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CounterExample_For_Least_Square_Model - Copy.tex
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\documentclass[11pt]{article}
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% Some characters . , ; ? ! / are not pygmentized.
% This macro makes them "active" and they will insert potential linebreaks
\newcommand*\Wrappedbreaksatpunct {%
\lccode`\~`\.\lowercase{\def~}{\discretionary{\hbox{\char`\.}}{\Wrappedafterbreak}{\hbox{\char`\.}}}%
\lccode`\~`\,\lowercase{\def~}{\discretionary{\hbox{\char`\,}}{\Wrappedafterbreak}{\hbox{\char`\,}}}%
\lccode`\~`\;\lowercase{\def~}{\discretionary{\hbox{\char`\;}}{\Wrappedafterbreak}{\hbox{\char`\;}}}%
\lccode`\~`\:\lowercase{\def~}{\discretionary{\hbox{\char`\:}}{\Wrappedafterbreak}{\hbox{\char`\:}}}%
\lccode`\~`\?\lowercase{\def~}{\discretionary{\hbox{\char`\?}}{\Wrappedafterbreak}{\hbox{\char`\?}}}%
\lccode`\~`\!\lowercase{\def~}{\discretionary{\hbox{\char`\!}}{\Wrappedafterbreak}{\hbox{\char`\!}}}%
\lccode`\~`\/\lowercase{\def~}{\discretionary{\hbox{\char`\/}}{\Wrappedafterbreak}{\hbox{\char`\/}}}%
\catcode`\.\active
\catcode`\,\active
\catcode`\;\active
\catcode`\:\active
\catcode`\?\active
\catcode`\!\active
\catcode`\/\active
\lccode`\~`\~
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\makeatother
\let\OriginalVerbatim=\Verbatim
\makeatletter
\renewcommand{\Verbatim}[1][1]{%
%\parskip\z@skip
\sbox\Wrappedcontinuationbox {\Wrappedcontinuationsymbol}%
\sbox\Wrappedvisiblespacebox {\FV@SetupFont\Wrappedvisiblespace}%
\def\FancyVerbFormatLine ##1{\hsize\linewidth
\vtop{\raggedright\hyphenpenalty\z@\exhyphenpenalty\z@
\doublehyphendemerits\z@\finalhyphendemerits\z@
\strut ##1\strut}%
}%
% If the linebreak is at a space, the latter will be displayed as visible
% space at end of first line, and a continuation symbol starts next line.
% Stretch/shrink are however usually zero for typewriter font.
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{\kern\fontdimen2\font}%
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% Allow breaks at special characters using \PYG... macros.
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% Breaks at punctuation characters . , ; ? ! and / need catcode=\active
\OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]%
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% Exact colors from NB
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\definecolor{outcolor}{HTML}{D84315}
\definecolor{cellborder}{HTML}{CFCFCF}
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% prompt
\makeatletter
\newcommand{\boxspacing}{\kern\kvtcb@left@rule\kern\kvtcb@boxsep}
\makeatother
\newcommand{\prompt}[4]{
{\ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip}}
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% Prevent overflowing lines due to hard-to-break entities
\sloppy
% Setup hyperref package
\hypersetup{
breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=urlcolor,
linkcolor=linkcolor,
citecolor=citecolor,
}
% Slightly bigger margins than the latex defaults
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\begin{document}
\maketitle
\subsection{Is it possible to indicate that in a least square
model:}\label{is-it-possible-to-indicate-that-in-a-least-square-model}
\begin{equation}
y = \beta_0 + \sum_{i = 1}^{p-1}\beta_i \implies \widehat{\beta_i} \perp\!\!\!\!\perp \widehat{\beta_j} \iff \sum_{k = 1}^n x_{i, k} x_{j, k} = 0, \quad \text{and} ~~i \ne j.
