Definition: Multiple linear regression
Definition: Let $y$ be an $n \times 1$ vector and let $X$ be an $n \times p$ matrix.
Then, a statement asserting a linear combination of $X$ into $y$
\[\label{eq:mlrmodel} y = X\beta + \varepsilon \; ,\]together with a statement asserting a normal distribution for $\varepsilon$
\[\label{eq:mlrnoise} \varepsilon \sim \mathcal{N}(0, \sigma^2 V)\]is called a univariate linear regression model or simply, “multiple linear regression”.

$y$ is called “measured data”, “dependent variable” or “measurements”;

$X$ is called “design matrix”, “set of independent variables” or “predictors”;

$V$ is called “covariance matrix” or “covariance structure”;

$\beta$ are called “regression coefficients” or “weights”;

$\varepsilon$ is called “noise”, “errors” or “error terms”;

$\sigma^2$ is called “noise variance” or “error variance”;

$n$ is the number of observations;

$p$ is the number of predictors.
Alternatively, the linear combination may also be written as
\[\label{eq:mlrmodelsum} y = \sum_{i=1}^{p} \beta_i x_i + \varepsilon\]or, when the model includes an intercept term, as
\[\label{eq:mlrmodelsumbase} y = \beta_0 + \sum_{i=1}^{p} \beta_i x_i + \varepsilon\]which is equivalent to adding a constant regressor $x_0 = 1_n$ to the design matrix $X$.
When the covariance structure $V$ is equal to the $n \times n$ identity matrix, this is called multiple linear regression with independent and identically distributed (i.i.d.) observations:
\[\label{eq:mlrnoiseiid} V = I_n \quad \Rightarrow \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n) \quad \Rightarrow \quad \varepsilon_i \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .\]Otherwise, it is called multiple linear regression with correlated observations.
 Wikipedia (2020): "Linear regression"; in: Wikipedia, the free encyclopedia, retrieved on 20200321; URL: https://en.wikipedia.org/wiki/Linear_regression#Simple_and_multiple_linear_regression.
Metadata: ID: D36  shortcut: mlr  author: JoramSoch  date: 20200321, 20:09.