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ulem is needed to support strikethroughs (\sout) + % normalem makes italics be italics, not underlines + \usepackage{soul} % strikethrough (\st) support for pandoc >= 3.0.0 + \usepackage{mathrsfs} + + + + % Colors for the hyperref package + \definecolor{urlcolor}{rgb}{0,.145,.698} + \definecolor{linkcolor}{rgb}{.71,0.21,0.01} + \definecolor{citecolor}{rgb}{.12,.54,.11} + + % ANSI colors + \definecolor{ansi-black}{HTML}{3E424D} + \definecolor{ansi-black-intense}{HTML}{282C36} + \definecolor{ansi-red}{HTML}{E75C58} + \definecolor{ansi-red-intense}{HTML}{B22B31} + \definecolor{ansi-green}{HTML}{00A250} + \definecolor{ansi-green-intense}{HTML}{007427} + \definecolor{ansi-yellow}{HTML}{DDB62B} + \definecolor{ansi-yellow-intense}{HTML}{B27D12} + \definecolor{ansi-blue}{HTML}{208FFB} + \definecolor{ansi-blue-intense}{HTML}{0065CA} + \definecolor{ansi-magenta}{HTML}{D160C4} + \definecolor{ansi-magenta-intense}{HTML}{A03196} + \definecolor{ansi-cyan}{HTML}{60C6C8} + \definecolor{ansi-cyan-intense}{HTML}{258F8F} + \definecolor{ansi-white}{HTML}{C5C1B4} + \definecolor{ansi-white-intense}{HTML}{A1A6B2} + \definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF} + \definecolor{ansi-default-inverse-bg}{HTML}{000000} + + % common color for the border for error outputs. + \definecolor{outerrorbackground}{HTML}{FFDFDF} + + % commands and environments needed by pandoc snippets + % extracted from the output of `pandoc -s` + \providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} + \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} + % Add ',fontsize=\small' for more characters per line + \newenvironment{Shaded}{}{} + \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}} + \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}} + \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}} + \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}} + \newcommand{\RegionMarkerTok}[1]{{#1}} + \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\NormalTok}[1]{{#1}} + + % Additional commands for more recent versions of Pandoc + \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} + \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}} + \newcommand{\ImportTok}[1]{{#1}} + \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}} + \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}} + \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}} + \newcommand{\BuiltInTok}[1]{{#1}} + \newcommand{\ExtensionTok}[1]{{#1}} + \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}} + \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}} + \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \makeatletter + \newsavebox\pandoc@box + \newcommand*\pandocbounded[1]{% + \sbox\pandoc@box{#1}% + % scaling factors for width and height + \Gscale@div\@tempa\textheight{\dimexpr\ht\pandoc@box+\dp\pandoc@box\relax}% + \Gscale@div\@tempb\linewidth{\wd\pandoc@box}% + % select the smaller of both + \ifdim\@tempb\p@<\@tempa\p@ + \let\@tempa\@tempb + \fi + % scaling accordingly (\@tempa < 1) + \ifdim\@tempa\p@<\p@ + \scalebox{\@tempa}{\usebox\pandoc@box}% + % scaling not needed, use as it is + \else + \usebox{\pandoc@box}% + \fi + } + \makeatother + + % Define a nice break command that doesn't care if a line doesn't already + % exist. + \def\br{\hspace*{\fill} \\* } + % Math Jax compatibility definitions + \def\gt{>} + \def\lt{<} + \let\Oldtex\TeX + \let\Oldlatex\LaTeX + \renewcommand{\TeX}{\textrm{\Oldtex}} + \renewcommand{\LaTeX}{\textrm{\Oldlatex}} + % Document parameters + % Document title + \title{Project1} + + + + + + + +% Pygments definitions +\makeatletter +\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax% + \let\PY@ul=\relax \let\PY@tc=\relax% + \let\PY@bc=\relax \let\PY@ff=\relax} +\def\PY@tok#1{\csname PY@tok@#1\endcsname} +\def\PY@toks#1+{\ifx\relax#1\empty\else% + \PY@tok{#1}\expandafter\PY@toks\fi} +\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{% + \PY@it{\PY@bf{\PY@ff{#1}}}}}}} +\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}} + +\@namedef{PY@tok@w}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}} +\@namedef{PY@tok@c}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cp}{\def\PY@tc##1{\textcolor[rgb]{0.61,0.40,0.00}{##1}}} +\@namedef{PY@tok@k}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kp}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kt}{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}} +\@namedef{PY@tok@o}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@ow}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\@namedef{PY@tok@nb}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@nf}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