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<!-- Title -->
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<a href="." >聚类性能评估指标</a>
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<h1 id="聚类模型性能评估指标">聚类模型性能评估指标</h1>
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<p>聚类的性能度量大致分为两类：一类是将聚类结果与某个参考模型作为参照进行比较，也就是所谓的<strong>外部指标</strong>；另一类是则是直接度量聚类的性能而不使用参考模型进行比较，也就是<strong>内部指标</strong>。</p>
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<h2 id="外部指标">外部指标</h2>
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<p><strong>外部指标通常使用 Jaccard Coefficient(JC系数)、Fowlkes and Mallows Index(FM指数)以及 Rand index（Rand指数）。</strong></p>
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<p>想要计算上述指标来度量聚类的性能，首先需要计算出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</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">e</span></span></span></span>。假设数据集<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>E</mi><mo>=</mo><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>m</mi></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">E=\{x_1,x_2,...,x_m\}</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="mord mathit" style="margin-right:0.05764em;">E</span><span class="mrel">=</span><span class="mopen">{</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 mathrm mtight">1</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="mpunct">,</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 mathrm mtight">2</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="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mpunct">,</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 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="mclose">}</span></span></span></span>。通过聚类模型给出的簇划分为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi><mo>=</mo><mo>{</mo><msub><mi>C</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>C</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>C</mi><mi>k</mi></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">C=\{C_1,C_2,...C_k\}</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="mord mathit" style="margin-right:0.07153em;">C</span><span class="mrel">=</span><span class="mopen">{</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 mathrm mtight">1</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="mpunct">,</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 mathrm mtight">2</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="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</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:0.03148em;">k</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="mclose">}</span></span></span></span>，参考模型给出的簇划分为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi><mo>=</mo><mo>{</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>D</mi><mi>s</mi></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">D=\{D_1,D_2,...D_s\}</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="mord mathit" style="margin-right:0.02778em;">D</span><span class="mrel">=</span><span class="mopen">{</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</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="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</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="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02778em;"><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">s</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="mclose">}</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">λ</span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\lambda^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;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="mbin mtight">∗</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>分别表示<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</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:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</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:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span></span>对应的簇标记，则有:</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mo>{</mo><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo>)</mo><mi mathvariant="normal">∣</mi><msub><mi>λ</mi><mi>i</mi></msub><mo>=</mo><msub><mi>λ</mi><mi>j</mi></msub><mo separator="true">,</mo><msubsup><mi>λ</mi><mi>i</mi><mo>∗</mo></msubsup><mo>=</mo><msubsup><mi>λ</mi><mi>j</mi><mo>∗</mo></msubsup><mo separator="true">,</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>}</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">
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a=|\{(x_i, x_j)|\lambda_i=\lambda_j, \lambda^*_i=\lambda^*_j,i < j\}|
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</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:1.133108em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mopen">{</span><span class="mopen">(</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 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="mpunct">,</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 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="mclose">)</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">λ</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 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="mrel">=</span><span class="mord"><span class="mord mathit">λ</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 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="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.247em;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:-0.363em;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="mbin mtight">∗</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"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.24700000000000003em;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" style="margin-right:0.05724em;">j</span></span></span><span style="top:-0.363em;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="mbin mtight">∗</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="mpunct">,</span><span class="mord mathit">i</span><span class="mrel"><</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">}</span><span class="mord mathrm">∣</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mo>{</mo><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo>)</mo><mi mathvariant="normal">∣</mi><msub><mi>λ</mi><mi>i</mi></msub><mo>=</mo><msub><mi>λ</mi><mi>j</mi></msub><mo separator="true">,</mo><msubsup><mi>λ</mi><mi>i</mi><mo>∗</mo></msubsup><mo>≠</mo><msubsup><mi>λ</mi><mi>j</mi><mo>∗</mo></msubsup><mo separator="true">,</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>}</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">
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b=|\{(x_i, x_j)|\lambda_i=\lambda_j, \lambda^*_i\neq\lambda^*_j, i < j\}|
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</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:1.133108em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mopen">{</span><span class="mopen">(</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 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="mpunct">,</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 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="mclose">)</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">λ</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 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="mrel">=</span><span class="mord"><span class="mord mathit">λ</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 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="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.247em;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:-0.363em;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="mbin mtight">∗</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"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.24700000000000003em;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" style="margin-right:0.05724em;">j</span></span></span><span style="top:-0.363em;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="mbin mtight">∗</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="mpunct">,</span><span class="mord mathit">i</span><span class="mrel"><</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">}</span><span class="mord mathrm">∣</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mo>{</mo><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo>)</mo><mi mathvariant="normal">∣</mi><msub><mi>λ</mi><mi>i</mi></msub><mo>≠</mo><msub><mi>λ</mi><mi>j</mi></msub><mo separator="true">,</mo><msubsup><mi>λ</mi><mi>i</mi><mo>∗</mo></msubsup><mo>=</mo><msubsup><mi>λ</mi><mi>j</mi><mo>∗</mo></msubsup><mo separator="true">,</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>}</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">
