Computational approaches to historical linguistics have been proposed for half a century. Within the last decade, this line of research has received a major boost, owing both to the transfer of ideas and software from computational biology and to the release of several large electronic data resources suitable for systematic comparative work. In this article, some of the central research topics of this new wave of computational historical linguistics are introduced and discussed. These are automatic assessment of genetic relatedness, automatic cognate detection, phylogenetic inference and ancestral state reconstruction. They will be demonstrated by means of a case study of automatically reconstructing a Proto-Romance word list from lexical data of 50 modern Romance languages and dialects. The results illustrate both the strengths and the weaknesses of the current state of the art of automating the comparative method.
Historical linguistics is the oldest sub-discipline of linguistics, and it constitutes an amazing success story. It gave us a clear idea of the laws governing language change, as well as detailed insights into the languages – and thus the cultures and living conditions – of prehistoric populations which left no written records. The diachronic dimension of languages is essential for a proper understanding of their synchronic properties. Also, the findings from historical linguistics are an important source of information for other fields of prehistory studies, such as archaeology, paleoanthropology and, in recent years, paleogenetics (Renfrew 1987; Pietrusewsky 2008; Anthony 2010; Haak et al. 2015, and many others).
The success of historical linguistics is owed to a large degree to a collection of very stringent methodological principles that go by the name of the comparative method (Meillet 1954; Weiss 2015). It can be summarized by the following workflow (from Ross and Durie 1996: 6–7):
Determine on the strength of diagnostic evidence that a set of languages are genetically related, that is, that they constitute a ‘family’.
Collect putative cognate sets for the family (both morphological paradigms and lexical items).
Work out the sound correspondences from the cognate sets, putting ‘irregular’ cognate sets on one side.
Reconstruct the protolanguage of the family as follows:
Reconstruct the protophonology from the sound correspondences worked out in (3), using conventional wisdom regarding the directions of sound changes.
Reconstruct protomorphemes (both morphological paradigms and lexical items) from the cognate sets collected in (2), using the protophonology reconstructed in (4a).
Establish innovations (phonological, lexical, semantic, morphological, morphosyntactic) shared by groups of languages within the family relative to the reconstructed protolanguage.
Tabulate the innovations established in (5) to arrive at an internal classification of the family, a ‘family tree’.
Construct an etymological dictionary, tracing borrowings, semantic change and so forth, for the lexicon of the family (or of one language of the family).
In practice, it is not applied in a linear, pipeline-like fashion. Rather, the results of each intermediate step are subsequently used to inform earlier as well as later steps. This workflow is graphically depicted in Figure 1.
The steps (2)–(7) each involve a systematic, almost mechanical comparison and evaluation of many options such as cognacy relations, proto-form reconstructions or family trees. The first step, establishing genetic relatedness, is less regimented, but it generally involves a systematic comparison of many variables from multiple languages as well. It is therefore not surprising that there have been many efforts to formalize parts of this workflow to a degree sufficient to implement it on a computer.
Lexicostatistics (e.g. Swadesh 1952; Swadesh 1955 and much subsequent work) can be seen as an early attempt to give an algorithmic rendering of step (6), even though it predates the computer age. Since the 1960s, several scholars applied computational methods within the overall framework of lexicostatistics (cf. e.g. Embleton 1986, inter alia). Likewise, there have been repeated efforts for computational treatments of other aspects of the comparative method, such as (Ringe 1992; Baxter and Manaster Ramer 2000; Kessler 2001) for step (1), (Kay 1964) for step (2), (Kondrak 2002) for steps (2) and (3), (Lowe and Mazaudon 1994) for steps (2) and (4), (Oakes 2000) for steps (2)–(7), (Covington 1996) for step (3), and (Ringe et al. 2002) for step (6), to mention just a few of the earlier contributions.
While the mentioned proposals mostly constitute isolated efforts of historical and computational linguists, the emerging field of computational historical linguistics received a major impetus since the early 2000s by the work of computational biologists such as Alexandre Bouchard-Côté, Russell Gray, Robert McMahon, Mark Pagel or Tandy Warnow and co-workers, who applied methods from their field to the problem of the reconstruction of language history, often in collaboration with linguists. This research trend might be dubbed computational phylogenetic linguistics as it heavily draws on techniques of phylogenetic inference from computational biology (Gray and Jordan 2000; Gray and Atkinson 2003; McMahon and McMahon 2005; Pagel et al. 2007; Atkinson et al. 2008; Gray et al. 2009; Dunn et al. 2011; Bouckaert et al. 2012; Bouchard-Côté et al. 2013; Pagel et al. 2013; Hruschka et al. 2015).
In recent years, more and more large collections of comparative linguistic data have become available in digital form, giving the field another boost. The following list gives a sample of the most commonly used databases; it is necessarily incomplete as new data sources are continuously made public.
Cognate-coded word lists
Indo-European Lexical Cognacy Database (IELex; ielex.mpi.nl): collection of 225-concept Swadesh lists from 163 Indo-European languages (based on Dyen et al. 1992). Entries are given in orthography with manually assigned cognate classes; for part of the entries, IPA transcriptions are given.
Austronesian Basic Vocabulary Database (ABVD; Greenhill et al. 2008; language.psy.auckland.ac.nz/austronesian): collection of 210-item Swadesh lists for 1,467 languages from the Pacific region, mostly belonging to the Austronesian language family. Entries are given in phonetic transcription with manually assigned cognate classes.
Phonetically transcribed word lists
ASJP database (compiled by the Automatic Similarity Judgment Program; Wichmann et al. 2016; asjp.clld.org): collection of word lists for 7,221 doculects (languages and dialects) over 40 concepts (100-item word lists for ca. 300 languages); entries are given in phonetic transcription.
Grammatical and typological classifications
World Atlas of Language Structure (Haspelmath et al. 2008; wals.info): manual expert classifications of 2,679 languages and dialects according to 192 typological features.
Syntactic Structures of the World’s Languages (sswl.railsplayground.net): Classification of 274 languages according to 148 syntactic features.
Expert language classifications
Glottolog (Hammarström et al. 2016; glottolog.org): genetic classification of 7,943 languages and dialects, alongside with information about geographic locations and extensive bibliographic references
Additionally there is a growing body of publicly available diachronic corpora of various languages, as well as studies using agent-based simulations of language change. The focus of this article is on computational work inspired by the comparative method, so this line of work will not further be covered here.
