공분산 행렬

공분산 행렬

두 변량 값들 간의 공분산을 행렬로 표기 한것이다.

데이터구조를 설명해주며, 특히 특징 쌍(feature pairs)들의 변동이 얼마나 닮았는가(다른말로는 얼마만큼이나 함께 변하는가)를 행렬에 나타낸다

5명의 사람의 키와 몸무게

$\begin{bmatrix}
170&&70\\
150&&45\\
160&&55\\
180&&60\\
170&&80
\end{bmatrix}$

X = D – mean(D) = $\begin{bmatrix}
170&&70\\
150&&45\\
160&&55\\
180&&60\\
170&&80
\end{bmatrix} –
\begin{bmatrix}
166&&62\\
166&&62\\
166&&62\\
166&&62\\
166&&62
\end{bmatrix}
=
\begin{bmatrix}
4&&8\\
-16&&-17\\
-6&&-7\\
14&&-2\\
6&&18
\end{bmatrix}$

$X^{T}X = \begin{bmatrix}
4&&-16&&-6&&14&&6\\
8&&-17&&-7&&-2&&18
\end{bmatrix}
\begin{bmatrix}
4&&8\\
-16&&-17\\
-6&&-7\\
14&&-2\\
6&&18
\end{bmatrix}
=
\begin{bmatrix}
540&&426\\
426&&730
\end{bmatrix}$

$X^{T}X$의 i행 j열의 성분 $(X^{T}X)_{ij}$ 는 d개의 feature 중 ‘i번째 feature와 j번째 feature가 얼마나 닮았는지’ 를 모든 사람으로 부터 값을 얻어서 내적 연산을 취함으로써 알려준다.

$\dfrac{X^{T}X}{n}$ : 여기서 n이 커질수록 내적 값은 계속 커져, n으로 나눠줘 해당문제를 해결한다.

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