Data Mining: Local Outlier Factor Explorer (2D)

Dataset

Neighbours k 6

Live Readouts

Selected LOF

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Selected 1-NN dist

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LOF flagged

Naive flagged

Mean LOF (dataset)

1.00

Disagreements (LOF ≠ naive)

LOF colour scale — click a point to inspect its k-nearest neighbours (white lines). A dashed ring marks the naive global-threshold verdict when comparison is on.

≈1 inlier≥1.8 outlier

How it works

The Local Outlier Factor (Breunig et al., 2000) flags a point as an outlier when its local neighbourhood is far sparser than its neighbours' neighbourhoods — a density ratio, not a global distance threshold — so it correctly separates a genuinely sparse-but-consistent cluster from a real local anomaly.

k-distance(A)      = distance to the k-th nearest neighbour of A
reach-dist_k(A,B)  = max( k-distance(B), dist(A,B) )
lrd_k(A)           = k / Σ_{B∈N_k(A)} reach-dist_k(A,B)   (local reachability density)
LOF_k(A)           = ( Σ_{B∈N_k(A)} lrd_k(B) ) / ( k · lrd_k(A) )

Dense cluster (left) — tightly packed; a healthy point here has LOF ≈ 1.
Sparse cluster (right) — spread far apart but internally consistent: every point's neighbours are roughly as far as it is, so LOF stays ≈ 1 even though the raw distances are large.
Planted local outlier — sits just off the dense cluster's edge. Its own neighbours (mostly dense-cluster points) are packed far tighter than its distance to them, so its density ratio — and LOF — spikes, even though it is not the farthest point on the plot.
Naive comparison — flags any point whose 1-nearest-neighbour distance exceeds mean + 1.2·σ of all 1-NN distances in the dataset. Being a single global cutoff, it typically over-flags the whole sparse cluster (false positives — their spacing alone crosses the global bar) and under-flags the planted local outlier (its absolute distance is still smaller than the sparse cluster's spacing, so it slips under the same bar) — exactly the failure mode LOF's local density ratio is built to avoid.

Real-world use: fraud detection, sensor-fault detection and rare-event mining all lean on this density-ratio idea to catch anomalies that a simple distance-from-centroid or fixed-radius rule would miss.

Click a point to inspect its k-nearest neighbours