day1/session3.tex
author Puneeth Chaganti <punchagan@fossee.in>
Mon, 26 Oct 2009 20:44:54 +0530
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%Tutorial slides on Python.
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%
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% Author: FOSSEE
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% Copyright (c) 2009, FOSSEE, IIT Bombay
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\documentclass[14pt,compress]{beamer}
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%\documentclass[draft]{beamer}
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%\documentclass[compress,handout]{beamer}
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%\usepackage{pgfpages} 
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%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
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% Modified from: generic-ornate-15min-45min.de.tex
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\mode<presentation>
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{
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  \usetheme{Warsaw}
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  \useoutertheme{split}
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  \setbeamercovered{transparent}
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}
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\usepackage[english]{babel}
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\usepackage[latin1]{inputenc}
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%\usepackage{times}
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\usepackage[T1]{fontenc}
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% Taken from Fernando's slides.
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\usepackage{ae,aecompl}
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\usepackage{mathpazo,courier,euler}
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\usepackage[scaled=.95]{helvet}
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\usepackage{amsmath}
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\definecolor{darkgreen}{rgb}{0,0.5,0}
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\usepackage{listings}
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\lstset{language=Python,
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    basicstyle=\ttfamily\bfseries,
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    commentstyle=\color{red}\itshape,
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  stringstyle=\color{darkgreen},
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  showstringspaces=false,
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  keywordstyle=\color{blue}\bfseries}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Macros
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\setbeamercolor{emphbar}{bg=blue!20, fg=black}
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\newcommand{\emphbar}[1]
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{\begin{beamercolorbox}[rounded=true]{emphbar} 
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      {#1}
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 \end{beamercolorbox}
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}
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\newcounter{time}
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\setcounter{time}{0}
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\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
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\newcommand{\typ}[1]{\lstinline{#1}}
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\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
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%%% This is from Fernando's setup.
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% \usepackage{color}
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% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
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% % Use and configure listings package for nicely formatted code
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% \usepackage{listings}
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% \lstset{
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%    language=Python,
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%    basicstyle=\small\ttfamily,
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%    commentstyle=\ttfamily\color{blue},
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%    stringstyle=\ttfamily\color{orange},
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%    showstringspaces=false,
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%    breaklines=true,
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%    postbreak = \space\dots
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% }
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Title page
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\title[]{Least Squares Fit\\Statistical Plotting}
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\author[FOSSEE] {FOSSEE}
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\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
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\date[] {31, October 2009\\Day 1, Session 3}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
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%\logo{\pgfuseimage{iitmlogo}}
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%% Delete this, if you do not want the table of contents to pop up at
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%% the beginning of each subsection:
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\AtBeginSubsection[]
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{
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  \begin{frame}<beamer>
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    \frametitle{Outline}
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    \tableofcontents[currentsection,currentsubsection]
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  \end{frame}
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}
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\AtBeginSection[]
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{
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  \begin{frame}<beamer>
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    \frametitle{Outline}
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    \tableofcontents[currentsection,currentsubsection]
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  \end{frame}
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}
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\newcommand{\num}{\texttt{numpy}}
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% If you wish to uncover everything in a step-wise fashion, uncomment
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% the following command: 
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%\beamerdefaultoverlayspecification{<+->}
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%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DOCUMENT STARTS
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\begin{document}
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\begin{frame}
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  \maketitle
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\end{frame}
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%% \begin{frame}
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%%   \frametitle{Outline}
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%%   \tableofcontents
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%%   % You might wish to add the option [pausesections]
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%% \end{frame}
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\begin{frame}
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  \frametitle{Statistical Analysis and Parsing}
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  Read the data supplied in \emph{sslc1.txt} and obtain the following statistics:
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  \begin{itemize}
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    \item Average total marks scored in each region
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    \item Subject wise average score of each region
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    \item \alert{??Subject wise average score for all regions combined??}
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    \item Find the subject wise standard deviation of scores for each region
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  \end{itemize}
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\end{frame}
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\begin{frame}
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  \frametitle{Statistical Analysis and Parsing \ldots}
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  Machinery Required -
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  \begin{itemize}
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    \item File reading and parsing
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    \item NumPy arrays - sum by rows and sum by coloumns
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    \item Dictionaries
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  \end{itemize}
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\end{frame}
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\begin{frame}
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  \frametitle{File reading and parsing}
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  Understanding the structure of sslc1.txt
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  \begin{itemize}
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    \item Each line in the file, i.e each row of a file is a single record.
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    \item Each record corresponds to a record of a single student
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    \item Each record consists of several fields separated by a ';'
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  \end{itemize}
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\end{frame}
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\begin{frame}
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  \frametitle{File reading and parsing \ldots}
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  Each record consists of:
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  \begin{itemize}
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    \item Region Code
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    \item Roll Number
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    \item Name
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    \item Marks of 5 subjects
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    \item Total marks
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    \item Pass (P)
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    \item Withdrawn (W)
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    \item Fail (F)
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{File reading and parsing \ldots}
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  \begin{lstlisting}
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for record in open('sslc1.txt'):
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    fields = record.split(';')
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data}
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  \begin{itemize}
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    \item Let the parsed data be stored in dictionary \typ{data}
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    \item \begin{lstlisting}
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data = {}  # is an empty dictionary
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\end{lstlisting}
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    \item Index of a dictionary is called a \emph{key}
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    \item \emph{Keys} of \typ{data} are strings - region codes
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    \item Value of a \emph{key} can be any Python object
120
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data...}
120
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  \begin{itemize}
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    \item In this problem let the value of a \emph{key} be another dictionary.
