Nonlinear least-squares fitting with two independent variables in C++: implementing GSL algorithm

Question

Following up to a previous question I asked in Fixing parameters of a fitting function in Nonlinear Least-Square GSL (successfully answered by @zkoza), I would like to implement an algorithm that can fit data to a non-linear function, by fixing some of its parameters while leaving other parameters to change for finding the best fit to the data. The difference to my previous question is that I want to have two independent variables instead of one independent variable.

Non-linear function used to fit the data

double gaussian(double x, double b, double a, double c)
{
    const double z = (x - b) / c;
    return a * std::exp(-0.5 * z * z);
}

In my previous question I was considering that x was the only independent variable . Now I would like to consider two independent variables, x and b .

The original algorithm used to fit a non-linear function using only one independent variable (while fixing variable a ) is a C++ wrapper of the GSL nonlinear least-squares algorithm (borrowed from https://github.com/Eleobert/gsl-curve-fit/blob/master/example.cpp ):

template <typename F, size_t... Is>
auto gen_tuple_impl(F func, std::index_sequence<Is...> )
{
    return std::make_tuple(func(Is)...);
}

template <size_t N, typename F>
auto gen_tuple(F func)
{
    return gen_tuple_impl(func, std::make_index_sequence<N>{} );
}

template <class R, class... ARGS>
struct function_ripper {
    static constexpr size_t n_args = sizeof...(ARGS);
};

template <class R, class... ARGS>
auto constexpr n_params(R (ARGS...) )
{
    return function_ripper<R, ARGS...>();
}


auto internal_solve_system(gsl_vector* initial_params, gsl_multifit_nlinear_fdf *fdf,
             gsl_multifit_nlinear_parameters *params) -> std::vector<double>
{
  // This specifies a trust region method
  const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
  const size_t max_iter = 200;
  const double xtol = 1.0e-8;
  const double gtol = 1.0e-8;
  const double ftol = 1.0e-8;

  auto *work = gsl_multifit_nlinear_alloc(T, params, fdf->n, fdf->p);
  int info;

  // initialize solver
  gsl_multifit_nlinear_init(initial_params, fdf, work);
  //iterate until convergence
  gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol, nullptr, nullptr, &info, work);

  // result will be stored here
  gsl_vector * y    = gsl_multifit_nlinear_position(work);
  auto result = std::vector<double>(initial_params->size);

  for(int i = 0; i < result.size(); i++)
  {
    result[i] = gsl_vector_get(y, i);
  }

  auto niter = gsl_multifit_nlinear_niter(work);
  auto nfev  = fdf->nevalf;
  auto njev  = fdf->nevaldf;
  auto naev  = fdf->nevalfvv;

  // nfev - number of function evaluations
  // njev - number of Jacobian evaluations
  // naev - number of f_vv evaluations
  //logger::debug("curve fitted after ", niter, " iterations {nfev = ", nfev, "} {njev = ", njev, "} {naev = ", naev, "}");

  gsl_multifit_nlinear_free(work);
  gsl_vector_free(initial_params);
  return result;
}

template<auto n>
auto internal_make_gsl_vector_ptr(const std::array<double, n>& vec) -> gsl_vector*
{
    auto* result = gsl_vector_alloc(vec.size());
    int i = 0;
    for(const auto e: vec)
    {
        gsl_vector_set(result, i, e);
        i++;
    }
    return result;
}


template<typename C1>
struct fit_data
{
    const std::vector<double>& t;
    const std::vector<double>& y;
    // the actual function to be fitted
    C1 f;
};


template<typename FitData, int n_params>
int internal_f(const gsl_vector* x, void* params, gsl_vector *f)
{
    auto* d  = static_cast<FitData*>(params);
    // Convert the parameter values from gsl_vector (in x) into std::tuple
    auto init_args = [x](int index)
    {
        return gsl_vector_get(x, index);
    };
    auto parameters = gen_tuple<n_params>(init_args);

    // Calculate the error for each...
    for (size_t i = 0; i < d->t.size(); ++i)
    {
        double ti = d->t[i];
        double yi = d->y[i];
        auto func = [ti, &d](auto ...xs)
        {
            // call the actual function to be fitted
            return d->f(ti, xs...);
        };
        auto y = std::apply(func, parameters);
        gsl_vector_set(f, i, yi - y);
    }
    return GSL_SUCCESS;
}

using func_f_type   = int (*) (const gsl_vector*, void*, gsl_vector*);
using func_df_type  = int (*) (const gsl_vector*, void*, gsl_matrix*);
using func_fvv_type = int (*) (const gsl_vector*, const gsl_vector *, void *, gsl_vector *);

template<auto n>
auto internal_make_gsl_vector_ptr(const std::array<double, n>& vec) -> gsl_vector*;


