A C++ library for useful tools on optimization, statistics and curve fitting.
These currently include:
- lsqlinear.hpp: linear regressions (by means of Eigen project)
- lsqnonlinear.hpp: non-linear regressions (by means of GNU GSL project)
- ttest.hpp: T test
- Student's t test and Welch's test for unequal variances (by means of statslib/GCEM project)
#include "lsqlinear.hpp"
// ...
std::vector<double> vx = {0, 1, 2};
std::vector<double> vy = {6, 0, 0};
// testing with mode 'default'
auto Mb = optstats::leastSquaresLinearRegression(vx, vy);
// least squares solution is: y = -3x + 5
assert(Mb.first == -3.0);
assert(Mb.second == 5.0);
#include "lsqlinear.hpp"
// ...
std::vector<double> vx = {-1, 1, 2, 3};
std::vector<double> vy = {1 / 2.0, -1, -1 / 2.0, 2};
// y = Bx²+Cx+D
std::vector<std::function<double(double)>> vf = {
[](double x) { return x * x; }, [](double x) { return x; }};
// testing with mode 'SVD'
auto vA = optstats::leastSquaresRegression(vx, vy, vf, LinearSolveMode::AccurateSVD);
//
// least squares solution is: y = 53/88 x² -379/440 x - 41/44
//
assert(53 / 88.0 == vA[0]);
assert(-379 / 440.0 == vA[1]);
assert(-41 / 44.0 == vA[2]);
See log transform strategies:
- https://math.stackexchange.com/questions/1488747/least-square-approximation-for-exponential-functions
- https://math.stackexchange.com/questions/2591061/exponential-least-squares-equation
There's also a test on tests/ that works on that using Eigen.
Note that error is greater than a real nonlinear approach (such as with Levenberg-Marquardt).
#include "lsqnonlinear.hpp"
// ...
// y = a*e^{-bx}
double model_exp_mi1(double x, double a, double b) {
return a * std::exp(-b * x);
}
// ...
// y = a*e^{-bx}
std::vector<double> ys = {8558, 5411, 2830, 2267, 760, 549, 249, 67, 47, 43};
std::vector<double> xs = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
auto res = optstats::leastSquaresNonLinearRegression(xs, ys, {0.0, 0.0}, model_exp_mi1);
// y = a*e^{-bx}
//
double realA = res[0];
double realB = res[1];
//
assert(8666.36934 == realA);
assert(0.52034 == realB);
//
double R2 = optstats::calcR2(
xs, ys, [realA, realB](double x) { return realA * ::exp(-realB * x); });
// expects very good fit (by means of GNU GSL)
assert(R2 >= 0.99);
In order to use linear regressions of lsqlinear.hpp, just #include "lsqlinear.hpp".
You will need to include Eigen support, just include -Ipath/to/eigen.
For nonlinear regressions, one needs to #include "lsqnonlinear.hpp".
In this case, GNU GSL will be required, so as flag -lgsl.
On Ubuntu 20.04, just apt install libgsl-dev.
Learn more about T vs Normal:
- https://www.statology.org/normal-distribution-vs-t-distribution/
- https://en.wikipedia.org/wiki/Student%27s_t-test
Two sided independent t-test (from Wikipedia).
#include "ttest.hpp"
//...
std::vector<double> a1 = {30.02, 29.99, 30.11, 29.97, 30.01, 29.99};
std::vector<double> a2 = {29.89, 29.93, 29.72, 29.98, 30.02, 29.98};
// Null Hypothesis: means of a1 and a2 are the same
double x1 = optstats::mean(a1);
double x2 = optstats::mean(a2);
assert(0.095 == (x1 - x2));
// test with unequal variances
auto [ttest, dof] = optstats::getIndependentTwoSampleTTest(a1, a2);
assert(1.959 == ttest); // check t-value
assert(7.031 == dof); // check degrees of freedom
// p-value for two-sided test (note the 'TestSides::Two')
double p = optstats::pIndependentTwoSampleTTest(a1, a2, TestSides::Both);
assert(0.09077 == p);
In order to use linear regressions of ttest.hpp, just #include "ttest.hpp".
You will need to include GCEM and stat library support, just include -Ipath/to/statlib (both are header-only).
There unit tests on tests/ folder, feel free to use them as examples.
- Levenberg-Marquardt algorithm used from GSL wrapper (provided by Eleobert) can also be found on "Eigen unsupported" (TODO: investigate this usage)
- TODO consider other statistic techniques, such as: https://en.wikipedia.org/wiki/Akaike_information_criterion
- TODO consider Armadillo for basic operations: http://arma.sourceforge.net/docs.html
Free Software - Feel free to use it and redistribute it
All novel code written here is MIT License, EXCEPT for dependencies (each .hpp has one).
Note that:
- Eigen: Mozilla Public License 2.0 (MPL2) - https://eigen.tuxfamily.org
- GNU GSL: GNU Public License 3.0 (GPL3) - https://www.gnu.org/software/gsl/
- stats library (statslib): Apache Version 2 (Apache 2.0) - https://github.com/kthohr/stats/
- gcem is a dependency from stats (library) - Apache 2.0 - https://github.com/kthohr/gcem/
Depending on the mix, it can be GPL-like or MIT-like (better explanations may come in the future...).
On short:
lsqlinear.hpp: MIT + Eigen => License MPL2lsqnonlinear.hpp: MIT + GNU GSL => License GPL3ttest.hpp: MIT + (stats library + GCEM) => MIT/Apache 2.0
Copyleft 2021
React
Typescript
Django
Laravel
Facebook
Microsoft
Google
Alibaba
D3
Tencent