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   데이터 사이언스 스쿨
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  파이썬 편
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     1.3 파이썬 패키지 설치하기
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     1.4 데이터 분석용 파이썬 패키지 소개
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     1.5 아이파이썬 및 주피터 설정
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   2장 파이썬 기초문법
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     2.2 부동소수점 실수 자료형
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     2.3 파이썬으로 글자를 출력하기
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     2.4 파이썬의 문자열 형식화
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     2.5 파이썬 조건문 기초
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     2.6 파이썬 함수
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     2.7 파이썬 for 반복문
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     2.8 여러 개의 자료를 한 변수에 담기
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     2.9 파이썬에서 리스트 자료형 다루기
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     2.10 리스트와 반복문을 사용하여 계산하기
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     2.11 파이썬에서 딕셔너리 자료형 다루기
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     2.12 파이썬 객체지향 프로그래밍
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     2.13 파이썬 패키지 사용하기
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     2.14 파이썬의 자료형
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     2.15 파이썬에서 날짜와 시간 다루기
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   3장 넘파이 배열 프로그래밍
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     3.1 넘파이 배열
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     3.2 배열의 생성과 변형
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     3.3 배열의 연산
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     3.4 기술 통계
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     3.5 난수 발생과 카운팅
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   4장 판다스 데이터 분석
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     4.1 판다스 패키지의 소개
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     4.2 데이터 입출력
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     4.3 데이터프레임 고급 인덱싱
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     4.4 데이터프레임의 데이터 조작
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     4.5 데이터프레임 인덱스 조작
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     4.6 데이터프레임 합성
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     4.7 피봇테이블과 그룹분석
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     4.8 시계열 자료 다루기
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   5장 데이터 시각화
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     5.1 시각화 패키지 맷플롯리브 소개
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     Matplotlib의 여러가지 플롯
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     Matplotlib의 triangular grid 사용법
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     Seaborn을 사용한 데이터 분포 시각화
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     Pandas의 시각화 기능
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<p class="caption">
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  수학 편
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   소개의 글
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   1장 수학 기호
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     1.1 그리스 문자
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     1.2 수열과 집합의 합과 곱
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   2장 넘파이(NumPy)로 공부하는 선형대수
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     2.1 데이터와 행렬
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     2.2 벡터와 행렬의 연산
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     2.3 행렬의 성질
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     2.4 선형 연립방정식과 역행렬
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   3장 고급 선형대수
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     3.1 선형대수와 해석기하의 기초
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     3.2 좌표와 변환
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     3.3 고윳값 분해
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     3.4 특잇값 분해
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     3.5 PCA
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   4장 심파이(SymPy)로 공부하는 미적분
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     4.1 함수
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     4.2 심파이를 사용한 함수 미분
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     4.3 적분
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     4.4 행렬의 미분
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     4.5 변분법
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   5장 사이파이(SciPy)로 공부하는 최적화
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     5.1 최적화 기초
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     5.2 제한조건이 있는 최적화 문제
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     5.3 선형계획법 문제와 이차계획법 문제
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   6장 피지엠파이(pgmpy)로 공부하는 확률론
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     6.1 집합
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     6.2 확률의 수학적 정의와 의미
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     6.3 확률의 성질
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     6.4 확률분포함수
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     6.5 결합확률과 조건부확률
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     6.6 베이즈 정리
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   7장 확률변수와 상관관계
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     7.1 확률적 데이터와 확률변수
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     7.2 기댓값과 확률변수의 변환
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     7.3 분산과 표준편차
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     7.4 다변수 확률변수
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     7.5 공분산과 상관계수
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     7.6 조건부기댓값과 예측 문제
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   8장 사이파이로 공부하는 확률분포
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     8.1 사이파이를 이용한 확률분포 분석
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     8.2 베르누이분포와 이항분포
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     8.3 카테고리분포와 다항분포
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     8.5 스튜던트 t분포, 카이제곱분포, F분포
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     8.6 다변수정규분포
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     8.7 베타분포, 감마분포, 디리클레분포
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     9.1 확률분포의 추정
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     9.2 최대가능도 추정법
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     9.3 베이즈 추정법
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     9.4 검정과 유의확률
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     10.2 조건부엔트로피
