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      <h1 class="site-logo" id="site-title">데이터 사이언스 스쿨</h1>
      
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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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     2.6 파이썬 함수
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     2.7 파이썬 for 반복문
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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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     3.4 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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     9.5 사이파이를 사용한 검정
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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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   5.1 확률론적 선형 회귀모형
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   5.2 회귀분석의 기하학
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   5.3 레버리지와 아웃라이어
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   6.1 모형 진단과 수정
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  <a class="reference internal nav-link" href="#id2">
   시그모이드함수
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   로지스틱함수
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   선형 판별함수
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   로지스틱 회귀분석 모형의 모수 추정
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   수치적 최적화
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   판별함수
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   로지스틱 회귀 성능 측정
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     연습 문제 1
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     연습 문제 2
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   로지스틱 회귀를 사용한 회귀분석
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  <div class="section" id="id1">
<h1>6.1 로지스틱 회귀분석<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
<p>로지스틱(Logistic) 회귀분석은 회귀분석이라는 명칭과 달리 회귀분석 문제와 분류문제 모두에 사용할 수 있다. 로지스틱 회귀분석 모형에서는 종속변수가 이항분포를 따르고 그 모수 <span class="math notranslate nohighlight">\(\mu\)</span>가 독립변수 <span class="math notranslate nohighlight">\(x\)</span>에 의존한다고 가정한다.</p>
<div class="math notranslate nohighlight">
\[ p(y \mid x) = \text{Bin} (y; \mu(x), N) \]</div>
<p>위 식에서 보듯이 로지스틱 함수는 <span class="math notranslate nohighlight">\(y\)</span>의 값이 특정한 구간내의 값(<span class="math notranslate nohighlight">\(0 \sim N\)</span>)만 가질 수 있기 때문에 종속변수가 이러한 특성을 가진 경우에 회귀분석 방법으로 쓸 수 있다.</p>
<p>또는 이항 분포의 특별한 경우(<span class="math notranslate nohighlight">\(N=1\)</span>)로 <span class="math notranslate nohighlight">\(y\)</span>가 베르누이 확률분포인 경우도 있을 수 있다. 여기에서는 베르누이 확률분포를 따르는 로지스틱 회귀분석만 고려하기로 한다.</p>
<div class="math notranslate nohighlight">
\[ p(y \mid x) = \text{Bern} (y; \mu(x) )\]</div>
<p>종속변수 <span class="math notranslate nohighlight">\(y\)</span>가 0또는 1인 분류 예측 문제를 풀 때는 <span class="math notranslate nohighlight">\(x\)</span> 값을 이용하여 <span class="math notranslate nohighlight">\(\mu(x)\)</span>를 예측한 후 다음 기준에 따라 <span class="math notranslate nohighlight">\(\hat{y}\)</span>값을 출력한다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} \hat{y} = \begin{cases} 1 &amp; \text{ if } \mu(x) \geq 0.5 \\ 0 &amp; \text{ if } \mu(x) &lt; 0.5 \end{cases} \end{split}\]</div>
<p>회귀분석을 할 때는 <span class="math notranslate nohighlight">\(\hat{y}\)</span>으로 <span class="math notranslate nohighlight">\(y=1\)</span>이 될 확률값 <span class="math notranslate nohighlight">\(\mu(x)\)</span>를 직접 사용한다.</p>
<div class="math notranslate nohighlight">
\[ \hat{y} = \mu(x) \]</div>
<div class="section" id="id2">
<h2>시그모이드함수<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
<p>로지스틱 회귀모형에서는 베르누이 확률분포의 모수 <span class="math notranslate nohighlight">\(\mu\)</span>가 <span class="math notranslate nohighlight">\(x\)</span>의 함수라고 가정한다. <span class="math notranslate nohighlight">\(\mu(x)\)</span>는 <span class="math notranslate nohighlight">\(x\)</span>에 대한 함수를 0부터 1사이의 값만 나올 수 있도록 **시그모이드함수(sigmoid function)**라는 함수를 사용하여 변형한 것을 사용한다.</p>
<p>시그모이드함수는 종속변수의 모든 실수 값에 대해</p>
<ul class="simple">
<li><p>유한한 구간 <span class="math notranslate nohighlight">\((a,b)\)</span> 사이의 한정된(bounded) 값을 가지고
$<span class="math notranslate nohighlight">\( a &lt; f(x) &lt; b \)</span>$</p></li>
<li><p>항상 양의 기울기를 가지는 단조증가하는
$<span class="math notranslate nohighlight">\( a &gt; b \; \rightarrow \; f(a) &gt; f(b) \)</span>$</p></li>
</ul>
<p>함수의 집합을 말한다. 실제로는 다음과 같은 함수들이 주로 사용된다.</p>
<ul class="simple">
<li><p>로지스틱(Logistic)함수</p></li>
</ul>
<div class="math notranslate nohighlight">
\[ \text{logitstic}(z) = \sigma(z) = \dfrac{1}{1+\exp{(-z)}} \]</div>
<ul class="simple">
<li><p>하이퍼볼릭탄젠트(Hyperbolic tangent)함수</p></li>
</ul>
<div class="math notranslate nohighlight">
\[ \tanh(z) = \frac{\sinh z}{\cosh z} = \frac{(e^z - e^{-z})/2}{(e^z + e^{-z})/2} = 2 \sigma(2z) - 1\]</div>
<ul class="simple">
<li><p>오차(Error)함수</p></li>
</ul>
<div class="math notranslate nohighlight">
\[ \text{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt \]</div>
<p>하이퍼볼릭탄젠트함수는 로지스틱함수를 위아래 방향으로 2배 늘리고 좌우 방향으로 1/2로 축소한 것과 같다.</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">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">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1000</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="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">xx</span><span class="p">)),</span> <span class="s1">&#39;r-&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;로지스틱함수&quot;</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">sp</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">erf</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">xx</span><span class="p">),</span> <span class="s1">&#39;g:&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;오차함수&quot;</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">np</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="s1">&#39;b--&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;하이퍼볼릭탄젠트함수&quot;</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="mf">1.1</span><span class="p">,</span> <span class="mf">1.1</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">loc</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">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">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_5_0.png" src="../_images/10.01 로지스틱 회귀분석_5_0.png" />
</div>
</div>
</div>
<div class="section" id="id3">
<h2>로지스틱함수<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
<p>로지스틱함수는 음의 무한대부터 양의 무한대까지의 실수값을 0부터 1사이의 실수값으로 1 대 1 대응시키는 시그모이드함수다. 보통 시그모이드함수라고 하면 로지스틱함수를 가리킨다. 로지스틱함수는 다음 과정을 통해 정의되었다.</p>
<p>베르누이 시도에서 1이 나올 확률 <span class="math notranslate nohighlight">\(\mu\)</span>와 0이 나올 확률 <span class="math notranslate nohighlight">\(1-\mu\)</span>의 비율(ratio)을 승산비(odds ratio)라고 한다.</p>
<div class="math notranslate nohighlight">
\[ \text{odds ratio} = \dfrac{\mu}{1-\mu} \]</div>
<p>0부터 1사이의 값만 가지는 <span class="math notranslate nohighlight">\(\mu\)</span>를 승산비로 변환하면 0부터 양의 무한대까지의 값을 가질 수 있다.</p>
<p>승산비를 로그 변환한 것이 로지트함수(Logit function)다.</p>
<div class="math notranslate nohighlight">
\[ z = \text{logit}(\text{odds ratio}) = \log \left(\dfrac{\mu}{1-\mu}\right) \]</div>
<p>로지트함수의 값은 로그 변환에 의해 음의 무한대(<span class="math notranslate nohighlight">\(-\infty\)</span>)부터 양의 무한대(<span class="math notranslate nohighlight">\(\infty\)</span>)까지의 값을 가질 수 있다.</p>
