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<!DOCTYPE html>
<html lang="en"><head>
<script src="index_files/libs/clipboard/clipboard.min.js"></script>
<script src="index_files/libs/quarto-html/tabby.min.js"></script>
<script src="index_files/libs/quarto-html/popper.min.js"></script>
<script src="index_files/libs/quarto-html/tippy.umd.min.js"></script>
<link href="index_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/light-border.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="index_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.4.550">
<meta name="author" content="Rob Wiederstein">
<title>Outlier Analysis</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reset.css">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ color: #003b4f; background-color: #f1f3f5; }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span { color: #003b4f; } /* Normal */
code span.al { color: #ad0000; } /* Alert */
code span.an { color: #5e5e5e; } /* Annotation */
code span.at { color: #657422; } /* Attribute */
code span.bn { color: #ad0000; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #003b4f; } /* ControlFlow */
code span.ch { color: #20794d; } /* Char */
code span.cn { color: #8f5902; } /* Constant */
code span.co { color: #5e5e5e; } /* Comment */
code span.cv { color: #5e5e5e; font-style: italic; } /* CommentVar */
code span.do { color: #5e5e5e; font-style: italic; } /* Documentation */
code span.dt { color: #ad0000; } /* DataType */
code span.dv { color: #ad0000; } /* DecVal */
code span.er { color: #ad0000; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #ad0000; } /* Float */
code span.fu { color: #4758ab; } /* Function */
code span.im { color: #00769e; } /* Import */
code span.in { color: #5e5e5e; } /* Information */
code span.kw { color: #003b4f; } /* Keyword */
code span.op { color: #5e5e5e; } /* Operator */
code span.ot { color: #003b4f; } /* Other */
code span.pp { color: #ad0000; } /* Preprocessor */
code span.sc { color: #5e5e5e; } /* SpecialChar */
code span.ss { color: #20794d; } /* SpecialString */
code span.st { color: #20794d; } /* String */
code span.va { color: #111111; } /* Variable */
code span.vs { color: #20794d; } /* VerbatimString */
code span.wa { color: #5e5e5e; font-style: italic; } /* Warning */
/* CSS for citations */
div.csl-bib-body { }
div.csl-entry {
clear: both;
margin-bottom: 0em;
}
.hanging-indent div.csl-entry {
margin-left:2em;
text-indent:-2em;
}
div.csl-left-margin {
min-width:2em;
float:left;
}
div.csl-right-inline {
margin-left:2em;
padding-left:1em;
}
div.csl-indent {
margin-left: 2em;
} </style>
<link rel="stylesheet" href="index_files/libs/revealjs/dist/theme/quarto.css">
<link href="index_files/libs/revealjs/plugin/quarto-line-highlight/line-highlight.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/quarto-menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-chalkboard/font-awesome/css/all.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-chalkboard/style.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
.callout.callout-style-simple {
padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
.callout.callout-style-default {
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
.callout .callout-body-container {
flex-grow: 1;
}
.callout.callout-style-simple .callout-body {
font-size: 1rem;
font-weight: 400;
}
.callout.callout-style-default .callout-body {
font-size: 0.9rem;
font-weight: 400;
}
.callout.callout-titled.callout-style-simple .callout-body {
margin-top: 0.2em;
}
.callout:not(.callout-titled) .callout-body {
display: flex;
}
.callout:not(.no-icon).callout-titled.callout-style-simple .callout-content {
padding-left: 1.6em;
}
.callout.callout-titled .callout-header {
padding-top: 0.2em;
margin-bottom: -0.2em;
}
.callout.callout-titled .callout-title p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.callout.callout-titled.callout-style-simple .callout-content p {
margin-top: 0;
}
.callout.callout-titled.callout-style-default .callout-content p {
margin-top: 0.7em;
}
.callout.callout-style-simple div.callout-title {
border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
.callout.callout-style-default div.callout-title {
border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-default div.callout-content {
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-simple .callout-icon::before {
height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
}
.callout.callout-style-default .callout-icon::before {
height: 0.9rem;
width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
}
.callout-title {
display: flex
}
.callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
.callout.no-icon::before {
display: none !important;
}
.callout.callout-titled .callout-body > .callout-content > :last-child {
padding-bottom: 0.5rem;
margin-bottom: 0;
}
.callout.callout-titled .callout-icon::before {
margin-top: .5rem;
padding-right: .5rem;
}
.callout:not(.callout-titled) .callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
/* Callout Types */
div.callout-note {
border-left-color: #4582ec !important;
}
div.callout-note .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
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<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Outlier Analysis</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Rob Wiederstein