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     10.3 교차엔트로피와 쿨백-라이블러 발산
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     10.4 상호정보량
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   사운드 프로세싱 기초
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   4.1 회귀분석 예제
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   6.1 모형 진단과 수정
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 <li class="toc-h2 nav-item toc-entry">
  <a class="reference internal nav-link" href="#id2">
   분류모형의 종류
  </a>
 </li>
 <li class="toc-h2 nav-item toc-entry">
  <a class="reference internal nav-link" href="#id3">
   확률적 모형
  </a>
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  <a class="reference internal nav-link" href="#id4">
   확률적 생성모형
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  <a class="reference internal nav-link" href="#qda">
   QDA
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  <a class="reference internal nav-link" href="#id5">
   나이브 베이지안 모형
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  <a class="reference internal nav-link" href="#id6">
   확률적 판별 모형
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  <a class="reference internal nav-link" href="#id7">
   로지스틱 회귀 모형
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  <a class="reference internal nav-link" href="#id8">
   판별함수 모형
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  <a class="reference internal nav-link" href="#id9">
   퍼셉트론
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   커널 SVM
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  <a class="reference internal nav-link" href="#id10">
   다중 클래스 분류
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   OvO 방법
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  <div class="section" id="id1">
<h1>5.3 분류모형<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
<p>분류(classification)문제는 독립변수값이 주어졌을 때 그 값과 가장 연관성이 큰 종속변수값(클래스)을 예측하는 문제이다. 현실적인 문제로 바꾸어 말하면 어떤 표본에 대한 데이터가 주어졌을 때 그 표본이 어떤 카테고리 혹은 클래스에 속하는지를 알아내는 문제이기도 하다. 선택해야 할 카테고리 혹은 클래스가 미리 주어졌다는 점에서 보기가 주어진 객관식 시험 문제를 푸는 것과 비슷하다고 말할 수 있다.</p>
<div class="section" id="id2">
<h2>분류모형의 종류<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
<p>분류문제를 푸는 방법을 분류모형이라고 한다. 분류모형은 크게 두 가지 종류로 나눌 수 있다.</p>
<p>하나는 주어진 데이터에 대해(conditionally) 각 카테고리 혹은 클래스가 정답일 조건부확률(conditional probability)를 계산하는 <strong>확률적 모형</strong>이다. 확률적 모형은 조건부확률을 계산하는 방법에 따라 직접 조건부확률 함수의 모양을 추정하는 <strong>확률적 판별(discriminative)모형</strong>과 베이즈 정리를 사용하여 간접적으로 조건부확률을 구하는 <strong>확률적 생성(generative)모형</strong>으로 나누어진다.</p>
<p>다른 하나는 주어진 데이터를 카테고리에 따라 서로 다른 영역으로 나누는 경계면(decision boundary)을 찾아낸 다음 이 경계면으로부터 주어진 데이터가 어느 위치에 있는지를 계산하는 판별함수(discriminant function)를 이용하는 <strong>판별함수 모형</strong>이다.</p>
<p>아래 표에 여러가지 분류모형과 각 분류모형이 어떤 방법론에 속하는지를 나타내었다.</p>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p>모형</p></th>
<th class="head"><p>방법론</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>LDA/QDA</p></td>
<td><p>확률적 생성모형</p></td>
</tr>
<tr class="row-odd"><td><p>나이브 베이지안</p></td>
<td><p>확률적 생성모형</p></td>
</tr>
<tr class="row-even"><td><p>로지스틱 회귀</p></td>
<td><p>확률적 판별모형</p></td>
</tr>
<tr class="row-odd"><td><p>의사결정나무</p></td>
<td><p>확률적 판별모형</p></td>
</tr>
<tr class="row-even"><td><p>퍼셉트론</p></td>
<td><p>판별함수 모형</p></td>
</tr>
<tr class="row-odd"><td><p>서포트벡터머신</p></td>
<td><p>판별함수 모형</p></td>
</tr>
<tr class="row-even"><td><p>인공신경망</p></td>
<td><p>판별함수 모형</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="id3">
<h2>확률적 모형<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
<p>출력변수 <span class="math notranslate nohighlight">\(y\)</span>가 <span class="math notranslate nohighlight">\(K\)</span>개의 클래스 <span class="math notranslate nohighlight">\(1, \cdots, K\)</span> 중의 하나의 값을 가진다고 가정하자. 확률적 모형은 다음과 같은 순서로 <span class="math notranslate nohighlight">\(x\)</span>에 대한 클래스를 예측한다.</p>
<p>(1) 입력 <span class="math notranslate nohighlight">\(x\)</span>가 주어졌을 때 <span class="math notranslate nohighlight">\(y\)</span>가 클래스 <span class="math notranslate nohighlight">\(k\)</span>가 될 확률 <span class="math notranslate nohighlight">\(P(y=k \mid x )\)</span>을 모두 계산하고,</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray}
P_1 &amp;=&amp; P(y=1 \mid x ) \\
\vdots &amp; &amp; \vdots \\
P_K &amp;=&amp; P(y=K \mid x )\\
\end{eqnarray}
\end{split}\]</div>
<p>(2) 이 중에서 가장 확률이 큰 클래스를 선택하는 방법이다.</p>
<div class="math notranslate nohighlight">
\[ \hat{y} = \arg\max_{k} P(y=k \mid x ) \]</div>
<p>사이킷런 패키지에서 조건부확률 <span class="math notranslate nohighlight">\(P(y=k|x)\)</span>을 사용하는 분류모형들은 모두 <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> 메서드와 <code class="docutils literal notranslate"><span class="pre">predict_log_proba</span></code> 메서드를 지원한다. 이 메서드들은 독립변수 <span class="math notranslate nohighlight">\(x\)</span>가 주어지면 종속변수 <span class="math notranslate nohighlight">\(y\)</span>의 모든 카테고리 값에 대해 조건부확률 또는 조건부확률의 로그값을 계산한다.</p>
<p>조건부확률을 계산하는 방법은 다음 두 가지가 있다.</p>
<ol class="simple">
<li><p>생성모형(generative model)</p></li>
<li><p>판별모형(discriminative model)</p></li>
</ol>
</div>
<div class="section" id="id4">
<h2>확률적 생성모형<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
<p>생성모형은 먼저 각 클래스 별 특징 데이터의 확률분포 <span class="math notranslate nohighlight">\(P(x \mid y=k)\)</span>을 추정한 다음 베이즈 정리를 사용하여 <span class="math notranslate nohighlight">\(P(y=k|x)\)</span>를 계산하는 방법이다.</p>
<div class="math notranslate nohighlight">
\[ P(y=k \mid x) = \dfrac{P(x \mid y=k)P(y=k)}{P(x)} \]</div>
<p>생성모형에서는 전체 확률의 법칙을 이용하여 특징 데이터 <span class="math notranslate nohighlight">\(x\)</span>의 무조건부 확률분포 <span class="math notranslate nohighlight">\(P(x)\)</span>를 구할 수 있다.</p>
<div class="math notranslate nohighlight">
\[ P(x) = \sum_{k=1}^K P(x \mid y=k)P(y=k) \]</div>
<p>따라서 새로운 가상의 특징 데이터를 생성해내거나 특징 데이터만으로도 아웃라이어를 판단할 수 있다.</p>
<p>하지만 클래스가 많을 경우 모든 클래스에 대해 <span class="math notranslate nohighlight">\(P(x|y=k)\)</span>를 추정하는 것은 많은 데이터를 필요로 할 뿐더러 최종적으로는 사용하지도 않을 확률분포를 계산하는데 계산량을 너무 많이 필요로 한다는 단점이 있다.</p>
</div>
<div class="section" id="qda">
<h2>QDA<a class="headerlink" href="#qda" title="Permalink to this headline">¶</a></h2>
<p>QDA(Quadratic Discriminant Analysis)는 조건부확률 기반 생성(generative) 모형의 하나이다. 예를 들어 다음과 같이 두 가지 클래스의 데이터가 있다고 하자. 스캐터 플롯의 데이터 색은 각 데이터의 클래스를 나타낸다. 검은색은 <span class="math notranslate nohighlight">\(y=0\)</span>인 클래스이고 흰색은 <span class="math notranslate nohighlight">\(y=1\)</span>인 클래스이다. 3개씩으로 이루어진 두 클래스 데이터가 서로 대칭을 이루고 있다는 점에 주의하자.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=0&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;학습용 데이터&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>다음 그림들은 이 분류문제를 QDA 방법으로 푼 결과이다. 모형의 자세한 내용과 사이킷런 사용법은 추후 설명하기로 하고 일단 다음과 같은 코드를 사용하면 이 분류문제를 풀 수 있다는 것만 기억하자.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.discriminant_analysis</span> <span class="kn">import</span> <span class="n">QuadraticDiscriminantAnalysis</span>

