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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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     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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     3.4 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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     8.4 정규분포와 중심극한정리
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     8.5 스튜던트 t분포, 카이제곱분포, F분포
    </a>
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     8.6 다변수정규분포
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     8.7 베타분포, 감마분포, 디리클레분포
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   </li>
  </ul>
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   9장 추정과 검정
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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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   10장 엔트로피
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     10.1 엔트로피
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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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<p class="caption">
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  머신러닝 편
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   1.1 데이터 분석의 소개
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   2.1 데이터 전처리 기초
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   NLTK 자연어 처리 패키지
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   KoNLPy 한국어 처리 패키지
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   Scikit-Learn의 문서 전처리 기능
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   확률론적 언어 모형
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   이미지 처리 기초
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   이미지 필터링
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   이미지 컨투어
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   이미지 변환
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   이미지 특징 추출
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   사운드 프로세싱 기초
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 </li>
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  <a class="reference internal" href="../03%20machine%20learning/03.03.02%20%ED%91%B8%EB%A6%AC%EC%97%90%20%EB%B3%80%ED%99%98%EA%B3%BC%20%EC%8A%A4%ED%8E%99%ED%8A%B8%EB%9F%BC.html">
   푸리에 변환과 스펙트럼
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   지리 정보 데이터 처리
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   Geocoding
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 <li class="toctree-l1">
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   4.1 회귀분석 예제
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   4.2 선형회귀분석의 기초
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   4.3 스케일링
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   4.4 범주형 독립변수
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   4.5 부분회귀
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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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   5.4 분산 분석과 모형 성능
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   6.1 모형 진단과 수정
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   6.2 기저함수 모형과 과최적화
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   6.3 교차검증
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   6.4 다중공선성과 변수 선택
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   6.5 정규화 선형회귀
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   13.1 추천 시스템
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   5.1 분류용 예제 데이터
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   5.2 분류용 가상 데이터 생성
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   5.3 분류모형
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   5.4 분류 성능평가
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   6.1 로지스틱 회귀분석
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   7.1 선형판별분석법과 이차판별분석법
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   7.2 나이브베이즈 분류모형
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   감성 분석
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   8.1 의사결정나무
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   12.02 모형 결합
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   부스팅 방법
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   서포트 벡터 머신
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   커널 서포트 벡터 머신
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 <li class="toc-h2 nav-item toc-entry">
  <a class="reference internal nav-link" href="#id2">
   함수
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   <li class="toc-h3 nav-item toc-entry">
    <a class="reference internal nav-link" href="#id3">
     예제
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  <a class="reference internal nav-link" href="#id4">
   표를 사용한 함수 구현
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    <a class="reference internal nav-link" href="#id5">
     예제
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   변수
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   연속과 불연속
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   부호함수
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   단위계단함수
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   지시함수
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   역함수
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     예제
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   함수의 그래프
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     연습 문제 4.1.1
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   역함수의 그래프
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   데이터 분석에서 많이 사용되는 함수들
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   다항식함수
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   최대함수와 최소함수
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   지수함수
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   로지스틱함수
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   로그함수
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    <a class="reference internal nav-link" href="#id23">
     로그함수의 성질 1: 로그함수는 곱하기를 더하기로 변환한다.
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     연습 문제 4.1.2
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     연습 문제 4.1.3
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     로그함수의 성질 2: 어떤 함수에 로그를 적용해도 함수의 최고점, 최저점의 위치는 변하지 않는다.
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     로그함수의 성질 3: 로그함수는 0부터 1사이의 작은 값을 확대시켜 보여준다.
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   소프트플러스함수
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   다변수함수
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     예제
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   분리 가능 다변수함수
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   함수의 평행이동
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   함수의 스케일링
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  <div class="section" id="id1">
<h1>4.1 함수<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h1>
<p>이 절에서는 함수와 역함수의 개념, 그리고 파이썬에서 어떻게 함수를 구현하는지를 익히고 데이터 분석에서 자주 쓰이는 다양한 함수와 그 특성에 대해 공부한다.</p>
<div class="section" id="id2">
<h2>함수<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
<p>**함수(function)**는 입력 값을 출력 값으로 바꾸어 출력하는 관계(relationship)를 말한다. 맵(map), 매핑(mapping), 또는 딕셔너리(dictionary)이라는 용어를 사용하기도 한다. 예를 들어 <span class="math notranslate nohighlight">\(1, 2, 3, \ldots\)</span> 란 숫자를 입력받아서 두 배가 되는  <span class="math notranslate nohighlight">\(2, 4, 6, \ldots\)</span>을 출력하는 것은 함수라고 할 수 있다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
1, 2, 3, \ldots \;\; \rightarrow \;\; \text{function} \;\; \rightarrow 2, 4, 6, \ldots
\tag{4.1.1}
\end{align}
\]</div>
<p>어떤 입력 값과 출력 값이 함수 관계를 이루기 위해서는 같은 입력 값에 대해서 항상 같은 출력 값이 나와야 한다. 예를 들어 똑같이 3을 입력했는데 어떤 경우에는 4가 나오고 어떤 경우에는 5가 나온다면 함수라고 할 수 없다.</p>
<p>함수에서 입력변수가 가질 수 있는 값의 집합을 <strong>정의역(domain)</strong>, 출력변수가 가질 수 있는 값의 집합을 **공역(range)**이라고 한다. 정의역은 실수 전체인 경우가 많다. 하지만 함수에 따라서는 일부 숫자만 정의역으로 가질 수도 있다.</p>
<div class="section" id="id3">
<h3>예제<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h3>
<p>서로 관계를 가지며 맺어지는 모든 숫자의 쌍은 함수라고 생각할 수 있다. 예를 들어,</p>
<ul class="simple">
<li><p>앰프의 조절 나사(knob)를 돌린 각도 <span class="math notranslate nohighlight">\(x\)</span>와 소리의 크기(volume) <span class="math notranslate nohighlight">\(y\)</span></p></li>
<li><p>등산을 할 때 출발점에서 간 거리 <span class="math notranslate nohighlight">\(x\)</span>와 그 위치의 해발 고도 <span class="math notranslate nohighlight">\(y\)</span></p></li>
</ul>
<p>등의 관계를 정량화할 때 함수를 쓸 수 있다.</p>
</div>
</div>
<div class="section" id="id4">
<h2>표를 사용한 함수 구현<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
<p>만약 정의역이 유한 개의 원소로만 이루어져 있다면 함수는 일종의 **표(lookup table)**가 된다. 정의역이 유한개여서 표로 나타낼 수 있는 함수는 **파이썬의 딕셔너리(dictionary)**로 구현할 수 있다.</p>
<div class="section" id="id5">
<h3>예제<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h3>
<p>다음은 정의역이</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
\{ 1, 2, 3, 4, 5 \}
\tag{4.1.2}
\end{align}
\]</div>
<p>이고 입력값의 2배를 출력하는 함수를 표로 나타낸 것이다.</p>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p>입력</p></th>
<th class="head"><p>출력</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>1</p></td>
<td><p>2</p></td>
</tr>
<tr class="row-odd"><td><p>2</p></td>
<td><p>4</p></td>
</tr>
<tr class="row-even"><td><p>3</p></td>
<td><p>6</p></td>
</tr>
<tr class="row-odd"><td><p>4</p></td>
<td><p>8</p></td>
</tr>
<tr class="row-even"><td><p>5</p></td>
<td><p>10</p></td>
</tr>
</tbody>
</table>
<p>이 함수를 파이썬으로 구현하면 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="c1"># 함수 정의</span>
<span class="n">f</span> <span class="o">=</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="mi">2</span><span class="p">:</span> <span class="mi">4</span><span class="p">,</span>
    <span class="mi">3</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
    <span class="mi">4</span><span class="p">:</span> <span class="mi">8</span><span class="p">,</span>
    <span class="mi">5</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</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="c1"># 함수 사용</span>
<span class="n">f</span><span class="p">[</span><span class="mi">1</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>2
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="id6">
<h2>변수<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
<p>만약 정의역이 무한 개의 원소로 이루어졌다면 더이상 표나 딕셔너리를 사용해서 함수를 구현할 수 없다. 이 때는 변수와 수식을 사용하여 입력과 출력간의 관계를 정의해야한다.</p>
<p>**변수(variable)**란 어떤 숫자를 대표하는 기호이다. 함수에서 입력 값이나 출력 값을 기호 즉 변수로 나타내기도 한다. 입력 값을 대표하는 변수를 **입력변수(input variable)**라 하고 출력 값을 대표하는 변수를 **출력변수(output variable)**라고 한다.  변수는 보통 <span class="math notranslate nohighlight">\(x\)</span>, <span class="math notranslate nohighlight">\(y\)</span>, <span class="math notranslate nohighlight">\(z\)</span> 등의 알파벳 소문자로 표시한다. 함수에서 기호를 사용하는 이유는 입력 값과 출력 값의 관계를 수식으로 표현하기 위해서이다. 예를 들어 어떤 숫자를 입력으로 받아 그 값의 2배가 되는 값을 출력으로 한다면 다음처럼 표현할 수 있다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
x \rightarrow 2x
\tag{4.1.3}
\end{align}
\]</div>
<p>또는</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
y = 2x
\tag{4.1.4}
\end{align}
\]</div>
<p>함수를 여러개 사용할 때는 이름을 붙여 구별한다. 변수와 마찬가지로 함수도 <span class="math notranslate nohighlight">\(f\)</span>, <span class="math notranslate nohighlight">\(g\)</span>, <span class="math notranslate nohighlight">\(h\)</span>와 같이 알파벳 소문자로 표시한다. 함수를 표기할 때는 다음처럼 입력변수를 붙여서 <span class="math notranslate nohighlight">\(f(x)\)</span>라고 표기한다. 이 기호는 “f of x” 라고 읽는다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) = 2x
\tag{4.1.5}
\end{align}
\]</div>
<p>파이썬의 함수 기능은 이러한 수학 함수의 개념을 구현한 것으로 볼 수 있다.</p>
<div class="section" id="id7">
<h3>예제<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h3>
<p>다음 파이썬 함수 <code class="docutils literal notranslate"><span class="pre">f</span></code>는 모든 실수의 집합을 정의역으로 가지고 입력변수의 값을 2배하여 출력한다.</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="c1"># 함수 정의</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span>

