1.1 Authorship Analysis

The analysis of language to determine who authored a text is a practice that has been around for millennia, most prominently in literary studies, where scholars of particular author styles tended to rely on their own expert judgement (Reference Coulthard, Johnson and WrightCoulthard et al., 2017). Work on the quantification of stylistic features began in the nineteenth century, leading to the emergence of the new field of stylometry. The initial quest of stylometry was to find author-invariant features that would remain constant for each single author over time and genre. Despite thousands of authorship analysis features being tested, however, so far no such feature has been found.

The most important breakthrough in stylometry is without any doubt Reference Mosteller and WallaceMosteller and Wallace’s (1963) analysis of the Federalist Papers. Approaching a text from the perspective of a statistician, Mosteller and Wallace focused on the frequencies of function words, such as articles, prepositions, or pronouns. Their reason for only considering the ‘filler words of the language’ was their obliviousness to topic and because ‘they provide fairly stable rates’ (Reference Mosteller and WallaceMosteller and Wallace, 1963, p. 275). This insight survived the test of time and to this day the analysis of word frequencies and especially of the frequency of function words is still among the most successful ways of carrying out authorship analysis (Reference GrieveGrieve, 2007; Reference StamatatosStamatatos, 2009; Reference ArgamonArgamon, 2018). The only other widely successful feature in modern stylometry has been the frequency of n-grams: sequences of length n collected using a moving window, in particular the frequency of character n-grams with small n (2 to 4) (Reference StamatatosStamatatos, 2013). This type of character n-gram actually ends up also capturing a lot of the information carried by function words and therefore some of its usefulness can still be reconducted to the same fundamental criteria introduced by Reference Mosteller and WallaceMosteller and Wallace (1963).

Although there is mostly agreement on the best features for authorship analysis, there is less consensus on how to use them. The basic distinction in the field is between similarity-based and machine learning approaches (Reference StamatatosStamatatos, 2009; Reference Koppel, Schler and ArgamonKoppel et al., 2013). In the first approach type, a similarity score is calculated for each text (or for an author’s profile, which is a combination of the author’s texts) and then this score is used to rank the candidate authors in terms of similarity to the questioned text, with the most similar being declared the correct author. In the second approach type, each text of known authorship is treated independently and a statistical model of how the candidate authors write is extracted. This model, generated using tools such as Principal Component Analysis, cluster analysis, multidimensional scaling, discriminant analysis, Support Vector Machine, or deep learning algorithms, is then used to determine who wrote the questioned text (Reference JuolaJuola, 2008; Reference StamatatosStamatatos, 2009; Reference ArgamonArgamon, 2018; Reference Kestemont, Manjavacas and MarkovKestemont et al., 2020). In the most specific case of authorship verification, that is when there is only one candidate author, the General Impostors algorithm has been found to be very effective. This algorithm is based on the idea of creating a line-up of fake candidates before applying a similarity-based analysis. If the similarity is greater for the candidate than for the impostors over multiple random tests with random sets of impostors and features, then the questioned text is assigned to the candidate author (Reference Seidman, Forner, Navigli, Tufis and FerroSeidman, 2013; Reference Koppel and WinterKoppel and Winter, 2014; Reference Kestemont, Stover, Koppel, Karsdorp and DaelemansKestemont et al., 2016).

1.2 Burrow’s Delta

In order to execute the General Impostors method or any similarity-based analysis, a score that expresses a measure of similarity or distance must be used. There are many ways of measuring similarity/distance but decades of research in stylometry have now firmly demonstrated that the most effective of them for authorship analysis is Delta (Reference BurrowsBurrows, 2002).

