Multivariate multidimensional polynomial. More...
#include <GenericPolynomial.hpp>
Inheritance diagram for TRTK::GenericPolynomial< T >: Collaboration diagram for TRTK::GenericPolynomial< T >:Public Types | |
| typedef super::MatrixXT | MatrixXT |
| typedef super::Point | Point |
| enum | Error { DIVISION_BY_ZERO, INVALID_ARGUMENT, NOT_ENOUGH_POINTS, UNEQUAL_NUMBER_OF_POINTS, UNKNOWN_ERROR, WRONG_COORDINATE_SIZE } |
Public Member Functions | |
| GenericPolynomial (unsigned n, unsigned m, unsigned r) | |
| Constructs an instance of GenericPolynomial. | |
| GenericPolynomial (const MatrixXT &, unsigned n, unsigned m, unsigned r) | |
| Constructs an instance of GenericPolynomial. | |
| GenericPolynomial (const GenericPolynomial &) | |
| Copy constructor. | |
| template<class U > | |
| GenericPolynomial (const GenericPolynomial< U > &, unsigned n, unsigned m, unsigned r) | |
| Copy constructor. | |
| virtual | ~GenericPolynomial () |
| Destroys the instance of GenericPolynomial. | |
| MatrixXT & | getCoefficients () |
| Returns the internal coefficients. | |
| const MatrixXT & | getCoefficients () const |
| Returns the internal coefficients. | |
| T | estimate (Range< Point > source_points, Range< Point > target_points, Range< T > weights=Range< T >()) |
| Estimates the polynomial function from two point sets. | |
| T | estimate (Iterator< Point > source_points_first, Iterator< Point > source_points_last, Iterator< Point > target_points_first, Iterator< Point > target_points_last, Iterator< T > weights_first=Iterator< T >(), Iterator< T > weights_last=Iterator< T >()) |
| Estimates the polynomial function from two point sets. | |
| Point | operator* (const Point &) const |
| Transforms a point with the internally saved transformation. | |
| Point | transform (const Point &) const |
| Transforms a point with the internally saved transformation. | |
| Polynomial< T > & | reset () |
| Resets the transformation. | |
| void | setDegree (unsigned degree) |
Public Attributes | |
|
EIGEN_MAKE_ALIGNED_OPERATOR_NEW typedef T | value_type |
Protected Types | |
| typedef super::VectorXT | VectorXT |
| typedef super::Vector3T | Vector3T |
| typedef super::coordinate_type | coordinate_type |
| typedef Eigen::Matrix< T, 2, 1 > | Vector2T |
Friends | |
| void | computeMonomials (unsigned, unsigned, unsigned, Iterator< Point >, std::vector< VectorXT > &) |
Related Functions | |
(Note that these are not member functions.) | |
| unsigned | numberMonomials (unsigned n, unsigned r) |
| template<class T > | |
| void | computeMonomials (unsigned n, unsigned r, unsigned number_points, Iterator< typename GenericPolynomial< T >::Point > it_source_points, std::vector< Eigen::Matrix< T, Eigen::Dynamic, 1 > > &monomials) |
Multivariate multidimensional polynomial.
This class provides means to estimate and compute a multivariate multidimensional polynomial of arbitrary degree.
The transformation is computed as follows (here, shown for a polynomial of degree two in three variables):
\[ \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} = f(x, y ,z) = \begin{pmatrix} a_{1,1} + a_{1,2} x + a_{1,3} y + \cdots + a_{1,8} y^2 + a_{1,9} yz + a_{1,10} z^2 \\ a_{2,1} + a_{2,2} x + a_{2,3} y + \cdots + a_{2,8} y^2 + a_{2,9} yz + a_{2,10} z^2 \\ a_{3,1} + a_{3,2} x + a_{3,3} y + \cdots + a_{3,8} y^2 + a_{3,9} yz + a_{3,10} z^2 \\ \end{pmatrix} \qquad f : \mathbb{R}^n \rightarrow \mathbb{R}^m \quad n = 3, \, m = 3, \, r = 2 \]
The number \( N \) of estimated coefficients is \( N = m \sum_{i=0}^{r} \binom{n + i - 1}{i} \) where \( n \) is the number of input variables, \( m \) is the dimension of the polynomial, and \( r \) is the degree of the polynomial. You will need at least as much as \( N / m \) point pairs to estimate the polynomial, otherwise the internal computation might suffer from a rank deficient matrix.
Most functions check for certain assertions (e.g., if the coordinate size is valid) and trigger an assertion failure if the assumption does not hold. This is meant for debugging purposes only and can be disabled by defining the macro NDEBUG.
