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333 lines (247 loc) · 8.26 KB
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#include <cmath>
#include <algorithm>
#include <stdexcept>
#include "csaps.h"
namespace csaps
{
DoubleArray Diff(const DoubleArray &vec)
{
const Size n = vec.size() - 1;
return vec.tail(n) - vec.head(n);
}
IndexArray Digitize(const DoubleArray &arr, const DoubleArray &bins)
{
// This code works if `arr` and `bins` are monotonically increasing
IndexArray indexes = IndexArray::Ones(arr.size());
auto IsInsideBin = [arr, bins](Index item, Index index)
{
const double prc = 1.e-8;
double a = arr(item);
double bl = bins(index - 1);
double br = bins(index);
// bins[i-1] <= a < bins[i]
return (a > bl || std::abs(a - bl) < std::abs(std::min(a, bl)) * prc) && a < br;
};
Index kstart = 1;
for (Index i = 0; i < arr.size(); ++i) {
for (Index k = kstart; k < bins.size(); ++k) {
if (IsInsideBin(i, k)) {
indexes(i) = k;
kstart = k;
break;
}
}
}
return indexes;
}
DoubleSparseMatrix MakeSparseDiagMatrix(const DoubleArray2D& diags, const IndexArray& offsets, Size rows, Size cols)
{
auto GetNumElemsAndIndex = [rows, cols](Index offset, Index &i, Index &j)
{
if (offset < 0) {
i = -offset;
j = 0;
}
else {
i = 0;
j = offset;
}
return std::min(rows - i, cols - j);
};
DoubleSparseMatrix m(rows, cols);
for (Index k = 0; k < offsets.size(); ++k) {
Index offset = offsets(k);
Index i, j;
Index n = GetNumElemsAndIndex(offset, i, j);
// When rows == cols or rows > cols, the function takes elements of the
// super-diagonal from the lower part of the corresponding diag array, and
// elements of the sub-diagonal from the upper part of the corresponding diag array.
//
// When rows < cols, the function does the opposite, taking elements of the
// super-diagonal from the upper part of the corresponding diag array, and
// elements of the sub-diagonal from the lower part of the corresponding diag array.
DoubleArray diag(n);
if (offset < 0) {
if (rows >= cols) {
diag = diags.row(k).head(n);
}
else {
diag = diags.row(k).tail(n);
}
}
else {
if (rows >= cols) {
diag = diags.row(k).tail(n);
}
else {
diag = diags.row(k).head(n);
}
}
for (Index l = 0; l < n; ++l) {
m.insert(i+l, j+l) = diag(l);
}
}
return m;
}
csaps::DoubleArray SolveLinearSystem(const DoubleSparseMatrix &A, const DoubleArray &b)
{
Eigen::SparseLU<DoubleSparseMatrix> solver;
// Compute the ordering permutation vector from the structural pattern of A
solver.analyzePattern(A);
// Compute the numerical factorization
solver.factorize(A);
// Use the factors to solve the linear system
DoubleArray x = solver.solve(b.matrix()).array();
return x;
}
UnivariateCubicSmoothingSpline::UnivariateCubicSmoothingSpline(const DoubleArray &xdata, const DoubleArray &ydata)
: UnivariateCubicSmoothingSpline(xdata, ydata, DoubleArray(), -1.0)
{
}
UnivariateCubicSmoothingSpline::UnivariateCubicSmoothingSpline(const DoubleArray &xdata, const DoubleArray &ydata, const DoubleArray &weights)
: UnivariateCubicSmoothingSpline(xdata, ydata, weights, -1.0)
{
}
UnivariateCubicSmoothingSpline::UnivariateCubicSmoothingSpline(const DoubleArray &xdata, const DoubleArray &ydata, double smooth)
: UnivariateCubicSmoothingSpline(xdata, ydata, DoubleArray(), smooth)
{
}
UnivariateCubicSmoothingSpline::UnivariateCubicSmoothingSpline(const DoubleArray &xdata, const DoubleArray &ydata, const DoubleArray &weights, double smooth)
: m_xdata(xdata)
, m_ydata(ydata)
, m_weights(weights)
, m_smooth(smooth)
{
if (m_xdata.size() < 2) {
throw std::runtime_error("There must be at least 2 data points");
}
if (m_weights.size() == 0) {
m_weights = DoubleArray::Constant(m_xdata.size(), 1.0);
}
if (m_smooth > 1.0) {
throw std::runtime_error("Smoothing parameter must be less than or equal 1.0");
}
if (m_xdata.size() != m_ydata.size() || m_xdata.size() != m_weights.size()) {
