```// The contents of this file are in the public domain. See LICENSE_FOR_EXAMPLE_PROGRAMS.txt
/*

This is an example illustrating the use the general purpose non-linear
least squares optimization routines from the dlib C++ Library.

This example program will demonstrate how these routines can be used for data fitting.
In particular, we will generate a set of data and then use the least squares
routines to infer the parameters of the model which generated the data.
*/

#include <dlib/optimization.h>
#include <iostream>
#include <vector>

using namespace std;
using namespace dlib;

// ----------------------------------------------------------------------------------------

typedef matrix<double,2,1> input_vector;
typedef matrix<double,3,1> parameter_vector;

// ----------------------------------------------------------------------------------------

// We will use this function to generate data.  It represents a function of 2 variables
// and 3 parameters.   The least squares procedure will be used to infer the values of
// the 3 parameters based on a set of input/output pairs.
double model (
const input_vector& input,
const parameter_vector& params
)
{
const double p0 = params(0);
const double p1 = params(1);
const double p2 = params(2);

const double i0 = input(0);
const double i1 = input(1);

const double temp = p0*i0 + p1*i1 + p2;

return temp*temp;
}

// ----------------------------------------------------------------------------------------

// This function is the "residual" for a least squares problem.   It takes an input/output
// pair and compares it to the output of our model and returns the amount of error.  The idea
// is to find the set of parameters which makes the residual small on all the data pairs.
double residual (
const std::pair<input_vector, double>& data,
const parameter_vector& params
)
{
return model(data.first, params) - data.second;
}

// ----------------------------------------------------------------------------------------

// This function is the derivative of the residual() function with respect to the parameters.
parameter_vector residual_derivative (
const std::pair<input_vector, double>& data,
const parameter_vector& params
)
{
parameter_vector der;

const double p0 = params(0);
const double p1 = params(1);
const double p2 = params(2);

const double i0 = data.first(0);
const double i1 = data.first(1);

const double temp = p0*i0 + p1*i1 + p2;

der(0) = i0*2*temp;
der(1) = i1*2*temp;
der(2) = 2*temp;

return der;
}

// ----------------------------------------------------------------------------------------

int main()
{
try
{
// randomly pick a set of parameters to use in this example
const parameter_vector params = 10*randm(3,1);
cout << "params: " << trans(params) << endl;

// Now let's generate a bunch of input/output pairs according to our model.
std::vector<std::pair<input_vector, double> > data_samples;
input_vector input;
for (int i = 0; i < 1000; ++i)
{
input = 10*randm(2,1);
const double output = model(input, params);

// save the pair
data_samples.push_back(make_pair(input, output));
}

// Before we do anything, let's make sure that our derivative function defined above matches
// the approximate derivative computed using central differences (via derivative()).
// If this value is big then it means we probably typed the derivative function incorrectly.
cout << "derivative error: " << length(residual_derivative(data_samples[0], params) -
derivative(residual)(data_samples[0], params) ) << endl;

// Now let's use the solve_least_squares_lm() routine to figure out what the
// parameters are based on just the data_samples.
parameter_vector x;
x = 1;

cout << "Use Levenberg-Marquardt" << endl;
// Use the Levenberg-Marquardt method to determine the parameters which
// minimize the sum of all squared residuals.
solve_least_squares_lm(objective_delta_stop_strategy(1e-7).be_verbose(),
residual,
residual_derivative,
data_samples,
x);

// Now x contains the solution.  If everything worked it will be equal to params.
cout << "inferred parameters: "<< trans(x) << endl;
cout << "solution error:      "<< length(x - params) << endl;
cout << endl;

x = 1;
cout << "Use Levenberg-Marquardt, approximate derivatives" << endl;
// If we didn't create the residual_derivative function then we could
// have used this method which numerically approximates the derivatives for you.
solve_least_squares_lm(objective_delta_stop_strategy(1e-7).be_verbose(),
residual,
derivative(residual),
data_samples,
x);

// Now x contains the solution.  If everything worked it will be equal to params.
cout << "inferred parameters: "<< trans(x) << endl;
cout << "solution error:      "<< length(x - params) << endl;
cout << endl;

x = 1;
cout << "Use Levenberg-Marquardt/quasi-newton hybrid" << endl;
// This version of the solver uses a method which is appropriate for problems
// where the residuals don't go to zero at the solution.  So in these cases
// it may provide a better answer.
solve_least_squares(objective_delta_stop_strategy(1e-7).be_verbose(),
residual,
residual_derivative,
data_samples,
x);

// Now x contains the solution.  If everything worked it will be equal to params.
cout << "inferred parameters: "<< trans(x) << endl;
cout << "solution error:      "<< length(x - params) << endl;

}
catch (std::exception& e)
{
cout << e.what() << endl;
}
}

// Example output:
/*
params: 8.40188 3.94383 7.83099

derivative error: 9.78267e-06
Use Levenberg-Marquardt
iteration: 0   objective: 2.14455e+10
iteration: 1   objective: 1.96248e+10
iteration: 2   objective: 1.39172e+10
iteration: 3   objective: 1.57036e+09
iteration: 4   objective: 2.66917e+07
iteration: 5   objective: 4741.9
iteration: 6   objective: 0.000238674
iteration: 7   objective: 7.8815e-19
iteration: 8   objective: 0
inferred parameters: 8.40188 3.94383 7.83099

solution error:      0

Use Levenberg-Marquardt, approximate derivatives
iteration: 0   objective: 2.14455e+10
iteration: 1   objective: 1.96248e+10
iteration: 2   objective: 1.39172e+10
iteration: 3   objective: 1.57036e+09
iteration: 4   objective: 2.66917e+07
iteration: 5   objective: 4741.87
iteration: 6   objective: 0.000238701
iteration: 7   objective: 1.0571e-18
iteration: 8   objective: 4.12469e-22
inferred parameters: 8.40188 3.94383 7.83099

solution error:      5.34754e-15

Use Levenberg-Marquardt/quasi-newton hybrid
iteration: 0   objective: 2.14455e+10
iteration: 1   objective: 1.96248e+10
iteration: 2   objective: 1.3917e+10
iteration: 3   objective: 1.5572e+09
iteration: 4   objective: 2.74139e+07
iteration: 5   objective: 5135.98
iteration: 6   objective: 0.000285539
iteration: 7   objective: 1.15441e-18
iteration: 8   objective: 3.38834e-23
inferred parameters: 8.40188 3.94383 7.83099

solution error:      1.77636e-15
*/

```