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<!-- Title -->
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<h1>
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<a href="." >分类性能评估指标</a>
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</h1>
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<h1 id="分类模型性能评估指标">分类模型性能评估指标</h1>
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<h2 id="准确度的缺陷">准确度的缺陷</h2>
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<p>准确度这个概念相信对于大家来说肯定并不陌生，就是正确率。例如模型的预测结果与数据真实结果如下表所示：</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>预测结果</th>
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<th>真实结果</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>1</td>
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<td>2</td>
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</tr>
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<tr>
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<td>2</td>
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<td>2</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>3</td>
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<td>3</td>
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</tr>
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<tr>
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<td>4</td>
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<td>1</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>2</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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<p>很明显，连小朋友都能算出来该模型的准确度为 3/5 。</p>
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<p>那么准确对越高就能说明模型的分类性能越好吗？非也！举个例子，现在我开发了一套癌症检测系统，只要输入你的一些基本健康信息，就能预测出你现在是否患有癌症，并且分类的准确度为 0.999 。您认为这样的系统的预测性能好不好呢？</p>
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<p>您可能会觉得，哇，这么高的准确度！这个系统肯定很牛逼！但是我们知道，一般年轻人患癌症的概率非常低，假设患癌症的概率为 0.001 ，那么其实我这个癌症检测系统只要一直输出您没有患癌症，准确度也可能能够达到 0.999 。</p>
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<p>假如现在有一个人本身已经患有癌症，但是他自己不知道自己患有癌症。这个时候用我的癌症检测系统检测发现他没有得癌症，那很显然我这个系统已经把他给坑了（耽误了治疗）。</p>
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<p>看到这里您应该已经体会到了，一个分类模型如果光看准确度是不够的，尤其是对这种样本<strong>极度不平衡</strong>的情况（ 10000 条健康信息数据中，只有 1 条的类别是患有癌症，其他的类别都是健康）。</p>
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<h2 id="混淆矩阵">混淆矩阵</h2>
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<p>想进一步的考量分类模型的性能如何，可以使用其他的一些性能指标，例如精准率和召回率。但这些指标计算的基础是<strong>混淆矩阵</strong>。</p>
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<p>继续以癌症检测系统为例，癌症检测系统的输出不是有癌症就是健康，这里为了方便，就用 1 表示患有癌症， 0 表示健康。假设现在拿 10000 条数据来进行测试，其中有 9978 条数据的真实类别是 0 ，系统预测的类别也是 0 ，有 2 条数据的真实类别是 1 却预测成了 0 ，有 12 条数据的真实类别是 0 但预测成了 1 ，有 8 条数据的真实类别是 1 ，预测结果也是 1 。</p>
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<p>如果我们把这些结果组成如下矩阵，则该矩阵就成为<strong>混淆矩阵</strong>。</p>
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<table>
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<thead>
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<tr>
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<th>真实\预测</th>
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<th>0</th>
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<th>1</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>0</td>
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<td>9978</td>
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<td>12</td>
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</tr>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>8</td>
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</tr>
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</tbody>
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</table>
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<p>混淆矩阵中每个格子所代表的的意义也很明显，意义如下：</p>
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<table>
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<thead>
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<tr>
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<th>真实\预测</th>
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<th>0</th>
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<th>1</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>0</td>
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<td>预测 0 正确的数量</td>
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<td>预测 1 错误的数量</td>
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</tr>
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<tr>
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<td>1</td>
