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decision-tree.html
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<h1 class="Header__Title">决策树</h1>
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<span class="DateTagBar__Item DateTagBar__Date">Posted on Tue, Mar 29, 2022</span>
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<article id="https://www.notion.so/6ad5794a568749a5a51bc92870641b34" class="PageRoot"><ul id="https://www.notion.so/b66a06e2bf504439838cd23e320366c5" class="ColorfulBlock ColorfulBlock--ColorGray TableOfContents"><li class="TableOfContents__Item"><a href="#https://www.notion.so/99d6a77102bd4760b1b4fc803754508e"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">分类</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/8853b79d7d5948f18bba2be005302c70"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">分类模式</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/8f73644986f54007877ffba390edd21f"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">分类问题评价方式</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/3a51e80a37a44985a2ce8ef752b621d1"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">决策树基本算法</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/3ca838cf784c4889a9cb67abb6f8a741"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">划分属性</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/5268b5c480ba41fda5a215c0c885622a"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">决策树的剪枝</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/43f2c41e1a2b4a9e8581c5b48c4cc223"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">预剪枝</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/31777d99c39e4bd0afd720b1e823de5e"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">后剪枝</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/8567830cc8f54e32a98ac32809a28383"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">回归决策树</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/19866ae798f04bcc9b8f88d117cf2388"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">属性缺失</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/aa02205905d849cc981abae487223323"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">多变量决策树</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/509ee9f7514e46b5b453277456fe303c"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">总结</span></span></div></a></li></ul><h2 id="https://www.notion.so/99d6a77102bd4760b1b4fc803754508e" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/99d6a77102bd4760b1b4fc803754508e"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">分类</span></span></h2><h3 id="https://www.notion.so/8853b79d7d5948f18bba2be005302c70" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/8853b79d7d5948f18bba2be005302c70"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">分类模式</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/3a659f6cd29a43d4acdc73dabf5b9595" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">binary: 二类问题，属于或不属于</span></span></li><li id="https://www.notion.so/a176a55e4f274208adf2cba09c0cd431" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">multi-class: 多类问题，有多个类别，可以拆分成二类问题</span></span></li><li id="https://www.notion.so/dd686a975aba45b5a294b0900b2e7b76" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">multi-label: 一个对象可以属于多类</span></span></li></ul><h3 id="https://www.notion.so/8f73644986f54007877ffba390edd21f" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/8f73644986f54007877ffba390edd21f"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">分类问题评价方式</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/d0387a8d8aad4681851fb9011dcf4386" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">准确率 (</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="P"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span></span></span></span><span class="SemanticString">, precision)</span></span></li><li id="https://www.notion.so/fd2e7f0b71e4480bb213d49cea5f4fc4" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">召回率 (</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="R"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></span></span><span class="SemanticString">, recall)</span></span></li><li id="https://www.notion.so/058d9ba052764b53b24f2c23b0ba0c69" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">F-Measure </span></span></li></ul><div id="https://www.notion.so/6c0139aa68314b5593ff76e7df049f4c" class="ColumnList"><div id="https://www.notion.so/9651c3d4387c420b9dc10b45dad7409d" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><p id="https://www.notion.so/3282396ffd904dddbb4af845e049f512" class="Equation" data-latex="\begin{aligned}
P &= \cfrac{a}{a+b} \\
R &= \cfrac{a}{a+c} \\
F &= \dfrac{1}{\alpha\frac{1}{P} + (1 - \alpha)\frac{1}{R}} \\
F_1 &= \cfrac{2PR}{P+R}
\end{aligned}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>R</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>F</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>α</mi><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mfrac><mn>1</mn><mi>R</mi></mfrac></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>F</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>P</mi><mi>R</mi></mrow><mrow><mi>P</mi><mo>+</mo><mi>R</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
P &= \cfrac{a}{a+b} \\
R &= \cfrac{a}{a+c} \\
F &= \dfrac{1}{\alpha\frac{1}{P} + (1 - \alpha)\frac{1}{R}} \\
F_1 &= \cfrac{2PR}{P+R}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.679538em;vertical-align:-5.0897689999999995em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.589769em;"><span style="top:-7.589769em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span><span style="top:-4.930439000000001em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span><span style="top:-2.539669000000001em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span><span style="top:0.43043899999999957em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.0897689999999995em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.589769em;"><span style="top:-7.589769em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-4.930439000000001em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-2.539669000000001em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.2648919999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">P</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.00773em;">R</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.080108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:0.43043899999999957em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.0897689999999995em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/ee8583c23ce641aabc6fb27a205ec874" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><div id="https://www.notion.so/d51766f9d7ad498d974a51984a24cb9e" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fcc685ce7-4bb0-4302-ac1b-b62e8b916550%2FUntitled.png?width=336&table=block&id=d51766f9-d7ad-498d-974a-51984a24cb9e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fcc685ce7-4bb0-4302-ac1b-b62e8b916550%2FUntitled.png?width=336&table=block&id=d51766f9-d7ad-498d-974a-51984a24cb9e" style="width:336px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><h2 id="https://www.notion.so/3a51e80a37a44985a2ce8ef752b621d1" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/3a51e80a37a44985a2ce8ef752b621d1"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">决策树基本算法</span></span></h2><div id="https://www.notion.so/fedb6bd6903243eb8090b47fcb42533d" class="ColumnList"><div id="https://www.notion.so/4efceea7910b4c58a36a825f58c703a5" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5625)"><div id="https://www.notion.so/797d03b4a710463997eb0133d4c2d67b" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F13bb299d-4912-47bd-a841-5683cd46e3fa%2FUntitled.png?width=591&table=block&id=797d03b4-a710-4639-97eb-0133d4c2d67b"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F13bb299d-4912-47bd-a841-5683cd46e3fa%2FUntitled.png?width=591&table=block&id=797d03b4-a710-4639-97eb-0133d4c2d67b" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div><div