corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

D/cov-samp.md

@@ -36,7 +36,7 @@ $$
 and the unbiased sample covariance of $x$ and $y$ is given by
 
 $$ \label{eq:cov-samp-unb}
-s^2_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y})
+s_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y})
 $$
 
 where $\bar{x}$ and $\bar{y}$ are the [sample means](/D/mean-samp).

D/regline.md

@@ -42,7 +42,7 @@ $$
 is called a "regression line" and the set
 
 $$ \label{eq:regline-ols}
-L(\hat{\beta}_0, \hat{\beta}_1) = \left\lbrace (x,y) \in \mathbb{R}^2 \mid y = \hat{\beta}_0 + \hat{\beta}_1 x \right\rbrace
+L(\hat{\beta}_0, \hat{\beta}_1)
 $$
 
 is called the "fitted regression line", with estimated regression coefficients $\hat{\beta}_0, \hat{\beta}_1$, e.g. obtained via [ordinary least squares](/P/slr-ols).

I/Table_of_Contents.md

@@ -490,7 +490,7 @@ title: "Table of Contents"
    &emsp;&ensp; 1.3.7. **[Regression line includes center of mass](/P/slr-comp)** <br>
    &emsp;&ensp; 1.3.8. **[Sum of residuals is zero](/P/slr-ressum)** <br>
    &emsp;&ensp; 1.3.9. **[Correlation with covariate is zero](/P/slr-rescorr)** <br>
-   &emsp;&ensp; 1.3.10. **[Residual variance in terms of sample variance](/P/slr-vars)** <br>
+   &emsp;&ensp; 1.3.10. **[Residual variance in terms of sample variance](/P/slr-resvar)** <br>
    &emsp;&ensp; 1.3.11. **[Correlation coefficient in terms of slope estimate](/P/slr-corr)** <br>
    &emsp;&ensp; 1.3.12. **[Coefficient of determination in terms of correlation coefficient](/P/slr-rsq)** <br>
 

P/mlr-ols2.md

@@ -49,7 +49,7 @@ $$
 **Proof:** The [residual sum of squares](/D/rss) is defined as
 
 $$ \label{eq:RSS}
-\mathrm{RSS}(\beta) = \sum_{i=1}^n \varepsilon_i = \varepsilon^\mathrm{T} \varepsilon = (y-X\beta)^\mathrm{T} (y-X\beta)
+\mathrm{RSS}(\beta) = \sum_{i=1}^n \varepsilon_i^2 = \varepsilon^\mathrm{T} \varepsilon = (y-X\beta)^\mathrm{T} (y-X\beta)
 $$
 
 which can be developed into

P/slr-corr.md

@@ -52,7 +52,7 @@ $$ \label{eq:slr-ols-sl}
 \hat{\beta}_1 = \frac{s_{xy}}{s_x^2} \; .
 $$
 
-Using the [relationship between covariance and correlation](/D/cov-corr)
+Using the [relationship between covariance and correlation](/P/cov-corr)
 
 $$ \label{eq:cov-corr}
 \mathrm{Cov}(X,Y) = \sigma_X \, \mathrm{Corr}(X,Y) \, \sigma_Y

P/slr-olsmean.md

@@ -81,6 +81,8 @@ $$ \label{eq:sum-ci}
 \end{split}
 $$
 
+and
+
 $$ \label{eq:sum-ci-xi}
 \begin{split}
 \sum_{i=1}^n c_i x_i &= \frac{\sum_{i=1}^n (x_i - \bar{x}) x_i}{\sum_{i=1}^n (x_i - \bar{x})^2} = \frac{\sum_{i=1}^n \left( x_i^2 - \bar{x} x_i \right)}{\sum_{i=1}^n (x_i - \bar{x})^2} \\

