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% Help prevent overflowing lines due to urls and other hard-to-break
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% Document level variables
\title{CSMATH-hw1}
\date{April 02, 2014}
\release{}
\author{Unknown Author}
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% TODO: Add option for the user to specify a logo for his/her export.
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% Import sphinx document type specifics.
% Body
% Start of the document
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\begin{alltt}Populating the interactive namespace from numpy and matplotlib
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\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{scipy} \PY{k+kn}{import} \PY{n}{optimize} \PY{k}{as} \PY{n}{opt}
\PY{k}{def} \PY{n+nf}{lossFunc}\PY{p}{(}\PY{n}{p}\PY{p}{,} \PY{n}{y}\PY{p}{,} \PY{n}{x}\PY{p}{,} \PY{n}{lamb}\PY{p}{)}\PY{p}{:}
\PY{n}{y}\PY{o}{=}\PY{n}{asarray}\PY{p}{(}\PY{n}{y}\PY{p}{)}
\PY{n}{x}\PY{o}{=}\PY{n}{asarray}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{n}{f}\PY{o}{=}\PY{n}{poly1d}\PY{p}{(}\PY{n}{p}\PY{p}{)}
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\PY{k}{def} \PY{n+nf}{convenience}\PY{p}{(}\PY{n}{degree}\PY{p}{,} \PY{n}{n}\PY{p}{,} \PY{n}{regular}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{lamb}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{ur\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ 一个便利函数,指定需要的多项式拟合曲线次数 degree, 以及生成的带噪声点个数 n, }
\PY{l+s+sd}{ 画拟合曲线, 不返回值}
\PY{l+s+sd}{ }
\PY{l+s+sd}{ PARAMS}
\PY{l+s+sd}{ \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{l+s+sd}{ degree: 多项式次数}
\PY{l+s+sd}{ n: 加噪声采样点个数}
\PY{l+s+sd}{ }
\PY{l+s+sd}{ RETURNS}
\PY{l+s+sd}{ \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{l+s+sd}{ None}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{x}\PY{o}{=}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{pi}\PY{p}{,} \PY{l+m+mi}{100}\PY{p}{)}
\PY{l+s}{\PYZsq{}}\PY{l+s}{原始 sin(x)}\PY{l+s}{\PYZsq{}}
\PY{n}{y}\PY{o}{=}\PY{n}{sin}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{l+s}{\PYZsq{}}\PY{l+s}{加了高斯噪声的 n 个采样点}\PY{l+s}{\PYZsq{}}
\PY{n}{sx}\PY{o}{=}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{pi}\PY{p}{,} \PY{n}{n}\PY{p}{)}
\PY{n}{sy}\PY{o}{=}\PY{n}{sin}\PY{p}{(}\PY{n}{sx}\PY{p}{)}\PY{o}{+}\PY{n}{randn}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{o}{*}\PY{o}{.}\PY{l+m+mi}{3} \PY{c}{\PYZsh{}sigma=0.3}
\PY{k}{if} \PY{n}{regular}\PY{o}{==}\PY{n+nb+bp}{False}\PY{p}{:}
\PY{n}{c}\PY{o}{=}\PY{n}{polyfit}\PY{p}{(}\PY{n}{sx}\PY{p}{,} \PY{n}{sy}\PY{p}{,} \PY{n}{degree}\PY{p}{)}
\PY{k}{else}\PY{p}{:}
\PY{n}{c}\PY{p}{,} \PY{n}{\PYZus{}}\PY{o}{=}\PY{n}{opt}\PY{o}{.}\PY{n}{leastsq}\PY{p}{(}\PY{n}{lossFunc}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{*}\PY{n}{n}\PY{p}{,} \PY{n}{args}\PY{o}{=}\PY{p}{(}\PY{n}{sy}\PY{p}{,} \PY{n}{sx}\PY{p}{,} \PY{n}{lamb}\PY{o}{*}\PY{o}{*}\PY{o}{.}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{)} \PY{c}{\PYZsh{}疑问: 这里用 e\PYZca{}\PYZhy{}18 反而效果很差?}
\PY{n}{p}\PY{o}{=}\PY{n}{poly1d}\PY{p}{(}\PY{n}{c}\PY{p}{)}
\PY{n}{yn}\PY{o}{=}\PY{n}{p}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{y}\PY{p}{,} \PY{l+s}{\PYZsq{}}\PY{l+s}{g}\PY{l+s}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s}{\PYZsq{}}\PY{l+s}{sin(x)}\PY{l+s}{\PYZsq{}}\PY{p}{)}
\PY{c}{\PYZsh{}plot(sx, sy, \PYZsq{}bo\PYZsq{}, label=\PYZsq{}sin(x) with noise\PYZsq{})}
\PY{c}{\PYZsh{}空心点哟:}
\PY{n}{plot}\PY{p}{(}\PY{n}{sx}\PY{p}{,} \PY{n}{sy}\PY{p}{,} \PY{l+s}{\PYZsq{}}\PY{l+s}{o}\PY{l+s}{\PYZsq{}}\PY{p}{,} \PY{n}{mfc}\PY{o}{=}\PY{l+s}{\PYZsq{}}\PY{l+s}{none}\PY{l+s}{\PYZsq{}}\PY{p}{,} \PY{n}{mec}\PY{o}{=}\PY{l+s}{\PYZsq{}}\PY{l+s}{b}\PY{l+s}{\PYZsq{}}\PY{p}{,} \PY{n}{mew}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s}{\PYZsq{}}\PY{l+s}{sin(x) with noise}\PY{l+s}{\PYZsq{}}\PY{p}{)}
\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{yn}\PY{p}{,} \PY{l+s}{\PYZsq{}}\PY{l+s}{r}\PY{l+s}{\PYZsq{}}\PY{p}{,} \PY{n}{lw}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s}{\PYZsq{}}\PY{l+s}{polynomial fit}\PY{l+s}{\PYZsq{}}\PY{p}{)}
\PY{n}{legend}\PY{p}{(}\PY{p}{)}
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\begin{ColorVerbatim}
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\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{3次多项式拟合 10 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{)}
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% If the first block is an image, minipage the image. Else
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\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_2_0.png}
\par
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\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}4{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{9次多项式拟合 10 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{9}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{)}
\end{Verbatim}
\vspace{-0.2\baselineskip}
\end{ColorVerbatim}
% If the first block is an image, minipage the image. Else
% request a certain amount of space for the input text.