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c=|\{(x_i, x_j)|\lambda_i\neq\lambda_j, \lambda^*_i=\lambda^*_j, i < j\}|
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</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:1.133108em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mopen">{</span><span class="mopen">(</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 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="mpunct">,</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 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="mclose">)</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">λ</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 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="mrel">≠</span><span class="mord"><span class="mord mathit">λ</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 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="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.247em;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:-0.363em;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="mbin mtight">∗</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"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.24700000000000003em;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" style="margin-right:0.05724em;">j</span></span></span><span style="top:-0.363em;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="mbin mtight">∗</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="mpunct">,</span><span class="mord mathit">i</span><span class="mrel"><</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">}</span><span class="mord mathrm">∣</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mo>{</mo><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo>)</mo><mi mathvariant="normal">∣</mi><msub><mi>λ</mi><mi>i</mi></msub><mo>≠</mo><msub><mi>λ</mi><mi>j</mi></msub><mo separator="true">,</mo><msubsup><mi>λ</mi><mi>i</mi><mo>∗</mo></msubsup><mo>≠</mo><msubsup><mi>λ</mi><mi>j</mi><mo>∗</mo></msubsup><mo separator="true">,</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>}</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">
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d=|\{(x_i, x_j)|\lambda_i\neq\lambda_j, \lambda^*_i\neq\lambda^*_j, i < j\}|
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</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:1.133108em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mopen">{</span><span class="mopen">(</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 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="mpunct">,</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 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="mclose">)</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">λ</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 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="mrel">≠</span><span class="mord"><span class="mord mathit">λ</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 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="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.247em;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:-0.363em;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="mbin mtight">∗</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"><span class="mord mathit">λ</span><span class="msupsub"><span class="vlist"><span style="top:0.24700000000000003em;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" style="margin-right:0.05724em;">j</span></span></span><span style="top:-0.363em;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="mbin mtight">∗</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="mpunct">,</span><span class="mord mathit">i</span><span class="mrel"><</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mclose">}</span><span class="mord mathrm">∣</span></span></span></span></p>
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<p>举个例子，参考模型给出的簇与聚类模型给出的簇划分如下：</p>
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<table>
|
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<thead>
|
|
<tr>
|
|
<th>编号</th>
|
|
<th>参考簇</th>
|
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<th>聚类簇</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>0</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>2</td>
|
|
<td>0</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>3</td>
|
|
<td>0</td>
|
|
<td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>4</td>
|
|
<td>1</td>
|
|
<td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>5</td>
|
|
<td>1</td>
|
|
<td>2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>6</td>
|
|
<td>1</td>
|
|
<td>2</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
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<p>那么满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>的样本对为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1, 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">1</span><span class="mpunct">,</span><span class="mord mathrm">2</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>号样本的参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span></span></span></span>，聚类簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span></span></span></span></strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>5</mn><mo separator="true">,</mo><mn>6</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(5, 6)</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">5</span><span class="mpunct">,</span><span class="mord mathrm">6</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">5</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>号样本的参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>，聚类簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span></strong>)。总共有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>个样本对满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>，因此<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">a=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span><span class="mrel">=</span><span class="mord mathrm">2</span></span></span></span>。</p>
|
|
<p>满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span></span></span></span>的样本对为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(3, 4)</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">3</span><span class="mpunct">,</span><span class="mord mathrm">4</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">3</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>号样本的参考簇不同，但聚类簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span></strong>)。总共有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>个样本对满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span></span></span></span>，因此<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span>。</p>
|
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<p>那么满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span>的样本对为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1, 3)</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">3</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">3</span></span></span></span>号样本的聚类簇不同，但参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span></span></span></span></strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2, 3)</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">3</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">3</span></span></span></span>号样本的聚类簇不同，但参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span></span></span></span></strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(4, 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">4</span><span class="mpunct">,</span><span class="mord mathrm">5</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">5</span></span></span></span>号样本的聚类簇不同，但参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span></strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>4</mn><mo separator="true">,</mo><mn>6</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(4, 6)</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">4</span><span class="mpunct">,</span><span class="mord mathrm">6</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>号样本的聚类簇不同，但参考簇都为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span></strong>)。总共有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>个样本对满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span>，因此<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">c=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mrel">=</span><span class="mord mathrm">4</span></span></span></span>。</p>