id="https://www.notion.so/5d72ffba3a974fc485ec97abe7bf6050" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.43749999999999994)"><div id="https://www.notion.so/9bb56707b4fa43aca277f785f2c3325c" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F0c624fee-3947-4f60-a1f0-fddf8ab8900c%2FUntitled.png?width=415&table=block&id=9bb56707-b4fa-43ac-a277-f785f2c3325c"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F0c624fee-3947-4f60-a1f0-fddf8ab8900c%2FUntitled.png?width=415&table=block&id=9bb56707-b4fa-43ac-a277-f785f2c3325c" style="width:415px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><div id="https://www.notion.so/4c9def022f2044aa9a0e65a427d653dc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">二分类学习任务</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/6a778f48197b4164b3a4ecc084fb0369" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">根结点：包含全部样本</span></span></li><li id="https://www.notion.so/a52c87fa91ca429698b5ae1a0be25153" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">叶结点：对应决策结果 “好瓜” “坏瓜”</span></span></li><li id="https://www.notion.so/0ec6779772704bb1b95cdfa7647ce6e3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">内部结点：对应属性测试</span></span></li></ul><div id="https://www.notion.so/832fcfc2b5724966aeb56b3b59b839d1" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5a3642dd-8cb9-40e7-9186-384fe4e452f5%2FUntitled.png?width=873&table=block&id=832fcfc2-b572-4966-aeb5-6b3b59b839d1"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5a3642dd-8cb9-40e7-9186-384fe4e452f5%2FUntitled.png?width=873&table=block&id=832fcfc2-b572-4966-aeb5-6b3b59b839d1" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/94a63513c5c4418790220ac1710b3f28" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">情形 (1) ：</span></span></p></div><div id="https://www.notion.so/2f87910073d94700b91f20f1fa6b0fb4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无需划分，样本都属于同一类</span></span></p></div><div id="https://www.notion.so/13d9025d08e442e5bdb16659d90cd5e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">情形 (2) ：</span></span></p></div><div id="https://www.notion.so/be5d8b98eeb14535826f6d58c9181eb3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无法划分，叶结点，设定为</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">该结点</mark></span><span class="SemanticString">所含样本最多的类别，利用当前结点的</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">后验分布</mark></span></span></p></div><div id="https://www.notion.so/5868f763ecd54ec2afcda2e9ca8d428f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">情形 (3) ：</span></span></p></div><div id="https://www.notion.so/88d24089b8a94d76959e65c93b760f3a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">划分后没有样本，设定为</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">其父结点</mark></span><span class="SemanticString">所含样本最多的类别，把父结点的样本分布作为当前结点的</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">先验分布</mark></span></span></p></div><h3 id="https://www.notion.so/3ca838cf784c4889a9cb67abb6f8a741" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/3ca838cf784c4889a9cb67abb6f8a741"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">划分属性</span></span></h3><div id="https://www.notion.so/7d7175908d21468f9f480b7e6364c788" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">希望决策树的分支结点所包含的样本尽可能属于同一类别，即</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">结点的“纯度”越来越高</mark></span><span class="SemanticString">，可以高效地从根结点到达叶结点。</span></span></p></div><div id="https://www.notion.so/52a9b33f478642cdbbf3354b205635d1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">三种度量结点“纯度”的指标：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/2ee2f6c88a9d44f18df0971649d98887" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">信息增益</strong></span></span><div id="https://www.notion.so/1abffa4ca97e4f58a856e6b77eff8490" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">“信息熵” (information entropy) 是度量样本集合纯度最常用的一种指标。假定当前样本集合</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">中第</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="k"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span></span></span><span class="SemanticString">类样本所占的比例为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="p_k \; (k=1,2,\dots,|\mathcal{Y}|)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_k \; (k=1,2,\dots,|\mathcal{Y}|)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">Y</span></span><span class="mord">∣</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> ，则</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">的信息熵定义为</span></span></p></div><p id="https://www.notion.so/74b21cbd3a914f45b660ffa9eef064ea" class="Equation" data-latex="\text{Ent}(D)=-\sum_{k=1}^{|\mathcal{Y}|}{p_k \log_2 p_k}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ent</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></munderover><mrow><msub><mi>p</mi><mi>k</mi></msub><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><msub><mi>p</mi><mi>k</mi></msub></mrow></mrow><annotation encoding="application/x-tex">\text{Ent}(D)=-\sum_{k=1}^{|\mathcal{Y}|}{p_k \log_2 p_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2631180000000004em;vertical-align:-1.302113em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/b96322721a194c588935204a00a438c7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Ent}(D)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ent</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Ent}(D)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的值越小，则</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">的纯度越高。（对于二分类任务，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="|\mathcal{Y}|=2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">|\mathcal{Y}|=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathcal" style="margin-right:0.08222em;">Y</span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span></span></span><span class="SemanticString">）</span></span></p></div><div id="https://www.notion.so/dfb14f61e913414a9d144f1773ccd84e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设离散属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">有</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString">个可能的取值</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{a^1,a^2,\dots,a^V\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi><mn>1</mn></msup><mo separator="true">,</mo><msup><mi>a</mi><mn>2</mn></msup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>a</mi><mi>V</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{a^1,a^2,\dots,a^V\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span></span></span><span class="SemanticString">，若使用</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">来对样本集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">进行划分，则会产生</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString">个分支结点，其中第</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></span><span class="SemanticString">个分支结点包含了</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">中所有在属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上取值为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a^v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>v</mi></msup></mrow><annotation encoding="application/x-tex">a^v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的样本，记为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D^v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>D</mi><mi>v</mi></msup></mrow><annotation encoding="application/x-tex">D^v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。