From c20fb45c4570e8bbad35f54a8c9d5628830c3d27 Mon Sep 17 00:00:00 2001 From: Davis King Date: Sat, 4 Dec 2010 19:37:58 +0000 Subject: [PATCH] Added an example program showing how to use the least squares optimization routines. --HG-- extra : convert_revision : svn%3Afdd8eb12-d10e-0410-9acb-85c331704f74/trunk%403945 --- examples/CMakeLists.txt | 1 + examples/least_squares_ex.cpp | 228 ++++++++++++++++++++++++++++++++++ 2 files changed, 229 insertions(+) create mode 100755 examples/least_squares_ex.cpp diff --git a/examples/CMakeLists.txt b/examples/CMakeLists.txt index ab462b92c..55c4cf07e 100644 --- a/examples/CMakeLists.txt +++ b/examples/CMakeLists.txt @@ -55,6 +55,7 @@ add_example(krls_ex) add_example(krls_filter_ex) add_example(krr_classification_ex) add_example(krr_regression_ex) +add_example(least_squares_ex) add_example(linear_manifold_regularizer_ex) add_example(logger_ex) add_example(logger_ex_2) diff --git a/examples/least_squares_ex.cpp b/examples/least_squares_ex.cpp new file mode 100755 index 000000000..92bb60db1 --- /dev/null +++ b/examples/least_squares_ex.cpp @@ -0,0 +1,228 @@ +// 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 +#include + + +using namespace std; +using namespace dlib; + +// ---------------------------------------------------------------------------------------- + +typedef matrix input_vector; +typedef matrix 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& 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& 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 lets generate a bunch of input/output pairs according to our model. + std::vector > 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, lets 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 lets 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 +*/