added 2 definitions

Author: JoramSoch

D/regline.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-27 07:30:00
+
+title: "Regression line"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Simple linear regression"
+definition: "Regression line"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Simple linear regression"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-10-27"
+    url: "https://en.wikipedia.org/wiki/Simple_linear_regression#Fitting_the_regression_line"
+
+def_id: "D164"
+shortcut: "regline"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let there be a [simple linear regression with independent observations](/D/slr) 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
+
+$$ \label{eq:regline-ols}
+L(\hat{\beta}_0, \hat{\beta}_1) = \left\lbrace (x,y) \in \mathbb{R}^2 \mid y = \hat{\beta}_0 + \hat{\beta}_1 x \right\rbrace
+$$
+
+is called the "fitted regression line", with estimated regression coefficients $\hat{\beta}_0, \hat{\beta}_1$, e.g. obtained via [ordinary least squares](/P/slr-ols).

D/slr.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-27 07:07:00
+
+title: "Simple linear regression"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Simple linear regression"
+definition: "Definition"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Simple linear regression"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-10-27"
+    url: "https://en.wikipedia.org/wiki/Simple_linear_regression#Fitting_the_regression_line"
+
+def_id: "D163"
+shortcut: "slr"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $y$ and $x$ be two $n \times 1$ vectors.
+
+Then, a statement asserting a linear relationship between $x$ and $y$
+
+$$ \label{eq:slr-model}
+y = \beta_0 + \beta_1 x + \varepsilon \; ,
+$$
+
+together with a statement asserting a [normal distribution](/D/mvn) for $\varepsilon$
+
+$$ \label{eq:slr-noise}
+\varepsilon \sim \mathcal{N}(0, \sigma^2 V)
+$$
+
+is called a univariate simple regression model or simply, "simple linear regression".
+
+* $y$ is called "dependent variable", "measured data" or "signal";
+
+* $x$ is called "independent variable", "predictor" or "covariate";
+
+* $V$ is called "covariance matrix" or "covariance structure";
+
+* $\beta_1$ is called "slope of the [regression line](/D/regline)";
+
+* $\beta_0$ is called "intercept of the [regression line](/D/regline)";
+
+* $\varepsilon$ is called "noise", "errors" or "error terms";
+
+* $\sigma^2$ is called "noise variance" or "error variance";
+
+* $n$ is the number of observations.
+
+When the covariance structure $V$ is equal to the $n \times n$ identity matrix, this is called simple linear regression with independent and identically distributed (i.i.d.) observations:
+
+$$ \label{eq:mlr-noise-iid}
+V = I_n \quad \Rightarrow \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n) \quad \Rightarrow \quad \varepsilon_i \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .
+$$
+
+In this case, the linear regression model can also be written as
+
+$$ \label{eq:slr-model-sum}
+y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2) \; .
+$$
+
+Otherwise, it is called simple linear regression with correlated observations.