Merge pull request #168 from JoramSoch/master

added 1 proof

Author: JoramSoch

I/ToC.md

@@ -550,25 +550,26 @@ title: "Table of Contents"
 
    1.4. Multiple linear regression <br>
    &emsp;&ensp; 1.4.1. *[Definition](/D/mlr)* <br>
-   &emsp;&ensp; 1.4.2. **[Ordinary least squares](/P/mlr-ols)** (1) <br>
-   &emsp;&ensp; 1.4.3. **[Ordinary least squares](/P/mlr-ols2)** (2) <br>
-   &emsp;&ensp; 1.4.4. *[Total sum of squares](/D/tss)* <br>
-   &emsp;&ensp; 1.4.5. *[Explained sum of squares](/D/ess)* <br>
-   &emsp;&ensp; 1.4.6. *[Residual sum of squares](/D/rss)* <br>
-   &emsp;&ensp; 1.4.7. **[Total, explained and residual sum of squares](/P/mlr-pss)** <br>
-   &emsp;&ensp; 1.4.8. *[Estimation matrix](/D/emat)* <br>
-   &emsp;&ensp; 1.4.9. *[Projection matrix](/D/pmat)* <br>
-   &emsp;&ensp; 1.4.10. *[Residual-forming matrix](/D/rfmat)* <br>
-   &emsp;&ensp; 1.4.11. **[Estimation, projection and residual-forming matrix](/P/mlr-mat)** <br>
-   &emsp;&ensp; 1.4.12. **[Idempotence of projection and residual-forming matrix](/P/mlr-idem)** <br>
-   &emsp;&ensp; 1.4.13. **[Weighted least squares](/P/mlr-wls)** (1) <br>
-   &emsp;&ensp; 1.4.14. **[Weighted least squares](/P/mlr-wls2)** (2) <br>
-   &emsp;&ensp; 1.4.15. **[Maximum likelihood estimation](/P/mlr-mle)** <br>
-   &emsp;&ensp; 1.4.16. **[Maximum log-likelihood](/P/mlr-mll)** <br>
-   &emsp;&ensp; 1.4.17. **[Deviance function](/P/mlr-dev)** <br>
-   &emsp;&ensp; 1.4.18. **[Akaike information criterion](/P/mlr-aic)** <br>
-   &emsp;&ensp; 1.4.19. **[Bayesian information criterion](/P/mlr-bic)** <br>
-   &emsp;&ensp; 1.4.20. **[Corrected Akaike information criterion](/P/mlr-aicc)** <br>
+   &emsp;&ensp; 1.4.2. **[Special case of general linear model](/P/mlr-glm)** <br>
+   &emsp;&ensp; 1.4.3. **[Ordinary least squares](/P/mlr-ols)** (1) <br>
+   &emsp;&ensp; 1.4.4. **[Ordinary least squares](/P/mlr-ols2)** (2) <br>
+   &emsp;&ensp; 1.4.5. *[Total sum of squares](/D/tss)* <br>
+   &emsp;&ensp; 1.4.6. *[Explained sum of squares](/D/ess)* <br>
+   &emsp;&ensp; 1.4.7. *[Residual sum of squares](/D/rss)* <br>
+   &emsp;&ensp; 1.4.8. **[Total, explained and residual sum of squares](/P/mlr-pss)** <br>
+   &emsp;&ensp; 1.4.9. *[Estimation matrix](/D/emat)* <br>
+   &emsp;&ensp; 1.4.10. *[Projection matrix](/D/pmat)* <br>
+   &emsp;&ensp; 1.4.11. *[Residual-forming matrix](/D/rfmat)* <br>
+   &emsp;&ensp; 1.4.12. **[Estimation, projection and residual-forming matrix](/P/mlr-mat)** <br>
+   &emsp;&ensp; 1.4.13. **[Idempotence of projection and residual-forming matrix](/P/mlr-idem)** <br>
+   &emsp;&ensp; 1.4.14. **[Weighted least squares](/P/mlr-wls)** (1) <br>
+   &emsp;&ensp; 1.4.15. **[Weighted least squares](/P/mlr-wls2)** (2) <br>
+   &emsp;&ensp; 1.4.16. **[Maximum likelihood estimation](/P/mlr-mle)** <br>
+   &emsp;&ensp; 1.4.17. **[Maximum log-likelihood](/P/mlr-mll)** <br>
+   &emsp;&ensp; 1.4.18. **[Deviance function](/P/mlr-dev)** <br>
+   &emsp;&ensp; 1.4.19. **[Akaike information criterion](/P/mlr-aic)** <br>
+   &emsp;&ensp; 1.4.20. **[Bayesian information criterion](/P/mlr-bic)** <br>
+   &emsp;&ensp; 1.4.21. **[Corrected Akaike information criterion](/P/mlr-aicc)** <br>
 
    1.5. Bayesian linear regression <br>
    &emsp;&ensp; 1.5.1. **[Conjugate prior distribution](/P/blr-prior)** <br>

P/mlr-glm.md

@@ -0,0 +1,66 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-07-21 08:28:00
+
+title: "Multiple linear regression is a special case of the general linear model"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Multiple linear regression"
+theorem: "Special case of general linear model"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "General linear model"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-07-21"
+    url: "https://en.wikipedia.org/wiki/General_linear_model#Comparison_to_multiple_linear_regression"
+
+proof_id: "P329"
+shortcut: "mlr-glm"
+username: "JoramSoch"
+---
+
+
+**Theorem:** [Multiple linear regression](/D/mlr) is a special case of the [general linear model](/D/mlr) with number of measurements $v = 1$, such that data matrix $Y$, regression coefficients $B$, noise matrix $E$ and noise covariance $\Sigma$ equate as
+
+$$ \label{eq:mlr-glm}
+Y = y, \quad B = \beta, \quad E = \varepsilon \quad \text{and} \quad \Sigma = \sigma^2
+$$
+
+where $y$, $\beta$, $\varepsilon$ and $\sigma^2$ are the data vector, regression coefficients, noise vector and noise variance from [multiple linear regression](/D/mlr).
+
+
+**Proof:** The [linear regression model with correlated errors](/D/mlr) is given by:
+
+$$ \label{eq:mlr}
+y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V) \; .
+$$
+
+Because $\varepsilon$ is an $n \times 1$ vector and $\sigma^2$ is scalar, we have the following identities:
+
+$$
+\begin{split}
+\mathrm{vec}(\varepsilon) &= \varepsilon \\
+\sigma^2 \otimes V &= \sigma^2 V \; .
+\end{split}
+$$
+
+Thus, using the [relationship between multivariate normal and matrix normal distribution](/P/matn-mvn), equation \eqref{eq:mlr} can also be written as
+
+$$ \label{eq:mlr-dev}
+y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{MN}(0, V, \sigma^2) \; .
+$$
+
+Comparing with the [general linear model with correlated observations](/D/glm)
+
+$$ \label{eq:glm}
+Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; ,
+$$
+
+we finally note the equivalences given in equation \eqref{eq:mlr-glm}.