可以计算出</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D^v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>D</mi><mi>v</mi></msup></mrow><annotation encoding="application/x-tex">D^v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的信息熵，再考虑到不同的分支结点所包含的样本数不同，给分支结点赋予权重</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="|D^v|/|D|"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|D^v|/|D|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord">/</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mord">∣</span></span></span></span></span></span><span class="SemanticString">，即样本数越多的分支结点的影响越大，于是可计算出用属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">对样本集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">进行划分所获得的“信息增益” (information gain)</span></span></p></div><p id="https://www.notion.so/0e8689f5e68d45a1a3631a196f689c78" class="Equation" data-latex="\text{Gain}(D,a)=\text{Ent}(D)-\sum_{v=1}^{V}\frac{|D^v|}{|D|}\text{Ent}(D^v)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>Ent</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><mfrac><mrow><mi mathvariant="normal">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mtext>Ent</mtext><mo stretchy="false">(</mo><msup><mi>D</mi><mi>v</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gain}(D,a)=\text{Ent}(D)-\sum_{v=1}^{V}\frac{|D^v|}{|D|}\text{Ent}(D^v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/d5db64a004b448c494c3c3723d0fdcee" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">信息增益越大，则意味着使用属性</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">来进行划分所获得的“纯度提升”越大。</mark></span></span></p></div><div id="https://www.notion.so/5962ac24047c4444a75069023b0c4f6f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">决策树算法第8行选择属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_*=\underset{a \in A}{\argmax}\text{Gain}(D,a)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mo>∗</mo></msub><mo>=</mo><mi><munder><mo><mi mathvariant="normal">arg max</mi><mo></mo></mo><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></munder></mi><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_*=\underset{a \in A}{\argmax}\text{Gain}(D,a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.175696em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.716141em;vertical-align:-0.966141em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.161229em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight">A</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mop"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">m</span><span class="mord mathrm">a</span><span class="mord mathrm">x</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.966141em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/a5b1f29b78074d27913b2beff2bccbac" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">著名的ID3决策树算法</mark></span></span></p></div><div id="https://www.notion.so/9e1b24b3273245f595052696f135c66c" class="Divider"></div><div id="https://www.notion.so/b305e9b3806646b4bf92210c1596ebab" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">若把“编号”也作为一个</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">候选划分属性</mark></span><span class="SemanticString">，则属性“编号”的信息增益远大于其他侯选属性。</span></span></p></div><div id="https://www.notion.so/f5f8d759c40143bfad0c1790c70a53d6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">信息增益准则对可取值数目较多的属性有所偏好。</mark></span></span></p></div></li><li id="https://www.notion.so/1f35b540914747f680c22a45f24689cd" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">增益率</strong></span></span><p id="https://www.notion.so/5ad17d23f3b1471ba553587d28ca4adb" class="Equation" data-latex="\text{Gain.ratio}(D,a)=\frac{\text{Gain}(D,a)}{\text{IV}(a)}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gain.ratio</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>IV</mtext><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{Gain.ratio}(D,a)=\frac{\text{Gain}(D,a)}{\text{IV}(a)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gain.ratio</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">IV</span></span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div id="https://www.notion.so/25b046aed3e14028a3f8cd3ec5be260b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span></span></p></div><p id="https://www.notion.so/ee7937d4cd474395a86cebd031e38459" class="Equation" data-latex="\text{IV}(a)=-\sum_{v=1}^{V}\log_2 \frac{|D^v|}{|D|}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>IV</mtext><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mfrac><mrow><mi mathvariant="normal">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{IV}(a)=-\sum_{v=1}^{V}\log_2 \frac{|D^v|}{|D|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">IV</span></span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div id="https://www.notion.so/1ac6057bc375403ebc323319bb2371e1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">称为属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">的“固有值” (intrinsic value)，属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">的可能取值数目越多（即</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="V"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span></span></span><span class="SemanticString">越大），则</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{IV}(a)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>IV</mtext><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{IV}(a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">IV</span></span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的值通常会越大。</span></span></p></div><div id="https://www.notion.so/72f762b196e848c89211ffc1ab07f560" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">增益率准则对可取值数目较少的属性有所偏好。</span></span></p></div><div id="https://www.notion.so/e7eb2d62aac6462d9581d088811797b7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">著名的C4.5决策树算法综合了信息增益准则和信息率准则的特点</mark></span><span class="SemanticString">：先从候选划分属性中找出信息增益高于平均水平的属性，再从中选择增益率最高的。</span></span></p></div></li><li id="https://www.notion.so/b376b449f0ff480b83378101484af0a7" class="NumberedList" value="3"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">基尼指数</strong></span></span><div id="https://www.notion.so/4dd8c271f2454123aa5f006234a3a8d9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">基尼值</span></span></p></div><p id="https://www.notion.so/707093b06fdf4913bc86c4f915481d91" class="Equation" data-latex="\begin{aligned}
\text{Gini}(D)&=\sum_{k=1}^{|\mathcal{Y}|}\sum_{k'\ne k}p_k p_{k'} \\
&=1-\sum_{k=1}^{|\mathcal{Y}|}p_k^2
\end{aligned}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Gini</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></munderover><munder><mo>∑</mo><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo mathvariant="normal">≠</mo><mi>k</mi></mrow></munder><msub><mi>p</mi><mi>k</mi></msub><msub><mi>p</mi><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msubsup><mi>p</mi><mi>k</mi><mn>2</mn></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\text{Gini}(D)&=\sum_{k=1}^{|\mathcal{Y}|}\sum_{k'\ne k}p_k p_{k'} \\
&=1-\sum_{k=1}^{|\mathcal{Y}|}p_k^2
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.262344000000001em;vertical-align:-3.3811720000000003em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8811720000000003em;"><span style="top:-5.881172em;"><span class="pstrut" style="height:3.961005em;"></span><span class="mord"><span class="mord text"><span class="mord">Gini</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span><span style="top:-2.181946em;"><span class="pstrut" style="height:3.961005em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.3811720000000003em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8811720000000003em;"><span style="top:-5.881172em;"><span class="pstrut" style="height:3.961005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight"><span class="mrel mtight"><span class="mord mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="rlap mtight"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel mtight"></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.438221em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.181946em;"><span class="pstrut" style="height:3.961005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.3811720000000003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/9fd3579338f44640add8940d711da3e0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">直观来说，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Gini}(D)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gini</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gini}(D)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gini</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">反映了从数据集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">中</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">随机抽取两个样本</mark></span><span class="SemanticString">，其类别标记不一致的概率。