idInterval)
{
// init time series
// ----------------
std::vector<Time> times;
for(UInt i=0; i<timeSeries.size(); i++)
if(timeSeries.at(i).isInInterval(timesInterval.at(idInterval), timesInterval.at(idInterval+1)))
times.push_back(timeSeries.at(i));
// create time series of potential coefficients
// --------------------------------------------
UInt count = times.size();
std::vector<SphericalHarmonics> harm(count);
for(UInt i=0; i<count; i++)
{
harm.at(i) = gravityfield->sphericalHarmonics(times.at(i), maxDegree, minDegree, GM, R);
maxDegree = harm.at(i).maxDegree();
}
// remove temporal mean
// --------------------
if(removeMean)
{
SphericalHarmonics harmMean;
for(UInt i=0; i<count; i++)
harmMean += harm.at(i);
harmMean *= 1./count;
for(UInt i=0; i<count; i++)
harm.at(i) -= harmMean;
}
// covariance function
// -------------------
count = harm.size()-delay;
for(UInt i=0; i<count; i++)
{
Matrix cnm1 = harm.at(i).cnm();
Matrix snm1 = harm.at(i).snm();
Matrix cnm2 = harm.at(i+delay).cnm();
Matrix snm2 = harm.at(i+delay).snm();
// sort coefficients into vectors
Vector x1(dim), x2(dim);
for(UInt n=0; n<=maxDegree; n++)
{
if(idxC[n][0]!=NULLINDEX) x1(idxC[n][0]) = cnm1(n,0);
if(idxC[n][0]!=NULLINDEX) x2(idxC[n][0]) = cnm2(n,0);
for(UInt m=1; m<=n; m++)
{
if(idxC[n][m]!=NULLINDEX) x1(idxC[n][m]) = cnm1(n,m);
if(idxC[n][m]!=NULLINDEX) x2(idxC[n][m]) = cnm2(n,m);
if(idxS[n][m]!=NULLINDEX) x1(idxS[n][m]) = snm1(n,m);
if(idxS[n][m]!=NULLINDEX) x2(idxS[n][m]) = snm2(n,m);
}
}
matMult(1., x1, x2.trans(), CovFull);
}
countTotal += count;
if(removeMean)
countTotal -= 1;
});
// ============================
if(delay==0)
CovFull.setType(Matrix::SYMMETRIC);
CovFull *= 1./countTotal;
Matrix Cov = CovFull;
Cov *= factorFullMatrix;
if(factorIsotropic!=0)
{
Vector kn(maxDegree+1);
for(UInt n=0; n<=maxDegree; n++)
{
if(idxC[n][0]!=NULLINDEX) kn(n) += CovFull(idxC[n][0],idxC[n][0]);
for(UInt m=1; m<=n; m++)
{
if(idxC[n][m]!=NULLINDEX) kn(n) += CovFull(idxC[n][m],idxC[n][m]);
if(idxS[n][m]!=NULLINDEX) kn(n) += CovFull(idxS[n][m],idxS[n][m]);
}
}
for(UInt n=0; n<=maxDegree; n++)
{
if(idxC[n][0]!=NULLINDEX) Cov(idxC[n][0],idxC[n][0]) += factorIsotropic * kn(n)/(2*n+1);
for(UInt m=1; m<=n; m++)
{
if(idxC[n][m]!=NULLINDEX) Cov(idxC[n][m],idxC[n][m]) += factorIsotropic * kn(n)/(2*n+1);
if(idxS[n][m]!=NULLINDEX) Cov(idxS[n][m],idxS[n][m]) += factorIsotropic * kn(n)/(2*n+1);
}
}
}
// ============================
// Write
// -----
logStatus << "save covariance matrix to <"<<outputName<<">"<< Log::endl;
if(delay == 0)
Cov.setType(Matrix::SYMMETRIC);
writeFileMatrix(outputName, Cov);
logInfo<<" number of fields : "<<countTotal<<Log::endl;
logInfo<<" minDegree : "<<minDegree<< Log::endl;
logInfo<<" maxDegree : "<<maxDegree<< Log::endl;
logInfo<<" size of state vector : "<<dim<< Log::endl;
logInfo<<" number of unknowns : "<<dim*dim<< Log::endl;
logInfo<<" redundancy : "<<countTotal/static_cast<Double>(dim) << Log::endl;
// Spherical Harmonics
// -------------------
if(!fileNamePotCoeff.empty())
{
logStatus<<"save potential coefficients to <"<<fileNamePotCoeff<<">"<< Log::endl;
Matrix cnm(maxDegree+1, Matrix::SYMMETRIC);
Matrix snm(maxDegree+1, Matrix::SYMMETRIC);
Matrix sigma2cnm(maxDegree+1, Matrix::SYMMETRIC);
Matrix sigma2snm(maxDegree+1, Matrix::SYMMETRIC);
for(UInt n=0; n<=maxDegree; n++)
{
if(idxC[n][0]!=NULLINDEX) cnm(n,0) = sqrt(Cov(idxC[n][0],idxC[n][0]));
if(idxC[n][0]!=NULLINDEX) sigma2cnm(n,0) = Cov(idxC[n][0],idxC[n][0]);
for(UInt m=1; m<=n; m++)
{
if(idxC[n][m]!=NULLINDEX) cnm(n,m) = sqrt(Cov(idxC[n][m],idxC[n][m]));
if(idxS[n][m]!=NULLINDEX) snm(n,m) = sqrt(Cov(idxS[n][m],idxS[n][m]));
if(idxC[n][m]!=NULLINDEX) sigma2cnm(n,m) = Cov(idxC[n][m],idxC[n][m]);
if(idxS[n][m]!=NULLINDEX) sigma2snm(n,m) = Cov(idxS[n][m],idxS[n][m]);
}
}
writeFileSphericalHarmonics(fileNamePotCoeff, SphericalHarmonics(GM, R, cnm, snm, sigma2cnm, sigma2snm));
}
// Test for positive definite
// --------------------------
try
{
if(delay==0)
cholesky(Cov);
}
catch(std::exception &/*e*/)
{
logWarning<<"covariance matrix is not positive definite!"<<Log::endl;
}
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
|
