Judging a Reading Against Its Neighbours

One pump runs under tightly controlled pressure, while another normally varies more. A single global distance threshold can flag the second pump’s ordinary readings and miss a gap inside the first pump’s tight group.

Let's compare each reading with the readings near it. Is it much more isolated than its neighbours are? We will start by finding those neighbours, then introduce reachability distance and use it to calculate a local density comparison.

A sparse area can be normal for the points that live there. Compare the point's density with its neighbours' densities, then change that local comparison.

2.25

Scale features using a training reference so units do not dominate distance. For each point, find its kth-neighbour distance after excluding itself. Include all neighbours within that distance when implementing the original tie convention.

The neighbourhood size sets the scale of comparison. Too small a neighbourhood can be unstable; too large one can mix different operating regimes.

The exercises use distinct nearest-neighbour distances to avoid an ambiguous tie rule.

The reachability distance from a point to a neighbour is the maximum of their distance and the neighbour’s kth-distance. Notice whose kth-distance is used: the neighbour’s.

This lower bound prevents an extremely close pair from producing arbitrarily large local density just because their mutual distance is tiny. It does not guarantee a positive denominator when duplicates dominate.

reachability(p,o) = max(distance(p,o), kth_distance(o))
local_density(p) = 1 / mean(reachability(p, neighbours))

Duplicate-only neighbourhoods need a declared numerical policy.

For each neighbour, divide its local density by the point’s density, then average. A value near one indicates density similar to the neighbourhood. A substantially larger value indicates the point is sparser than its neighbours.

Point 10 has local density 1/9
Its nearest neighbour at 1 has density 1
LOF = 1 / (1/9) = 9

LOF is not an anomaly probability, and one is not a universal hard decision threshold.

import numpy as np
reachability = np.array([2., 2.])
neighbour_density = np.array([0.5, 1.0])
own_density = 1.0 / reachability.mean()
print(float(own_density))
print(float(np.mean(neighbour_density / own_density)))

0.5
1.5

The classic training-set calculation includes each reference point’s neighbourhood. Scoring a new observation against a fixed reference is a related novelty-detection procedure and must avoid silently rebuilding the reference with the new point.

Inspect sensitivity to k, distance scaling, and contamination. The method can miss a group of anomalies that forms a dense group of its own.

The workshop combines supplied reachability distances and neighbour densities. The Python example computes them from raw points.

For each reference point, find its distance to its kth neighbour. When comparing a query with a neighbour, the reachability distance is at least that neighbour's kth-distance. This prevents a tiny pairwise distance inside a dense group from dominating the density estimate.

Average the reachability distances, take their reciprocal to form local reachability density, then compare the query density with its neighbours' densities. Each stage has a separate role; an ordinary mean distance ratio is not automatically LOF.

The supplied neighbour radii and query distances below let us see the floor directly. A close query-neighbour pair can still receive a larger reachability distance because of the neighbour's surrounding spacing.

Neighbour selection, distance units and tie handling must be consistent throughout the calculation. The example supplies those neighbours rather than pretending to fit a complete local-density model.

import numpy as np
distance = np.array([.1, 1.2])
neighbour_radius = np.array([.5, .7])
reachability = np.maximum(distance, neighbour_radius)
density = 1/reachability.mean()
print(reachability.tolist())
print(round(float(density), 4))

[0.5, 1.2]
1.1765

Novelty detection scores new observations against a fitted reference collection. Including all new observations in the neighbourhood fit can make a cluster of unusual incoming points support each other and appear ordinary.

Distinguish that task from detecting outliers inside one supplied dataset. Both can be legitimate, but their fitted neighbourhoods and score interpretations differ. Test duplicates, isolated points and varying density before trusting a numerical threshold.