# Sampling the Swadesh List to Identify Similar Languages with Tree Spaces

Garett Ordway
Florida State University, Florida, U.S.A  Vic Patrangenaru
Florida State University, Florida, U.S.A

(April 9, 2024)

###### Abstract

Communication plays a vital role in human interaction. Studying language is a
worthwhile task and more recently has become quantitative in nature with
developments of fields like quantitative comparative linguistics and
lexicostatistics. With respect to the authors own native languages, the
ancestry of the English language and the Latin alphabet are of the primary
interest. The Indo-European Tree traces many modern languages back to the
Proto-Indo-European root. Swadesh’s cognates played a large role in developing
that historical perspective where some of the primary branches are Germanic,
Celtic, Italic, and Balto-Slavic. This paper will use data analysis on open
books where the simplest singular space is the 3-spider - a union
$\mathcal{T}_{3}$ of three rays with their endpoints glued at a point 0 -
which can represent these tree spaces for language clustering. These trees are
built using a single linkage method for clustering based on distances between
samples from languages which use the Latin Script. Taking three languages at a
time, the barycenter is determined. Some initial results have found both non-
sticky and sticky sample means. If the mean exhibits non-sticky properties,
then one language may come from a different ancestor than the other two. If
the mean is considered sticky, then the languages may share a common ancestor
or all languages may have different ancestry.

Keywords: Linguistics, Tree Spaces, Phylogenetics

##  1 Introduction

Whether you believe the story that God confused the languages of mankind at
the Tower of Babel, that the multitude of modern-day languages evolved from
one common language as migration transpired (monogenesis), or that distinct
languages have existed since the beginning of the human species (polygenesis),
language is ubiquitous throughout history. Woodard summarizes many scholars
with his introduction in The Ancient Languages of Europe [38]. Several
languages have records beginning more than 4500 years ago (c. 2600 BC). Algeo
in The Origins and Development of the English Language argues that human
language is “fundamentally different” from animal communication. While apes
have shown the most progress of animals in learning communication, “it is a
far cry from the fullness of a human language.” Human language is special[1].
As different people groups began to interact, translation of languages
certainly began quickly. The Sumerian Epic of Gilgamesh is one of the most
commonly known early literary works that has been found in multiple languages.
A more modern and certainly more famous ancient translation is the Rosetta
Stone believed to be from 196 BC with an Ancient Greek and two Ancient
Egyptian languages. Augustine of Hippo in AD c.420 wrote in City of God about
the Tower of Babel and discussed the many languages that needed translations
of the Bible [5]. Campbell’s Historical Linguistics: An Introduction connects
philology to linguistics which encapsulates the modern-day study of language.
Fields such as quantitative comparative linguistics, historical linguistics,
glottochronology, and lexicostatistics have searched for insights with varying
methods. While Morris Swadesh was not the first linguist (the pioneer of
monogenesis was Alfredo Trombetti [36] in 1905), he has made perhaps the
biggest contribution. Swadesh’s initial study was of America’s indigenous
languages. His most famous contributions began in the 1950s with the
publication of the first Swadesh List [33]. Subsequent Swadesh lists contained
more or fewer or different cognate words. Crystal’s A Dictionary of
Linguistics and Phonetics defines a cognate as “a language or linguistic form
which is historically derived from the same source as another
language/form”[11]. Modern languages identify which ancient language they
received words from or which other modern languages they are similar to using
these cognates. In English, some of these cognate words from Swadesh’s 207
list are “where,” “big,” “child,” and “to think.” From a monogenesis
perspective, the search continues for a connection of all world languages to a
“Proto-Human” language in a famous work from Merritt Ruhlen [31]. As
computational power increases and archaeology continues uncovering ancient
civilizations, hope grows.
In the search for the original language, scholars identify several Proto-
languages such as Proto-Indo-European and Proto-Afroasiatic. While these most
ancient languages are only hypothesized, they do allow for adding structure to
the study of language. The Indo-European language family’s prominence in
Western Civilization and preserved writings for millennia has produced a
fairly settled phylogenetic tree with several main branches being Germanic,
Latin (Italic), Hellenic (Greek), Celtic, Balto-Slavic, and Indo-Iranian. The
phylogenetic tree should depict language origins and groupings of similar
languages. Trees can be developed through a variety of methods ranging from
completely subjectively to expert opinion to data modeling.

These trees have their own roots from scientists’ studies of natural world in
the 1800s trying to trace the lineage of plants and animals [12] [3]. A
phylogenetic tree attempts to trace the evolutionary ancestry of plant and
animal species. The species is the final branch of the tree, and branches
connect at nodes. In the animal kingdom, a taxonomic rank is imposed to
distinguish the different species within a genus from other genera and
different genera within a family from other families and so on [26]. While
taxonomy provides organization for the animal and plant kingdoms,
phylogenetics attempts to provide answers for the evolutionary ancestry. How
the phylogenetic tree is built depends on the method of analysis used. Early
trees (mid-19th century) were built using only similar characteristics, but as
the field of phylogenetics developed, trees grew in complexity using
techniques from statistics such as Maximum Likelihood [15] [27], Nearest
Neighbor Interchange [30], and Bayesian Inference [32][25][29]. Phylogenetic
trees have thus become very well-known even to laymen through their
development in systematic biology and use in science textbooks and, more
recently, investigations related to the SARS-Cov-2 virus outbreak [24][9][4]
[39]. A SARS-Cov-2 phylogenetic tree can be made since the approximately
30,000 letter sequence is widely studied illustrating the rapid effects of
microevolution.

