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author Jordi Gutiérrez Hermoso <jordigh@octave.org>
date Tue, 17 May 2016 23:35:36 -0400
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1 %%% BEGIN BEAMER PREAMBLE %%%
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2 \documentclass[blue]{beamer}
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3 \usepackage{bm, fourier, anyfontsize, xcolor}
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4 \newcommand{\MC}{\operatorname{MC}}
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5 \newcommand{\signum}{\operatorname{signum}}
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6 \newcommand{\IQR}{\operatorname{IQR}}
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8 \theoremstyle{definition}
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9 \newtheorem*{defn}{Definition}
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12 \mode<presentation>
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13 {
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14 \usetheme{boxes}
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15 \usecolortheme{crane}
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16 }
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17 \beamertemplatenavigationsymbolsempty
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19 \AtBeginSection[]
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20 {
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21 \begin{frame}<beamer>
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22 \frametitle{Outline}
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23 \tableofcontents[currentsection]
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24 \end{frame}
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25 }
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28 \usepackage{times}
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29 %%% END BEAMER PREAMBLE %%%
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32 %%% BEGIN METADATA %%%
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34 \author{Jordi G. H. $\langle$jordigh@octave.org$\rangle$ }
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37 \title{The Medcouple}
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38 \subtitle{A robust measure of skewness}
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39 \date{}
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41 %%% END METADATA%%%
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43 \begin{document}
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45 \begin{frame}
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46 \titlepage
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47 \end{frame}
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47
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49 \begin{frame}
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50 \frametitle{Outline}
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51 \tableofcontents
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52 \end{frame}
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53
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54 \section{Outliers and Boxplots}
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55
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56 \begin{frame}{What is an outlier?}
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57 \pause
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58 \begin{center}
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59 \pgfimage[height=2.5in]{img/normal-boxhistplot.pdf}
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60 \end{center}
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61 A simple answer: Tukey's boxplots
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62 \end{frame}
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63
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64 \begin{frame}{Anatomy of a boxplot}
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65 \begin{overlayarea}{\textwidth}{8cm}
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66 \only<1>{\pgfimage[width=4in]{img/normal-points}}
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67 \only<2>{\pgfimage[width=4in]{img/normal-boxplot}}
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68 \only<3>{\pgfimage[width=4in]{img/normal-boxplot-bare/base}}
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69 \only<4>{\pgfimage[width=4in]{img/normal-boxplot-bare/median}}
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70 \only<5>{\pgfimage[width=4in]{img/normal-boxplot-bare/q1q3}}
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71 \only<6>{\pgfimage[width=4in]{img/normal-boxplot-bare/IQR}}
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72 \only<7>{\pgfimage[width=4in]{img/normal-boxplot-bare/whiskers}}
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73 \only<8>{\pgfimage[width=4in]{img/normal-boxplot-bare/15}}
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74 \only<9>{\pgfimage[width=4in]{img/normal-boxplot-bare/outliers}}
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75 \end{overlayarea}
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76 \end{frame}
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77
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78 \begin{frame}{Anatomy of a boxplot}
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79 \begin{itemize}
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80 \item Why 1.5?
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81 \pause
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82 \item Tukey responded: ``it's less than 2 and more than 1''
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83 \end{itemize}
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84 \end{frame}
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85
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86 \begin{frame}{Outliers}
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87 \pause
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88 \begin{center}
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89 \pgfimage[height=2.5in]{img/normal-boxhistplot}
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90 \end{center}
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91
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92 The boxplot identifies $10$ outliers out of $1000$ points ($1\%$)
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93 \end{frame}
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94
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95 \begin{frame}{Skew distributions}
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96 Remember:
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97 \begin{center}
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98 \pgfimage[width=4in]{img/skew-distributions}
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99 \end{center}
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100 \end{frame}
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101
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102 \begin{frame}
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103 For skew distributions...
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104 \end{frame}
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105
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106 \begin{frame}
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107 \begin{overlayarea}{\textwidth}{8cm}
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108 \only<1>{
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109 \pgfimage[height=3in]{img/geometric-boxhistplot}
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110
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111 $433$ outliers out of $10 000$ points ($4.3\%$)
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112 }
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113 \only<2>{
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114 \pgfimage[height=3in]{img/boys-and-girls}
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115
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116 $578$ and $644$ outliers for actors and actresses respectively
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117 ($1.2\%$ and $3\%$)
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118 }
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119 \end{overlayarea}
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120 \end{frame}
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121
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122 \begin{frame}
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123 \begin{itemize}
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124 \item Too many outliers...