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<p>满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span></span></span></span>的样本对为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>4</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1, 4)</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">4</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(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="mpunct">,</span><span class="mord mathrm">5</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">5</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>6</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1, 6)</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">6</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2, 4)</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">4</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2, 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">2</span><span class="mpunct">,</span><span class="mord mathrm">5</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">5</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>6</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2, 6)</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">6</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>3</mn><mo separator="true">,</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(3, 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">3</span><span class="mpunct">,</span><span class="mord mathrm">5</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">3</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">5</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>3</mn><mo separator="true">,</mo><mn>6</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(3, 6)</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">3</span><span class="mpunct">,</span><span class="mord mathrm">6</span><span class="mclose">)</span></span></span></span>(<strong>因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">3</span></span></span></span>号样本与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>号样本的参考簇不同，聚类簇也不同</strong>)。总共有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>8</mn></mrow><annotation encoding="application/x-tex">8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">8</span></span></span></span>个样本对满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span></span></span></span>，因此<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">d=8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span><span class="mrel">=</span><span class="mord mathrm">8</span></span></span></span>。</p>
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<h3 id="jc系数">JC系数</h3>
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<p><strong>JC系数</strong>根据上面所提到的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span>来计算，并且值域为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo>]</mo></mrow><annotation encoding="application/x-tex">[0, 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">0</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">]</span></span></span></span>，值越大说明聚类性能越好，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>J</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
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JC=\frac{a}{a+b+c}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.695392em;"></span><span class="strut bottom" style="height:1.0987230000000001em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mrel">=</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">a</span><span class="mbin mtight">+</span><span class="mord mathit mtight">b</span><span class="mbin mtight">+</span><span class="mord mathit mtight">c</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:-0.394em;"><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="mord mathit mtight">a</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></span></span></p>
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<p>因此刚刚的例子中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>J</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>7</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">JC=\frac{2}{2+1+4}=\frac{2}{7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mrel">=</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 mathrm mtight">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">4</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:-0.394em;"><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="mord mathrm mtight">2</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="mrel">=</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 mathrm mtight">7</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:-0.394em;"><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="mord mathrm mtight">2</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></span></span></p>
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<h3 id="fm指数">FM指数</h3>
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<p><strong>FM指数</strong>根据上面所提到的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span>来计算，并且值域为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo>]</mo></mrow><annotation encoding="application/x-tex">[0, 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">0</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">]</span></span></span></span>，值越大说明聚类性能越好，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mi>M</mi><mi>I</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfrac><mo>∗</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">
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FMI=\sqrt{\frac{a}{a+b}*\frac{a}{a+c}}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8310355000000003em;"></span><span class="strut bottom" style="height:1.2400100000000003em;vertical-align:-0.40897449999999996em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:0.058964499999999975em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord reset-textstyle textstyle cramped"><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">a</span><span class="mbin mtight">+</span><span class="mord mathit mtight">b</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:-0.394em;"><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">a</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="mbin">∗</span><span class="mord reset-textstyle textstyle cramped"><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">a</span><span class="mbin mtight">+</span><span class="mord mathit mtight">c</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:-0.394em;"><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">a</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></span><span style="top:-0.7510355000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></p>