Phylogenetic trees are being extended to include work on branch length and
phylogenetic networks. Phylogenetic trees must then have topological
properties and can be studied from that perspective. Billera, Holmes, and
Vogtmann’s critical work “Geometry of the Space of Phylogenetic Trees” gave
significant mathematical insight for future studies [7]. They gave the idea
that the usual Euclidean distance can be used to calculate distances for
points in the same orthant while adding two straight segments for points in
different orthants can give a distance as well. Additionally, they showed that
a centroid can give a consensus of set of trees. Later, Hotz et al extended
this work to the open book and proved the Sticky Central Limit Theorem [18].
Paradis and Binet et al both discuss the idea of branch length carrying some
importance on the phylogenetic tree [28] [8]. For evolutionary theory, a
longer branch should indicate more genetic distance. Huson and Bryant consider
phylogenetic trees to be one type of phylogenetic network extending the idea
to allow branches to interact with one another separately from connecting to
the node [19]. Phylogenetics continues to develop with key contributions over
the last several decades.

But biology is not the only possible application of phylogenetics. Human
language has also undergone significant change over the past millennia. While
Swadesh was one of the first linguists to popularize the language tree to show
modern languages connection to ancient languages, some work had been done
previously (prior to 1950), and much work has been done since in applying a
wide variety of techniques. Early work by Kroeber calculated similarity
coefficients for languages based on a plus or minus approach denoting whether
a characteristic was present or absent and counting some number of times the
characteristic was present in both, present in one, or absent in both [22].
Swadesh’s cognates stirred much discussion from the 1950s onward prompting
discussion of choice of native words and synonyms and gave birth to the
lexicostatistical approach of which Dyen’s 1965 work is paramount [13]. Dyen
published with Kruskal and Black in 1992 a thorough analysis of the
IndoEuropean tree including the new _box diagram_ [14]. They used hierarchical
clustering with lexicostatistical percentages calculated from similar or
different cognate words. Glottochronology attempts to determine dates of
linguistic evolution through rates of change. Swadesh, Lees [23], and Hymes
[20] all provided early contributions in the field with Chretien emphasizing
the mathematics of the function [10]. Modern studies by Gray and Atkinson
attempt to go farther back in history to original emergence of the Indo-
European languages [16]. Other approaches to language evolution involve
concern with borrowed words due to cultural interactions and phonetic changes
over time, among others, necessitating a variety of methods similar to those
mentioned for biological phylogenetics earlier.

Some examples of phylogenetic trees for language are presented next. Algeo
gives the following tree for the Indo-European family based on comparative
linguistics [1].

Figure 1: Indo-European Languages Tree-part I Figure 2: Indo-European
Languages Tree-part II

Johnson and Wichern’s use of the first letter of the ten numerals gives this
tree diagram [21].

Figure 3: Partial Indo-European Languages Tree from numbers

A tree for Indo-European languages, due to an anonymous author, is displayed
below can be found at [2]

Figure 4: Indo-European Languages Tree Tree

Additional trees that are more informative or more visually appealing can be
found with a quick Google search. If these trees are based on statistical
methods, then they necessitate some mathematical background. The more leaves a
tree has, the more complicated the space becomes.
Billera, Holmes, and Vogtmann [7] formalized a phylogenetic tree saying a tree
with $p$ leaves is an equivalence class based on a certain equivalence, of a
DNA-based connected directed graph of species with no loops, having an
unobserved root (common ancestor) and $p$ observed leaves (current observed
species of a certain family of living creatures). In our case, the root is the
ancestor language is no longer spoken while the leaves are the modern
languages. A tree with $p$ leaves is a simply connected graph with a
distinguished vertex, labeled $o$, called the root, and $p$ vertices of degree
1, called leaves, that are labeled from 1 to p. In addition, we assume that
all interior edges have positive lengths. An edge of a p-tree is called
interior if it is not connected to a leaf. Now consider a tree T, with
interior edges $e_{1},\dots,e_{r}$ of lengths $l_{1},\dots,l_{r}$
respectively. If T is binary, then $r=p-2,$ otherwise $r<n-2.$ The vector
$(l_{1},\dots,l_{r})^{T}$ specifies a point in the positive open orthant
$(0,\infty)^{r}.$ That is to say that a binary $p$-tree has the maximal
possible number of interior edges and thus determines the largest possible
dimensional orthant; in this case the orthant is $p-2$-dimensional. The
orthant corresponding to each non-binary tree appears as a boundary face of
the orthants corresponding to at least three binary trees. In particular, the
origin of each orthant corresponds to the (unique) tree with no interior
edges, which is known as the star tree. The space $T_{p}$ is constructed by
taking one $p-2$-dimensional orthant for each of the $(2p-3)!!$ possible
binary trees and gluing them together along their common faces. Note that tree
spaces are not manifolds. Singularities (points where the space does not have
a tangent space) are present in the tree space structure. For further detail
on phylogenetic trees and the construction of the tree space, see Billera et
al.(2001)[7]. Phylogenetic trees with $p$ leaves are points on a metric space
$T_{p}$ that has $p-2$ dimensional stratification. In particular, the space of
trees with 3 leaves is $T_{3}=S_{3}$, a $3$-spider, which is the union of
three line segments with a common end (see Figure 5, next page). For a
probability measure on $S_{p},$ if none of the the “legs” of the $p$-spider
has a dominant expected mean distance to the center of the spider, then the
Fréchet mean is the star tree. This result will be stated more formally in the
following section and extended to more general spaces. $T_{4}$ is a two
dimensional stratified space obtained from $15=(2\times 4-3)!!$ 2D quadrants
glued according to tree identification rules (see Billera et. al.(2001)[7]).
Interior points of these quadrants are combinatorial binary trees with four
leaves, the coordinates of an interior point being given by the two interior
edges of a binary tree in one of these combinatorial binary trees. Points on
the boundaries of the quadrants, correspond to combinatorial trees with four
leaves, which are obtained from a combinatorial binary tree by shrinking one
of the interior edges to zero length.