因此，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Gini}(D)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gini</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gini}(D)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gini</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">越小，则数据集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">的纯度越高。</span></span></p></div><div id="https://www.notion.so/72dfb8fb8d654e72a3f07649be7c4088" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">基尼指数</span></span></p></div><p id="https://www.notion.so/6d704b4c4be143de97e5be64487569fb" class="Equation" data-latex="\text{Gini\_index}(D,a)=\sum_{v=1}^V\frac{|D^v|}{|D|}\text{Gini}(D^v)"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gini_index</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><mfrac><mrow><mi mathvariant="normal">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mtext>Gini</mtext><mo stretchy="false">(</mo><msup><mi>D</mi><mi>v</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gini\_index}(D,a)=\sum_{v=1}^V\frac{|D^v|}{|D|}\text{Gini}(D^v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.06em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">Gini_index</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord text"><span class="mord">Gini</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><div id="https://www.notion.so/fca79640433c48f88b8e9167cd4ffda5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">在侯选属性集合</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="A"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span></span></span><span class="SemanticString">中，选择那个使得划分后</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">基尼指数最小的属性</mark></span><span class="SemanticString">作为最优划分属性，即</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a_*=\underset{a\in A}{\argmin}\;\text{Gini\_index}(D,a)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mo>∗</mo></msub><mo>=</mo><mi><munder><mo><mi mathvariant="normal">arg min</mi><mo></mo></mo><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></munder></mi><mtext> </mtext><mtext>Gini_index</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_*=\underset{a\in A}{\argmin}\;\text{Gini\_index}(D,a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.175696em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.716141em;vertical-align:-0.966141em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-2.161229em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight">A</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mop"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">m</span><span class="mord mathrm">i</span><span class="mord mathrm">n</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.966141em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord">Gini_index</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/7519575aaa8f4c6ab33791a2e243cc56" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">著名的CART决策树算法</mark></span></span></p></div></li></ol><h2 id="https://www.notion.so/5268b5c480ba41fda5a215c0c885622a" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/5268b5c480ba41fda5a215c0c885622a"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">决策树的剪枝</span></span></h2><div id="https://www.notion.so/32acd4312121400ba4033d8f34b7b1ec" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">剪枝：通过主动去掉一些分支来降低过拟合的风险。</span></span></p></div><div id="https://www.notion.so/6fb46f5acee5461a9153b8159e297f1f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">将数据集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">划分为两个互斥的集合：训练集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="S"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span></span></span><span class="SemanticString">和测试集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="T"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f7d237a8a03248a9a67966dd6871a76b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">评估精度：正确分类的样本占所有样本的比例</span></span></p></div><div id="https://www.notion.so/1e35fa03c2854a0f9cf07d0a8d4c7bff" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">决策树的剪枝策略有两种：</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">预剪枝</strong></span><span class="SemanticString">和</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">后剪枝</strong></span><span class="SemanticString">。</span></span></p></div><h3 id="https://www.notion.so/43f2c41e1a2b4a9e8581c5b48c4cc223" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/43f2c41e1a2b4a9e8581c5b48c4cc223"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">预剪枝</span></span></h3><div id="https://www.notion.so/662838de3ae448fbba0966f72fe916bc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">预剪枝</strong></span><span class="SemanticString">：在决策树生成过程中，</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">对每个结点在划分前先进行估计</mark></span><span class="SemanticString">，若当前结点的划分不能带来决策树泛化性能提升，则停止划分并将当前结点标记为叶结点。</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/4639db77a00e41fbb50f0d86f6c848f2" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">优点：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/b848bb6e4acb4276bf72fd082861d5b7" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">降低过拟合的风险</span></span></li><li id="https://www.notion.so/f2681ffc1daa49aba4f3490573a3aa2b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">减少了训练时间开销和测试时间开销</span></span></li></ul></li><li id="https://www.notion.so/4010799d8f584945874d0acc008c2e4b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">不足：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/120c15d68e1145eeb555586415c779b3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">基于“贪心”本质禁止某些分支展开，带来了欠拟合的风险</span></span></li></ul></li></ul><h3 id="https://www.notion.so/31777d99c39e4bd0afd720b1e823de5e" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/31777d99c39e4bd0afd720b1e823de5e"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">后剪枝</span></span></h3><div id="https://www.notion.so/b183a593765e46d1a0876a4652b29175" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">后剪枝</strong></span><span class="SemanticString">：先从训练集</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">生成一棵完整的决策树</mark></span><span class="SemanticString">，</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">然后自底向上地对非叶结点进行考察</mark></span><span class="SemanticString">，若将该结点对应的子树</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">替换为叶结点</mark></span><span class="SemanticString">能带来决策树泛化性能提升，则将该子树替换为叶结点。</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/9eb50b41d61449669c2c309c530848ab" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">优点：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/5097f2ee31e04d3fad0f9d292c93758e" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">保留了更多的分支</span></span></li><li id="https://www.notion.so/77314cfe6cd14be7b79553f74fa1d51e" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">欠拟合风险很小</span></span></li><li id="https://www.notion.so/905ce09a0306473ea9f9ae89a971dd04" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">泛化能力优于预剪枝决策树</span></span></li></ul></li><li id="https://www.notion.so/b76cb3078d9f41bf8d3585794c218b3f" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">缺点：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/3ae18525c4264d49a9ce0771f3c06733" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">训练时间比</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">未剪枝和预剪枝</mark></span><span class="SemanticString">决策树大很多</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/3c2810fd626f43b6a29bb231ff2004b5" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">生产完全决策树</span></span></li><li id="https://www.notion.so/12e8314485e74ce4848d68f883176a39" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">所有非叶结点逐一考察</span></span></li></ul></li></ul></li></ul><h2 id="https://www.notion.so/8567830cc8f54e32a98ac32809a28383" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/8567830cc8f54e32a98ac32809a28383"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">回归决策树</span></span></h2><div