2 A program for computational historical linguistics
Conceived in a broad sense, computational historical linguistics comprises all efforts deploying computational methods to answer questions about the history of natural languages. As spelled out above, there is a decade-old tradition of this kind of research.
In this article, however, the term will be used in a rather narrower sense to describe an emerging subfield which has reached a certain degree of convergence regarding research goals, suitable data source and computational methods and tools to be deployed. I will use the abbreviation CHL to refer to computational historical linguistics in this narrow sense. The following remarks strive to describe this emerging consensus. They are partially programmatic in nature though; not all researchers active in this domain will agree with all of them.
CHL is informed by three intellectual traditions:
the comparative method of classical historical linguistics,
computational linguistics in general, especially modern statistical natural language processing (NLP).
CHL shares, to a large degree, the research objectives of the comparative method. The goal is to reconstruct the historical processes that led to the observed diversity of extant or documented ancient languages. This involves, inter alia establishing cognacy relations between words and morphemes, identifying regular sound correspondences, inferring family trees (phylogenetic trees or simply phylogenies in the biology-inspired terminology common in CHL), reconstructing proto-forms and historical processes such as sound laws and lexical innovations. CHL also shares some limitations of the comparative method, e.g. the idealized assumptions that language diversification proceeds in a tree-like fashion and that language contact is not systematically modeled.
CHL’s guiding model is adapted from computational biology, drawing on the observation going back to Schleicher and Darwin that the processes of biological and of linguistic diversification are analogous. The history of a group of languages is represented by a phylogenetic tree (including branch lengths), with observed linguistic varieties at the leafs of the tree. Splits in a tree represent diversification events, i.e. the separation of an ancient language into daughter lineages. Language change is conceptualized as a continuous-time Markov process applying to discrete, finite-values characters. (Details will be spelled out below.) Inference amounts to finding the model (a phylogenetic tree plus a parameterization of the Markov process) that best explains the observed data.
Last but not least, CHL adopts techniques and methodological guidelines from statistical NLP. The pertinent computational tools, such as string comparison algorithms, to a certain degree overlap with those inspired by computational biology. Equally important are certain methodological standards from NLP and machine learning.
Generally, work in CHL is a kind of inference, where a collection of data is used as input (premises) to produce output data (conclusions). Input data can be phonetically or orthographically transcribed word lists, pairwise or multiply aligned word lists, grammatical feature vectors etc. Output data are for instance cognate class labels, alignments, phylogenies or proto-form reconstructions. Inference is performed by constructing a model and training its parameters. Following the standards in statistical NLP, the following guiding principles are desirable when performing inference:
Replicability. All data used in a study, including all manual pre-processing steps, are available to the scientific community. Likewise, each computational inference step is either documented in sufficient detail to enable re-implementation, or made available as source code.
Rigorous evaluation. The quality of the inference, or goodness of fit of the trained model, is evaluated by applying a well-defined quantitative measure to the output of the inference. This measure is applicable to competing models for the same inference task, facilitating model comparison and model selection.
Separation of training and test data. Different data sets are used for training and evaluating a model.
Only raw data as input. Only such data are used as input for inference that can be obtained without making prior assumptions about the inference task. For instance, word lists in orthographic or phonetic transcription are suitable as input if the transcriptions were produced without using diachronic information.
The final criterion is perhaps the most contentious one. It excludes, for instance, the use of orthographic information in languages such as English or French for training purposes, as the orthographic conventions of those languages reflect the phonetics of earlier stages. Also, it follows that the cognate class labels from databases such as IELex or ABVD, as well as expert classifications such as Ethnologue or Glottolog, are unsuitable as input for inference and should only be used as gold standard for training and testing.
Conceived this way, CHL is much narrower in scope than, e.g. computational phylogenetic linguistics. For instance, inference about the time depth and homeland of language families (such as Gray and Atkinson 2003; Bouckaert et al. 2012) is hard to fit into this framework as long as there are no independent test data to evaluate models against (but see Rama 2013). Also, it is common practice in computational phylogenetic linguistics to use manually collected cognate classifications as input for inference (Gray and Jordan 2000; Gray and Atkinson 2003; Pagel et al. 2007; Atkinson et al. 2008; Gray et al. 2009; Dunn et al. 2011; Bouckaert et al. 2012; Bouchard-Côté et al. 2013; Pagel et al. 2013; Hruschka et al. 2015). While the results obtained this way are highly valuable and insightful, they are not fully replicable, since expert cognacy judgments are necessarily subjective and variable. Also, the methods used in the work mentioned do not generalize easily to understudied language families, since correctly identifying cognates between distantly related languages requires the prior application of the classical comparative method, and the necessary research has not been done with equal intensity for all language families.
3 A case study: reconstructing Proto-Romance
In this section, a case study will be presented that illustrates many of the techniques common in current CHL. Training data are 40-item word lists from 50 Romance (excluding Latin) and 3 Albanian languages and dialects in phonetic transcription from the ASJP database (Wichmann et al. 2016) (version 17, accessed on August 2, 2016 from asjp.clld.org/static/download/asjp-dataset.tab.zip). The inference goal is the reconstruction of the corresponding word list from the latest common ancestor of the Romance languages and dialects (Proto-Romance, i.e. some version of Vulgar Latin). The results will be tested against the Latin word lists from ASJP. A subset of the data used is shown in Table 1 for illustration.
The phonetic transcriptions use the 41 ASJP sound classes (cf. Brown et al. 2013). Diacritics are removed. If the database lists more than one translation for a concept in a given language, only the first one is used.
The following steps will be performed (mirroring to a large degree the steps of the comparative method):
Demonstrate that the Romance languages and dialects are related.
Compute pairwise string alignments and string similarities between synonymous words from different languages/dialects.
Cluster the words for each concept into automatically inferred cognate classes.
Infer a phylogenetic tree (or a collection of trees).
Perform ancestral state reconstruction for cognate classes to infer the cognate class of the Proto-Romance word for each concept.
Perform multiple sequence alignment of the reflexes of those cognate classes within the Romance languages and dialects.
Perform ancestral state reconstruction to infer the state (sound class or gap) of each column in the multiple sequence alignments.
Compare the results to the Latin ASJP word list.
3.1 Demonstration of genetic relationship
In (Jäger 2013), a dissimilarity measure between ASJP word lists is developed. Space does not permit to explain it in any detail here. Suffice it to say that this measure is based on the average string similarity between the corresponding elements of two word lists while controlling for the possibility of chance similarities. Let us call this dissimilarity measure between two word lists the PMI distance, since it makes crucial use of the pointwise mutual information (PMI) between phonetic strings.