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125 \pause
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126 \item Idea: adjust whisker lengths taking into account skewness:
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127 \end{itemize}
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128 \emph{M. Hubert; E. Vandervieren (2008). "An adjusted boxplot for skewed
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129 distributions". Computational Statistics and Data Analysis 52
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130 (12): 5186-5201. doi:10.1016/j.csda.2007.11.008.}
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131 \end{frame}
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132
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133 \begin{frame}{Adjusted boxplot}
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134 \begin{overlayarea}{\textwidth}{3cm}
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135 \only<1>{
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136 Recall normal whiskers:
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137 % Trick to hide medcouple, use whiteout, so that the text gets
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138 % positioned the same with or without it.
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139 \begin{align*}
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140 \text{lower} &= Q_1 - 1.5 \IQR\textcolor{white}{e^{a \MC}} \\
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141 \text{higher} &= Q_3 + 1.5 \IQR\textcolor{white}{e^{b \MC}}
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142 \end{align*}
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143 }
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144 \only<2>{
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145 Instead, use adjusted whiskers:
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146 \begin{align*}
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147 \text{lower} &= Q_1 - 1.5 \IQR\textcolor{red}{e^{a \MC}} \\
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148 \text{higher} &= Q_3 + 1.5 \IQR\textcolor{red}{e^{b \MC}}
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149 \end{align*}
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150 \begin{itemize}
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151 \item[$\MC$] -- the \emph{medcouple}, a measure of skewness
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152 \item[$a, b$] -- parameters to fit across some sample distributions
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153 \end{itemize}
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154 }
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155 \end{overlayarea}
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156 \end{frame}
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157
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158 \begin{frame}{Adjusted boxplot}
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159 For the whiskers, Hubert and Vandervieren recommend:
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160 \[
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161 \begin{cases}
61
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162 [Q_1 - 1.5 \IQR e^{-3 \MC}, Q_3 + 1.5 \IQR e^{4 \MC}] &\text{if } \MC > 0 \\
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163 [Q_1 - 1.5 \IQR e^{-4 \MC}, Q_3 + 1.5 \IQR e^{3 \MC}] &\text{if } \MC < 0
60
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164 \end{cases}
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165 \]
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166 \pause
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167 Of course, if $\MC = 0$ (no skewness) then no adjustment
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168 \end{frame}
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169
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170 \begin{frame}
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171 Let's see some adjusted boxplots...
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172 \end{frame}
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173
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174 \begin{frame}
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175 \begin{overlayarea}{\textwidth}{8cm}
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176 \only<1>{
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177 \pgfimage[height=3in]{img/geometric-boxhistplot}
61
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178
60
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179 $433$ outliers out of $10 000$ points ($4.3\%$)
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180 }
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181 \only<2>{
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182 \pgfimage[height=3in]{img/geometric-boxhistplot-adjusted}
61
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183
60
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184 \textcolor{red}{$25$ outliers} out of $10 000$ points
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185 (\textcolor{red}{$0.25\%$}) (\textcolor{blue}{$\MC = 0.25$})
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186 }
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187 \end{overlayarea}
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188 \end{frame}
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189
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190 \begin{frame}
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191 \begin{overlayarea}{\textwidth}{8cm}
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192 \only<1>{
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193 \pgfimage[height=3in]{img/normal-boxhistplot}
61
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194
60
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195 $10$ outliers out of $1 000$ points ($1\%$)
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196 }
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197 \only<2>{
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198 \pgfimage[height=3in]{img/normal-boxhistplot-adjusted}
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199
60
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200 \textcolor{red}{$10$ outliers} out of $1 000$ points
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201 (\textcolor{red}{$1\%$}) (\textcolor{blue}{$\MC = 0.0006$})
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202 }
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203 \end{overlayarea}
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204 \end{frame}
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205
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206 \begin{frame}
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207 \begin{overlayarea}{\textwidth}{8cm}
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208 \only<1>{
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209 \pgfimage[height=3in]{img/boys-and-girls}
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210
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211 $578$ and $644$ outliers for actors and actresses respectively
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212 ($1.2\%$ and $3\%$)
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213 }
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214 \only<2>{
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215 \pgfimage[height=3in]{img/boys-and-girls-adjusted}
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216
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217 \textcolor{red}{$346$} and \textcolor{red}{$657$} outliers for
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218 actors and actresses respectively
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219 (\textcolor{red}{$0.69\%$} and \textcolor{red}{$3\%$})
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220 (\textcolor{blue}{$\MC = 0.12$} and \textcolor{blue}{$\MC = 0.231$})
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221
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222 }
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223 \end{overlayarea}
47
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224 \end{frame}
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225
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226 \section{The Medcouple}
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227
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228 \begin{frame}
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229 \emph{G. Brys; M. Hubert; A. Struyf (November 2004). "A Robust
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230 Measure of Skewness". Journal of Computational and Graphical
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231 Statistics 13 (4): 996-1017. doi:10.1198/106186004X12632.}
