corrected some pages

In equations, "\left[ 1_n, \, x \right]" was corrected to "\left[ \begin{matrix} 1_n & x \end{matrix} \right]".

Author: JoramSoch

P/slr-mat.md

@@ -37,7 +37,7 @@ where $1_n$ is an $n \times 1$ vector of ones, $x$ is the $n \times 1$ single pr
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,
 $$
 
 such that the simple linear regression model can also be written as
@@ -64,7 +64,7 @@ which is a $2 \times n$ matrix and can be reformulated as follows:
 $$ \label{eq:E-qed}
 \begin{split}
 E &= (X^\mathrm{T} X)^{-1} X^\mathrm{T} \\
-&= \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ 1_n, \, x \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
+&= \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ \begin{matrix} 1_n & x \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
 &= \left( \left[ \begin{matrix} n & n\bar{x} \\ n\bar{x} & x^\mathrm{T} x \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
 &= \frac{1}{n x^\mathrm{T} x - (n\bar{x})^2} \left[ \begin{matrix} x^\mathrm{T} x & -n\bar{x} \\ -n\bar{x} & n \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
 &= \frac{1}{x^\mathrm{T} x - n\bar{x}^2} \left[ \begin{matrix} x^\mathrm{T} x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
@@ -83,7 +83,7 @@ which is an $n \times n$ matrix and can be reformulated as follows:
 
 $$ \label{eq:P-qed}
 \begin{split}
-P &= X \, E = \left[ 1_n, \, x \right] \left[ \begin{matrix} e_1 \\ e_2 \end{matrix} \right] \\
+P &= X \, E = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \left[ \begin{matrix} e_1 \\ e_2 \end{matrix} \right] \\
 &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \end{matrix} \right] \left[ \begin{matrix} (x^\mathrm{T} x/n) - \bar{x} x_1 & \cdots & (x^\mathrm{T} x/n) - \bar{x} x_n \\ -\bar{x} + x_1 & \cdots & -\bar{x} + x_n \end{matrix} \right] \\
 &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \; .
 \end{split}

P/slr-mle2.md

@@ -43,7 +43,7 @@ where $\bar{x}$ and $\bar{y}$ are the [sample means](/D/mean-samp), $s_x^2$ is t
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
 and [weighted least sqaures estimates](/P/mlr-mle) are given by

P/slr-mlr.md

@@ -24,7 +24,7 @@ username: "JoramSoch"
 **Theorem:** [Simple linear regression](/D/slr) is a special case of [multiple linear regression](/D/mlr) with design matrix $X$ and regression coefficients $\beta$
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
 where $1_n$ is an $n \times 1$ vector of ones, $x$ is the $n \times 1$ single predictor variable, $\beta_0$ is the intercept and $\beta_1$ is the slope of the [regression line](/D/regline).
@@ -41,14 +41,14 @@ In matrix notation and using the [multivariate normal distribution](/D/mvn), thi
 $$ \label{eq:slr-mlr-s1}
 \begin{split}
 y &= \beta_0 1_n + \beta_1 x + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, I_n) \\
-y &= \left[ 1_n, \, x \right] \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, I_n) \; .
+y &= \left[ \begin{matrix} 1_n & x \end{matrix} \right] \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, I_n) \; .
 \end{split}
 $$
 
 Comparing with the [multiple linear regression equations for uncorrelated errors](/D/mlr), we finally note:
 
 $$ \label{eq:slr-mlr-s3}
-y = X\beta + \varepsilon \quad \text{with} \quad X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; .
+y = X\beta + \varepsilon \quad \text{with} \quad X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; .
 $$
 
 In the [case of correlated observations](/D/slr), the [error distribution changes to](/D/mlr):

P/slr-ols2.md

@@ -42,7 +42,7 @@ where $\bar{x}$ and $\bar{y}$ are the [sample means](/D/mean-samp), $s_x^2$ is t
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
 and [ordinary least sqaures estimates](/P/mlr-ols) are given by

P/slr-olsdist.md

@@ -45,7 +45,7 @@ where $\bar{x}$ is the [sample mean](/D/mean-samp) and $s_x^2$ is the [sample va
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] \; ,
 $$
 
 such that \eqref{eq:slr} can also be written as
@@ -79,7 +79,7 @@ Applying \eqref{eq:slr-mlr}, the [covariance matrix](/D/mvn) can be further deve
 
 $$ \label{eq:b-est-cov}
 \begin{split}
-\sigma^2 (X^\mathrm{T} X)^{-1} &= \sigma^2 \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ 1_n, \, x \right] \right)^{-1} \\
+\sigma^2 (X^\mathrm{T} X)^{-1} &= \sigma^2 \left( \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \left[ \begin{matrix} 1_n & x \end{matrix} \right] \right)^{-1} \\
 &= \sigma^2 \left( \left[ \begin{matrix} n & n\bar{x} \\ n\bar{x} & x^\mathrm{T} x \end{matrix} \right] \right)^{-1} \\
 &= \frac{\sigma^2}{n x^\mathrm{T} x - (n\bar{x})^2} \left[ \begin{matrix} x^\mathrm{T} x & -n\bar{x} \\ -n\bar{x} & n \end{matrix} \right] \\
 &= \frac{\sigma^2}{x^\mathrm{T} x - n\bar{x}^2} \left[ \begin{matrix} x^\mathrm{T} x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \; .

P/slr-wls2.md

@@ -42,7 +42,7 @@ where $1_n$ is an $n \times 1$ vector of ones.
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with
 
 $$ \label{eq:slr-mlr}
-X = \left[ 1_n, \, x \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
+X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
 and [weighted least sqaures estimates](/P/mlr-wls) are given by