<p>로지스틱함수(Logistic function)는 로지트함수의 역함수이다. 즉 <strong>음의 무한대(<span class="math notranslate nohighlight">\(-\infty\)</span>)부터 양의 무한대(<span class="math notranslate nohighlight">\(\infty\)</span>)까지의 값을 가지는 입력변수를 0부터 1사의 값을 가지는 출력변수로 변환</strong>한 것이다.</p>
<div class="math notranslate nohighlight">
\[ \text{logitstic}(z) = \mu(z) = \dfrac{1}{1+\exp{(-z)}} \]</div>
</div>
<div class="section" id="id4">
<h2>선형 판별함수<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
<p>로지스틱함수 <span class="math notranslate nohighlight">\(\sigma(z)\)</span>를 사용하는 경우에는 <span class="math notranslate nohighlight">\(z\)</span>값과 <span class="math notranslate nohighlight">\(\mu\)</span>값은 다음과 같은 관계가 있다.</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(z = 0\)</span>일 때 <span class="math notranslate nohighlight">\(\mu = 0.5\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(z &gt; 0\)</span>일 때 <span class="math notranslate nohighlight">\(\mu &gt; 0.5 \; \rightarrow \hat{y} = 1\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(z &lt; 0\)</span>일 때 <span class="math notranslate nohighlight">\(\mu &lt; 0.5 \; \rightarrow \hat{y} = 0\)</span></p></li>
</ul>
<p>즉 <span class="math notranslate nohighlight">\(z\)</span>가 분류 모형의 판별함수(decision function)의 역할을 한다. 로지스틱 회귀분석에서는 판별함수 수식으로 선형함수를 사용한다.</p>
<div class="math notranslate nohighlight">
\[ z = w^Tx  \]</div>
<p>따라서 판별 경계면도 선형이 된다.</p>
</div>
<div class="section" id="id5">
<h2>로지스틱 회귀분석 모형의 모수 추정<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h2>
<p>로지스틱 회귀분석 모형의 모수 <span class="math notranslate nohighlight">\(w\)</span>는 최대가능도(Maximum Likelihood Estimation, MLE)방법으로 추정할 수 있다.</p>
<p>우선 베르누이분포의 확률밀도함수는 다음과 같다.</p>
<div class="math notranslate nohighlight">
\[ p(y \mid x) = \text{Bern} (y;\mu(x;w) ) = \mu(x;w)^y ( 1 - \mu(x;w) )^{1-y} \]</div>
<p><span class="math notranslate nohighlight">\(\mu\)</span>는 <span class="math notranslate nohighlight">\(w^Tx\)</span>에 로지스틱함수를 적용한 값이다.</p>
<div class="math notranslate nohighlight">
\[
\mu(x;w) = \dfrac{1}{1 + \exp{(-w^Tx)}}
\]</div>
<p>이 식을 대입하면 조건부 확률은 다음과 같다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{eqnarray}
p(y \mid x) 
&amp;=&amp; \left(  \dfrac{1}{1 + \exp{(-w^Tx)}} \right) ^y \left(  1 - \dfrac{1}{1 + \exp{(-w^Tx)}} \right) ^{1-y} \\
&amp;=&amp; \left(  \dfrac{1}{1 + \exp{(-w^Tx)}} \right) ^y \left( \dfrac{\exp{(-w^Tx)}}{1 + \exp{(-w^Tx)}} \right) ^{1-y} \\
\end{eqnarray}
\end{split}\]</div>
<p>데이터 표본이 <span class="math notranslate nohighlight">\(\{ x_i, y_i \}_{1:N}\)</span>로 여러 개 있는 경우 전체 데이터의 로그가능도 <span class="math notranslate nohighlight">\({LL}\)</span>를 구하면 다음과 같다.</p>
<p>베르누이 확률분포의 정의에서</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray}
{LL} 
&amp;=&amp; \log \prod_{i=1}^N \mu(x_i;w)^{y_i} (1-\mu(x_i;w))^{1-y_i} \\
&amp;=&amp; \sum_{i=1}^N \left( y_i \log\mu(x_i;w) +  (1-y_i)\log(1-\mu(x_i;w)) \right) \\
&amp;=&amp; \sum_{i=1}^N \left( y_i \log\left(\dfrac{1}{1 + \exp{(-w^Tx_i)}}\right) + (1-y_i)\log\left(\dfrac{\exp{(-w^Tx_i)}}{1 + \exp{(-w^Tx_i)}}\right) \right) \\
\end{eqnarray}
\end{split}\]</div>
<p>가 된다.</p>
<p>로그가능도를 최대화하는 <span class="math notranslate nohighlight">\(w\)</span> 값을 구하기 위해 모수로 미분한다.</p>
<div class="math notranslate nohighlight">
\[ \dfrac{\partial{LL}}{\partial w}  = \sum_{i=1}^N \dfrac{\partial{LL}}{\partial \mu(x_i;w)} \dfrac{\partial\mu(x_i;w)}{\partial w} \]</div>
<p><span class="math notranslate nohighlight">\(LL\)</span>을 <span class="math notranslate nohighlight">\(\mu\)</span>로 미분하면</p>
<div class="math notranslate nohighlight">
\[
\dfrac{\partial{LL}}{\partial \mu(x_i;w)} =  \left( y_i \dfrac{1}{\mu(x_i;w)} - (1-y_i)\dfrac{1}{1-\mu(x_i;w)} \right) 
\]</div>
<p><span class="math notranslate nohighlight">\(\mu\)</span>를 <span class="math notranslate nohighlight">\(w\)</span>로 미분하면</p>
<div class="math notranslate nohighlight">
\[ 
\dfrac{\partial \mu(x_i;w)}{\partial w} 
= \dfrac{\partial}{\partial w} \dfrac{1}{1 + \exp{(-w^Tx_i)}} \ 
= \dfrac{\exp{(-w^Tx_i)}}{(1 + \exp{(-w^Tx_i)})^2} x_i \ 
= \mu(x_i;w)(1-\mu(x_i;w)) x_i 
\]</div>
<p>두 식을 곱하면 그레디언트 벡터의 수식을 구할 수 있다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{eqnarray}
\dfrac{\partial {LL}}{\partial w} 
&amp;=&amp; \sum_{i=1}^N \left( y_i \dfrac{1}{\mu(x_i;w)} - (1-y_i)\dfrac{1}{1-\mu(x_i;w)} \right) \mu(x_i;w)(1-\mu(x_i;w)) x_i   \\
&amp;=&amp; \sum_{i=1}^N \big( y_i (1-\mu(x_i;w)) - (1-y_i)\mu(x_i;w)  \big)  x_i \\
&amp;=&amp; \sum_{i=1}^N \big( y_i  - \mu(x_i;w) \big) x_i \\
\end{eqnarray}
\end{split}\]</div>
<p>그레디언트 벡터가 영벡터가 되는 모수의 값이 로그가능도를 최대화하는 값이다. 하지만 그레디언트 벡터 수식이 <span class="math notranslate nohighlight">\(w\)</span>에 대한 비선형 함수이므로 선형 모형과 같이 간단하게 그레디언트가 0이 되는 모수 <span class="math notranslate nohighlight">\(w\)</span> 값에 대한 수식을 구할 수 없으며 수치적인 최적화 방법(numerical optimization)을 통해 반복적으로 최적 모수 <span class="math notranslate nohighlight">\(w\)</span>의 값을 구해야 한다.</p>
</div>
<div class="section" id="id6">
<h2>수치적 최적화<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
<p>로그가능도함수 <span class="math notranslate nohighlight">\(LL\)</span>을 최대화하는 것은 다음 목적함수를 최소화하는 것과 같다.</p>
<div class="math notranslate nohighlight">
\[ J = -LL \]</div>
<p>최대경사도(Steepest Gradient Descent)방법을 사용하자.</p>
<p>그레디언트 벡터는
$<span class="math notranslate nohighlight">\(
g_k = \dfrac{d}{dw}(-LL)
\)</span>$</p>
<p>이고, 이 방향으로 스텝사이즈 <span class="math notranslate nohighlight">\(\eta_k\)</span>만큼 이동한다.</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray}
w_{k+1} 
&amp;=&amp; w_{k} - \eta_k g_k \\
&amp;=&amp; w_{k} + \eta_k \sum_{i=1}^N \big( y_i  - \mu(x_i; w_k) \big) x_i\\
\end{eqnarray}
\end{split}\]</div>
</div>
<div class="section" id="statsmodels">
<h2>StatsModels 패키지의 로지스틱 회귀<a class="headerlink" href="#statsmodels" title="Permalink to this headline">¶</a></h2>
<p>다음과 같은 1차원 독립변수를 가지는 분류문제를 풀어보자.</p>
<div class="cell docutils container">
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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="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">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">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="s2">&quot;k&quot;</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">sns</span><span class="o">.</span><span class="n">distplot</span><span class="p">(</span><span class="n">X0</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</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">hist</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">sns</span><span class="o">.</span><span class="n">distplot</span><span class="p">(</span><span class="n">X0</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">1</span><span class="p">,</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">hist</span><span class="o">=</span><span class="kc">False</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="mf">0.2</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">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_18_0.png" src="../_images/10.01 로지스틱 회귀분석_18_0.png" />
</div>
</div>
<p>StatsModels 패키지는 베르누이 분포를 따르는 로지스틱 회귀 모형 <code class="docutils literal notranslate"><span class="pre">Logit</span></code> 를 제공한다. 사용방법은 <code class="docutils literal notranslate"><span class="pre">OLS</span></code> 클래스 사용법과 동일하다. 종속변수와 독립변수 데이터를 넣어 모형을 만들고 <code class="docutils literal notranslate"><span class="pre">fit</span></code> 메서드로 학습을 시킨다. <code class="docutils literal notranslate"><span class="pre">fit</span></code> 메서드의 <code class="docutils literal notranslate"><span class="pre">disp=0</span></code> 인수는 최적화 과정에서 문자열 메세지를 나타내지 않는 역할을 한다.</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="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">X0</span><span class="p">)</span>