@nc}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@nn}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@ne}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.80,0.25,0.22}{##1}}} +\@namedef{PY@tok@nv}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@no}{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}} +\@namedef{PY@tok@nl}{\def\PY@tc##1{\textcolor[rgb]{0.46,0.46,0.00}{##1}}} +\@namedef{PY@tok@ni}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.44,0.44,0.44}{##1}}} +\@namedef{PY@tok@na}{\def\PY@tc##1{\textcolor[rgb]{0.41,0.47,0.13}{##1}}} +\@namedef{PY@tok@nt}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@nd}{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\@namedef{PY@tok@s}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sd}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@si}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.64,0.35,0.47}{##1}}} +\@namedef{PY@tok@se}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.36,0.12}{##1}}} +\@namedef{PY@tok@sr}{\def\PY@tc##1{\textcolor[rgb]{0.64,0.35,0.47}{##1}}} +\@namedef{PY@tok@ss}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@sx}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@m}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@gh}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\@namedef{PY@tok@gu}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}} +\@namedef{PY@tok@gd}{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}} +\@namedef{PY@tok@gi}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.52,0.00}{##1}}} +\@namedef{PY@tok@gr}{\def\PY@tc##1{\textcolor[rgb]{0.89,0.00,0.00}{##1}}} +\@namedef{PY@tok@ge}{\let\PY@it=\textit} +\@namedef{PY@tok@gs}{\let\PY@bf=\textbf} +\@namedef{PY@tok@ges}{\let\PY@bf=\textbf\let\PY@it=\textit} +\@namedef{PY@tok@gp}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\@namedef{PY@tok@go}{\def\PY@tc##1{\textcolor[rgb]{0.44,0.44,0.44}{##1}}} +\@namedef{PY@tok@gt}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} +\@namedef{PY@tok@err}{\def\PY@bc##1{{\setlength{\fboxsep}{\string -\fboxrule}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}} +\@namedef{PY@tok@kc}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kd}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kn}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kr}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@bp}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@fm}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@vc}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vg}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vi}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vm}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@sa}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sb}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sc}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@dl}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@s2}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sh}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@s1}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@mb}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mf}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mh}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mi}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@il}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mo}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@ch}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cm}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cpf}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@c1}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cs}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} + +\def\PYZbs{\char`\\} +\def\PYZus{\char`\_} +\def\PYZob{\char`\{} +\def\PYZcb{\char`\}} +\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % For linebreaks inside Verbatim environment from package fancyvrb. + \makeatletter + \newbox\Wrappedcontinuationbox + \newbox\Wrappedvisiblespacebox + \newcommand*\Wrappedvisiblespace {\textcolor{red}{\textvisiblespace}} + \newcommand*\Wrappedcontinuationsymbol {\textcolor{red}{\llap{\tiny$\m@th\hookrightarrow$}}} + \newcommand*\Wrappedcontinuationindent {3ex } + \newcommand*\Wrappedafterbreak {\kern\Wrappedcontinuationindent\copy\Wrappedcontinuationbox} + % Take advantage of the already applied Pygments mark-up to insert + % potential linebreaks for TeX processing. + % {, <, #, %, $, ' and ": go to next line. + % _, }, ^, &, >, - and ~: stay at end of broken line. + % Use of \textquotesingle for straight quote. + \newcommand*\Wrappedbreaksatspecials {% + \def\PYGZus{\discretionary{\char`\_}{\Wrappedafterbreak}{\char`\_}}% + \def\PYGZob{\discretionary{}{\Wrappedafterbreak\char`\{}{\char`\{}}% + \def\PYGZcb{\discretionary{\char`\}}{\Wrappedafterbreak}{\char`\}}}% + \def\PYGZca{\discretionary{\char`\^}{\Wrappedafterbreak}{\char`\^}}% + \def\PYGZam{\discretionary{\char`\&}{\Wrappedafterbreak}{\char`\&}}% + \def\PYGZlt{\discretionary{}{\Wrappedafterbreak\char`\<}{\char`\<}}% + \def\PYGZgt{\discretionary{\char`\>}{\Wrappedafterbreak}{\char`\>}}% + \def\PYGZsh{\discretionary{}{\Wrappedafterbreak\char`\#}{\char`\#}}% + \def\PYGZpc{\discretionary{}{\Wrappedafterbreak\char`\%}{\char`\%}}% + \def\PYGZdl{\discretionary{}{\Wrappedafterbreak\char`\$}{\char`\$}}% + \def\PYGZhy{\discretionary{\char`\-}{\Wrappedafterbreak}{\char`\-}}% + \def\PYGZsq{\discretionary{}{\Wrappedafterbreak\textquotesingle}{\textquotesingle}}% + \def\PYGZdq{\discretionary{}{\Wrappedafterbreak\char`\"}{\char`\"}}% + \def\PYGZti{\discretionary{\char`\~}{\Wrappedafterbreak}{\char`\~}}% + } + % 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`\~`\~ + } + \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. + \def\FV@Space {% + \nobreak\hskip\z@ plus\fontdimen3\font minus\fontdimen4\font + \discretionary{\copy\Wrappedvisiblespacebox}{\Wrappedafterbreak} + {\kern\fontdimen2\font}% + }% + + % Allow breaks at special characters using \PYG... macros. + \Wrappedbreaksatspecials + % Breaks at punctuation characters . , ; ? ! and / need catcode=\active + \OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]% + } + \makeatother + + % Exact colors from NB + \definecolor{incolor}{HTML}{303F9F} + \definecolor{outcolor}{HTML}{D84315} + \definecolor{cellborder}{HTML}{CFCFCF} + \definecolor{cellbackground}{HTML}{F7F7F7} + + % 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}} + } + + + + % 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 + + + + + + + \hypertarget{project-1-on-machine-learning-deadline-october-6-midnight-2025}{% +\section*{Project 1 on Machine Learning, deadline October 6 (midnight), +2025}\label{project-1-on-machine-learning-deadline-october-6-midnight-2025}} + +\textbf{Data Analysis and Machine Learning FYS-STK3155/FYS4155}, +University of Oslo, Norway + +Date: \textbf{September 2} + + \hypertarget{preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools}{% +\subsection*{Preamble: Note on writing reports, using reference material, +AI and other +tools}\label{preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools}} + +We want you to answer the three different projects by handing in reports +written like a standard scientific/technical report. The links at +\url{https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects} +contain more information. There you can find examples of previous +reports, the projects themselves, how we rade reports etc. How to write +reports will also be discussed during the various lab sessions. Please +do ask us if you are in doubt. + +When using codes and material from other sources, you should refer to +these in the bibliography of your report, indicating wherefrom you for +example got the code, whether this is from the lecture notes, softwares +like Scikit-Learn, TensorFlow, PyTorch or other sources. These sources +should always be cited correctly. How to cite some of the libraries is +often indicated from their corresponding GitHub sites or websites, see +for example how to cite Scikit-Learn at +\url{https://scikit-learn.org/dev/about.html}. + +We enocurage you to use tools like +\href{https://openai.com/chatgpt/}{ChatGPT} or similar in writing the +report. If you use for example ChatGPT, please do cite it properly and +include (if possible) your questions and answers as an addition to the +report. This can be uploaded to for example your website, GitHub/GitLab +or similar as supplemental material. + +If you would like to study other data sets, feel free to propose other +sets. What we have proposed here are mere suggestions from our side. If +you opt for another data set, consider using a set which has been +studied in the scientific literature. This makes it easier for you to +compare and analyze your results. Comparing with existing results from +the scientific literature is also an essential element of the scientific +discussion. The University of California at Irvine with its Machine +Learning repository at \url{https://archive.ics.uci.edu/ml/index.php} is +an excellent site to look up for examples and inspiration. +\href{https://www.kaggle.com/}{Kaggle.com} is an equally interesting +site. Feel free to explore these sites. When selecting other data sets, +make sure these are sets used for regression problems (not +classification). + + \hypertarget{regression-analysis-and-resampling-methods}{% +\subsection*{Regression