Figure 5: Tree spaces $T_{3},T_{4},T_{5}$.

Therefore a representation of $T_{4}$ as a surface with singularities can be
obtained from the polyhedral surface given in Figure 6 by identifying the
edges labeled with the same letter. While this is a 3D pictorial
representation only, in fact, as mentioned in Section 3, given that
$2^{4}-4-2=10,$ $T_{4}$ is embedded in $\mathbb{R}^{10}$ having the star tree
at the origin.

Figure 6: A 2D stratified space - $T_{4}$, space of trees with four leaves

In this representation, the intersection of a small sphere in
$\mathbb{R}^{10}$ centered at the origin with $T_{4}$ is the so called
Petersen graph. An edge of Petersen graph is the transverse intersection of
one quadrant with a sphere, thus there are $15$ edges, and a vertex is the
point where one of the coordinate axes pierces the sphere, therefore there are
$10$ vertices ( see figure 7).

Figure 7: Petersen graph

For this work, topics like Clustering, Fréchet means, and Convergence must be
considered to develop some analytical techniques for trees. A brief analysis
of two different types of means on tree spaces with 3 leaves will be
considered in the context of language ancestry. Finally, new frontiers for
analysis of tree spaces will be explored.

##  2 Intrinsic Moments on a Metric Space

###  2.1 General Notions

Frèchet’s random element in a metric space from 1948 opened new analysis
options in the field of statistics. A separable metric space
$(\mathcal{M},\rho)$ has some mean with the consistency of the sample mean
when the space is compact being shown by Ziezold in 1977. In 2003, the Frèchet
sample mean and total sample variance on a complete separable metric space
were proven to be consistent by Bhattacharya and Patrangenaru [6]. One
application of interest here is the random object on a complete space with
manifold stratification which will be the phylogenetic tree. Specifically, the
intrinsic moments on a manifold $\mathcal{M}$ with probability measure $Q$
result from the Riemannian distance $\rho$ induced by a Riemannian structure
$g$ on $\mathcal{M}$.

###  2.2 On Stratified Spaces: Open Books

Hotz, et. al., says the open book is the “simplest singular topologically
stratified space” and defines an open book as the disjoint union of the spine
and the interior of the leaves [18]. An open book must contain at least 3
leaves ($K\geq 3$). Each leaf contains the non-negative reals; and, when all
leaves are “glued” together at the spine, the leaves comprise just the
positive reals with the spine being just the number 0. Hotz, et. al. [18]
gives the following definition.

######  Definition 1.

The open book $\mathcal{O}$ consists of $K\geq 3$ leaves $L_{k}$, for
$k=1,...,K,$ each of dimension $d+1$ and defined by

$L_{k}=\bar{H}_{+}\times\\{k\\}$

where $\bar{H}_{+}=\mathbb{R}_{\geq 0}\times S$ with $S=\mathbb{R}^{d}$. The
leaves are joined together along the spine $L_{0}$ which comprises the
equivalence classes in $\bigcup_{k=1}^{K}(H\times\\{k\\})$,i.e. $L_{0}$ can be
identified with the hyperplane $H=\\{0\\}\times S$ or with the space
$S=\mathbb{R}^{d}$.

Furthermore, the open book $\mathcal{O}$ is the disjoint union of the spine
and the interiors $L_{k}^{+}=L_{k}\setminus L_{0}$ of the leaves so
$\mathcal{O}=L_{0}\cup L_{1}^{+}\cup\cdots\cup L_{K}^{+}$. On this open book,
a metric $d$ (distance) between two points is found by taking the absolute
value of the difference between two points if they are on the same or by
reflecting ($R$) one of the two points across the hyperplane to a negative
value if the two points lie on difference leaves. This reflection is given for
$x\in\bar{H}_{+}$ as $Rx\in\bar{H}_{-}=\mathbb{R}_{\geq
0}\times\mathbb{R}^{d}$ Hotz et al give two points $p,q\in\mathcal{O}$ with
$p=(x,k)$ and $q=(y,j)$ with $k,j$ denoting the specific leaf of points $x,y$
respectively [18].

