LeastSqFitter2d.h

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// -*- LSST-C++ -*-

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#ifndef LEAST_SQ_FITTER_2D
#define LEAST_SQ_FITTER_2D

#include <cstdio>
#include <memory>
#include <vector>

#include "Eigen/Core"
#include "Eigen/SVD"

#include "lsst/pex/exceptions/Runtime.h"
#include "lsst/afw/math/FunctionLibrary.h"

namespace lsst {
namespace meas {
namespace astrom {
namespace sip {

template <class FittingFunc>
class LeastSqFitter2d {
public:
    LeastSqFitter2d(const std::vector<double> &x, const std::vector<double> &y, const std::vector<double> &z,
                    const std::vector<double> &s, int order);

    Eigen::MatrixXd getParams();
    Eigen::MatrixXd getErrors();
    double valueAt(double x, double y);
    std::vector<double> residuals();

    double getChiSq();
    double getReducedChiSq();

private:
    void initFunctions();

    Eigen::MatrixXd expandParams(Eigen::VectorXd const &input) const;

    double func2d(double x, double y, int term);
    double func1d(double value, int exponent);

    std::vector<double> _x, _y, _z, _s;
    int _order;  // Degree of polynomial to fit, e.g 4=> cubic
    int _nPar;   // Number of parameters in fitting eqn, e.g x^2, xy, y^2, x^3,
    int _nData;  // Number of data points, == _x.size()

    Eigen::JacobiSVD<Eigen::MatrixXd> _svd;
    Eigen::VectorXd _par;

    std::vector<std::shared_ptr<FittingFunc> > _funcArray;
};

// The .cc part

template <class FittingFunc>
LeastSqFitter2d<FittingFunc>::LeastSqFitter2d(const std::vector<double> &x, const std::vector<double> &y,
                                              const std::vector<double> &z, const std::vector<double> &s,
                                              int order)
        : _x(x), _y(y), _z(z), _s(s), _order(order), _nPar(0), _par(1) {
    //_nPar, the number of terms to fix (x^2, xy, y^2 etc.) is \Sigma^(order+1) 1
    _nPar = 0;
    for (int i = 0; i < order; ++i) {
        _nPar += i + 1;
    }

    // Check input vectors are the same size
    _nData = _x.size();
    if (_nData != static_cast<int>(_y.size())) {
        throw LSST_EXCEPT(pex::exceptions::RuntimeError, "x and y vectors of different lengths");
    }
    if (_nData != static_cast<int>(_s.size())) {
        throw LSST_EXCEPT(pex::exceptions::RuntimeError, "x and s vectors of different lengths");
    }
    if (_nData != static_cast<int>(_z.size())) {
        throw LSST_EXCEPT(pex::exceptions::RuntimeError, "x and z vectors of different lengths");
    }

    for (int i = 0; i < _nData; ++i) {
        if (_s[i] == 0.0) {
            std::string msg = "Illegal zero value for fit weight encountered.";
            throw LSST_EXCEPT(pex::exceptions::RuntimeError, msg);
        }
    }

    if (_nData < _order) {
        throw LSST_EXCEPT(pex::exceptions::RuntimeError, "Fewer data points than parameters");
    }

    initFunctions();

    Eigen::MatrixXd design(_nData, _nPar);
    Eigen::VectorXd rhs(_nData);
    for (int i = 0; i < _nData; ++i) {
        rhs[i] = z[i] / s[i];
        for (int j = 0; j < _nPar; ++j) {
            design(i, j) = func2d(_x[i], _y[i], j) / _s[i];
        }
    }
    _svd.compute(design, Eigen::ComputeThinU | Eigen::ComputeThinV);
    _par = _svd.solve(rhs);
}

template <class FittingFunc>
Eigen::MatrixXd LeastSqFitter2d<FittingFunc>::getParams() {
    return expandParams(_par);
}

template <class FittingFunc>
Eigen::MatrixXd LeastSqFitter2d<FittingFunc>::expandParams(Eigen::VectorXd const &input) const {
    Eigen::MatrixXd out = Eigen::MatrixXd::Zero(_order, _order);
    int count = 0;
    for (int i = 0; i < _order; ++i) {
        for (int j = 0; j < _order - i; ++j) {
            out(i, j) = input[count++];
        }
    }
    return out;
}

template <class FittingFunc>
double LeastSqFitter2d<FittingFunc>::getChiSq() {
    double chisq = 0;
    for (int i = 0; i < _nData; ++i) {
        double val = _z[i] - valueAt(_x[i], _y[i]);
        val /= _s[i];
        chisq += pow(val, 2);
    }

    return chisq;
}

template <class FittingFunc>
double LeastSqFitter2d<FittingFunc>::getReducedChiSq() {
    return getChiSq() / (double)(_nData - _nPar);
}

template <class FittingFunc>
double LeastSqFitter2d<FittingFunc>::valueAt(double x, double y) {
    double val = 0;

    // Sum the values of the different orders to get the value of the fitted function
    for (int i = 0; i < _nPar; ++i) {
        val += _par[i] * func2d(x, y, i);
    }
    return val;
}

template <class FittingFunc>
std::vector<double> LeastSqFitter2d<FittingFunc>::residuals() {
    std::vector<double> out;
    out.reserve(_nData);

    for (int i = 0; i < _nData; ++i) {
        out.push_back(_z[i] - valueAt(_x[i], _y[i]));
    }

    return out;
}

template <class FittingFunc>
Eigen::MatrixXd LeastSqFitter2d<FittingFunc>::getErrors() {
    Eigen::ArrayXd variance(_nPar);
    for (int i = 0; i < _nPar; ++i) {
        variance[i] = _svd.matrixV().row(i).dot(
                (_svd.singularValues().array().inverse().square() * _svd.matrixV().col(i).array()).matrix());
    }
    return expandParams(variance.sqrt().matrix());
}

template <class FittingFunc>
void LeastSqFitter2d<FittingFunc>::initFunctions() {
    // Initialise the array of functions. _funcArray[i] is a object of type math::Function1 of order i
    _funcArray.reserve(_order);

    std::vector<double> coeff;
    coeff.reserve(_order);

    coeff.push_back(1);
    for (int i = 0; i < _order; ++i) {
        std::shared_ptr<FittingFunc> p(new FittingFunc(coeff));
        _funcArray.push_back(p);

        coeff[i] = 0;
        coeff.push_back(1);  // coeff now looks like [0,0,...,0,1]
    }
}

// The ith term in the fitting polynomial is of the form x^a * y^b. This function figures
// out the value of a and b, then calculates the value of the ith term at the given x and y
template <class FittingFunc>
double LeastSqFitter2d<FittingFunc>::func2d(double x, double y, int term) {
    int yexp = 0;  // y exponent
    int xexp = 0;  // x exponent

    for (int i = 0; i < term; ++i) {
        yexp = (yexp + 1) % (_order - xexp);
        if (yexp == 0) {
            xexp++;
        }
    }

    double xcomp = func1d(x, xexp);  // x component of polynomial
    double ycomp = func1d(y, yexp);  // y component

#if 0  // A useful debugging printf statement
        printf("The %i(th) function: x^%i * y^%i = %.1f * %.1f\n", term, xexp, yexp, xcomp, ycomp);
#endif

    return xcomp * ycomp;
}

template <class FittingFunc>
double LeastSqFitter2d<FittingFunc>::func1d(double value, int exponent) {
    return (*_funcArray[exponent])(value);
}

}  // namespace sip
}  // namespace astrom
}  // namespace meas
}  // namespace lsst

#endif