regline.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	definition	sources	def_id	shortcut	username
definition	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2021-10-27 00:30:00 -0700	Regression line	Statistical Models	Univariate normal data	Simple linear regression	Regression line	authors year title in pages url Wikipedia 2021 Simple linear regression Wikipedia, the free encyclopedia retrieved on 2021-10-27 https://en.wikipedia.org/wiki/Simple_linear_regression#Fitting_the_regression_line	D164	regline	JoramSoch

Definition: Let there be a simple linear regression with independent observations using dependent variable $y$ and independent variable $x$:

$$ \label{eq:slr} y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, ; \varepsilon_i \sim \mathcal{N}(0, \sigma^2) ; . $$

Then, given some parameters $\beta_0, \beta_1 \in \mathbb{R}$, the set

$$ \label{eq:regline} L(\beta_0, \beta_1) = \left\lbrace (x,y) \in \mathbb{R}^2 \mid y = \beta_0 + \beta_1 x \right\rbrace $$

is called a "regression line" and the set

$$ \label{eq:regline-ols} L(\hat{\beta}_0, \hat{\beta}_1) $$

is called the "fitted regression line", with estimated regression coefficients $\hat{\beta}_0, \hat{\beta}_1$, e.g. obtained via ordinary least squares.