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684 lines
46 KiB
5 years ago
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简介
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<li class="chapter " data-level="1.2" data-path="machine_learning.html">
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<a href="machine_learning.html">
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机器学习概述
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</a>
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常见机器学习算法
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.3.1" data-path="kNN.html">
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<a href="kNN.html">
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近朱者赤近墨者黑-kNN
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</a>
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</li>
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<li class="chapter " data-level="1.3.2" data-path="linear_regression.html">
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<a href="linear_regression.html">
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最简单的回归算法-线性回归
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</a>
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</li>
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<li class="chapter " data-level="1.3.3" data-path="logistic_regression.html">
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<a href="logistic_regression.html">
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使用回归的思想进行分类-逻辑回归
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</a>
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</li>
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<li class="chapter " data-level="1.3.4" data-path="decision_tree.html">
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<a href="decision_tree.html">
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最接近人类思维的算法-决策树
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</a>
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</li>
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<li class="chapter " data-level="1.3.5" data-path="random_forest.html">
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<a href="random_forest.html">
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群众的力量是伟大的-随机森林
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</a>
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</li>
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<li class="chapter " data-level="1.3.6" data-path="kMeans.html">
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<a href="kMeans.html">
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物以类聚人以群分-kMeans
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</a>
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</li>
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<li class="chapter active" data-level="1.3.7" data-path="AGNES.html">
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<a href="AGNES.html">
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以距离为尺-AGNES
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.4" data-path="metrics.html">
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<a href="metrics.html">
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模型评估指标
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.4.1" data-path="classification_metrics.html">
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<a href="classification_metrics.html">
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分类性能评估指标
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</a>
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</li>
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<li class="chapter " data-level="1.4.2" data-path="regression_metrics.html">
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<a href="regression_metrics.html">
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回归性能评估指标
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</a>
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</li>
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<li class="chapter " data-level="1.4.3" data-path="cluster_metrics.html">
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<a href="cluster_metrics.html">
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聚类性能评估指标
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.5" data-path="sklearn.html">
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<a href="sklearn.html">
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使用sklearn进行机器学习
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</a>
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</li>
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<li class="chapter " data-level="1.6" >
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<span>
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综合实战案例
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</span>
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<ul class="articles">
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<li class="chapter " data-level="1.6.1" >
