Index: The Book of Statistical ProofsStatistical Models ▷ Multivariate normal data ▷ Inverse general linear model ▷ Corresponding forward model

Definition: Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ estimated from $Y$ and $X$, such that right-multiplying $Y$ with the weight matrix gives an estimate or prediction of $X$:

\[\label{eq:bda} \hat{X} = Y W \; .\]

Given that the columns of $\hat{X}$ are linearly independent, then

\[\label{eq:cfm} Y = \hat{X} A^\mathrm{T} + E \quad \text{with} \quad \hat{X}^\mathrm{T} E = 0\]

is called the corresponding forward model relative to the weight matrix $W$.


Metadata: ID: D162 | shortcut: cfm | author: JoramSoch | date: 2021-10-21, 17:01.