# Tutorial

The aim of Historical Glottometry is to provide a representation of language genealogy in which

1. subgroups are defined by exclusively-shared innovations (following Leskien’s principle [1876], one of the mainstays of the Comparative Method);
2. subgroups are allowed to intersect (as expected under the Wave model);
3. the “strength” of each subgroup is measured on a continuous scale (rather than subgroups simply being absent or present).

How might one go about creating such a representation? One option would be to use an isogloss map, as in dialectology, but with the following properties:

1. Each isogloss delimits a set of languages defined by one or more exclusively-shared innovations — that is, a genealogical subgroup (in the sense of the Comparative Method);
2. The graphic representation of each isogloss can be used to represent the strength of the subgroup. For example, the contour’s thickness could be made proportional to the number of exclusively-shared innovations.

However, this would fail to capture the intuition that the strength of a subgroup is not just defined by the number of innovations that confirm it, but is also affected by the number of innovations that contradict it (that is, the number of isoglosses that cross-cut the subgroup). If there are no innovations that cross-cut a subgroup, then its strength should be equal to the number of exclusively-shared innovations defining it; but as the number of cross-cutting innovations grows, the strength of the subgroup should be reduced accordingly, and ultimately tend to zero.

# Cohesiveness

The definition of subgroupiness that we use captures this intuition by weighting the number of exclusively-shared innovations (called ε ‘epsilon’) by a “cohesiveness coëfficient” (κ ‘kappa’). Consider a candidate subgroup. If the number of cross-cutting innovations is equal to zero, this means we have a “perfect” subgroup which is never contradicted: in this case, the coefficient κ is equal to 1.  Conversely, as the number of cross-cutting innovations tends to infinity, κ should tend to zero.

Given a candidate subgroup S, let us call p the number of innovations that properly confirm its cohesion (i.e., those which contain the whole of the subgroup as well as at least one language outside the subgroup);  and q the number of cross-cutting innovations.  Together, p and q constitute all the relevant evidence needed to assess a subgroup’s cohesiveness.  We define κ as follows:

```      p + 1
κ = —————————
p + q + 1```

Note that κ is always between 0 and 1, and can be expressed as a percentage.

The intuition behind this definition of cohesiveness is as follows. Consider a perfectly tree-like language family of the form [a [b c]]. In such a family, if we find an innovation that b shares with a, we can be absolutely certain that it is also shared with c—otherwise the innovation would be incompatible with the tree. Indeed, this is the essential property of a tree: that if a member of a subgroup shares an innovation with a non-member, it always “invites” all of the other members. Cohesiveness simply measures the proportion of the time that subgroup members are “loyal” in inviting all of their co-members (p), relative to the number of times that they are “disloyal” in excluding some of their co-members (q).

# Subgroupiness

Given this definition of cohesiveness, we define the subgroupiness of a subgroup  (called ς ‘sigma’).  In principle, subgroupiness could simply be conceived as the product of epsilon and kappa — that is, the number of exclusively shared innovations, weighted by the subgroup’s cohesiveness rating:

```
(  p + 1  )
ς = ε · κ  = ε · (——————————)
(p + q + 1)

```

However, we wish to add a final refinement to the results, by adding a strictness parameter n, that would set the “penalty” for overlapping subgroups:

```                            n
n       (  p + 1  )
ς = ε · κ  = ε · (——————————)
(p + q + 1)

```

The strictness parameter n ranges from 0 to ∞. As n → ∞, κn → 0 for all 0 ≤ κ < 1—which means that the only subgroups that are “left standing” are those that are perfectly cohesive, i.e., those that are never cross-cut, and thus conform to the “ideal” of a branch in a tree. Conversely, as n → 0, κn → 1 for all 0 < κ ≤ 1; in other words, ε·κn → ε, meaning that subgroupiness is measured simply by counting exclusively-shared innovations, no matter how often the subgroup may be cross-cut. By default, we set n = 1, though this is purely a matter of convenience, and carries no theoretical significance.

# Glottometric diagrams

Having measured the subgroupiness of every subgroup that is defined by at least one exclusively-shared innovation, we retain only those subgroups for which ς > 1. The reason for this is that since 0 < κn ≤ 1, the only condition under which ς > 1 is when ε > 1, i.e., when the subgroup is defined by more than one exclusively-shared innovation. Thus, this criterion excludes “hapaxes” (subgroups that are defined by just one innovation), as well as subgroups that are “no better than a hapax”: those which, despite being defined by more than one exclusively-shared innovation, have too low a cohesiveness value. “Better-than-a-hapax” subgroups may be seen as well-supported genealogical subgroups, knowing that the quality of that support can be made more demanding by adjusting the strictness parameter n.

Finally, well-supported subgroups can be represented on a schematic isogloss map in which line thickness is proportional to subgroupiness (ς); this is what we call a glottometric diagram. The isoglosses in a glottometric diagram may form one or more “connected sets”; we say that each connected set of isoglosses delimits a linkage.

Here is an example of a glottometric diagram: it represents the genealogy of the 18 northernmost languages of Vanuatu (see also our Case studies page), after the breakup of their shared ancestor Proto Oceanic.

Glottometric diagram of Northern Vanuatu languages, strictness = 1