Class implements robust linear regression. More...

#include <robustRegression.hpp>

Inheritance diagram for julian::RobustRegression:

Public Types
enum	type { BISQUARE, CAUCHY, FAIR, HUBER, OLS, WELSCH }
	Robust regression types. More...

Public Member Functions
	RobustRegression ()
	constructor More...

	RobustRegression (int order, RobustRegression::type t)
	constructor More...

void	estimate (const std::vector< double > &x, const std::vector< double > &y)
	estimates the parameters basing on provided data More...

std::vector< double >	getCoefficient () const
	return coefficients of the regression More...

double	operator() (double) const
	Operator performing calculation. More...

Public Member Functions inherited from julian::DeeplyCopyableRegression< RobustRegression >
virtual Regression *	clone () const
	virtual copy constructor More...

Public Member Functions inherited from julian::Regression
	Regression ()
	Constructor. More...

virtual	~Regression ()
	Destructor. More...

Private Attributes
int	order_
	Order of polynomial. More...

RobustRegression::type	type_
	Type of robust regression. More...

std::vector< double >	coefs_
	Vector of coefficients. More...

Detailed Description

Class implements robust linear regression.

Robust regression is introduced because OLS models are often heavily influenced by the presence of outliers. Generally robust regression is estimated by minimizing the objective function

$\sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))$

where $e_i = y_i - Y(c, x_i)$ is the residual of the ith data point, and $\rho(e_i)$ is a function that:

1) $\rho(e) \ge 0$

2) $\rho(0) = 0$

3) $\rho(-e) = \rho(e)$

4) $\rho(e_1) > \rho(e_2) for |e_1| > |e_2|$

When $\rho(e) = e^2$ we obtain OLS estimator. To find the coefficients minimizing the objective function Iteratively Reweighed Least Squares (IRLS) algorithm is used. In this algorithm the problem of minimizing above sum is reduced to standard OLS with weighting, where weights $w(e)$ depends on error term.

Apart from OLS following functions can be used:

Tukey's bi-square function

$w(e) = \left\{ \begin{array}{cc} (1 - e^2)^2, & |e| \le 1 \\ 0, & |e| > 1 \end{array}\right.$
Cauchy’s function (aka Lorentzian function)

$w(e) = \frac{1}{1 + e^2}$
fair regression

$w(e) = frac{1}{ 1 + |e|}$
OLS

$w(e) = 1$
Welsch function

$w(e) = \exp{(-e^2)}$