<span class="n">logit_mod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
<span class="n">logit_res</span> <span class="o">=</span> <span class="n">logit_mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">disp</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">logit_res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>                           Logit Regression Results                           
==============================================================================
Dep. Variable:                      y   No. Observations:                  100
Model:                          Logit   Df Residuals:                       98
Method:                           MLE   Df Model:                            1
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.7679
Time:                        10:01:05   Log-Likelihood:                -16.084
converged:                       True   LL-Null:                       -69.295
Covariance Type:            nonrobust   LLR p-value:                 5.963e-25
==============================================================================
                 coef    std err          z      P&gt;|z|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.2515      0.477      0.527      0.598      -0.683       1.186
x1             4.2382      0.902      4.699      0.000       2.470       6.006
==============================================================================
</pre></div>
</div>
</div>
</div>
<p>결과 객체에서 <code class="docutils literal notranslate"><span class="pre">summary</span></code> 메서드를 사용하여 리포트를 출력할 수 있다. 결과 리포트에서 판별함수의 수식이 다음과 같다는 것을 알 수 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\mu(x) = \sigma(4.2382x + 0.2515)
\]</div>
<p>따라서 <span class="math notranslate nohighlight">\(z\)</span>값의 부호를 나누는 기준값은 <span class="math notranslate nohighlight">\(4.2382x + 0.2515 = 0.5\)</span>가 되는 <span class="math notranslate nohighlight">\(x\)</span>값 즉, <span class="math notranslate nohighlight">\((0.5-0.2515)/4.2382\)</span>다.</p>
<p><code class="docutils literal notranslate"><span class="pre">predict</span></code> 메서드를 사용하면 <span class="math notranslate nohighlight">\(\mu(x)\)</span>값을 출력한다.</p>
<p>유의확률을 감안하면 상수항의 값은 0과 마찬가지이므로 <span class="math notranslate nohighlight">\(\mu(x)\)</span>가 다음과 같다고 볼 수도 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\mu(x) = \sigma(4.2382x)
\]</div>
<p>이렇게 생각하면 <span class="math notranslate nohighlight">\(z\)</span>값의 부호를 나누는 기준값은 실질적으로는 <span class="math notranslate nohighlight">\(0.5/4.2382=0.118\)</span>이다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></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">mu</span> <span class="o">=</span> <span class="n">logit_res</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">xx</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">mu</span><span class="p">,</span> <span class="n">lw</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">X0</span><span class="p">,</span> <span class="n">y</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="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">lw</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">X0</span><span class="p">,</span> <span class="n">logit_res</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="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$\hat</span><span class="si">{y}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;s&#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="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</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">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="sa">r</span><span class="s2">&quot;$\mu$&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="sa">r</span><span class="s2">&quot;$\hat</span><span class="si">{y}</span><span class="s2"> = \mu(x)$&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>
</pre></div>
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<div class="section" id="id7">
<h2>판별함수<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">Logit</span></code> 모형의 결과 객체에는 <code class="docutils literal notranslate"><span class="pre">fittedvalues</span></code>라는 속성으로 판별함수 <span class="math notranslate nohighlight">\(z=w^Tx\)</span> 값이 들어가 있다. 이 값을 이용하여 분류문제를 풀 수도 있다.</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">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">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="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">lw</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;데이터&quot;</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">X0</span><span class="p">,</span> <span class="n">logit_res</span><span class="o">.</span><span class="n">fittedvalues</span> <span class="o">*</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;판별함수값&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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<div class="section" id="id8">
<h2>로지스틱 회귀 성능 측정<a class="headerlink" href="#id8" title="Permalink to this headline">¶</a></h2>
<p>로지스틱 회귀 성능은 맥파든 의사결정계수(McFadden pseudo R square)값으로 측정한다.</p>
<div class="math notranslate nohighlight">
\[ R^2_{\text{pseudo}} = 1 - \dfrac{G^2}{G^2_0} \]</div>
<p><span class="math notranslate nohighlight">\(G^2\)</span>는 이탈도(deviance)라고 하는 양으로 다음과 같이 정의된다.</p>
<div class="math notranslate nohighlight">
\[ G^2 = 2\sum_{i=1}^N \left( y_i\log\dfrac{y_i}{\hat{y}_i} + (1-y_i)\log\dfrac{1-y_i}{1-\hat{y}_i} \right) \]</div>
<p>여기에서 <span class="math notranslate nohighlight">\(\hat{y}\)</span>는 <span class="math notranslate nohighlight">\(y=1\)</span>일 확률 <span class="math notranslate nohighlight">\(\mu\)</span>를 뜻한다.</p>
<div class="math notranslate nohighlight">
\[ \hat{y}_i = \mu(x_i) \]</div>
<p>이탈도는 모형이 100% 정확한 경우에는 0이 되고 모형의 성능이 나빠질수록 값이 커진다.</p>
<p>또한 이탈도는 로그 가능도에 음수를 취한 값과 같다.</p>
<div class="math notranslate nohighlight">
\[
G^2 = -LL
\]</div>
<p><span class="math notranslate nohighlight">\(G^2\)</span>는 현재 이탈도이고 <span class="math notranslate nohighlight">\(G^2_0\)</span>는 귀무모형(null model)으로 측정한 이탈도다.</p>
<p>귀무모형이란 모든 <span class="math notranslate nohighlight">\(x\)</span>가 <span class="math notranslate nohighlight">\(y\)</span>를 예측하는데 전혀 영향을 미치지 않는 모형을 말한다. 즉, 무조건부 확률 <span class="math notranslate nohighlight">\(p(y)\)</span>에 따라 <span class="math notranslate nohighlight">\(x\)</span>에 상관없이 동일하게 <span class="math notranslate nohighlight">\(y\)</span>를 예측하는 모형을 말한다. 결국 우리가 만들 수 있는 가장 성능이 나쁜 모형이 된다.</p>
<div class="math notranslate nohighlight">
\[ \mu_{\text{null}} = \dfrac{\text{number of $Y=1$ data}}{\text{number of all data}} \]</div>
<p>따라서 맥파든 의사결정계수는 가장 성능이 좋을 때는 1이 되고 가장 성능이 나쁠 때는 0이 된다.</p>
<p>scikit-learn 패키지의 metric 서브패키지에는 로그 손실을 계산하는 <code class="docutils literal notranslate"><span class="pre">log_loss</span></code> 함수가 있다. <code class="docutils literal notranslate"><span class="pre">normalize=False</span></code>로 놓으면 이탈도와 같은 값을 구한다</p>
<p>위 예제에서 최적 모형의 로그 손실은 약 16.08로 계산된다.</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.metrics</span> <span class="kn">import</span> <span class="n">log_loss</span>