<span class="c1"># 함수 사용</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>

<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;f(</span><span class="si">{}</span><span class="s2">) = </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</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>f(10) = 20
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="id8">
<h2>연속과 불연속<a class="headerlink" href="#id8" title="Permalink to this headline">¶</a></h2>
<p>함수의 값이 중간에 갑자기 변하는 것을 **불연속(discontinuous)**이라고 하고 그렇지 않으면 **연속(continuous)**이라고 한다. 다음에 설명할 부호함수, 단위계단함수, 지시함수는 데이터 분석에서 자주 등장하는 불연속함수다.</p>
</div>
<div class="section" id="id9">
<h2>부호함수<a class="headerlink" href="#id9" title="Permalink to this headline">¶</a></h2>
<p>부호함수(sign function)는 입력이 양수이면 1, 음수이면 -1, 0이면 0을 출력하는 <span class="math notranslate nohighlight">\(x=0\)</span>에서 불연속인 함수이다. 넘파이에서는 <code class="docutils literal notranslate"><span class="pre">sign()</span></code> 명령으로 구현되어 있다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
\text{sgn}(x) 
=
\begin{cases} 
1, &amp; x &gt; 0, \\ 
0, &amp; x = 0, \\ 
-1, &amp; x &lt; 0 
\end{cases} 
\tag{4.1.6}
\end{align}
\end{split}\]</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">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="o">-</span><span class="mf">0.0001</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="mf">0.0001</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>(-1.0, 0, 1.0)
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id10">
<h2>단위계단함수<a class="headerlink" href="#id10" title="Permalink to this headline">¶</a></h2>
<p>단위계단함수(Heaviside step function)도 <span class="math notranslate nohighlight">\(x=0\)</span>에서 불연속인 함수이다. 넘파이 구현이 없으므로 직접 구현해야 한다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
H(x) =
\begin{cases} 
1, &amp; x \ge 0, \\ 
0, &amp; x &lt; 0  
\end{cases}
\tag{4.1.7}
\end{align}
\end{split}\]</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="k">def</span> <span class="nf">heaviside_step</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">0</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="k">else</span><span class="p">:</span>
        <span class="k">return</span> <span class="mf">1.0</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="k">else</span> <span class="mf">0.0</span> 