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<span>
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泰坦尼克生还预测
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</span>
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<ul class="articles">
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<li class="chapter " data-level="1.6.1.1" data-path="titanic/introduction.html">
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<a href="titanic/introduction.html">
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简介
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</a>
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</li>
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<li class="chapter " data-level="1.6.1.2" data-path="titanic/EDA.html">
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<a href="titanic/EDA.html">
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探索性数据分析(EDA)
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</a>
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</li>
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<li class="chapter " data-level="1.6.1.3" data-path="titanic/feature engerning.html">
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<a href="titanic/feature engerning.html">
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特征工程
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</a>
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</li>
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<li class="chapter " data-level="1.6.1.4" data-path="titanic/fit and predict.html">
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<a href="titanic/fit and predict.html">
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构建模型进行预测
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</a>
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</li>
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<li class="chapter " data-level="1.6.1.5" data-path="titanic/tuning.html">
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<a href="titanic/tuning.html">
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调参
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.6.2" >
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<span>
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使用强化学习玩乒乓球游戏
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</span>
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<ul class="articles">
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<li class="chapter " data-level="1.6.2.1" data-path="pingpong/what is reinforce learning.html">
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<a href="pingpong/what is reinforce learning.html">
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什么是强化学习
|
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</a>
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</li>
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<li class="chapter " data-level="1.6.2.2" data-path="pingpong/Policy Gradient.html">
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<a href="pingpong/Policy Gradient.html">
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Policy Gradient原理
|
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</a>
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</li>
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<li class="chapter " data-level="1.6.2.3" data-path="pingpong/coding.html">
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<a href="pingpong/coding.html">
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使用Policy Gradient玩乒乓球游戏
|
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</a>
|
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.7" data-path="recommand.html">
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<a href="recommand.html">
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实训推荐
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</a>
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<li class="divider"></li>
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<li>
|
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<a href="https://www.gitbook.com" target="blank" class="gitbook-link">
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Published with GitBook
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<!-- Title -->
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<h1>
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<i class="fa fa-circle-o-notch fa-spin"></i>
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<a href="." >以距离为尺-AGNES</a>
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</h1>
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</div>
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<section class="normal markdown-section">
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<h1 id="以距离为尺-agnes算法">以距离为尺-AGNES算法</h1>
|
||
|
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<p>AGNES 算法是一种聚类算法，最初将每个对象作为一个簇，然后这些簇根据某些距离准则被一步步地合并。两个簇间的相似度有多种不同的计算方法。聚类的合并过程反复进行直到所有的对象最终满足簇数目。所以理解 AGNES 算法前需要先理解一些距离准则。</p>
|
||