<span class="n">y_hat</span> <span class="o">=</span> <span class="n">logit_res</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="n">log_loss</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_hat</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>16.084355200413036
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<p>귀무 모형의 모수값을 구하면 0.51이고 이 값으로 로그 손실을 계산하면 약 69이다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mu_null</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">mu_null</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.51
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">y_null</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">mu_null</span>
<span class="n">log_loss</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_null</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>69.29471672244784
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<p>두 값을 이용하여 맥파든 의사 결정계수 값을 계산할 수 있다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">log_loss</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_hat</span><span class="p">)</span> <span class="o">/</span> <span class="n">log_loss</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_null</span><span class="p">))</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.7678848264170398
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<div class="section" id="scikit-learn">
<h2>Scikit-Learn 패키지의 로지스틱 회귀<a class="headerlink" href="#scikit-learn" title="Permalink to this headline">¶</a></h2>
<p>Scikit-Learn 패키지는 로지스틱 회귀 모형 <code class="docutils literal notranslate"><span class="pre">LogisticRegression</span></code> 를 제공한다.</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">LogisticRegression</span>

<span class="n">model_sk</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">mu</span> <span class="o">=</span> <span class="mf">1.0</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">model_sk</span><span class="o">.</span><span class="n">coef_</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="o">*</span><span class="n">xx</span> <span class="o">-</span> <span class="n">model_sk</span><span class="o">.</span><span class="n">intercept_</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">plot</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">mu</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">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="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">lw</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">X0</span><span class="p">,</span> <span class="n">model_sk</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$\hat</span><span class="si">{y}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;s&#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="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</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="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">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="sa">r</span><span class="s2">&quot;$\mu$&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="sa">r</span><span class="s2">&quot;$\hat</span><span class="si">{y}</span><span class="s2">$ = sign $\mu(x)$&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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<div class="section" id="id9">
<h3>연습 문제 1<a class="headerlink" href="#id9" title="Permalink to this headline">¶</a></h3>
<ol class="simple">
<li><p>붓꽃 분류문제에서 클래스가 세토사와 베르시칼라 데이터만 사용하고 (setosa=0, versicolor=1) 독립변수로는 꽃받침 길이(Sepal Length)와 상수항만 사용하여 StatsModels 패키지의 로지스틱 회귀모형으로 결과를 예측하고 보고서를 출력한다. 이 보고서에서 어떤 값이 세토사와 베르시칼라를 구분하는 기준값(threshold)으로 사용되고 있는가?</p></li>
<li><p>위 결과를 분류결과표(confusion matrix)와 분류결과보고서(classification report)로 나타내라.</p></li>
<li><p>이 모형에 대해 ROC커브를 그리고 AUC를 구한다. 이 때 Scikit-Learn의 <code class="docutils literal notranslate"><span class="pre">LogisticRegression</span></code>을 사용하지 않고 위에서 StatsModels로 구한 모형을 사용한다.</p></li>
</ol>
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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">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="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span>
<span class="n">dfX</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">dfy</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;species&quot;</span><span class="p">])</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">dfX</span><span class="p">,</span> <span class="n">dfy</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="p">[[</span><span class="s2">&quot;sepal length (cm)&quot;</span><span class="p">,</span> <span class="s2">&quot;species&quot;</span><span class="p">]]</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="o">.</span><span class="n">isin</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">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="n">columns</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;sepal length (cm)&quot;</span><span class="p">:</span> <span class="s2">&quot;sepal_length&quot;</span> <span class="p">})</span>