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232 \end{frame}
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233
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234 \begin{frame}{Motivation}
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235 Consider the quartile skewness:
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236 \[
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237 B_1 = \frac{(Q_3 - Q_2) - (Q_2 - Q_1)}{Q_3 - Q_1}
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238 \]
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239 $Q_2 = \text{median}$
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240 \end{frame}
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241
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242 \begin{frame}{Definition}
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243 Idea: compute this kernel over all couples split along the median:
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244 \[
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245 h(x_i, x_j) =
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246 \begin{cases}
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247 \frac{(x_i - m) - (m - x_j)}{x_i - x_j} \\
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248 \signum(p - 1 - i - j) & \text{if } x_i = m = x_j
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249 \end{cases}
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250 \]
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251 \pause
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252 where
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253 \begin{itemize}
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254 \item $m = \text{median}$
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255 \item $x_i \geq m \geq x_j$
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256 \item $p = |\{x_i \geq m\}|$
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257 \end{itemize}
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258 \pause
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259 \begin{defn}
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260 The \emph{medcouple} is the median of the kernel of all couples
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261 above.
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262 \end{defn}
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263 \end{frame}
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264
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265 \begin{frame}{Properties}
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266 It is easy to see that medcouple is
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267 \pause
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268 \begin{itemize}
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269 \item location-invariant
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270 \pause
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271 \item scale-invariant
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272 \pause
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273 \item between $-1$ and $1$
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274 \pause
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275 \item a measure of skewness
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276 \end{itemize}
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277 \end{frame}
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278
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279 \begin{frame}{Properties}
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280 The medcouple is a \emph{robust} measure of skewness.
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281 \pause
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282 \begin{defn}
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283 A statistic is \emph{robust} if it does not depend on the values
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284 of extreme values (outliers).
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285 \end{defn}
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286 \pause
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287 \begin{itemize}
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288 \item The median has maximum robustness. Its breakdown point is $50\%$.
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289 \pause
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290 \item The medcouple's breakdown point is $25\%$.
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291 \end{itemize}
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292 \end{frame}
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293
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294
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295 \begin{frame}{Computing the medcouple}
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296 \begin{center}
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297 \pgfimage[width=4in]{img/naive/x-orig.png}
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298 \end{center}
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299
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300 Take some $X$ random numbers.
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301 \end{frame}
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302
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303 \begin{frame}{Computing the medcouple}
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304 \begin{center}
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305 \pgfimage[width=4in]{img/naive/x-sorted.png}
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306 \end{center}
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307
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308 Sort them.
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309 \end{frame}
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310
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311 \begin{frame}{Computing the medcouple}
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312 \begin{center}
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313 \pgfimage[width=4in]{img/naive/sortx-red.png}
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314 \end{center}
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315
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316 Pick the median.
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317 \end{frame}
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318
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319 \begin{frame}{Computing the medcouple}
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320 \begin{overlayarea}{\textwidth}{8cm}
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321 \only<1>{
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322 \begin{center}
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323 \pgfimage[height=2in]{img/naive/medc-computation-init.png}
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324 \end{center}
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325 Split up $X$ into $X^+$ and $X^-$ along the median.
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326 }
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327 \only<2>{
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328 \begin{center}
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329 \pgfimage[height=2in]{img/naive/medc-computation.png}
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330 \end{center}
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331 Evaluate the kernel for all couples.
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332 \[
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333 \frac{ (x_i - x_m) - (x_m - x_j)}{x_i - x_j},
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334 \quad x_i \in X^+, \quad x_j \in X^-
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335 \]
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336 }
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337 \only<3>{
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338 \begin{center}
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339 \pgfimage[height=2in]{img/naive/medc-computation-done.png}
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340 \end{center}
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341 The median of this matrix is the medcouple.
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342 }
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343 \end{overlayarea}
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344 \end{frame}
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345
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346 \begin{frame}
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347 Problem: this algorithm is $O(n^2)$!!