The original Delta introduced by Reference BurrowsBurrows (2002) is calculated starting from the relative frequencies, $f$ , of the top $N$ most frequent words in the corpus considered, a relative frequency being the number of times the word is found in the text divided by the total number of word tokens in the text. For each of these most frequent words, firstly a z-score is calculated, which is a measure of deviation from the mean calculated by subtracting the mean, $\mu$ , of the relative frequency of the word considered from its relative frequency, and dividing this quantity by the standard deviation, $\sigma$ . Once this is done for all the words considered, the Delta distance between texts $A$ and $B$ , ${\Delta }_{\left(A,B\right)}$ , is calculated by taking the average of the absolute differences across all z-scores:

${\Delta }_{\left(A,B\right)}=\frac{1}{N}\sum _{i=1}^N\left|z_i\left(A\right)-z_i\left(B\right)\right|\text{.}$

For example, suppose we have only two words, the and of, so $N=2$ , and the table of counts for these two features for two texts $A$ and $B$ is Table 1.

In this example, the relative frequencies would be (multiplied by 100 to make them easier to read):

$f_{the}\left(A\right)=\frac{56}{800}*100=7$ and $f_{the}\left(B\right)=\frac{27}{900}*100=3$

$f_{of}\left(A\right)=\frac{32}{800}*100=4$ and $f_{of}\left(B\right)=\frac{90}{900}*100=10\text{.}$

Let’s imagine that we have more texts and that we calculate the mean and standard deviations for the relative frequencies of the and of, which are as follows:

${\mu }_{the}=2$ and ${\mu }_{of}=2$

${\sigma }_{the}=1$ and ${\sigma }_{of}=2\text{.}$

At this point it is possible to transform the relative frequencies, $f_i$ , into z-scores, $z_i$ , which indicate how different each relative frequency is from the mean.

Table 2 Relative frequencies from Table 1 transformed into z-scores.

	$z_{the}=\frac{f_{the}-{\mu }_{the}}{{\sigma }_{the}}$	$z_{of}=\frac{f_{of}-{\mu }_{of}}{{\sigma }_{of}}$
Text A	5	1
Text B	1	4

With the z-scores it is now possible to plug in the values in the Delta formula to calculate the Delta distance between these two texts:

$\begin{array}{ccc}{\text{Δ}}_{\left(\text{A},\text{B}\right)}& =& \frac{1}{2}\sum _{\text{i}=1}^2\left|z_i\left(A\right)-z_i\left(B\right)\right|=\\ & =& \frac{\left|z_{the}\left(A\right)-z_{the}\left(B\right)\right|+\left|z_{of}\left(A\right)-z_{of}\left(B\right)\right|}{2}=\\ & =& \frac{\left|5-1\right|+\left|1-4\right|}{2}=\frac{4+3}{2}=3.5\end{array}$

To put this in non-mathematical terms, by quantifying how much each author deviates from the norm, Delta is a measure of how much two texts are similar to each other in respect to how much they differ from a norm.

This simple formula has solidly passed the test of time and is one of the greatest achievements in authorship analysis (e.g. Reference HooverHoover, 2004; Reference JuolaJuola, 2012; Reference EderEder, 2015; Reference Evert, Proisl, Vitt, Feldman, Kazantseva and SzpakowiczEvert et al., 2015, Reference Evert, Proisl and Jannidis2017). Repeated experiments have demonstrated that if enough register-compatible data for each author is collected then the values of Delta will be relatively small for texts that are written by the same author and relatively large for texts written by different authors. Delta can be used to attribute a text or to narrow down a large list of candidates because candidate authors can be ranked from more distant to less distant from a questioned text.

Even in the age of neural networks and deep learning algorithms, simple variations of this method are still extremely reliable. For example, Reference Halvani, Graner and RegevHalvani et al. (2020) used a method that is very similar to Delta and that still performed very well compared to artificial intelligence methods. Similarly, Reference Narayanan, Paskov and GongNarayanan et al. (2012) and Reference Pokhriyal, Tayal, Nwogu and GovindarajuPokhriyal et al. (2017) show how methods based on distances similar to Delta are optimal even in scenarios including thousands of authors and relatively short texts. In sum, Delta is a simple and largely transparent technique that has been found to work in various independent replications. However, despite this success, relatively little has been done to untangle the inner mechanisms responsible for this performance.

1.2.1 The Key Profiles Hypothesis

The first significant advance in the mathematical understanding of Delta is Reference ArgamonArgamon’s (2008) geometric reinterpretation. Let’s now imagine Table 2 not as a table of z-scored frequencies but as a Cartesian bidimensional space where each word is a dimension, like in Figure 1.