$d(p,q)=\begin{cases}|x-y|&\text{if }k=j,\\\ |x-Ry|&\text{if }k\neq
j.\end{cases}$

###  2.3 Probability Measures and Moments

Let $Q=P_{X}$ be a Borel probability measure on $\mathcal{O}$ such that
$d(0,X)$ has bounded expectation and square expectation. For a Borel set
$A\subseteq\mathcal{O}$,

$Q(A)=w_{0}Q_{0}(A\cap L_{0})+\sum_{k=1}^{K}w_{k}Q_{k}(A\cap L_{k}^{+})$

where $w$ are weights with $0\geq w_{k}\geq 1$ and $\sum_{k=1}^{K}w_{k}=1$.
To go leaf by leaf, a Folding map
$F_{k}:\mathcal{O}\rightarrow\mathbb{R}^{d+1}$ is used where for
$k\in\\{1,\ldots,K\\}$ and $p\in\mathcal{O}$

$F_{k}p=\begin{cases}x&\text{if }p=(x,k)\in L_{k},\\\ Rx&\text{if }p=(x,j)\in
L_{j}\text{ and }j\neq k.\end{cases}$

This allows for the pushforward measure $\tilde{Q}_{k}=Q\circ F_{k}^{-1}$ on
the $k^{\text{th}}$ leaf to be

$\tilde{Q}_{k}(A)=w_{k}Q_{k}(A\cap\bar{H}_{+})+w_{0}Q_{0}(A\cap S)+\sum_{j\geq
1,j\neq k}w_{j}Q_{j}(A\cap H_{-}).$

With the measures given, moments can be found for each leaf. Hotz et al
defines the first moment on a leaf [18].

######  Definition 2.

Let $x^{(0)},x^{(1)},\ldots,x^{(d)}$ be the coordinate functions on
$\mathbb{R}^{d+1}$. The first moment of the measure Q on the $k^{\text{th}}$
leaf $L_{k}$ is the real number

$m_{k}=\int_{\mathbb{R}^{d+1}}x^{(0)}d\tilde{Q}_{k}(x)=\int_{\mathcal{O}}(\pi_{0}F_{k}p)dQ(p),$

where $\pi_{0}:\mathbb{R}^{d+1}\rightarrow\mathbb{R}$ is the orthogonal
projection with kernel $H=\\{0\\}\times\mathbb{R}^{d}$.

This first moment can only be nonnegative on at most one leaf as proven by
Hotz et al. (2013)[18], where the following definition is given.

######  Definition 3.

Under integrability and nondegeneracy, the mean of the measure $Q$ is either

  1. 1.

nonsticky if $m_{k}>0$ for some (unique) $k\in\\{1,\ldots,K\\}$, or

  2. 2.

partly sticky if $m_{k}=0$ for some (unique) $k\in\\{1,\ldots,K\\}$, or

  3. 3.

sticky if $m_{j}<0$ for all indices $j\in\\{1,\ldots,K\\}$.

When the dimensionality of the data is one, the open book becomes a spider.
When $k=3$, it is the simplest open book called a 3-spider.

##  3 Sample Mean on $\mathcal{T}_{3}$

Results in this section are presenting an elementary proof of the Sticky CLT
in Hotz et al(2013)[17]. To define a Spider, consider an arbitrary nonempty
set $K$ of size at least $3$ and, for each of its elements, $i$, define the
ray(leg) $L_{i}=\\{(i,x):x\in[0,\infty)\\}$. The Spider is formed by
identifying the origins $(i,0),$ for all $i\in K,$ which corresponds to
joining the rays together at a common center $C$ (see figure 8).

$S_{K}=\\{(i,x):i\in K,x\in[0,\infty)\\}$

where $(i,0),\dots,(j,0)$ for all $i,j\in K$, the equivalence class of all
points of the form $(i,0)$, we denote by $C$, named center of the spider.

Figure 8: Spider $S_{p}$

Assume $X_{i},i=1,\dots,n$ are i.i.d. random objects on a spider $S_{K}$,
having legs $L_{i},i=1,\dots,K$ and center $C$. Further, assume the intrinsic
mean $\mu_{I}$ exists and the intrinsic variance is finite. The sample mean of
each leaf must be found by finding what Hotz et al calls the $k^{t}h$ folded
average

$\eta_{k,N}=\frac{1}{N}\sum_{n=1}^{N}F_{k}p_{n}$

for $N$ points $\\{p_{n}\\}_{n=1}^{N}\subset\mathcal{O}$. Following from the
theory presented earlier, at most one of these folded averages can be
nonnegative.
Any probability measure $Q$ on $S_{K}$ then decomposes uniquely as a weighted
sum of probability measures $Q_{K}$ on the legs $L_{K}$ and an atom $Q_{0}$ at
$C$. More precisely, there are nonnegative real numbers
$\\{w_{k}\\}_{k=0}^{p}$ (probability weights) summing to 1 such that, for any
Borel set $A\subseteq S_{p}$ , the measure $Q$ takes the value