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<p>因此刚刚的例子中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mi>M</mi><mi>I</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mo>∗</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>1</mn><mn>8</mn></mrow></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">FMI=\sqrt{\frac{2}{2+1}*\frac{2}{2+4}}=\sqrt{\frac{4}{18}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.235064em;"></span><span class="strut bottom" style="height:1.8691855em;vertical-align:-0.6341215em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.015898500000000038em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size2">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord reset-textstyle textstyle cramped"><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 mathrm mtight">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</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:-0.394em;"><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 mathrm mtight">2</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="mbin">∗</span><span class="mord reset-textstyle textstyle cramped"><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 mathrm mtight">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">4</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:-0.394em;"><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 mathrm mtight">2</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></span><span style="top:-1.1258985em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04506399999999999em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size2">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord reset-textstyle textstyle cramped"><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 mathrm mtight">1</span><span class="mord mathrm mtight">8</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:-0.394em;"><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 mathrm mtight">4</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></span><span style="top:-1.1550639999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></p>
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<h3 id="rand指数">Rand指数</h3>
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<p><strong>Rand指数</strong>根据上面所提到的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</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">a</span></span></span></span>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span></span></span></span>来计算，并且值域为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo>]</mo></mrow><annotation encoding="application/x-tex">[0, 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">0</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">]</span></span></span></span>，值越大说明聚类性能越好，假设<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>为样本数量，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>I</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">
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RandI=\frac{2(a+d)}{m(m-1)}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.53em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">a</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</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:0.34500000000000003em;"><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="mopen mtight">(</span><span class="mord mathit mtight">m</span><span class="mbin mtight">−</span><span class="mord mathrm mtight">1</span><span class="mclose mtight">)</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:-0.485em;"><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="mord mathrm mtight">2</span><span class="mopen mtight">(</span><span class="mord mathit mtight">a</span><span class="mbin mtight">+</span><span class="mord mathit mtight">d</span><span class="mclose mtight">)</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></span></span></p>
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<p>因此刚刚的例子中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>I</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∗</mo><mo>(</mo><mn>2</mn><mo>+</mo><mn>8</mn><mo>)</mo></mrow><mrow><mn>6</mn><mo>∗</mo><mo>(</mo><mn>6</mn><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">RandI=\frac{2*(2+8)}{6*(6-1)}=\frac{2}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.53em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">a</span><span class="mord mathit">n</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</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:0.34500000000000003em;"><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 mathrm mtight">6</span><span class="mbin mtight">∗</span><span class="mopen mtight">(</span><span class="mord mathrm mtight">6</span><span class="mbin mtight">−</span><span class="mord mathrm mtight">1</span><span class="mclose mtight">)</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:-0.485em;"><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="mord mathrm mtight">2</span><span class="mbin mtight">∗</span><span class="mopen mtight">(</span><span class="mord mathrm mtight">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">8</span><span class="mclose mtight">)</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="mrel">=</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 mathrm mtight">3</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:-0.394em;"><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="mord mathrm mtight">2</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></span></span>。</p>
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<h2 id="内部指标">内部指标</h2>
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<p><strong>内部指标通常使用 Davies-Bouldin Index (DB指数)以及 Dunn Index（Dunn指数）。</strong></p>
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<h5 id="db指数">DB指数</h5>
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<p><strong>DB指数</strong>又称 DBI ，计算公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi><mi>B</mi><mi>I</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mi>m</mi><mi>a</mi><mi>x</mi><mo>(</mo><mfrac><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mi>i</mi></msub><mo>)</mo><mo>+</mo><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mi>j</mi></msub><mo>)</mo></mrow><mrow><msub><mi>d</mi><mi>c</mi></msub><mo>(</mo><msub><mi>μ</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>μ</mi><mi>j</mi></msub><mo>)</mo></mrow></mfrac><mo>)</mo><mo separator="true">,</mo><mi>i</mi><mo>≠</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">
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DBI=\frac{1}{k}\sum_{i=1}^kmax(\frac{avg(C_i)+avg(C_j)}{d_c(\mu_i,\mu_j)}), i \neq j
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.03232em;"></span><span class="strut bottom" style="height:1.5746399999999998em;vertical-align:-0.5423199999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</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" style="margin-right:0.03148em;">k</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:-0.394em;"><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="mord mathrm mtight">1</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="mop"><span class="mop op-symbol small-op" style="top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.30001em;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 scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.364em;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:0.03148em;">k</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 mathit">m</span><span class="mord mathit">a</span><span class="mord mathit">x</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:0.3449999999999999em;"><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 mtight"><span class="mord mathit mtight">d</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">c</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="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight">μ</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">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="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathit mtight">μ</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:0em;"><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" 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="mclose mtight">)</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:-0.50732em;"><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="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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">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="mclose mtight">)</span><span class="mbin mtight">+</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" 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="mclose mtight">)</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 class="mpunct">,</span><span class="mord mathit">i</span><span class="mrel">≠</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span></p>
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<p>公式中的表达式其实很好理解，其中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>代表聚类有多少个簇，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>μ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mu_i</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:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">μ</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 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>代表第<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>个簇的中心点，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">avg(C_i)</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="mord mathit">a</span><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</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="mclose">)</span></span></span></span>代表<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>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>个簇中所有数据与第<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>个簇的中心点的平均距离。<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>d</mi><mi>c</mi></msub><mo>(</mo><msub><mi>μ</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>μ</mi><mi>j</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">d_c(\mu_i, \mu_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"><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 mathit mtight">c</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="mopen">(</span><span class="mord"><span class="mord mathit">μ</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 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="mpunct">,</span><span class="mord"><span class="mord mathit">μ</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 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="mclose">)</span></span></span></span>代表第<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>个簇的中心点与第<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>个簇的中心点的距离。</p>
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<p>举个例子，现在有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">6</span></span></span></span>条西瓜数据<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mn>6</mn></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">\{x_1,x_2,...,x_6\}</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"><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 mathrm mtight">1</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="mpunct">,</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 mathrm mtight">2</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="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mpunct">,</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 mathrm mtight">6</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="mclose">}</span></span></span></span>，这些数据已经聚类成了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span></span></span></span>个簇。</p>
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<table>
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<thead>
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<tr>
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<th>编号</th>
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<th>体积</th>
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<th>重量</th>
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<th>簇</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>3</td>
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<td>4</td>
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<td>1</td>
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</tr>
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<tr>
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<td>2</td>
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<td>6</td>
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<td>9</td>
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<td>2</td>
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</tr>
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<tr>
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<td>3</td>
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<td>2</td>
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<td>3</td>
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<td>1</td>
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</tr>
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<tr>
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<td>4</td>
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<td>3</td>
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<td>4</td>
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<td>1</td>
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</tr>
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<tr>
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<td>5</td>
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<td>7</td>
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<td>10</td>
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<td>2</td>
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</tr>
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<tr>
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<td>6</td>
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<td>8</td>
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<td>11</td>
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<td>2</td>
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</tr>
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</tbody>
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</table>
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<p>从表格可以看出：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">
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k=2
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">=</span><span class="mord mathrm">2</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>μ</mi><mn>1</mn></msub><mo>=</mo><mo>(</mo><mfrac><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mo>(</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>4</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><mo separator="true">,</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">
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\mu_1=(\frac{(3+2+3)}{3}, \frac{(4+3+4)}{3})=(2.67,3.67)
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">μ</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 mathrm mtight">1</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="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 mathrm mtight">3</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:-0.485em;"><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="mopen mtight">(</span><span class="mord mathrm mtight">3</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">3</span><span class="mclose mtight">)</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 mathrm mtight">3</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:-0.485em;"><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="mopen mtight">(</span><span class="mord mathrm mtight">4</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">3</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">4</span><span class="mclose mtight">)</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 class="mrel">=</span><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mpunct">,</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose">)</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>μ</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><mfrac><mrow><mo>(</mo><mn>6</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>8</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mo>(</mo><mn>9</mn><mo>+</mo><mn>1</mn><mn>0</mn><mo>+</mo><mn>1</mn><mn>1</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo><mo>=</mo><mo>(</mo><mn>7</mn><mo separator="true">,</mo><mn>1</mn><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">
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\mu_2=(\frac{(6+7+8)}{3}, \frac{(9+10+11)}{3})=(7,10)