analysis and resampling +methods}\label{regression-analysis-and-resampling-methods}} + +The main aim of this project is to study in more detail various +regression methods, including Ordinary Least Squares (OLS) reegression, +Ridge regression and LASSO regression. In addition to the scientific +part, in this course we want also to give you an experience in writing +scientific reports. + +We will study how to fit polynomials to specific one-dimensional +functions (feel free to replace the suggested function with more +complicated ones). + +We will use Runge's function (see +\url{https://en.wikipedia.org/wiki/Runge\%27s_phenomenon} for a +discussion). The one-dimensional function we will study is + + \[ +f(x) = \frac{1}{1+25x^2}. +\] + + Our first step will be to perform an OLS regression analysis of this +function, trying out a polynomial fit with an \(x\) dependence of the +form \([x,x^2,\dots]\). You can use a uniform distribution to set up the +arrays of values for \(x \in [-1,1]\), or alternatively use a fixed step +size. Thereafter we will repeat many of the same steps when using the +Ridge and Lasso regression methods, introducing thereby a dependence on +the hyperparameter (penalty) \(\lambda\). + +We will also include bootstrap as a resampling technique in order to +study the so-called \textbf{bias-variance tradeoff}. After that we will +include the so-called cross-validation technique. + + \hypertarget{part-a-ordinary-least-square-ols-for-the-runge-function}{% +\subsubsection*{Part a : Ordinary Least Square (OLS) for the Runge +function}\label{part-a-ordinary-least-square-ols-for-the-runge-function}} + +We will generate our own dataset for abovementioned function +\(\mathrm{Runge}(x)\) function with \(x\in [-1,1]\). You should explore +also the addition of an added stochastic noise to this function using +the normal distribution \(N(0,1)\). + +\emph{Write your own code} (using for example the pseudoinverse function +\textbf{pinv} from \textbf{Numpy} ) and perform a standard +\textbf{ordinary least square regression} analysis using polynomials in +\(x\) up to order \(15\) or higher. Explore the dependence on the number +of data points and the polynomial degree. + +Evaluate the mean Squared error (MSE) + + \[ +MSE(\boldsymbol{y},\tilde{\boldsymbol{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\] + + and the \(R^2\) score function. If \(\tilde{\boldsymbol{y}}_i\) is the +predicted value of the \(i-th\) sample and \(y_i\) is the corresponding +true value, then the score \(R^2\) is defined as + + \[ +R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +\] + + where we have defined the mean value of \(\boldsymbol{y}\) as + + \[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\] + + Plot the resulting scores (MSE and R\(^2\)) as functions of the +polynomial degree (here up to polymial degree 15). Plot also the +parameters \(\theta\) as you increase the order of the polynomial. +Comment your results. + +Your code has to include a scaling/centering of the data (for example by +subtracting the mean value), and a split of the data in training and +test data. For the scaling you can either write your own code or use for +example the function for splitting training data provided by the library +\textbf{Scikit-Learn} (make sure you have installed it). This function +is called \(train\_test\_split\). \textbf{You should present a critical +discussion of why and how you have scaled or not scaled the data}. + +It is normal in essentially all Machine Learning studies to split the +data in a training set and a test set (eventually also an additional +validation set). There is no explicit recipe for how much data should be +included as training data and say test data. An accepted rule of thumb +is to use approximately \(2/3\) to \(4/5\) of the data as training data. + +You can easily reuse the solutions to your exercises from week 35. See +also the lecture slides from week 35 and week 36. + +On scaling, we recommend reading the following section from the +scikit-learn software description, see +\url{https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html\#plot-all-scaling-standard-scaler-section}. + + \hypertarget{part-b-adding-ridge-regression-for-the-runge-function}{% +\subsubsection*{Part b: Adding Ridge regression for the Runge +function}\label{part-b-adding-ridge-regression-for-the-runge-function}} + +Write your own code for the Ridge method as done in the previous +exercise. The lecture notes from week 35 and 36 contain more +information. Furthermore, the results from the exercise set from week 36 +is something you can reuse here. + +Perform the same analysis as you did in the previous exercise but now +for