# HG changeset patch # User Puneeth Chaganti # Date 1256570094 -19800 # Node ID 4ec0418ba0416bb8875cd987ed3000d0e34e7e3f # Parent 5e2925eed57e64346097fb2aa0803a76dd33a5eb# Parent d922270c85bbcc67f961555d182610747689b1df Merged with Mainline. diff -r 5e2925eed57e -r 4ec0418ba041 day1/session3.tex --- a/day1/session3.tex Mon Oct 26 20:28:58 2009 +0530 +++ b/day1/session3.tex Mon Oct 26 20:44:54 2009 +0530 @@ -126,118 +126,6 @@ %% % You might wish to add the option [pausesections] %% \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} - \begin{frame} \frametitle{Statistical Analysis and Parsing} Read the data supplied in \emph{sslc1.txt} and obtain the following statistics: @@ -419,4 +307,116 @@ \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}