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2013-03-15-Advanced-Cluster.html
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<body>
<div id="preamble">
</div>
<div id="content">
<h1 class="title">2013-03-15-Advanced-Cluster</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#sec-1">1 Advanced Clustering</a>
<ul>
<li><a href="#sec-1-1">1.1 Questions from last week</a></li>
</ul>
</li>
<li><a href="#sec-2">2 Review</a>
<ul>
<li><a href="#sec-2-1">2.1 Types</a></li>
</ul>
</li>
<li><a href="#sec-3">3 Topics?</a>
<ul>
<li><a href="#sec-3-1">3.1 Model by topic</a></li>
</ul>
</li>
<li><a href="#sec-4">4 Fuzzy Clusters</a>
<ul>
<li><a href="#sec-4-1">4.1 Restrictions</a></li>
</ul>
</li>
<li><a href="#sec-5">5 Generative Model</a>
<ul>
<li><a href="#sec-5-1">5.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-6">6 HW: 1-D Clustering</a>
<ul>
<li><a href="#sec-6-1">6.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-7">7 Parameters</a>
<ul>
<li><a href="#sec-7-1">7.1 Translation</a></li>
</ul>
</li>
<li><a href="#sec-8">8 Generative Model</a>
<ul>
<li><a href="#sec-8-1">8.1 3 Clusters</a></li>
</ul>
</li>
<li><a href="#sec-9">9 Best Fit?</a>
<ul>
<li><a href="#sec-9-1">9.1 Choose</a></li>
</ul>
</li>
<li><a href="#sec-10">10 Revisit k-means</a>
<ul>
<li><a href="#sec-10-1">10.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-11">11 Expectation-Maximization</a>
<ul>
<li><a href="#sec-11-1">11.1 k-means translation</a></li>
</ul>
</li>
<li><a href="#sec-12">12 Revisit k-means</a>
<ul>
<li><a href="#sec-12-1">12.1 Questions</a></li>
</ul>
</li>
<li><a href="#sec-13">13 Fuzzy Clustering</a>
<ul>
<li><a href="#sec-13-1">13.1 Change</a></li>
</ul>
</li>
<li><a href="#sec-14">14 Distance</a>
<ul>
<li><a href="#sec-14-1">14.1 Similarity</a></li>
</ul>
</li>
<li><a href="#sec-15">15 Distance<sup>2</sup></a>
<ul>
<li><a href="#sec-15-1">15.1 Similarity</a></li>
</ul>
</li>
<li><a href="#sec-16">16 Reset Center</a>
<ul>
<li><a href="#sec-16-1">16.1 Weighting by distance</a></li>
</ul>
</li>
<li><a href="#sec-17">17 Stability</a>
<ul>
<li><a href="#sec-17-1">17.1 Uses</a></li>
</ul>
</li>
<li><a href="#sec-18">18 <b>Break</b></a></li>
</ul>
</div>
</div>
<div id="outline-container-1" class="outline-2">
<h2 id="sec-1"><span class="section-number-2">1</span> Advanced Clustering <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-1-1" class="outline-3">
<h3 id="sec-1-1"><span class="section-number-3">1.1</span> Questions from last week <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-1-1">
<ul>
<li>Heuristic for choosing # of clusters? root(n/2)
</li>
<li>k-means: how to calculate the centroid?
</li>
<li>k-medoids: how to calculate the medoid?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-2" class="outline-2">
<h2 id="sec-2"><span class="section-number-2">2</span> Review <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-2">
<ul>
<li>Clustering groups points by using similarity
</li>
<li>Build up, or break down groups
</li>
<li>Each point belongs to 1 cluster
</li>
</ul>
</div>
<div id="outline-container-2-1" class="outline-3">
<h3 id="sec-2-1"><span class="section-number-3">2.1</span> Types <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-2-1">
<ul>
<li>Agglomerative, Divisive
</li>
<li>Assign each point to 1 centroid (k-means)
</li>
<li>or assign group clusters together, starting with every point as a cluster
(hierarchical)
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-3" class="outline-2">
<h2 id="sec-3"><span class="section-number-2">3</span> Topics? <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-3">
<ul>
<li>"This place is great. I've been here for meetings, to get work done, to hang out with friends, and on dates, and it's fit the bill every time. In the summer, there's a wonderful patio, and in the winter it's cozy and warm. All the food I've had is delicious – especially the salad dressing!"