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">μ</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 mathrm mtight">2</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="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 mathrm mtight">3</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:-0.485em;"><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="mopen mtight">(</span><span class="mord mathrm mtight">6</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">7</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">8</span><span class="mclose mtight">)</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 mathrm mtight">3</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:-0.485em;"><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="mopen mtight">(</span><span class="mord mathrm mtight">9</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">0</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">1</span><span class="mclose mtight">)</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 class="mrel">=</span><span class="mopen">(</span><span class="mord mathrm">7</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>d</mi><mi>c</mi></msub><mo>(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>μ</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><msqrt><mrow><mo>(</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><mo>−</mo><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><mo>−</mo><mn>1</mn><mn>0</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>7</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><mn>3</mn><mn>9</mn><mn>1</mn></mrow><annotation encoding="application/x-tex">
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d_c(\mu_1, \mu_2)=\sqrt{(2.67-7)^2+(3.67-10)^2}=7.67391
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></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 mathit mtight">c</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="mopen">(</span><span class="mord"><span class="mord mathit">μ</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 mathrm mtight">1</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="mpunct">,</span><span class="mord"><span class="mord mathit">μ</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 mathrm mtight">2</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="mclose">)</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mbin">−</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">7</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mord mathrm">3</span><span class="mord mathrm">9</span><span class="mord mathrm">1</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>)</mo><mo>=</mo><mo>(</mo><msqrt><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>4</mn><mo>−</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>3</mn><mo>−</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>4</mn><mo>−</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>)</mo><mi mathvariant="normal">/</mi><mn>3</mn><mo>=</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>2</mn><mn>8</mn><mn>5</mn><mn>3</mn><mn>9</mn></mrow><annotation encoding="application/x-tex">
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avg(C_1)=(\sqrt{(3-2.67)^2+(4-3.67)^2}+\sqrt{(2-2.67)^2+(3-3.67)^2}+\sqrt{(3-2.67)^2+(4-3.67)^2})/3=0.628539
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</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 mathrm mtight">1</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="mclose">)</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">4</span><span class="mbin">−</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mbin">−</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">4</span><span class="mbin">−</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mord mathrm">/</span><span class="mord mathrm">3</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">2</span><span class="mord mathrm">8</span><span class="mord mathrm">5</span><span class="mord mathrm">3</span><span class="mord mathrm">9</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><mo>(</mo><msqrt><mrow><mo>(</mo><mn>6</mn><mo>−</mo><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>9</mn><mo>−</mo><mn>1</mn><mn>0</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>(</mo><mn>7</mn><mo>−</mo><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>1</mn><mn>0</mn><mo>−</mo><mn>1</mn><mn>0</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mrow><mo>(</mo><mn>8</mn><mo>−</mo><mn>7</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>1</mn><mn>1</mn><mo>−</mo><mn>1</mn><mn>0</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>)</mo><mi mathvariant="normal">/</mi><mn>3</mn><mo>=</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>9</mn><mn>4</mn><mn>2</mn><mn>8</mn><mn>1</mn></mrow><annotation encoding="application/x-tex">
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avg(C_2)=(\sqrt{(6-7)^2+(9-10)^2}+\sqrt{(7-7)^2+(10-10)^2}+\sqrt{(8-7)^2+(11-10)^2})/3=0.94281
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit">a</span><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</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 mathrm mtight">2</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="mclose">)</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">6</span><span class="mbin">−</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">9</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">7</span><span class="mbin">−</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">8</span><span class="mbin">−</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mord mathrm">/</span><span class="mord mathrm">3</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">9</span><span class="mord mathrm">4</span><span class="mord mathrm">2</span><span class="mord mathrm">8</span><span class="mord mathrm">1</span></span></span></span></p>
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<p>因此有：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi><mi>B</mi><mi>I</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mi>m</mi><mi>a</mi><mi>x</mi><mo>(</mo><mfrac><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mi>i</mi></msub><mo>)</mo><mo>+</mo><mi>a</mi><mi>v</mi><mi>g</mi><mo>(</mo><msub><mi>C</mi><mi>j</mi></msub><mo>)</mo></mrow><mrow><msub><mi>d</mi><mi>c</mi></msub><mo>(</mo><msub><mi>μ</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>μ</mi><mi>j</mi></msub><mo>)</mo></mrow></mfrac><mo>)</mo><mo>=</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>2</mn><mn>0</mn><mn>4</mn><mn>7</mn><mn>6</mn><mn>5</mn></mrow><annotation encoding="application/x-tex">
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DBI=\frac{1}{k}\sum_{i=1}^kmax(\frac{avg(C_i)+avg(C_j)}{d_c(\mu_i,\mu_j)})=0.204765
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.03232em;"></span><span class="strut bottom" style="height:1.5746399999999998em;vertical-align:-0.5423199999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mrel">=</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" style="margin-right:0.03148em;">k</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:-0.394em;"><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="mord mathrm mtight">1</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="mop"><span class="mop op-symbol small-op" style="top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.30001em;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 scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.364em;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:0.03148em;">k</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 mathit">m</span><span class="mord mathit">a</span><span class="mord mathit">x</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:0.3449999999999999em;"><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 mtight"><span class="mord mathit mtight">d</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">c</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="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight">μ</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">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="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathit mtight">μ</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:0em;"><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" 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="mclose mtight">)</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:-0.50732em;"><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="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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">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="mclose mtight">)</span><span class="mbin mtight">+</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" 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="mclose mtight">)</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 class="mrel">=</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord mathrm">0</span><span class="mord mathrm">4</span><span class="mord mathrm">7</span><span class="mord mathrm">6</span><span class="mord mathrm">5</span></span></span></span></p>
