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Local Outlier Factor: LOF

  • Reachability distance from o’ to o:

\[ reachdist_{k}(o\leftarrow o^{'})=max\left \{ dist_{k}(o),dist(o,o^{'}) \right \} \]

    • where k is a user-specified parameter
  • Local reachability density of o:

 \[ lrdk_{k}(o)=\frac{||N_{k}(o)||}{\sum_{o^{'}\epsilon N_{k}(o)}reachdist_{k}(o^{'}\leftarrow o)} \]

  • LOF (Local outlier factor) of an object o is the average of the ratio of local reachability of o and those of o’s k-nearest neighbors
\[ LOF_{k}(o)=\frac{\sum_{o^{'}\epsilon N_{k}(o)}\frac{lrd_{k}(o^{'})}{lrd_{k}(o)}}{||N_{k}(o)||}=\sum_{o^{'}\epsilon N_{k}(o)}lrd_{k}(o^{'}).\sum_{o^{'}\epsilon N_{k}(o)}reachdist_{k}(o^{'}\leftarrow o) \]

  • The lower the local reachability density of o, and the higher the local reachability density of the kNN of o, the higher LOF
  • This captures a local outlier whose local density is relatively low comparing to the local densities of its kNN

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