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%Tutorial slides on Python.

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% Author: FOSSEE

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[]{Least Squares Fit\\Statistical Plotting}

\author[FOSSEE] {FOSSEE}

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

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

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    \frametitle{Outline}

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% 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{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    \item Withdrawn (W)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]

  \frametitle{Dictionary - Building parsed data}

  \begin{itemize}

    \item Let the parsed data be stored in dictionary \typ{data}

    \item \begin{lstlisting}

data = {}  # is an empty dictionary

\end{lstlisting}

    \item Index of a dictionary is called a \emph{key}

    \item \emph{Keys} of \typ{data} are strings - region codes

    \item Value of a \emph{key} can be any Python object

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data...}

  \begin{itemize}

    \item In this problem let the value of a \emph{key} be another dictionary.

    \item This dictionary contains:

    \begin{itemize}

      \item 'marks': A \emph{List} of \emph{Lists} containing all marks

      \item 'total': A \emph{List} of 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]] = {

            'marks': [],

            'total': [],

            '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)

  \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{NumPy arrays}

  \centerline{\alert{But I lied!?!?!?}}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Calculations}

  \begin{lstlisting}

for k in data:

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

        data[k]['marks'])

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

        data[k]['total'])

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Calculations}

  \small

  \begin{lstlisting}

    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}

\begin{frame}[fragile]

  \frametitle{New Concepts}

  \begin{itemize}

   \item Dictionaries

   \item Slicing lists

   \item New type of conditional

   \item NumPy arrays

   \item Slicing NumPy arrays

   \item NumPy array functions - square, average, sqrt

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Least Squares Fit}

\vspace{-0.15in}

\begin{figure}

\includegraphics[width=4in]{data/least-sq-fit.png}

\end{figure}

\end{frame}

\begin{frame}[fragile]

\frametitle{Calculating $T^2$ Efficiently}

\begin{lstlisting}

In []: for t in T:

 ....:     Tsq.append(t*t)

\end{lstlisting}

\begin{itemize}

\item This is not very efficient

\item We use arrays to make it efficient

\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 \typ{T} and \typ{L} are now arrays

\item 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}

\frametitle{Least Square Fit Curve}

\begin{itemize}

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

\item Hence, Least Square Fit Curve is a line

\item we shall use the \typ{lstsq} function

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{lstsq}}

\begin{itemize}

\item We need to fit a line through points for the equation $T^2 = m \cdot L+c$

\item The equation can be re-written as $T^2 = A \cdot p$

\item where A is   

  $\begin{bmatrix}

  L_1 & 1 \\

  L_2 & 1 \\

  \vdots & \vdots\\

  L_N & 1 \\

  \end{bmatrix}$

  and p is 

  $\begin{bmatrix}

  m\\

  c\\

  \end{bmatrix}$

\item We need to find $p$ to plot the line

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Van der Monde Matrix}

\begin{itemize}

\item A is also called a Van der Monde matrix

\item It can be generated using \typ{vander}

\end{itemize}

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{\typ{lstsq} \ldots}

\begin{itemize}

\item Now use the \typ{lstsq} function

\item Along with a lot of things, it returns the least squares solution

\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}

We get the points of the line from \typ{coef}

\begin{lstlisting}

In []: Tline = coef[0]*L + coef[1]

\end{lstlisting}

\begin{itemize}

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

\end{itemize}

\begin{lstlisting}

In []: plot(L, Tline)

\end{lstlisting}

\end{frame}

\end{document}