<span class="n">model</span> <span class="o">=</span> <span class="n">QuadraticDiscriminantAnalysis</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
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<p>윗쪽 그림에서 X자로 표시한 것이 테스트 데이터이고, 아래쪽 그램은 이 테스트 데이터가 각 클래스에 속할 조건부확률을 나타낸다.
만약 테스트 데이터가  두 데이터 클래스의 정중앙에 있다면 이 테스트 데이터가 어느 클래스에 속할지 알 수 없으므로 확률이 0.5로 같은 값을 가진다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=0&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;테스트 데이터&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;조건부 확률분포&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="p">[</span><span class="s2">&quot;$P(y=0|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">,</span> <span class="s2">&quot;$P(y=1|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>만약 테스트 데이터가 <span class="math notranslate nohighlight">\(y=0\)</span>인 클래스에 더 가깝다면 <span class="math notranslate nohighlight">\(P(y=0|x_{\text{test}})\)</span>가 0.5보다 커지게 된다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.1</span><span class="p">]]</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=0&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;테스트 데이터&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;조건부 확률분포&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="p">[</span><span class="s2">&quot;$P(y=0|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">,</span> <span class="s2">&quot;$P(y=1|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<img alt="../_images/09.03 분류모형_19_0.png" src="../_images/09.03 분류모형_19_0.png" />
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<p>반대로 테스트 데이터가 <span class="math notranslate nohighlight">\(y=1\)</span>인 클래스에 더 가깝다면 <span class="math notranslate nohighlight">\(P(y=1|x_{\text{test}})\)</span>가 0.5보다 커지게 된다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">]]</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=0&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="mi">3</span><span class="p">:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y=1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;테스트 데이터&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;조건부 확률분포&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="p">[</span><span class="s2">&quot;$P(y=0|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">,</span> <span class="s2">&quot;$P(y=1|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="id5">
<h2>나이브 베이지안 모형<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h2>
<p>조건부확률 기반 생성 모형의 장점 중 하나는 클래스가 3개 이상인 경우에도 바로 적용할 수 있다는 점이다. 나이브 베이지안(Naive Bayesian) 모형도 조건부확률 모형의 일종이다. 아래는 20 뉴스그룹 분류문제를 나이브 베이지안 모형을 사용하여 푼 결과이다.</p>
<p>다음 코드에서는 사이킷런의 <code class="docutils literal notranslate"><span class="pre">TfidfVectorizer</span></code> 전처리기와 <code class="docutils literal notranslate"><span class="pre">MultinomialNB</span></code> 모형을 사용하여 20 뉴스그룹 분류문제를 풀었다. <code class="docutils literal notranslate"><span class="pre">TfidfVectorizer</span></code> 전처리기는 텍스트 데이터를 정해진 크기의 실수 벡터로 변환한다. <code class="docutils literal notranslate"><span class="pre">MultinomialNB</span></code> 모형은 나이브 베이즈 방법으로 분류문제를 예측한다. 이 두 클래스 객체는 <code class="docutils literal notranslate"><span class="pre">Pipeline</span></code>을 사용하여 하나의 모형으로 합쳐놓았다. <code class="docutils literal notranslate"><span class="pre">Pipeline</span></code>으로 합쳐진 모형은 일반적인 모형처럼 <code class="docutils literal notranslate"><span class="pre">fit</span></code>, <code class="docutils literal notranslate"><span class="pre">predict</span></code> 등의 메서드를 제공하며 내부의 전처리기의 메소드를 상황에 따라 적절히 호출한다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">fetch_20newsgroups</span>
<span class="kn">from</span> <span class="nn">sklearn.feature_extraction.text</span> <span class="kn">import</span> <span class="n">TfidfVectorizer</span>
<span class="kn">from</span> <span class="nn">sklearn.naive_bayes</span> <span class="kn">import</span> <span class="n">MultinomialNB</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">Pipeline</span>