different values of \(\lambda\). Compare and analyze your results +with those obtained in part a) with the OLS method. Study the dependence +on \(\lambda\). + + \hypertarget{part-c-writing-your-own-gradient-descent-code}{% +\subsubsection*{Part c: Writing your own gradient descent +code}\label{part-c-writing-your-own-gradient-descent-code}} + +Replace now the analytical expressions for the optimal parameters +\(\boldsymbol{\theta}\) with your own gradient descent code. In this +exercise we focus only on the simplest gradient descent approach with a +fixed learning rate (see the exercises from week 37 and the lecture +notes from week 36). + +Study and compare your results from parts a) and b) with your gradient +descent approch. Discuss in particular the role of the learning rate. + + \hypertarget{part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate}{% +\subsubsection*{Part d: Including momentum and more advanced ways to +update the learning the +rate}\label{part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate}} + +We keep our focus on OLS and Ridge regression and update our code for +the gradient descent method by including \textbf{momentum}, +\textbf{ADAgrad}, \textbf{RMSprop} and \textbf{ADAM} as methods fro +iteratively updating your learning rate. Discuss the results and compare +the different methods applied to the one-dimensional Runge function. The +lecture notes from week 37 contain several examples on how to implement +these methods. + + \hypertarget{part-e-writing-our-own-code-for-lasso-regression}{% +\subsubsection*{Part e: Writing our own code for Lasso +regression}\label{part-e-writing-our-own-code-for-lasso-regression}} + +LASSO regression (see lecture slides from week 36 and week 37) +represents our first encounter with a machine learning method which +cannot be solved through analytical expressions (as in OLS and Ridge +regression). Use the gradient descent methods you developed in parts c) +and d) to solve the LASSO optimization problem. You can compare your +results with the functionalities of \textbf{Scikit-Learn}. + +Discuss (critically) your results for the Runge function from OLS, Ridge +and LASSO regression using the various gradient descent approaches. + + \hypertarget{part-f-stochastic-gradient-descent}{% +\subsubsection*{Part f: Stochastic gradient +descent}\label{part-f-stochastic-gradient-descent}} + +Our last gradient step is to include stochastic gradient descent using +the same methods to update the learning rates as in parts c-e). Compare +and discuss your results with and without stochastic gradient and give a +critical assessment of the various methods. + + \hypertarget{part-g-bias-variance-trade-off-and-resampling-techniques}{% +\subsubsection*{Part g: Bias-variance trade-off and resampling +techniques}\label{part-g-bias-variance-trade-off-and-resampling-techniques}} + +Our aim here is to study the bias-variance trade-off by implementing the +\textbf{bootstrap} resampling technique. \textbf{We will only use the +simpler ordinary least squares here}. + +With a code which does OLS and includes resampling techniques, we will +now discuss the bias-variance trade-off in the context of continuous +predictions such as regression. However, many of the intuitions and +ideas discussed here also carry over to classification tasks and +basically all Machine Learning algorithms. + +Before you perform an analysis of the bias-variance trade-off on your +test data, make first a figure similar to Fig. 2.11 of Hastie, +Tibshirani, and Friedman. Figure 2.11 of this reference displays only +the test and training MSEs. The test MSE can be used to indicate +possible regions of low/high bias and variance. You will most likely not +get an equally smooth curve! You may also need to increase the +polynomial order and play around with the number of data points as well +(see also the exercise set from week 35). + +With this result we move on to the bias-variance trade-off analysis. + +Consider a dataset \(\mathcal{L}\) consisting of the data +\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\). + +We assume that the true data is generated from a noisy model + + \[ +\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}. +\] + + Here \(\epsilon\) is normally distributed with mean zero and standard +deviation \(\sigma^2\). + +In our derivation of the ordinary least squares method we defined then +an approximation to the function \(f\) in terms of the parameters +\(\boldsymbol{\theta}\) and the design matrix \(\boldsymbol{X}\) which +embody our model, that is +\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\theta}\). + +The parameters \(\boldsymbol{\theta}\) are in turn found by optimizing +the mean