throw std::runtime_error("Lenghts of the input data vectors are not equal");
}
MakeSpline();
}
DoubleArray UnivariateCubicSmoothingSpline::operator()(const DoubleArray &xidata)
{
if (xidata.size() < 2) {
throw std::runtime_error("There must be at least 2 data points");
}
return Evaluate(xidata);
}
DoubleArray UnivariateCubicSmoothingSpline::operator()(const Size pcount, DoubleArray &xidata)
{
if (pcount < 2) {
throw std::runtime_error("There must be at least 2 data points");
}
xidata.resize(pcount);
xidata << DoubleArray::LinSpaced(pcount, m_xdata(0), m_xdata(m_xdata.size()-1));
return Evaluate(xidata);
}
void UnivariateCubicSmoothingSpline::MakeSpline()
{
const Size pcount = m_xdata.size();
const Size pcount_m1 = pcount - 1;
const Size pcount_m2 = pcount - 2;
DoubleArray dx = Diff(m_xdata);
DoubleArray dy = Diff(m_ydata);
DoubleArray divdydx = dy / dx;
double p = m_smooth;
if (pcount > 2) {
// Create diagonal sparse matrices
const Size n = dx.size() - 1;
DoubleArray2D diags(3, n);
DoubleArray head_r = dx.head(n);
DoubleArray tail_r = dx.tail(n);
diags.row(0) = tail_r;
diags.row(1) = 2 * (tail_r + head_r);
diags.row(2) = head_r;
IndexArray offsets(3);
offsets << -1, 0, 1;
DoubleSparseMatrix r = MakeSparseDiagMatrix(diags, offsets, pcount_m2, pcount_m2);
DoubleArray odx = 1. / dx;
DoubleArray head_qt = odx.head(n);
DoubleArray tail_qt = odx.tail(n);
diags.row(0) = head_qt;
diags.row(1) = -(tail_qt + head_qt);
diags.row(2) = tail_qt;
offsets << 0, 1, 2;
DoubleSparseMatrix qt = MakeSparseDiagMatrix(diags, offsets, pcount_m2, pcount);
DoubleArray ow = 1. / m_weights;
DoubleArray osqw = 1. / m_weights.sqrt();
offsets.resize(1);
offsets << 0;
DoubleSparseMatrix w = MakeSparseDiagMatrix(ow.transpose(), offsets, pcount, pcount);
DoubleSparseMatrix qw = MakeSparseDiagMatrix(osqw.transpose(), offsets, pcount, pcount);
DoubleSparseMatrix qtw = qt * qw;
DoubleSparseMatrix qtwq = qtw * qtw.transpose();
auto Trace = [](const DoubleSparseMatrix &m)
{
return m.diagonal().sum();
};
double p = m_smooth;
if (p < 0) {
p = 1. / (1. + Trace(r) / (6. * Trace(qtwq)));
}
DoubleSparseMatrix A = ((6. * (1. - p)) * qtwq) + (p * r);
A.makeCompressed();
DoubleArray b = Diff(divdydx);
// Solve linear system Ab = u
DoubleArray u = SolveLinearSystem(A, b);
DoubleArray d1 = DoubleArray::Zero(u.size() + 2);
d1.segment(1, u.size()) = u; d1 = Diff(d1) / dx;
DoubleArray d2 = DoubleArray::Zero(d1.size() + 2);
d2.segment(1, d1.size()) = d1; d2 = Diff(d2);
DoubleArray yi = m_ydata - ((6. * (1. - p)) * w * d2.matrix()).array();
DoubleArray c3 = DoubleArray::Zero(u.size() + 2);
c3.segment(1, u.size()) = p * u;
DoubleArray c2 = Diff(yi) / dx - dx * (2. * c3.head(pcount_m1) + c3.tail(pcount_m1));
m_coeffs.resize(pcount_m1, 4);
m_coeffs.col(0) = Diff(c3) / dx;
m_coeffs.col(1) = 3. * c3.head(pcount_m1);
m_coeffs.col(2) = c2;
m_coeffs.col(3) = yi.head(pcount_m1);
}
else {
p = 1.0;
m_coeffs.resize(1, 2);
m_coeffs(0, 0) = divdydx(0);
m_coeffs(0, 1) = m_ydata(0);
}
m_smooth = p;
}
DoubleArray UnivariateCubicSmoothingSpline::Evaluate(const DoubleArray & xidata)
{
const Size x_size = m_xdata.size();
DoubleArray mesh = m_xdata.segment(1, x_size - 2);
DoubleArray edges(x_size);
edges(0) = -DoubleLimits::infinity();
edges.segment(1, x_size - 2) = mesh;
edges(x_size - 1) = DoubleLimits::infinity();
IndexArray indexes = Digitize(xidata, edges);
// Use 0 as the start index
indexes -= 1;
const Size xi_size = xidata.size();
DoubleArray xidata_loc(xi_size);
DoubleArray yidata(xi_size);
for (Index i = 0; i < xi_size; ++i) {
Index index = indexes(i);
// Go to local coordinates
xidata_loc(i) = xidata(i) - m_xdata(index);
// Initial values
yidata(i) = m_coeffs(index, 0);
}
DoubleArray coeffs(xi_size);
for (Index i = 1; i < m_coeffs.cols(); ++i) {
for (Index k = 0; k < xi_size; ++k) {
coeffs(k) = m_coeffs(indexes(k), i);
}
yidata = xidata_loc * yidata + coeffs;
}
return yidata;
}
} // namespace csaps