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/***********************************************/
/**
* @file polynomial.cpp
*
* @brief Interpolation by polynomial.
*
* @author Torsten Mayer-Guerr
* @date 2017-05-27
*
*/
/***********************************************/
#include "base/importStd.h"
#include "base/polynomial.h"
/***********************************************/
void Polynomial::init(const std::vector<Time> ×, UInt degree, Bool throwException,
Bool leastSquares, Double range, Double extrapolation, Double margin)
{
try
{
this->times = times;
this->degree = degree;
this->throwException = throwException;
this->isLeastSquares = leastSquares;
this->sampling = medianSampling(times).seconds();
this->range = range * ((range < 0) ? -sampling : 1.);
this->extrapolation = extrapolation * ((extrapolation < 0) ? -sampling : 1.);
this->margin = margin;
if(times.size() < degree+1)
throw(Exception("Not enough data points ("+times.size()%"%i) to interpolate with polynomial degree "s+degree%"%i"s));
if(std::adjacent_find(times.begin(), times.end(), [margin](const Time &t1, const Time &t2){return (t2-t1).seconds() <= margin;}) != times.end())
throw(Exception("Input time series is unordered or contains duplicates"));
std::vector<Bool> isConstInterval(times.size()-1);
for(UInt i=0; i<isConstInterval.size(); i++)
isConstInterval.at(i) = (std::fabs((times.at(i+1)-times.at(i)).seconds()-sampling) < margin);
isPrecomputed.clear();
isPrecomputed.resize(times.size()-degree, TRUE);
if(degree > 0)
for(UInt i=0; i<isPrecomputed.size(); i++)
isPrecomputed.at(i) = !std::any_of(isConstInterval.begin()+i, isConstInterval.begin()+(i+degree), [](Bool b){return !b;});
// precomputed polynomial interpolation matrix
W = Matrix(degree+1, degree+1);
for(UInt i=0; i<W.rows(); i++)
{
W(0,i) = 1.0;
for(UInt n=1; n<W.columns(); n++)
W(n,i) = (i-degree/2.) * W(n-1,i);
}
inverse(W);
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
Matrix Polynomial::interpolate(const std::vector<Time> ×New, const_MatrixSliceRef A, UInt rowsPerEpoch, UInt derivative, Bool adjoint) const
{
try
{
if(!derivative && !isLeastSquares && (timesNew == times)) // need interpolation?
return A;
Matrix B(rowsPerEpoch*(adjoint ? times.size() : timesNew.size()), A.columns());
auto searchStart = times.begin();
for(UInt i=0; i<timesNew.size(); i++)
{
// find interval
// -------------
UInt idx = NULLINDEX;
UInt count = degree+1;
if(!isLeastSquares)
{
searchStart = std::lower_bound(searchStart, times.end(), timesNew.at(i)); // first epoch greater or equal than interpolation point
const UInt idxFirstRight = static_cast<UInt>(std::distance(times.begin(), searchStart));
// same time? -> no interpolation needed
if(!adjoint && !derivative && (idxFirstRight < times.size()) && (std::fabs((*searchStart-timesNew.at(i)).seconds()) <= margin))
{
copy(A.row(rowsPerEpoch*idxFirstRight, rowsPerEpoch), B.row(rowsPerEpoch*i, rowsPerEpoch));
continue;
}
// interpolation possible?
