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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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   2장 파이썬 기초문법
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     2.4 파이썬의 문자열 형식화
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     2.5 파이썬 조건문 기초
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     2.6 파이썬 함수
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     2.7 파이썬 for 반복문
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     2.8 여러 개의 자료를 한 변수에 담기
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     2.9 파이썬에서 리스트 자료형 다루기
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     2.11 파이썬에서 딕셔너리 자료형 다루기
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     2.12 파이썬 객체지향 프로그래밍
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     2.13 파이썬 패키지 사용하기
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     2.14 파이썬의 자료형
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     2.15 파이썬에서 날짜와 시간 다루기
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   3장 넘파이 배열 프로그래밍
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     3.2 배열의 생성과 변형
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     3.4 기술 통계
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     3.5 난수 발생과 카운팅
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   4장 판다스 데이터 분석
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     4.1 판다스 패키지의 소개
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     4.2 데이터 입출력
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     4.3 데이터프레임 고급 인덱싱
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     4.4 데이터프레임의 데이터 조작
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     4.5 데이터프레임 인덱스 조작
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     4.6 데이터프레임 합성
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     4.7 피봇테이블과 그룹분석
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     4.8 시계열 자료 다루기
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   5장 데이터 시각화
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     5.1 시각화 패키지 맷플롯리브 소개
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     Matplotlib의 여러가지 플롯
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     Matplotlib의 triangular grid 사용법
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     Seaborn을 사용한 데이터 분포 시각화
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     Pandas의 시각화 기능
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<p class="caption">
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  수학 편
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   소개의 글
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   1장 수학 기호
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     1.1 그리스 문자
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     1.2 수열과 집합의 합과 곱
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   2장 넘파이(NumPy)로 공부하는 선형대수
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     2.1 데이터와 행렬
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     2.2 벡터와 행렬의 연산
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     2.3 행렬의 성질
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     2.4 선형 연립방정식과 역행렬
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   3장 고급 선형대수
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     3.1 선형대수와 해석기하의 기초
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     3.2 좌표와 변환
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     3.3 고윳값 분해
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     3.4 특잇값 분해
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     3.5 PCA
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   4장 심파이(SymPy)로 공부하는 미적분
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     4.1 함수
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     4.2 심파이를 사용한 함수 미분
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     4.3 적분
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     4.4 행렬의 미분
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     4.5 변분법
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   5장 사이파이(SciPy)로 공부하는 최적화
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     5.1 최적화 기초
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     5.2 제한조건이 있는 최적화 문제
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     5.3 선형계획법 문제와 이차계획법 문제
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   6장 피지엠파이(pgmpy)로 공부하는 확률론
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     6.1 집합
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     6.2 확률의 수학적 정의와 의미
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     6.3 확률의 성질
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     6.4 확률분포함수
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     6.5 결합확률과 조건부확률