</li>
</ul>
</div>
<div id="outline-container-3-1" class="outline-3">
<h3 id="sec-3-1"><span class="section-number-3">3.1</span> Model by topic <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-3-1">
<ul>
<li>ambiance,
</li>
<li>romantic
</li>
<li>good for work
</li>
<li>food quality
</li>
<li><b>Many</b>
</li>
<li>So how do we model?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-4" class="outline-2">
<h2 id="sec-4"><span class="section-number-2">4</span> Fuzzy Clusters <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-4">
<ul>
<li>Membership is a degree in [0-1]
</li>
<li><code>1 = sum(membership(v,c) for c in clusters)</code>
</li>
<li>Every point belongs to at least 1 cluster
</li>
</ul>
</div>
<div id="outline-container-4-1" class="outline-3">
<h3 id="sec-4-1"><span class="section-number-3">4.1</span> Restrictions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-4-1">
<ul>
<li>Every point still must be in a cluster
</li>
<li>Think of the degree as a probability
</li>
<li>Probabilities must add up to 100% (1)
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-5" class="outline-2">
<h2 id="sec-5"><span class="section-number-2">5</span> Generative Model <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-5">
<ul>
<li>"Real" model that produced original data points
</li>
<li>Our mission is to reproduce the original model
</li>
<li>Thus we have different techniques that can model different behavior
</li>
</ul>
</div>
<div id="outline-container-5-1" class="outline-3">
<h3 id="sec-5-1"><span class="section-number-3">5.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-5-1">
<ul>
<li>What is a "generative model"?
</li>
<li>What is data mining trying to discover? What is machine learning hoping to
reproduce?
</li>
<li>Why have different classifiers? Decision tree, Naive Bayes, etc?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-6" class="outline-2">
<h2 id="sec-6"><span class="section-number-2">6</span> HW: 1-D Clustering <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-6">
<ul>
<li>Imagine plotting the points on the number line
</li>
<li>I generated these points with a process
</li>
<li>Two Gaussian arrays, concatenated
</li>
</ul>
</div>
<div id="outline-container-6-1" class="outline-3">
<h3 id="sec-6-1"><span class="section-number-3">6.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-6-1">
<ul>
<li>Draw number line
</li>
<li>How did I come up with these numbers? I wanted two clusters.
</li>
<li>What parameters did I use, in the code to generate this specific set of N
numbers?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-7" class="outline-2">
<h2 id="sec-7"><span class="section-number-2">7</span> Parameters <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-7">
<ul>
<li>Median
</li>
<li>Standard Deviation
</li>
<li>How many points to generate from each
</li>
</ul>
<p> <img src="img/gaussian-simple.png" alt="img/gaussian-simple.png" />
</p>
</div>
<div id="outline-container-7-1" class="outline-3">
<h3 id="sec-7-1"><span class="section-number-3">7.1</span> Translation <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-7-1">
<ul>
<li>I had two distributions (median, stddev)
</li>
<li>Now I picked one or the other with a certain probability
</li>
<li>Then generated a number from it
</li>
<li>In reality, just generated 10 from A, 10 from B, but you can imagine that
being 50% 50%
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-8" class="outline-2">
<h2 id="sec-8"><span class="section-number-2">8</span> Generative Model <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-8">
<p> <img src="img/gaussian.png" alt="img/gaussian.png" />
</p>
</div>
<div id="outline-container-8-1" class="outline-3">
<h3 id="sec-8-1"><span class="section-number-3">8.1</span> 3 Clusters <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-8-1">
<ul>
<li>We have all three parameters:
</li>
<li>median
</li>
<li>stddev
</li>
<li>probability of choosing distribution (height)
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-9" class="outline-2">
<h2 id="sec-9"><span class="section-number-2">9</span> Best Fit? <span class="tag"><span class="slide">slide</span> <span class="two_col">two_col</span></span></h2>
<div class="outline-text-2" id="text-9">
<p> <img src="img/gaussian-badfit.png" alt="img/gaussian-badfit.png" />
</p>
<p>
<img src="img/gaussian-goodfit.png" alt="img/gaussian-goodfit.png" />
</p>
</div>
<div id="outline-container-9-1" class="outline-3">
<h3 id="sec-9-1"><span class="section-number-3">9.1</span> Choose <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-9-1">
<ul>
<li>These letters are points on our number line
</li>
<li>Which is more likely to be generated by our real model?
