%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Tutorial slides on Python.

%

% Author: FOSSEE

% Copyright (c) 2009, FOSSEE, IIT Bombay

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[14pt,compress]{beamer}

%\documentclass[draft]{beamer}

%\documentclass[compress,handout]{beamer}

%\usepackage{pgfpages} 

%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]

% Modified from: generic-ornate-15min-45min.de.tex

\mode<presentation>

{

  \usetheme{Warsaw}

  \useoutertheme{split}

  \setbeamercovered{transparent}

}

\usepackage[english]{babel}

\usepackage[latin1]{inputenc}

%\usepackage{times}

\usepackage[T1]{fontenc}

% Taken from Fernando's slides.

\usepackage{ae,aecompl}

\usepackage{mathpazo,courier,euler}

\usepackage[scaled=.95]{helvet}

\usepackage{amsmath}

\definecolor{darkgreen}{rgb}{0,0.5,0}

\usepackage{listings}

\lstset{language=Python,

    basicstyle=\ttfamily\bfseries,

    commentstyle=\color{red}\itshape,

  stringstyle=\color{darkgreen},

  showstringspaces=false,

  keywordstyle=\color{blue}\bfseries}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Macros

\setbeamercolor{emphbar}{bg=blue!20, fg=black}

\newcommand{\emphbar}[1]

{\begin{beamercolorbox}[rounded=true]{emphbar} 

      {#1}

 \end{beamercolorbox}

}

\newcounter{time}

\setcounter{time}{0}

\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}

\newcommand{\typ}[1]{\lstinline{#1}}

\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }

%%% This is from Fernando's setup.

% \usepackage{color}

% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}

% % Use and configure listings package for nicely formatted code

% \usepackage{listings}

% \lstset{

%    language=Python,

%    basicstyle=\small\ttfamily,

%    commentstyle=\ttfamily\color{blue},

%    stringstyle=\ttfamily\color{orange},

%    showstringspaces=false,

%    breaklines=true,

%    postbreak = \space\dots

% }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Title page

\title[]{Arrays \& Least Squares Fit}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}

%\logo{\pgfuseimage{iitmlogo}}

%% Delete this, if you do not want the table of contents to pop up at

%% the beginning of each subsection:

\AtBeginSubsection[]

{

  \begin{frame}<beamer>

    \frametitle{Outline}

    \tableofcontents[currentsection,currentsubsection]

  \end{frame}

}

\AtBeginSection[]

{

  \begin{frame}<beamer>

    \frametitle{Outline}

    \tableofcontents[currentsection,currentsubsection]

  \end{frame}

}

\newcommand{\num}{\texttt{numpy}}

% If you wish to uncover everything in a step-wise fashion, uncomment

% the following command: 

%\beamerdefaultoverlayspecification{<+->}

%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% DOCUMENT STARTS

\begin{document}

\begin{frame}

  \maketitle

\end{frame}

%% \begin{frame}

%%   \frametitle{Outline}

%%   \tableofcontents

%%   % You might wish to add the option [pausesections]

%% \end{frame}

\begin{frame}

\frametitle{Least Squares Fit}

\begin{itemize}

\item Plot a least squares fit line

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Squares Fit \ldots}

Machinery Required -

\begin{itemize}

\item Reading files and parsing data

\item Plotting points, lines

\item Calculating the Coefficients of the Least Squares Fit curve

\begin{itemize}

  \item Arrays

\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Calculating $T^2$}

\begin{itemize}

\item Each element of the list T must be squared

\item Iterating over each element of the list works

\item But very slow \ldots

\item Instead, we use arrays

\end{itemize}

\begin{lstlisting}

In []: L = array(L)

In []: T = array(T)

In []: Tsq = T*T

In []: plot(L, Tsq, 'o')

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Arrays}

\begin{itemize}

\item T is now a \typ{numpy array}

\item \typ{numpy} arrays are very efficient and powerful 

\item Very easy to perform element-wise operations

\item \typ{+, -, *, /, \%}

\item More about arrays later

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Polynomial}

\begin{enumerate}

\item $T^2 = \frac{4\pi^2}{g}L$

\item $T^2$ and $L$ have a linear relationship

\item We find an approximate solution to $Ax = y$, where A is the Van der Monde matrix to get coefficients of the least squares fit line. 