To demonstrate that all Romance languages and dialects used in this study are mutually related, I will use the ASJP word lists from Papunesia, i.e. “all islands between Sumatra and the Americas, excluding islands off Australia and excluding Japan and islands to the North of it” (Hammarström et al. 2016) as training data and the ASJP word lists from Africa as test data. Input for inference are PMI distances between pairs of languages/dialect, and the output is the classification of this pair as related or unrelated, where two doculects count as related if they belong to the same language family according to the Glottolog classification. The distribution of PMI distances are shown in Figure 2. The graphics illustrates that all doculect pairs with a PMI distance ≤ 0.75 are, with a very high probability, related. The largest PMI distance among Romance dialects (between Aromanian and Nones) is 0.65.
A statistical test confirms this impression. I fitted a cumulative density estimation for the PMI distances of the unrelated doculect pairs from the training data, using the R-package logspline (Kooperberg 2016). If a pair of doculects has a PMI distance d, the value of the cumulative density function for d can then be interpreted as the (one-sided) p-values for the null hypothesis that the doculects are unrelated.
Using a threshold of α = 0.0001, I say that a doculect pair is predicted to be related if the model predicts it to be unrelated with a probability ≤ α. In Table 2, the predictions are tabulated against the Glottolog gold standard.
These results amount to ca. 0.3% of false positives and ca. 84% of false negatives. This test ensures that the chosen model and threshold is sufficiently conservative to keep the risk of wrongly assessing doculects to be related small. Since the method is so conservative, it produces a large amount of false negatives.
In the next step, I compute the probability of all pairs of Romance doculects to be unrelated, using the model obtained from the training data. Using the Holm–Bonferroni method to control for multiple tests, the highest p-value for the null hypothesis that the doculect pair in question is unrelated is 1.8 × 10 - 5, i.e. all adjusted p-values are < α. We can therefore reject the null hypothesis for all Romance doculect pairs.
3.2 Pairwise string comparison
All subsequent steps rely on a systematic comparison of words for the same concept from different doculects. Let us consider as an example the words for water from Catalan and Italian from ASJP, “aigua” and “acqua”. Both are descendants of the Latin “aqua” (Meyer-Lübke 1935: 46). In ASJP transcription, these are aixw~3 and akwa. The sequence w~ in the Catalan word encodes a diacritic (indicating labialization of the preceding segment) and is removed in the subsequent processing steps.
A pairwise sequence alignment of two strings arranges them in such a way that corresponding segments are aligned, possibly inserting gap symbols for segments in one string that have no correspondent in the other string. For the example, the historically correct alignment would arguably be as follows:
In this study, the quality of a pairwise alignment is quantified as its aggregate pointwise mutual information (PMI). (See List 2014 for a different approach.) The PMI between two sound classes a,b is defined as
Here s(a,b) is the probability that a is aligned to b in a correct alignment, and q(a),q(b) are the probabilities of occurrence of a and b in a string. If one of the two symbols is a gap, the PMI score is a gap penalty. I use affine gap penalties, i.e. the gap penalty is reduced if the gap is preceded by another gap. The captures the heuristics that both addition and deletion of phonetic material often targets several consecutive segments.
If the PMI scores for each pair of sound classes and the gap penalties are known, the best alignment between two strings (i.e. the alignment maximizing the aggregate PMI score) can efficiently be computed using the Needleman–Wunsch algorithm (Needleman and Wunsch 1970).
The quantities s(a,b) and q(a),q(b) must be estimated from the data. Here I follow a simplified version of the parameter estimation technique from (Jäger 2013). In a first step, I set
Also, I set the initial gap penalties to - 1. (This amounts to Levenshtein alignment.) For instance, the alignment for the example given above would come out as
This alignment has one identity (the initial a) and three non-identities, so the PMI0 score is - 3.
Using these parameters, all pairs of words for the same concept from different doculects are aligned.
From those alignments, s(a,b) is estimated as the relative frequency of a and b being aligned among all non-gap alignment pairs, while q(a) is estimated as the relative frequency of sound class a in the data. The PMI scores are then estimated using equation (1). For the gap penalties I used the values from (Jäger 2013), i.e. - 2.49 for opening gaps and - 1.70 for extending gaps (i.e. for gaps preceded by another gap). Using those parameters, all synonymous word pairs are realigned.
In the next step, only word pairs with an aggregate PMI score ≥ 4.45 are used. (This threshold is taken from Jäger 2013 as well.) Those word pairs are realigned and the PMI scores are reestimated. This step is repeated ten times.
The threshold of 4.45 is rather strict; almost all word pairs above this threshold are either cognates or loans. For instance, for the language pair Italian/Albanian, the only translation pair with a higher PMI score is Italian peSe/Albanian peSk (“fish”), where the former is a descendant and the latter a loan from Latin piscis (cf. http://ielex.mpi.nl). For Spanish/Romanian, two rather divergent Romance languages, we find eight such word pairs. They are shown alongside with the inferred alignments in Table 3.
The aggregate PMI score for the best alignment between two strings is a measure for the degree of similarity between the strings. I will call it the PMI similarity henceforth.
3.3 Cognate clustering
Automatic cognate detection is an area of active investigation in CHL (Dolgopolsky 1986; Bergsma and Kondrak 2007; Hall and Klein 2010; Turchin et al. 2010; Hauer and Kondrak 2011; List 2012; List 2014; Rama 2015; Jäger and Sofroniev 2016; Jäger et al. 2017, inter alia). For the present study, I chose a rather simple approach based on unsupervised learning.
Figure 3 shows the PMI similarities for words from different doculects having different or identical meanings. Within our data, synonymous word pairs are, on average, more similar to each other than non-synonymous ones. The most plausible explanation for this effect is that the synonymous word pairs contain a large proportion of cognate pairs. Therefore “identity of meaning” will be used as a proxy for “being cognate”. The implicit assumption underlying this procedure is that cognate words always have the same meaning. This is evidently false when considering the entire lexicon. There is a plethora of examples, such as English deer vs. German Tier “animal”, which are cognate cf. (Kroonen 2013: 94) without being synonyms. However, within the 40-concept core vocabulary space covered by ASJP, such cross-concept cognate pairs are arguably very rare.