UInt optimalCentricity = MAX_UINT; // primary metric (number of points left and right)
Double optimalDistance = 1e99; // secondary metric
for(UInt k=std::max(idxFirstRight, count)-count; k<=std::min(idxFirstRight, times.size()-count); k++)
{
const Double dt1 = (times.at(k)-timesNew.at(i)).seconds();
const Double dt2 = (times.at(k+degree)-timesNew.at(i)).seconds();
if((dt2-dt1 <= range) && (dt2 >= -extrapolation) && (dt1 <= extrapolation))
{
const UInt centricity = std::max(idxFirstRight-k, k+count-idxFirstRight); // max(number of points left and right)
if(centricity > optimalCentricity)
break; // there won't be any better polynomial to the right of the current optimal anymore if centricity is increasing
if((centricity < optimalCentricity) || (std::max(std::fabs(dt1), std::fabs(dt2)) < optimalDistance))
{
idx = k;
optimalCentricity = centricity;
optimalDistance = std::max(std::fabs(dt1), std::fabs(dt2));
}
}
}
}
else // least squares
{
auto searchEnd = std::upper_bound(searchStart, times.end(), timesNew.at(i)+seconds2time(range)); // first epoch outside search interval
searchStart = std::lower_bound(searchStart, searchEnd, timesNew.at(i)-seconds2time(range)); // first epoch greater or equal than search interval
idx = std::distance(times.begin(), searchStart);
count = static_cast<UInt>(std::distance(searchStart, searchEnd));
if((count < degree+1) || ((times.at(idx) - timesNew.at(i)).seconds() > extrapolation) || // all points are after newTime and we are too far away
((timesNew.at(i) - times.at(idx+count-1)).seconds() > extrapolation)) // all points are before newTime and we are too far away
idx = NULLINDEX;
}
// check if we are allowed to predict
// ----------------------------------
if(idx == NULLINDEX)
{
if(throwException || adjoint)
throw(Exception("cannot interpolate at "+timesNew.at(i).dateTimeStr()));
B.row(i*rowsPerEpoch, rowsPerEpoch).fill(NAN_EXPR);
continue;
}
// compute interpolation coefficients
// ----------------------------------
Vector coeff;
if(!isLeastSquares && isPrecomputed.at(idx))
{
const Double tau = (timesNew.at(i)-times.at(idx)).seconds()/sampling - degree/2.;
coeff = Vector(W.rows());
Double t = 1.0;
for(UInt n=derivative; n<=degree; n++)
{
Double d = 1.0;
for(UInt k=0; k<derivative; k++)
d *= (n-k)/sampling;
axpy(d*t, W.column(n), coeff);
t *= tau;
}
}
else
{
// polynomial matrix
Matrix P(count, degree+1, Matrix::NOFILL);
for(UInt k=0; k<count; k++)
{
const Double factor = (timesNew.at(i)-times.at(idx+k)).seconds()/sampling;
P(k,0) = 1.0;
for(UInt n=1; n<=degree; n++)
P(k,n) = factor * P(k,n-1);
}
coeff = Vector(count);
coeff(derivative) = 1.;
for(UInt n=1; n<=derivative; n++)
coeff(derivative) *= n/sampling;
if(P.rows() > P.columns()) // solve with QR-decomposition
{
const Vector tau = QR_decomposition(P);
triangularSolve(1., P.row(0, P.columns()).trans(), coeff.row(0, P.columns())); // R^(-T)*coeff
QMult(P, tau, coeff); // coeff := Q*R^(-T)*coeff
}
else
{
solveInPlace(Matrix(P.trans()), coeff);
}
}
// interpolate
// -----------
if(!adjoint)
{
if(rowsPerEpoch == 1)
matMult(1., coeff.trans(), A.row(idx, coeff.rows()), B.row(i, rowsPerEpoch));
else
for(UInt k=0; k<coeff.rows(); k++)
axpy(coeff(k), A.row(rowsPerEpoch*(idx+k), rowsPerEpoch), B.row(rowsPerEpoch*i, rowsPerEpoch));
}
else
{
if(rowsPerEpoch == 1)
matMult(1., coeff, A.row(i, rowsPerEpoch), B.row(idx, coeff.rows()));
else
for(UInt k=0; k<coeff.rows(); k++)
axpy(coeff(k), A.row(rowsPerEpoch*i, rowsPerEpoch), B.row(rowsPerEpoch*(idx+k), rowsPerEpoch));
}
}
return B;
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
|