squared error via the so-called cost function + + \[ +C(\boldsymbol{X},\boldsymbol{\theta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. +\] + + Here the expected value \(\mathbb{E}\) is the sample value. + +Show that you can rewrite this in terms of a term which contains the +variance of the model itself (the so-called variance term), a term which +measures the deviation from the true data and the mean value of the +model (the bias term) and finally the variance of the noise. + +That is, show that + + \[ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2, +\] + + with (we approximate \(f(\boldsymbol{x})\approx \boldsymbol{y}\)) + + \[ +\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right], +\] + + and + + \[ +\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. +\] + + \textbf{Important note}: Since the function \(f(x)\) is unknown, in +order to be able to evalute the bias, we replace \(f(\boldsymbol{x})\) +in the expression for the bias with \(\boldsymbol{y}\). + +The answer to this exercise should be included in the theory part of the +report. This exercise is also part of the weekly exercises of week 38. +Explain what the terms mean and discuss their interpretations. + +Perform then a bias-variance analysis of the Runge function by studying +the MSE value as function of the complexity of your model. + +Discuss the bias and variance trade-off as function of your model +complexity (the degree of the polynomial) and the number of data points, +and possibly also your training and test data using the +\textbf{bootstrap} resampling method. You can follow the code example in +the jupyter-book at +\url{https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html\#the-bias-variance-tradeoff}. + + \hypertarget{part-h-cross-validation-as-resampling-techniques-adding-more-complexity}{% +\subsubsection*{Part h): Cross-validation as resampling techniques, +adding more +complexity}\label{part-h-cross-validation-as-resampling-techniques-adding-more-complexity}} + +The aim here is to implement another widely popular resampling +technique, the so-called cross-validation method. + +Implement the \(k\)-fold cross-validation algorithm (feel free to use +the functionality of \textbf{Scikit-Learn} or write your own code) and +evaluate again the MSE function resulting from the test folds. + +Compare the MSE you get from your cross-validation code with the one you +got from your \textbf{bootstrap} code from the previous exercise. +Comment and interpret your results. + +In addition to using the ordinary least squares method, you should +include both Ridge and Lasso regression in the final analysis. + + \hypertarget{background-literature}{% +\subsection*{Background literature}\label{background-literature}} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\item + For a discussion and derivation of the variances and mean squared + errors using linear regression, see the + \href{https://arxiv.org/abs/1509.09169}{Lecture notes on ridge + regression by Wessel N. van Wieringen} +\item + The textbook of + \href{https://www.springer.com/gp/book/9780387848570}{Trevor Hastie, + Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical + Learning, Springer}, chapters 3 and 7 are the most relevant ones for + the analysis of parts g) and h). +\end{enumerate} + + \hypertarget{introduction-to-numerical-projects}{% +\subsection*{Introduction to numerical +projects}\label{introduction-to-numerical-projects}} + +Here follows a brief recipe and recommendation on how to answer the +various questions when preparing your answers. + +\begin{itemize} +\item + Give a short description of the nature of the problem and the eventual + numerical methods you have used. +\item + Describe the algorithm you have used and/or developed. Here you may + find it convenient to use pseudocoding. In many cases you can describe + the algorithm in the program itself. +\item + Include the source code of your program. Comment your program + properly. You should have the code at your GitHub/GitLab link. You can + also place the code in an appendix of your report. +\item + If possible, try to find analytic solutions, or known limits in order + to test your program when developing the code. +\item + Include your results either in figure form or in a table. Remember to + label your results. All tables and figures should have relevant + captions and labels on the axes. +\item + Try to evaluate the reliabilty and numerical stability/precision of + your results. If possible, include a qualitative and/or quantitative + discussion of the numerical stability, eventual loss of precision etc. +\item + Try to give an interpretation of you results in your answers to the + problems. +\item + Critique: if possible include your comments and reflections about the + exercise, whether you felt you learnt something, ideas for + improvements and other thoughts you've made when solving the exercise. + We wish to keep this course at the interactive level and your comments + can help us improve it. +\item + Try to establish a practice where you log your work at the + computerlab. You may find such a logbook very handy at later stages in + your work, especially when you don't properly remember what a previous + test version of your program did. Here you could also record the time + spent on solving the exercise, various algorithms you may have tested + or other topics which you feel worthy of mentioning. +\end{itemize} + + \hypertarget{format-for-electronic-delivery-of-report-and-programs}{% +\subsection*{Format for electronic delivery of report and +programs}\label{format-for-electronic-delivery-of-report-and-programs}} + +The preferred format for the report is a PDF file. You can also use DOC +or postscript formats or as an ipython notebook file. As programming +language we prefer that you choose between C/C++, Fortran2008, Julia or +Python. The following prescription should be followed when preparing the +report: + +\begin{itemize} +\item + Use Canvas to hand in your projects, log in at + \url{https://www.uio.no/english/services/it/education/canvas/} with + your normal UiO username and password. +\item + Upload \textbf{only} the report file or the link to your GitHub/GitLab + or similar typo of repos! For the source code file(s) you have + developed please provide us with your link to your GitHub/GitLab or + similar domain. The report file should include all of your discussions + and a list of the codes you have developed. Do not include library + files which are available at the course homepage, unless you have made + specific changes to them. +\item + In your GitHub/GitLab or similar repository, please include a folder + which contains selected results. These can be in the form of output + from your code for a selected set of runs and input parameters. +\end{itemize} + +Finally, we encourage you to collaborate. Optimal working groups consist +of 2-3 students. You can then hand in a common report. + + \hypertarget{software-and-needed-installations}{% +\subsection*{Software and needed +installations}\label{software-and-needed-installations}} + +If you have Python installed (we recommend Python3) and you feel pretty +familiar with installing different packages, we recommend that you +install the following Python packages via \textbf{pip} as 1. pip install +numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas +pillow + +For Python3, replace \textbf{pip} with \textbf{pip3}. + +See below for a discussion of \textbf{tensorflow} and +\textbf{scikit-learn}. + +For OSX users we recommend also, after having installed Xcode, to +install \textbf{brew}. Brew allows for a seamless installation of +additional software via for example 1. brew install python3 + +For Linux users, with its variety of distributions like for example the +widely popular Ubuntu distribution you can use \textbf{pip} as well and +simply install Python as 1. sudo apt-get install python3 (or python for +python2.7) + +etc etc. + +If you don't want to install various Python packages with their +dependencies separately, we recommend two widely used distrubutions +which set up all relevant dependencies for Python, namely 1. +\href{https://docs.anaconda.com/}{Anaconda} Anaconda is an open source +distribution of the Python and R programming languages for large-scale +data processing, predictive analytics, and scientific computing, that +aims to simplify package management and deployment. Package versions are +managed by the package management system \textbf{conda} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\setcounter{enumi}{1} +\tightlist +\item + \href{https://www.enthought.com/product/canopy/}{Enthought canopy} is + a Python distribution for scientific and analytic computing + distribution and analysis environment, available for free and under a + commercial license. +\end{enumerate} + +Popular software packages written in Python for ML are + +\begin{itemize} +\item + \href{http://scikit-learn.org/stable/}{Scikit-learn}, +\item + \href{https://www.tensorflow.org/}{Tensorflow}, +\item + \href{http://pytorch.org/}{PyTorch} and +\item + \href{https://keras.io/}{Keras}. +\end{itemize} + +These are all freely available at their respective GitHub sites. They +encompass communities of developers in the thousands or more. And the +number of code developers and contributors keeps increasing. + + + % Add a bibliography block to the postdoc + + + +\end{document}