For each concept, a weighted graph is constructed, with the words denoting this concept as vertices. Two vertices are connected if the predicted probability of these words to be synonymous (based on their PMI similarity and the logistic regression model) is ≥ 0.5. The weight of each edge equals the predicted probabilities. The nodes of the graph are clustered using the weighted version of the Label Propagation algorithm (Raghavan et al. 2007) as implemented in the igraph software (Csardi and Nepusz 2006). As a result, a class label is assigned to each word. Non-synonymous words never carry the same class label. Table 4 illustrates the resulting clustering for the concept “person” and a subset of the doculects. A manual inspection reveals that the automatic classification does not completely coincide with the cognate classification a human expert would assume. For instance, the descendants of Latin homo are split into classes 1, 2, 5, and 7. Also, Gheg Albanian 5eri and Sardinian omini have the same label but are not cognate.
Based on evaluations against manually assembled cognacy judgments for different but similar data (Jäger and Sofroniev 2016; Jäger et al. 2017), we can expect an average F-score of 60%–80% for automatic cognate detection. This means that on average, for each word, 60%–80% of its true cognates are assigned the same label, and 60%–80% of the words carrying the same label are genuine cognates.
3.4 Phylogenetic inference
3.4.1 General remarks
A phylogenetic tree (or simply phylogeny) is a data structure similar to family trees according to the comparative method, but there are some subtle but important differences between those concepts. Like a family tree, a phylogeny is a tree graph, i.e. an acyclic graph. If one node in the graph is identified as root, the phylogeny is rooted; otherwise it is unrooted. The branches (or edges) of a phylogeny have non-negative branch lengths. A phylogeny without branch length is called topology.
Nodes with a degree 1 (i.e. nodes which are the endpoint of exactly one branch) are called leaves or tips. They are usually labeled with the names of observed entities, such as documented languages. Nodes with a degree > 1 are the internal nodes. If the root (if present) has degree 2 and all other internal nodes have degree 3 (i.e. one mother node and two daughter nodes), the phylogeny is binary-branching. Most algorithms for phylogenetic inference produce binary-branching trees.
Like a linguistic family tree, a rooted phylogeny is a model of the historic process leading to the observed diversity between the objects at the leaves. Time flows from the root to the leaves. Internal nodes represent unobserved historical objects, such as ancient languages. Branching nodes represent diversification events, i.e. the splitting of a lineage into several daughter lineages.
The most important difference between family trees and phylogenies is the fact that the latter have branch lengths. Depending on the context, these lengths may represent two different quantities. They may capture the historic time (measured in years) between diversification events, or they indicate the amount of change along the branch, measured for instance as the expected number of lexical replacements or the expected number of sound changes. The two interpretations only coincide if the rate of change is constant. This assumption is known to be invalid for language change (cf. e.g. the discussion in McMahon and McMahon 2006).
Another major difference, at least in practice, between family trees and phylogenies concerns the type of justification that is expected for the stipulation of an internal node. According to the comparative method, such a node is justified if and only if a shared innovation can be reconstructed for all daughter lineages of this node. Consequently, family trees obtained via the comparative method often contain multiply branching nodes because the required evidence for further subgrouping is not available. Phylogenies, in contradistinction, are mostly binary branching, at least in practice. Partially this is a matter of computational convenience. Since the set of binary branching trees is a proper subset of the set of all trees, this reduces the search space. Also, algorithms working recursively leaves-to-root can be formulated in a more efficient way if all internal nodes are known to have at most two daughters. Furthermore, the degree of justification of a certain topology is evaluated globally, not for each internal node individually. In the context of phylogenetic inference, it is therefore not required to identify shared innovations for individual nodes.
There is a large variety of algorithms from computational biology to infer phylogenies from observed data. The overarching theme of phylogenetic inference is that a phylogeny represents (or is part of) a mathematical model explaining the observed variety. There are criteria quantifying how good an explanation a phylogeny provides for observed data. Generally speaking, the goal is to find a phylogeny that provides an optimal explanation for the observed data. The most commonly used algorithms are (in ascending order of sophistication and computational costs) Neighbor Joining (Saitou and Nei 1987) and its variant BIONJ (Gascuel 1997), FastMe (Desper and Gascuel 2002), Fitch-Margoliash (Fitch and Margoliash 1967), Maximum Parsimony (Fitch 1971), Maximum Likelihood and Bayesian Phylogenetic Inference (cf. Chen et al. 2014 for an overview).
The latter two approaches, Maximum Likelihood and Bayesian Phylogenetic Inference, are based on a probabilistic model of language change. To apply them, a language has to be represented as a character vector. A character is a feature with a finite number of possible values, such as “order of verb and object”, “the first person plural pronoun contains a dental consonant” or what have you. In most applications, characters are binary, with “0” and “1” as possible values. In what follows, we will assume all characters are binary.
Diachronic change of a character value is modeled as a continuous time Markov process. At each point in time, a character can spontaneously switch to the other value with a fixed probability density. A two-state process is characterized by two parameters, r and s, where r is the rate of change of (the probability density of a switch to 1 if the current state is 0) and s the rate of change for . For a given time interval of length t, the probability of being in state i at the start of the interval and in state j at the end is then given by P(t)ij, where
For instance, the cell in the upper right corner of the matrix, multiplied with the coefficient , gives the probability that the system is in state 1 after a time interval of length t if it is in state 0 at the beginning of the interval. This probability is a function of t that starts at 0 for t = 0 and exponentially converges towards as t gets longer. The cell in the upper left corner gives a probability that the system is still or again in state 0 after time t if it starts in state 0. (The possibility of multiple switches occurring during the interval is factored in.) For t = 0, this probability is 1, and it exponentially converges to as t grows to infinity. The same holds, mutatis mutandis, for the second row.
If the state at the end of the interval t is known – for instance by observation –, one can try to probabilistically estimate the state at the beginning of the interval. However, the longer t is, the less precise will this estimate be. If t grows to infinity, the amount of information that can be inferred about the initial state converges to 0.
A probabilistic model for a given set of character vectors is a phylogenetic tree (with the leaves indexed by the characters vectors) plus a mapping from edges to rates (r,s) for each character and a probability distribution over character values at the root for each character.
Suppose we know not only the character states at the leaves of the phylogeny but also at all internal nodes. The likelihood of a given branch is then given by P(t)ij, where i and j are the states at the top and the bottom of the branch respectively, and t is the length of the branch. The likelihood of the entire phylogeny for a given character is then the product of all branch likelihoods, multiplied with the probability of the root state. The total likelihood of the phylogeny is the product of its likelihoods for all characters.
