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.
Basic idea
The local outlier factor is based on a concept of a local density, where locality is given by 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. The "reachability distance" used by LOF has some subtle details that are often found incorrect in secondary sources, e.g., in the textbook of Ethem Alpaydin.
Formal definition Let k -distance ( A ) {\displaystyle k{\text{-distance}}(A)} be the distance of the object 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 N k ( A ) {\displaystyle N_{k}(A)} .
This distance is used to define what is called reachability distance:
reachability-distance k ( A , B ) = max { k -distance ( B ) , d ( A , B ) } {\displaystyle {\text{reachability-distance}}_{k}(A,B)=\max\{k{\text{-distance}}(B),d(A,B)\}}
In words, the reachability distance of an object A from B is the true distance between the two objects, but at least the k -distance {\displaystyle k{\text{-distance}}} of B. Objects that belong to the k nearest neighbors of B (the "core" of B, see DBSCAN cluster analysis) are considered to be equally distant. The reason for this is to reduce the statistical fluctuations between all points A close to B, where increasing the value for k increases the smoothing effect. Note that this is not a distance in the mathematical definition, since it is not symmetric. (While it is a common mistake to always use the k -distance ( A ) {\displaystyle k{\text{-distance}}(A)} , this yields a slightly different method, referred to as Simplified-LOF) The local reachability density of an object A is defined by
lrd k ( A ) := | N k ( A ) | ∑ B ∈ N k ( A ) reachability-distance k ( A , B ) {\displaystyle {\text{lrd}}_{k}(A):={\frac {|N_{k}(A)|}{\sum _{B\in N_{k}(A)}{\text{reachability-distance}}_{k}(A,B)}}}
which is the inverse of the average reachability distance of the object A from its neighbors. Note that it is not the average reachability of the neighbors from A (which by definition would be the k -distance ( A ) {\displaystyle k{\text{-distance}}(A)} ), but the distance at which A can be "reached" from its neighbors. With duplicate points, this value can become infinite. The local reachability densities are then compared with those of the neighbors using
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