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<!-- Title -->
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<h1>
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<a href="." >物以类聚人以群分-kMeans</a>
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<h1 id="物以类聚人以群分-k-means">物以类聚人以群分-k Means</h1>
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<p><strong>k Means</strong>是属于机器学习里面的非监督学习，通常是大家接触到的第一个聚类算法，其原理非常简单，是一种典型的基于<strong>距离</strong>的聚类算法。<strong>距离</strong>指的是每个样本到质心的距离。那么，这里所说的质心是什么呢？</p>
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<p>其实，质心指的是样本每个特征的均值所构成的一个坐标。举个例子：假如有两个数据 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1,1)</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:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span> 和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2,2)</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:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathrm">2</span><span class="mclose">)</span></span></span></span> 则这两个样本的质心为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mi mathvariant="normal">.</mi><mn>5</mn><mo separator="true">,</mo><mn>1</mn><mi mathvariant="normal">.</mi><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1.5,1.5)</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:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">5</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">5</span><span class="mclose">)</span></span></span></span>。</p>
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<p>同样的，如果一份数据有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">m</span></span></span></span> 个样本，每个样本有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span> 个特征，用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>x</mi><mi>i</mi><mi>j</mi></msubsup></mrow><annotation encoding="application/x-tex">x_i^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.942572em;"></span><span class="strut bottom" style="height:1.219436em;vertical-align:-0.276864em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.276864em;margin-left:0em;margin-right:0.05em;"><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 style="top:-0.480908em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped 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> 来表示第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span> 个样本的第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span> 个特征，则它们的质心为：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi><mi>m</mi><mi>a</mi><mi>s</mi><mi>s</mi><mo>=</mo><mo>(</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msubsup><mi>x</mi><mn>1</mn><mi>j</mi></msubsup></mrow><mrow><mi>m</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msubsup><mi>x</mi><mn>2</mn><mi>j</mi></msubsup></mrow><mrow><mi>m</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msubsup><mi>x</mi><mi>n</mi><mi>j</mi></msubsup></mrow><mrow><mi>m</mi></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">Cmass=(\frac{\sum_{j=1}^mx_1^j}{m},\frac{\sum_{j=1}^mx_2^j}{m},...,\frac{\sum_{j=1}^mx_n^j}{m})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.3002669999999998em;"></span><span class="strut bottom" style="height:1.6452669999999998em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mord mathit">m</span><span class="mord mathit">a</span><span class="mord mathit">s</span><span class="mord mathit">s</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord reset-textstyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><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></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.631927em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="top:0.074995em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.32101em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord scriptscriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.43100000000000005em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord mathit mtight">m</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 mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist"><span style="top:0.27551428571428577em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span style="top:-0.48368571428571433em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped 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="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><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></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.631927em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="top:0.074995em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.32101em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord scriptscriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.43100000000000005em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord mathit mtight">m</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 mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist"><span style="top:0.27551428571428577em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span style="top:-0.48368571428571433em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped 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="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><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></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.631927em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="top:0.074995em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.32101em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord scriptscriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.43100000000000005em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord mathit mtight">m</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 mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist"><span style="top:0.286em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span style="top:-0.431em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped 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="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>。</p>
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<p>知道什么是质心后，就可以看看<strong>k Means算法</strong>的流程了。</p>
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<h2 id="k-means算法流程">k Means算法流程</h2>
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<p>使用<strong>k Means</strong>来聚类时需要首先定义参数<strong>k</strong>，<strong>k</strong>的意思是我想将数据聚成几个类别。假设<strong>k=3</strong>，就是将数据划分成<strong>3</strong>个类别。接下来就可以开始<strong>k Means</strong>算法的流程了，流程如下：</p>
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<p><code>1.</code>随机初始<strong>k</strong>个样本，作为类别中心。
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<code>2.</code>对每个样本将其标记为距离类别中心最近的类别。
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<code>3.</code>将每个类别的质心更新为新的类别中心。
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<code>4.</code>重复步骤<code>2</code>、<code>3</code>，直到类别中心的变化小于阈值。</p>
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<p>过程示意图如下（其中 X 表示类别的中心，数据点的颜色代表不同的类别）：</p>
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<p><img src="img/29.gif" alt=""></p>
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