Here is an example of how to use the GenericPolynomial class:
#include <iomanip> #include <iostream> #include <vector> #include <TRTK/Tools.hpp> #include <TRTK/GenericPolynomial.hpp> using namespace std; using namespace TRTK; using namespace TRTK::Tools; typedef TRTK::Coordinate<double> Point; Point f(const Point & point) { double x = point.x(); double y = point.y(); double z = point.z(); return Point(-x, x + 20 * y, z + 5 * x * z); }; int main() { // Generate two point sets that relate to each other via the // trivariate quadratic polynomial function f. vector<Point> source_points; vector<Point> target_points; for (unsigned i = 0; i < 100; ++i) { Point source_point(0, 0, 0); source_point.x() = rand(-10.0, 10.0); source_point.y() = rand(-10.0, 10.0); source_point.z() = rand(-10.0, 10.0); Point target_point = f(source_point); // Add noise. target_point.x() += 0.1 * randn(); target_point.y() += 0.1 * randn(); target_point.z() += 0.1 * randn(); source_points.push_back(source_point); target_points.push_back(target_point); } // Estimate the polynomial function f from the above two sets. GenericPolynomial<double> polynomial(3, 3, 2); // double rmse = polynomial.estimate(make_iterator(source_points.begin()), // make_iterator(source_points.end()), // make_iterator(target_points.begin()), // make_iterator(target_points.end())); // // or simply: double rmse = polynomial.estimate(make_range(source_points), make_range(target_points)); // Print out the results. cout << "Root Mean Square Error: " << rmse << endl; cout << "Coefficients: " << endl << setprecision(4) << fixed << polynomial.getCoefficients().transpose() << endl; return 0; }
Output:
Root Mean Square Error: 0.162183
Coefficients:
0.0409 0.0047 0.0017
-0.9991 0.9957 0.0000
-0.0019 19.9994 0.0009
-0.0006 -0.0000 1.0001
-0.0000 0.0002 -0.0002
-0.0002 -0.0001 -0.0000
-0.0005 0.0001 5.0002
-0.0008 -0.0005 0.0002
0.0001 -0.0001 -0.0004
0.0003 0.0001 0.0002
This algorithm estimates a linear model
\[ y_i = x_i^T \beta + \epsilon_i \]
where the parameter vector \( \beta \) contains the sought coefficients and where \( \epsilon_i \) represents the unknown error. The parameter vector is estimated using weighted least squares
\[ \begin{align} \hat\beta &= \arg_{\beta}\min \sum_i w_i (x_i^T \beta - y_i)^2 \\ &= \arg_{\beta}\min || W^{1/2} (A \beta - y) ||^2 \\[0.75em] &= [A^T W A]^{-1} A^T W y \end{align} \]
If no weights are given \( w_i \) is assumed to be equal to 1.
The algorithm returns the (weighted) root-mean-square error RMSE \( = \sqrt{\frac{1}{N} \sum_{i=1}^N w_i (f(x_i) - y_i)^2} \) of the estimation.
Given constant weights \( w_i = w \) the RMSE is proportional to the RMSE of an ordinary least squares estimation ( \( w_i = 1 \)) by a factor of \( w^{-1/2} \). It follows, a smaller RMSE does not necessarily imply a better estimation result. You might want to compare results by scaling the RMSE with \( \bar{w}^{-1/2} \) where \( \bar{w} \) is the mean of all weights.
Definition at line 426 of file GenericPolynomial.hpp.
| TRTK::GenericPolynomial< T >::GenericPolynomial | ( | unsigned | n, |
| unsigned | m, | ||
| unsigned | r | ||
| ) |
Constructs an instance of GenericPolynomial.
| T | scalar type |
| [in] | n | Number of input variables (dimension of the domain). |
| [in] | m | Dimension of the output vector (dimension of the codomain). |
| [in] | r | Degree of the polynomial. |
The transformation is initialized to be the identity function.
Definition at line 512 of file GenericPolynomial.hpp.
| TRTK::GenericPolynomial< T >::GenericPolynomial | ( | const MatrixXT & | matrix, |
| unsigned | n, | ||
| unsigned | m, | ||
| unsigned | r | ||
| ) |
Constructs an instance of GenericPolynomial.
| T | scalar type |
| [in] | matrix | A matrix containing the coefficients. |
| [in] | n | Number of input variables (dimension of the domain). |
| [in] | m | Dimension of the output vector (dimension of the codomain). |
| [in] | r | Degree of the polynomial. |
Sets the internal coefficients to the given matrix. The matrix must have m rows and \( \sum_{i=0}^{r} \binom{n + i - 1}{i} \) columns.
