Definition: Corresponding forward model
Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Multivariate normal data ▷ Inverse general linear model ▷ Corresponding forward model
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Metadata: ID: D162 | shortcut: cfm | author: JoramSoch | date: 2021-10-21, 17:01.
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 $W$.
- Haufe S, Meinecke F, Görgen K, Dähne S, Haynes JD, Blankertz B, Bießmann F (2014): "On the interpretation of weight vectors of linear models in multivariate neuroimaging"; in: NeuroImage, vol. 87, pp. 96–110, eq. 3; URL: https://www.sciencedirect.com/science/article/pii/S1053811913010914; DOI: 10.1016/j.neuroimage.2013.10.067.
Metadata: ID: D162 | shortcut: cfm | author: JoramSoch | date: 2021-10-21, 17:01.