Figure 1 Simple representation of a two-dimensional vector space containing two texts, $A$ and $B$ .

In this space, we can represent the two texts $A$ and $B$ as two points with their coordinates being the z-scores of the words the and of, thus:

$A=\left[\begin{array}{c}1\\ 5\end{array}\right],B=\left[\begin{array}{c}4\\ 1\end{array}\right]\text{,}$

where the top row is for of and the bottom row is for the. These mathematical objects are called vectors and they can be represented as arrows that start from the origin of the space and reach the points. In this two-dimensional space, Delta is simply the Manhattan distance between these two points:

$d_M=\sum _{\text{i}=1}^{\text{N}}\left|A_i-B_i\right|\text{.}$(1)

The Manhattan distance is so called because it is the distance between two points if we imagine driving or walking in a city like Manhattan, where it would be impossible to reach a destination by taking the quickest trajectory, a straight line. This straight distance, which is called the Euclidean distance, is represented in Figure 1 as the dashed line connecting $A$ and $B$ . The Manhattan distance between $A$ and $B$ is instead the L-shaped path between $A$ and $B$ following the dotted line. Using the formula above, the calculation of the Manhattan distance in this case would be $\left|1-4\right|+\left|5-1\right|=3+4=7$ , which is the same result obtained above for Delta, just without the division by 2. Because $N$ is a constant number for any pair of texts involved, dividing by it is an unnecessary step and therefore the Manhattan distance between these two vectors is the same as Delta. When calculating Delta, instead of two dimensions there are hundreds or thousands of words and, thus, hundreds or thousands of dimensions but the formula remains the same. Reference ArgamonArgamon’s (2008) insight is useful and increases our mathematical understanding of Delta but it is more like a rephrasing than an explanation. We still do not know why or how the Manhattan distance between two vectors of frequencies works so well to identify authors.

Further explanatory advancements were made a few years later by Reference Evert, Proisl, Vitt, Feldman, Kazantseva and SzpakowiczEvert et al. (2015, Reference Evert, Proisl and Jannidis2017). As a first step, Reference Evert, Proisl and JannidisEvert et al. (2017) compared various versions of Delta, such as traditional Delta (with the Manhattan distance), Quadratic Delta (with the Euclidean distance), and Cosine Delta (with the Cosine distance) using the refcor corpus, a multilingual corpus of novels collected from Project Gutenberg.Footnote ¹ Looking back at Figure 1, Quadratic Delta would be the straight line between the two points, while Cosine Delta would be the angle α. Reference Evert, Proisl and JannidisEvert et al. (2017) found that Cosine Delta, originally introduced by Reference Smith and AldridgeSmith and Aldridge (2011), is superior to the other ones as well as more robust when increasing or decreasing the number of features considered.

The fact that Cosine Delta performs better than classic Delta is crucial for our understanding of its mechanisms. When the measure of distance between two vectors is the angle, then their length does not matter. For example, if the $A$ point was not (1, 5) but another point on the imaginary continuation of that same line, say (2, 10), then the distance between $A$ and $B$ would be the same because the angle between the vectors would not change. Because Cosine Delta outperforms other types of Delta we can infer that the lengths of the vectors do not matter (e.g. whether the author of $A$ uses the much more often than the author of $B$ ) as much as their orientation in relation to each other, or the angle that they form (e.g. the fact that the author of $A$ uses the more frequently than the average and of less frequently while $B$ does the opposite). This leads Reference Evert, Proisl and JannidisEvert et al. (2017) to formulate the following hypothesis:

Key Profiles Hypothesis: authors are distinguished by the overall pattern of variation of all features in relation to the norm and not by the magnitude of these variations.

To test this hypothesis, Reference Evert, Proisl and JannidisEvert et al. (2017) transformed all ‘above average’ z-scores to +1, all ‘below average’ z-scores to −1, and all values in between to 0. By doing that, they removed any effect of the length of the vectors. Because this operation consistently made all other Deltas almost as good as Cosine Delta, Reference Evert, Proisl and JannidisEvert et al. (2017) concluded that there is strong evidence for the key profiles hypothesis. An important implication of this hypothesis is that when it comes to authorship analysis it is not just a few remarkable features that matter but rather the patterning of the conjunction of many features that by themselves are not necessarily distinctive. It is the profile of the covariation that makes them distinctive, not their own single magnitude.