Definition at line 540 of file GenericPolynomial.hpp.
| TRTK::GenericPolynomial< T >::GenericPolynomial | ( | const GenericPolynomial< T > & | other ) |
Copy constructor.
| T | scalar type of the newly created instance |
Definition at line 560 of file GenericPolynomial.hpp.
| TRTK::GenericPolynomial< T >::GenericPolynomial | ( | const GenericPolynomial< U > & | other, |
| unsigned | n, | ||
| unsigned | m, | ||
| unsigned | r | ||
| ) |
Copy constructor.
| T | scalar type of the newly created instance |
| U | scalar type of the copied instance |
| [in] | other | An instance of GenericPolynomial with differing template parameter. |
| [in] | n | Number of input variables (dimension of the domain). |
| [in] | m | Dimension of the output vector (dimension of the codomain). |
| [in] | r | Degree of the polynomial. |
Definition at line 584 of file GenericPolynomial.hpp.
| TRTK::GenericPolynomial< T >::~GenericPolynomial | ( | ) | [virtual] |
Destroys the instance of GenericPolynomial.
| T | scalar type |
Definition at line 599 of file GenericPolynomial.hpp.
| T TRTK::GenericPolynomial< T >::estimate | ( | Range< Point > | source_points, |
| Range< Point > | target_points, | ||
| Range< T > | weights = Range<T>() |
||
| ) | [virtual] |
Estimates the polynomial function from two point sets.
| T | scalar type |
The input parameters are ranges [first, last) of a sequence (e.g. a STL container like vector). The last elements are not included in the ranges which coincides with the convention in the STL.
Source and target points as well as weights must correspond to each other. If no weights are given, the weights are assumed to be equal to one.
| ErrorObj | An error object is thrown if the point sets differ in their sizes. |
| ErrorObj | An error object is thrown if not enough (less than ten) points are given. |
Implements TRTK::Polynomial< T >.
Definition at line 649 of file GenericPolynomial.hpp.
| T TRTK::GenericPolynomial< T >::estimate | ( | Iterator< Point > | source_points_first, |
| Iterator< Point > | source_points_last, | ||
| Iterator< Point > | target_points_first, | ||
| Iterator< Point > | target_points_last, | ||
| Iterator< T > | weights_first = Iterator<T>(), |
||
| Iterator< T > | weights_last = Iterator<T>() |
||
| ) | [virtual] |
Estimates the polynomial function from two point sets.
| T | scalar type |
The input parameters are ranges [first, last) of a sequence (e.g. a STL container like vector). The last elements are not included in the ranges which coincides with the convention in the STL.
Source and target points as well as weights must correspond to each other. If no weights are given, the weights are assumed to be equal to one.
| ErrorObj | An error object is thrown if the point sets differ in their sizes. |
| ErrorObj | An error object is thrown if not enough (less than ten) points are given. |
Implements TRTK::Polynomial< T >.
Definition at line 681 of file GenericPolynomial.hpp.
| const GenericPolynomial< T >::MatrixXT & TRTK::GenericPolynomial< T >::getCoefficients | ( | ) | const [virtual] |
Returns the internal coefficients.
| T | scalar type |
Implements TRTK::Polynomial< T >.
Definition at line 622 of file GenericPolynomial.hpp.
| GenericPolynomial< T >::MatrixXT & TRTK::GenericPolynomial< T >::getCoefficients | ( | ) | [virtual] |
Returns the internal coefficients.
| T | scalar type |
Implements TRTK::Polynomial< T >.
Definition at line 610 of file GenericPolynomial.hpp.
| GenericPolynomial< T >::Point TRTK::GenericPolynomial< T >::operator* | ( | const Point & | point ) | const [virtual] |
Transforms a point with the internally saved transformation.
| T | scalar type |
| [in] | point | Point to be transformed. |
Implements TRTK::Polynomial< T >.
Definition at line 864 of file GenericPolynomial.hpp.
| Polynomial< T > & TRTK::GenericPolynomial< T >::reset | ( | ) | [virtual] |
Resets the transformation.
| T | scalar type |
Sets the internal coefficients such that the transformation is similar to the identity function. That is, the i-th component of the input vector is mapped to the i-th component of the output vector. If the dimension of the codomain is less than the dimension of the domain, the values are dropped. The other way round the output vector is padded with zeros.
Example:
(1, 2, 3) maps to (1, 2) (n = 3, m = 2)
(1, 2, 3) maps to (1, 2, 3) (n = 3, m = 3)
(1, 2, 3) maps to (1, 2, 3, 0) (n = 3, m = 4)
*this Implements TRTK::Polynomial< T >.
Definition at line 891 of file GenericPolynomial.hpp.
| GenericPolynomial< T >::Point TRTK::GenericPolynomial< T >::transform | ( | const Point & | point ) | const [virtual] |
Transforms a point with the internally saved transformation.
| T | scalar type |
| [in] | point | Point to be transformed. |
Implements TRTK::Polynomial< T >.
Definition at line 914 of file GenericPolynomial.hpp.
Documentation generated by Doxygen