</div>
</div>
</div>
</section>
<section>
<section id="overview" class="title-slide slide level1 center">
<h1>Overview</h1>
</section>
<section id="illustration" class="slide level2">
<h2>Illustration</h2>
<div class="center-xy">
<table>
<tbody>
<tr class="odd">
<td style="text-align: left;"><strong>Princess Fiona: </strong></td>
<td style="text-align: left;">“What kind of knight are you?”</td>
</tr>
<tr class="even">
<td style="text-align: left;"><strong>Shrek:</strong></td>
<td style="text-align: left;">“One of a kind.”</td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="based-upon" class="slide level2">
<h2>Based Upon</h2>
<img data-src="./img/talagala_anomaly_detection.png" width="849" class="r-stretch quarto-figure-center"><p class="caption">Article<span class="citation" data-cites="talagala2021"><a href="#/bibliography" role="doc-biblioref" onclick="">[1]</a></span></p><aside class="notes">
<p>“We applied our stray algorithm to a dataset obtained from an automated pedestrian counting system with 43 sensors in the city of Melbourne, Australia (City of Melbourne 2019; Wang 2018), to identify unusual pedestrian activities within the municipality.” The article uses the KNN algorith and scagonostics to identify days of unusual activity.</p>
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</section>
<section id="roadmap" class="slide level2">
<h2>Roadmap</h2>
<ul>
<li>Basics</li>
<li>Distributions</li>
<li>Models (KNN)</li>
<li>Part B Claims Data</li>
<li>Scagnostics</li>
<li>Interactive Display</li>
</ul>
</section>
<section id="also-known-as" class="slide level2">
<h2>Also Known As</h2>
<p>“outliers, novelty, faults, deviants, discordant observations, extreme values/cases, change points, rare events, intrusions, misuses, exceptions, aberrations, surprises, peculiarities, odd values and contaminants”<span class="citation" data-cites="talagala2021"><a href="#/bibliography" role="doc-biblioref" onclick="">[1]</a></span></p>
</section>
<section id="definitions" class="slide level2">
<h2>Definitions</h2>
<ul>
<li><p><strong>Kurtosis</strong> – is a measure of the <span class="fragment highlight-red">tailedness</span> of a distribution. Tailedness is how often outliers occur.</p></li>
<li><p><strong>Outlier</strong> – “An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.”<span class="citation" data-cites="hawkins1980identification"><a href="#/bibliography" role="doc-biblioref" onclick="">[2]</a></span></p></li>
<li><p><strong>Skewness</strong> is a measure of the <span class="fragment highlight-red">asymetry</span> of the probability distribution of a real-valued random variable about its mean.</p></li>
<li><p><strong>Standardize</strong> scale all of the values in the dataset such that the mean value is 0 and the standard deviation is 1.</p></li>
</ul>
</section>
<section id="symbols" class="slide level2">
<h2>Symbols</h2>
<table>
<thead>
<tr class="header">
<th style="text-align: center;">Symbol</th>
<th style="text-align: center;">Short</th>
<th style="text-align: center;">Meaning</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(\mu\)</span></td>
<td style="text-align: center;">“mew”</td>
<td style="text-align: center;">mean</td>
</tr>
<tr class="even">
<td style="text-align: center;"><span class="math inline">\(\sigma\)</span></td>
<td style="text-align: center;">“sigma”</td>
<td style="text-align: center;">std. dev.</td>
</tr>
</tbody>
</table>
<aside class="notes">
<p>None.</p>
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</section></section>
<section>
<section id="basics" class="title-slide slide level1 center">
<h1>Basics</h1>
</section>
<section id="outliers-classified" class="slide level2">
<h2>Outliers Classified</h2>
<img data-src="./img/outliers_classified.png" width="644" class="r-stretch quarto-figure-center"><p class="caption"><span class="math inline">\(c_1\)</span> and <span class="math inline">\(c_2\)</span> are clusters; <span class="math inline">\(x_1\)</span> and <span class="math inline">\(x_2\)</span> are global anomalies; <span class="math inline">\(x_3\)</span> is a local anomaly; and <span class="math inline">\(c_3\)</span> is potentially ambiguous. <span class="citation" data-cites="goldstein2016"><a href="#/bibliography" role="doc-biblioref" onclick="">[3]</a></span></p><aside class="notes">
<p>“Two anomalies can be easily identified by eye: x1 and x2 are very different from the dense areas with respect to their attributes and are therefore called global anomalies. When looking at the dataset globally, x3 can be seen as a normal record since it is not too far away from the cluster c2. However, when we focus only on the cluster c2 and compare it with x3 while neglecting all the other instances, it can be seen as an anomaly. Therefore, x3 is called a local anomaly, since it is only anomalous when compared with its close-by neighborhood.”<span class="citation" data-cites="goldstein2016"><a href="#/bibliography" role="doc-biblioref" onclick="">[3]</a></span></p>
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</section>
<section id="continuum-of-outlierness" class="slide level2">
<h2>Continuum of Outlierness</h2>
<img data-src="img/outlierness.png" width="858" class="r-stretch"></section>
<section id="univariate-outliers" class="slide level2">