$Q(A)=w_{0}Q_{0}(A\cap C)+\sum_{k=1}^{p}w_{k}Q_{k}(A\cap L_{k}).$

Assume $w_{0}=0$ and $x\in L_{a}$, the Fréchet function
$F_{X}(x)=F(x)=E(\rho^{2}(X,x)).$ If there exists an unique minimizer for the
Fréchet function $F(x)$, the minimizer is called intrinsic mean $\mu_{I}$
(based on the intrinsic distance $\rho$). If on each leg, the distance is
Euclidean, $F(x)$ is defined as follows,

$\displaystyle F(x)$ $\displaystyle=\sum_{i=1,i\neq
a}^{K}\int_{0}^{\infty}(x+u)^{2}w_{i}Q_{i}(du)+\int_{0}^{\infty}(x-u)^{2}w_{a}Q_{a}(du)$
$\displaystyle=x^{2}\sum_{i=1}^{K}\int_{0}^{\infty}w_{i}Q_{i}(du)+2x[\sum_{i=1,i\neq
a}^{K}\int_{0}^{\infty}uw_{i}Q_{i}(du)-\int_{0}^{\infty}uw_{a}Q_{a}(du)]+\sum_{i=1}^{K}\int_{0}^{\infty}u^{2}w_{i}Q_{i}(du)$
$\displaystyle=x^{2}+2[\sum_{i=1,i\neq a}^{K}v_{i}-v_{a}]x+const.$

where $v_{i}=\int_{0}^{\infty}uw_{i}Q_{i}(du)$.

The minimizer of the quadratic form is the first sample moment denoted
$x^{*}_{a}=v_{a}-\sum_{i\neq a}v_{i}$, where $x_{a}\in
L_{a}=\\{(a,u):u\in[0,\infty)\\}$. Thus, we have three situations:(i)
$x^{*}_{a}>0$ for some unique $a\in\\{1,...,K\\}$, (ii) $x^{*}_{a}=0$ for some
unique $a\in\\{1,...,K\\}$, and (iii) $x^{*}_{a}<0$ $\forall a$. In case (i),
we have $\mu_{I}$ well defined on $L_{a}$, then the classical C.L.T is
applied. In case (ii), we can fold the other legs into that half line opposite
to $L_{a}$ and then apply C.L.T. Since the negative part is undefined, so the
result goes to a positive truncated normal distribution. In case (iii), for
any $a\in\\{1,\dots,K\\}$, we have $\mu_{I}=C$, which shows that intrinsic
mean $\mu_{I}$ sticks to the center $C$.

  1. 1.

$x^{*}_{a}>0$ for some (unique) $a\in\\{1,\dots,K\\}$, then $\mu_{I}\in L_{a}$
and due to the consistency of the intrinsic sample mean, for $n$ large enough
$\bar{X}_{I,n}\in L_{a}$ and $\sqrt{n}(\bar{X}_{I,n}-\mu_{I})$ has
asymptotically a normal distribution.

  2. 2.

$x^{*}_{a}=0$ for some (unique) $a\in\\{1,\dots,K\\}$, then after folding the
legs $L_{i},i\neq a$, into one half line opposite to $L_{a}$,
$\sqrt{n}(\bar{X_{n}}-\mu_{I})$ has asymptotically a positive truncated normal
distribution.

  3. 3.

$x^{*}_{a}<0$ for all $a\in\\{1,\dots,K\\}$, then $\mu_{I}=C$ and there is
$n_{0}$ s.t. $\forall n\geq n_{0}$, then $\bar{X}_{I,n}=C$ a.s.

##  4 Applications to Language Ancestry

###  4.1 Data Collection

####  4.1.1 Words

Of the varieties of cognate lists for language ancestry, one of the original
lists developed by Morris Swadesh contains 207 words [33][34][35][37]. These
words range from familial relations and numbers to action verbs and anatomy.
Each language has a representation of these words. The lists were accessed
through the Wikipedia Module “Swadesh modules” which contains the 207 words in
hundreds of languages. The languages chosen for analysis are English, German,
French, Italian, Spanish, and Irish (Gaelic). Words were input into a data
frame in R for analysis. To imagine two populations, words were randomly
sampled into groups of 102 and 105. Words were then randomly sampled 3 at a
time to create 34 samples from population 1 and 35 from population 2. Three
analyses will be run involving English, German, and French first then French,
Italian, and Spanish second, and German, French, and Irish last.

####  4.1.2 Calculate Distances

Per Johnson and Wichern’s example [21], the first letter of each word is used
to check for similarity. If the letter is the same, then a distance of zero is
assigned. If the letter is different, then a distance of one is assigned. This
is done for each of the 207 words between each pair of languages to be
analyzed: English and German, English and French, German and French, etc. Sum
the distances within each of the 69 samples for two languages at a time. The
minimum possible distance is 0 when all 3 languages begin with the same letter
for all 3 words and the maximum possible distance is 3 when all 3 languages
begin with different letters for all 3 words. This produces a $3\times 3$
distance matrix.
Some examples of the distance calculations are given next.