\needspace{4\baselineskip}
% Add document contents.
\begin{InvisibleVerbatim}
\vspace{-0.5\baselineskip}
\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_3_0.png}
\par
\end{center}
\end{InvisibleVerbatim}
% Make sure that atleast 4 lines are below the HR
\needspace{4\baselineskip}
\vspace{6pt}
\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}5{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{9次多项式拟合 15 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{9}\PY{p}{,} \PY{l+m+mi}{15}\PY{p}{)}
\end{Verbatim}
\vspace{-0.2\baselineskip}
\end{ColorVerbatim}
% If the first block is an image, minipage the image. Else
% request a certain amount of space for the input text.
\needspace{4\baselineskip}
% Add document contents.
\begin{InvisibleVerbatim}
\vspace{-0.5\baselineskip}
\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_4_0.png}
\par
\end{center}
\end{InvisibleVerbatim}
% Make sure that atleast 4 lines are below the HR
\needspace{4\baselineskip}
\vspace{6pt}
\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}6{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{9次多项式拟合 100 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{9}\PY{p}{,} \PY{l+m+mi}{100}\PY{p}{)}
\end{Verbatim}
\vspace{-0.2\baselineskip}
\end{ColorVerbatim}
% If the first block is an image, minipage the image. Else
% request a certain amount of space for the input text.
\needspace{4\baselineskip}
% Add document contents.
\begin{InvisibleVerbatim}
\vspace{-0.5\baselineskip}
\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_5_0.png}
\par
\end{center}
\end{InvisibleVerbatim}
% Make sure that atleast 4 lines are below the HR
\needspace{4\baselineskip}
\vspace{6pt}
\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}7{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{引入正则化, 9次多项式拟合 10 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{9}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{,} \PY{n}{regular}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\end{Verbatim}
\vspace{-0.2\baselineskip}
\end{ColorVerbatim}
% If the first block is an image, minipage the image. Else
% request a certain amount of space for the input text.
\needspace{4\baselineskip}
% Add document contents.
\begin{InvisibleVerbatim}
\vspace{-0.5\baselineskip}
\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_6_0.png}
\par
\end{center}
\end{InvisibleVerbatim}
% Make sure that atleast 4 lines are below the HR
\needspace{4\baselineskip}
\vspace{6pt}
\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}8{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{l+s}{\PYZsq{}}\PY{l+s}{引入正则化, 9次多项式拟合 10 点}\PY{l+s}{\PYZsq{}}
\PY{n}{convenience}\PY{p}{(}\PY{l+m+mi}{9}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{,} \PY{n}{regular}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{,} \PY{n}{lamb}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{e}\PY{o}{*}\PY{o}{*}\PY{o}{\PYZhy{}}\PY{l+m+mi}{18}\PY{p}{)}
\end{Verbatim}
\vspace{-0.2\baselineskip}
\end{ColorVerbatim}
% If the first block is an image, minipage the image. Else
% request a certain amount of space for the input text.
\needspace{4\baselineskip}
% Add document contents.
\begin{InvisibleVerbatim}
\vspace{-0.5\baselineskip}
\begin{center}
\includegraphics[max size={\textwidth}{\textheight}]{CSMATH-hw1_files/CSMATH-hw1_7_0.png}
\par
\end{center}
\end{InvisibleVerbatim}
% Make sure that atleast 4 lines are below the HR
\needspace{4\baselineskip}
\vspace{6pt}
\makebox[0.1\linewidth]{\smaller\hfill\tt\color{nbframe-in-prompt}In\hspace{4pt}{[}{]}:\hspace{4pt}}\\*
\vspace{-2.65\baselineskip}
\begin{ColorVerbatim}
\vspace{-0.7\baselineskip}
\begin{Verbatim}[commandchars=\\\{\}]
\end{Verbatim}
\vspace{0.3\baselineskip}
\end{ColorVerbatim}
\renewcommand{\indexname}{Index}
\printindex
% End of document
\end{document}