<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>

<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;species ~ sepal_length&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.321056
         Iterations 8
                           Logit Regression Results                           
==============================================================================
Dep. Variable:                species   No. Observations:                  100
Model:                          Logit   Df Residuals:                       98
Method:                           MLE   Df Model:                            1
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.5368
Time:                        10:01:22   Log-Likelihood:                -32.106
converged:                       True   LL-Null:                       -69.315
Covariance Type:            nonrobust   LLR p-value:                 6.320e-18
================================================================================
                   coef    std err          z      P&gt;|z|      [0.025      0.975]
--------------------------------------------------------------------------------
Intercept      -27.8315      5.434     -5.122      0.000     -38.481     -17.182
sepal_length     5.1403      1.007      5.107      0.000       3.168       7.113
================================================================================
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># 기준값</span>
<span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="mf">27.8315</span><span class="p">)</span> <span class="o">/</span>  <span class="mf">5.1403</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>5.511643289302181
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">y_pred</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">sepal_length</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mf">0.5</span>

<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>

<span class="n">confusion_matrix</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[45,  5],
       [ 6, 44]])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">classification_report</span>

<span class="nb">print</span><span class="p">(</span><span class="n">classification_report</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>              precision    recall  f1-score   support

           0       0.88      0.90      0.89        50
           1       0.90      0.88      0.89        50

    accuracy                           0.89       100
   macro avg       0.89      0.89      0.89       100
weighted avg       0.89      0.89      0.89       100
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">roc_curve</span>

<span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="n">thresholds</span> <span class="o">=</span> <span class="n">roc_curve</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">sepal_length</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">fpr</span><span class="p">,</span> <span class="n">tpr</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>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_44_0.png" src="../_images/10.01 로지스틱 회귀분석_44_0.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">auc</span>
<span class="n">auc</span><span class="p">(</span><span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9326
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="id10">
<h2>로지스틱 회귀를 사용한 이진 분류의 예<a class="headerlink" href="#id10" title="Permalink to this headline">¶</a></h2>
<p>다음 데이터는 미국 의대생의 입학관련 데이터이다. 데이터의 의미는 다음과 같다.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">Acceptance</span></code>: 0이면 불합격, 1이면 합격</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">BCPM</span></code>: Bio/Chem/Physics/Math 과목의 학점 평균</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">GPA</span></code>: 전체과목 학점 평균</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">VR</span></code>: MCAT Verbal reasoning 과목 점수</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">PS</span></code>: MCAT Physical sciences 과목 점수</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">WS</span></code>: MCAT Writing sample 과목 점수</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">BS</span></code>: MCAT Biological sciences 과목 점수</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">MCAT</span></code>: MCAT 촘점</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">Apps</span></code>: 의대 지원 횟수</p></li>
</ul>
<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">data_med</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">&quot;MedGPA&quot;</span><span class="p">,</span> <span class="n">package</span><span class="o">=</span><span class="s2">&quot;Stat2Data&quot;</span><span class="p">)</span>
<span class="n">df_med</span> <span class="o">=</span> <span class="n">data_med</span><span class="o">.</span><span class="n">data</span>
<span class="n">df_med</span><span class="o">.</span><span class="n">tail</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><div>
<style scoped>
    .dataframe tbody tr th:only-of-type {
        vertical-align: middle;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }

    .dataframe thead th {
        text-align: right;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>Accept</th>
      <th>Acceptance</th>
      <th>Sex</th>
      <th>BCPM</th>
      <th>GPA</th>
      <th>VR</th>
      <th>PS</th>
      <th>WS</th>
      <th>BS</th>
      <th>MCAT</th>
      <th>Apps</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>50</th>
      <td>D</td>
      <td>0</td>
      <td>M</td>
      <td>2.41</td>
      <td>2.72</td>
      <td>8</td>
      <td>8</td>
      <td>8.0</td>
      <td>8</td>
      <td>32</td>
      <td>7</td>
    </tr>
    <tr>
      <th>51</th>
      <td>D</td>
      <td>0</td>
      <td>M</td>
      <td>3.51</td>
      <td>3.56</td>
      <td>11</td>
      <td>8</td>
      <td>6.0</td>
      <td>9</td>
      <td>34</td>
      <td>6</td>
    </tr>
    <tr>
      <th>52</th>
      <td>A</td>
      <td>1</td>
      <td>F</td>
      <td>3.43</td>
      <td>3.48</td>
      <td>7</td>
      <td>10</td>
      <td>7.0</td>
      <td>10</td>
      <td>34</td>
      <td>14</td>
    </tr>
    <tr>
      <th>53</th>
      <td>D</td>
      <td>0</td>
      <td>M</td>
      <td>2.61</td>
      <td>2.80</td>
      <td>7</td>
      <td>5</td>
      <td>NaN</td>
      <td>6</td>
      <td>18</td>
      <td>6</td>
    </tr>
    <tr>
      <th>54</th>
      <td>D</td>
      <td>0</td>
      <td>M</td>
      <td>3.36</td>
      <td>3.44</td>
      <td>11</td>
      <td>11</td>
      <td>8.0</td>
      <td>9</td>
      <td>39</td>
      <td>1</td>
    </tr>
  </tbody>
</table>
</div></div></div>
</div>
<p>일단 학점(GPA)과 합격여부의 관계를 살펴보자.</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">sns</span><span class="o">.</span><span class="n">stripplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">&quot;GPA&quot;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s2">&quot;Acceptance&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_med</span><span class="p">,</span>
              <span class="n">jitter</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">orient</span><span class="o">=</span><span class="s1">&#39;h&#39;</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">[</span><span class="mi">1</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">grid</span><span class="p">(</span><span class="kc">True</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>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_50_0.png" src="../_images/10.01 로지스틱 회귀분석_50_0.png" />
</div>
</div>
<p>로지스틱 회귀분석을 실시한다. <code class="docutils literal notranslate"><span class="pre">MCAT</span> <span class="pre">=</span> <span class="pre">VR</span> <span class="pre">+</span> <span class="pre">PS</span> <span class="pre">+</span> <span class="pre">WS</span> <span class="pre">+</span> <span class="pre">BS</span></code>이므로 이 <code class="docutils literal notranslate"><span class="pre">MCAT</span></code>은 독립 변수에서 제외해야 한다.</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">model_med</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;Acceptance ~ Sex + BCPM + GPA + VR + PS + WS + BS + Apps&quot;</span><span class="p">,</span> <span class="n">df_med</span><span class="p">)</span>
<span class="n">result_med</span> <span class="o">=</span> <span class="n">model_med</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result_med</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.280736
         Iterations 9
                           Logit Regression Results                           
==============================================================================
Dep. Variable:             Acceptance   No. Observations:                   54
Model:                          Logit   Df Residuals:                       45
Method:                           MLE   Df Model:                            8
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.5913
Time:                        10:01:33   Log-Likelihood:                -15.160
converged:                       True   LL-Null:                       -37.096
Covariance Type:            nonrobust   LLR p-value:                 6.014e-07
==============================================================================
                 coef    std err          z      P&gt;|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    -46.6414     15.600     -2.990      0.003     -77.216     -16.067
Sex[T.M]      -2.2835      1.429     -1.597      0.110      -5.085       0.518
BCPM          -6.1633      6.963     -0.885      0.376     -19.811       7.484
GPA           12.3973      8.611      1.440      0.150      -4.479      29.274
VR             0.0790      0.311      0.254      0.799      -0.530       0.688
PS             1.1673      0.539      2.164      0.030       0.110       2.225
WS            -0.7784      0.396     -1.968      0.049      -1.554      -0.003
BS             1.9184      0.682      2.814      0.005       0.582       3.255
Apps           0.0512      0.147      0.348      0.728      -0.237       0.340
==============================================================================
</pre></div>
</div>
</div>
</div>
<p>예측 결과와 실제 결과를 비교하면 다음과 같다.</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">df_med</span><span class="p">[</span><span class="s2">&quot;Prediction&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">result_med</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df_med</span><span class="p">)</span>
<span class="n">sns</span><span class="o">.</span><span class="n">boxplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">&quot;Acceptance&quot;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s2">&quot;Prediction&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_med</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>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_54_0.png" src="../_images/10.01 로지스틱 회귀분석_54_0.png" />
</div>
</div>
<p>위 분석 결과를 토대로 유의하지 않은 변수들을 제외하고 PS와 BS 점수만을 이용하여 다시 회귀분석하면 다음과 같다.</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">model_med</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;Acceptance ~ PS + BS&quot;</span><span class="p">,</span> <span class="n">df_med</span><span class="p">)</span>
<span class="n">result_med</span> <span class="o">=</span> <span class="n">model_med</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result_med</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.460609
         Iterations 7
                           Logit Regression Results                           
==============================================================================
Dep. Variable:             Acceptance   No. Observations:                   55
Model:                          Logit   Df Residuals:                       52
Method:                           MLE   Df Model:                            2
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.3315
Time:                        10:01:36   Log-Likelihood:                -25.333
converged:                       True   LL-Null:                       -37.896
Covariance Type:            nonrobust   LLR p-value:                 3.503e-06
==============================================================================
                 coef    std err          z      P&gt;|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    -15.5427      4.684     -3.318      0.001     -24.723      -6.362
PS             0.4798      0.316      1.518      0.129      -0.140       1.099
BS             1.1464      0.387      2.959      0.003       0.387       1.906
==============================================================================
</pre></div>
</div>
</div>
</div>
<p>위 결과를 바탕으로 다음 점수가 <span class="math notranslate nohighlight">\(15.5427+0.5\)</span>보다 크면 합격이라고 예측할 수 있다.</p>
<div class="math notranslate nohighlight">
\[
0.4798 \text{PS} + 1.1464 \text{BS}
\]</div>
<div class="section" id="id11">
<h3>연습 문제 2<a class="headerlink" href="#id11" title="Permalink to this headline">¶</a></h3>
<ol class="simple">
<li><p>붓꽃 분류문제에서 클래스가 베르시칼라(versicolor)와 버지니카(virginica) 데이터만 사용하여(versicolor=1, virginica=2) 로지스틱 회귀모형으로 결과를 예측하고 보고서를 출력한다. 독립변수는 모두 사용한다. 이 보고서에서 버지니카와 베르시칼라를 구분하는 경계면의 방정식을 찾아라.</p></li>
<li><p>위 결과를 분류결과표와 분류결과보고서로 나타내라.</p></li>
<li><p>이 모형에 대해 ROC커브를 그리고 AUC를 구하라.  이 때 Scikit-Learn의 <code class="docutils literal notranslate"><span class="pre">LogisticRegression</span></code>을 사용하지 않고 위에서 StatsModels로 구한 모형을 사용한다.</p></li>
</ol>
<div class="cell docutils container">
<div class="cell_input docutils container">
<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">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="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span>
<span class="n">dfX</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">dfy</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;species&quot;</span><span class="p">])</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">dfX</span><span class="p">,</span> <span class="n">dfy</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="o">.</span><span class="n">isin</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])]</span>
<span class="n">df</span><span class="p">[</span><span class="s2">&quot;species&quot;</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span>
    <span class="n">columns</span><span class="o">=</span><span class="p">{</span>
        <span class="s2">&quot;sepal length (cm)&quot;</span><span class="p">:</span> <span class="s2">&quot;sepal_length&quot;</span><span class="p">,</span>
        <span class="s2">&quot;sepal width (cm)&quot;</span><span class="p">:</span> <span class="s2">&quot;sepal_width&quot;</span><span class="p">,</span>
        <span class="s2">&quot;petal length (cm)&quot;</span><span class="p">:</span> <span class="s2">&quot;petal_length&quot;</span><span class="p">,</span>
        <span class="s2">&quot;petal width (cm)&quot;</span><span class="p">:</span> <span class="s2">&quot;petal_width&quot;</span><span class="p">,</span>
    <span class="p">}</span>
<span class="p">)</span>

<span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="k">as</span> <span class="nn">sm</span>