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     6.6 베이즈 정리
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   7장 확률변수와 상관관계
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     7.1 확률적 데이터와 확률변수
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     7.2 기댓값과 확률변수의 변환
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     7.3 분산과 표준편차
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     7.4 다변수 확률변수
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     7.5 공분산과 상관계수
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     7.6 조건부기댓값과 예측 문제
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   8장 사이파이로 공부하는 확률분포
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     8.1 사이파이를 이용한 확률분포 분석
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     8.2 베르누이분포와 이항분포
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     8.3 카테고리분포와 다항분포
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     8.5 스튜던트 t분포, 카이제곱분포, F분포
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     8.6 다변수정규분포
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     8.7 베타분포, 감마분포, 디리클레분포
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     9.1 확률분포의 추정
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     9.2 최대가능도 추정법
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     9.3 베이즈 추정법
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     9.4 검정과 유의확률
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     10.4 상호정보량
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   사운드 프로세싱 기초
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   4.1 회귀분석 예제
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  <div class="section" id="id1">
<h1>5.4 분산 분석과 모형 성능<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
<div class="section" id="id2">
<h2>분산 분석<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
<p>선형회귀분석의 결과가 얼마나 좋은지는 단순히 잔차제곱합(RSS: Residula Sum of Square)으로 평가할 수 없다. 변수의 단위 즉, 스케일이 달라지면 회귀분석과 상관없이 잔차제곱합도 달라지기 때문이다.</p>
<p>분산 분석(ANOVA: Analysis of Variance)은 종속변수의 분산과 독립변수의 분산간의 관계를 사용하여 선형회귀분석의 성능을 평가하고자 하는 방법이다. 분산 분석은 서로 다른 두 개의 선형회귀분석의 성능 비교에 응용할 수 있으며 독립변수가 카테고리 변수인 경우 각 카테고리 값에 따른 영향을 정량적으로 분석하는데도 사용된다.</p>
<p><span class="math notranslate nohighlight">\(\bar{y}\)</span>를 종속 변수 <span class="math notranslate nohighlight">\(y\)</span>의 샘플 평균이라고 하자.</p>
<div class="math notranslate nohighlight">
\[\bar{y}=\frac{1}{N}\sum_{i=1}^N y_i \]</div>
<p>종속 변수 <span class="math notranslate nohighlight">\(y\)</span>의 분산(샘플의 갯수로 나누지 않았으므로 정확하게는 분산이 아니지만 여기에서는 분산이라는 용어를 사용하자)을 나타내는 **TSS(total sum of square)**라는 값을 정의한다. <strong>TSS는 종속변수값의 움직임의 범위</strong>를 나타낸다.</p>
<div class="math notranslate nohighlight">
\[\text{TSS} = \sum_{i=1}^N (y_i-\bar{y})^2 = (y - \bar{y}1_N)^T(y - \bar{y}1_N)\]</div>
<p>위 식에서 <span class="math notranslate nohighlight">\(\bar{y}1_N\)</span>는 <span class="math notranslate nohighlight">\(\bar{y}\)</span>이라는 스칼라가 <span class="math notranslate nohighlight">\(N\)</span>번 반복된 브로드캐스팅 벡터다.</p>
<p>마찬가지로 회귀 분석에 의해 예측한 값 <span class="math notranslate nohighlight">\(\hat{y}\)</span>의 분산을 나타내는 <strong>ESS(explained sum of squares)</strong>,</p>
<div class="math notranslate nohighlight">
\[\text{ESS}=\sum_{i=1}^N (\hat{y}_i -\bar{\hat{y}})^2 = (\hat{y} - \bar{\hat{y}}1_N)^T(\hat{y} - \bar{\hat{y}}1_N)\]</div>
<p>잔차 <span class="math notranslate nohighlight">\(e\)</span>의 분산을 나타내는 **RSS(residual sum of squares)**도 정의할 수 있다.</p>
<div class="math notranslate nohighlight">
\[\text{RSS}=\sum_{i=1}^N (y_i - \hat{y}_i)^2\ = e^Te\]</div>
<p>위 식에서 <span class="math notranslate nohighlight">\(\bar{\hat{y}}\)</span>는 모형 예측값 <span class="math notranslate nohighlight">\(\hat{y}\)</span>의 평균이다.</p>
<p>또한 <strong>ESS는 모형에서 나온 예측값의 움직임의 범위</strong>, <strong>RSS는 잔차의 움직임의 범위, 즉 오차의 크기</strong>를 뜻한다고 볼 수 있다.</p>
<p>만약 회귀모형이 상수항을 포함하여 올바르게 정의되었다면 잔차의 평균이 0이 된다. 즉, 종속변수의 평균과 모형 예측값의 평균이 같다.</p>
<div class="math notranslate nohighlight">
\[ \bar{e} = \bar{y} - \bar{\hat{y}} = 0\]</div>
<div class="math notranslate nohighlight">
\[ \bar{y} = \bar{\hat{y}} \]</div>
<p>그리고 이 분산값들 간에는 다음과 같은 관계가 성립한다.</p>
<div class="math notranslate nohighlight">
\[\text{TSS} = \text{ESS} + \text{RSS}\]</div>
<p>이는 다음과 같이 증명할 수 있다.</p>
<p>우선 회귀 분석으로 구한 가중치 벡터를 <span class="math notranslate nohighlight">\(\hat{w}\)</span>, 독립 변수(설명 변수) <span class="math notranslate nohighlight">\(x\)</span>에 의한 종속 변수의 추정값을 <span class="math notranslate nohighlight">\(\hat{y}\)</span>, 잔차를 <span class="math notranslate nohighlight">\(e\)</span> 라고 하면 다음 식이 성립한다.</p>
<div class="math notranslate nohighlight">
\[ y = X\hat{w} + e = \hat{y} + e \]</div>
<p>그리고 <span class="math notranslate nohighlight">\(X\)</span>의 평균 데이터 <span class="math notranslate nohighlight">\(\bar{x}\)</span></p>
<div class="math notranslate nohighlight">