<span class="n">heaviside_step</span><span class="p">(</span><span class="o">-</span><span class="mf">0.0001</span><span class="p">),</span> <span class="n">heaviside_step</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">heaviside_step</span><span class="p">(</span><span class="mf">0.0001</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>(0.0, 1.0, 1.0)
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id11">
<h2>지시함수<a class="headerlink" href="#id11" title="Permalink to this headline">¶</a></h2>
<p>다음 함수는 지시함수(indicator function)라는 함수로 함수 이름에 아래 첨자로 미리 지정된 값이 들어오면 출력이 1이 되고 아니면 출력이 0이 된다.</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
\mathbb{I}_i(x)
=
\delta_{ix} 
=
\begin{cases}
1 &amp; \text{ if } x = i \\ 
0 &amp; \text{ if } x \neq i \\ 
\end{cases}
\tag{4.1.8}
\end{align}
\end{split}\]</div>
<p>지시함수는 다음처럼 표기하기도 한다.</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
\mathbb{I}(x=i)
=
\begin{cases}
1 &amp; \text{ if } x = i \\ 
0 &amp; \text{ if } x \neq i \\ 
\end{cases}
\tag{4.1.9}
\end{align}
\end{split}\]</div>
<p>지시함수는 데이터 중에서 특정한 데이터만 선택하여 갯수를 세는데 사용된다. 예를 들어 데이터 <span class="math notranslate nohighlight">\(x_1, x_2, \ldots, x_N\)</span> 중에서 값이 0인 데이터의 갯수와 값이 1인 데이터의 갯수는 지시함수를 사용하여 다음처럼 표시한다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
N_0 = \sum_{i=1}^N \mathbb{I} (x_i=0)
\tag{4.1.10}
\end{align}
\]</div>
<div class="math notranslate nohighlight">
\[
\begin{align}
N_1 = \sum_{i=1}^N \mathbb{I} (x_i=1)
\tag{4.1.11}
\end{align}
\]</div>
</div>
<div class="section" id="id12">
<h2>역함수<a class="headerlink" href="#id12" title="Permalink to this headline">¶</a></h2>
<p>어떤 함수의 입력/출력 관계와 정반대의 입출력 관계를 갖는 함수를 **역함수(inverse function)**라고 한다.</p>
<p>예를 들어 어떤 함수 <span class="math notranslate nohighlight">\(f\)</span>가 <span class="math notranslate nohighlight">\(1, 2, 3, \ldots\)</span> 를 입력받아서  <span class="math notranslate nohighlight">\(1, 4, 9, \ldots\)</span> 을 출력하고
다른 함수 <span class="math notranslate nohighlight">\(g\)</span>는 반대로 <span class="math notranslate nohighlight">\(1, 4, 9, \ldots\)</span> 를 입력받아서  <span class="math notranslate nohighlight">\(1, 2, 3, \ldots\)</span> 을 출력하면 두 함수 <span class="math notranslate nohighlight">\(f\)</span>, <span class="math notranslate nohighlight">\(g\)</span>는 서로 역함수이다.</p>
<p>함수 <span class="math notranslate nohighlight">\(f\)</span>의 역함수는 <span class="math notranslate nohighlight">\(f^{-1}\)</span> 기호로 표시한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = f(x), \;\;\; \rightarrow \;\;\; x = f^{-1}(y) 
\tag{4.1.12}
\end{align}
\]</div>
<p>함수의 “역함수”와 함수의 “역수”는 기호와 의미가 모두 다르다는 점에 주의한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f^{-1}(x) \neq f(x)^{-1} = \dfrac{1}{f(x)} 
\tag{4.1.13}
\end{align}
\]</div>
<p>함수가 존재한다고 해서 항상 역함수가 존재할 수 있는 것은 아니다. 예를 들어 두 개의 서로 다른 <span class="math notranslate nohighlight">\(x_1\)</span>, <span class="math notranslate nohighlight">\(x_2\)</span>값에 대해 <span class="math notranslate nohighlight">\(y\)</span>의 값이 <span class="math notranslate nohighlight">\(y^{\ast}\)</span>로 같다고 하자.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x_1) = y^{\ast}
\tag{4.1.14}
\end{align}
\]</div>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x_2) = y^{\ast}
\tag{4.1.15}
\end{align}
\]</div>
<p>그러면 반대로 <span class="math notranslate nohighlight">\(y^{\ast}\)</span>에 대응하는 <span class="math notranslate nohighlight">\(x\)</span> 값을 선택할 수 없기 때문에 역함수가 존재하지 않는다.</p>
<div class="section" id="id13">
<h3>예제<a class="headerlink" href="#id13" title="Permalink to this headline">¶</a></h3>
<p>다음 함수는 역함수가 존재하지 않는다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) = x^2
\tag{4.1.16}
\end{align}
\]</div>
<p>이 함수의 역함수가 존재하려면 정의역을 양수로 제한해야 한다 (<span class="math notranslate nohighlight">\(x &gt; 0\)</span>) 이 때는 다음과 같이 역함수가 존재한다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f^{-1}(x) = \sqrt{x}
\tag{4.1.17}
\end{align}
\]</div>
</div>
</div>
<div class="section" id="id14">
<h2>함수의 그래프<a class="headerlink" href="#id14" title="Permalink to this headline">¶</a></h2>
<p>함수의 형상을 시각적으로 파악하기 위해 <strong>그래프(graph)</strong> 또는 **플롯(plot)**를 사용하기도 한다.</p>
<p>파이썬에서는 맷플롯립의 라인 플롯을 사용하여 그래프를 만들 수 있다. 다만 맷플롯립에서는 그래프를 작성할 영역을 작은 구간으로 나눈 독립변숫값 벡터 <span class="math notranslate nohighlight">\(x\)</span>를 생성하고 이 위치에 대한 함숫값 벡터 <span class="math notranslate nohighlight">\(y\)</span>를 하나하나 계산하여 이렇게 만들어진 점을 직선으로 연결하는 방식으로 플롯을 그린다. 따라서 구간 간격이 너무 크면 그래프가 부정확해지고, 구간 간격이 너무 작으면 계산 시간이 증가하며 메모리 등의 리소스가 낭비된다.</p>
<p>맷플롯립으로 다음과 같은 함수의 그래프를 그려보자.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f(x) = x^3 - 3x^2 + x 
\tag{4.1.18}
\end{align}
\]</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="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span>
</pre></div>
</div>
</div>
</div>
<p>우선 <span class="math notranslate nohighlight">\(x=-1\)</span> 부터 <span class="math notranslate nohighlight">\(x=3\)</span>까지 0.5 간격으로 9개의 점을 찍는다</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
<span class="n">x</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([-1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ,  2.5,  3. ])
</pre></div>
</div>
</div>
</div>
<p>그리고 이 위치에 대해 <span class="math notranslate nohighlight">\(y = f(x)\)</span>값을 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">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">y</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([-5.   , -1.375,  0.   , -0.125, -1.   , -1.875, -2.   , -0.625,
        3.   ])
</pre></div>
</div>
</div>
</div>
<p>이를 표로 나타내면 다음과 같다.</p>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(x\)</span></p></th>
<th class="head"><p>-1</p></th>
<th class="head"><p>-0.5</p></th>
<th class="head"><p>0</p></th>
<th class="head"><p>0.5</p></th>
<th class="head"><p>1</p></th>
<th class="head"><p>1.5</p></th>
<th class="head"><p>2</p></th>
<th class="head"><p>2.5</p></th>
<th class="head"><p>3</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(y\)</span></p></td>
<td><p>-5</p></td>
<td><p>-1.375</p></td>
<td><p>0</p></td>
<td><p>-0.125</p></td>
<td><p>-1</p></td>
<td><p>-1.875</p></td>
<td><p>-2</p></td>
<td><p>-0.625</p></td>
<td><p>3</p></td>
</tr>
</tbody>
</table>
<p>맷플롯립이 그리는 그래프는 이 표에 나타난 <span class="math notranslate nohighlight">\(x,y\)</span> 쌍을 직선으로 연결한 그림이다.</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">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">&#39;ro-&#39;</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">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;함수 $f(x) = x^3 - 3x^2 + x$의 그래프&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</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/04.01 함수_37_0.png" src="../_images/04.01 함수_37_0.png" />
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<p>이 <span class="math notranslate nohighlight">\(x\)</span>, <span class="math notranslate nohighlight">\(y\)</span> 벡터의 구간을 더 조밀하게 만들면 그래프는 곡선으로 보이게 된다. 다음 그래프는 곡선으로 보이지만 실제로는 약 400개의 직선이 연결된 것이다.</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">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">400</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</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">x</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">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;함수 $f(x) = x^3 - 3x^2 + x$의 그래프&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</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/04.01 함수_39_0.png" src="../_images/04.01 함수_39_0.png" />
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</div>
<div class="section" id="id15">
<h3>연습 문제 4.1.1<a class="headerlink" href="#id15" title="Permalink to this headline">¶</a></h3>
<p>맷플롯립으로 부호함수와 단위계단함수의 라인플롯을 그려라.</p>
</div>
</div>
<div class="section" id="id16">
<h2>역함수의 그래프<a class="headerlink" href="#id16" title="Permalink to this headline">¶</a></h2>
<p>역함수의 그래프는 원래의 함수에서 <span class="math notranslate nohighlight">\(x\)</span>축과 <span class="math notranslate nohighlight">\(y\)</span>축이 바뀐 것이므로 <span class="math notranslate nohighlight">\(y=x\)</span>가 나타내는 직선(원점을 통과하는 기울기 1인 직선)을 대칭축으로 대칭인 함수의 그래프와 같다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">x</span> <span class="o">**</span> <span class="mi">2</span>


