Given a set X of n feature vectors x 1= (x 11,…,x 1p ), ..., x n = (x n1,…,x np ) of dimension p, the problem is to identify the vectors that do not belong to the underlying distribution using the BACON method (see [Billor2000]).

In the iterative method, each iteration involves several steps:

  1. Identify an initial basic subset of m > p feature vectors that can be assumed as not containing outliers. The constant m is set to 5p. The library supports two approaches to selecting the initial subset:
    1. Based on distances from the medians ||x i - med||, where:
      • med is the vector of coordinate-wise medians
      • ||.|| is the vector norm
      • i=1, ..., n
    2. Based on the Mahalanobis distance

      , where:
      • mean and S are the mean and the covariance matrix, respectively, of n feature vectors
      • i=1, ..., n

    Each method chooses m feature vectors with the smallest values of distances.

  2. Compute the discrepancies using the Mahalanobis distance above, where mean and S are the mean and the covariance matrix, respectively, computed for the feature vectors contained in the basic subset.
  3. Set the new basic subset to all feature vectors with the discrepancy less than

    , where:

    1. is the (1 - α) percentile of the Chi2 distribution with p degrees of freedom

    2. where

      • r is the size of the current basic subset

      • , where

        and [ ] is the integer part of a number

  4. Iterate steps 2 and 3 until the size of the basic subset no longer changes.
  5. Nominate the feature vectors that are not part of the final basic subset as outliers.

For more complete information about compiler optimizations, see our Optimization Notice.
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