| layout |
mathjax |
author |
affiliation |
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chapter |
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topic |
theorem |
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proof_id |
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proof |
true |
Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2021-10-21 10:43:00 -0700 |
Existence of a corresponding forward model |
Statistical Models |
Multivariate normal data |
Inverse general linear model |
Proof of existence |
| authors |
year |
title |
in |
pages |
url |
doi |
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 |
NeuroImage |
vol. 87, pp. 96–110, Appendix B |
|
10.1016/j.neuroimage.2013.10.067 |
|
|
P270 |
cfm-exist |
JoramSoch |
Theorem: Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
$$ \label{eq:bda}
\hat{X} = Y W ; .
$$
Then, there exists a corresponding forward model.
Proof: The corresponding forward model is defined as
$$ \label{eq:cfm}
Y = \hat{X} A^\mathrm{T} + E \quad \text{with} \quad \hat{X}^\mathrm{T} E = 0
$$
and the parameters of the corresponding forward model are equal to
$$ \label{eq:cfm-para}
A = \Sigma_y W \Sigma_x^{-1} \quad \text{where} \quad \Sigma_x = \hat{X}^\mathrm{T} \hat{X} \quad \text{and} \quad \Sigma_y = Y^\mathrm{T} Y ; .
$$
1) Because the columns of $\hat{X}$ are assumed to be linearly independent [by definition of the corresponding forward model](/D/cfm), the matrix $\Sigma_x = \hat{X}^\mathrm{T} \hat{X}$ is invertible, such that $A$ in \eqref{eq:cfm-para} is well-defined.
2) Moreover, the solution for the matrix $A$ satisfies the [constraint of the corresponding forward model](/D/cfm) for predicted $X$ and errors $E$ to be uncorrelated which can be shown as follows:
$$ \label{eq:X-E-0}
\begin{split}
\hat{X}^\mathrm{T} E &\overset{\eqref{eq:cfm}}{=} \hat{X}^\mathrm{T} \left( Y - \hat{X} A^\mathrm{T} \right) \\
&\overset{\eqref{eq:cfm-para}}{=} \hat{X}^\mathrm{T} \left( Y - \hat{X} , \Sigma_x^{-1} W^\mathrm{T} \Sigma_y \right) \\
&= \hat{X}^\mathrm{T} Y - \hat{X}^\mathrm{T} \hat{X} , \Sigma_x^{-1} W^\mathrm{T} \Sigma_y \\
&\overset{\eqref{eq:cfm-para}}{=} \hat{X}^\mathrm{T} Y - \hat{X}^\mathrm{T} \hat{X} \left( \hat{X}^\mathrm{T} \hat{X} \right)^{-1} W^\mathrm{T} \left( Y^\mathrm{T} Y \right) \\
% &= \hat{X}^\mathrm{T} Y - W^\mathrm{T} \left( Y^\mathrm{T} Y \right) \\
&\overset{\eqref{eq:bda}}{=} (Y W)^\mathrm{T} Y - W^\mathrm{T} \left( Y^\mathrm{T} Y \right) \\
&= W^\mathrm{T} Y^\mathrm{T} Y - W^\mathrm{T} Y^\mathrm{T} Y \\
&= 0 ; .
\end{split}
$$
This completes the proof.