\end{equation}
\subsection{We know that in a least square model the vector of
coefficients estimated by below
relation:}\label{we-know-that-in-a-least-square-model-the-vector-of-coefficients-estimated-by-below-relation}
\begin{align*}
& \widehat{\beta} = (X'X)^{-1} X' y \implies \text{Var}(\widehat{\beta}) = \sigma^2 \times (X'X)^{-1} \implies \\
& \widehat{\beta} \sim \mathcal{N}_p (\beta, ~~\sigma^2 (X'X)^{-1}) \implies \\
& \text{if}\quad \mathbb{V} = (X'X)^{-1} \implies \mathbb{Cov}(\widehat{\beta_i}, \widehat{\beta_j}) = \sigma^2 \times \mathbb{V}[i, j] \implies \\
& \widehat{\beta_i} \perp\!\!\!\!\perp \widehat{\beta_j} \iff \mathbb{V}[i+1, j+1] = 0
\end{align*}
\subsection{I give an Counterexample for the relationship
(1)!}\label{i-give-an-counterexample-for-the-relationship-1}
let
\begin{align*}
& x_1 = \begin{bmatrix}
1 \\ 2 \\ 3 \\ 4 \\ 5
\end{bmatrix} \quad x_2 = \begin{bmatrix}
2 \\ 4 \\ 7 \\ 1 \\ -7
\end{bmatrix} \implies \\
& X = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 4 \\ 1 & 3 & 7 \\ 4 & 1 & 1 \\ 1 & 5 & -7
\end{bmatrix} \\
& \implies n = 5, ~~ p = 3, \\
& X' X = \begin{bmatrix}
5 & 15 & 7 \\
15 & 55 & 0 \\
7 & 0 & 119
\end{bmatrix} \implies \sum_{k = 1}^5 x_{1, k} x_{2, k} = 0 \\
& \mathbb{V} = (X'X)^{-1} = \begin{bmatrix}
\frac{187}{93} & -\frac{17}{31} & -\frac{11}{93} \\
-\frac{17}{31} & \frac{26}{155} & \frac{1}{31} \\
-\frac{11}{93} & \frac{1}{31} & \frac{10}{651}
\end{bmatrix} \implies \\
& \mathbb{Cov}(\widehat{\beta_1}, ~~\widehat{\beta_2}) = \sigma^2 \times \mathbb{V}[2, 3] = \sigma^2 \times \frac{1}{31} \ne 0
\end{align*}
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\subsection{Using Python}\label{using-python}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{o}{*}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{13}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{x1} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{]}
\PY{n}{x2} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{7}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{l+m+mi}{7}\PY{p}{]}
\PY{n}{X} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{7}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{l+m+mi}{7}\PY{p}{]}\PY{p}{)}\PY{o}{.}\PY{n}{reshape}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{)}\PY{o}{.}\PY{n}{T}\PY{p}{)}
\PY{n}{X}
\end{Verbatim}
\end{tcolorbox}
\prompt{Out}{outcolor}{13}{}
$\displaystyle \left[\begin{matrix}1 & 1 & 2\\1 & 2 & 4\\1 & 3 & 7\\1 & 4 & 1\\1 & 5 & -7\end{matrix}\right]$
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{14}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{XX} \PY{o}{=} \PY{n}{X}\PY{o}{.}\PY{n}{T} \PY{o}{@} \PY{n}{X}
\PY{n}{XX}
\end{Verbatim}
\end{tcolorbox}
\prompt{Out}{outcolor}{14}{}
$\displaystyle \left[\begin{matrix}5 & 15 & 7\\15 & 55 & 0\\7 & 0 & 119\end{matrix}\right]$
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{15}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{xxinverse} \PY{o}{=} \PY{n}{XX}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}
\PY{n}{xxinverse}
\end{Verbatim}
\end{tcolorbox}
\prompt{Out}{outcolor}{15}{}
$\displaystyle \left[\begin{matrix}\frac{187}{93} & - \frac{17}{31} & - \frac{11}{93}\\- \frac{17}{31} & \frac{26}{155} & \frac{1}{31}\\- \frac{11}{93} & \frac{1}{31} & \frac{10}{651}\end{matrix}\right]$
% Add a bibliography block to the postdoc
\end{document}