\[ 
\bar{x} = \frac{1}{N}X^T1_N
\]</div>
<p>에 대한 예측값은 <span class="math notranslate nohighlight">\(y\)</span>의 평균데이터 <span class="math notranslate nohighlight">\(\bar{y}\)</span>가 되므로</p>
<div class="math notranslate nohighlight">
\[
\bar{x}^T\hat{w} = \bar{y}
\]</div>
<p>각 행의 값이 평균 데이터 <span class="math notranslate nohighlight">\(\bar{x}\)</span>로 반복되는 행렬 <span class="math notranslate nohighlight">\(\bar{X}\)</span></p>
<div class="math notranslate nohighlight">
\[
\bar{X} = \frac{1}{N}X^T1_N 1_N^T
\]</div>
<p>에 대한 예측값 벡터는 <span class="math notranslate nohighlight">\(\bar{y}\)</span>값이 반복되는 벡터가 된다.</p>
<div class="math notranslate nohighlight">
\[
\bar{X}\hat{w} = \bar{y}1_N
\]</div>
<p>이를 위 식에 대입하면</p>
<div class="math notranslate nohighlight">
\[
\hat{y} - \bar{y}1_N  = (X- \bar{X})\hat{w} 
\]</div>
<p>가 된다.</p>
<p>그런데 <span class="math notranslate nohighlight">\(\bar{X}\)</span>와 잔차 <span class="math notranslate nohighlight">\(e\)</span>는 다음과 같은 직교 관계가 성립한다.</p>
<div class="math notranslate nohighlight">
\[ \bar{X}^Te = \frac{1}{N}X^T1_N 1_N^Te = \frac{1}{N}X^T1_N 0 = 0 \]</div>
<p>직교방정식</p>
<div class="math notranslate nohighlight">
\[
X^Te = 0
\]</div>
<p>과 합치면 다음 식이 성립한다.</p>
<div class="math notranslate nohighlight">
\[ \bar{X}^Te - X^Te = (\bar{X} - X)^Te \]</div>
<p>따라서</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{eqnarray}
\text{TSS} 
&amp;=&amp; (y - \bar{y}1_N)^T(y - \bar{y}1_N) \\
&amp;=&amp; (\hat{y} - \bar{y}1_N + e)^T(\hat{y} - \bar{y}1_N + e) \\
&amp;=&amp; (\hat{y} - \bar{y}1_N)^T(\hat{y} - \bar{y}1_N) + e^Te + 2(\hat{y} - \bar{y}1_N)^Te \\
&amp;=&amp; (\hat{y} - \bar{y}1_N)^T(\hat{y} - \bar{y}1_N) + e^Te + 2\hat{w}^T(X - \bar{X})^Te \\
&amp;=&amp; (\hat{y} - \bar{y}1_N)^T(\hat{y} - \bar{y}1_N) + e^Te \\
&amp;=&amp; \text{ESS} + \text{RSS}
\end{eqnarray}
\end{split}\]</div>
<p>위 식이 말하는 바는 다음과 같다.</p>
<blockquote>
<div><p>모형 예측치의 움직임의 크기(분산)은 종속변수의 움직임의 크기(분산)보다 클 수 없다.</p>
</div></blockquote>
<blockquote>
<div><p>모형의 성능이 좋을수록 모형 예측치의 움직임의 크기는 종속변수의 움직임의 크기와 비슷해진다.</p>
</div></blockquote>
<p>간단한 1차원 데이터와 모형을 사용하여 이 식이 성립하는지 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="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_regression</span>

<span class="n">X0</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">coef</span> <span class="o">=</span> <span class="n">make_regression</span><span class="p">(</span>
    <span class="n">n_samples</span><span class="o">=</span><span class="mi">100</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">noise</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">coef</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">dfX0</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">X0</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;X&quot;</span><span class="p">])</span>
<span class="n">dfX</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">dfX0</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;Y&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">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;Y ~ X&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>
</pre></div>
</div>
</div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">RegressionResult</span></code> 타입 객체는 다음과 같이 분산분석과 관련된 속성값을 가진다.</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;TSS = &quot;</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">uncentered_tss</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;ESS = &quot;</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">mse_model</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;RSS = &quot;</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">ssr</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;ESS + RSS = &quot;</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">mse_model</span> <span class="o">+</span> <span class="n">result</span><span class="o">.</span><span class="n">ssr</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R squared = &quot;</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">rsquared</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>TSS =  291345.7578983061
ESS =  188589.61349210917
RSS =  102754.33755137533
ESS + RSS =  291343.9510434845
R squared =  0.6473091780922586
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sns</span><span class="o">.</span><span class="n">distplot</span><span class="p">(</span><span class="n">y</span><span class="p">,</span>
             <span class="n">kde</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fit</span><span class="o">=</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">hist_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="s2">&quot;alpha&quot;</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">},</span> <span class="n">fit_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;r&quot;</span><span class="p">},</span>
