const ArrayXd inverseGaussianDist::variance(const ArrayXd& mu) const {return mu.cube();}
ArrayXd Functions::cubicSplineInterpolation(RefArrayXd const observedAbscissa, RefArrayXd const observedOrdinate,
RefArrayXd const interpolatedAbscissaUntruncated)
{
// Number of data points
int size = observedAbscissa.size();
// Number of interpolation grid points.
int interpolatedSize = interpolatedAbscissaUntruncated.size();
// Since the formula requires at least 2 data points, check if array size is not below 2,
// if the interpolated grid has at least one point, and if input abscissa and ordinate
// have same number of elements.
assert(size >= 2);
assert(interpolatedSize >= 1);
assert(observedOrdinate.size() == size);
// Check if lower bound set by observed abscissa is respected by interpolated abscissa
assert(observedAbscissa(0) <= interpolatedAbscissaUntruncated(0));
// Compare upper bound of observed and interpolated abscissas and truncate the latter if it exceeds the observed upper limit
double largestObservedAbscissa = observedAbscissa(size-1);
double largestInterpolatedAbscissa = interpolatedAbscissaUntruncated(interpolatedSize-1);
ArrayXd interpolatedAbscissa;
if (largestObservedAbscissa < largestInterpolatedAbscissa)
{
// Since upper bound of observed abscissa is lower, and the routine is not doing any extrapolation, truncate the array of
// interpolated abscissa at this upper bound.
int extraSize = Functions::countArrayIndicesWithinBoundaries(interpolatedAbscissaUntruncated, largestObservedAbscissa, largestInterpolatedAbscissa);
interpolatedSize = interpolatedSize - extraSize;
interpolatedAbscissa = interpolatedAbscissaUntruncated.segment(0,interpolatedSize);
}
else
interpolatedAbscissa = interpolatedAbscissaUntruncated;
// Define some array differences
ArrayXd differenceOrdinate = observedOrdinate.segment(1,size-1) - observedOrdinate.segment(0,size-1);
ArrayXd differenceAbscissa = observedAbscissa.segment(1,size-1) - observedAbscissa.segment(0,size-1);
ArrayXd differenceAbscissa2 = observedAbscissa.segment(2,size-2) - observedAbscissa.segment(0,size-2);
// Lower bound condition for natural spline
vector<double> secondDerivatives(size);
vector<double> partialSolution(size-1);
secondDerivatives[0] = 0.0;
partialSolution[0] = 0.0;
// Do tridiagonal decomposition for computing second derivatives of observed ordinate
ArrayXd sigma = differenceAbscissa.segment(0,size-2)/differenceAbscissa2.segment(0,size-2);
double beta;
// Forward computation of partial solutions in tridiagonal system
for (int i = 1; i < size-1; ++i)
{
beta = sigma(i-1) * secondDerivatives[i-1] + 2.0;
secondDerivatives[i] = (sigma(i-1) - 1.0)/beta;
partialSolution[i] = differenceOrdinate(i)/differenceAbscissa(i) - differenceOrdinate(i-1)/differenceAbscissa(i-1);
partialSolution[i] = (6.0*partialSolution[i]/differenceAbscissa2(i-1)-sigma(i-1)*partialSolution[i-1])/beta;
}
// Upper bound condition for natural spline
secondDerivatives[size-1] = 0.0;
// Backward substitution
for (int k = (size-2); k >= 0; --k)
{
secondDerivatives[k] = secondDerivatives[k]*secondDerivatives[k+1]+partialSolution[k];
}
// Initialize arrays of differences in both ordinate and abscissa
ArrayXd interpolatedOrdinate = ArrayXd::Zero(interpolatedSize);
ArrayXd remainingInterpolatedAbscissa = interpolatedAbscissa; // The remaining part of the array of interpolated abscissa
int cumulatedBinSize = 0; // The cumulated number of interpolated points from the beginning
int i = 0; // Bin counter
while ((i < size-1) && (cumulatedBinSize < interpolatedSize))
{
// Find which values of interpolatedAbscissa are containined within the selected bin of observedAbscissa.
// Since elements in interpolatedAbscissa are monotonically increasing, we cut the input array each time
// we identify the elements of the current bin. This allows to speed up the process.
double lowerAbscissa = observedAbscissa(i);
double upperAbscissa = observedAbscissa(i+1);
// Find total number of interpolated points falling in the current bin
int binSize = Functions::countArrayIndicesWithinBoundaries(remainingInterpolatedAbscissa, lowerAbscissa, upperAbscissa);
// Do interpolation only if interpolated points are found within the bin
if (binSize > 0)
{
double lowerOrdinate = observedOrdinate(i);
double upperOrdinate = observedOrdinate(i+1);
double denominator = differenceAbscissa(i);
double upperSecondDerivative = secondDerivatives[i+1];
double lowerSecondDerivative = secondDerivatives[i];
ArrayXd interpolatedAbscissaInCurrentBin = remainingInterpolatedAbscissa.segment(0, binSize);
ArrayXd interpolatedOrdinateInCurrentBin = ArrayXd::Zero(binSize);
// Compute coefficients for cubic spline interpolation function
ArrayXd a = (upperAbscissa - interpolatedAbscissaInCurrentBin) / denominator;
ArrayXd b = 1.0 - a;
ArrayXd c = (1.0/6.0) * (a.cube() - a)*denominator*denominator;
ArrayXd d = (1.0/6.0) * (b.cube() - b)*denominator*denominator;
interpolatedOrdinateInCurrentBin = a*lowerOrdinate + b*upperOrdinate + c*lowerSecondDerivative + d*upperSecondDerivative;
// Merge bin ordinate into total array of ordinate
interpolatedOrdinate.segment(cumulatedBinSize, binSize) = interpolatedOrdinateInCurrentBin;
// Reduce size of array remainingInterpolatedAbscissa by binSize elements and initialize the array
// with remaining part of interpolatedAbscissa
int currentRemainingSize = interpolatedSize - cumulatedBinSize;
remainingInterpolatedAbscissa.resize(currentRemainingSize - binSize);
cumulatedBinSize += binSize;
remainingInterpolatedAbscissa = interpolatedAbscissa.segment(cumulatedBinSize, interpolatedSize-cumulatedBinSize);
}
// Move to next bin
++i;
}
return interpolatedOrdinate;
}