<span class="k">def</span> <span class="nf">f1inv</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>


<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">300</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">x</span><span class="p">,</span> <span class="n">f1</span><span class="p">(</span><span class="n">x</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;함수 $f(x) = x^2$&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">x</span><span class="p">,</span> <span class="n">f1inv</span><span class="p">(</span><span class="n">x</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;역함수 $f^{-1}(x) = \sqrt</span><span class="si">{x}</span><span class="s2">$&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">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="s2">&quot;g--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</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="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">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">title</span><span class="p">(</span><span class="s2">&quot;역함수의 그래프&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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</div>
<div class="section" id="id17">
<h2>데이터 분석에서 많이 사용되는 함수들<a class="headerlink" href="#id17" title="Permalink to this headline">¶</a></h2>
<p>데이터 분석에는 다양한 함수가 사용된다. 여기에서는 몇 가지 간단한 예를 보인다.</p>
</div>
<div class="section" id="id18">
<h2>다항식함수<a class="headerlink" href="#id18" title="Permalink to this headline">¶</a></h2>
<p><strong>다항식(polynomial)함수</strong>는 상수항 <span class="math notranslate nohighlight">\(c_0\)</span>, 일차항 <span class="math notranslate nohighlight">\(c_1x\)</span>, 이차항 <span class="math notranslate nohighlight">\(c_2x^2\)</span>, <span class="math notranslate nohighlight">\(\cdots\)</span> 등의 거듭제곱 항의 선형조합으로 이루어진 함수다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f(x) = c_0 + c_1 x + c_2 x^2 + \cdots + c_n x^n 
\tag{4.1.19}
\end{align}
\]</div>
</div>
<div class="section" id="id19">
<h2>최대함수와 최소함수<a class="headerlink" href="#id19" title="Permalink to this headline">¶</a></h2>
<p>최대함수는 두 개의 인수 중에서 큰 값을 출력하는 함수이다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
\max(x, y) 
=
\begin{cases}
x &amp; \text{ if } x \geq y \\
y &amp; \text{ if } x &lt; y 
\end{cases}
\tag{4.1.20}
\end{align}
\end{split}\]</div>
<p>최소함수는 최대함수와 반대로 두 개의 인수 중 작은 값을 출력하는 함수이다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
\min(x, y) 
=
\begin{cases}
x &amp; \text{ if } x \leq y \\
y &amp; \text{ if } x &gt; y 
\end{cases}
\tag{4.1.21}
\end{align}
\end{split}\]</div>
<p>최대함수는 원래 두 개의 입력을 갖는 함수이지만 보통 <span class="math notranslate nohighlight">\(y=0\)</span>으로 고정해서 입력값 <span class="math notranslate nohighlight">\(x\)</span>가 양수이면 그대로 출력하고 음수일 때는 0으로 만들 때 주로 사용한다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
\max(x, 0) 
=
\begin{cases}
x &amp; \text{ if } x \geq 0 \\
0 &amp; \text{ if } x &lt; 0 
\end{cases}
\tag{4.1.22}
\end{align}
\end{split}\]</div>
<p>인공신경망에서는 이 함수를 **ReLU(Rectified Linear Unit)**라고 부른다.</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">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</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">maximum</span><span class="p">(</span><span class="n">xx</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">title</span><span class="p">(</span><span class="s2">&quot;max(x,0) 또는 ReLU&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$ReLU(x)$&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>
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<div class="cell_output docutils container">
<img alt="../_images/04.01 함수_50_0.png" src="../_images/04.01 함수_50_0.png" />
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</div>
</div>
<div class="section" id="id20">
<h2>지수함수<a class="headerlink" href="#id20" title="Permalink to this headline">¶</a></h2>
<p>정해진 어떤 값을 입력 값으로 거듭제곱하여 출력을 구하는 함수를 생각해보자.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = a^x 
\tag{4.1.23}
\end{align}
\]</div>
<p>거듭제곱을 할 때 아래에 있는 수를 밑(base)라고 한다. 밑을 오일러 수 <span class="math notranslate nohighlight">\(e\)</span>(약 2.718)로 하여 거듭제곱을 하는 함수를 **지수함수(exponential function)**라고 한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = e^x 
\tag{4.1.24}
\end{align}
\]</div>
<p>지수함수는 다음처럼 표시할 수도 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = \exp (x) =\exp x 
\tag{4.1.25}
\end{align}
\]</div>
<p>넘파이에서는 <code class="docutils literal notranslate"><span class="pre">e</span></code> 상수로 오일러 수를, <code class="docutils literal notranslate"><span class="pre">exp()</span></code> 명령으로 지수함수를 계산한다. 여러가지 값을 대입하여 지수함수의 값을 구해 위 성질이 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="n">np</span><span class="o">.</span><span class="n">e</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>2.718281828459045
</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">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">),</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="mi">1</span><span class="p">),</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="mf">0.1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mf">0.1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">10</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>(4.5399929762484854e-05,
 0.36787944117144233,
 0.9048374180359595,
 1.0,
 1.1051709180756477,
 2.718281828459045,
 22026.465794806718)
</pre></div>
</div>
</div>
</div>
<p>지수함수는 다음과 같은 특성을 갖는다.</p>
<ul class="simple">
<li><p>양수(<span class="math notranslate nohighlight">\(e\)</span>)를 거듭제곱한 값이므로 항상 양수다.</p></li>
<li><p><span class="math notranslate nohighlight">\(x=0\)</span>일 때 1이 된다.</p></li>
<li><p><span class="math notranslate nohighlight">\(x\)</span>가 양의 무한대로 가면(<span class="math notranslate nohighlight">\(x \rightarrow \infty\)</span>), 양의 무한대로 다가간다.</p></li>
<li><p><span class="math notranslate nohighlight">\(x\)</span>가 음의 무한대로 가면(<span class="math notranslate nohighlight">\(x \rightarrow -\infty\)</span>), 0으로 다가간다.</p></li>
<li><p><span class="math notranslate nohighlight">\(x_1 &gt; x_2\)</span>이면 <span class="math notranslate nohighlight">\(\exp{x_1} &gt; \exp{x_2}\)</span>이다.</p></li>
</ul>
<p>위 특성에 의해 지수함수의 그래프는 다음처럼 그려진다.</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">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</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">title</span><span class="p">(</span><span class="s2">&quot;지수함수&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">yy</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$\exp(x)$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>두 지수함수의 곱의 값은 입력값 합의 지수함수 값과 같다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f(x_1) \cdot f(x_2) = e^{x_1} \cdot e^{x_2} = e^{x_1 + x_2} = f(x_1 + x_2)
\tag{4.1.26}
\end{align}
\]</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="p">)</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="mi">3</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(148.4131591025766, 148.4131591025766)
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</div>
<div class="section" id="id21">
<h2>로지스틱함수<a class="headerlink" href="#id21" title="Permalink to this headline">¶</a></h2>
<p>로지스틱(logistic) 함수는 지수함수를 변형한 함수로 회귀 분석이나 인공신경망에서 자주 사용된다. 로지스틱함수는 원래 시그모이드(sigmoid) 함수라는 종류의 하나다. 하지만 시그모이드 함수 중에서는 로지스틱함수가 가장 널리 쓰이기 때문에 보통 시그모이드 함수라고 하면 이 로지스틱함수를 뜻한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\sigma(x) = \dfrac{1}{1 + \exp(-x)} 
\tag{4.1.27}
\end{align}
\]</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">logistic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</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">x</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">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</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">logistic</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">title</span><span class="p">(</span><span class="s2">&quot;로지스틱함수&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$\sigma(x)$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="id22">
<h2>로그함수<a class="headerlink" href="#id22" title="Permalink to this headline">¶</a></h2>
<p><span class="math notranslate nohighlight">\(e\)</span>를 거듭제곱하여 특정한 값이 되는 수를 찾아야 할 때가 있다. 예를 들어 <span class="math notranslate nohighlight">\(e\)</span>를 제곱하면 약 7.4가 되고 세제곱하면 약 20이 된다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(7.38905609893065, 20.085536923187668)