             <span class="n">label</span><span class="o">=</span><span class="s2">&quot;TSS&quot;</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">result</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span>
             <span class="n">kde</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">hist_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;g&quot;</span><span class="p">,</span> <span class="s2">&quot;alpha&quot;</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">},</span> <span class="n">fit</span><span class="o">=</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">fit_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;g&quot;</span><span class="p">},</span>
             <span class="n">label</span><span class="o">=</span><span class="s2">&quot;ESS&quot;</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">result</span><span class="o">.</span><span class="n">resid</span><span class="p">,</span>
             <span class="n">kde</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">hist_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="s2">&quot;alpha&quot;</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">},</span> <span class="n">fit</span><span class="o">=</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">fit_kws</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;color&quot;</span><span class="p">:</span> <span class="s2">&quot;b&quot;</span><span class="p">},</span>
             <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RSS&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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<img alt="../_images/05.04 분산 분석과 모형 성능_12_0.png" src="../_images/05.04 분산 분석과 모형 성능_12_0.png" />
</div>
</div>
</div>
<div class="section" id="coefficient-of-determination">
<h2>결정계수(Coefficient of Determination)<a class="headerlink" href="#coefficient-of-determination" title="Permalink to this headline">¶</a></h2>
<p>위의 분산 관계식에서 모형의 성능을 나타내는 결정계수(Coefficient of Determination) <span class="math notranslate nohighlight">\(R^2\)</span>를 정의할 수 있다.</p>
<div class="math notranslate nohighlight">
\[R^2 \equiv 1 - \dfrac{\text{RSS}}{\text{TSS}}\ = \dfrac{\text{ESS}}{\text{TSS}}\ \]</div>
<p>분산 관계식과 모든 분산값이 0보다 크다는 점을 이용하면 <span class="math notranslate nohighlight">\(R^2\)</span>의 값은 다음과 같은 조건을 만족함을 알 수 있다.</p>
<div class="math notranslate nohighlight">
\[0 \leq R^2  \leq 1\]</div>
<p>여기에서 <span class="math notranslate nohighlight">\(R^2\)</span>가 0이라는 것은 오차의 분산 RSS가 최대이고 회귀분석 예측값의 분산 ESS가 0인 경우이므로 회귀분석 결과가 아무런 의미가 없다는 뜻이다.
반대로 <span class="math notranslate nohighlight">\(R^2\)</span>가 1이라는 것은 오차의 분산 RSS가 0이고 회귀분석 예측의 분산 ESS가 TSS와 같은 경우이므로 회귀분석 결과가 완벽하다는 뜻이다.
따라서 결정계수값은 회귀분석의 성능을 나타내는 수치라고 할 수 있다.</p>
</div>
<div class="section" id="id3">
<h2>분산 분석표<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
<p>분산 분석의 결과는 보통 다음과 같은 분산 분석표를 사용하여 표시한다. 아래의 표에서 <span class="math notranslate nohighlight">\(N\)</span>은 데이터의 갯수, <span class="math notranslate nohighlight">\(K\)</span>는 모수의 갯수를 뜻한다.</p>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p>source</p></th>
<th class="head"><p>degree of freedom</p></th>
<th class="head"><p>sum of square</p></th>
<th class="head"><p>mean square</p></th>
<th class="head"><p>F test-statstics</p></th>
<th class="head"><p>p-value</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Regression</p></td>
<td><p>$<span class="math notranslate nohighlight">\(K-1\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(\text{ESS}\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(s_{\hat{y}}^2 = \dfrac{\text{ESS}}{K-1}\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(F=\dfrac{s_{\hat{y}}^2}{s_e^2} \)</span>$</p></td>
<td><p>p-value</p></td>
</tr>
<tr class="row-odd"><td><p>Residual</p></td>
<td><p>$<span class="math notranslate nohighlight">\(N-K\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(\text{RSS}\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(s_e^2= \dfrac{\text{RSS}}{N-K}\)</span>$</p></td>
<td><p></p></td>
<td><p></p></td>
</tr>
<tr class="row-even"><td><p>Total</p></td>
<td><p>$<span class="math notranslate nohighlight">\(N-1\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(\text{TSS}\)</span>$</p></td>
<td><p>$<span class="math notranslate nohighlight">\(s_y^2= \dfrac{\text{TSS}}{N-1}\)</span>$</p></td>
<td><p></p></td>
<td><p></p></td>
</tr>
<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(R^2\)</span></p></td>
<td><p></p></td>
<td><p>$<span class="math notranslate nohighlight">\(\text{ESS} / \text{TSS}\)</span>$</p></td>
<td><p></p></td>
<td><p></p></td>
<td><p></p></td>
</tr>
</tbody>
</table>
<p>표 29.1 : 분산 분석표</p>
</div>
<div class="section" id="f">