P/slr-olsvar.md

@@ -92,7 +92,7 @@ $$ \label{eq:slr-ols-sl}
 \end{split}
 $$
 
-and with \eqref{eq:Var-yi} and \eqref{eq:sum-ci2} as well as [invariance](/P/var-inv), [scaling](/P/var-scal) and [additivity](/P/var-add) of the variance, the variance of $\hat{\beta}_1$ becomes:
+and with \eqref{eq:Var-yi} and \eqref{eq:sum-ci2} as well as [invariance](/P/var-inv), [scaling](/P/var-scal) and [additivity](/P/var-add) of the variance, the variance of $\hat{\beta}_1$ is:
 
 $$ \label{eq:Var-b1}
 \begin{split}

P/slr-vars.md P/slr-resvar.mdP/slr-vars.md renamed to P/slr-resvar.md

@@ -1,87 +1,87 @@
----
-layout: proof
-mathjax: true
-
-author: "Joram Soch"
-affiliation: "BCCN Berlin"
-e_mail: "joram.soch@bccn-berlin.de"
-date: 2021-10-27 14:37:00
-
-title: "Relationship between residual variance and sample variance in simple linear regression"
-chapter: "Statistical Models"
-section: "Univariate normal data"
-topic: "Simple linear regression"
-theorem: "Residual variance in terms of sample variance"
-
-sources:
-  - authors: "Penny, William"
-    year: 2006
-    title: "Relation to correlation"
-    in: "Mathematics for Brain Imaging"
-    pages: "ch. 1.2.3, p. 18, eq. 1.28"
-    url: "https://ueapsylabs.co.uk/sites/wpenny/mbi/mbi_course.pdf"
-  - authors: "Wikipedia"
-    year: 2021
-    title: "Simple linear regression"
-    in: "Wikipedia, the free encyclopedia"
-    pages: "retrieved on 2021-10-27"
-    url: "https://en.wikipedia.org/wiki/Simple_linear_regression#Numerical_properties"
-
-proof_id: "P278"
-shortcut: "slr-vars"
-username: "JoramSoch"
----
-
-
-**Theorem:** Assume a [simple linear regression model](/D/slr) with independent observations
-
-$$ \label{eq:slr}
-y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n
-$$
-
-and consider estimation using [ordinary least squares](/P/slr-ols). Then, [residual variance](/D/resvar) and [sample variance](/D/var-samp) are related to each other via the [correlation coefficient](/D/corr):
-
-$$ \label{eq:slr-vars}
-\hat{\sigma}^2 = \left( 1 - r_{xy}^2 \right) s_y^2 \; .
-$$
-
-
-**Proof:** The [residual variance](/D/resvar) can be expressed in terms of the [residual sum of squares](/D/rss):
-
-$$ \label{eq:slr-res}
-\hat{\sigma}^2 = \frac{1}{n-1} \, \mathrm{RSS}(\hat{\beta}_0,\hat{\beta}_1)
-$$
-
-and [the residual sum of squares for simple linear regression](/P/slr-sss) is
-
-$$ \label{eq:slr-rss}
-\mathrm{RSS}(\hat{\beta}_0,\hat{\beta}_1) = (n-1) \left( s_y^2 - \frac{s_{xy}^2}{s_x^2} \right) \; .
-$$
-
-Combining \eqref{eq:slr-res} and \eqref{eq:slr-rss}, we obtain:
-
-$$ \label{eq:slr-vars-s1}
-\begin{split}
-\hat{\sigma}^2 &= \left( s_y^2 - \frac{s_{xy}^2}{s_x^2} \right) \\
-&= \left( 1 - \frac{s_{xy}^2}{s_x^2 s_y^2} \right) s_y^2 \\
-&= \left( 1 - \left( \frac{s_{xy}}{s_x \, s_y} \right)^2 \right) s_y^2 \; .
-\end{split}
-$$
-
-Using the [relationship between correlation, covariance and standard deviation](/D/corr)
-
-$$ \label{eq:corr-cov-std}
-\mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}}
-$$
-
-which also holds for sample correlation, [sample covariance](/D/cov-samp) and sample [standard deviation](/D/std)
-
-$$ \label{eq:corr-cov-std-samp}
-r_{xy} = \frac{s_{xy}}{s_x \, s_y} \; ,
-$$
-
-we get the final result:
-
-$$ \label{eq:slr-vars-s2}
-\hat{\sigma}^2 = \left( 1 - r_{xy}^2 \right) s_y^2 \; .
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-10-27 14:37:00