auto internal_solve_system(gsl_vector* initial_params, gsl_multifit_nlinear_fdf *fdf,
             gsl_multifit_nlinear_parameters *params) -> std::vector<double>;

template<typename C1>
auto curve_fit_impl(func_f_type f, func_df_type df, func_fvv_type fvv, gsl_vector* initial_params, fit_data<C1>& fd) -> std::vector<double>
{
    assert(fd.t.size() == fd.y.size());

    auto fdf = gsl_multifit_nlinear_fdf();
    auto fdf_params = gsl_multifit_nlinear_default_parameters();

    fdf.f   = f;
    fdf.df  = df;
    fdf.fvv = fvv;
    fdf.n   = fd.t.size();
    fdf.p   = initial_params->size;
    fdf.params = &fd;

    // "This selects the Levenberg-Marquardt algorithm with geodesic acceleration."
    fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
    return internal_solve_system(initial_params, &fdf, &fdf_params);
}


template <typename Callable, auto n>
auto curve_fit(Callable f, const std::array<double, n>& initial_params, const std::vector<double>& x, const std::vector<double>& y) -> std::vector<double>
{
    // We can't pass lambdas without convert to std::function.
    //constexpr auto n = 3;//decltype(n_params(f))::n_args - 5;
    //constexpr auto n = 2;
    assert(initial_params.size() == n);

    auto params = internal_make_gsl_vector_ptr(initial_params);
    auto fd = fit_data<Callable>{x, y, f};
    return curve_fit_impl(internal_f<decltype(fd), n>, nullptr, nullptr, params,  fd);
}

In order to fix one of the parameters of the gaussian function, @zkoza proposed to use functors :

struct gaussian_fixed_a
{
    double a;
    gaussian_fixed_a(double a) : a{a} {}
    double operator()(double x, double b, double c) const { return gaussian(x, b, a, c); }
};

And these last lines show how I would create a fake dataset of observed data (with some noise which is normally distributed) and test the fitting curve function with two independent variables, given by the vectors xs and bs .

    int main()
    {
        auto device = std::random_device();
        auto gen    = std::mt19937(device());
    
        auto xs = linspace<std::vector<double>>(0.0, 1.0, 300);
        auto bs = linspace<std::vector<double>>(0.4, 1.4, 300);
        auto ys = std::vector<double>(xs.size());
    
        double a = 5.0, c = 0.15;
    
        for(size_t i = 0; i < xs.size(); i++)
        {

            auto y =  gaussian(xs[i], a, bs[i], c);
            auto dist  = std::normal_distribution(0.0, 0.1 * y);
            ys[i] = y + dist(gen);
        }
        gaussian_fixed_a g(a);
        auto r = curve_fit(g, std::array{0.11}, xs, bs, ys);
    
        std::cout << "result: " << r[0] << ' ' << '\n';
        std::cout << "error : " << r[0] - c << '\n';
    
    }

Do you have any idea on how I could implement the two-independent variables non-linear fitting?

Answer 1

The solution, as suggested in the comments by @BenVoigt, is to replace the x and b independent variables in the gaussian function with 'one independent variable' given as a vector, whose first element is x and the second element is b .

Also the backbone of the nonlinear fitting needs to be slightly edited. The edits consist:

1. Replace the fit_data functor with:

    struct fit_data { const std::vector< vector<double> > &t; const std::vector<double>& y; // the actual function to be fitted C1 f; };

Such that, the independent variable is no longer a vector but rather a vector of a vector (aka a matrix ).

2. Replace within the function internal_f .

a) double ti = d->t[i] with std::vector<double> ti = d->t[i] b) auto func = [ti, &d](auto ...xs) with auto func = [ti, &d](auto ...xs_matrix) c) return d->f(ti, xs...) with return d->f(ti, xs_matrix...)

3. Replace within curve_fit function:

a) const std::vector<double>& x with const std::vector< vector<double> > &xs_matrix b) auto fd = fit_data<Callable>{x, y, f} with auto fd = fit_data<Callable>{xs_matrix, y, f}

Whereas the gaussian function, gaussian_fixed_a functor and the fitting function looks like:

double gaussian(std::vector<double> x_vector, double a, double c)
    {
        const double z = (x_vector[0] - x_vector[1]) / c;
        return a * std::exp(-0.5 * z * z);
    }

struct gaussian_fixed_a
{
    double a;
    gaussian_fixed_a(double a) : a{a} {}
    double operator()(std::vector<double> x_vector, double c) const { return gaussian(x_vector, a, c); }
};

double fittingTest(const std::vector< vector<double> > &xs_matrix, const std::vector<double> ys, const double a){

      gaussian_fixed_a g(a);
      auto r = curve_fit(g, std::array{3.0}, xs_matrix, ys);

        return r[0]);

        }