<h2>회귀 분석 F-검정과 분산 분석의 관계<a class="headerlink" href="#f" title="Permalink to this headline">¶</a></h2>
<p>이러한 모양의 표를 사용하는 이유는 분산 분석의 결과를 이용하여 회귀 분석 F-검정에 필요한 검정통계량을 구할 수 있기 때문이다.</p>
<p>회귀 분석 F-검정의 원래 귀무 가설은 모든 계수 <span class="math notranslate nohighlight">\(w_i\)</span>가 <span class="math notranslate nohighlight">\(0\)</span> 이라는 것이지만 이 때는 모형이 아무런 의미가 없으므로 결정계수 값도 0이 된다</p>
<div class="math notranslate nohighlight">
\[ H_0: R^2 = 0 \]</div>
<p>이 때 <span class="math notranslate nohighlight">\(\hat{w}\)</span>값은 기대값이 0인 정규 분포에서 나온 표본이므로 예측값 <span class="math notranslate nohighlight">\(\hat{y} = \hat{w}^T x\)</span>는 정규 분포의 선형 조합이라서 마찬가지로 정규 분포를 따른다. 그리고 잔차(residual)는 오차(disturbance)의 선형 변환으로 정규 분포를 따르므로 ESS와 RSS의 비율은 F 분포를 따른다.</p>
<div class="math notranslate nohighlight">
\[ \dfrac{\text{ESS}}{K-1} \div  \dfrac{\text{RSS}}{N-K} \sim F(K-1, N-K) \]</div>
<p>따라서 이 값을 회귀 분석 F-검정의 검정통계량으로 사용할 수 있다.</p>
<p>statsmodels 에서는 다음과 같이 <code class="docutils literal notranslate"><span class="pre">anova_lm</span></code> 명령을 사용하여 분산 분석표를 출력할 수 있다. 다만 이 명령을 사용하기 위해서는 모형을 <code class="docutils literal notranslate"><span class="pre">from_formula</span></code> 메서드로 생성하여야 한다.</p>
<p><code class="docutils literal notranslate"><span class="pre">anova_lm</span></code> 명령으로 구한 F 검정통계량과 유의확률은 모형 <code class="docutils literal notranslate"><span class="pre">summary</span></code> 명령으로 구한 <code class="docutils literal notranslate"><span class="pre">F-statistic</span></code> 및 <code class="docutils literal notranslate"><span class="pre">Prob</span> <span class="pre">(F-statistic)</span></code>과 일치한다.</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">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">anova_lm</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
</pre></div>
</div>
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<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>df</th>
      <th>sum_sq</th>
      <th>mean_sq</th>
      <th>F</th>
      <th>PR(&gt;F)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>X</th>
      <td>1.0</td>
      <td>188589.613492</td>
      <td>188589.613492</td>
      <td>179.863766</td>
      <td>6.601482e-24</td>
    </tr>
    <tr>
      <th>Residual</th>
      <td>98.0</td>
      <td>102754.337551</td>
      <td>1048.513648</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
  </tbody>
</table>
</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="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>                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      Y   R-squared:                       0.647
Model:                            OLS   Adj. R-squared:                  0.644
Method:                 Least Squares   F-statistic:                     179.9
Date:                Sat, 03 Nov 2018   Prob (F-statistic):           6.60e-24
Time:                        10:03:56   Log-Likelihood:                -488.64
No. Observations:                 100   AIC:                             981.3
Df Residuals:                      98   BIC:                             986.5
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P&gt;|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -2.4425      3.244     -0.753      0.453      -8.880       3.995
X             43.0873      3.213     13.411      0.000      36.712      49.463
==============================================================================
Omnibus:                        3.523   Durbin-Watson:                   1.984
Prob(Omnibus):                  0.172   Jarque-Bera (JB):                2.059
Skew:                          -0.073   Prob(JB):                        0.357
Kurtosis:                       2.312   Cond. No.                         1.06
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id4">
<h2>결정 계수와 상관 계수<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
<p><span class="math notranslate nohighlight">\(y\)</span>와 <span class="math notranslate nohighlight">\(\hat{y}\)</span>의 샘플 상관계수 <span class="math notranslate nohighlight">\(r\)</span>의 제곱은 결정 계수 <span class="math notranslate nohighlight">\(R^2\)</span>와 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">sns</span><span class="o">.</span><span class="n">jointplot</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">fittedvalues</span><span class="p">,</span> <span class="n">y</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/05.04 분산 분석과 모형 성능_28_0.png" src="../_images/05.04 분산 분석과 모형 성능_28_0.png" />
</div>
</div>
</div>
<div class="section" id="id5">
<h2>상수항이 없는 모형의 경우<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h2>
<p>모형에서 상수항을 지정하지 않은 경우에는 결정계수의 정의에 사용되는 TSS의 정의가 다음과 같이 달라진다.</p>
<div class="math notranslate nohighlight">
\[\text{TSS} = \sum_i y_i^2 = y^Ty \]</div>