If only the character values for the leaves are known, the likelihood of the phylogeny given the character vectors at the leaves is the sum of its likelihoods for all possible state combinations at the internal nodes.
This general model is very parameter rich since for each branch and each character, a pair of rates has to be specified. There are various ways to reduce the degrees of freedom. The simplest method is to assume that rates are constant across branches and characters, and that the root probability of each character equals the equilibrium probabilities of the Markov process: . More sophisticated approaches assume that rates vary across characters and across branches according to some parameter-poor probability distribution, and the expected likelihood of the tree is obtained by integrating over this distribution. For a detailed mathematical exposition, the interested reader is referred to the relevant literature from computational biology, such as Ewens and Grant (2005).
A parameterized model, i.e. a phylogeny plus rate specifications for all characters and branches, and root probabilities for each character, assigns a certain likelihood to the observed character vectors. Maximum Likelihood (ML) inference searches for the model that maximizes this likelihood given the observations. While the optimal numerical parameters of a model, i.e. branch lengths, rates and root probabilities, can efficiently be found by standard optimization techniques, finding the topology that gives rise to the ML-model is computationally hard. Existing implementations use various heuristics to search the tree space and find some local optimum, but there is no guarantee that the globally optimal topology is found.
Bayesian phylogenetic inference requires some suitable prior probability distributions over models (i.e. topologies, branch lengths, rates, possibly rate variations across characters and rate variation across branches) and produces a sample of the posterior distribution over models via a Markov Chain Monte Carlo simulation.
3.4.2 Application to the case study
For the case study, doculects were represented by two types of binary characters:
Inferred class label characters (cf. Subsection 3.3). Each inferred class label is a character. A doculect has value 1 for such a character if and only if its word list contains a word carrying this label.
Soundclass-concept characters. There is one character for each pair (s,c) of a sound class s and a concept c. A doculect has value 1 for that character if and only if its word list contains a word w for c that contains s in its transcription.
Both types of characters carry a diachronic signal. For instance, the mutation for class label 6/concept person (cf. Table 4) represents a lexical replacement of Latin “homo” or “persona” by descendants of Latin “christianus” in some Romance dialects (Meyer-Lübke 1935: 179). The mutation for the soundclass-concept character k/person represents the same historical process. Soundclass-concept characters, however, also capture sound shifts. For instance, the mutation for b/person reflects the epenthetic insertion of b in descendants of Latin “homo” in some Iberian dialects.
I performed Bayesian phylogenetic inference on those characters. The inference was carried out using the Software MrBayes (Ronquist and Huelsenbeck 2003). Separate rate models were inferred for the two character types. Rate variation across characters was modeled by a discretized Gamma distribution using four rate categories. I assumed no rate variation across edges. Root probabilities were identified with equilibrium probabilities. An ascertainment correction for missing all-0 characters was used.
I assumed rates to be constant across branches. This entails that the fitted branch lengths reflect the expected amount of change (i.e. the expected number of mutations) along that branch.
In such a model, the likelihood of a phylogeny does not depend on the location of the root (the assumed Markov process is time reversible.) Therefore, phylogenetic inference provides no information about the location of the root. This motivates the inclusion of the Albanian doculects. Those doculects were used as outgroup, i.e. the root was placed on the branch separating the Albanian and the Romance doculects.
I obtained a sample of the posterior distribution containing 2,000 phylogenies. Figure 4 displays a representative member of this sample (the maximum clade credibility tree). The labels at the nodes indicate posterior probabilities of that node, i.e. the proportion of the phylogenies in the posterior sample having the same sub-group.
These posterior probabilities are mostly rather low, indicating a large degree of topological variation in the posterior sample. Some subgroups, such as Balkan Romance or the Piemontese dialects, achieve high posterior probabilities though.
Notably, branch lengths carry information about the amount of change. According to the phylogeny in Figure 4, for instance, the Tuscan dialect of Italian (ITALIAN_GROSSETO_TUSCAN) is predicted to be the most conservative Romance dialect (since its distance to the latest common ancestors of all Romance dialects is shortest) and French the most innovative one.
These results indicate that the data only contain a weak tree-like signal. This is unsurprising since the Romance languages and dialects form a dialect continuum where horizontal transfer of innovations is an important factor.
Phylogenetic trees, like traditional family trees, only model vertical descent, not horizontal diffusion. They are therefore only an approximation of the historical truth. But nevertheless, they are useful as statistical models for further inference steps.
3.5 Ancestral state reconstruction
If a fully specified model is given, it is possible to estimate the probability distributions over character states for each internal node.
Let be a model, i.e. a phylogeny plus further parameters (rates and root probabilities, possibly specifications of rate variation). Let i be a character and n a node within .
The parameters specify a Markov process, including rates, for the branch leading to n. Let be the equilibrium probabilities of that process. (If n is the root, are directly given by .)
Let M(ni = x) be the same model as M, except that the value of character i at node n is fixed to the value x. is the likelihood of model M given the observed character vectors for the leaves.
The probability distribution over values of character i at node n, given M, is determined by Bayes’ rule:
Figure 5 illustrates this principle with the Romance part of the tree from Figure 4 and the character person:3 (cf. Table 4). The pie charts at the nodes display the probability distribution for that node, where white represents 0 and red 1.
This kind of computation was carried out for each class label character and each tree in the posterior sample for the latest common ancestor of the Romance doculects. For each concept, the class label for that concept with the highest average probability for value 1 at the root of the Romance subtree was inferred to represent the cognate class of the Proto-Romance word for that concept. For the concept person, e.g. character person:3 (representing the descendants of Latin “persona”), comes out as the best reconstruction.
3.6 Multiple sequence alignment
In the previous step, for the concept eye, the class label 6 was reconstructed for Proto-Romance. Its reflexes are given in Table 5.
A multiple sequence alignment (MSA) is a generalization of pairwise alignment to more than two sequences. Ideally, all segments within a column are descendants of the same sound in some common ancestor.
MSA, as applied to DNA or protein sequences, is a major research topic in bioinformatics. The techniques developed in this field are mutatis mutandis also applicable to MSA of phonetic strings. In this subsection, one approach will briefly be sketched. For a wider discussion and proposals for related but different approaches, see (List 2014).