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<p>그렇다면 몇 제곱을 해야 <span class="math notranslate nohighlight">\(10\)</span>이라는 값이 될까?  <span class="math notranslate nohighlight">\(e\)</span>를 약 2.3025851 거듭제곱을 하면 10이 되는 것을 확인할 수 있다. 이렇게 <span class="math notranslate nohighlight">\(e\)</span>를 거듭제곱하여 특정한 수 <span class="math notranslate nohighlight">\(a\)</span>가 되도록 하는 수를 <span class="math notranslate nohighlight">\(\log a\)</span>라 표기하고 로그(log)라고 읽는다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mf">2.3025851</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>10.000000070059542
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<p>넘파이에서 로그함수 명령은 <code class="docutils literal notranslate"><span class="pre">log</span></code>이다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.302585092994046
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<p>지수함수에서 밑이 <span class="math notranslate nohighlight">\(e\)</span>가 아닌 경우에는 다음처럼 로그를 이용하여 계산할 수 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = a^x = (e^{\log a})^x = e^{x \log a} 
\tag{4.1.28}
\end{align}
\]</div>
<p>로그는 지수함수의 출력이 특정한 값이 되게 하는 입력을 찾는 것이므로 지수함수의 역함수이다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
y = \log x 
\tag{4.1.29}
\end{align}
\]</div>
<p>로그함수는 다음과 같은 특징을 가진다.</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(x\)</span>값, 즉 입력변수값이 양수이어야 한다. 0이거나 음수이면 정의되지 않는다.</p></li>
<li><p><span class="math notranslate nohighlight">\(x &gt; 1\)</span>면 <span class="math notranslate nohighlight">\(y &gt; 0\)</span> (양수)</p></li>
<li><p><span class="math notranslate nohighlight">\(x = 1\)</span>이면 <span class="math notranslate nohighlight">\(y = 0\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(0 &lt; x &lt; 1\)</span>면 <span class="math notranslate nohighlight">\(y &lt; 0\)</span> (음수)</p></li>
<li><p><span class="math notranslate nohighlight">\(x_1 &gt; x_2\)</span>면 <span class="math notranslate nohighlight">\(\log{x_1} &gt; \log{x_2}\)</span>이다.</p></li>
</ul>
<p>로그함수의 그래프는 다음과 같은 형태를 띈다.</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="mf">0.01</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</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">title</span><span class="p">(</span><span class="s2">&quot;로그함수&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">yy</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$\log(x)$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>로그함수는 확률론에서 가장 많이 사용되는 함수다. 그 이유는 로그함수가 갖는 다음과 같은 특이한 성질 때문이다.</p>
<div class="section" id="id23">
<h3>로그함수의 성질 1: 로그함수는 곱하기를 더하기로 변환한다.<a class="headerlink" href="#id23" title="Permalink to this headline">¶</a></h3>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\log{(x_1 \cdot x_2)} = \log{x_1} + \log{x_2} 
\tag{4.1.30}
\end{align}
\]</div>
<p>이 성질을 확장하면 다음과 같이 표기한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\log{\left(\prod_i x_i\right)} = \sum_i \left(\log{x_i}\right) 
\tag{4.1.31}
\end{align}
\]</div>
<p>위 식에서 <span class="math notranslate nohighlight">\(x_1 = x_2 = \cdots = x\)</span> 라면</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\log x^n = n \log x 
\tag{4.1.32}
\end{align}
\]</div>
</div>
<div class="section" id="id24">
<h3>연습 문제 4.1.2<a class="headerlink" href="#id24" title="Permalink to this headline">¶</a></h3>
<p><span class="math notranslate nohighlight">\(\log 2\)</span>의 값은 약 0.69, <span class="math notranslate nohighlight">\(\log 3\)</span>의 값은 약 1.10 이다. 이 때 <span class="math notranslate nohighlight">\(\log 12\)</span>의 값을 구하라.</p>
</div>
<div class="section" id="id25">
<h3>연습 문제 4.1.3<a class="headerlink" href="#id25" title="Permalink to this headline">¶</a></h3>
<p>로지스틱함수의 역함수를 구하라.</p>
</div>
<div class="section" id="id26">
<h3>로그함수의 성질 2: 어떤 함수에 로그를 적용해도 함수의 최고점, 최저점의 위치는 변하지 않는다.<a class="headerlink" href="#id26" title="Permalink to this headline">¶</a></h3>
<p>만약 양수값만을 갖는 함수가 있을 때 이 함수에 다시 로그함수를 적용하면 높낮이는 바뀌지만 최고점이나 최저점의 위치는 바뀌지 않는다. 따라서 최적화할 때 원래의 함수가 아니라 로그를 취한 함수에 대해 최적화를 하는 경우가 많다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\arg\max_x f(x) = \arg\max_x \log f(x)
\tag{4.1.33}
\end{align}
\]</div>
<p>위 식에서 <span class="math notranslate nohighlight">\(\arg\max_x f(x)\)</span>는 <span class="math notranslate nohighlight">\(f\)</span>값을 최대로 만들어주는 <span class="math notranslate nohighlight">\(x\)</span>의 값을 뜻한다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">ff</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">20</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">7</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">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">ff</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">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$f(x)$&quot;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">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">log</span><span class="p">(</span><span class="n">yy</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$log f(x)$&quot;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class="section" id="id27">
<h3>로그함수의 성질 3: 로그함수는 0부터 1사이의 작은 값을 확대시켜 보여준다.<a class="headerlink" href="#id27" title="Permalink to this headline">¶</a></h3>
<p>로그함수는 0부터 1사이의 구간을 음의 무한대부터 0까지로 확장시켜주기 때문에 확률값처럼 0과 1사이에 있는 작은 값을 더 확실하게 비교할 수 있도록 한다.</p>
<img src="https://datascienceschool.net/upfiles/66ee613cb3c14fc4b3090e5f7e1e1490.png" style="width:100%;"><p>그림 4.1.1 : 로그함수의 정의역과 치역의 비교</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">/</span> <span class="n">x</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;0, 1 사이 숫자들의 $\log$ 변환&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">x</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</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">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;log 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>
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<div class="section" id="id28">
<h2>소프트플러스함수<a class="headerlink" href="#id28" title="Permalink to this headline">¶</a></h2>
<p>**소프트플러스(softplus)**함수는 지수함수와 로그함수를 결합하여 만든 함수로 0을 인수로 갖는 최대함수와 비슷하지만 <span class="math notranslate nohighlight">\(x=0\)</span> 근처에서 값이 부드럽게 변한다는 장점이 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
\zeta(x) = \log( 1 + \exp(x))
\tag{4.1.34}
\end{align}
\]</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="k">def</span> <span class="nf">softplus</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">log</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="n">x</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">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</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">softplus</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">title</span><span class="p">(</span><span class="s2">&quot;소프트플러스함수&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Softplus($x$)&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>
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<div class="section" id="id29">
<h2>다변수함수<a class="headerlink" href="#id29" title="Permalink to this headline">¶</a></h2>
<p>함수는 복수의 입력변수를 가질 수도 있다. 이러한 함수를 **다변수함수(multivariate function)**라고 한다. 예를 들어 다음 수식은 함수 <span class="math notranslate nohighlight">\(f\)</span>가 2개의 입력변수 <span class="math notranslate nohighlight">\(x\)</span>, <span class="math notranslate nohighlight">\(y\)</span>를 받아서 변수 <span class="math notranslate nohighlight">\(z\)</span>를 출력하는 다변수함수라는 뜻이다. 2차원 (다변수)함수라고도 한다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
z = f(x, y) 
\tag{4.1.35}
\end{align}
\]</div>
<p>두 개의 독립변수를 갖는 함수 <span class="math notranslate nohighlight">\(f(x, y)\)</span>의 예로는 위도 <span class="math notranslate nohighlight">\(x\)</span>와 경도 <span class="math notranslate nohighlight">\(y\)</span>를 입력받아 해발 고도 <span class="math notranslate nohighlight">\(z\)</span>를 출력하는 함수이다.</p>
<p>따라서 2차원 함수는 평면상의 지형과 같기 때문에 3차원 <strong>서피스 플롯(surface plot)</strong>, 또는 **컨투어 플롯(contour plot)**으로 나타낼 수 있다.</p>
<div class="section" id="id30">
<h3>예제<a class="headerlink" href="#id30" title="Permalink to this headline">¶</a></h3>
<p>다음은 2차원 다변수함수의 예다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f(x, y) = 2x^2 + 6xy + 7y^2 - 26x - 54y + 107
\tag{4.1.36}
\end{align}
\]</div>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span> <span class="o">*</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">7</span> <span class="o">*</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">26</span> <span class="o">*</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">54</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">107</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">7</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">)</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>