<p>즉, 실제 샘플평균과 상관없이 <span class="math notranslate nohighlight">\(\bar{y} = \)</span>이라는 가정하에 TSS를 계산한다. 이렇게 정의하지 않으면 TSS = RSS + ESS 관계식이 성립하지 않아서 결정계수의 값이 1보다 커지게 된다.</p>
<p>따라서 모형의 결정계수를 비교할 때 <strong>상수항이 없는 모형과 상수항이 있는 모형은 직접 비교하면 안된다.</strong></p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X0</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">coef</span> <span class="o">=</span> <span class="n">make_regression</span><span class="p">(</span>
    <span class="n">n_samples</span><span class="o">=</span><span class="mi">100</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">noise</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">coef</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</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">X0</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;X&quot;</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;Y&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">model2</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s2">&quot;Y ~ X + 0&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">result2</span> <span class="o">=</span> <span class="n">model2</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">result2</span><span class="o">.</span><span class="n">rsquared</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.18768724705943896
</pre></div>
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</div>
<div class="section" id="id6">
<h2>F 검정을 이용한 모형 비교<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
<p>F 검정을 이용하면 다음과 같이 포함관계(nesting)에 있는 두 모형의 성능을 비교할 수 있다.</p>
<ul class="simple">
<li><p>전체 모형(Full Model): $<span class="math notranslate nohighlight">\( y = w_0 + w_1 x_1 + w_2 x_2 + w_3 x_3 \)</span>$</p></li>
<li><p>축소 모형(Reduced Model): $<span class="math notranslate nohighlight">\( y = w_0 + w_1 x_1 \)</span>$</p></li>
</ul>
<p>다음과 같은 귀무 가설을 검정하는 것은 위의 두 모형이 실질적으로 같은 모형이라는 가설을 검장하는 것과 같다.</p>
<div class="math notranslate nohighlight">
\[ H_0: w_2 = w_3 = 0 \]</div>
<p>이 검정도 F 검정을 사용하여 할 수 있다. StatsModels에서는 <code class="docutils literal notranslate"><span class="pre">anova_lm</span></code> 명령에 두 모형의 result 객체를 인수로 넣어주면 이러한 검정을 할 수 있다. 인수를 넣어줄 때는 축소 모형(reduced model), 전체 모형(full model)의 순서로 넣어준다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_boston</span>

<span class="n">boston</span> <span class="o">=</span> <span class="n">load_boston</span><span class="p">()</span>
<span class="n">dfX0_boston</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">boston</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">boston</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">dfy_boston</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">boston</span><span class="o">.</span><span class="n">target</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;MEDV&quot;</span><span class="p">])</span>
<span class="n">dfX_boston</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">dfX0_boston</span><span class="p">)</span>
<span class="n">df_boston</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_boston</span><span class="p">,</span> <span class="n">dfy_boston</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_full</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;MEDV ~ CRIM + ZN + INDUS + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT + CHAS&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_boston</span><span class="p">)</span>
<span class="n">model_reduced</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;MEDV ~ CRIM + ZN + NOX + RM + DIS + RAD + TAX + PTRATIO + B + LSTAT + CHAS&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_boston</span><span class="p">)</span>

<span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">anova_lm</span><span class="p">(</span><span class="n">model_reduced</span><span class="o">.</span><span class="n">fit</span><span class="p">(),</span> <span class="n">model_full</span><span class="o">.</span><span class="n">fit</span><span class="p">())</span>
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      <th></th>
      <th>df_resid</th>
      <th>ssr</th>
      <th>df_diff</th>
      <th>ss_diff</th>
      <th>F</th>
      <th>Pr(&gt;F)</th>
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  <tbody>
    <tr>
      <th>0</th>
      <td>494.0</td>
      <td>11081.363952</td>
      <td>0.0</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
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      <th>1</th>
      <td>492.0</td>
      <td>11078.784578</td>
      <td>2.0</td>
      <td>2.579374</td>
      <td>0.057274</td>
      <td>0.944342</td>
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<div class="section" id="id7">
<h2>F 검정을 사용한 변수 중요도 비교<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h2>
<p>F 검정은 각 독립변수의 중요도를 비교하기 위해 사용할 수 있다.