######  Example 1.

This is an example where all three languages are equidistant in the sample.

In this example, the distance between English and French is 3, between English
and German is 3, and between French and German is 3.

######  Example 2.

This is an example where one language is furthest away and equidistant from
the other two.

In this example, the distance between English and French is 3, between English
and German is 1, and between French and German is 3.

######  Example 3.

This is an example where one language is furthest away, but is closer to one
language than it is to the other.

In this example, the distance between English and French is 2, between English
and German is 1, and between French and German is 3.

######  Example 4.

This is an example where two groups of languages tie for the minimum distance.
No graphs are shown since more than one graph could be created depending on
the random choice between clustering English and French first or English and
German first.

English | French | German
---|---|---
road | route | straße
freeze | geler | frieren
because | parce que | weil

In this example, the distance between English and French is 2, between English
and German is 2, and between French and German is 3.

####  4.1.3 Points on a 3-spider

The distances are used to cluster as discussed in the Preliminaries for single
linkage and then given as points on a 3-spider. Here, the three individuals
are three languages with the following four possible ways to cluster and
assign points on leaves.

  1. 1.

All languages are equidistant so only one cluster is needed. In the context of
a 3-spider, the point is at the origin. See 1.

  2. 2.

One language is furthest away and is equidistant from the other two. For a
3-spider, the point is on that language’s leaf with a distance equal to the
distance to the other languages. See 2. The point is at 3 on the French leaf.

  3. 3.

One language is furthest away, but is closer to one language than the other.
For a 3-spider, the point is on that language’s leaf with a distance equal to
the minimum distance from that language to the other two. See 3. The point is
2 on the French leaf.

  4. 4.

Two groups of languages tie for the minimum distance. For example, Language 1
is distance $c_{1}$ from Language 2 and Language 2 is also distance $c_{1}$
from Language 3, but Language 1 is a distance greater than $c_{1}$ (say
$c_{2}$) from Language 3. In this case, randomly decide whether to cluster 1
with 2 or 2 with 3. For a 3-spider, then use the $c_{2}$ distance as the point
on the Language’s leaf that was not clustered originally (the point would be
at $c_{2}$ on leaf 1 or leaf 3). See 4. The point is 2 and was randomly
assigned to the French leaf.

The distances are then points on a 3-spider where a point is on a particular
leaf $L_{i}$ where $i=E,G,F$ or $i=F,I,S$ if that language is the furthest
(added to the cluster last).

###  4.2 Non-sticky Sample Mean

English, French, and German are the first three chosen languages. English is
classified as a Germanic language while French is classified as a Romance
(Italic or Latin) language. As discussed in the previous sections, only one
leaf may have a positive mean (although none are guaranteed). In this case,
one language seems to be more different than the other two.

Language | Mean Distance 1 | Mean Distance 2
---|---|---
English | -2.00 | -2.41
German | -1.66 | -1.41
French | 1.49 | 1.41

Since French has a positive mean distance, it is considered the sample mean
(barycenter) and is used to calculate the measures of variability. This is the
non-sticky case for the mean. The variance of the first sample is 4.14 and the
second is 4.61. Using the Non-sticky Central Limit Theorem, traditional
asymptotic inference can be performed. The pooled sample variance is 4.43.
These statistics yield the two sample t-statistic value of 0.16 which is far
from significant. This is to be expected since the samples had no necessary
distinction; they were randomly assigned. If two groups of words were
different for some reason, say medical terms versus agricultural terms, then
perhaps some significant difference may be found.

###  4.3 Sticky Sample Mean

French, Spanish, and Italian are 3 popular Romance languages sharing many
commonalities. Latin was the language of the ancient boot peninsula and used
prominently by the Roman Catholic church for centuries. However, with no
speakers today, Latin is considered a dead language. In this case, we may
expect many observations at the origin and no starkly different language.

Language | Mean Distance 1 | Mean Distance 2
---|---|---
French | -0.51 | -0.76
Spanish | -0.74 | -0.35
Italian | -0.11 | -0.05

Since none of the means are positive, the sample mean is considered to be
sticky, and the barycenter is at 0. From that mean of zero, the variances of
each sample are calculated to be 2.88 and 2.30 respectively.

###  4.4 Another Sticky Sample Mean

So far, two cases have been considered where one language has been from a
different ancestor and where all three languages are from the same ancestor.
This case will consider all three languages from different ancestors. German,
French, and Irish (Gaelic, from the Celtic ancestor) words from the Swadesh
207 list were used from the earlier procedures to find the following mean
distances.

Language | Mean Distance 1 | Mean Distance 2
---|---|---
German | -0.74 | -0.03
French | -0.46 | -0.97
Irish | -0.40 | -0.85

In both samples, all 3 means are negative values indicating a sticky sample
mean to be placed at the origin. From that mean of zero, the sample variances
are calculated to be 4.88 and 5.45, respectively. Unlike the previous case of
a sticky sample mean where the languages were considered to be of the same
ancestor, here, the languages are believed to all be of different ancestors.