|
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<h1 id="距离准则">距离准则</h1>
|
||
|
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<h2 id="为什么需要距离">为什么需要距离</h2>
|
||
|
|
<p>AGNES 算法是一种自底向上聚合的层次聚类算法，它先会将数据集中的每个样本看作一个初始簇，然后在算法运行的每一步中找出距离最近的两个簇进行合并，直至达到预设的簇的数量。所以AGNES算法需要不断的计算簇之间的距离，这也符合聚类的核心思想（物以类聚，人以群分），因此怎样度量两个簇之间的距离成为了关键。</p>
|
||
|
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<h2 id="距离的计算">距离的计算</h2>
|
||
|
|
<p>衡量两个簇之间的距离通常分为最小距离、最大距离和平均距离。在 AGNES 算法中可根据具体业务选择其中一种距离作为度量标准。</p>
|
||
|
|
<h3 id="最小距离">最小距离</h3>
|
||
|
|
<p>最小距离描述的是两个簇之间距离最近的两个样本所对应的距离。例如下图中圆圈和菱形分别代表两个簇，两个簇之间离得最近的样本的<strong>欧式距离</strong>为 3.3 ，则最小距离为 3.3。</p>
|
||
|
|
<p><img src="img/59.jpg" alt=""></p>
|
||
|
|
<p>假设给定簇<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">C_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>，则最小距离为：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>d</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mi>x</mi><mo>∈</mo><mi>i</mi><mo separator="true">,</mo><mi>z</mi><mo>∈</mo><mi>j</mi></mrow></msub><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">d_{min}=min_{x\in i,z\in j}dist(x,z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">=</span><span class="mord mathit">m</span><span class="mord mathit">i</span><span class="mord"><span class="mord mathit">n</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">x</span><span class="mrel
|
||
|
|
<h3 id="最大距离">最大距离</h3>
|
||
|
|
<p>最大距离描述的是两个簇之间距离最远的两个样本所对应的距离。例如下图中圆圈和菱形分别代表两个簇，两个簇之间离得最远的样本的<strong>欧式距离</strong>为 23.3 ，则最大距离为 23.3 。</p>
|
||
|
|
<p><img src="img/60.jpg" alt=""></p>
|
||
|
|
<p>假设给定簇<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">C_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>，则最大距离为：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>d</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><mi>m</mi><mi>a</mi><msub><mi>x</mi><mrow><mi>x</mi><mo>∈</mo><mi>i</mi><mo separator="true">,</mo><mi>z</mi><mo>∈</mo><mi>j</mi></mrow></msub><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">d_{min}=max_{x\in i,z\in j}dist(x,z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">=</span><span class="mord mathit">m</span><span class="mord mathit">a</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">x</span><span class="mrel
|
||
|
|
<h3 id="平均距离">平均距离</h3>
|
||
|
|
<p>平均距离描述的是两个簇之间样本的平均距离。例如下图中圆圈和菱形分别代表两个簇，计算两个簇之间的所有样本之间的欧式距离并求其平均值。</p>
|
||
|
|
<p><img src="img/61.jpg" alt=""></p>
|
||
|
|
<p>假设给定簇<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">C_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|C_i|,|C_j|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord mathrm">∣</span><span class="mpunct">,</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord mathrm">"
|
||
|
|
<h1 id="agnes-算法流程">AGNES 算法流程</h1>
|
||
|
|
<p>AGNES 算法是一种自底向上聚合的层次聚类算法，它先会将数据集中的每个样本看作一个<strong>初始簇</strong>，然后在算法运行的每一步中找出距离最近的两个簇进行合并，直至达到预设的簇的数量。</p>
|
||
|
|
<p>举个例子，现在先要将西瓜数据聚成两类，数据如下表所示：</p>
|
||
|
|
<table>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th>编号</th>
|
||
|
|
<th>体积</th>
|
||
|
|
<th>重量</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>1</td>
|
||
|
|
<td>1.2</td>
|
||
|
|
<td>2.3</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>2</td>
|
||
|
|
<td>3.6</td>
|
||
|
|
<td>7.1</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>3</td>
|
||
|
|
<td>1.1</td>
|
||
|
|
<td>2.2</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>4</td>
|
||
|
|
<td>3.5</td>
|
||
|
|
<td>6.9</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>5</td>
|
||
|
|
<td>1.5</td>
|
||
|
|
<td>2.5</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<p>一开始，每个样本都看成是一个簇( 1 号样本看成是 1 号簇， 2 号样本看成是 2 号簇，...， 5 号样本看成是 5 号簇)，假设簇的集合为 C=[[1], [2], [3], [4], [5]] 。</p>
|
||
|
|
<p>假设使用簇间最小距离来度量两个簇之间的远近，从表中可以看出 1 号簇与 3 号簇的簇间最小距离最小。因此需要将 1 号簇和 3 号簇合并，那么此时簇的集合 C=[[1, 3], [2], [4], [5]]。</p>
|
||
|
|
<p>然后继续看这 4 个簇中哪两个簇之间的最小距离最小，我们发现 2 号簇与 4 号簇的最小距离最小，因此我们要进行合并，合并之后 C=[[1, 3], [2, 4], [5]]。</p>
|
||
|
|
<p>然后继续看这 3 个簇中哪两个簇之间的最小距离最小，我们发现 5 号簇与 [1, 3] 簇的最小距离最小，因此我们要进行合并，合并之后 C=[[1, 3, 5], [2, 4]]。</p>
|
||
|
|
<p>这个时候 C 中只有两个簇了，达到了我们的预期目标（想要聚成两类），所以算法停止。算法停止后会发现，我们已经将 5 个西瓜，聚成了两类，一类是小西瓜，另一类是大西瓜。</p>
|
||
|
|
<p>如果将整个聚类过程中的合并，与合并的次序可视化出来，就能看出为什么说 AGNES 是自底向上的层次聚类算法了。</p>
|
||
|
|
<p><img src="img/62.jpg" alt=""></p>
|
||
|
|
<p>所以 AGNES 伪代码如下：</p>
|
||
|
|
<pre><code class="lang-python"><span class="hljs-comment">#假设数据集为D，想要聚成的簇的数量为k</span>
|
||
|
|
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">AGNES</span><span class="hljs-params">(D, k)</span>:</span>
|
||
|
|
<span class="hljs-comment">#C为聚类结果</span>
|
||
|
|
C = []
|
||
|
|
<span class="hljs-comment">#将每个样本看成一个簇</span>
|
||
|
|
<span class="hljs-keyword">for</span> d <span class="hljs-keyword">in</span> D:
|
||
|
|
C.append(d)
|
||
|
|
|
||
|
|
<span class="hljs-comment">#C中簇的数量</span>
|
||
|
|
q=len(C)
|
||
|
|
<span class="hljs-keyword">while</span> q > k:
|
||
|
|
寻找距离最小的两个簇a和b
|
||
|
|
将a和b合并，并修改C
|
||
|
|
q = len(C)
|
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<span class="hljs-keyword">return</span> C
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