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<td>预测 0 错误的数量</td>
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<td>预测 1 正确的数量</td>
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</tr>
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</tbody>
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</table>
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<p>如果将正确看成是 True ，错误看成是 False ， 0 看成是 Negtive ， 1 看成是 Positive 。然后将上表中的文字替换掉，混淆矩阵如下：</p>
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<table>
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<thead>
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<tr>
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<th>真实\预测</th>
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<th>0</th>
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<th>1</th>
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</tr>
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</thead>
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<tbody>
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<tr>
|
|
<td>0</td>
|
|
<td>TN</td>
|
|
<td>FP</td>
|
|
</tr>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>FN</td>
|
|
<td>TP</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>因此 TN 表示真实类别是 Negtive ，预测结果也是 Negtive 的数量； FP 表示真实类别是 Negtive ，预测结果是 Positive 的数量； FN 表示真实类别是 Positive ，预测结果是Negtive 的数量； TP 表示真实类别是 Positive ，预测结果也是 Positive 的数量。</p>
|
|
<p>很明显，当 FN 和 FP 都等于 0 时，模型的性能应该是最好的，因为模型并没有在预测的时候犯错误。即如下混淆矩阵：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>真实\预测</th>
|
|
<th>0</th>
|
|
<th>1</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>0</td>
|
|
<td>9978</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>0</td>
|
|
<td>22</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p><strong>所以模型分类性能越好，混淆矩阵中非对角线上的数值越小。</strong></p>
|
|
<h2 id="精准率">精准率</h2>
|
|
<p><strong>精准率(Precision)</strong>指的是模型预测为 Positive 时的预测准确度，其计算公式如下：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mi>r</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>i</mi><mi>n</mi><mo>=</mo><mfrac><mrow><mi>T</mi><mi>P</mi></mrow><mrow><mi>T</mi><mi>P</mi><mo>+</mo><mi>F</mi><mi>P</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
Precisioin=\frac{TP}{TP+FP}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.275662em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">e</span><span class="mord mathit">c</span><span class="mord mathit">i</span><span class="mord mathit">s</span><span class="mord mathit">i</span><span class="mord mathit">o</span><span class="mord mathit">i</span><span class="mord mathit">n</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:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</span><span class="mbin mtight">+</span><span class="mord mathit mtight" style="margin-right:0.13889em;">F</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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" style="margin-right:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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>
|
|
<p>假如癌症检测系统的混淆矩阵如下：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>真实\预测</th>
|
|
<th>0</th>
|
|
<th>1</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>0</td>
|
|
<td>9978</td>
|
|
<td>12</td>
|
|
</tr>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>2</td>
|
|
<td>8</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>则该系统的精准率=8/(8+12)=0.4 。</p>
|
|
<p>0.4 这个值表示癌症检测系统的预测结果中如果有 100 个人被预测成患有癌症，那么其中有 40 人是真的患有癌症。<strong>也就是说，精准率越高，那么癌症检测系统预测某人患有癌症的可信度就越高。</strong></p>
|
|
<h2 id="召回率">召回率</h2>
|
|
<p><strong>召回率(Recall)</strong>指的是我们关注的事件发生了，并且模型预测正确了的比值，其计算公式如下：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>l</mi><mo>=</mo><mfrac><mrow><mi>T</mi><mi>P</mi></mrow><mrow><mi>F</mi><mi>N</mi><mo>+</mo><mi>T</mi><mi>P</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
Recall=\frac{TP}{FN+TP}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.275662em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">e</span><span class="mord mathit">c</span><span class="mord mathit">a</span><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit" style="margin-right:0.01968em;">l</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:0.13889em;">F</span><span class="mord mathit mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">+</span><span class="mord mathit mtight" style="margin-right:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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" style="margin-right:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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>