##  5 Conclusions

The clustering of language may be done through the use of the 3-spider. In the
three applications shown, a non-sticky mean was observed when one language was
from a different family while a sticky mean was observed when all languages
have the same ancestry or all languages has different ancestry. English and
German are from the Germanic family while French, Italian, and Spanish are
from the Latin (Romance, Italic) family while German, French, and Irish are
from different families. For these examples, the samples representing two
populations were simply randomly assigned, but if two groups of words were
different for some reason, say medical terms versus agricultural terms, then
perhaps some significant difference may be found in the ancestry as a
different leaf may dominate in each case. Other applications may be more
apparent to a studied linguist. While these results may not apply to all
languages, it provides a start. Furthermore, inference may be done using the
Central Limit Theorems for each case.
The distance calculated between each language was also exceptionally simple.
An edit distance for full words may provide additional insights. While the
Swadesh list of words has been the standard for linguistics, phrases or
sentences introducing structure could be of interest for investigation. The
languages in this study all used the Latin alphabet which is extremely common
in Western civilization, but it is not the only alphabet as Cyrillic, Greek,
and Arabic among others could be considered. Paradoxically, although while
here compare Indo-European languages, our primary tool, the Latin alphabet
sources from the Proto-Sinaitic script developed from the popular non Indo-
European language of Semitic-speaking workers and slaves in the antique Egypt,
as they were unskilled to handle the hieroglyphic system used in the Egyptian
language, that required a large number of pictograms, as opposed to the
semantic values, of the own Canaanite language of slaves. This Semitic
alphabet, ancestor of multiple writing systems across the Middle East, imposed
itself having only about half as many letters as the Vedic Sanskrit alphabet,
who appeared around the same period. The written language only was considered
regardless of pronunciation which would need the International Phonetic
Alphabet (IPA) instead of the Latin Script. With the addition of more than 3
languages, additional types of phylogenetic trees beyond the 3-Spider would
need to be considered. This grows the number of possible trees quickly
following from the work by Billera and Holmes necessitating additional
computational power and theory.

##  6 Acknowledgements

The authors acknowledge support by the National Science Foundation under Grant
DMS:2311059.

##  7 R Code with Data

Please access the R file at the following github repository:
https://github.com/GarettO9/lang_tree/tree/main.

The code contains the data and commands with a set seed for random selection
to reproduce the results.