<span class="n">news</span> <span class="o">=</span> <span class="n">fetch_20newsgroups</span><span class="p">(</span><span class="n">subset</span><span class="o">=</span><span class="s2">&quot;all&quot;</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Pipeline</span><span class="p">([</span>
    <span class="p">(</span><span class="s1">&#39;vect&#39;</span><span class="p">,</span> <span class="n">TfidfVectorizer</span><span class="p">(</span><span class="n">stop_words</span><span class="o">=</span><span class="s2">&quot;english&quot;</span><span class="p">)),</span>
    <span class="p">(</span><span class="s1">&#39;nb&#39;</span><span class="p">,</span> <span class="n">MultinomialNB</span><span class="p">()),</span>
<span class="p">])</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">news</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">news</span><span class="o">.</span><span class="n">target</span><span class="p">)</span>

<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">news</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">:</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;=&quot;</span> <span class="o">*</span> <span class="mi">80</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;실제 클래스:&quot;</span><span class="p">,</span> <span class="n">news</span><span class="o">.</span><span class="n">target_names</span><span class="p">[</span><span class="n">news</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">n</span><span class="p">]])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;예측 클래스:&quot;</span><span class="p">,</span> <span class="n">news</span><span class="o">.</span><span class="n">target_names</span><span class="p">[</span><span class="n">y</span><span class="p">])</span>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Downloading 20news dataset. This may take a few minutes.
Downloading dataset from https://ndownloader.figshare.com/files/5975967 (14 MB)
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>From: mblawson@midway.ecn.uoknor.edu (Matthew B Lawson)
Subject: Which high-performance VLB video card?
Summary: Seek recommendations for VLB video card
Nntp-Posting-Host: midway.ecn.uoknor.edu
Organization: Engineering Computer Network, University of Oklahoma, Norman, OK, USA
Keywords: orchid, stealth, vlb
Lines: 21