|
|
<p>假如癌症检测系统的混淆矩阵如下：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>真实\预测</th>
|
|
<th>0</th>
|
|
<th>1</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>0</td>
|
|
<td>9978</td>
|
|
<td>12</td>
|
|
</tr>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>2</td>
|
|
<td>8</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>则该系统的召回率=8/(8+2)=0.8。</p>
|
|
<p>从计算出的召回率可以看出，假设有 100 个患有癌症的病人使用这个系统进行癌症检测，系统能够检测出 80 人是患有癌症的。<strong>也就是说，召回率越高，那么我们感兴趣的对象成为漏网之鱼的可能性越低。</strong></p>
|
|
<h2 id="精准率与召回率之间的关系">精准率与召回率之间的关系</h2>
|
|
<p>假设有这么一组数据，菱形代表 Positive ，圆形代表 Negtive 。</p>
|
|
<p><img src="img/66.jpg" alt=""></p>
|
|
<p>现在需要训练一个模型对数据进行分类，假如该模型非常简单，就是在数据上画一条线作为分类边界。模型认为边界的左边是 Negtive ，右边是 Positive 。如果该模型的分类边界向左或者向右移动的话，模型所对应的精准率和召回率如下图所示：</p>
|
|
<p><img src="img/67.jpg" alt=""></p>
|
|
<p>从上图可知，<strong>模型的精准率变高，召回率会变低，精准率变低，召回率会变高。</strong></p>
|
|
<h2 id="f1-score">F1 Score</h2>
|
|
<p>上一关中提到了精准率变高，召回率会变低，精准率变低，召回率会变高。那如果想要同时兼顾精准率和召回率，这个时候就可以使用<strong>F1 Score</strong>来作为性能度量指标了。</p>
|
|
<p>F1 Score 是统计学中用来衡量二分类模型精确度的一种指标。它同时兼顾了分类模型的准确率和召回率。F1 Score 可以看作是模型准确率和召回率的一种加权平均，它的最大值是 1 ，最小值是0 。其公式如下：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mn>1</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∗</mo><mi>p</mi><mi>r</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>∗</mo><mi>r</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>l</mi></mrow><mrow><mi>p</mi><mi>r</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>+</mo><mi>r</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>l</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
F1=\frac{2*precision*recall}{precision+recall}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9322159999999999em;"></span><span class="strut bottom" style="height:1.4133239999999998em;vertical-align:-0.481108em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mord mathrm">1</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">p</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">c</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">s</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">o</span><span class="mord mathit mtight">n</span><span class="mbin mtight">+</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">c</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</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:-0.44610799999999995em;"><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="mord mathit mtight">p</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">c</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">s</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">o</span><span class="mord mathit mtight">n</span><span class="mbin mtight">∗</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">c</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</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>
|
|
<ul>
|
|
<li><p>假设模型 A 的精准率为 0.2 ，召回率为 0.7 ，那么模型 A 的 F1 Score 为 0.31111 。</p>
|
|
</li>
|
|
<li><p>假设模型 B 的精准率为 0.7 ，召回率为 0.2 ，那么模型 B 的 F1 Score 为 0.31111 。</p>
|
|
</li>
|
|
<li><p>假设模型 C 的精准率为 0.8 ，召回率为 0.7 ，那么模型 C 的 F1 Score 为 0.74667 。</p>
|
|
</li>
|
|
<li><p>假设模型 D 的精准率为 0.2 ，召回率为 0.3 ，那么模型 D 的 F1 Score 为 0.24 。</p>
|
|
</li>
|
|
</ul>
|
|
<p>从上述 4 个模型的各种性能可以看出，模型C的精准率和召回率都比较高，因此它的 F1 Score 也比较高。而其他模型的精准率和召回率要么都比较低，要么一个低一个高，所以它们的 F1 Score 比较低。</p>
|
|
<p>这也说明了只有当模型的精准率和召回率都比较高时 F1 Score 才会比较高。这也是 F1 Score 能够同时兼顾精准率和召回率的原因。</p>
|
|
<h2 id="roc曲线">ROC曲线</h2>
|
|
<p>ROC曲线(Receiver Operating Characteristic Curve)描述的 TPR（True Positive Rate）与 FPR（False Positive Rate）之间关系的曲线。</p>
|
|
<p>TPR 与 FPR 的计算公式如下：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi><mi>P</mi><mi>R</mi><mo>=</mo><mfrac><mrow><mi>T</mi><mi>P</mi></mrow><mrow><mi>T</mi><mi>P</mi><mo>+</mo><mi>F</mi><mi>N</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
TPR=\frac{TP}{TP+FN}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.275662em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="mord mathit" style="margin-right:0.00773em;">R</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:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</span><span class="mbin mtight">+</span><span class="mord mathit mtight" style="margin-right:0.13889em;">F</span><span class="mord mathit mtight" style="margin-right:0.10903em;">N</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" style="margin-right:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mi>P</mi><mi>R</mi><mo>=</mo><mfrac><mrow><mi>F</mi><mi>P</mi></mrow><mrow><mi>F</mi><mi>P</mi><mo>+</mo><mi>T</mi><mi>N</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
