Index: The Book of Statistical ProofsStatistical Models ▷ Multivariate normal data ▷ General linear model ▷ Definition

Definition: Let $Y$ be an $n \times v$ matrix and let $X$ be an $n \times p$ matrix. Then, a statement asserting a linear mapping from $X$ to $Y$ with parameters $B$ and matrix-normally distributed errors $E$

\[\label{eq:glm} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma)\]

is called a multivariate linear regression model or simply, “general linear model”.

  • $Y$ is called “data matrix”, “set of dependent variables” or “measurements”;

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

  • $B$ are called “regression coefficients” or “weights”;

  • $E$ is called “noise matrix” or “error terms”;

  • $V$ is called “covariance across rows”;

  • $\Sigma$ is called “covariance across columns”;

  • $n$ is the number of observations;

  • $v$ is the number of measurements;

  • $p$ is the number of predictors.

When rows of $Y$ correspond to units of time, e.g. subsequent measurements, $V$ is called “temporal covariance”. When columns of $Y$ correspond to units of space, e.g. measurement channels, $\Sigma$ is called “spatial covariance”.

When the covariance matrix $V$ is a scalar multiple of the $n \times n$ identity matrix, this is called a general linear model with independent and identically distributed (i.i.d.) observations:

\[\label{eq:glm-iid} V = \lambda I_n \quad \Rightarrow \quad E \sim \mathcal{MN}(0, \lambda I_n, \Sigma) \quad \Rightarrow \quad \varepsilon_i \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, \lambda \Sigma) \; .\]

Otherwise, it is called a general linear model with correlated observations.


Metadata: ID: D40 | shortcut: glm | author: JoramSoch | date: 2020-03-21, 22:24.