Here I will follow the overall approach from (Notredame et al. 2000) and combine it with the PMI-based method for pairwise alignment described in Subsection 3.2. (Notredame et al. 2000) dub their approach T-Coffee (“Tree-based Consistency Objective Function For alignment Evaluation”), and I will use this name for the method sketched here as well.
In a first step, all pairwise alignments between words from the list to be multiply aligned are collected. For this purpose, I use PMI pairwise alignment. Some examples would be
The last row shows the score of the alignment, i.e. the proportion of identical matches (disregarding gaps).
In a second step, all indirect alignments between a given word pair are collected, which are obtained via relation composition with a third word. Some examples for indirect alignments between okiu and oky would be:
The direct pairwise alignment matches the i in okiu with the y in oky. Most indirect alignments pair these two positions as well, but not all of them. In the last columns, the i from okiu is related to the k of oky, and the y from oky with a gap. For each pair of positions in two strings, the relative frequency of them being indirectly aligned, weighted by the score of the two pairwise alignments relating them, is summed. They form the extended score between those positions.
The optimal MSA for the entire group of words is the one where the sum of the pairwise extended scores per column is maximized. Finding this global optimum is computationally not feasible though, since the complexity of this task grows exponentially with the number of sequences. Progressive alignment (Hogeweg and Hesper 1984) is a method to obtain possibly sub-optimal but good MSAs in polynomial time. Using a guide tree with sequences at the leaves, MSAs are obtained recursively leaves-to-root. For each internal node, the MSAs at the daughter nodes are combined via the Needleman–Wunsch algorithm while respecting all partial alignments from the daughter nodes.
Using this method, MSAs were computed for each inferred class label that was inferred to be present in Proto-Romance via Ancestral State Reconstruction.
3.7 Proto-form reconstruction
A final step toward the reconstruction of Proto-Romance forms, Ancestral State Reconstruction is performed for the sound classes in each column, for each MSA obtained in the previous step.
Consider the first column of the MSA in Table 6. It contains three possible states, v, w, and the gap symbols -. For each of these states, a binary presence–absence character is constructed. For doculects which do not occur in the MSA in question, this character is undefined.
The method for ancestral state reconstruction described in Subsection 3.5 was applied to these characters. For phylogeny in the posterior sample, the probabilities for state 1 at the Proto-Romance node was computed for each character. For each column of an MSA, the state with the highest average probability was considered as reconstructed.
The reconstructed proto-form for a given concept is then obtained by concatenating the reconstructed states for the corresponding MSA and deleting all gap symbols. The results are given in Table 7.
To evaluate the quality of the automatic reconstructions, they were compared to the corresponding elements of the Latin word list. For each reconstructed word, the normalized Levenshtein distance (i.e. the Levenshtein distance divided by the length of the longer string) to each Latin word (without diacritics) for that concept was computed. The smallest such value counts as the score for that concept. The average score was 0.484. The extant Romance doculects have an average score of 0.627. The most conservative doculect, Sardinian, has a score of 0.502, and the least conservative, Arpitan, 0.742. The evaluation results are depicted in Figure 6.
These findings indicate that the automatic reconstruction does in fact capture a historical signal. Manual inspection of the reconstructed word list reveals that, to a large degree, the discrepancies to Latin actually reflect language change between Classical Latin and the latest common ancestor of the modern Romance doculects, namely Vulgar Latin. To mention just a few points: (1) Modern Romance nouns are mostly derived from the (Vulgar-)Latin accusative form (Herman 2000: 3), while the word lists contains the nominative form. For instance, the common ancestor forms for “tooth” and “night” are dentem and noctem. The reconstructed t in the corresponding reconstructed forms are therefore historically correct. (2) Some Vulgar Latin words are morphologically derived from their Classical Latin counterparts, such as mons montanea “mountain” (Meyer-Lübke 1935: 464) or genus genukulum “knee” (Meyer-Lübke 1935: 319). This is likewise partially reflected in the reconstructions. (3) For some concepts, lexical replacement by non-cognate words took place between Classical and Vulgar Latin, such as via strata “path”,ignis focus “fire” (Meyer-Lübke 1935 293), or iecur ficatum “liver” (Herman 2000: 106). Again, this is reflected in the reconstruction.
On the negative side, the reconstructions occasionally reflect sound changes that only took place in the Western Romania, such as the voicing of plosives between vowels (Herman 2000: 46), as in figat “liver” ( ficatum).
Let me conclude this section with some reflections on how the reconstructions were obtained and how this relates to the comparative method.
A major difference to the traditional approach is the stochastic nature of the workflow sketched here. Both phylogenetic inference and ancestral state reconstruction are based on probabilities rather than categorical decisions. The results shown in Table 7 propose a unique reconstruction for each concept, but it would be a minor modification of the workflow only to derive a probability distribution over reconstructions instead. This probabilistic approach is arguably an advantage since it allows to utilize uncertain and inconclusive information while taking this uncertainty properly into account.
Another major difference concerns the multiple independence assumptions implicit in the probabilistic model sketched in Subsection 3.4. The likelihood of a phylogeny is the product of its likelihoods for the individual characters. This amounts to the assumptions that the characters are mutually stochastically independent.
The characters used here (and generally in computational phylogenetics as applied to historical linguistics) are mutually dependent in manifold ways though. For instance, the loss of a cognate class makes it more likely that the affected lineage will acquire another cognate class for the same semantic slot and vice versa.
This problem is even more severe for phonetic change. Since the work of the Neogrammarians in the nineteenth century, it is recognized that many sound changes are regular, i.e. they apply to all instances of a certain sound (given contextual conditions) throughout the lexicon. Furthermore, both regular and irregular sound changes are usually dependent on their syntagmatic phonetic context, and sometimes on the paradigmatic context within inflectional paradigms as well. Bouchard-Côté et al. (2013) and Hruschka et al. (2015) propose more sophisticated probabilistic models of language change than the one used here to take these dependencies into account.
Last but not least, the treatment of borrowing (and language contact in general) is an unsolved problem for computational historical linguistics. Automatic cognate clustering does not distinguish between genuine cognates (related via unbroken chains of vertical descent) and (descendants of) loanwords. This introduces a potential bias for phylogenetic inference and ancestral state reconstruction, since borrowed items might be misconstrued as shared retentions.