방법은 전체 모형과 각 변수 하나만을 뺀 모형들의 성능을 비교하는 것이다. 이는 간접적으로 각 독립 변수의 영향력을 측정하는 것과 같다. 예를 들어 보스턴 집값 데이터에서 CRIM이란 변수를 뺀 모델과 전체 모델의 비교하는 검정을 하면 이 검정 결과는 CRIM변수의 중요도를 나타낸다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_full</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;MEDV ~ CRIM + ZN + INDUS + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT + CHAS&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_boston</span><span class="p">)</span>
<span class="n">model_reduced</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;MEDV ~ ZN + INDUS + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT + CHAS&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_boston</span><span class="p">)</span>

<span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">anova_lm</span><span class="p">(</span><span class="n">model_reduced</span><span class="o">.</span><span class="n">fit</span><span class="p">(),</span> <span class="n">model_full</span><span class="o">.</span><span class="n">fit</span><span class="p">())</span>
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      <th></th>
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      <th>df_diff</th>
      <th>ss_diff</th>
      <th>F</th>
      <th>Pr(&gt;F)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>493.0</td>
      <td>11322.004277</td>
      <td>0.0</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
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    <tr>
      <th>1</th>
      <td>492.0</td>
      <td>11078.784578</td>
      <td>1.0</td>
      <td>243.219699</td>
      <td>10.801193</td>
      <td>0.001087</td>
    </tr>
  </tbody>
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<p><code class="docutils literal notranslate"><span class="pre">anova_lm</span></code> 명령에서는 <code class="docutils literal notranslate"><span class="pre">typ</span></code> 인수를 <code class="docutils literal notranslate"><span class="pre">2</span></code>로 지정하면 하나 하나의 변수를 뺀 축소 모형에서의 F 검정값을 한꺼번에 계산할 수 있다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_boston</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span>
    <span class="s2">&quot;MEDV ~ CRIM + ZN + INDUS + NOX + RM + AGE + DIS + RAD + TAX + PTRATIO + B + LSTAT + CHAS&quot;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_boston</span><span class="p">)</span>
<span class="n">result_boston</span> <span class="o">=</span> <span class="n">model_boston</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="n">sm</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">anova_lm</span><span class="p">(</span><span class="n">result_boston</span><span class="p">,</span> <span class="n">typ</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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      <th></th>
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  </thead>
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    <tr>
      <th>CRIM</th>
      <td>243.219699</td>
      <td>1.0</td>
      <td>10.801193</td>
      <td>1.086810e-03</td>
    </tr>
    <tr>
      <th>ZN</th>
      <td>257.492979</td>
      <td>1.0</td>
      <td>11.435058</td>
      <td>7.781097e-04</td>
    </tr>
    <tr>
      <th>INDUS</th>
      <td>2.516668</td>
      <td>1.0</td>
      <td>0.111763</td>
      <td>7.382881e-01</td>
    </tr>
    <tr>
      <th>NOX</th>
      <td>487.155674</td>
      <td>1.0</td>
      <td>21.634196</td>
      <td>4.245644e-06</td>
    </tr>
    <tr>
      <th>RM</th>
      <td>1871.324082</td>
      <td>1.0</td>
      <td>83.104012</td>
      <td>1.979441e-18</td>
    </tr>
    <tr>
      <th>AGE</th>
      <td>0.061834</td>
      <td>1.0</td>
      <td>0.002746</td>
