GaussNewtonNlls.cpp

Description

A Templatized Header Only C++ Implementation of the Python Numpy Library

Author

David Pilger dpilg.nosp@m.er26.nosp@m.@gmai.nosp@m.l.co.nosp@m.m GitHub Repository

License

Copyright 2018-2023 David Pilger

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files(the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions :

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Testing

C++ Standards: C++14 C++17 C++20

Compilers: Visual Studio: 2017, 2019 GNU: 6.5, 7.5, 8.4, 9.3, 10.1 Clang: 6, 7, 8, 9, 10

Boost Versions: 1.68, 1.70, 1.72, and 1.73

Example for using the linalg::gaussNewtonNlls function

#include "NumCpp.hpp"
 
#include <cstdlib>
#include <iostream>
 
using FunctionType = std::function<double(const nc::NdArray<double>&, const nc::NdArray<double>&)>;
 
void wikipediaExample()
{
    // https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm
 
    // In a biology experiment studying the relation between substrate concentration [S] and reaction rate in
    // an enzyme-mediated reaction, the data in the following table were obtained.
    nc::NdArray<double> sMeasured    = { 0.038, 0.194, 0.425, 0.626, 1.253, 2.5, 3.74 };
    nc::NdArray<double> rateMeasured = { 0.05, 0.127, 0.094, 0.2122, 0.2729, 0.2665, 0.3317 };
 
    // It is desired to find a curve (model function) of the form
    FunctionType function = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double s     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
 
        return (beta1 * s) / (beta2 + s);
    };
 
    // partial derivative of function with respect to beta1
    FunctionType delFdelBeta1 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double s     = coordinates.at(0);
        const double beta2 = betas.at(1);
 
        return s / (beta2 + s);
    };
 
    // partial derivative of function with respect to beta2
    FunctionType delFdelBeta2 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double s     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
 
        return -(beta1 * s) / nc::square(beta2 + s);
    };
 
    // starting with the initial estimates of beta1Guess and beta2Guess and calculating after 5 iterations
    const nc::uint32 numIterations = 5;
    const double     beta1Guess    = 0.9;
    const double     beta2Guess    = 0.2;
 
    auto [betas, rms] = nc::linalg::gaussNewtonNlls(numIterations,
                                                    sMeasured.transpose(),
                                                    rateMeasured,
                                                    function,
                                                    { delFdelBeta1, delFdelBeta2 },
                                                    beta1Guess,
                                                    beta2Guess);
 
    std::cout << "==========Wikipedia Example==========\n";
    std::cout << "beta values = " << betas;
    std::cout << "RMS = " << rms << '\n';
}
 
void exponentialExample()
{
    // United States population (in millions) and the corresponding year:
    nc::NdArray<double> year       = nc::arange<double>(1.0, 9.0); // just use time points rather than the year
    nc::NdArray<double> population = { 8.3, 11.0, 14.7, 19.7, 26.7, 35.2, 44.4, 55.9 };
 
    // It is desired to find a curve (model function) of the form
    FunctionType exponentialFunction = [](const nc::NdArray<double>& coordinates,
                                          const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
 
        return beta1 * nc::exp(beta2 * t);
    };
 
    // partial derivative of function with respect to beta1
    FunctionType delFdelBeta1 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta2 = betas.at(1);
 
        return nc::exp(beta2 * t);
    };
 
    // partial derivative of function with respect to beta2
    FunctionType delFdelBeta2 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
 
        return beta1 * t * nc::exp(beta2 * t);
    };
 
    // starting with the initial estimates of beta1Guess and beta2Guess and calculating after 5 iterations
    const nc::uint32 numIterations = 5;
    const double     beta1Guess    = 6.0;
    const double     beta2Guess    = 0.3;
 
    auto [betas, rms] = nc::linalg::gaussNewtonNlls(numIterations,
                                                    year.transpose(),
                                                    population,
                                                    exponentialFunction,
                                                    { delFdelBeta1, delFdelBeta2 },
                                                    beta1Guess,
                                                    beta2Guess);
 
    std::cout << "==========Exponential Population Example==========\n";
    std::cout << "beta values = " << betas;
    std::cout << "RMS = " << rms << '\n';
}
 
void sinusoidalExample()
{
    // Average monthly high temperatures for Baton Rouge, LA:
    nc::NdArray<double> month       = nc::arange<double>(1.0, 13.0);
    nc::NdArray<double> temperature = { 61.0, 65.0, 72.0, 78.0, 85.0, 90.0, 92.0, 92.0, 88.0, 81.0, 72.0, 63.0 };
 