<h2>Univariate Outliers</h2>
<blockquote>
<p>The detection of outliers in the observed distribution of a single variable spans the entire history of outlier detection. It spans this history not only because it is the simplest formulation of the problem, but also because it is deceptively simple.<span class="citation" data-cites="wilkinson2018"><a href="#/bibliography" role="doc-biblioref" onclick="">[4]</a></span></p>
</blockquote>
<p><span class="math display">\[
\{1, 2, 3, 4, 50, 97, 98, 99\}
\]</span></p>
</section>
<section id="distance-from-the-center-rule" class="slide level2">
<h2>Distance from the Center Rule</h2>
<blockquote>
<p>“The word outlier implies lying at an extreme end of a set of ordered values – far away from the center of those values. The modern history of outlier detection emerged with methods that depend on a measure of centrality and a distance from that measure of centrality.” <span class="citation" data-cites="wilkinson2018"><a href="#/bibliography" role="doc-biblioref" onclick="">[4]</a></span></p>
</blockquote>
<p><span class="math display">\[
\{1, 47, 47, 49, 51, 52, 55, 100\}
\]</span></p>
</section>
<section id="common-outlier-definitions" class="slide level2">
<h2>Common Outlier Definitions</h2>
<ul>
<li><p>1.5 x the inter quartile range - Tukey</p></li>
<li><p>3.0 x the standard deviation</p></li>
<li><p><span class="fragment highlight-red">Percentile?</span></p></li>
</ul>
<aside class="notes">
<p>1.5 IQR -</p>
<p>3x st dev - this rule uses the mean and the standard deviation, more appropriate for symetric distributions.</p>
<p>All rules for identifying outliers are arbitrary</p>
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</section>
<section id="four-methods-to-identify-outliers" class="slide level2">
<h2>Four Methods to Identify Outliers</h2>
<ol type="1">
<li>Extreme Value Analysis</li>
<li>Probabilistic and Statistic Models</li>
<li>Linear Models</li>
<li>Proximity-Based Models
<ul>
<li>Cluster</li>
<li>Density</li>
<li><span class="fragment highlight-red">Distance</span> <==(We are here!)</li>
</ul></li>
</ol>
<aside class="notes">
<p>EVA: “The most basic form of outlier detection is extreme-value analysis of 1-dimensional data. These are very specific types of outliers in which it is assumed that the values that are either too large or too small are outliers.” Singh and Upadhyaya 2012 “The key is to determine the statistical tails of the underlying distribution.” PSA: “In probabilistic and statistical models, the data is modeled in the form of a closed-form probability distribution, and the parameters of this model are learned.” LM: These methods model the data along lower-dimensional subspaces with the use of linear correlations. PB: “Proximity-based methods are among the most popular class of methods used in outlier analysis. Proximity-based methods may be applied in one of three ways, which are clustering methods, density-based methods”</p>
<p>-Proximity based. “Proximity-based techniques define a data point as an outlier when its locality (or proximity) is sparsely populated.”<span class="citation" data-cites="aggarwal2017"><a href="#/bibliography" role="doc-biblioref" onclick="">[5]</a></span> Cluster, Distance and Density based. “The distance of a data point to its k-nearest neighbor (or other variant) is used in order to define proximity.”<span class="citation" data-cites="aggarwal2017"><a href="#/bibliography" role="doc-biblioref" onclick="">[5]</a></span></p>
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</section>
<section id="tools-density-plot" class="slide level2">
<h2>Tools Density Plot</h2>
<img data-src="./img/Standard_deviation_diagram.svg" class="r-stretch"><aside class="notes">
<p>About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations.[6] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.</p>
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</section>
<section id="tools-histogram-binning" class="slide level2">
<h2>Tools Histogram (Binning)</h2>
<img data-src="index_files/figure-revealjs/histogram-example-1.png" width="960" class="r-stretch"></section>
<section id="tools-boxplots" class="slide level2">
<h2>Tools Boxplots</h2>
<img data-src="./img/boxplot_explained.png" width="751" class="r-stretch"><aside class="notes">
<p>“In descriptive statistics, a box plot or boxplot is a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles.[1] In addition to the box on a box plot, there can be lines (which are called whiskers) extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also called the box-and-whisker plot and the box-and-whisker diagram.”</p>
<p>“The range-bar method was first introduced by Mary Eleanor Spear in her book”Charting Statistics” in 1952[4] and again in her book “Practical Charting Techniques” in 1969.[5] The box-and-whisker plot was first introduced in 1970 by John Tukey, who later published on the subject in his book “Exploratory Data Analysis” in 1977.”</p>
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</section></section>
<section>
<section id="distributions" class="title-slide slide level1 center">
<h1>Distributions</h1>
</section>
<section id="normal" class="slide level2">
<h2>Normal</h2>
<img data-src="index_files/figure-revealjs/normal-dist-plot-1.png" width="960" class="r-stretch"><div class="cell">
<div class="cell-output-display">
