theplu::yat::regression::Linear Class Reference

#include <yat/regression/Linear.h>

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

List of all members.

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 $\beta$

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::Linear::s2 ( void ) const [virtual]

$\frac{\sum \epsilon_i^2}{N-2}$

Returns:: variance of residuals

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

double theplu::yat::regression::Linear::standard_error2 ( const double x ) const [virtual]

The error of the model is estimated as $\textrm{alpha\_err}^2+\textrm{beta\_err}^2*(x-m_x)*(x-m_x)$

Returns:: estimated error of model in x

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

double theplu::yat::regression::OneDimensional::chisq ( void ) const [inherited]

Chi-squared.

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

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

double theplu::yat::regression::OneDimensional::r2 ( void ) const [inherited]

r2 is defined as $1 - \frac{Var(Y|x)}{Var(Y)}$ or the fraction of the variance explained by the regression model.

See also:: s2()

double theplu::yat::regression::OneDimensional::variance ( void ) const [protected, inherited]

Variance of y

Member Data Documentation

statistics::AveragerPair theplu::yat::regression::OneDimensional::ap_ [protected, inherited]

Averager for pair of x and y

double theplu::yat::regression::OneDimensional::chisq_ [protected, inherited]

See also:: chisq()

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

yat/regression/Linear.h


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_