  My brother is in the market for a high-performance video card that supports
VESA local bus with 1-2MB RAM.  Does anyone have suggestions/ideas on:

  - Diamond Stealth Pro Local Bus

  - Orchid Farenheit 1280

  - ATI Graphics Ultra Pro

  - Any other high-performance VLB card


Please post or email.  Thank you!

  - Matt

-- 
    |  Matthew B. Lawson &lt;------------&gt; (mblawson@essex.ecn.uoknor.edu)  |   
  --+-- &quot;Now I, Nebuchadnezzar, praise and exalt and glorify the King  --+-- 
    |   of heaven, because everything he does is right and all his ways  |   
    |   are just.&quot; - Nebuchadnezzar, king of Babylon, 562 B.C.           |   

================================================================================
실제 클래스: comp.sys.ibm.pc.hardware
예측 클래스: comp.sys.ibm.pc.hardware
</pre></div>
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<p>다음 그림은 나이브 베이지안 모형으로 특정한 텍스트 데이터가 20개의 카테고리에 속할 확률을 각각 계산한 것이다. 위쪽 플롯은 확률값이고 아래쪽 플롯은 로그 변환한 값이다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;카테고리(클래스)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_log_proba</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;카테고리(클래스)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률의 로그값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;조건부 확률분포&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="id6">
<h2>확률적 판별 모형<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
<p>위에서 보인 QDA나 나이브 베이지안 방법은 베이즈 정리를 사용하는 조건부확률 기반 생성 모형이었다. 조건부확률 기반 생성 모형은 조건부확률 <span class="math notranslate nohighlight">\(p(y \mid x)\)</span>를 구하기 위해 우선 likelihood <span class="math notranslate nohighlight">\(p(x \mid y)\)</span>를 구하고 베이즈 정리를 사용하여 조건부확률을 계산한다.</p>
<p>하지만 확률론적 판별 모형(probabilistic discriminative model)은 조건부확률 <span class="math notranslate nohighlight">\(p(y=1 \mid x)\)</span>이 <span class="math notranslate nohighlight">\(x\)</span>에 대한 함수 <span class="math notranslate nohighlight">\(f(x)\)</span>로 표시될 수 있다고 가정하고 그 함수를 직접 찾아내는 방법이다.</p>
<div class="math notranslate nohighlight">
\[ p(y = k \mid x) = f(x) \]</div>
<p>단, 이 함수 <span class="math notranslate nohighlight">\(f(x)\)</span>는 0보다 같거나 크고 1보다 같거나 작다는 조건을 만족해야 한다.</p>
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<div class="section" id="id7">
<h2>로지스틱 회귀 모형<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h2>
<p>로지스틱 회귀 모형은 확률론적 판별 모형에 속한다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_classification</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>