<div>
<table class="table table-striped table-hover table-condensed" data-quarto-postprocess="true" style="font-size: 20px; margin-left: auto; margin-right: auto;">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th"></th>
<th style="text-align: right;" data-quarto-table-cell-role="th">vars</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">n</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">mean</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">sd</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">median</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">min</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">max</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">skew</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">kurtosis</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">se</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">y1</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">0.11</td>
<td style="text-align: right;">0.90</td>
<td style="text-align: right;">0.11</td>
<td style="text-align: right;">-2.21</td>
<td style="text-align: right;">2.40</td>
<td style="text-align: right;">-0.07</td>
<td style="text-align: right;">-0.05</td>
<td style="text-align: right;">0.09</td>
</tr>
<tr class="even">
<td style="text-align: left;">y2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">-0.08</td>
<td style="text-align: right;">1.92</td>
<td style="text-align: right;">-0.35</td>
<td style="text-align: right;">-3.83</td>
<td style="text-align: right;">4.62</td>
<td style="text-align: right;">0.44</td>
<td style="text-align: right;">-0.31</td>
<td style="text-align: right;">0.19</td>
</tr>
<tr class="odd">
<td style="text-align: left;">y3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">0.09</td>
<td style="text-align: right;">3.10</td>
<td style="text-align: right;">0.00</td>
<td style="text-align: right;">-8.67</td>
<td style="text-align: right;">7.95</td>
<td style="text-align: right;">-0.24</td>
<td style="text-align: right;">0.26</td>
<td style="text-align: right;">0.31</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
<section id="zipf" class="slide level2">
<h2>Zipf</h2>
<img data-src="index_files/figure-revealjs/zipf-distribution-1.png" width="960" class="r-stretch"><div class="cell">
<div class="cell-output-display">
<div>
<table class="table table-striped table-hover table-condensed" data-quarto-postprocess="true" style="font-size: 20px; margin-left: auto; margin-right: auto;">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th"></th>
<th style="text-align: right;" data-quarto-table-cell-role="th">vars</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">n</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">mean</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">sd</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">median</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">min</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">max</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">skew</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">kurtosis</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">se</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">y1</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">139.14</td>
<td style="text-align: right;">198.89</td>
<td style="text-align: right;">30.5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">951</td>
<td style="text-align: right;">1.83</td>
<td style="text-align: right;">3.07</td>
<td style="text-align: right;">19.89</td>
</tr>
<tr class="even">
<td style="text-align: left;">y2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">237.34</td>
<td style="text-align: right;">270.50</td>
<td style="text-align: right;">123.0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">976</td>
<td style="text-align: right;">1.23</td>
<td style="text-align: right;">0.40</td>
<td style="text-align: right;">27.05</td>
</tr>
<tr class="odd">
<td style="text-align: left;">y3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">274.25</td>
<td style="text-align: right;">287.68</td>
<td style="text-align: right;">154.0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">965</td>
<td style="text-align: right;">1.08</td>
<td style="text-align: right;">-0.11</td>
<td style="text-align: right;">28.77</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<aside class="notes">
<p>Zipf’s law is an empirical law that often holds, approximately, when a list of measured values is sorted in decreasing order. It states that the value of the nth entry is inversely proportional to n. The best known instance of Zipf’s law applies to the frequency table of words in a text or corpus of natural language: word frequency ∝ 1 word rank . {.}</p>
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</section>
<section id="log" class="slide level2">
<h2>Log</h2>
<img data-src="index_files/figure-revealjs/log-distribution-1.png" width="960" class="r-stretch"><div class="cell">
<div class="cell-output-display">
<div>
<table class="table table-striped table-hover table-condensed" data-quarto-postprocess="true" style="font-size: 20px; margin-left: auto; margin-right: auto;">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th"></th>
