pr_ml_dm.latex

\documentclass[a4paper,12pt]{book}
\usepackage{xeCJK,xunicode}
\usepackage{syntonly,fontspec,graphicx}
\usepackage{enumitem,verbatim,array,multirow}
\usepackage[bookmarks]{hyperref}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{scalerel,booktabs}
\hypersetup{
	colorlinks,
	linkcolor=blue
}
%\punctstyle{kaiming}
\makeatletter
\let\@afterindentfalse\@afterindenttrue
\@afterindenttrue
\makeatother
\setlength{\parindent}{2em}%中文缩进两个汉字位
\setCJKmainfont[BoldFont=SimHei,ItalicFont=KaiTi]{SimSun}
\setCJKmonofont{FangSong}
\newcommand*\conj[1]{\bar{#1}}
\newcommand*\mean[1]{\bar{#1}}
\newtheorem{theorem}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\theoremstyle{plain}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}{Example}[section]


\begin{document}
\part{Data Mining,Concepts and Techniques}
\chapter{认识数据}
\section{Basic Statistical Descriptions of Data}
当观察值太多时，计算中位数是一个耗时的任务，此时可以\emph{approximate}。
假设数据按照间隔分组，并且每组的个数已知。可用如下近似公式求中位数\[
	median = L_1 + \left( \frac{N/2 - (\sum freq)_l}{freq_{median}}
	\right)width
\]
其中，$L_1$表示中位区间的lower bound，$N$表示样本大小，$(\sum freq)_l$表
示所有小于中位区间的样本数目，$freq_{median}$表示中位区间的样本数目，
$width$表示中位区间的宽度

真实数据经常不是对称的，要么正偏，要么负偏，\emph{偏度不能决定mean和
median之间的关系}

\subsection{Pythagorean means}
\begin{itemize}
	\item $AM(x_1, \ldots, x_n) = \frac{1}{n}(x_1+ \cdots + x_n)$
	\item $GM(x_1, \ldots, x_n) = \sqrt[n]{x_1 \cdots x_n}$
	\item $HM(x_1, \ldots, x_n) = \frac{n}{\frac{1}{x_1} + \cdots +
		\frac{1}{x_n}}$
\end{itemize}
每种均值满足一下性质
\begin{itemize}
	\item 值保持 $M(x, \ldots, x) = x$
	\item 一阶线性 $M(bx_1, \ldots, bx_n) = bM(x_1, \ldots, x_n)$
	\item 均值 $min(x_1, \ldots, x_n) \leq M(x_1, \ldots, x_n) \leq
		max(x_1, \ldots, x_n)$
\end{itemize}
当所有的$x_i$正定时，\[
	min \leq HM \leq GM \leq AM \leq max
\]
\subsubsection{Weighted Arithmetic Mean}
$\bar{x} = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}$
\subsubsection{Harmonic Mean}
$min(x_1, \ldots, x_n) \leq H(x_1, \ldots, x_n) \leq n\, min(x_1,
\ldots, x_n)$


	
\section{Measuring Data Similarity and Dissimilarity}
\subsection{Proximity Measures for Binary Attributes}
\begin{table}
	\begin{tabular}{c c c c c }
		\toprule \\
		&\multicolumn{4}{c}{\textbf{\emph{Object}} $j$} \\ \cmidrule{2-5}
		\multirow{4}{*}{\textbf{\emph{Object}} $i$} && 1 & 0 & sum \\
				&1 & $q$ & $r$ & $q+r$\\
			   &0 & $s$ & $t$ & $s+t$\\
			&sum & $q+s$ & $r+t$ & $p$\\
		\bottomrule\\
		\label{tab:contingency_table}
	\end{tabular}
	\caption{Contingency Table for Binary Attributes}
\end{table}

\emph{Simple Matching Coefficient} = $\frac{Number of Matching
Attributes}{Number of Attributes} = \frac{q+t}{q+r+s+t}$

\emph{Jaccard Coefficient} = $\frac{q}{q+r+s}$


\end{document}