## References

  * [1] John Algeo and Thomas Pyles.  The Origins and Development of the English Language.  Wadsworth, 6th edition, 2010.
  * [2] Anonymous.  The indo-european language tree.  2024\.
  * [3] J. David Archibald.  Edward hitchcock’s pre-darwinian (1840) “tree of life”.  Journal of the History of Biology, 42(3):561–592, Sep 2008.
  * [4] Stephen W. Attwood, Sarah C. Hill, David M. Aanensen, Thomas R. Connor, and Oliver G. Pybus.  Phylogenetic and phylodynamic approaches to understanding and combating the early SARS-COV-2 pandemic.  Nature Reviews Genetics, 23(9):547–562, Apr 2022.
  * [5] Augustine, Balthasar Hans Urs von, William S. Babcock, Rachel Coleman, and Augustine.  The City of God.  Ignatius Press, 2021.
  * [6] R.N. Bhattacharya and V. Patrangenaru.  Large sample theory of intrinsic and extrinsic sample means on manifolds-part one.  The Annals of Statistics, 31:1–29, 2003.
  * [7] Louis J. Billera, Susan P. Holmes, and Karen Vogtmann.  Geometry of the space of phylogenetic trees.  Advances in Applied Mathematics, 27(4):733–767, 2001.
  * [8] Manuel Binet, Olivier Gascuel, Celine Scornavacca, Emmanuel J. P. Douzery, and Fabio Pardi.  Fast and accurate branch lengths estimation for phylogenomic trees.  BMC Bioinformatics, 17(1), Jan 2016.
  * [9] Yen-Ju Chen, Jason C. Huang, Ching-Ping Yang, Kuo-Feng Hsu, and Hsin-Fu Liu.  A comprehensive phylogenetic analysis of SARS-COV-2: Utilizing a novel and convenient in-house rt-pcr method for characterization without virus culture and bsl-3 facilities.  Viruses, 15(7):1562, Jul 2023.
  * [10] C. Douglas Chrétien.  The mathematical models of glottochronology.  Language, 38(1):11–37, 1962.
  * [11] David Crystal and Yu Alan C L.  A dictionary of linguistics and phonetics.  Wiley and Sons, Inc, 2024.
  * [12] Charles Darwin.  The origin of species, by means of natural selection or the preservation of favoured races in the struggle for life.  William Collins Classics, 2024.
  * [13] I. Dyen.  A Lexicostatistical Classification of the Austronesian Languages.  Number no. 19 in A Lexicostatistical Classification of the Austronesian Languages. Waverly Press, 1965.
  * [14] Isidore Dyen, Joseph B. Kruskal, and Paul Black.  An indoeuropean classification: A lexicostatistical experiment.  Transactions of the American Philosophical Society, 82(5):iii–132, 1992.
  * [15] A.W.F. Edwards.  Abstracts of papers.  Annals of Human Genetics, 27(1):103–105, 1963.
  * [16] Russell D. Gray and Quentin D. Atkinson.  Language-tree divergence times support the anatolian theory of indo-european origin.  Nature, 426(6965):435–439, Nov 2003.
  * [17] Thomas Hotz, Stephan Huckemann, Huiling Le, James S. Marron, Jonathan C. Mattingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru, and Sean Skwerer.  Sticky central limit theorems on open books.  Annals of Applied Probability, 23:2238–2258, 2013.
  * [18] Thomas Hotz, Sean Skwerer, Stephan Huckemann, Huiling Le, J. S. Marron, Jonathan C. Mattingly, Ezra Miller, James Nolen, Megan Owen, and Vic Patrangenaru.  Sticky central limit theorems on open books.  The Annals of Applied Probability, 23(6):2238 – 2258, 2013.
  * [19] Daniel H. Huson and David Bryant.  Application of Phylogenetic Networks in Evolutionary Studies.  Molecular Biology and Evolution, 23(2):254–267, 10 2005.
  * [20] Dell Hymes.  Speech and language: On the origins and foundations of inequality among speakers.  Daedalus, 102(3):59–85, 1973.
  * [21] Richard Arnold Johnson and Dean W. Wichern.  Applied Multivariate Statistical Analysis.  Pearson, 6th edition, 2019.
  * [22] A. L. Kroeber and C. D. Chrétien.  Quantitative classification of indo-european languages.  Language, 13(2):83–103, 1937.
  * [23] Robert B. Lees.  The basis of glottochronology.  Language, 29(2):113–127, Apr 1953.
  * [24] Tingting Li, Dongxia Liu, Yadi Yang, Jiali Guo, Yujie Feng, Xinmo Zhang, Shilong Cheng, and Jie Feng.  Phylogenetic supertree reveals detailed evolution of SARS-COV-2.  Scientific Reports, 10(1), Dec 2020.
  * [25] Bob Mau, Michael A. Newton, and Bret Larget.  Bayesian Phylogenetic Inference via Markov Chain Monte Carlo Methods.  Biometrics, 55(1):1–12, 05 2004.
  * [26] Brent D. Mishler and John S. Wilkins.  The hunting of the snarc: A snarky solution to the species problem.  Philosophy, Theory, and Practice in Biology, 10(20220112), Mar 2018.
  * [27] Jerzy Neyman.  Molecular studies of evolution: A source of novel statistical problems**this investigation was supported in part by research grant gm 10525-08 from the national institutes of health, public health service.  In Shanti S. Gupta and James Yackel, editors, Statistical Decision Theory and Related Topics, pages 1–27. Academic Press, 1971.
  * [28] Emmanuel Paradis.  The distribution of branch lengths in phylogenetic trees.  Molecular Phylogenetics and Evolution, 94:136–145, 2016.
  * [29] Bruce Rannala and Ziheng Yang.  Probability distribution of molecular evolutionary trees: A new method of phylogenetic inference.  Journal of Molecular Evolution, 43(3):304–311, Sep 1996.
  * [30] D.F Robinson.  Comparison of labeled trees with valency three.  Journal of Combinatorial Theory, Series B, 11(2):105–119, 1971.
  * [31] Merritt Ruhlen.  The origin of language: Tracing the evolution of the mother tongue.  John Wiley &; Sons, 1994.
  * [32] Dennis K. Pearl Shuying Li and Hani Doss.  Phylogenetic tree construction using markov chain monte carlo.  Journal of the American Statistical Association, 95(450):493–508, 2000.
  * [33] Morris Swadesh.  Lexico-statistic dating of prehistoric ethnic contacts: With special reference to north american indians and eskimos.  Proceedings of the American Philosophical Society, 96(4):452–463, 1952.
  * [34] Morris Swadesh.  Towards greater accuracy in lexicostatistic dating.  International Journal of American Linguistics, 21(2):121–137, 1955.
  * [35] Morris Swadesh, Dell H. Hymes, and Joel Sherzer.  The origin and diversification of language.  Aldine-Atherton, 1971.
  * [36] Alfredo Trombetti.  L’unita d’origine del Linguaggio.  Civitas Dei, 1962.
  * [37] unknown.  Category:swadesh modules, 2021.  [Online; accessed 10-October-2023].
  * [38] Roger D. Woodard.  The ancient languages of Europe.  Cambridge Univ. Press, 2010.
  * [39] Ning Zhao, Nan Zhou, Huafeng Fan, Jie Ding, Xingyu Xu, Xiaoqing Dong, Xiaoxiao Dong, Dandan Xu, Xiaoyu Min, Yan Yu, and et al.  Mutations and phylogenetic analyses of SARS-COV-2 among imported covid-19 from abroad in Nanjing, China.  Frontiers in Microbiology, 13, Mar 2022.