<th style="text-align: right;" data-quarto-table-cell-role="th">vars</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">n</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">mean</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">sd</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">median</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">min</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">max</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">skew</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">kurtosis</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">se</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">y1</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">1.65</td>
<td style="text-align: right;">1.72</td>
<td style="text-align: right;">1.12</td>
<td style="text-align: right;">0.11</td>
<td style="text-align: right;">11.04</td>
<td style="text-align: right;">2.86</td>
<td style="text-align: right;">10.49</td>
<td style="text-align: right;">0.17</td>
</tr>
<tr class="even">
<td style="text-align: left;">y2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">6.25</td>
<td style="text-align: right;">16.53</td>
<td style="text-align: right;">0.70</td>
<td style="text-align: right;">0.02</td>
<td style="text-align: right;">101.08</td>
<td style="text-align: right;">3.89</td>
<td style="text-align: right;">15.74</td>
<td style="text-align: right;">1.65</td>
</tr>
<tr class="odd">
<td style="text-align: left;">y3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">60.71</td>
<td style="text-align: right;">333.89</td>
<td style="text-align: right;">1.00</td>
<td style="text-align: right;">0.00</td>
<td style="text-align: right;">2828.50</td>
<td style="text-align: right;">7.10</td>
<td style="text-align: right;">51.51</td>
<td style="text-align: right;">33.39</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
<section id="z-value-test" class="slide level2">
<h2>Z-value Test</h2>
<p><span class="math display">\[
Z_1 = \frac{| X_1 - \mu |} \sigma
\]</span></p>
<p>where <span class="math inline">\(X_1\)</span> = observation, <span class="math inline">\(\mu\)</span> = mean, and <span class="math inline">\(\sigma\)</span> = standard deviation</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href=""></a><span class="co"># in R</span></span>
<span id="cb1-2"><a href=""></a>df<span class="sc">$</span>z <span class="ot"><-</span> (df<span class="sc">$</span>points<span class="sc">-</span><span class="fu">mean</span>(df<span class="sc">$</span>points))<span class="sc">/</span><span class="fu">sd</span>(df<span class="sc">$</span>points)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<aside class="notes">
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</section>
<section id="normalizedstandardized" class="slide level2">
<h2>Normalized/Standardized</h2>
<img data-src="index_files/figure-revealjs/uniform-example-1.png" width="960" class="r-stretch"><div class="cell">
<div class="cell-output-display">
<div>
<table class="table table-striped table-hover table-condensed" data-quarto-postprocess="true" style="font-size: 20px; margin-left: auto; margin-right: auto;">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th"></th>
<th style="text-align: right;" data-quarto-table-cell-role="th">vars</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">n</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">mean</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">sd</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">median</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">min</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">max</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">skew</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">kurtosis</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">se</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">uniform</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">25.97</td>
<td style="text-align: right;">13.94</td>
<td style="text-align: right;">25.94</td>
<td style="text-align: right;">0.59</td>
<td style="text-align: right;">48.78</td>
<td style="text-align: right;">-0.03</td>
<td style="text-align: right;">-1.17</td>
<td style="text-align: right;">1.39</td>
</tr>
<tr class="even">
<td style="text-align: left;">transformed</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">0.00</td>
<td style="text-align: right;">1.00</td>
<td style="text-align: right;">0.00</td>
<td style="text-align: right;">-1.82</td>
<td style="text-align: right;">1.64</td>
<td style="text-align: right;">-0.03</td>
<td style="text-align: right;">-1.17</td>
<td style="text-align: right;">0.10</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</section></section>
<section>
<section id="boxplots" class="title-slide slide level1 center">
<h1>Boxplots</h1>
</section>
<section id="ziph-box-plots" class="slide level2">
<h2>Ziph Box Plots</h2>
<img data-src="index_files/figure-revealjs/zipf-box-plots-1.png" width="960" class="r-stretch"></section>
<section id="normal-box-plot" class="slide level2">
<h2>Normal Box Plot</h2>
<img data-src="index_files/figure-revealjs/nd-box-plots-1.png" width="960" class="r-stretch"></section></section>
<section>
<section id="models" class="title-slide slide level1 center">
<h1>Models</h1>
</section>
<section id="knn" class="slide level2">