id="https://www.notion.so/d3b371afb4404fc4a40d035b25e71aab" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">连续属性离散化技术</mark></span><span class="SemanticString">：二分法 C4.5决策树算法</span></span></p></div><div id="https://www.notion.so/54d059d39be54e03b9a51e0a6c069b86" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于样本集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">，连续属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">有</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></span></span><span class="SemanticString">个不同的取值，将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></span></span><span class="SemanticString">个取值从小到大排序：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{a^1,a^2,\dots,a^n\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi><mn>1</mn></msup><mo separator="true">,</mo><msup><mi>a</mi><mn>2</mn></msup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>a</mi><mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{a^1,a^2,\dots,a^n\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f44d53cabf284e6c95bdb3c010facad4" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">划分点</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span></span></span><span class="SemanticString">（数值）将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">划分为两个子集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D_t^-"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>D</mi><mi>t</mi><mo>−</mo></msubsup></mrow><annotation encoding="application/x-tex">D_t^-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.057218em;vertical-align:-0.24575599999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.454244em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24575599999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D_t^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>D</mi><mi>t</mi><mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">D_t^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.057218em;vertical-align:-0.24575599999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.454244em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24575599999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{\underbrace{a^1,a^2,\dots,a^i}_{D_t^-},\underbrace{a^{i+1},\dots,a^n}_{D_t^+}\}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><munder><munder><mrow><msup><mi>a</mi><mn>1</mn></msup><mo separator="true">,</mo><msup><mi>a</mi><mn>2</mn></msup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>a</mi><mi>i</mi></msup></mrow><mo stretchy="true">⏟</mo></munder><msubsup><mi>D</mi><mi>t</mi><mo>−</mo></msubsup></munder><mo separator="true">,</mo><munder><munder><mrow><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msup><mi>a</mi><mi>n</mi></msup></mrow><mo stretchy="true">⏟</mo></munder><msubsup><mi>D</mi><mi>t</mi><mo>+</mo></msubsup></munder><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\underbrace{a^1,a^2,\dots,a^i}_{D_t^-},\underbrace{a^{i+1},\dots,a^n}_{D_t^+}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6951739999999997em;vertical-align:-1.82051em;"></span><span class="mopen">{</span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8746639999999999em;"><span style="top:-1.38287em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8209857142857142em;"><span style="top:-2.209457142857143em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-2.9043214285714285em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29054285714285716em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8746639999999998em;"><span class="svg-align" style="top:-2.15756em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMinYMin slice'><path d='M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z'/></svg></span><span class="brace-center" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMidYMin slice'><path d='M199572 214
c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z'/></svg></span><span class="brace-right" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMaxYMin slice'><path d='M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z'/></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8746639999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8424400000000001em;"><span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.82051em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8746639999999999em;"><span style="top:-1.38287em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8209857142857142em;"><span style="top:-2.209457142857143em;margin-left:-0.02778em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-2.9043214285714285em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29054285714285716em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.874664em;"><span class="svg-align" style="top:-2.15756em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMinYMin slice'><path d='M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z'/></svg></span><span class="brace-center" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMidYMin slice'><path d='M199572 214
c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z'/></svg></span><span class="brace-right" style="height:0.548em;"><svg width='400em' height='0.548em' viewBox='0 0 400000 548' preserveAspectRatio='xMaxYMin slice'><path d='M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z'/></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.874664em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8424400000000001em;"><span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.82051em;"><span></span></span></span></span></span><span class="mclose">}</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/6872c041b53a4aa2bb7b879cc53b7bb3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对相邻的属性取值</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a^i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">a^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a^{i+1}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">来说，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span></span></span><span class="SemanticString">在区间</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="[a^i,a^{i+1})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>a</mi><mi>i</mi></msup><mo separator="true">,</mo><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[a^i,a^{i+1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">中取任何值所产生的划分结果都相同。</span></span></p></div><div id="https://www.notion.so/18523c8ae2fb466d9f590294aebf51ca" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">可考察包含</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="n-1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">个元素的候选划分点集合</span></span></p></div><p id="https://www.notion.so/a801e769fa654747bbb8cc87e1c2cf33" class="Equation" data-latex="T_a=\{\frac{a^i+a^{i+1}}{2}\;|\;1\le i \le n-1\}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>a</mi></msub><mo>=</mo><mo stretchy="false">{</mo><mfrac><mrow><msup><mi>a</mi><mi>i</mi></msup><mo>+</mo><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mn>2</mn></mfrac><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">T_a=\{\frac{a^i+a^{i+1}}{2}\;|\;1\le i \le n-1\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.187664em;vertical-align:-0.686em;"></span><span class="mopen">{</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5016639999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79549em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span></span></p><div id="https://www.notion.so/3ed852f35ffe4c3e92a3518671caaeee" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">即</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">把区间</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="[a^i,a^{i+1})"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>a</mi><mi>i</mi></msup><mo separator="true">,</mo><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[a^i,a^{i+1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">的中位点</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\frac{a^i+a^{i+1}}{2}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msup><mi>a</mi><mi>i</mi></msup><mo>+</mo><msup><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{a^i+a^{i+1}}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3704599999999998em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0254599999999998em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9020857142857143em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9020857142857143em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">作为候选划分点</mark></span><span class="SemanticString">。然后，我们就可像离散属性值一样来考察这些划分点，选取最优的划分点进行样本集合的划分。例如，</span></span></p></div><p id="https://www.notion.so/7c97cc1337e542e39fe4634b3e6bff0d" class="Equation" data-latex="\begin{aligned}
