linear regression. More...

#include <yat/regression/Linear.h>

Inheritance diagram for theplu::yat::regression::Linear:

Public Member Functions
	Linear (void)
	The default constructor.

virtual	~Linear (void)
	The destructor.

double	alpha (void) const

double	alpha_var (void) const

double	beta (void) const

double	beta_var (void) const

void	fit (const utility::VectorBase &x, const utility::VectorBase &y)

double	predict (const double x) const

double	s2 (void) const

double	standard_error2 (const double x) const

double	chisq (void) const
	Chi-squared.

double	prediction_error2 (const double x) const

std::ostream &	print (std::ostream &os, const double min, double max, const unsigned int n) const
	print output to ostream os

double	r2 (void) const

Protected Member Functions
double	variance (void) const

Protected Attributes
statistics::AveragerPair	ap_

double	chisq_

Detailed Description

linear regression.

Data are modeled as $y_i = \alpha + \beta (x_i-m_x) + \epsilon_i$ .

Member Function Documentation

double theplu::yat::regression::Linear::alpha ( void ) const

The parameter $\alpha$ is estimated as $\frac{1}{n}\sum y_i$

Returns: the parameter $\alpha$

double theplu::yat::regression::Linear::alpha_var ( void ) const

The variance is estimated as $\frac{s^2}{n}$ where $s^2 = \frac{\sum \epsilon^2}{n-2}$

Returns: variance of parameter $\alpha$

double theplu::yat::regression::Linear::beta ( void ) const

The parameter $\beta$ is estimated as $\frac{\textrm{Cov}(x,y)}{\textrm{Var}(x)}$

Returns: the parameter $\beta$

double theplu::yat::regression::Linear::beta_var ( void ) const

The variance is estimated as $\frac{s^2}{\sum (x-m_x)^2}$ where $s^2 = \frac{\sum \epsilon^2}{n-2}$

Returns: variance of parameter $\beta$

double theplu::yat::regression::OneDimensional::chisq ( void ) const

inherited

Chi-squared.

Chi-squared is defined as the $\sum{(\hat{y_i}-y_i)^2}$

void theplu::yat::regression::Linear::fit	(	const utility::VectorBase &	x,
		const utility::VectorBase &	y
	)

virtual

Model is fitted by minimizing $\sum{(y_i - \alpha - \beta (x-m_x))^2}$ . By construction $\alpha$ and $\beta$ are independent.

Implements theplu::yat::regression::OneDimensional.

double theplu::yat::regression::Linear::predict ( const double x ) const

virtual

Returns: $\alpha + \beta x$

Implements theplu::yat::regression::OneDimensional.

double theplu::yat::regression::OneDimensional::prediction_error2 ( const double x ) const

inherited

The prediction error is defined as the expected squared deviation a new data point will have from value the model provides: $E(Y|x - \hat{y}(x))^2$ and is typically divided into the conditional variance ( see s2() ) given $ x $ and the squared standard error ( see standard_error2() ) of the model estimation in $ x $ .

Returns: expected squared prediction error for a new data point in x

std::ostream& theplu::yat::regression::OneDimensional::print	(	std::ostream &	os,
		const double	min,
		double	max,
		const unsigned int	n
	)		const

inherited

print output to ostream os

Printing estimated model to os in the points defined by min, max, and n. The values printed for each point is the x-value, the estimated y-value, and the estimated standard deviation of a new data poiunt will have from the y-value given the x-value (see prediction_error()).

Parameters

os	Ostream printout is sent to
n	number of points printed
min	smallest x-value for which the model is printed
max	largest x-value for which the model is printed