<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">view_init</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="o">-</span><span class="mi">110</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="s1">&#39;x&#39;</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="s1">&#39;y&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s1">&#39;z&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;서피스 플롯(Surface Plot)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">CS</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">clabel</span><span class="p">(</span><span class="n">CS</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s2">&quot;</span><span class="si">%d</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;컨투어 플롯&#39;</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="id31">
<h2>분리 가능 다변수함수<a class="headerlink" href="#id31" title="Permalink to this headline">¶</a></h2>
<p>다변수함수 중에는 단변수함수의 곱으로 표현 가능한 다변수함수도 있다.</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
f(x, y) = f_1(x) f_2(y)
\tag{4.1.37}
\end{align}
\]</div>
<p>이러한 다변수함수를 **분리 가능 다변수함수(separable multivariate function)**이라고 하며 추후 확률변수에서 중요한 역할을 한다.</p>
<p>2차원 함수는 3차원 공간상의 지형에 비유할 수 있다. 2차원 함수에서 <span class="math notranslate nohighlight">\(x\)</span> 값을 어떤 상수 <span class="math notranslate nohighlight">\(x_0\)</span>로 고정하거나</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
x=x_0
\tag{4.1.38}
\end{align}
\]</div>
<p>또는 <span class="math notranslate nohighlight">\(y\)</span> 값을 어떤 상수 <span class="math notranslate nohighlight">\(y_0\)</span>로 고정하자.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
y=y_0
\tag{4.1.39}
\end{align}
\]</div>
<p>그러면 움직일 수 있는 변수가 이제 하나이므로 이 함수는 1차원 단변수함수가 된다. 이 단변수함수의 모양은 3차원 공간상의 지형을 케이크 자르듯이 나눌 때 생기는 단면의 모양과 같다.</p>
<p>그런데 분리 가능 다변수함수는 단면의 모양이 모두 같다.  예를 들어 <span class="math notranslate nohighlight">\(x=x_0\)</span>로 고정하면 단면 함수는</p>
<div class="math notranslate nohighlight">
\[ 
\begin{align}
g(x_0,y) = g(x_0)g(y) = k_0 g(y)
\tag{4.1.40}
\end{align}
\]</div>
<p>이므로 <span class="math notranslate nohighlight">\(x_0\)</span>의 값과 상관없이 <span class="math notranslate nohighlight">\(g(y)\)</span>의 높이만 조절한 모양이 된다.</p>
<div class="section" id="id32">
<h3>예제<a class="headerlink" href="#id32" title="Permalink to this headline">¶</a></h3>
<p>다음 함수는 분리 가능 다변수함수의 한 예이다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
g(x, y) = \exp(-x^2 - 16y^2) = \exp(-x^2)\exp(-16y^2) = g_1(x)g_2(y)
\tag{4.1.41}
\end{align}
\]</div>
<p>아래의 그림은 <span class="math notranslate nohighlight">\(y=0, 0.1, 0.2, 0.3\)</span>로 고정했을 때 생기는 단면의 모양을 나타낸 것이다.</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">matplotlib</span> <span class="kn">import</span> <span class="n">transforms</span>
<span class="kn">from</span> <span class="nn">matplotlib.ticker</span> <span class="kn">import</span> <span class="n">NullFormatter</span>