\text{Gain}(D,a)&=\max_{t\in T_a}\text{Gain}(D,a,t)\\
&=\max_{t\in T_a}\text{Ent}(D)-\sum_{\lambda\in\{-,+\}}\frac{|D_t^\lambda|}{|D|}\text{Ent}(D_t^\lambda)
\end{aligned}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo><mi>max</mi><mo></mo></mo><mrow><mi>t</mi><mo>∈</mo><msub><mi>T</mi><mi>a</mi></msub></mrow></munder><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo><mi>max</mi><mo></mo></mo><mrow><mi>t</mi><mo>∈</mo><msub><mi>T</mi><mi>a</mi></msub></mrow></munder><mtext>Ent</mtext><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>−</mo><munder><mo>∑</mo><mrow><mi>λ</mi><mo>∈</mo><mo stretchy="false">{</mo><mo>−</mo><mo separator="true">,</mo><mo>+</mo><mo stretchy="false">}</mo></mrow></munder><mfrac><mrow><mi mathvariant="normal">∣</mi><msubsup><mi>D</mi><mi>t</mi><mi>λ</mi></msubsup><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mtext>Ent</mtext><mo stretchy="false">(</mo><msubsup><mi>D</mi><mi>t</mi><mi>λ</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\text{Gain}(D,a)&=\max_{t\in T_a}\text{Gain}(D,a,t)\\
&=\max_{t\in T_a}\text{Ent}(D)-\sum_{\lambda\in\{-,+\}}\frac{|D_t^\lambda|}{|D|}\text{Ent}(D_t^\lambda)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.326544em;vertical-align:-2.413272em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.913272em;"><span style="top:-5.59938em;"><span class="pstrut" style="height:3.526108em;"></span><span class="mord"><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-2.928841em;"><span class="pstrut" style="height:3.526108em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.413272em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.913272em;"><span style="top:-5.59938em;"><span class="pstrut" style="height:3.526108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.355669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.13889em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8444309999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span><span style="top:-2.928841em;"><span class="pstrut" style="height:3.526108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.43055999999999994em;"><span style="top:-2.355669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.13889em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8444309999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">λ</span><span class="mrel mtight">∈</span><span class="mopen mtight">{</span><span class="mord mtight">−</span><span class="mpunct mtight">,</span><span class="mord mtight">+</span><span class="mclose mtight">}</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">λ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.413272em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/8a9008dac87742f69867736e48dc8eef" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Gain}(D,a,t)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gain}(D,a,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">是样本集</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">基于划分点</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="t"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span></span></span><span class="SemanticString">二分后的信息增益。于是，我们就可选择使</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\text{Gain}(D,a,t)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Gain}(D,a,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">最大化的划分点。</span></span></p></div><div id="https://www.notion.so/6440c1dd105d45c8bbab776c1731a953" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">与离散属性不同，若当前节点划分属性为连续属性，该连续属性还可被再次选作后代结点的最优划分属性。</span></span></p></div><h2 id="https://www.notion.so/19866ae798f04bcc9b8f88d117cf2388" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/19866ae798f04bcc9b8f88d117cf2388"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">属性缺失</span></span></h2><div id="https://www.notion.so/cbac6e53cd354962b0980278f6cb5638" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">问题1</strong></span><span class="SemanticString">：属性值缺失时，如何进行划分属性选择？</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">如何计算信息增益？</mark></span></span></p></div><div id="https://www.notion.so/87cec2bef31d4e96a995e730ac5b4859" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">问题2</strong></span><span class="SemanticString">：给定划分属性，若样本在该属性上的值确实，</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">如何对样本进行划分？</mark></span></span></p></div><div id="https://www.notion.so/df1437a2dccb4e9b85e1a8aced008763" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="D"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span></span></span><span class="SemanticString">：训练集</span></span></p></div><div id="https://www.notion.so/8772d883c64b49dabaccbe02300ee5bf" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{D}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9201899999999998em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">：训练集中在属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上没有缺失值的样本子集</span></span></p></div><div id="https://www.notion.so/cb3958c5f88449668362a47e1839c9c6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{D}^v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>v</mi></msup></mrow><annotation encoding="application/x-tex">\tilde{D}^v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9201899999999998em;vertical-align:0em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{D}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9201899999999998em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">被属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">划分后的样本子集</span></span></p></div><div id="https://www.notion.so/f408285b30d84702932b2999852bc867" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{D}_k"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{D}_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0701899999999998em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{D}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9201899999999998em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">中属于第</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="k"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span></span></span><span class="SemanticString">类的样本子集</span></span></p></div><div id="https://www.notion.so/13d2f88b7e7b4d849704588dd0d2f130" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假定我们为每个样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">赋予一个权重</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w_{\boldsymbol{x}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><annotation encoding="application/x-tex">w_{\boldsymbol{x}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，并定义</span></span></p></div><div id="https://www.notion.so/7308f4d2e7dd4e69b51db5d0b9560685" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无缺失值样本所占比例 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\rho=\dfrac{\sum_{\boldsymbol{x}\in \tilde{D}}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in D}w_{\boldsymbol{x}}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><mi>D</mi></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\rho=\dfrac{\sum_{\boldsymbol{x}\in \tilde{D}}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in D}w_{\boldsymbol{x}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ρ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.48016em;vertical-align:-1.01308em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4670800000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.32708000000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.71708em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3444229999999998em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.3023300000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord