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% Author: FOSSEE

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% 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}

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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{Least Squares Fit}

In this session - 

\begin{itemize}

\item We shall plot a least squares fit curve for time-period(T) squared vs. length(L) plot of a Simple Pendulum. 

\item Given a file containing L and T values

\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{Reading pendulum.txt}

\begin{itemize}

  \item The file has two columns

  \item Column1 - L; Column2 - T

\end{itemize}

\begin{lstlisting}

In []: L = []

In []: T = []

In []: for line in open('pendulum.txt'):

  ....     T.append(float(t))

\end{lstlisting}

We now have two lists L and T

\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]

  \frametitle{Dictionary}

  \begin{itemize}

    \item lists index: 0 \ldots n

    \item dictionaries index using any hashable objects

    \item d = \{ ``Hitchhiker's guide'' : 42, ``Terminator'' : ``I'll be back''\}

    \item d[``Terminator''] => ``I'll be back''

    \item ``Terminator'' is called the key of \typ{d}

    \item ``I'll be back'' is called the value of the key ``Terminator''

  \end{itemize}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Dictionary - Building parsed data}

  \begin{itemize}

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

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

    \item Value of the key is another dictionary.

    \item This dictionary contains:

    \begin{itemize}

      \item 'marks': A list of NumPy arrays

      \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]] = {

            'marks': array([]),

            'total': array([]),

            '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}