

![0_image_0.png](0_image_0.png)

Figure 6: The strong triangular inequality defines an ultrametric: every triplet of points satisfies the relationship: d(*x, z*) ≤ max{d(x, y), d(*y, z*)} for distance d. Cf. by reading off the hierarchy, how this is verified for all x, y, z:
d(*x, z*) = 3.5; d(*x, y*) = 3.5; d(*y, z*) = 1.0. In addition the symmetry and positive definiteness conditions hold for any pair of points.

What sort of explanation does this provide for our conundrum? It means that the query is a novel, or anomalous, or unusual "document". It is up to us to decide how to treat such new, innovative cases. It raises though the interesting perspective that here we have a way to model and subsequently handle the semantics of anomaly or innocuousness.

The strong triangular inequality, or ultrametric inequality, holds for tree distances: see Figure 6. The closest common ancestor distance is such an ultrametric.

## 1.6 Casablanca Narrative: Illustrative Analysis Continued

Figure 7 uses a sequence-constrained complete link agglomerative algorithm.

It shows up scenes 9 to 10, and progressing from 39, to 40 and 41, as major changes. The sequence constrained algorithm, i.e. agglomerations are permitted between adjacent segments of scenes only, is described in an Appendix to this article, and in greater detail in [7]. The agglomerative criterion used, that is subject to this sequence constraint, is a complete link one.