LeastSquares module

Functions

template<typename VectorType , typename HyperplaneType >
void fitHyperplane (int numPoints, VectorType **points, HyperplaneType *result, typename NumTraits< typename VectorType::Scalar >::Real *soundness=0)

template<typename VectorType >
void linearRegression (int numPoints, VectorType **points, VectorType *result, int funcOfOthers)

Detailed Description

This module provides linear regression and related features.

#include <Eigen/LeastSquares>

Function Documentation

 void Eigen::fitHyperplane ( int numPoints, VectorType ** points, HyperplaneType * result, typename NumTraits< typename VectorType::Scalar >::Real * soundness = `0` )

This is defined in the LeastSquares module.

#include <Eigen/LeastSquares>

This function is quite similar to linearRegression(), so we refer to the documentation of this function and only list here the differences.

The main difference from linearRegression() is that this function doesn't take a funcOfOthers argument. Instead, it finds a general equation of the form

where , , and we denote by the n coordinates in the n-dimensional space.

Thus, the vector retCoefficients has size , which is another difference from linearRegression().

In practice, this function performs an hyper-plane fit in a total least square sense via the following steps: 1 - center the data to the mean 2 - compute the covariance matrix 3 - pick the eigenvector corresponding to the smallest eigenvalue of the covariance matrix The ratio of the smallest eigenvalue and the second one gives us a hint about the relevance of the solution. This value is optionally returned in soundness.

linearRegression()
 void Eigen::linearRegression ( int numPoints, VectorType ** points, VectorType * result, int funcOfOthers )

This is defined in the LeastSquares module.

#include <Eigen/LeastSquares>

For a set of points, this function tries to express one of the coords as a linear (affine) function of the other coords.

This is best explained by an example. This function works in full generality, for points in a space of arbitrary dimension, and also over the complex numbers, but for this example we will work in dimension 3 over the real numbers (doubles).

So let us work with the following set of 5 points given by their coordinates:

Vector3d points[5];
points[0] = Vector3d( 3.02, 6.89, -4.32 );
points[1] = Vector3d( 2.01, 5.39, -3.79 );
points[2] = Vector3d( 2.41, 6.01, -4.01 );
points[3] = Vector3d( 2.09, 5.55, -3.86 );
points[4] = Vector3d( 2.58, 6.32, -4.10 );

Suppose that we want to express the second coordinate ( ) as a linear expression in and , that is,

for some constants . Thus, we want to find the best possible constants so that the plane of equation fits best the five above points. To do that, call this function as follows:

// create a vector of pointers to the points
std::vector<Vector3d> points_ptrs(5);
for(int k=0; k<5; ++k) points_ptrs[k] = &points[k];
Vector3d coeffs; // will store the coefficients a, b, c
5,
&(points_ptrs[0]),
&coeffs,
1 // the coord to express as a function of
// the other ones. 0 means x, 1 means y, 2 means z.
);

Now the vector coeffs is approximately . Thus, we get . Let us check for instance how near points[0] is from the plane of equation . Looking at the coords of points[0], we see that:

On the other hand, we have . We see that the values and are near, so points[0] is very near the plane of equation .

Let's now describe precisely the parameters:

Parameters
 numPoints the number of points points the array of pointers to the points on which to perform the linear regression result pointer to the vector in which to store the result. This vector must be of the same type and size as the data points. The meaning of its coords is as follows. For brevity, let , , and . Denote by the n coordinates in the n-dimensional space. Then the resulting equation is: funcOfOthers Determines which coord to express as a function of the others. Coords are numbered starting from 0, so that a value of 0 means , 1 means , 2 means , ...