Functions
LeastSquares.h File Reference

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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 
)

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 143 of file LeastSquares.h.

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

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:

    Vector3d coeffs; // will store the coefficients a, b, c
    linearRegression(
      5,
      &points,
      &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:
numPointsthe number of points
pointsthe array of pointers to the points on which to perform the linear regression
resultpointer 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 $n=Size$, $r_i=result[i]$, and $f=funcOfOthers$. Denote by $x_0,\ldots,x_{n-1}$ the n coordinates in the n-dimensional space. Then the resulting equation is:

\[ x_f = r_0 x_0 + \cdots + r_{f-1}x_{f-1} + r_{f+1}x_{f+1} + \cdots + r_{n-1}x_{n-1} + r_n. \]

funcOfOthersDetermines which coord to express as a function of the others. Coords are numbered starting from 0, so that a value of 0 means $x$, 1 means $y$, 2 means $z$, ...
See also:
fitHyperplane()

Definition at line 98 of file LeastSquares.h.



libicr
Author(s): Robert Krug
autogenerated on Mon Jan 6 2014 11:34:05