FPR=\frac{FP}{FP+TN}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.872331em;"></span><span class="strut bottom" style="height:1.275662em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="mord mathit" style="margin-right:0.00773em;">R</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:0.13889em;">F</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</span><span class="mbin mtight">+</span><span class="mord mathit mtight" style="margin-right:0.13889em;">T</span><span class="mord mathit mtight" style="margin-right:0.10903em;">N</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" style="margin-right:0.13889em;">F</span><span class="mord mathit mtight" style="margin-right:0.13889em;">P</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>
|
|
<p>其中 TPR 的计算公式您可能有点眼熟，没错！就是召回率的计算公式。<strong>也就是说 TPR 就是召回率</strong>。<strong>所以 TPR 描述的是模型预测 Positive 并且预测正确的数量占真实类别为 Positive 样本的比例。而 FPR 描述的模型预测 Positive 并且预测错了的数量占真实类别为 Negtive 样本的比例。</strong></p>
|
|
<p>和精准率与召回率一样， TPR 与 FPR 之间也存在关系。假设有这么一组数据，菱形代表 Positive ，圆形代表 Negtive 。</p>
|
|
<p><img src="img/66.jpg" alt=""></p>
|
|
<p>现在需要训练一个逻辑回归的模型对数据进行分类，假如将从 0 到 1 中的一些值作为模型的分类阈值。若模型认为当前数据是 Positive 的概率<strong>小于</strong>分类阈值则分类为 Negtive ，<strong>否则</strong>就分类为 Positive （<strong>假设分类阈值为 0.8 ，模型认为这条数据是 Positive 的概率为 0.7 ， 0.7 小于 0.8 ，那么模型就认为这条数据是 Negtive</strong>）。在不同的分类阈值下，模型所对应的 TPR 与 FPR 如下图所示（竖线代表分类阈值，模型会将竖线左边的数据分类成 Negtive ，竖线右边的分类成 Positive ）：</p>
|
|
<p><img src="img/68.jpg" alt=""></p>
|
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<p>从图中可以看出，<strong>当模型的 TPR 越高 FPR 也会越高， TPR 越低 FPR 也会越低。这与精准率和召回率之间的关系刚好相反。</strong>并且，模型的分类阈值一但改变，就有一组对应的 TPR 与 FPR 。假设该模型在不同的分类阈值下其对应的 TPR 与 FPR 如下表所示：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>TPR</th>
|
|
<th>FPR</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>0.2</td>
|
|
<td>0.08</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.35</td>
|
|
<td>0.1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.37</td>
|
|
<td>0.111</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.51</td>
|
|
<td>0.12</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.53</td>
|
|
<td>0.13</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.56</td>
|
|
<td>0.14</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.71</td>
|
|
<td>0.21</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.82</td>
|
|
<td>0.26</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.92</td>
|
|
<td>0.41</td>
|
|
</tr>
|
|
<tr>
|
|
<td>0.93</td>
|
|
<td>0.42</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>若将 FPR 作为横轴， TPR 作为纵轴，将上面的表格以折线图的形式画出来就是 ROC曲线 。</p>
|
|
<p><img src="img/69.jpg" alt=""></p>
|
|
<p>假设现在有模型 A 和模型 B ，它们的 ROC 曲线如下图所示(其中模型 A 的 ROC曲线 为黄色，模型 B 的 ROC曲线 为蓝色)：</p>
|
|
<p><img src="img/70.jpg" alt=""></p>
|
|
<p>那么模型 A 的性能比模型 B 的性能好，因为模型 A 当 FPR 较低时所对应的 TPR 比模型 B 的低 FPR 所对应的 TPR 更高。由由于随着 FPR 的增大， TPR 也会增大。所以 ROC 曲线与横轴所围成的面积越大，模型的分类性能就越高。而 ROC曲线 的面积称为 AUC。</p>
|
|
<h5 id="auc">AUC</h5>
|
|
<p>很明显模型的 AUC 越高，模型的二分类性能就越强。 AUC 的计算公式如下：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>U</mi><mi>C</mi><mo>=</mo><mfrac><mrow><msub><mo>∑</mo><mrow><mi>i</mi><mi>e</mi><mi>p</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>c</mi><mi>l</mi><mi>a</mi><mi>s</mi><mi>s</mi></mrow></msub><mi>r</mi><mi>a</mi><mi>n</mi><msub><mi>k</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><mi>M</mi><mo>(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mi>M</mi><mo>∗</mo><mi>N</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
AUC=\frac{\sum_{ie positive class}rank_i-\frac{M(M+1)}{2}}{M*N}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.366927em;"></span><span class="strut bottom" style="height:1.711927em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.10903em;">U</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" style="margin-right:0.10903em;">M</span><span class="mbin mtight">∗</span><span class="mord mathit mtight" style="margin-right:0.10903em;">N</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.631927em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="top:0.074995em;">∑</span><span class="msupsub"><span class="vlist"><span style="top:0.32101em;margin-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 mathit mtight">i</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">p</span><span