      <td>9.582293e-01</td>
    </tr>
    <tr>
      <th>DIS</th>
      <td>1232.412493</td>
      <td>1.0</td>
      <td>54.730457</td>
      <td>6.013491e-13</td>
    </tr>
    <tr>
      <th>RAD</th>
      <td>479.153926</td>
      <td>1.0</td>
      <td>21.278844</td>
      <td>5.070529e-06</td>
    </tr>
    <tr>
      <th>TAX</th>
      <td>242.257440</td>
      <td>1.0</td>
      <td>10.758460</td>
      <td>1.111637e-03</td>
    </tr>
    <tr>
      <th>PTRATIO</th>
      <td>1194.233533</td>
      <td>1.0</td>
      <td>53.034960</td>
      <td>1.308835e-12</td>
    </tr>
    <tr>
      <th>B</th>
      <td>270.634230</td>
      <td>1.0</td>
      <td>12.018651</td>
      <td>5.728592e-04</td>
    </tr>
    <tr>
      <th>LSTAT</th>
      <td>2410.838689</td>
      <td>1.0</td>
      <td>107.063426</td>
      <td>7.776912e-23</td>
    </tr>
    <tr>
      <th>CHAS</th>
      <td>218.970357</td>
      <td>1.0</td>
      <td>9.724299</td>
      <td>1.925030e-03</td>
    </tr>
    <tr>
      <th>Residual</th>
      <td>11078.784578</td>
      <td>492.0</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
  </tbody>
</table>
</div></div></div>
</div>
<p>이 값은 단일 계수 t 검정의 유의확률과 동일하다. 그 이유는 다음과 같은 t 분포와 F 분포의 동치 성질 때문이다.</p>
<div class="math notranslate nohighlight">
\[ t_n^2 = F_{(1, n)} \]</div>
</div>
<div class="section" id="id8">
<h2>조정 결정 계수<a class="headerlink" href="#id8" title="Permalink to this headline">¶</a></h2>
<p>선형 회귀 모형에서 독립 변수가 추가되면 결정 계수의 값은 항상 증가한다. 이는 다음과 같이 확인할 수 있다.</p>
<p>종속 변수 <span class="math notranslate nohighlight">\(y\)</span>를 회귀 분석하기 위한 기존의 독립 변수가 <span class="math notranslate nohighlight">\(X\)</span>이고 여기에 추가적인 독립 변수 <span class="math notranslate nohighlight">\(z\)</span>가 더해졌을 때, 다음과 같은 관계가 성립한다.</p>
<div class="math notranslate nohighlight">
\[
R^2_{Xz} = R^2_{X} + (1-R^2_{X})r^{\ast 2}_{yz}
\]</div>
<p>여기에서</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(R^2_{X}\)</span>: 기존의 독립 변수 <span class="math notranslate nohighlight">\(X\)</span>를 사용한 경우의 결정 계수</p></li>
<li><p><span class="math notranslate nohighlight">\(R^2_{Xz}\)</span>: 기존의 독립 변수 <span class="math notranslate nohighlight">\(X\)</span>와 추가적인 독립 변수 <span class="math notranslate nohighlight">\(z\)</span>를 모두 사용한 경우의 결정 계수</p></li>
<li><p><span class="math notranslate nohighlight">\(r^{\ast 2}_{yz}\)</span>: 추가적인 독립 변수 <span class="math notranslate nohighlight">\(z\)</span>와 종속 변수 <span class="math notranslate nohighlight">\(y\)</span>간의 상관 관계 계수</p></li>
</ul>
<p>이고 이 항목들은 모두 양수이므로</p>
<div class="math notranslate nohighlight">
\[
R^2_{Xz} \geq R^2_{X}
\]</div>
<p>이러한 독립 변수 추가 효과를 상쇄시키기 위한 다양한 기준들이 제시되었다. 그 중 하나가 다음과 같이 독립 변수의 갯수 <span class="math notranslate nohighlight">\(K\)</span>에 따라 결정 계수의 값을 조정하는 조정 결정 계수이다</p>
<div class="math notranslate nohighlight">
\[
R_{adj}^2 = 1 - \frac{n-1}{n-K}(1-R^2) = \dfrac{(n-1)R^2 +1-K}{n-K}
\]</div>
</div>
<div class="section" id="id9">
<h2>정보량 규준<a class="headerlink" href="#id9" title="Permalink to this headline">¶</a></h2>
<p>조정 결정 계수와 함께 많이 쓰이는 모형 비교 기준은 최대 우도에 독립 변수의 갯수에 대한 손실(penalty)분을 반영하는 방법이다.
이를 정보량 규준(information criterion)이라고 하며 손실 가중치의 계산 법에 따라 AIC (Akaike Information Criterion)와 BIC (Bayesian Information Criterion) 두 가지를 사용한다.</p>
<p>AIC는 모형과 데이터의 확률 분포 사이의 Kullback-Leibler 수준을 가장 크게하기 위한 시도에서 나왔다. BIC는 데이터가 exponential family라는 가정하에 주어진 데이터에서 모형의 likelihood를 측정하기 위한 값에서 유도되었다. 둘 다 값이 작을 수록 올바른 모형에 가깝다.</p>
<div class="math notranslate nohighlight">
\[
\text{AIC} = -2\log L + 2K
\]</div>
<div class="math notranslate nohighlight">
\[
\text{BIC} = -2\log L + K\log n
\]</div>
</div>
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