author | Christopher Burns <chris.d.burns@gmail.com> |
Tue, 29 Jun 2010 00:59:26 -0500 | |
branch | scipy2010 |
changeset 432 | 13e5d0e2cd40 |
parent 343 | adb85e126341 |
permissions | -rwxr-xr-x |
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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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|
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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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|
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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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|
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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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|
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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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|
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In []: C[:2,:] |
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Out[]: |
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array([[1, 1, 2], |
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108 |
[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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112 |
In []: C[1:,:] |
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113 |
Out[]: |
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114 |
array([[ 0, 0, 0], |
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115 |
[-1, 3, 7]]) |
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116 |
|
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117 |
In []: C[1:,:2] |
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118 |
Out[]: |
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119 |
array([[ 0, 0], |
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120 |
[-1, 3]]) |
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121 |
\end{lstlisting} |
315
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122 |
\typ{'1:'} => Start from second row, till last row\\ |
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123 |
\typ{':2'} => Start from first column, till and excluding third column. |
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124 |
\newpage |
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125 |
\subsection{Striding} |
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126 |
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. \\ |
315
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127 |
Following method will return Matrix with alternate rows. |
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128 |
\begin{lstlisting} |
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129 |
In []: C[::2,:] |
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130 |
Out[]: |
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131 |
array([[ 1, 1, 2], |
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[-1, 3, 7]]) |
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133 |
\end{lstlisting} |
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134 |
\typ{C[startR:stopR:stepR,startC:stopC:stepC]} => Syntax of mentioning starting index, ending index, and step to jump.\\ |
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135 |
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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136 |
\begin{lstlisting} |
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137 |
In []: C[:,::2] |
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138 |
Out[]: |
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139 |
array([[ 1, 2], |
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140 |
[ 0, 0], |
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141 |
[-1, 7]]) |
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142 |
\end{lstlisting} |
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143 |
Same as above, just that here we get matrix with each alternate column and all rows. |
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144 |
\begin{lstlisting} |
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145 |
In []: C[::2,::2] |
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146 |
Out[]: |
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147 |
array([[ 1, 2], |
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148 |
[-1, 7]]) |
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149 |
\end{lstlisting} |
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150 |
\section{Matrix Operations} |
315
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151 |
For a Matrix A and B of equal shapes. |
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\begin{lstlisting} |
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153 |
In []: A.T # Transpose |
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154 |
In []: sum(A) # Sum of all elements |
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155 |
In []: A+B # Addition |
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156 |
In []: A*B # Element wise product |
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157 |
In []: dot(A,b) #Matrix multiplication |
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158 |
In []: inv(A) # Inverse |
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159 |
In []: det(A) # Determinant |
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160 |
\end{lstlisting} |
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161 |
Eigen Values and Eigen Vectors |
295
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162 |
\begin{lstlisting} |
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163 |
In []: eig(A) #Eigen Values and Vectors |
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In []: eigvals(A) #Eigen Values |
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165 |
\end{lstlisting} |
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166 |
%% Norm |
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167 |
%% \begin{lstlisting} |
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168 |
%% In []: norm(A) |
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169 |
%% \end{lstlisting} |
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170 |
%% Single Value Decomposition |
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171 |
%% \begin{lstlisting} |
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172 |
%% In []: svd(A) |
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173 |
%% \end{lstlisting} |
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174 |
\section{Least Square Fit Line} |
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175 |
\begin{lstlisting} |
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176 |
L = [] |
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177 |
T = [] |
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178 |
for line in open('pendulum.txt'): |
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179 |
point = line.split() |
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180 |
L.append(float(point[0])) |
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181 |
T.append(float(point[1])) |
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182 |
Tsq = [] |
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183 |
for time in T: |
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184 |
Tsq.append(time*time) |
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185 |
plot(L, Tsq, '.') |
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186 |
\end{lstlisting} |
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187 |
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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188 |
$L=m*Tsq+c$ (something similar to $y=m*x+c$)\\ |
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189 |
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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190 |
For finding Least Square Fit line for this particular data we have to do following steps:\\ |
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191 |
\typ{In []: A = array([L, ones\_like(L)])}\\ |
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192 |
A is 2x(Length of array L) array. |
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193 |
\begin{lstlisting} |
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194 |
In []: A = A.T #now A.shape = (Length of array L)x2 |
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195 |
In []: result = lstsq(A,TSq) |
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196 |
In []: coef = result[0] |
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197 |
In []: Tline = coef[0]*L + coef[1] |
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198 |
\end{lstlisting} |
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199 |
\typ{coef[0]} is array with all $m$ values, and \typ{coef[1]} contains $c$.\\ |
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200 |
To get the final plot.\\ |
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201 |
\typ{In []: plot(L, Tline)} |
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202 |
|
284
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203 |
\end{document} |