mtight">~</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.32708000000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.01308em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/ab357119f4874b5a926653ef0af0acd6" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无缺失值样本中第</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="k"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span></span></span><span class="SemanticString">类所占比例 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{p}_k=\dfrac{\sum_{\boldsymbol{x}\in \tilde{D}_k}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in \tilde{D}}w_{\boldsymbol{x}}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>k</mi></msub></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\tilde{p}_k=\dfrac{\sum_{\boldsymbol{x}\in \tilde{D}_k}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in \tilde{D}}w_{\boldsymbol{x}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8622999999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.55865em;vertical-align:-1.01308em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.54557em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3444229999999998em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.3023300000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord mtight">~</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.32708000000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.79557em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3444229999999998em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.3023300000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord mtight">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.40557em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.01308em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/e351200814b649af9bd7dd5186907e93" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无缺失值样本中在属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上取值</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a^v"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>v</mi></msup></mrow><annotation encoding="application/x-tex">a^v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">的样本所占比例 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{r}_v=\dfrac{\sum_{\boldsymbol{x}\in\tilde{D}^v}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in\tilde{D}}w_{\boldsymbol{x}}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>v</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><msup><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>v</mi></msup></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><mrow><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover></mrow></msub><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\tilde{r}_v=\dfrac{\sum_{\boldsymbol{x}\in\tilde{D}^v}w_{\boldsymbol{x}}}{\sum_{\boldsymbol{x}\in\tilde{D}}w_{\boldsymbol{x}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8178599999999999em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.48016em;vertical-align:-1.01308em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4670800000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3444229999999998em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.3023300000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord mtight">~</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.32708000000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.71708em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3444229999999998em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.3023300000000004em;"><span class="pstrut" style="height:2.7em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord mtight">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5935428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.32708000000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.01308em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/5aefc8da3ca74de08de12b22e6261383" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">问题1</strong></span><span class="SemanticString">：</span></span></p></div><div id="https://www.notion.so/6b6aecefe875409f9b8eda43ca0f365c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">显然，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></msubsup><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3461079999999999em;vertical-align:-0.30130799999999996em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.398692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2197999999999998em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\sum_{v=1}^{V}\tilde{r}_v=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></msubsup><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>v</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum_{v=1}^{V}\tilde{r}_v=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">.</span></span></p></div><div id="https://www.notion.so/c108427974aa49698c7e8c91d899397e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">基于上述定义，可将信息增益的计算式推广为</span></span></p></div><p id="https://www.notion.so/ba34f0d18c964c62b48603c4eef4d872" class="Equation" data-latex="\begin{aligned}
\text{Gain}(D,a)&=\rho\times\text{Gain}(\tilde{D},a) \\
&=\rho\times\Big(\text{Ent}(\tilde{D})-\sum_{v=1}^V\tilde{r}_v\text{Ent}(\tilde{D}^v)\Big)
\end{aligned}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Gain</mtext><mo stretchy="false">(</mo><mi>D</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>ρ</mi><mo>×</mo><mtext>Gain</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>ρ</mi><mo>×</mo><mo fence="false">(</mo><mtext>Ent</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></munderover><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>v</mi></msub><mtext>Ent</mtext><mo stretchy="false">(</mo><msup><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>v</mi></msup><mo stretchy="false">)</mo><mo fence="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\text{Gain}(D,a)&=\rho\times\text{Gain}(\tilde{D},a) \\
&=\rho\times\Big(\text{Ent}(\tilde{D})-\sum_{v=1}^V\tilde{r}_v\text{Ent}(\tilde{D}^v)\Big)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.975639em;vertical-align:-2.2378195em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7378195em;"><span style="top:-5.645965500000001em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-3.1576295em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.2378195em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7378195em;"><span style="top:-5.645965500000001em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">ρ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text"><span class="mord">Gain</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span><span style="top:-3.1576295em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">ρ</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.2378195em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/ce5a7c26fd7a420c8ac1f99976ff88cf" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span></span></p></div><p id="https://www.notion.so/5760c6d7896e4d578f0d25ebd374d45d" class="Equation" data-latex="\text{Ent}(\tilde{D})=-\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k\log_2 \tilde{p}_k"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ent</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>D</mi><mo>~</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="script">Y</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><msub><mover accent="true"><mi>p</mi><mo>~</mo></mover><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\text{Ent}(\tilde{D})=-\sum_{k=1}^{|\mathcal{Y}|}\tilde{p}_k\log_2 \tilde{p}_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1701899999999998em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ent</span></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2631180000000004em;vertical-align:-1.302113em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mtight"><span class="mord mathcal mtight" style="margin-right:0.08222em;">Y</span></span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">p</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">~</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><div id="https://www.notion.so/23c587a734784db38462c0d9a1f4d58f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于</span><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">问题2</strong></span><span class="SemanticString">：</span></span></p></div><div id="https://www.notion.so/c1e82c4816954394a232b2968d33a746" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">对于缺失属性值的样本如何将它从父结点划分到子结点中？