<span class="k">def</span> <span class="nf">g1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</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">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>


<span class="k">def</span> <span class="nf">g2</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
    <span class="k">return</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="mi">16</span> <span class="o">*</span> <span class="n">y</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>


<span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="k">return</span> <span class="n">g1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">g2</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">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">)</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>

<span class="n">left</span><span class="p">,</span> <span class="n">width</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.65</span>
<span class="n">bottom</span><span class="p">,</span> <span class="n">height</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.65</span>
<span class="n">bottom_h</span> <span class="o">=</span> <span class="n">left_h</span> <span class="o">=</span> <span class="n">left</span> <span class="o">+</span> <span class="n">width</span> <span class="o">+</span> <span class="mf">0.02</span>
<span class="n">rect</span> <span class="o">=</span> <span class="p">[</span><span class="n">left</span><span class="p">,</span> <span class="n">bottom</span><span class="p">,</span> <span class="n">width</span><span class="p">,</span> <span class="n">height</span><span class="p">]</span>
<span class="n">rectx</span> <span class="o">=</span> <span class="p">[</span><span class="n">left</span><span class="p">,</span> <span class="n">bottom_h</span><span class="p">,</span> <span class="n">width</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">]</span>
<span class="n">recty</span> <span class="o">=</span> <span class="p">[</span><span class="n">left_h</span><span class="p">,</span> <span class="n">bottom</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="n">height</span><span class="p">]</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">rect</span><span class="p">)</span>
<span class="n">axx</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">rectx</span><span class="p">)</span>
<span class="n">axy</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">recty</span><span class="p">)</span>

<span class="n">nullfmt</span> <span class="o">=</span> <span class="n">NullFormatter</span><span class="p">()</span>
<span class="n">axx</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">nullfmt</span><span class="p">)</span>
<span class="n">axy</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">nullfmt</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-.&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;:&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>

<span class="n">axx</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">g1</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="n">axx</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">g2</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span> <span class="o">*</span> <span class="n">g1</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">axx</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">g2</span><span class="p">(</span><span class="mf">0.2</span><span class="p">)</span> <span class="o">*</span> <span class="n">g1</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-.&quot;</span><span class="p">)</span>
<span class="n">axx</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">g2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span> <span class="o">*</span> <span class="n">g1</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;:&quot;</span><span class="p">)</span>
<span class="n">axx</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;g1(x)&quot;</span><span class="p">)</span>
<span class="n">axx</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="s2">&quot;y=0.3로 자른 단면&quot;</span><span class="p">)</span>
<span class="n">axx</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="s2">&quot;y=0.2으로 자른 단면&quot;</span><span class="p">)</span>
<span class="n">axx</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="s2">&quot;y=0.1로 자른 단면&quot;</span><span class="p">)</span>

<span class="n">base</span> <span class="o">=</span> <span class="n">axy</span><span class="o">.</span><span class="n">transData</span>
<span class="n">rot</span> <span class="o">=</span> <span class="n">transforms</span><span class="o">.</span><span class="n">Affine2D</span><span class="p">()</span><span class="o">.</span><span class="n">rotate_deg</span><span class="p">(</span><span class="o">-</span><span class="mi">90</span><span class="p">)</span>
<span class="n">axy</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">yy</span><span class="p">,</span> <span class="n">g2</span><span class="p">(</span><span class="n">yy</span><span class="p">),</span> <span class="n">transform</span><span class="o">=</span><span class="n">rot</span> <span class="o">+</span> <span class="n">base</span><span class="p">)</span>
<span class="n">axy</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;g2(y)&quot;</span><span class="p">)</span>
<span class="n">axy</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="n">g2</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.02</span><span class="p">,</span> <span class="s2">&quot;$g_2$(0)=</span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">g2</span><span class="p">(</span><span class="mi">0</span><span class="p">)))</span>
<span class="n">axy</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.1</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.12</span><span class="p">,</span> <span class="s2">&quot;$g_2$(0.1)=</span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)))</span>
<span class="n">axy</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-.&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.22</span><span class="p">,</span> <span class="s2">&quot;$g_2$(0.2)=</span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.2</span><span class="p">)))</span>
<span class="n">axy</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">xmax</span><span class="o">=</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;:&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.32</span><span class="p">,</span> <span class="s2">&quot;$g_2$(0.3)=</span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">g2</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)))</span>