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<p><strong>DB指数越小就越就意味着簇内距离越小同时簇间距离越大，也就是说DB指数越小越好。</strong></p>
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<h3 id="dunn指数">Dunn指数</h3>
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<p><strong>Dunn指数</strong>又称DI，计算公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi><mi>I</mi><mo>=</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></msub><mo>{</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow></msub><mo>(</mo><mfrac><mrow><msub><mi>d</mi><mi>m</mi></msub><mi>i</mi><mi>n</mi><mo>(</mo><msub><mi>C</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>C</mi><mi>j</mi></msub><mo>)</mo></mrow><mrow><mi>m</mi><mi>a</mi><msub><mi>x</mi><mrow><mn>1</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>k</mi></mrow></msub><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mo>(</mo><msub><mi>C</mi><mi>l</mi></msub><mo>)</mo></mrow></mfrac><mo>)</mo><mo>}</mo></mrow><annotation encoding="application/x-tex">
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DI=min_{1\leq i\leq k}\{min_{i\neq j}(\frac{d_min(C_i,C_j)}{max_{1\leq l\leq k}diam(C_l)})\}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.03232em;"></span><span class="strut bottom" style="height:1.55232em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.07847em;">I</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:0.14999999999999997em;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 mathrm mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathit mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</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="mopen">{</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:0.15639999999999996em;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">i</span><span class="mrel mtight">≠</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</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="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:0.34500000000000003em;"><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">a</span><span class="mord mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15122857142857138em;margin-right:0.07142857142857144em;margin-left:0em;"><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 mathrm mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</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="mord mathit mtight">d</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight">m</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15122857142857138em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" style="margin-right:0.01968em;">l</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="mclose mtight">)</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:-0.50732em;"><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="mord mtight"><span class="mord mathit mtight">d</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">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 mathit mtight">i</span><span class="mord mathit mtight">n</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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">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="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" 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="mclose mtight">)</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 class="mclose">}</span></span></span></span></p>
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<p>公式中的表达式其实很好理解，其中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>代表聚类有多少个簇，<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><msub><mi>C</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>C</mi><mi>j</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">d_{min}(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"><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="mopen">(</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="mpunct">,</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="mclose">)</span></span></span></span>代表第<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>个簇中的样本与第<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>个簇中的样本之间的最短距离，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mo>(</mo><msub><mi>C</mi><mi>l</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">diam(C_l)</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="mord mathit">d</span><span class="mord mathit">i</span><span class="mord mathit">a</span><span class="mord mathit">m</span><span class="mopen">(</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:0.01968em;">l</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="mclose">)</span></span></span></span>代表第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.01968em;">l</span></span></span></span>个簇中相距最远的样本之间的距离。</p>
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<p>还是这个例子，现在有 6 条西瓜数据<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mn>6</mn></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">\{x_1,x_2,...,x_6\}</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"><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 mathrm mtight">1</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="mpunct">,</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 mathrm mtight">2</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="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mpunct">,</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 mathrm mtight">6</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="mclose">}</span></span></span></span>，这些数据已经聚类成了 2 个簇。</p>
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<table>
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<thead>
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<tr>
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<th>编号</th>
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<th>体积</th>
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<th>重量</th>
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<th>簇</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>3</td>
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<td>4</td>
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<td>1</td>
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</tr>
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<tr>
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<td>2</td>
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<td>6</td>
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<td>9</td>
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<td>2</td>
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</tr>
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<tr>
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<td>3</td>
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<td>2</td>
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<td>3</td>
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<td>1</td>
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</tr>
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<tr>
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<td>4</td>
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<td>3</td>
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<td>4</td>
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<td>1</td>
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</tr>
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<tr>
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<td>5</td>
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<td>7</td>
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<td>10</td>
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<td>2</td>
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</tr>
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<tr>
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<td>6</td>
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<td>8</td>
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<td>11</td>
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<td>2</td>
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</tr>
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</tbody>
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</table>