+
+title: "Relationship between residual variance and sample variance in simple linear regression"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Simple linear regression"
+theorem: "Residual variance in terms of sample variance"
+
+sources:
+  - authors: "Penny, William"
+    year: 2006
+    title: "Relation to correlation"
+    in: "Mathematics for Brain Imaging"
+    pages: "ch. 1.2.3, p. 18, eq. 1.28"
+    url: "https://ueapsylabs.co.uk/sites/wpenny/mbi/mbi_course.pdf"
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Simple linear regression"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-10-27"
+    url: "https://en.wikipedia.org/wiki/Simple_linear_regression#Numerical_properties"
+
+proof_id: "P278"
+shortcut: "slr-resvar"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Assume a [simple linear regression model](/D/slr) with independent observations
+
+$$ \label{eq:slr}
+y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n
+$$
+
+and consider estimation using [ordinary least squares](/P/slr-ols). Then, [residual variance](/D/resvar) and [sample variance](/D/var-samp) are related to each other via the [correlation coefficient](/D/corr):
+
+$$ \label{eq:slr-vars}
+\hat{\sigma}^2 = \left( 1 - r_{xy}^2 \right) s_y^2 \; .
+$$
+
+
+**Proof:** The [residual variance](/D/resvar) can be expressed in terms of the [residual sum of squares](/D/rss):
+
+$$ \label{eq:slr-res}
+\hat{\sigma}^2 = \frac{1}{n-1} \, \mathrm{RSS}(\hat{\beta}_0,\hat{\beta}_1)
+$$
+
+and [the residual sum of squares for simple linear regression](/P/slr-sss) is
+
+$$ \label{eq:slr-rss}
+\mathrm{RSS}(\hat{\beta}_0,\hat{\beta}_1) = (n-1) \left( s_y^2 - \frac{s_{xy}^2}{s_x^2} \right) \; .
+$$
+
+Combining \eqref{eq:slr-res} and \eqref{eq:slr-rss}, we obtain:
+
+$$ \label{eq:slr-vars-s1}
+\begin{split}
+\hat{\sigma}^2 &= \left( s_y^2 - \frac{s_{xy}^2}{s_x^2} \right) \\
+&= \left( 1 - \frac{s_{xy}^2}{s_x^2 s_y^2} \right) s_y^2 \\
+&= \left( 1 - \left( \frac{s_{xy}}{s_x \, s_y} \right)^2 \right) s_y^2 \; .
+\end{split}
+$$
+
+Using the [relationship between correlation, covariance and standard deviation](/D/corr)
+
+$$ \label{eq:corr-cov-std}
+\mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}}
+$$
+
+which also holds for sample correlation, [sample covariance](/D/cov-samp) and sample [standard deviation](/D/std)
+
+$$ \label{eq:corr-cov-std-samp}
+r_{xy} = \frac{s_{xy}}{s_x \, s_y} \; ,
+$$
+
+we get the final result:
+
+$$ \label{eq:slr-vars-s2}
+\hat{\sigma}^2 = \left( 1 - r_{xy}^2 \right) s_y^2 \; .
 $$

P/slr-rsq.md

@@ -88,7 +88,7 @@ R^2 &= \frac{\sum_{i=1}^{n} (\bar{y} - \hat{\beta}_1 \bar{x} + \hat{\beta}_1 x_i
 \end{split}
 $$
 
-Using the [relationship between correlation coefficient and slope estimate](/D/slr-corr), we conclude:
+Using the [relationship between correlation coefficient and slope estimate](/P/slr-corr), we conclude:
 
 $$ \label{eq:slr-R2-qed}
 R^2 = \left( \frac{s_x}{s_y} \, \hat{\beta}_1 \right)^2 = r_{xy}^2 \; .

P/tglm-para.md

@@ -70,7 +70,7 @@ $$
 The [covariance across rows for the transformed general linear model](/P/tglm-dist) is equal to
 
 $$ \label{eq:U}
-U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}
+U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \; .
 $$
 
 Applying \eqref{eq:U}, \eqref{eq:X-Xt-T} and \eqref{eq:glm2-wls}, the estimates in \eqref{eq:tglm-wls} can be developed into