This article gives a brief sketch of the state of the art in computational historical linguistics, a relatively young subfield at the interface between historical linguistics, computational linguistics and computational biology. The case study discussed in the previous section serves to illustrate some of the major research topics in this domain: identification of genetic relationships between languages, phylogenetic inference, automatic cognate detection and ancestral state recognition. These concern the core issues of the field; the information obtained by these methods is suitable to address questions of greater generality, pertaining to general patterns of language change as well as the relationship between the linguistic and non-linguistic history of specific human populations.
I thank Johann-Mattis List, Wiltrud Mihatsch, the co-authors of this issue Anne Breitbarth, Ailis Cournane and George Walkden, as well as the editor Hans-Martin Gärtner, for helpful feedback on draft versions of this article.
Anthony, D. W. 2010. The horse, the wheel, and language: How Bronze-Age riders from the Eurasian steppes shaped the modern world. Princeton: PUB. Search in Google Scholar
Atkinson, Q. D. & R. Gray. 2005. Curious parallels and curious connections — phylogenetic thinking in biology and historical linguistics. Systematic Biology 54(4). 513–526.10.1080/10635150590950317 Search in Google Scholar
Baxter, W. H. & A. Manaster Ramer. 2000. Beyond lumping and splitting. Probabilistic issues in historical linguistics. In C. Renfrew et al. (eds.), Time depth in historical linguistics, vol. 1, 167–188. Cambridge: McDonald Institute for Archaeological Research. Search in Google Scholar
Bergsma, S. & G. Kondrak. 2007. Multilingual cognate identification using integer linear programming. In Proceedings of the RANLP Workshop, 656–663. Search in Google Scholar
Bouchard-Côté, A., D. Hall, T. L. Griffiths & D. Klein. 2013. Automated reconstruction of ancient languages using probabilistic models of sound change. Proceedings of the National Academy of Sciences 36(2). 141–150. Search in Google Scholar
Campbell, L. 2013. Historical linguistics. An introduction. Edinburgh: EUB. Search in Google Scholar
Chen, M.-H., L. Kuo & P. O. Lewis. 2014. Bayesian phylogenetics. Methods, algorithms and applications. Abingdon: CRC Press. Search in Google Scholar
Covington, M. A. 1996. An algorithm to align words for historical comparison. Computational Linguistics 22(4). 481–496. Search in Google Scholar
Csardi, G. & T. Nepusz. 2006. The igraph software package for complex network research. InterJournal, Complex Systems 1695(5). 1–9. Search in Google Scholar
Desper, R. & O. Gascuel. 2002. Fast and accurate phylogeny reconstruction algorithms based on the minimum-evolution principle. Journal of computational biology 9(5). 687–705.10.1089/106652702761034136 Search in Google Scholar
Dolgopolsky, A. B. 1986. A probabilistic hypothesis concerning the oldest relationships among the language families of Northern Eurasia. In V. V. Shevoroshkin (ed.), Typology, relationship and time: A collection of papers on language change and relationship by Soviet linguists, 27–50. Ann Arbor: Karoma Publisher. Search in Google Scholar
Dunn, M., S. J. Greenhill, S. Levinson & R. D. Gray. 2011. Evolved structure of language shows lineage-specific trends in word-order universals. Nature 473(7345). 79–82.10.1038/nature09923 Search in Google Scholar
Durbin, R., S. R. Eddy, A. Krogh & G. Mitchison. 1989. Biological Sequence Analysis. Cambridge, UK: CUP. Search in Google Scholar
Dyen, I., J. B. Kruskal & P. Black. 1992. An Indoeuropean classification: A lexicostatistical experiment. Transactions of the American Philosophical Society 82(5). 1–132. Search in Google Scholar
Edwards, A. W. F. & L. L. Cavalli-Sforza. 1964. Reconstruction of evolutionary trees. In V. H. Heywood & J. R. McNeill (eds.), Phenetic and phylogenetic classification, 67–76. London: Systematics Association Publisher. Search in Google Scholar
Embleton, S. M. 1986. Statistics in historical linguistics. Bochum: Brockmeyer. Search in Google Scholar
Ewens, W. & G. Grant. 2005. Statistical methods in bioinformatics: An introduction. New York: Springer. Search in Google Scholar
François, A. 2015. Trees, waves and linkages: Models of language diversification. In C. Bowern & B. Evans (eds.), The Routledge handbook of historical linguistics, 179–207. Abingdon: Routledge. Search in Google Scholar
Gascuel, O. 1997. BIONJ: An improved version of the NJ algorithm based on a simple model of sequence data. Molecular Biology and Evolution 14(7). 685–695.10.1093/oxfordjournals.molbev.a025808 Search in Google Scholar
Gray, R. D., A. J. Drummond & S. J. Greenhill. 2009. Language phylogenies reveal expansion pulses and pauses in Pacific settlement. Science 323(5913). 479–483.10.1126/science.1166858 Search in Google Scholar
Greenhill, S. J., R. Blust & R. D. Gray. 2008. The Austronesian basic vocabulary database: From bioinformatics to lexomics. Evolutionary Bioinformatics 4. 271–283. Search in Google Scholar
Hall, D. & D. Klein. 2010. Finding cognate groups using phylogenies. In Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics, 1030–1039. ACL. Search in Google Scholar
Hammarström, H., R. Forkel, M. Haspelmath & S. Bank. 2016. Glottolog 2.7. Max Planck Institute for the Science of Human History, Jena. Available online at http://glottolog.org accessed 29 January 2017).( Search in Google Scholar
Hauer, B. & G. Kondrak. 2011. Clustering semantically equivalent words into cognate sets in multilingual lists. In Proceedings of the 5th International Joint NLP conference, 865–873. Search in Google Scholar
Heggarty, P., W. Maguire & A. McMahon. 2010. Splits or waves? Trees or webs? How divergence measures and network analysis can unravel language histories. Philosophical Transactions of the Royal Society B: Biological Sciences 365(1559). 3829–3843.10.1098/rstb.2010.0099 Search in Google Scholar
Herman, J. 2000. Vulgar Latin. University Park, PA: The Pennsylvania State University Press. Search in Google Scholar
Hogeweg, P. & B. Hesper. 1984. The alignment of sets of sequences and the construction of phyletic trees: An integrated method. Journal of molecular evolution 20(2). 175–186.10.1007/BF02257378 Search in Google Scholar
