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Parametric Methods II: Detection of Multivariate Outliers

  • Multivariate data: A data set involving two or more attributes or variables
  • Transform the multivariate outlier detection task into a univariate outlier detection problem
  • Method 1. Compute Mahalaobis distance
    • Let ō be the mean vector for a multivariate data set. Mahalaobis distance for an object o to ō is
      \[ MDist(o, \bar{o}) = (o-\bar{o})^{T} S^{(-1)}(o-\bar{o}) \]
      where S is the covariance matrix
    • Use the Grubb's test on this measure to detect outliers
  • Method 2. Use χ2 –statistic:

\[ X^{2} = \sum_{i=1}^{n}\frac{(o_{i}-E_{i})^{2}}{E_{i}} \]

    • where Ei is the mean of the i-dimension among all objects, and n is the dimensionality
    • If χ2 –statistic is large, then object oi is an outlier

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