<span class="n">X0</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_classification</span><span class="p">(</span><span class="n">n_features</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_redundant</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
                            <span class="n">n_informative</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_clusters_per_class</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>

<span class="n">xx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">XX</span> <span class="o">=</span> <span class="n">xx</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
<span class="n">prob</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">XX</span><span class="p">)[:,</span> <span class="mi">1</span><span class="p">]</span>
<span class="c1"># prob = 1.0/(1 + np.exp(-model.coef_[0][0]*xx - model.intercept_[0]))</span>

<span class="n">x_test</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">]]</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">prob</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x_test</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x_test</span><span class="p">)[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-.</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y=0 또는 y=1일 확률&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s2">&quot;$P(y=1|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">x_test</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">classes_</span><span class="p">,</span> <span class="p">[</span><span class="s2">&quot;$P(y=0|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">,</span> <span class="s2">&quot;$P(y=1|x_</span><span class="si">{test}</span><span class="s2">)$&quot;</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;조건부 확률분포&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;카테고리(클래스)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;확률값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="id8">
<h2>판별함수 모형<a class="headerlink" href="#id8" title="Permalink to this headline">¶</a></h2>
<p>또 다른 분류 방법은 동일한 클래스가 모여 있는 영역과 그 영역을 나누는 경계면(boundary plane)을 정의하는 것이다.
이 경계면은 경계면으로부터의 거리를 계산하는 <span class="math notranslate nohighlight">\(f(x)\)</span> 형태의 함수인 **판별함수(discriminant function)**로 정의된다. 판별함수의 값의 부호에 따라 클래스가 나뉘어진다.</p>
<div class="math notranslate nohighlight">
\[ \text{판별 경계선}: \;\; f(x) = 0 \]</div>
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\[ \text{클래스 1}: \;\; f(x) &gt; 0 \]</div>
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\[ \text{클래스 0}: \;\; f(x) &lt; 0 \]</div>
<p>사이킷런에서 판별함수 모형은 판별함수값을 출력하는 <code class="docutils literal notranslate"><span class="pre">decision_function</span></code> 메서드를 제공한다.</p>
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<div class="section" id="id9">
<h2>퍼셉트론<a class="headerlink" href="#id9" title="Permalink to this headline">¶</a></h2>
<p>퍼셉트론(Perceptron)은 가장 단순한 판별함수 모형이다. 다음 그림과 같이 직선이 경계선(boundary line)으로 데이터 영역을 나눈다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Perceptron</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
<span class="n">idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">in1d</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span>

<span class="n">model</span> <span class="o">=</span> <span class="n">Perceptron</span><span class="p">(</span><span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">eta0</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">XX_min</span><span class="p">,</span> <span class="n">XX_max</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">YY_min</span><span class="p">,</span> <span class="n">YY_max</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">XX</span><span class="p">,</span> <span class="n">YY</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">XX_min</span><span class="p">,</span> <span class="n">XX_max</span><span class="p">,</span> <span class="mi">1000</span><span class="p">),</span>
                     <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">YY_min</span><span class="p">,</span> <span class="n">YY_max</span><span class="p">,</span> <span class="mi">1000</span><span class="p">))</span>
<span class="n">ZZ</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">XX</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">YY</span><span class="o">.</span><span class="n">ravel</span><span class="p">()])</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">XX</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">XX</span><span class="p">,</span> <span class="n">YY</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>

<span class="n">idx</span> <span class="o">=</span> <span class="p">[</span><span class="mi">22</span><span class="p">,</span> <span class="mi">36</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">80</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">idx</span><span class="p">:</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">xy</span><span class="o">=</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.1</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;퍼셉트론의 판별영역&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idx</span><span class="p">)),</span> <span class="n">model</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idx</span><span class="p">)),</span> <span class="n">idx</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;각 데이터의 판별함수 값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;표본 번호&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;판별함수값 f(x)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>만약 데이터의 차원이 3차원이라면 다음과 같이 경계면(boundary surface)을 가지게 된다. 이러한 경계면이나 경계선을 의사결정 하이퍼 플레인(decision hyperplane)이라고 한다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>

<span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span>
<span class="n">idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span>

<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">Axes3D</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">elev</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">azim</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">mpl</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">jet</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.5</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.5</span><span class="p">]]),</span>
                <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]),</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">3</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;3차원 특징데이터의 판별경계&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;z&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="section" id="svm">
<h2>커널 SVM<a class="headerlink" href="#svm" title="Permalink to this headline">¶</a></h2>
<p>커널 SVM(Kernel Support Vector Machine)을 사용하면 다음과 같이 복잡한 형태의 경계선을 생성할 수도 있다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span>

<span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">500</span><span class="p">),</span>
                     <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">500</span><span class="p">))</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">300</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logical_xor</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span>

<span class="n">model</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">NuSVC</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">xx</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">yy</span><span class="o">.</span><span class="n">ravel</span><span class="p">()])</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">Z</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">Z</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">,</span>
           <span class="n">extent</span><span class="o">=</span><span class="p">(</span><span class="n">xx</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">xx</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="n">yy</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">yy</span><span class="o">.</span><span class="n">max</span><span class="p">()),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span>
           <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">PuOr_r</span><span class="p">)</span>
<span class="n">contours</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">Y</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Paired</span><span class="p">)</span>
<span class="n">idx</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">60</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">Y</span><span class="p">[</span><span class="n">idx</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">idx</span><span class="p">:</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">xy</span><span class="o">=</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mf">0.15</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;white&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;x2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;커널 SVM의 판별영역&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idx</span><span class="p">)),</span> <span class="n">model</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">idx</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idx</span><span class="p">)),</span> <span class="n">idx</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;표본 번호&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;판별함수값 f(x)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;각 데이터의 판별함수 값&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="section" id="id10">
<h2>다중 클래스 분류<a class="headerlink" href="#id10" title="Permalink to this headline">¶</a></h2>
<p>확률적 모형은 클래스가 3개 이상인 경우를 다중 클래스(Multi-Class) 분류문제도 풀 수 있지만 판별함수 모형은 종속변수의 클래스가 2개인 경우를 이진(Binary Class) 분류문제밖에는 풀지 못한다.</p>
<p>이때는 OvO (One-Vs-One) 방법이나  OvR (One-vs-the-Rest) 방법 등을 이용하여 여러개의 이진 클래스 분류문제로 변환하여 푼다.</p>
</div>
<div class="section" id="ovo">
<h2>OvO 방법<a class="headerlink" href="#ovo" title="Permalink to this headline">¶</a></h2>
<p>OvO (One-Vs-One) 방법은 <span class="math notranslate nohighlight">\(K\)</span>개의 클래스가 존재하는 경우, 이 중 2개의 클래스 조합을 선택하여 <span class="math notranslate nohighlight">\(K(K - 1)/2\)</span>개의 이진 클래스 분류문제를 풀고  이진 분류문제를 풀어 가장 많은 결과가 나온 클래스를 선택하는 방법이다. 선택받은 횟수로 선택하면 횟수가 같은 경우도 나올 수 있기 때문에 각 클래스가 얻은 조건부 확률값을 모두 더한 값을 비교하여 가장 큰 조건부 확률 총합을 가진 클래스를 선택한다.</p>
</div>
<div class="section" id="ovr">
<h2>OvR 방법<a class="headerlink" href="#ovr" title="Permalink to this headline">¶</a></h2>
<p>OvO 방법은 클래스의 수가 많아지면 실행해야 할 이진 분류문제의 수가 너무 많아진다.</p>
<p>OvR(One-vs-the-Rest) 방법은 <span class="math notranslate nohighlight">\(K\)</span>개의 클래스가 존재하는 경우, 각각의 클래스에 대해 표본이 속하는지(y=1) 속하지 않는지(y=0)의 이진 분류문제를 푼다. OvO와 달리 클래스 수만큼의 이진 분류문제를 풀면 된다.</p>
<p>OvR에서도 판별 결과의 수가 같은 동점 문제가 발생할 수가 있기 때문에 각 클래스가 얻은 조건부 확률값을 더해서 이 값이 가장 큰 클래스를 선택한다.</p>
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