</li>
<li>But we don't know the generative model, so how do we discover it?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-10" class="outline-2">
<h2 id="sec-10"><span class="section-number-2">10</span> Revisit k-means <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-10">
<ul>
<li>Each object assigned to closest cluster
</li>
<li>Reset center of the cluster to average
</li>
<li>Repeat until steady
</li>
</ul>
</div>
<div id="outline-container-10-1" class="outline-3">
<h3 id="sec-10-1"><span class="section-number-3">10.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-10-1">
<ul>
<li>What are the steps of k-means?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-11" class="outline-2">
<h2 id="sec-11"><span class="section-number-2">11</span> Expectation-Maximization <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-11">
<dl>
<dt>Expectation</dt><dd>Given current state, create a solution that fits our
expectations
</dd>
<dt>Maximization</dt><dd>Adjust the state to maximize the likelihood of the solution
being true
</dd>
<dt>Terminate</dt><dd>When adjustments do not change
</dd>
</dl>
</div>
<div id="outline-container-11-1" class="outline-3">
<h3 id="sec-11-1"><span class="section-number-3">11.1</span> k-means translation <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-11-1">
<ul>
<li>Our expectation in k-means is that points belong to the cluster closest to
them
</li>
<li>Our state or parameters for our model are the locations of the centers of
those clusters
</li>
<li>The maximization step therefore moves the centers to maximize the
likelihood of their being the true center
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-12" class="outline-2">
<h2 id="sec-12"><span class="section-number-2">12</span> Revisit k-means <span class="tag"><span class="slide">slide</span> <span class="animate">animate</span></span></h2>
<div class="outline-text-2" id="text-12">
<ul>
<li>Each object assigned to closest cluster
</li>
<li>Reset center of the cluster to average
</li>
<li>Repeat until steady
</li>
</ul>
</div>
<div id="outline-container-12-1" class="outline-3">
<h3 id="sec-12-1"><span class="section-number-3">12.1</span> Questions <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-12-1">
<ul>
<li>What are the steps of k-means?
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-13" class="outline-2">
<h2 id="sec-13"><span class="section-number-2">13</span> Fuzzy Clustering <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-13">
<ul>
<li><del>Each object assigned to closest cluster</del>
</li>
<li>Each object assigned <b>probability</b> of cluster
</li>
<li><del>Reset center of the cluster to average</del>
</li>
<li>Reset center of the cluster to <b>weighted</b> average
</li>
<li>Repeat until steady
</li>
</ul>
</div>
<div id="outline-container-13-1" class="outline-3">
<h3 id="sec-13-1"><span class="section-number-3">13.1</span> Change <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-13-1">
<ul>
<li>Only difference here is that we're calculating the probability of a point
belonging to a cluster
</li>
<li>What should we base that probability off of? distance
</li>
<li>If point A has a high probability of belonging to cluster C, what can you
say about A and C? Close
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-14" class="outline-2">
<h2 id="sec-14"><span class="section-number-2">14</span> Distance <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-14">
<ul>
<li><code>dist(o,C) / sum(dist(o,c) for c in clusters))</code>
</li>
</ul>
</div>
<div id="outline-container-14-1" class="outline-3">
<h3 id="sec-14-1"><span class="section-number-3">14.1</span> Similarity <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-14-1">
<ul>
<li>Our old friend distance
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-15" class="outline-2">
<h2 id="sec-15"><span class="section-number-2">15</span> Distance<sup>2</sup> <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-15">
<ul>
<li><code>dist(o,C)**2 / sum(dist(o,c)**2 for c in clusters))</code>
</li>
<li>Squared distance to primary cluster, divided by squared distance to all
clusters
</li>
</ul>
</div>
<div id="outline-container-15-1" class="outline-3">
<h3 id="sec-15-1"><span class="section-number-3">15.1</span> Similarity <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-15-1">
<ul>
<li>Our old friend distance
</li>
<li>And squared, to make sure we stay positive
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-16" class="outline-2">
<h2 id="sec-16"><span class="section-number-2">16</span> Reset Center <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-16">
<ul>
<li><code>sum(weight[c][p]**2 * p for p in points)</code>
</li>
<li>Squared weight for this point in this cluster, multiplied by point
coordinates
</li>
</ul>
</div>
<div id="outline-container-16-1" class="outline-3">
<h3 id="sec-16-1"><span class="section-number-3">16.1</span> Weighting by distance <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-16-1">
<ul>
<li>Distance of point affects how much a cluster center is pulled toward it
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-17" class="outline-2">
<h2 id="sec-17"><span class="section-number-2">17</span> Stability <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-17">
<ul>
<li>When our centroids stabilize, we can estimate the parameters of our
distributions
</li>
<li>Or use probabilities of points directly
</li>
</ul>
</div>
<div id="outline-container-17-1" class="outline-3">
<h3 id="sec-17-1"><span class="section-number-3">17.1</span> Uses <span class="tag"><span class="notes">notes</span></span></h3>
<div class="outline-text-3" id="text-17-1">
<ul>
<li>Sometimes you may not need original parameters
</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-18" class="outline-2">
<h2 id="sec-18"><span class="section-number-2">18</span> <b>Break</b> <span class="tag"><span class="slide">slide</span></span></h2>
<div class="outline-text-2" id="text-18">
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</div>
<div id="postamble">
<p class="date">Date: 2013-03-15 13:43:10 PDT</p>
<p class="author">Author: Jim Blomo</p>
<p class="creator">Org version 7.8.02 with Emacs version 23</p>
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