在 C++ 中拟合两个自变量的非线性最小二乘：实现 GSL 算法

Question

继我在Fixing parameters of afitting function in Nonlinear Least-Square GSL （@zkoza 成功回答）中提出的上一个问题之后，我想通过修复一些来实现一种可以将数据拟合到非线性函数的算法保留其参数，同时更改其他参数以找到最适合数据的参数。 与我之前的问题的不同之处在于我想要两个自变量而不是一个自变量。

用于拟合数据的非线性函数

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

在我之前的问题中，我认为x是唯一的independent variable 。 现在我想考虑两个自变量x和b 。

用于仅使用一个independent variable （同时固定变量a ）拟合非线性函数的原始算法是 GSL 非线性最小二乘算法的 C++ 包装器（借自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);
}

为了修复高斯函数的参数之一，@zkoza 建议使用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); }
};

最后几行显示了我将如何创建一个观察数据的假数据集（带有一些正态分布的噪声），并使用向量xs和bs给出的两个自变量测试拟合曲线函数。

    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';
    
    }

您对如何实现two-independent variables非线性拟合有任何想法吗？

Answer 1

正如@BenVoigt 在评论中所建议的那样，解决方案是将高斯函数中的x和b自变量替换为作为向量给出的“一个自变量”，其第一个元素是x ，第二个元素是b 。

非线性拟合的主干也需要稍微编辑。 编辑包括：

1. 将fit_data函子替换为：

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

这样，自变量不再是vector ，而是vector of a vector （又名matrix ）。

2. 在函数internal_f替换。

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. 在curve_fit函数内替换：

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}

而gaussian函数、 gaussian_fixed_a函子和拟合函数如下所示：

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]);

        }