<span class="n">axx</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">ax</span><span class="o">.</span><span class="n">get_xlim</span><span class="p">())</span>
<span class="n">axx</span><span class="o">.</span><span class="n">set_ylim</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">axy</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">ax</span><span class="o">.</span><span class="n">get_ylim</span><span class="p">())</span>
<span class="n">axy</span><span class="o">.</span><span class="n">set_xlim</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">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;$g(x, y) = \exp(-x^2)\exp(-16y^2) = g_1(x)g_2(y)$의 그래프&quot;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">1.04</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/04.01 함수_100_0.png" src="../_images/04.01 함수_100_0.png" />
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<div class="section" id="id33">
<h2>다변수 다출력 함수<a class="headerlink" href="#id33" title="Permalink to this headline">¶</a></h2>
<p>어떤 함수는 입력변수 뿐 아니라 출력변수도 여러개일 수 있다. 이 경우에는 출력을 벡터나 행렬로 나타낼 수 있다.</p>
<p>예를 들어  **소프트맥스(softmax)**함수는 다차원 벡터를 입력받아 다차원 벡터를 출력한다. 다음은 3차원 소프트맥스 함수이다.</p>
<div class="math notranslate nohighlight">
\[\begin{split} 
\begin{align}
y =
\begin{bmatrix}
y_1 \\ y_2 \\ y_3
\end{bmatrix}
=S(x_1, x_2, x_3) = 
\begin{bmatrix}
\dfrac{\exp(w_1x_1)}{\exp(w_1x_1) + \exp(w_2x_2) + \exp(w_3x_3)} \\
\dfrac{\exp(w_2x_2)}{\exp(w_1x_1) + \exp(w_2x_2) + \exp(w_3x_3)} \\
\dfrac{\exp(w_3x_3)}{\exp(w_1x_1) + \exp(w_2x_2) + \exp(w_3x_3)} \\
\end{bmatrix}
\tag{4.1.42}
\end{align}
\end{split}\]</div>
<p>출력 벡터는 다음과 같은 특성을 갖는다.</p>
<ul class="simple">
<li><p>모든 출력 원소는 0와 1 사잇값을 갖는다.</p></li>
<li><p>모든 출력 원소의 합은 1이다.</p></li>
<li><p>입력 원소의 크기 순서와 출력 원소의 크기 순서가 같다.</p></li>
</ul>
<p>소프트맥스함수는 <strong>다변수 입력을 확률(probability)처럼 보이게 출력한다</strong>. 이러한 특성 때문에 인공신경망의 마지막단에서 출력을 조건부확률로 변형하는 데 사용된다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">):</span>  <span class="c1"># x는 1차원 배열, w는 가중치 벡터</span>
    <span class="n">e</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="n">w</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">w</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">e</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>


<span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="n">y</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([0.62853172, 0.2312239 , 0.14024438])
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></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>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.0
</pre></div>
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<p>가중치가 커지면 최대값과 최소값의 차이가 더 벌어진다.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
</pre></div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([0.97962921, 0.01794253, 0.00242826])
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<div class="section" id="id34">
<h2>함수의 평행이동<a class="headerlink" href="#id34" title="Permalink to this headline">¶</a></h2>
<p>단변수함수를 그래프 상에서 오른쪽으로 <span class="math notranslate nohighlight">\(a\)</span>만큼 평행이동하려면 다음과 같이 함수식을 바꾸면 된다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) \;\; \rightarrow \;\; f(x-a)
\tag{4.1.43}
\end{align}
\]</div>
<p>즉 함수식에서 <span class="math notranslate nohighlight">\(x\)</span>로 표기된 부분을 모두 <span class="math notranslate nohighlight">\(x-a\)</span>라는 글자로 바꾼다.</p>
<p>예를 들어 함수</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) = x^2 + 2x
\tag{4.1.44}
\end{align}
\]</div>
<p>를 오른쪽으로 2만큼 이동하려면</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x-2) = (x-2)^2 + 2(x-2)
\tag{4.1.45}
\end{align}
\]</div>
<p>로 바꾼다.</p>
<p>단변수함수를 그래프 상에서 위로 <span class="math notranslate nohighlight">\(b\)</span>만큼 평행이동하려면 다음과 같이 함수식을 바꾼다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) \;\; \rightarrow \;\; f(x) + b
\tag{4.1.46}
\end{align}
\]</div>
<p>다음 그래프는 로지스틱함수를 오른쪽으로 5, 아래로 1만큼 이동한 것이다.</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">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</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">logistic</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$\sigma(x)$&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-&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">logistic</span><span class="p">(</span><span class="n">xx</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$\sigma(x-5)-1$&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&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">title</span><span class="p">(</span><span class="s2">&quot;로지스틱함수를 오른쪽으로 5, 아래로 1만큼 이동&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<p>변수가 두 개인 다변수함수를 오른쪽으로 a만큼, 위로 b만큼 평행이동할 때는 함수를 다음처럼 변형한다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x, y) \;\; \rightarrow \;\; f(x-a, y-b)
\tag{4.1.47}
\end{align}
\]</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
    <span class="k">return</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">x</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span><span class="mi">16</span> <span class="o">*</span> <span class="n">y</span> <span class="o">**</span> <span class="mi">2</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">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">)</span>
<span class="n">Z1</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="n">Z2</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">X</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">Y</span><span class="o">-</span><span class="mf">0.75</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z2</span><span class="p">,</span> <span class="n">linestyles</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.02</span><span class="p">,</span> <span class="s2">&quot;f(x,y)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.35</span><span class="p">,</span> <span class="mf">0.73</span><span class="p">,</span> <span class="s2">&quot;f(x-0.5,y-0.75)&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">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;다변수함수의 평행이동&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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<div class="section" id="id35">
<h2>함수의 스케일링<a class="headerlink" href="#id35" title="Permalink to this headline">¶</a></h2>
<p>단변수함수를 x축 방향으로 a배만큼 늘릴려면 함수를 다음처럼 변형한다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) \;\; \rightarrow \;\; f\left(\frac{x}{a}\right)
\tag{4.1.48}
\end{align}
\]</div>
<p>단변수함수를 y축 방향으로 b배만큼 늘릴려면 함수를 다음처럼 변형한다.</p>
<div class="math notranslate nohighlight">
\[
\begin{align}
f(x) \;\; \rightarrow \;\; bf(x)
\tag{4.1.49}
\end{align}
\]</div>
<p>다음 그래프는 로지스틱함수를 x축 방향으로 1/2배, y축 방향으로 2배한 것이다.</p>
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<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">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">100</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">logistic</span><span class="p">(</span><span class="n">xx</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$\sigma(x)$&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;-&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="mi">2</span><span class="o">*</span><span class="n">logistic</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">xx</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;2$\sigma(2x)$&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&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">title</span><span class="p">(</span><span class="s2">&quot;로지스틱함수를 x축 방향으로 1/2배, y축 방향으로 2배&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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