</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/31b5a4ebe483484c976e46f2e82581a1" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">在划分属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上的取值已知，则将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">划入</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">与其取值对应的子结点</mark></span><span class="SemanticString">，且样本权值在子结点中保持为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w_{\boldsymbol{x}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><annotation encoding="application/x-tex">w_{\boldsymbol{x}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">.</span></span></li><li id="https://www.notion.so/0f425116b7e14f2a8b2470078d289276" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">若样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">在划分属性</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><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="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">上的取值未知，则将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\boldsymbol{x}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">x</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span></span></span><span class="SemanticString">同时划入所有子结点，且样本权值在子结点中调整为</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\tilde{r}_v\cdot w_{\boldsymbol{x}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>~</mo></mover><mi>v</mi></msub><mo>⋅</mo><msub><mi>w</mi><mi mathvariant="bold-italic">x</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{r}_v\cdot w_{\boldsymbol{x}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8178599999999999em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">x</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">，就是</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">让同一个样本以不同的概率划入不同的子结点中</mark></span><span class="SemanticString">。</span></span></li></ul><h2 id="https://www.notion.so/aa02205905d849cc981abae487223323" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/aa02205905d849cc981abae487223323"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">多变量决策树</span></span></h2><div id="https://www.notion.so/aa63e0277c874d4b8dc5d305721c908e" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6af50e21-0feb-47af-bdf6-1516cb23de26%2FUntitled.png?width=576&table=block&id=aa63e027-7c87-4d4b-8dc5-d305721c908e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F6af50e21-0feb-47af-bdf6-1516cb23de26%2FUntitled.png?width=576&table=block&id=aa63e027-7c87-4d4b-8dc5-d305721c908e" style="width:576px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/9551f6ee34164d6398874a88b0cc8c38" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">决策树形成的分类边界的明显特点：轴平行，分类边界由若干个与坐标轴平行的分段组成。</span></span></p></div><div id="https://www.notion.so/5869c969eadb403489adc42664ccf7d8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">优点：学习结果可解释性强，每个划分都对应一个属性取值</span></span></p></div><div id="https://www.notion.so/b4a305626262484e97dbdf34d7e5b86e" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ffd2dc29d-5a41-4473-b5ad-a642d3aaf15b%2FUntitled.png?width=1033&table=block&id=b4a30562-6262-484e-97db-df34d7e5b86e"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ffd2dc29d-5a41-4473-b5ad-a642d3aaf15b%2FUntitled.png?width=1033&table=block&id=b4a30562-6262-484e-97db-df34d7e5b86e" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/50c5d60224ba42d6831939a2d24d9d3c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">不足：决策树对复杂分类使用分段近似，此时的决策树会相当复杂，由于要进行大量的属性测试，预测时间开销会很大。</span></span></p></div><div id="https://www.notion.so/fbe14c934c3f488187e760b00821399b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">若能</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">使用斜的划分边界</mark></span><span class="SemanticString">，如图中红色线段所示，则决策树模型将大为简化。“多变量决策树” (multivariate decision tree) 就是能实现这样的“斜划分”甚至更复杂划分的决策树。</span></span></p></div><div id="https://www.notion.so/fc482234ac344077bb8942e6648a530b" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1326ff5d-74fb-4307-8749-1b0ae1f7a815%2FUntitled.png?width=240&table=block&id=fc482234-ac34-4077-bb89-42e6648a530b"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1326ff5d-74fb-4307-8749-1b0ae1f7a815%2FUntitled.png?width=240&table=block&id=fc482234-ac34-4077-bb89-42e6648a530b" style="width:240px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/b9f4e3e429fc44128a374da2b87daa49" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc2f6f4e6-228e-43d1-a83c-a727740aa6d0%2FUntitled.png?width=624&table=block&id=b9f4e3e4-29fc-4412-8a37-4da2b87daa49"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc2f6f4e6-228e-43d1-a83c-a727740aa6d0%2FUntitled.png?width=624&table=block&id=b9f4e3e4-29fc-4412-8a37-4da2b87daa49" style="width:624px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/3be485e8c55342dcb848816788b4e320" class="ColumnList"><div id="https://www.notion.so/0f95f8f737354c228eecbcc8580eecd6" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><div id="https://www.notion.so/24e27daa5aea4ffaba763021421c05e2" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc314ddaa-ba85-4b22-b10b-da047511dd10%2FUntitled.png?width=549&table=block&id=24e27daa-5aea-4ffa-ba76-3021421c05e2"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fc314ddaa-ba85-4b22-b10b-da047511dd10%2FUntitled.png?width=549&table=block&id=24e27daa-5aea-4ffa-ba76-3021421c05e2" style="width:549px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div><div id="https://www.notion.so/46364b4e8dc243f6b3b6a0e443111aee" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5)"><div id="https://www.notion.so/760070a717c14cdc85622b6d695b0283" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fe537e962-395a-4541-982a-dd9be8a907fe%2FUntitled.png?width=465&table=block&id=760070a7-17c1-4cdc-8562-2b6d695b0283"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fe537e962-395a-4541-982a-dd9be8a907fe%2FUntitled.png?width=465&table=block&id=760070a7-17c1-4cdc-8562-2b6d695b0283" style="width:465px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><h2 id="https://www.notion.so/509ee9f7514e46b5b453277456fe303c" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/509ee9f7514e46b5b453277456fe303c"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">总结</span></span></h2><ul class="BulletedListWrapper"><li id="https://www.notion.so/2e72eda600104ae0bd7959a19f103743" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策树是一种</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">基于规则的方法</mark></span><span class="SemanticString">，嵌套规则进行预测。根据判断结果进入一个分支，反复执行这个操作直到叶子结点，得到预测结果。这些规则通过训练得到，而非人工制定。</span></span></li><li id="https://www.notion.so/e65646b1bcff4b4695f20b890e7a3b98" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">本质上决策树是通过一系列规则对数据进行分类的过程。首先对数据进行处理，利用归纳算法生成可读的规则和决策树，然后使用决策树对新数据进行分析。</span></span></li><li id="https://www.notion.so/ddc9d94ba56344d4b7bc2a602c94f2be" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策树的优点：</span></span><ul class="BulletedListWrapper"><li id="https://www.notion.so/956a39f85a2f45a196d48aea1e019ce3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">推理过程容易理解，决策推理过程可以表示成 If Then 形式；</span></span></li><li id="https://www.notion.so/4cd276b4ba56467e94aa654b59976be5" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">推理过程完全依赖于属性变量的取值特点；</span></span></li><li id="https://www.notion.so/bd5ac8f622fd4e868362ce9d56d5c698" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">可自动忽略目标变量没有贡献的属性变量，也为判断属性变量的重要性、减少变量的数目提供参考。</span></span></li></ul></li></ul></article>
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