120
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    \item This dictionary contains:
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    \begin{itemize}
170
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      \item 'marks': A \emph{List} of \emph{Lists} containing all marks
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      \item 'total': A \emph{List} of total marks of each student
120
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      \item 'P': Number of passes
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      \item 'F': Number of failures
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      \item 'W': Number of withdrawls
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    \end{itemize}
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data \ldots}
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  \small
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  \begin{lstlisting}
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data = {}
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for record in open('sslc1.txt'):
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    fields = record.split(';')
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    if fields[0] not in data:
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        data[fields[0]] = {
166
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            'marks': [],
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            'total': [],
120
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            'P': 0,
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            'F': 0,
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            'W': 0
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            }
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data \ldots}
125
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  \begin{lstlisting}
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marks = []
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for field in fields[3:8]:
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    score_str = field.strip()
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    score = 0 if score_str == 'AA'
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        or score_str == 'AAA'
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        or score_str == ''
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        else int(score_str)
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    marks.append(score)
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data[fields[0]]['marks'].append(marks)
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data \ldots}
120
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  \begin{lstlisting}
125
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total = 0 if score_str == 'AA'
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    or score_str == 'AAA'
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    or score_str == ''
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    else int(fields[8])
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data[fields[0]]['total'].append(total)
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  \end{lstlisting}
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\end{frame}
120
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125
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\begin{frame}[fragile]
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  \frametitle{Dictionary - Building parsed data \ldots}
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  \begin{lstlisting}
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pfw_key = fields[9]
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    or fields[10]
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    or 'F'
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data[fields[0]][pfw_key] += 1
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{NumPy arrays}
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  \centerline{\alert{But I lied!?!?!?}}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Calculations}
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  \begin{lstlisting}
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for k in data:
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    data[k]['marks'] = array(
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        data[k]['marks'])
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    data[k]['total'] = array(
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        data[k]['total'])
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Calculations}
125
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  \small
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  \begin{lstlisting}
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    data[k]['avg'] = average(
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        data[k]['total'])
125
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    marks = data[k]['marks']
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    sub_avg = average(marks, axis=1)
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    sub_std = sqrt(sum(square(
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        sub_avg[:,newaxis] - marks), axis=0) /
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        len(marks))
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    data[k]['sub_avg'] = sub_avg
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    data[k]['sub_std'] = sub_std
120
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  \end{lstlisting}
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\end{frame}
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166
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\begin{frame}[fragile]
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  \frametitle{New Concepts}
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  \begin{itemize}
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   \item Dictionaries
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   \item Slicing lists
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   \item New type of conditional
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   \item NumPy arrays
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   \item Slicing NumPy arrays
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   \item NumPy array functions - square, average, sqrt
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Least Squares Fit}
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\vspace{-0.15in}
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\begin{figure}
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\includegraphics[width=4in]{data/least-sq-fit.png}
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\end{figure}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Calculating $T^2$ Efficiently}
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\begin{lstlisting}
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In []: for t in T:
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 ....:     Tsq.append(t*t)
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\end{lstlisting}
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\begin{itemize}
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\item This is not very efficient
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\item We use arrays to make it efficient
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\end{itemize}
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\begin{lstlisting}
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In []: L = array(L)
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In []: T = array(T)
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In []: Tsq = T*T
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In []: plot(L, Tsq, 'o')
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Arrays}
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\begin{itemize}
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\item \typ{T} and \typ{L} are now arrays
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\item arrays are very efficient and powerful 
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\item Very easy to perform element-wise operations
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\item \typ{+, -, *, /, \%}
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\item More about arrays later
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Least Square Fit Curve}
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\begin{itemize}
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\item $T^2$ and $L$ have a linear relationship
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\item Hence, Least Square Fit Curve is a line
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\item we shall use the \typ{lstsq} function
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{\typ{lstsq}}
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\begin{itemize}
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\item We need to fit a line through points for the equation $T^2 = m \cdot L+c$
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\item The equation can be re-written as $T^2 = A \cdot p$
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\item where A is   
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  $\begin{bmatrix}
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  L_1 & 1 \\
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  L_2 & 1 \\
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  \vdots & \vdots\\
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  L_N & 1 \\
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  \end{bmatrix}$
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  and p is 
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  $\begin{bmatrix}
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  m\\
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  c\\
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  \end{bmatrix}$
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\item We need to find $p$ to plot the line
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Van der Monde Matrix}
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\begin{itemize}
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\item A is also called a Van der Monde matrix
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\item It can be generated using \typ{vander}
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\end{itemize}
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Van der Monde matrix of order M
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\begin{equation*}
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  \begin{bmatrix}
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  l_1^{M-1} & \ldots & l_1 & 1 \\
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  l_2^{M-1} & \ldots &l_2 & 1 \\
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  \vdots & \ldots & \vdots & \vdots\\
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  l_N^{M-1} & \ldots & l_N & 1 \\
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  \end{bmatrix}
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\end{equation*}
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\begin{lstlisting}
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In []: A = vander(L,2)
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{\typ{lstsq} \ldots}
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\begin{itemize}
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\item Now use the \typ{lstsq} function
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\item Along with a lot of things, it returns the least squares solution
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\end{itemize}
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\begin{lstlisting}
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In []: coef, res, r, s = lstsq(A,Tsq)
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\end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Least Square Fit Line \ldots}
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We get the points of the line from \typ{coef}
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\begin{lstlisting}
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In []: Tline = coef[0]*L + coef[1]
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\end{lstlisting}
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\begin{itemize}
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\item Now plot Tline vs. L, to get the Least squares fit line. 
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\end{itemize}
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\begin{lstlisting}
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In []: plot(L, Tline)
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\end{lstlisting}
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\end{frame}
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115
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\end{document}