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348 \end{frame}
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349
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350 \section{Computation of the Medcouple}
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351
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352 \begin{frame}
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353 Now pure CS:
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354 \pause
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355 \begin{problem}
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356 Find the median of a matrix with sorted rows and sorted columns.
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357 \end{problem}
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358 \end{frame}
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359
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360 \begin{frame}
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361 \emph{Donald B. Johnson; Tetsuo Mizoguchi (May 1978). "Selecting The
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362 Kth Element In X + Y And X1 + X2 +...+ Xm". SIAM Journal of
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363 Computing 7 (2): 147-153. doi:10.1137/0207013.}
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364 \pause
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365 \begin{itemize}
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366 \item Old paper, hard to read. But cool ideas!
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367 \pause
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368 \item Paper generalises to $n$-dimensional array with sorted slices
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369 \end{itemize}
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370 \end{frame}
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371
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372 \begin{frame}{Preliminaries}
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373 Helper algorithms:
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374 \pause
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375 \begin{description}
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376 \item[Selection Algorithm:] e.g. quickselect,
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377 \texttt{nth\_element} in C++ or \texttt{partition} in numpy, $O(n)$ time
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378 \pause
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379 \item[Weighted Median:] binary search using
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380 selection algorithm, $O(n)$ time.
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381 \end{description}
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382 \end{frame}
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383
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384 \begin{frame}
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385 We now make two important observations
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386 \end{frame}
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387
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388 \begin{frame}{Two observations}
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389 Compare against whole matrix in $O(n)$ time
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390 \begin{overlayarea}{\textwidth}{6cm}
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391 \only<1>{
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392 \begin{center}
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393 \pgfimage[width=2in]{img/kth-pair/greater-than}
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394 \end{center}
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395
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396 Bottom-up.
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397 }
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398 \only<2>{
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399 \begin{center}
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400 \pgfimage[width=2in]{img/kth-pair/less-than}
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401 \end{center}
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402
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403 Top-down.
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404 }
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405 \end{overlayarea}
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406 \end{frame}
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407
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408 \begin{frame}{Two observations}
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409 Compare against half the matrix in $O(1)$ time
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410 \begin{center}
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411 \pgfimage[width=2in]{img/kth-pair/middle-of-middle}
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412 \end{center}
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413 \end{frame}
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414
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415 \begin{frame}{Putting it together}
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416 Iteratively, we have bounds for the global median
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417 \begin{center}
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418 \pgfimage[width=2in]{img/kth-pair/row-medians}
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419 \end{center}
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420 \end{frame}
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421
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422 \begin{frame}{Putting it together}
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423 If we align the row medians...
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424 \begin{center}
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425 \pgfimage[width=2in]{img/kth-pair/row-medians-aligned}
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426 \end{center}
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427 \end{frame}
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428
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429 \begin{frame}{Putting it together}
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430 With the weighted median (in $O(n)$ time), we compare to at least half the
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431 remaining entries, discarding at least $1/4$.
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432 \begin{center}
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433 \pgfimage[width=2in]{img/kth-pair/row-medians-compared}
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434 \end{center}
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435 \end{frame}
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436
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437 \begin{frame}
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438 So, final algorithm!
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439 \end{frame}
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440
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441 \begin{frame}{Grand finale}
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442 Procedure (serves one medcouple)
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443
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444 \begin{enumerate}
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445 \item Compute ingredients for medcouple matrix: $O(n \log n)$
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446 \pause
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447 \begin{enumerate}[(A)]
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448 \item \label{loop} Compute $t = \text{median of row medians}$: $O(n)$
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449 \pause
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450 \item \label{compare} Compare $t$ to the whole matrix: $O(n)$
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451 \pause
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452 \item Found the median? Found the medcouple! DONE.
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453 \pause
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454 \item \label{compare} Use comparison from \eqref{compare} to
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455 throw out entries.
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456 \pause
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457 \item More than $n$ entries remaining? LOOP TO \eqref{loop}.
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458 \end{enumerate}
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459 \pause
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460 \item Use selection algorithm on remaining entries: $O(n)$.
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461 \end{enumerate}
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462 \pause
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463
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464 Step \eqref{compare} throws out at least $1/4$ of all entries, so
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465 the whole algorithm is $O(n \log n)$.
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466 \end{frame}
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467
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468 \begin{frame}
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469 \begin{center}
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470 \LARGE{Questions?}
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471 \end{center}
47
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472 \end{frame}
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473
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474 \end{document}