Linear Regression Task

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A Linear Regression Task is a regression analysis task that is based on linear predictor functions.

[math]y_i=f(x_i,\boldsymbol\beta)+\varepsilon_i\quad[/math] with [math]f(x_i,\beta_j )=\sum _{j=0}^{m}\beta _{j}\phi_{j}(x_i)[/math] for [math]\quad i=1,\cdots,n \;[/math] and [math]j=0,\cdots, p[/math]
in this regression function [math]f(X)=f(x_i,\boldsymbol\beta)[/math] is a parametric regression function which is a linear combination between [math]p[/math] regression coefficients [math]\beta_j[/math] (parameters) and basis functions [math]\phi_j(x_i)[/math]. Usually, the basis function are polynomial, i.e. [math]\phi_j(x^i)=x_i^{j}[/math]
[math]\mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{U},[/math] where [math]\mathbf{Y}=y_i[/math] is the measurement matrix, [math]\mathbf{X}=\phi _{j}(x_i)[/math] is design matrix, [math]\mathbf{B}=\boldsymbol{\beta}=\beta_j [/math] the parameters matrix and [math]\mathbf{U}=\varepsilon_i[/math] a errors matrix. This is,

[math]\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} \phi_0(x_1) & \phi_1(x_1) & \cdots & \phi_p(x_1) \\ \phi_0(x_2) & \phi_1(x_2) & \cdots & \phi_p{x_2} \\ \vdots & \vdots & \ddots & \vdots \\ \phi_0(x_n) & \phi_1(x_n) & \cdots & \phi_p(x_n) \end{pmatrix}\begin{pmatrix} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_p \end{pmatrix}+\begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_n \end{pmatrix}[/math]

by estimating the best-fitting [math]\beta[/math] parameters that optimizes the following objective function:

[math]E(f)=\sum _{i=1}^{n}L(y_{i},f(x_{i},{\boldsymbol \beta }))[/math]

[math]L(\cdot)[/math] is an error function that may be derived as a loss function or the negative of a likelihood function.




  • (Wikipedia, 2014) ⇒ Retrieved:2014-11-23.


      Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

