added 3 definitions

Author: JoramSoch

D/cfm.md

@@ -0,0 +1,43 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-21 17:01
+
+title: "General linear model"
+chapter: "Statistical Models"
+section: "Multivariate normal data"
+topic: "Inverse general linear model"
+definition: "Corresponding forward model"
+
+sources:
+  - authors: "Haufe S, Meinecke F, Görgen K, Dähne S, Haynes JD, Blankertz B, Bießmann F"
+    year: 2014
+    title: "On the interpretation of weight vectors of linear models in multivariate neuroimaging"
+    in: "NeuroImage"
+    pages: "vol. 87, pp. 96–110, eq. 3"
+    url: "https://www.sciencedirect.com/science/article/pii/S1053811913010914"
+    doi: "10.1016/j.neuroimage.2013.10.067"
+
+def_id: "D162"
+shortcut: "cfm"
+username: "JoramSoch"
+---
+
+
+**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 $W$.

D/iglm.md

@@ -0,0 +1,37 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-21 15:31:00
+
+title: "General linear model"
+chapter: "Statistical Models"
+section: "Multivariate normal data"
+topic: "Inverse general linear model"
+definition: "Definition"
+
+sources:
+  - authors: "Soch J, Allefeld C, Haynes JD"
+    year: 2020
+    title: "Inverse transformed encoding models – a solution to the problem of correlated trial-by-trial parameter estimates in fMRI decoding"
+    in: "NeuroImage"
+    pages: "vol. 209, art. 116449, Appendix C"
+    url: "https://www.sciencedirect.com/science/article/pii/S1053811919310407"
+    doi: "10.1016/j.neuroimage.2019.116449"
+
+def_id: "D161"
+shortcut: "iglm"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let there be a [general linear models](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ in terms of the [design matrix](/D/glm) $X \in \mathbb{R}^{n \times p}$:
+
+$$ \label{eq:glm}
+Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; .
+$$
+
+Then, a [linear model](/D/glm) of $X$ in terms of $Y$, under the assumption of \eqref{eq:glm}, is called an inverse general linear model.

D/tglm.md

@@ -0,0 +1,49 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-21 14:43:00
+
+title: "General linear model"
+chapter: "Statistical Models"
+section: "Multivariate normal data"
+topic: "Transformed general linear model"
+definition: "Definition"
+
+sources:
+  - authors: "Soch J, Allefeld C, Haynes JD"
+    year: 2020
+    title: "Inverse transformed encoding models – a solution to the problem of correlated trial-by-trial parameter estimates in fMRI decoding"
+    in: "NeuroImage"
+    pages: "vol. 209, art. 116449, Appendix A"
+    url: "https://www.sciencedirect.com/science/article/pii/S1053811919310407"
+    doi: "10.1016/j.neuroimage.2019.116449"
+
+def_id: "D160"
+shortcut: "tglm"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let there be two [general linear models](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ using [design matrices](/D/glm) $X \in \mathbb{R}^{n \times p}$ and $X_t \in \mathbb{R}^{n \times t}$
+
+$$ \label{eq:glm1}
+Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma)
+$$
+
+$$ \label{eq:glm2}
+Y = X_t \Gamma + E_t, \; E_t \sim \mathcal{MN}(0, V, \Sigma_t)
+$$
+
+and assume that $X_t$ can be transformed into $X$ using a transformation matrix $T \in \mathbb{R}^{t \times p}$
+
+$$ \label{eq:X-Xt-T}
+X = X_t \, T
+$$
+
+where $p < t$ and $X$, $X_t$ and $T$ have full ranks $\mathrm{rk}(X) = p$, $\mathrm{rk}(X_t) = t$ and $\mathrm{rk}(T) = p$.
+
+Then, a [linear model](/D/glm) of the parameter estimates from \eqref{eq:glm2}, under the assumption of \eqref{eq:glm1}, is called a transformed general linear model.