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<p>从表格可以看出：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">
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k=2
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">=</span><span class="mord mathrm">2</span></span></span></span></p>
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<p><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><msub><mi>C</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><msqrt><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>6</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>4</mn><mo>−</mo><mn>9</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>5</mn><mi mathvariant="normal">.</mi><mn>8</mn><mn>3</mn><mn>1</mn></mrow><annotation encoding="application/x-tex">
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d_{min}(C_1,C_2)=\sqrt{(3-6)^2+(4-9)^2}=5.831
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></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="mopen">(</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 mathrm mtight">1</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="mpunct">,</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 mathrm mtight">2</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="mclose">)</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mbin">−</span><span class="mord mathrm">6</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">4</span><span class="mbin">−</span><span class="mord mathrm">9</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">5</span><span class="mord mathrm">.</span><span class="mord mathrm">8</span><span class="mord mathrm">3</span><span class="mord mathrm">1</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mo>(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>)</mo><mo>=</mo><msqrt><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>4</mn><mo>−</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>1</mn><mi mathvariant="normal">.</mi><mn>4</mn><mn>1</mn><mn>4</mn></mrow><annotation encoding="application/x-tex">
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diam(C_1)=\sqrt{(3-2)^2+(4-2)^2}=1.414
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span><span class="mord mathit">i</span><span class="mord mathit">a</span><span class="mord mathit">m</span><span class="mopen">(</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 mathrm mtight">1</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="mclose">)</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">3</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">4</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">4</span><span class="mord mathrm">1</span><span class="mord mathrm">4</span></span></span></span></p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mo>(</mo><msub><mi>C</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><msqrt><mrow><mo>(</mo><mn>6</mn><mo>−</mo><mn>8</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>9</mn><mo>−</mo><mn>1</mn><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>8</mn><mn>2</mn><mn>8</mn></mrow><annotation encoding="application/x-tex">
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diam(C_2)=\sqrt{(6-8)^2+(9-11)^2}=2.828
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></span><span class="base textstyle uncramped"><span class="mord mathit">d</span><span class="mord mathit">i</span><span class="mord mathit">a</span><span class="mord mathit">m</span><span class="mopen">(</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 mathrm mtight">2</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="mclose">)</span><span class="mrel">=</span><span class="mord sqrt"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mopen">(</span><span class="mord mathrm">6</span><span class="mbin">−</span><span class="mord mathrm">8</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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="mbin">+</span><span class="mopen">(</span><span class="mord mathrm">9</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.289em;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 mathrm mtight">2</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 style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">8</span><span class="mord mathrm">2</span><span class="mord mathrm">8</span></span></span></span></p>
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<p>因此有：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi><mi>I</mi><mo>=</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></msub><mo>{</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow></msub><mo>(</mo><mfrac><mrow><msub><mi>d</mi><mi>m</mi></msub><mi>i</mi><mi>n</mi><mo>(</mo><msub><mi>C</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>C</mi><mi>j</mi></msub><mo>)</mo></mrow><mrow><mi>m</mi><mi>a</mi><msub><mi>x</mi><mrow><mn>1</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>k</mi></mrow></msub><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mo>(</mo><msub><mi>C</mi><mi>l</mi></msub><mo>)</mo></mrow></mfrac><mo>)</mo><mo>}</mo><mo>=</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>0</mn><mn>6</mn><mn>1</mn><mn>5</mn><mn>5</mn><mn>3</mn></mrow><annotation encoding="application/x-tex">
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DI=min_{1\leq i\leq k}\{min_{i\neq j}(\frac{d_min(C_i,C_j)}{max_{1\leq l\leq k}diam(C_l)})\}=2.061553
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.03232em;"></span><span class="strut bottom" style="height:1.55232em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.07847em;">I</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:0.14999999999999997em;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 mathrm mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathit mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</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="mopen">{</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:0.15639999999999996em;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">i</span><span class="mrel mtight">≠</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</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="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:0.34500000000000003em;"><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">a</span><span class="mord mtight"><span class="mord mathit mtight">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15122857142857138em;margin-right:0.07142857142857144em;margin-left:0em;"><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 mathrm mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mrel mtight">≤</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</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="mord mathit mtight">d</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight">m</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15122857142857138em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" style="margin-right:0.01968em;">l</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="mclose mtight">)</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:-0.50732em;"><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="mord mtight"><span class="mord mathit mtight">d</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:0em;"><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">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 mathit mtight">i</span><span class="mord mathit mtight">n</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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">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="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.14300000000000002em;margin-right:0.07142857142857144em;margin-left:-0.07153em;"><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" 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="mclose mtight">)</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 class="mclose">}</span><span class="mrel">=</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">0</span><span class="mord mathrm">6</span><span class="mord mathrm">1</span><span class="mord mathrm">5</span><span class="mord mathrm">5</span><span class="mord mathrm">3</span></span></span></span></p>
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<p><strong>Dunn指数越大意味着簇内距离越小同时簇间距离越大，也就是说Dunn指数越大越好。</strong></p>
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