class="mord mathit mtight">o</span><span class="mord mathit mtight">s</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight">t</span><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">c</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight">s</span><span class="mord mathit mtight">s</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" style="margin-right:0.02778em;">r</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight">n</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.143em;margin-right:0.07142857142857144em;margin-left:-0.03148em;"><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="mbin mtight">−</span><span class="mord reset-scriptstyle scriptstyle uncramped mtight"><span class="mopen sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.344em;"><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">2</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.5142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord scriptscriptstyle uncramped mtight"><span class="mord mathit mtight" style="margin-right:0.10903em;">M</span><span class="mopen mtight">(</span><span class="mord mathit mtight" style="margin-right:0.10903em;">M</span><span class="mbin mtight">+</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-scriptstyle textstyle uncramped nulldelimiter"></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p>
|
|
<p>其中 M 为真实类别为 Positive 的样本数量， N 为真实类别为 Negtive 的样本数量。 ranki 代表了真实类别为 Positive 的样本点额预测概率从小到大排序后，该预测概率排在第几。</p>
|
|
<p>举个例子，现有预测概率与真实类别的表格如下所示（其中 0 表示 Negtive ， 1 表示 Positive ）：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>编号</th>
|
|
<th>预测概率</th>
|
|
<th>真实类别</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>0.1</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>2</td>
|
|
<td>0.4</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>3</td>
|
|
<td>0.3</td>
|
|
<td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>4</td>
|
|
<td>0.8</td>
|
|
<td>1</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>想要得到公式中的 rank ，就需要将预测概率从小到大排序，排序后如下：</p>
|
|
<table>
|
|
<thead>
|
|
<tr>
|
|
<th>编号</th>
|
|
<th>预测概率</th>
|
|
<th>真实类别</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>1</td>
|
|
<td>0.1</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>3</td>
|
|
<td>0.3</td>
|
|
<td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td>2</td>
|
|
<td>0.4</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr>
|
|
<td>4</td>
|
|
<td>0.8</td>
|
|
<td>1</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>排序后的表格中，真实类别为 Positive 只有编号为 3 和编号为 4 的数据，并且编号为 3 的数据排在第 2 ，编号为 4 的数据排在第 4 。所以 rank=[2, 4]。又因表格中真是类别为 Positive 的数据有 2 条，Negtive 的数据有 2 条。因此 M 为2，N 为2。所以根据 AUC 的计算公式可知：</p>
|
|
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>U</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mn>4</mn><mo>)</mo><mo>−</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn><mo>∗</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>7</mn><mn>5</mn></mrow><annotation encoding="application/x-tex">
|
|
AUC=\frac{(2+4)-\frac{2(2+1)}{2}}{2*2}=0.75
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.2858em;"></span><span class="strut bottom" style="height:1.6308em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.10903em;">U</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">2</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:-0.5508em;"><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">2</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">4</span><span class="mclose mtight">)</span><span class="mbin mtight">−</span><span class="mord reset-scriptstyle scriptstyle uncramped mtight"><span class="mopen sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.344em;"><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">2</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.5142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord scriptscriptstyle uncramped mtight"><span class="mord mathrm mtight">2</span><span class="mopen mtight">(</span><span class="mord mathrm mtight">2</span><span class="mbin mtight">+</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-scriptstyle textstyle uncramped nulldelimiter"></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">7</span><span class="mord mathrm">5</span></span></span></span>。</p>
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</body>
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</html>
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