Basic terms for multivariate analysis technique ( 多変量解析のための基本用語 )

• Mean (平均)
  • x_mean = ( \sigma_{1}^{n} ( x_i ) ) / n
  • y_mean = ( \sigma_{1}^{n} ( y_i ) ) / n

• Variance (分散)
  • s_x^2 = ( \sigma_{1}^{n} ( x_i - x_mean )^2 ) / n
  • s_y^2 = ( \sigma_{1}^{n} ( y_i - y_mean )^2 ) / n

• Root-mean-square deviation (標準偏差)
  • s_x = sqrt( s_x^2 )
  • s_y = sqrt( s_y^2 )
  • This operation is done to match the unit with mean.
  • この処理は, 平均との単位を揃えるために行います.

• Standardization (データの標準化)
  • standard_x = ( x - x_mean ) / s_x
  • standard_y = ( y - y_mean ) / s_y
  • After this operation, the mean of the data is 0 and the variance of the data is 1.
  • データの標準化の後は, データの平均は 0, 分散は 1 になります.

• Covariance ( 共分散 )
  • s_xy = ( \sigma_{1}^{n} ( x_i - x_mean )( y_i - y_mean ) ) / n

• Variance-Covariance matrix (分散共分散行列)

  ( s_x^2    s_xy  
    s_xy     s_y^2 )

• Correlation function (相関関数)
  • r = s_xy / ( s_x * s_y )

• Correlation Matrix (相関行列)

  ( 1        r
    r        1 )

• • If you compute the variance-covariance matrix by using standard data, you can get the correlation matrix.
  • 標準化したデータで, 分散共分散行列を作ると相関行列が得られます.

• 参考文献
  • 入門 はじめての多変量解析. 石村貞夫, 石村光資郎.