day1/cheatsheet4.tex
author Santosh G. Vattam <vattam.santosh@gmail.com>
Mon, 08 Mar 2010 20:45:33 +0530
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\documentclass[12pt]{article}
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\title{Matrices and Least Square Fit}
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\author{FOSSEE}
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\usepackage{listings}
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\lstset{language=Python,
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    basicstyle=\ttfamily,
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  commentstyle=\itshape\bfseries,
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  showstringspaces=false,
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}
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\newcommand{\typ}[1]{\lstinline{#1}}
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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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\usepackage{ae,aecompl}
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\usepackage{mathpazo,courier,euler}
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\usepackage[scaled=.95]{helvet}
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\begin{document}
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\date{}
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\vspace{-1in}
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\begin{center}
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\LARGE{Matrices and Least Square Fit}\\
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\large{FOSSEE}
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\end{center}
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\section{Matrices}
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\subsection{Basics}
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Matrix Creation\\
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\typ{In []: C = array([[1,1,2], [2,4,1], [-1,3,7]])}\\
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It creates C matrix of shape 3x3\\
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Shape is dimensions of given array.
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\begin{lstlisting}
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In []: C.shape 
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Out[]: (3, 3)
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In []: shape([[1,2],[4,5],[3,0]])
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Out[]: (3, 2)
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\end{lstlisting}
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\typ{In []: B = ones_like(C)} \\
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B would be array of ones with the same shape and type as C.\\
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\typ{In []: A = ones((3,2))} \\
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A would be new matrix of given shape(arguments), filled with ones.\\ 
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\typ{In []: I = identity(3)}\\
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I would be identity matrix of shape 3x3
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\subsection{Accessing Elements}
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\begin{lstlisting}
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In []: C
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Out[]: 
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array([[ 1,  1,  2],
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       [ 2,  4,  1],
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       [-1,  3,  7]])
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In []: C[1,2]
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Out[]: 1
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\end{lstlisting}
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Two indexes separated by \typ{','} specifies [row, column]. So \typ{C[1,2]} gets third element of second row(indices starts from 0).
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\newpage
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\begin{lstlisting}
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In []: C[1]
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Out[]: array([2, 4, 1])
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\end{lstlisting}
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Single index implies complete row.
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\subsection{Changing elements}
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\begin{lstlisting}
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In []: C[1,1] = -2
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In []: C
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Out[]: 
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array([[ 1,  1,  2],
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       [ 2, -2,  1],
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       [-1,  3,  7]])
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In []: C[1] = [0,0,0]
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In []: C
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Out[]: 
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array([[ 1,  1,  2],
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       [ 0,  0,  0],
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       [-1,  3,  7]])
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\end{lstlisting}
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\subsection{Slicing}
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Accessing rows with Matrices is straightforward. But If one wants to access particular Column, or want a sub-matrix, Slicing is the way to go.
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\begin{lstlisting}
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In []: C[:,1]
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Out[]: array([1, 0, 3])
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\end{lstlisting}
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First index(:) specifies row(':' implies all the rows) and second index(1) specifies column(second column).
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\begin{lstlisting}
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In []: C[1,:]
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Out[]: array([0, 0, 0])
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\end{lstlisting}
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Here we get second row(1), all columns(':') of C matrix.
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\newpage
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\begin{lstlisting}
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In []: C[0:2,:]
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Out[]: 
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array([[1, 1, 2],
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       [0, 0, 0]])
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\end{lstlisting}
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Result is sub-matrix with first and second row(endpoint is excluded), and all columns from C.
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\begin{lstlisting}
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In []: C[1:3,:]
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Out[]: 
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array([[ 0,  0,  0],
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       [-1,  3,  7]])
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In []: C[:2,:]
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Out[]: 
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array([[1, 1, 2],
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       [0, 0, 0]])
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\end{lstlisting}
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\typ{':2'} => start from first row, till and excluding third row.
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\begin{lstlisting}
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In []: C[1:,:]
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Out[]: 
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array([[ 0,  0,  0],
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       [-1,  3,  7]])
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In []: C[1:,:2]
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Out[]: 
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array([[ 0,  0],
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       [-1,  3]])
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\end{lstlisting}
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\typ{'1:'} => Start from second row, till last row\\
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\typ{':2'} => Start from first column, till and excluding third column.
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\newpage
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\subsection{Striding}
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Often apart from sub-matrix, one needs to get some mechanism to jump a step. For example, how can we have all alternate rows of a Matrix. \\
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Following method will return Matrix with alternate rows.
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\begin{lstlisting}
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In []: C[::2,:]
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Out[]: 
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array([[ 1,  1,  2],
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       [-1,  3,  7]])
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\end{lstlisting}
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\typ{C[startR:stopR:stepR,startC:stopC:stepC]} => Syntax of mentioning starting index, ending index, and step to jump.\\
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In above mentioned case, \typ{'::2'} means, start from first row, till last row(both are blank), with step of 2, that is, skipping alternate row. After first row, C[startR], next row would be C[startR+stepR] and so on.
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\begin{lstlisting}
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In []: C[:,::2]
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Out[]: 
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array([[ 1,  2],
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       [ 0,  0],
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       [-1,  7]])
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\end{lstlisting}
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Same as above, just that here we get matrix with each alternate column and all rows.
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\begin{lstlisting}
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In []: C[::2,::2]
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Out[]: 
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array([[ 1,  2],
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       [-1,  7]])
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\end{lstlisting}
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\section{Matrix Operations}
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For a Matrix A and B of equal shapes.
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\begin{lstlisting}
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In []: A.T # Transpose
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In []: sum(A) # Sum of all elements
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In []: A+B # Addition
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In []: A*B # Element wise product
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In []: dot(A,b) #Matrix multiplication
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In []: inv(A) # Inverse
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In []: det(A) # Determinant
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\end{lstlisting}
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Eigen Values and Eigen Vectors
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\begin{lstlisting}
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In []: eig(A) #Eigen Values and Vectors
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In []: eigvals(A) #Eigen Values 
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\end{lstlisting}
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%% Norm
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%% \begin{lstlisting}
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%% In []: norm(A)
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%% \end{lstlisting}
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%% Single Value Decomposition
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%% \begin{lstlisting}
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%% In []: svd(A)
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%% \end{lstlisting}
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\section{Least Square Fit Line}
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\begin{lstlisting}
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L = []
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T = []
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for line in open('pendulum.txt'):
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    point = line.split()
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    L.append(float(point[0]))
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    T.append(float(point[1]))
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Tsq = []
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for time in T:
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    Tsq.append(time*time)
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plot(L, Tsq, '.')
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\end{lstlisting}
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This is exact curve we get from L Vs Tsq from data.This relation among L and Tsq is not of straight line. For getting Least Square Fit line, we have to solve the relations:\\
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$L=m*Tsq+c$ (something similar to $y=m*x+c$)\\
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For present scenario, we have L and corresponding Tsq values. For finding m and c at given points we use \typ{lstlq} function provided by pylab. It returns the least-squares solution to an equation. \\
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For finding Least Square Fit line for this particular data we have to do following steps:\\
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\typ{In []: A = array([L, ones\_like(L)])}\\
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A is 2x(Length of array L) array.
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\begin{lstlisting}
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In []: A = A.T #now A.shape = (Length of array L)x2
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In []: result = lstsq(A,TSq)
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In []: coef = result[0]
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In []: Tline = coef[0]*L + coef[1]
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\end{lstlisting}
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\typ{coef[0]} is array with all $m$ values, and \typ{coef[1]} contains $c$.\\
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To get the final plot.\\
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\typ{In []: plot(L, Tline)}
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\end{document}