\end{enumerate}

\end{frame}

\begin{frame}[fragile]

\frametitle{Van der Monde Matrix}

Van der Monde matrix of order M

\begin{equation*}

  \begin{bmatrix}

  l_1^{M-1} & \ldots & l_1 & 1 \\

  l_2^{M-1} & \ldots &l_2 & 1 \\

  \vdots & \ldots & \vdots & \vdots\\

  l_N^{M-1} & \ldots & l_N & 1 \\

  \end{bmatrix}

\end{equation*}

\begin{lstlisting}

In []: A=vander(L,2)

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Fit Line}

\begin{itemize}

\item We use the \typ{lstsq} function of pylab

\item It returns the 

\begin{enumerate}

\item Least squares solution

\item Sum of residues

\item Rank of matrix A

\item Singular values of A

\end{enumerate}

\end{itemize}

\begin{lstlisting}

In []: coef, res, r, s = lstsq(A,Tsq)

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Square Fit Line \ldots}

\begin{itemize}

\item Use the poly1d function of pylab, to create a function for the line equation using the coefficients obtained

\begin{lstlisting}

In []: p=poly1d(coef)

\end{lstlisting}

\item Get new $T^2$ values using the function \typ{p} obtained

\begin{lstlisting}

In []: Tline = p(L)

\end{lstlisting}

\item Now plot Tline vs. L, to get the Least squares fit line. 

\begin{lstlisting}

In []: plot(L, Tline)

\end{lstlisting}

\end{itemize}

\end{frame}

\begin{frame}

  \frametitle{Statistical Analysis and Parsing}

  Read the data supplied in \emph{sslc1.txt} and obtain the following statistics:

  \begin{itemize}

    \item Average total marks scored in each region

    \item Subject wise average score of each region

    \item \alert{??Subject wise average score for all regions combined??}

    \item Find the subject wise standard deviation of scores for each region

  \end{itemize}

\end{frame}

\begin{frame}

  \frametitle{Statistical Analysis and Parsing \ldots}

  Machinery Required -

  \begin{itemize}

    \item File reading and parsing

    \item NumPy arrays - sum by rows and sum by coloumns

    \item Dictionaries

  \end{itemize}

\end{frame}

\begin{frame}

  \frametitle{File reading and parsing}

  Understanding the structure of sslc1.txt

  \begin{itemize}

    \item Each line in the file, i.e each row of a file is a single record.

    \item Each record corresponds to a record of a single student

    \item Each record consists of several fields separated by a ';'

  \end{itemize}

\end{frame}

\begin{frame}

  \frametitle{File reading and parsing \ldots}

  Each record consists of:

  \begin{itemize}

    \item Region Code

    \item Roll Number

    \item Name

    \item Marks of 5 subjects

    \item Total marks

    \item Pass (P)

    \item Withdrawn (W)

    \item Fail (F)

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{File reading and parsing \ldots}

  \begin{lstlisting}

for record in open('sslc1.txt'):

    fields = record.split(';')

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \begin{itemize}

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \begin{itemize}

    \item This dictionary contains:

    \begin{itemize}

      \item 'total': Total marks of each student

      \item 'P': Number of passes

      \item 'F': Number of failures

      \item 'W': Number of withdrawls

    \end{itemize}

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data \ldots}

  \small

  \begin{lstlisting}

data = {}

for record in open('sslc1.txt'):

    fields = record.split(';')

    if fields[0] not in data:

        data[fields[0]] = {

            'P': 0,

            'F': 0,

            'W': 0

            }

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data \ldots}

  \begin{lstlisting}

marks = []

for field in fields[3:8]:

    score_str = field.strip()

    score = 0 if score_str == 'AA'

        or score_str == 'AAA'

        or score_str == ''

        else int(score_str)

    marks.append(score)

data[fields[0]]['marks'].append(marks)

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data \ldots}

  \begin{lstlisting}

total = 0 if score_str == 'AA'

    or score_str == 'AAA'

    or score_str == ''

    else int(fields[8])

data[fields[0]]['total'].append(total)

pfw_key = fields[9]

    or fields[10]

    or 'F'

data[fields[0]][pfw_key] += 1

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data \ldots}

  \begin{lstlisting}

pfw_key = fields[9]

    or fields[10]

    or 'F'

data[fields[0]][pfw_key] += 1

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Calculations}

  \small

  \begin{lstlisting}

for k in data:

    data[k]['marks'] = array(data[k]['marks'])

    data[k]['total'] = array(data[k]['total'])

    data[k]['avg'] = average(

        data[k]['total'])

    marks = data[k]['marks']

    sub_avg = average(marks, axis=1)

    sub_std = sqrt(sum(square(

        sub_avg[:,newaxis] - marks), axis=0) /

        len(marks))

    data[k]['sub_avg'] = sub_avg

    data[k]['sub_std'] = sub_std

  \end{lstlisting}

\end{frame}

\end{document}