相関係数

２つのデータ列

$ X = \{x_1,x_2, \ldots, x_N\}

$ Y = \{y_1,y_2,\ldots,y_N\}

分散(variance)

$ Vx = \frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})^2

$ Vy = \frac{1}{N} \sum_{i=1}^{N}(y_i - \bar{y})^2

標準偏差(standard deviation)

$ S_x = \sqrt{V_x}

$ S_y = \sqrt{V_y}

共分散(covariance)

$ S_{xy} = S_{yx} = \frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})

$ \Sigma = \left[\begin{array}{cc}S_{x}&S_{xy}\\S_{xy}&S_{y}\end{array}\right]

相関係数(correlation coefficient)

$ r_{xy} = r_{yx} = \frac{S_{xy}}{S_x S_y}

相関行列(correlation matrix)

$ \left[\begin{array}{cc} 1 & r_{xy} \\ r_{xy} & 1 \end{array}\right]

プログラムを示す。

code:cm01.py

import numpy as np

print('定義式--------------')

Vx = np.sum((x - np.mean(x))**2) / N

Vy = np.sum((y - np.mean(y))**2) / N

print('分散',Vx, Vy)

Sx1 = np.sqrt(Vx1)

Sy1 = np.sqrt(Vy1)

print('標準偏差', Sx1, Sy1)

Sxy1 = np.sum((x- np.mean(x))*(y - np.mean(y))) / N

print('共分散', Sxy1)

r1 = Sxy1 / (Sx1 * Sy1)

print('相関係数', r1)

code:result.txt

定義式--------------

分散 256.0 176.0

標準偏差 16.0 13.2664991614216

共分散 126.0

相関係数 0.5936004596374721

code:（続き）.py

print('NumPy--------------')

Sxy2 = np.cov(x, y, bias=True)

print('分散共分散行列\n', Sxy2)

r2 = np.corrcoef(x, y)

print('相関行列: \n', r2)

code:実行結果.txt

NumPy--------------

分散共分散行列

相関行列: