Local Outlier Factor

Computes the local outlier factor (LOF) of Breunig et al. (2000) for each observation. The LOF compares the local density around an observation with the densities around its k nearest neighbours, so that an observation in a sparse region surrounded by dense ones scores high even when it is not globally extreme.

Usage

lof(x, k)

Arguments

x: a numeric matrix or data frame, observations in rows.
k: the number of neighbours defining the local neighbourhood, not counting the observation itself.

Value

a numeric vector with the local outlier factor of each observation, in the order of the rows of x.

Details

Scores near 1 indicate that the density around the observation matches that of its neighbours. Substantially larger values indicate a locally sparse neighbourhood and thus an outlier; the user decides on the threshold, as its scale depends on the data. Values below 1 arise in unusually dense regions and are not normally of interest.

The computation uses dbscan::lof() from the dbscan package, which builds a kd-tree and therefore scales to large data. Note that its minPts counts the observation itself, whereas k here counts neighbours only, following Breunig et al.; the translation minPts = k + 1 is applied internally.

Duplicated observations need care. Where an observation has more than k exact duplicates, every neighbour distance is zero, the local reachability density is infinite and the LOF is formally \(0/0\). dbscan reports 1 in that case, on the grounds that the coincident points already supply enough density for the observation not to be an outlier. Breunig et al. instead assume genuine duplicates have been removed beforehand; do that if you need results matching the paper exactly.

Note that the infinite density is not confined to the duplicates themselves. Any observation that has one of them among its own k neighbours inherits an infinite score, so a block of duplicates lying inside the data can render its whole surroundings Inf. This follows from the definition rather than from the implementation, and is a further reason to remove exact duplicates first.

Only complete observations can be scored, since the distances are otherwise undefined.

References

Breunig, M. M., Kriegel, H.-P., Ng, R. T., & Sander, J. (2000). LOF: Identifying density-based local outliers. Proceedings of the ACM SIGMOD International Conference on Management of Data, 93-104. doi:10.1145/335191.335388

Examples

# \donttest{
if(requireNamespace("dbscan", quietly = TRUE)) {

  res <- lof(iris[, -5], k = 10)

  # the ten most outlying observations
  head(order(res, decreasing = TRUE), 10)

  # scores near 1 are unremarkable, large ones are not
  summary(res)
}
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.9539  0.9904  1.0325  1.0978  1.1454  2.1783 
# }