    // It is desired to find a curve (model function) of the form
    FunctionType sinusodialFunction = [](const nc::NdArray<double>& coordinates,
                                         const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
        const double beta3 = betas.at(2);
        const double beta4 = betas.at(3);
 
        return beta1 * nc::sin(beta2 * t + beta3) + beta4;
    };
 
    // partial derivative of function with respect to beta1
    FunctionType delFdelBeta1 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta2 = betas.at(1);
        const double beta3 = betas.at(2);
 
        return nc::sin(beta2 * t + beta3);
    };
 
    // partial derivative of function with respect to beta2
    FunctionType delFdelBeta2 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
        const double beta3 = betas.at(2);
 
        return beta1 * t * nc::cos(beta2 * t + beta3);
    };
 
    // partial derivative of function with respect to beta3
    FunctionType delFdelBeta3 = [](const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas) -> double
    {
        const double t     = coordinates.at(0);
        const double beta1 = betas.at(0);
        const double beta2 = betas.at(1);
        const double beta3 = betas.at(2);
 
        return beta1 * nc::cos(beta2 * t + beta3);
    };
 
    // partial derivative of function with respect to beta4
    FunctionType delFdelBeta4 = [](const nc::NdArray<double>& /*coordinates*/,
                                   const nc::NdArray<double>& /*betas*/) -> double { return 1.0; };
 
    // starting with the initial estimates and calculating after 5 iterations
    const nc::uint32 numIterations = 5;
    const double     beta1Guess    = 17.0;
    const double     beta2Guess    = 0.5;
    const double     beta3Guess    = 10.5;
    const double     beta4Guess    = 77.0;
 
    auto [betas, rms] = nc::linalg::gaussNewtonNlls(numIterations,
                                                    month.transpose(),
                                                    temperature,
                                                    sinusodialFunction,
                                                    { delFdelBeta1, delFdelBeta2, delFdelBeta3, delFdelBeta4 },
                                                    beta1Guess,
                                                    beta2Guess,
                                                    beta3Guess,
                                                    beta4Guess);
 
    std::cout << "==========Sinusodial Temperature Example==========\n";
    std::cout << "beta values = " << betas;
    std::cout << "RMS = " << rms << '\n';
}
 
// https://en.wikipedia.org/wiki/Gaussian_function
double baseGaussianFunction(const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas)
{
    const auto x = coordinates.at(0);
    const auto y = coordinates.at(1);
 
    const auto a      = betas.at(0);
    const auto x0     = betas.at(2);
    const auto y0     = betas.at(3);
    const auto sigmaX = betas.at(4);
    const auto sigmaY = betas.at(5);
 
    return a * nc::exp(-(nc::square(x - x0) / (2.0 * nc::square(sigmaX)) +
                         nc::square(y - y0) / (2.0 * nc::square(sigmaY))));
}
 
// It is desired to find a curve (model function) of the form
double gaussianFunction(const nc::NdArray<double>& coordinates, const nc::NdArray<double>& betas)
{
    const auto dcOffset = betas.at(1);
    return baseGaussianFunction(coordinates, betas) + dcOffset;
}
 
void twoDimensionalGaussianExample()
{
    // create some data points to describe a two-dimensional gaussian function
    auto coords      = nc::arange<double>(-3.0, 3.1, 0.1);
    auto [y, x]      = nc::meshgrid<double>(coords, coords);
    auto coordinates = nc::vstack({ x.flatten(), y.flatten() }).transpose();
 
    // randomize some truth values for the model parameters
    nc::random::seed(666); // set the random seed for repeatability
 
    // make some random "truth" parameters for generating data
    const auto aTruth        = nc::random::randFloat<double>(900.0, 1100.0);
    const auto dcOffsetTruth = nc::random::randFloat<double>(90.0, 100.0);
    const auto x0Truth       = nc::random::randFloat<double>(-0.5, 0.5);
    const auto y0Truth       = nc::random::randFloat<double>(-0.5, 0.5);
    const auto sigmaXTruth   = nc::random::randFloat<double>(0.9, 1.1);
    const auto sigmaYTruth   = nc::random::randFloat<double>(0.9, 1.1);
 
    nc::NdArray<double> betasTruth = { aTruth, dcOffsetTruth, x0Truth, y0Truth, sigmaXTruth, sigmaYTruth };
 
    // make some "measurements"
    nc::NdArray<double> measurements(1, coordinates.shape().rows);
    const auto          cSlice = coordinates.cSlice();
    for (nc::uint32 i = 0; i < measurements.size(); ++i)
    {
        measurements[i] = gaussianFunction(coordinates(i, cSlice), betasTruth);
    }
 
    // add some noise to the measurements
    auto noise = nc::random::randN<double>(measurements.shape()) * 10.0; // STD = 10 of noise
    measurements += noise;
 
    // partial derivative of gaussianFunction with respect to a
    FunctionType delFdelA = [](const nc::NdArray<double>& coordinates_, const nc::NdArray<double>& betas) -> double
    {
        const auto a = betas.at(0);
        return baseGaussianFunction(coordinates_, betas) / a;
    };
 
    // partial derivative of gaussianFunction with respect to dcOffset
    FunctionType delFdelDcOffset = [](const nc::NdArray<double>& /*coordinates*/,
                                      const nc::NdArray<double>& /*betas*/) -> double { return 1.0; };
 
    // partial derivative of gaussianFunction with respect to x0
    FunctionType delFdelX0 = [](const nc::NdArray<double>& coordinates_, const nc::NdArray<double>& betas) -> double
    {
        const auto x_ = coordinates_.at(0);
 
        const auto x0     = betas.at(2);
        const auto sigmaX = betas.at(4);
 
        return (x_ - x0) / nc::square(sigmaX) * baseGaussianFunction(coordinates_, betas);
    };
 
    // partial derivative of gaussianFunction with respect to y0
    FunctionType delFdelY0 = [](const nc::NdArray<double>& coordinates_, const nc::NdArray<double>& betas) -> double
    {
        const auto y_ = coordinates_.at(1);
 
        const auto y0     = betas.at(3);
        const auto sigmaY = betas.at(5);
 
        return (y_ - y0) / nc::square(sigmaY) * baseGaussianFunction(coordinates_, betas);
    };
 
    // partial derivative of gaussianFunction with respect to sigmaX
    FunctionType delFdelSigmaX = [](const nc::NdArray<double>& coordinates_, const nc::NdArray<double>& betas) -> double
    {
        const auto x_ = coordinates_.at(0);
 
        const auto x0     = betas.at(2);
        const auto sigmaX = betas.at(4);
 
        return (nc::square(x_ - x0) / nc::power(sigmaX, 3)) * baseGaussianFunction(coordinates_, betas);
    };
 
    // partial derivative of gaussianFunction with respect to sigmaY
    FunctionType delFdelSigmaY = [](const nc::NdArray<double>& coordinates_, const nc::NdArray<double>& betas) -> double
    {
        const auto y_ = coordinates_.at(1);
 
        const auto y0     = betas.at(2);
        const auto sigmaY = betas.at(4);
 
        return (nc::square(y_ - y0) / nc::power(sigmaY, 3)) * baseGaussianFunction(coordinates_, betas);
    };
 
    // starting with the initial estimates of and calculating after 5 iterations
    const nc::uint32 numIterations = 5;
    const double     aGuess        = aTruth + nc::random::randN<double>() * 5.0;
    const double     dcOffsetGuess = dcOffsetTruth + nc::random::randN<double>() * 5.0;
    const double     x0Guess       = x0Truth + nc::random::randN<double>() * 0.2;
    const double     y0Guess       = y0Truth + nc::random::randN<double>() * 0.2;
    const double     sigmaXGuess   = sigmaXTruth + nc::random::randN<double>() * 0.2;
    const double     sigmaYGuess   = sigmaYTruth + nc::random::randN<double>() * 0.2;
 
    auto [betas, rms] =
        nc::linalg::gaussNewtonNlls(numIterations,
                                    coordinates,
                                    measurements,
                                    FunctionType(gaussianFunction),
                                    { delFdelA, delFdelDcOffset, delFdelX0, delFdelY0, delFdelSigmaX, delFdelSigmaY },
                                    aGuess,
                                    dcOffsetGuess,
                                    x0Guess,
                                    y0Guess,
                                    sigmaXGuess,
                                    sigmaYGuess);
 
    nc::NdArray<double> initialGuess = { aGuess, dcOffsetGuess, x0Guess, y0Guess, sigmaXGuess, sigmaYGuess };
 
    std::cout << "==========2D Gaussian Example==========\n";
    std::cout << "truth values  = " << betasTruth;
    std::cout << "initial guess = " << initialGuess;
    std::cout << "beta values   = " << betas;
    std::cout << "RMS = " << rms << '\n';
}
 
int main()
{
    wikipediaExample();
    exponentialExample();
    sinusoidalExample();
    twoDimensionalGaussianExample();
 
    return EXIT_SUCCESS;
}