<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;species ~ sepal_length + sepal_width + petal_length + petal_width&quot;</span><span class="p">,</span> 
    <span class="n">data</span><span class="o">=</span><span class="n">df</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.059493
         Iterations 12
                           Logit Regression Results                           
==============================================================================
Dep. Variable:                species   No. Observations:                  100
Model:                          Logit   Df Residuals:                       95
Method:                           MLE   Df Model:                            4
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.9142
Time:                        10:01:37   Log-Likelihood:                -5.9493
converged:                       True   LL-Null:                       -69.315
Covariance Type:            nonrobust   LLR p-value:                 1.947e-26
================================================================================
                   coef    std err          z      P&gt;|z|      [0.025      0.975]
--------------------------------------------------------------------------------
Intercept      -42.6378     25.708     -1.659      0.097     -93.024       7.748
sepal_length    -2.4652      2.394     -1.030      0.303      -7.158       2.228
sepal_width     -6.6809      4.480     -1.491      0.136     -15.461       2.099
petal_length     9.4294      4.737      1.990      0.047       0.145      18.714
petal_width     18.2861      9.743      1.877      0.061      -0.809      37.381
================================================================================

Possibly complete quasi-separation: A fraction 0.60 of observations can be
perfectly predicted. This might indicate that there is complete
quasi-separation. In this case some parameters will not be identified.
</pre></div>
</div>
</div>
</div>
<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">y_pred</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mf">0.5</span>

<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>

<span class="n">confusion_matrix</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[49,  1],
       [ 1, 49]])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">classification_report</span>

<span class="nb">print</span><span class="p">(</span><span class="n">classification_report</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>              precision    recall  f1-score   support

           0       0.98      0.98      0.98        50
           1       0.98      0.98      0.98        50

    accuracy                           0.98       100
   macro avg       0.98      0.98      0.98       100
weighted avg       0.98      0.98      0.98       100
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">roc_curve</span>

<span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="n">thresholds</span> <span class="o">=</span> <span class="n">roc_curve</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">species</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df</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">fpr</span><span class="p">,</span> <span class="n">tpr</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>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_62_0.png" src="../_images/10.01 로지스틱 회귀분석_62_0.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">auc</span>
<span class="n">auc</span><span class="p">(</span><span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9972000000000001
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="id12">
<h2>로지스틱 회귀를 사용한 회귀분석<a class="headerlink" href="#id12" title="Permalink to this headline">¶</a></h2>
<p>로지스틱 회귀는 분류문제뿐만 아니라 종속변수 <span class="math notranslate nohighlight">\(y\)</span>가 0부터 1까지 막혀있는 회귀분석 문제에도 사용할 수 있다. 이때는 다음처럼 <span class="math notranslate nohighlight">\(\mu\)</span> 값을 종속변수 y의 예측값으로 사용한다.</p>
<div class="math notranslate nohighlight">
\[ \hat{y} = \mu(x) \]</div>
<p>만약 실제 y의 범위가 0부터 1이 아니면 스케일링을 통해 바꿔야 한다.</p>
<div class="section" id="id13">
<h3>예제<a class="headerlink" href="#id13" title="Permalink to this headline">¶</a></h3>
<p>다음 데이터는 1974년도에 “여성은 가정을 보살피고 국가를 운영하는 일은 남자에게 맡겨두어야 한다.”라는 주장에 대한 찬성, 반대 입장을 조사한 결과이다. 각 열은 다음을 뜻한다.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">education</span></code>: 교육 기간</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">sex</span></code>: 성별</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">agree</span></code>: 찬성 인원</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">disagree</span></code>: 반대 인원</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">ratio</span></code>: 찬성 비율</p></li>
</ul>
<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">data_wrole</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">get_rdataset</span><span class="p">(</span><span class="s2">&quot;womensrole&quot;</span><span class="p">,</span> <span class="n">package</span><span class="o">=</span><span class="s2">&quot;HSAUR&quot;</span><span class="p">)</span>
<span class="n">df_wrole</span> <span class="o">=</span> <span class="n">data_wrole</span><span class="o">.</span><span class="n">data</span>
<span class="n">df_wrole</span><span class="p">[</span><span class="s2">&quot;ratio&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df_wrole</span><span class="o">.</span><span class="n">agree</span> <span class="o">/</span> <span class="p">(</span><span class="n">df_wrole</span><span class="o">.</span><span class="n">agree</span> <span class="o">+</span> <span class="n">df_wrole</span><span class="o">.</span><span class="n">disagree</span><span class="p">)</span>
<span class="n">df_wrole</span><span class="o">.</span><span class="n">tail</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><div>
<style scoped>
    .dataframe tbody tr th:only-of-type {
        vertical-align: middle;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }

    .dataframe thead th {
        text-align: right;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>education</th>
      <th>sex</th>
      <th>agree</th>
      <th>disagree</th>
      <th>ratio</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>37</th>
      <td>16</td>
      <td>Female</td>
      <td>13</td>
      <td>115</td>
      <td>0.101562</td>
    </tr>
    <tr>
      <th>38</th>
      <td>17</td>
      <td>Female</td>
      <td>3</td>
      <td>28</td>
      <td>0.096774</td>
    </tr>
    <tr>
      <th>39</th>
      <td>18</td>
      <td>Female</td>
      <td>0</td>
      <td>21</td>
      <td>0.000000</td>
    </tr>
    <tr>
      <th>40</th>
      <td>19</td>
      <td>Female</td>
      <td>1</td>
      <td>2</td>
      <td>0.333333</td>
    </tr>
    <tr>
      <th>41</th>
      <td>20</td>
      <td>Female</td>
      <td>2</td>
      <td>4</td>
      <td>0.333333</td>
    </tr>
  </tbody>
</table>
</div></div></div>
</div>
<p>교육을 많이 받은 사람일수록 찬성 비율이 감소하는 것을 볼 수 있다.</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">sns</span><span class="o">.</span><span class="n">scatterplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">&quot;education&quot;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s2">&quot;ratio&quot;</span><span class="p">,</span> <span class="n">style</span><span class="o">=</span><span class="s2">&quot;sex&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_wrole</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">True</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>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_70_0.png" src="../_images/10.01 로지스틱 회귀분석_70_0.png" />
</div>
</div>
<p>분석 결과는 다음과 같다.</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">model_wrole</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;ratio ~ education + sex&quot;</span><span class="p">,</span> <span class="n">df_wrole</span><span class="p">)</span>
<span class="n">result_wrole</span> <span class="o">=</span> <span class="n">model_wrole</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result_wrole</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.448292
         Iterations 6
                           Logit Regression Results                           
==============================================================================
Dep. Variable:                  ratio   No. Observations:                   41
Model:                          Logit   Df Residuals:                       38
Method:                           MLE   Df Model:                            2
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.3435
Time:                        10:01:57   Log-Likelihood:                -18.380
converged:                       True   LL-Null:                       -27.997
Covariance Type:            nonrobust   LLR p-value:                 6.660e-05
===============================================================================
                  coef    std err          z      P&gt;|z|      [0.025      0.975]
-------------------------------------------------------------------------------
Intercept       2.0442      0.889      2.299      0.022       0.302       3.787
sex[T.Male]    -0.1968      0.736     -0.267      0.789      -1.640       1.247
education      -0.2127      0.071     -2.987      0.003      -0.352      -0.073
===============================================================================
</pre></div>
</div>
</div>
</div>
<p>성별은 유의하지 않다는 것을 알게되었으므로 성별을 제외하고 다시 모형을 구한다.</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">model_wrole2</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;ratio ~ education&quot;</span><span class="p">,</span> <span class="n">df_wrole</span><span class="p">)</span>
<span class="n">result_wrole2</span> <span class="o">=</span> <span class="n">model_wrole2</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">result_wrole2</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.449186
         Iterations 6
                           Logit Regression Results                           
==============================================================================
Dep. Variable:                  ratio   No. Observations:                   41
Model:                          Logit   Df Residuals:                       39
Method:                           MLE   Df Model:                            1
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.3422
Time:                        10:01:58   Log-Likelihood:                -18.417
converged:                       True   LL-Null:                       -27.997
Covariance Type:            nonrobust   LLR p-value:                 1.202e-05
==============================================================================
                 coef    std err          z      P&gt;|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      1.9345      0.781      2.478      0.013       0.405       3.464
education     -0.2117      0.071     -2.983      0.003      -0.351      -0.073
==============================================================================
</pre></div>
</div>
</div>
</div>
<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">sns</span><span class="o">.</span><span class="n">scatterplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="s2">&quot;education&quot;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s2">&quot;ratio&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_wrole</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="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">df_wrole_p</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span><span class="s2">&quot;education&quot;</span><span class="p">:</span> <span class="n">xx</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">result_wrole2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">df_wrole_p</span><span class="p">),</span> <span class="s2">&quot;r-&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;예측&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>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/10.01 로지스틱 회귀분석_75_0.png" src="../_images/10.01 로지스틱 회귀분석_75_0.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># 연습문제 1 답</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<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">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">1</span><span class="p">])</span>
<span class="n">X0</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">1</span><span class="p">]</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">X0</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>
</pre></div>
</div>
</div>
</div>
<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">logit_mod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
<span class="n">logit_res</span> <span class="o">=</span> <span class="n">logit_mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">disp</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">logit_res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>                           Logit Regression Results                           
==============================================================================
Dep. Variable:                      y   No. Observations:                  100
Model:                          Logit   Df Residuals:                       98
Method:                           MLE   Df Model:                            1
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.5368
Time:                        10:02:05   Log-Likelihood:                -32.106
converged:                       True   LL-Null:                       -69.315
Covariance Type:            nonrobust   LLR p-value:                 6.320e-18
==============================================================================
                 coef    std err          z      P&gt;|z|      [0.025      0.975]
------------------------------------------------------------------------------
const        -27.8315      5.434     -5.122      0.000     -38.481     -17.182
x1             5.1403      1.007      5.107      0.000       3.168       7.113
==============================================================================
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">logit_res</span><span class="o">.</span><span class="n">params</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([-27.83145099,   5.14033614])
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">-</span><span class="n">logit_res</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">logit_res</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>5.41432510257189
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">y_pred</span> <span class="o">=</span> <span class="n">logit_res</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="o">&gt;=</span> <span class="mf">0.5</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>

<span class="n">confusion_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[45,  5],
       [ 6, 44]])
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">classification_report</span>

<span class="nb">print</span><span class="p">(</span><span class="n">classification_report</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>              precision    recall  f1-score   support

           0       0.88      0.90      0.89        50
           1       0.90      0.88      0.89        50

    accuracy                           0.89       100
   macro avg       0.89      0.89      0.89       100
weighted avg       0.89      0.89      0.89       100
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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">plot</span><span class="p">(</span><span class="n">logit_res</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="s2">&quot;ro-&quot;</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">y_pred</span><span class="p">,</span> <span class="s2">&quot;bs-&quot;</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&lt;matplotlib.lines.Line2D at 0x7f653c0d97f0&gt;]
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<img alt="../_images/10.01 로지스틱 회귀분석_84_1.png" src="../_images/10.01 로지스틱 회귀분석_84_1.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">roc_curve</span>

<span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="n">thresholds</span> <span class="o">=</span> <span class="n">roc_curve</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">logit_res</span><span class="o">.</span><span class="n">fittedvalues</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">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="s1">&#39;o-&#39;</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/10.01 로지스틱 회귀분석_85_0.png" src="../_images/10.01 로지스틱 회귀분석_85_0.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># 연습문제 2 답</span>
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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">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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="n">X0</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</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="n">columns</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[:])</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">add_constant</span><span class="p">(</span><span class="n">X0</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="o">-</span> <span class="mi">1</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">logit_mod</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">Logit</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
<span class="n">logit_res</span> <span class="o">=</span> <span class="n">logit_mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">disp</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">logit_res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimization terminated successfully.
         Current function value: 0.059493
         Iterations 12
                           Logit Regression Results                           
==============================================================================
Dep. Variable:                      y   No. Observations:                  100
Model:                          Logit   Df Residuals:                       95
Method:                           MLE   Df Model:                            4
Date:                Sat, 06 Jun 2020   Pseudo R-squ.:                  0.9142
Time:                        10:02:15   Log-Likelihood:                -5.9493
converged:                       True   LL-Null:                       -69.315
Covariance Type:            nonrobust   LLR p-value:                 1.947e-26
=====================================================================================
                        coef    std err          z      P&gt;|z|      [0.025      0.975]
-------------------------------------------------------------------------------------
const               -42.6378     25.708     -1.659      0.097     -93.024       7.748
sepal length (cm)    -2.4652      2.394     -1.030      0.303      -7.158       2.228
sepal width (cm)     -6.6809      4.480     -1.491      0.136     -15.461       2.099
petal length (cm)     9.4294      4.737      1.990      0.047       0.145      18.714
petal width (cm)     18.2861      9.743      1.877      0.061      -0.809      37.381
=====================================================================================

Possibly complete quasi-separation: A fraction 0.60 of observations can be
perfectly predicted. This might indicate that there is complete
quasi-separation. In this case some parameters will not be identified.
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">y_pred</span> <span class="o">=</span> <span class="n">logit_res</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="o">&gt;=</span> <span class="mf">0.5</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>

<span class="n">confusion_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[49,  1],
       [ 1, 49]])
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">classification_report</span>

<span class="nb">print</span><span class="p">(</span><span class="n">classification_report</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>              precision    recall  f1-score   support

           0       0.98      0.98      0.98        50
           1       0.98      0.98      0.98        50

    accuracy                           0.98       100
   macro avg       0.98      0.98      0.98       100
weighted avg       0.98      0.98      0.98       100
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