Strict Geometric Definition (Strong Isolation)
Let (M,d) be a metric space, and let S⊂M be a finite set of observations. Consider an element x∈S, and let S′=S∖{x} denote the remainder of the set.
The point x is defined as a strong outlier if the following inequality holds:
dist(x,S′)>diam(S′)
Where the distance and diameter are defined as follows:
Point-to-Set Distance (distance to the nearest neighbor):
dist(x,S′)=y∈S′infd(x,y)
(The distance from x to the closest element in the remaining set.)
Diameter of the Set (maximal internal distance):
diam(S′)=y,z∈S′supd(y,z)
(The distance between the two farthest points within the remaining set.)