Hruschka, D. J., S. Branford, E. D. Smitch, J. Wilkins, A. Meade, M. Pagel & T. Bhattachary. 2015 Detecting regular sound changes in linguistics as events of concerted evolution. Current Biology 25(1). 1–9.10.1016/j.cub.2014.10.064 Search in Google Scholar
Jäger, G. 2013. Phylogenetic inference from word lists using weighted alignment with empirically determined weights. Language Dynamics and Change 3(2). 245–291.10.1163/22105832-13030204 Search in Google Scholar
Jäger, G. & J.-M. List. 2017. Using ancestral state reconstruction methods for onomasiological reconstruction in multilingual word lists. Language Dynamics and Change 8(1). 22–54. Search in Google Scholar
Jäger, G. & P. Sofroniev. 2016. Automatic cognate classification with a Support Vector Machine. In S. Dipper et al. (eds.), Proceedings of the 13th Conference on Natural Language Processing, 128–134. Bochum: RUB. Search in Google Scholar
Jäger, G., J.-M. List & P. Sofroniev. 2017. Using support vector machines and state-of-the-art algorithms for phonetic alignment to identify cognates in multi-lingual wordlists. In Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics. ACL. Search in Google Scholar
Kay, M. 1964. The logic of cognate recognition in historical linguistics. Santa Monica, CA: Rand Corporation. Search in Google Scholar
Kessler, B. 2001. The significance of word lists. Stanford: CSLI Publications. Search in Google Scholar
Kondrak, G. 2002. Algorithms for language reconstruction. University of Toronto PhD thesis. Search in Google Scholar
Kooperberg, C. 2016. Package ‘logspline’. https://cran.r-project.org/web/packages/logspline/index.html. version 2.1.9. Search in Google Scholar
Kroonen, G. 2013. Etymological dictionary of Proto-Germanic. Leiden, Boston: Brill. Search in Google Scholar
Lewis, M. P., G. F. Simons & C. D. Fennig (eds.). 2016. Ethnologue: Languages of the world, 9th edn. Dallas, Texas: SIL International. Search in Google Scholar
List, J.-M. 2012. Lexstat: Automatic detection of cognates in multilingual wordlists. In M. Butt & J. Prokić (eds.), Proceedings of LINGVIS & UNCLH, Workshop at EACL 2012, 117–125, Avignon. Search in Google Scholar
List, J.-M. 2014. Sequence comparison in historical linguistics. Düsseldorf: DUP. Search in Google Scholar
Lowe, J. B. & M. Mazaudon. 1994. The reconstruction engine: A computer implementation of the comparative method. Computational Linguistics 20(3). 381–417. Search in Google Scholar
McMahon, A. & R. McMahon. 2005. Language classification by numbers. Oxford: OUP. Search in Google Scholar
McMahon, A. & R. McMahon. 2006 Why linguists don’t do dates: Evidence from Indo-European and Australian languages. In P. Forster & C. Renfrew (eds.), Phylogenetic methods and the prehistory of languages, 153–160. Cambridge, UK: McDonald Institute for Archaeological Research. Search in Google Scholar
Meillet, A. 1954. La méthode comparative en linguistique historique. Paris: Honoré Champion. Search in Google Scholar
Meyer-Lübke, W. 1935. Romanisches etymologisches Wörterbuch. Heidelberg: Carl Winters Universitätsbuchhandlung. 3. Auflage. Search in Google Scholar
Needleman, S. B. & C. D. Wunsch. 1970. A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of Molecular Biology 48. 443–453.10.1016/0022-2836(70)90057-4 Search in Google Scholar
Nguyen, L.-T., H. A. Schmidt, A. von Haeseler & B. Q. Minh. 2015. IQ-TREE: A fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies. Molecular Biology and Evolution 32(1). 268–274.10.1093/molbev/msu300 Search in Google Scholar
Notredame, C., D. G. Higgins & J. Heringa. 2000. T-Coffee: A novel method for fast and accurate multiple sequence alignment. Journal of Molecular Biology 302(1). 205–217.10.1006/jmbi.2000.4042 Search in Google Scholar
Oakes, M. P. 2000. Computer estimation of vocabulary in a protolanguage from word lists in four daughter languages. Journal of Quantitative Linguistics 7(3). 233–243.10.1076/jqul.18.104.22.16805 Search in Google Scholar
Pagel, M., Q. D. Atkinson & A. Meade. 2007 Frequency of word-use predicts rates of lexical evolution throughout Indo-European history. Nature 449(7163). 717–720.10.1038/nature06176 Search in Google Scholar
Pagel, M., Q. D. Atkinson, A. S. Calude & A. Meade. 2013. Ultraconserved words point to deep language ancestry across Eurasia. Proceedings of the National Academy of Sciences 110(21). 8471–8476.10.1073/pnas.1218726110 Search in Google Scholar
Pietrusewsky, M. 2008. Craniometric variation in Southeast Asia and neighboring regions: a multivariate analysis of cranial measurements. Human Evolution 23(1–2). 49–86. Search in Google Scholar
Raghavan, U. N., R. Albert & S. Kumara. 2007. Near linear time algorithm to detect community structures in large-scale networks. Physical Review E 76(3). 036106.10.1103/PhysRevE.76.036106 Search in Google Scholar
Rama, T. 2015. Automatic cognate identification with gap-weighted string subsequences. In Proceedings of the North American Association for Computational Linguistics, 1227–1231. ACL. Search in Google Scholar
Renfrew, C. 1987. Archaeology and language: The puzzle of Indo-European origins. Cambridge, UK: CUP. Search in Google Scholar
Ross, M. & M. Durie. 1996. Introduction. In Mark Durie & Malcolm Ross (eds.), The comparative method reviewed. Regularity and irregularity in language change, 3–38. Oxford: OUP. Search in Google Scholar
Saitou, N. & M. Nei. 1987. The neighbor-joining method: A new method for reconstructing phylogenetic trees. Molecular Biology and Evolution 4(4). 406–425. Search in Google Scholar
Swadesh, M. 1952. Lexico-statistic dating of prehistoric ethnic contacts. Proceedings of the American Philosophical Society 96(4). 452–463. Search in Google Scholar
Turchin, P., I. Peiros & M. Gell-Mann. 2010. Analyzing genetic connections between languages by matching consonant classes. Journal of Language Relationship 3. 117–126. Search in Google Scholar
Weiss, M. 2015. The comparative method. In C. Bowern & B. Evans (eds.), The Routledge handbook of historical linguistics, 119–121. London: Routledge. Search in Google Scholar
© 2019 Jäger, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.