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<!-- Title -->
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<a href="." >回归性能评估指标</a>
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</h1>
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<h1 id="回归模型性能评估指标">回归模型性能评估指标</h1>
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<h2 id="mse">MSE</h2>
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<p>MSE（Mean Squared Error）叫做均方误差，其实就是线性回归的损失函数。公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mo>(</mo><msup><mi>y</mi><mi>i</mi></msup><mo>−</mo><msup><mi>p</mi><mi>i</mi></msup><msup><mo>)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">
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\frac{1}{m}\sum_{i=1}^m(y^i-p^i)^2
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</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:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><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">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="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</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="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">−</span><span class="mord"><span class="mord mathit">p</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="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 class="mclose">)</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="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></p>
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<p>其中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>y</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">y^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.824664em;"></span><span class="strut bottom" style="height:1.019104em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</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="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>表示第 i 个样本的真实标签，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>p</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">p^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.824664em;"></span><span class="strut bottom" style="height:1.019104em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">p</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="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>表示模型对第 i 个样本的预测标签。线性回归的目的就是让损失函数最小。那么模型训练出来了，我们在测试集上用损失函数来评估模型就行了。</p>
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<h2 id="rmse">RMSE</h2>
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<p>RMSE（Root Mean Squard Error）均方根误差，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msqrt><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mo>(</mo><msup><mi>y</mi><mi>i</mi></msup><mo>−</mo><msup><mi>p</mi><mi>i</mi></msup><msup><mo>)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">
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\sqrt{\frac{1}{m}\sum_{i=1}^m(y^i-p^i)^2}
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</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:1.84002em;vertical-align:-0.604956em;"></span><span class="base textstyle uncramped"><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 mathit mtight">m</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><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">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 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="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</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 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">−</span><span class="mord"><span class="mord mathit">p</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 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 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:-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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<p>RMSE 其实就是<strong>MSE</strong>开个根号。有什么意义呢？其实实质是一样的。只不过用于数据更好的描述。</p>
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<p>例如：要做房价预测，每平方是万元，我们预测结果也是万元。那么差值的平方单位应该是千万级别的。那我们不太好描述自己做的模型效果。怎么说呢？我们的模型误差是多少千万？于是干脆就开个根号就好了。我们误差的结果就跟我们数据是一个级别的了，在描述模型的时候就说，我们模型的误差是多少万元。</p>
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<h2 id="mae">MAE</h2>
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<p>MAE(Mean Aboslute Error)，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mi mathvariant="normal">∣</mi><msup><mi>y</mi><mi>i</mi></msup><mo>−</mo><msup><mi>p</mi><mi>i</mi></msup><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">
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\frac{1}{m}\sum_{i=1}^m|y^i-p^i|
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</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:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><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">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 mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</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="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">−</span><span class="mord"><span class="mord mathit">p</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="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord mathrm">∣</span></span></span></span></p>
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<p>MAE 虽然不作为损失函数，确是一个非常直观的评估指标，它表示每个样本的预测标签值与真实标签值的 L1 距离。</p>
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<h5 id="r-squared">R-Squared</h5>
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<p>上面的几种衡量标准针对不同的模型会有不同的值。比如说预测房价 那么误差单位就是万元。数子可能是 3 ， 4 ， 5 之类的。那么预测身高就可能是 0.1 ，0.6 之类的。没有什么可读性，到底多少才算好呢？不知道，那要根据模型的应用场景来。 看看分类算法的衡量标准就是正确率，而正确率又在 0～1 之间，最高百分之百。最低 0 。如果是负数，则考虑非线性相关。很直观，而且不同模型一样的。那么线性回归有没有这样的衡量标准呢？</p>
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<p>R-Squared 就是这么一个指标，公式如下：</p>
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<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mo>∑</mo><mi>i</mi></msub><mo>(</mo><msup><mi>p</mi><mi>i</mi></msup><mo>−</mo><msup><mi>y</mi><mi>i</mi></msup><msup><mo>)</mo><mn>2</mn></msup></mrow><mrow><msub><mo>∑</mo><mi>i</mi></msub><mo>(</mo><msubsup><mi>y</mi><mrow><mi>m</mi><mi>e</mi><mi>a</mi><mi>n</mi></mrow><mi>i</mi></msubsup><mo>−</mo><msup><mi>y</mi><mi>i</mi></msup><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">
|
|
R^2=1-\frac{\sum_i(p^i-y^i)^2}{\sum_i(y_{mean}^i-y^i)^2}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.166167em;"></span><span class="strut bottom" style="height:1.735874em;vertical-align:-0.569707em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.00773em;">R</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="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="mord mathrm">1</span><span class="mbin">−</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="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 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="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:0.286em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord scriptscriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathit mtight">e</span><span class="mord mathit mtight">a</span><span class="mord mathit mtight">n</span></span></span></span><span style="top:-0.286em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathit mtight">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 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:-0.286em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathit mtight">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 class="mclose mtight">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.286em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span><span 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:-0.534707em;"><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 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="mopen mtight">(</span><span class="mord mtight"><span class="mord mathit mtight">p</span><span class="msupsub"><span class="vlist"><span style="top:-0.431em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord mathit mtight">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 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:-0.431em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord mathit mtight">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 class="mclose mtight">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.431em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped mtight"><span class="mord 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><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>其中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>y</mi><mrow><mi>m</mi><mi>e</mi><mi>a</mi><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">y_{mean}</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" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><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">e</span><span class="mord mathit mtight">a</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></span></span>表示所有测试样本标签值的均值。为什么这个指标会有刚刚我们提到的性能呢？我们分析下公式：</p>
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<p><img src="img/71.jpg" alt=""></p>
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<p>其实分子表示的是模型预测时产生的误差，分母表示的是对任意样本都预测为所有标签均值时产生的误差，由此可知：</p>
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<ol>
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<li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">R^2 \leq1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.950078em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.00773em;">R</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="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="mord mathrm">1</span></span></span></span>,当我们的模型不犯任何错误时，取最大值 1 。</p>
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</li>
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<li><p>当我们的模型性能跟基模型性能相同时，取 0 。</p>
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</li>
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<li><p>如果为负数，则说明我们训练出来的模型还不如基准模型，此时，很有可能我们的数据不存在任何线性关系。</p>
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</li>
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</ol>
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