LeastSquares.h File Reference

Go to the source code of this file.

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)

Function Documentation

template<typename VectorType , typename HyperplaneType >

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

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

$r_0 x_0 + \cdots + r_{n-1}x_{n-1} + r_n = 0,$

where $n=Size$ , $r_i=retCoefficients[i]$ , and we denote by $x_0,\ldots,x_{n-1}$ the n coordinates in the n-dimensional space.

Thus, the vector retCoefficients has size $n+1$ , 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.

See also:: linearRegression()

Definition at line 146 of file LeastSquares.h.

template<typename VectorType >

void linearRegression	(	int	numPoints,
		VectorType **	points,
		VectorType *	result,
		int	funcOfOthers
	)			`[inline]`

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 $(x,y,z)$ 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 ( $y$ ) as a linear expression in $x$ and $z$ , that is,

$y=ax+bz+c$

for some constants $a,b,c$ . Thus, we want to find the best possible constants $a,b,c$ so that the plane of equation $y=ax+bz+c$ 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
    linearRegression(
      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 $( 0.495 , -1.927 , -2.906 )$ . Thus, we get $a=0.495, b = -1.927, c = -2.906$ . Let us check for instance how near points[0] is from the plane of equation $y=ax+bz+c$ . Looking at the coords of points[0], we see that:

$ax+bz+c = 0.495 * 3.02 + (-1.927) * (-4.32) + (-2.906) = 6.91.$

On the other hand, we have $y=6.89$ . We see that the values $6.91$ and $6.89$ are near, so points[0] is very near the plane of equation $y=ax+bz+c$ .

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 $x_0,\ldots,x_{n-1}$ the n coordinates in the n-dimensional space. Then the resulting equation is: $x_f = r_0 x_0 + \cdots + r_{f-1}x_{f-1} + r_{f+1}x_{f+1} + \cdots + r_{n-1}x_{n-1} + r_n.$
	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 , ...

See also:: fitHyperplane()

Definition at line 101 of file LeastSquares.h.

LeastSquares.h File Reference

Functions

Function Documentation