honey::LinearLeastSqr< Real > Class Template Reference

Linear least squares solver. More...

#include <LinearLeastSqr.h>

Public Types
typedef Matrix< matrix::dynamic, matrix::dynamic, Real >	Matrix
typedef Vec< matrix::dynamic, Real >	Vec

Public Member Functions
	LinearLeastSqr ()
void	calc (const Matrix &x, const Vec &y, Vec &b)
	Linear least squares. More...
void	calc (const Matrix &x, const Vec &y, const Vec &w, Vec &b)
	Weighted linear least squares. Each of the m equations in has an associated weight. A relatively low weight corresponds to high uncertainty. More...
void	calc (const Matrix &x, const Vec &y, const Vec &w, const Matrix &c, const Vec &d, Vec &b)
	Constrained weighted linear least squares. Get a best-fit solution to , subject to the constraints . More...

Detailed Description

template<class Real>
class honey::LinearLeastSqr< Real >

Linear least squares solver.

Member Typedef Documentation

template<class Real >

typedef Matrix<matrix::dynamic, matrix::dynamic, Real> honey::LinearLeastSqr< Real >::Matrix

template<class Real >

typedef Vec<matrix::dynamic, Real> honey::LinearLeastSqr< Real >::Vec

Constructor & Destructor Documentation

template<class Real >

honey::LinearLeastSqr< Real >::LinearLeastSqr ( )

inline

Member Function Documentation

template<class Real >

void honey::LinearLeastSqr< Real >::calc	(	const Matrix &	x,
		const Vec &	y,
		Vec &	b
	)

Linear least squares.

Get a best-fit solution to the system: where the rows of (m x n) X and m-dim form a system of m linear equations, and n-dim contains unknowns assumed to be linearly related.

Given the residual (error) as , the least squares approach is to minimize the residual's Euclidean -norm (aka. magnitude) .
Thus, the least squared problem becomes:
It can be shown that this minimization problem can be uniquely solved using the SVD and the pseudo-inverse.

Example: Curve Fitting

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Consider a quadratic curve:
Let's say we have a series of samples on a graph that we want to fit to our curve. If we plug in our samples directly we will find the left/right-hand sides don't agree, there are errors and the best we can do is minimize them.

Let's introduce the linearly related unknowns into the curve equation:

We want to find the coefficients of which minimize the errors across all samples (a best-fit curve).

With the model formulated we must now work it into a linear system for the solver. For each sample we append a row to X, and a row (y) to . We now have all the parameters needed to solve for .

Parameters

x	A (m x n) matrix of function coefficients. In the usual case m > n, ie. there are more samples than unknowns.
y	A m-dim vector of function results
b	Result: a n-dim vector of coefficients that best approximates the solution

template<class Real >

void honey::LinearLeastSqr< Real >::calc	(	const Matrix &	x,
		const Vec &	y,
		const Vec &	w,
		Vec &	b
	)

Weighted linear least squares. Each of the m equations in has an associated weight. A relatively low weight corresponds to high uncertainty.

Ideally, a sample's weight should be the inverse of its variance. The residuals that least squares minimizes are multiplied by the weights.

Parameters

x
y
w	A m-dim vector of sample weights
b

template<class Real >

void honey::LinearLeastSqr< Real >::calc	(	const Matrix &	x,
		const Vec &	y,
		const Vec &	w,
		const Matrix &	c,
		const Vec &	d,
		Vec &	b
	)

Constrained weighted linear least squares. Get a best-fit solution to , subject to the constraints .

This is similar to solving with k additional equations that are infinitely weighted.

Parameters

x
y
w
c	A (k x n) matrix of constraint coefficients, where k < n. X and C col sizes must match.
d	A k-dim vector of constraint results.
b

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The documentation for this class was generated from the following files:

• src/common/Honey/Math/NumAnalysis/LinearLeastSqr.h
• src/common/Honey/Math/NumAnalysis/LinearLeastSqr.cpp