Local Outlier Factor

In anomaly detection, the local outlier factor (LOF) is an algorithm proposed by Markus M. Breunig, Hans-Peter Kriegel, Raymond T. Ng and Jörg Sander in 2000 for finding anomalous data points by measuring the local deviation of a given data point with respect to its neighbours.

LOF shares some concepts with DBSCAN and OPTICS such as the concepts of "core distance" and "reachability distance", which are used for local density estimation.

The local outlier factor is based on a concept of a local density, where locality is given by ${\displaystyle k}$ nearest neighbors, whose distance is used to estimate the density. By comparing the local density of an object to the local densities of its neighbors, one can identify regions of similar density, and points that have a substantially lower density than their neighbors. These are considered to be outliers.

The local density is estimated by the typical distance at which a point can be "reached" from its neighbors. The definition of "reachability distance" used in LOF is an additional measure to produce more stable results within clusters.

Let ${\displaystyle {\mbox{k-distance}}(A)}$ be the distance of the object ${\displaystyle A}$ to the k-th nearest neighbor. Note that the set of the k nearest neighbors includes all objects at this distance, which can in the case of a "tie" be more than k objects. We denote the set of k nearest neighbors as ${\displaystyle N_{k}(A)}$.

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