This conclusion is rather significant and, despite being purely mathematical, it advances our understanding of Delta. Accepting the Key Profiles Hypothesis means that given a multidimensional space of (function) words, each individual has its own place in it based on the pattern of over- or under-used (function) words. This is a fascinating picture yet it is not a full explanation because there are no immediately obvious linguistic reasons for the existence of these key profiles.

1.3 Fundamentals of Cognitive Linguistics

A framework for linguistically realistic authorship analysis should meet two desiderata: (1) it must be able to handle real-life language usage; and (2) it must be scientifically plausible, meaning that it should be consistent with other well-established facts about the human mind. The proposal put forward in the present work is that these desiderata are met by usage-based linguistics and, more specifically, cognitive linguistics.

Firstly, what is needed for authorship analysis is a theory of language that can explain usage events. Therefore, purely formal and idealised frameworks such as mainstream generative grammar, which by definition rejects the study of usage, must be excluded. This is not a value judgement: such a framework is clearly designed for purposes that do not match the reality of dealing with language in use. In contrast, cognitive linguistics subscribes to Reference Beckner, Ellis and BlytheBeckner et al.’s (2009) main tenet that language is a complex adaptive system from which the structures of language emerge spontaneously out of individuals’ interactions in context. This in turn means that studies in cognitive linguistics are always concerned with and oriented towards language usage.

Secondly, authorship analysis should be based as much as possible on models of language that see language as a property of the human mind because, ultimately, the act of text creation is a process that happens in the mind of the author. Thus, the optimal point of departure for a science of authorship analysis must be what is currently known about language production. Different varieties of cognitive linguistics meet this desideratum because they subscribe to the Cognitive Commitment to ‘make one’s account of human language accord with what is generally known about the mind and the brain’ (Reference LakoffLakoff, 1990, p. 40).

Among the various frameworks or theories that can be classified as cognitive linguistics we can name Cognitive Grammar (Reference LangackerLangacker, 1987), Construction Grammar (Reference GoldbergGoldberg, 1995, Reference Goldberg2003, Reference Goldberg2006), the theory of Lexical Priming (Reference HoeyHoey, 2005), Word Grammar (Reference HudsonHudson, 1996, Reference Hudson2010), and Radical Construction Grammar (Reference CroftCroft, 2001). In the rest of this section, I will outline four key foundational concepts that cognitive linguistics usage-based frameworks agree on and that are useful to build the foundations for a theory of individuality for authorship analysis, which is then introduced in Section 2.

1.4 Idiolect and Linguistic Individuality

Before moving on to the outlining of a theory of individuality, this section reviews the available theories and evidence regarding individuality and uniqueness in language. This topic is not something that has been thoroughly investigated in linguistics and the reason for this gap is understandable. For the scientific study of human language to progress, it is natural that the focus at first should be on what is shared across individuals rather than on differences and uniqueness. Although linguists from different theoretical standpoints tacitly agree that each person has their own idiolect, a notion first introduced by Reference BlochBloch (1948) indicating a personal variety of a language, the magnitude of these differences has simply not been investigated so far. Indeed, until very recent modern work that will be reviewed below, individual differences have largely been ignored in linguistics. This is a similar situation to cognitive science more generally. Reference Kidd, Donnelly and ChristiansenKidd et al. (2018, p. 154), for example, comment that the large individual differences in cognition are ‘an inconvenient truth’: although we know that they exist, we treat them as noise in the data in order to generalise our findings.

Among the first systematic theoretical accounts of individuality and idiolect in language for the purpose of authorship analysis is Reference McMenaminMcMenamin (2002). In his theory, Reference McMenaminMcMenamin (2002, p. 53) explains that ‘no two individuals use and perceive language in exactly the same way’ as the result of the fact that language is a discrete combinatorial system (Reference PinkerPinker, 1994). Language, like DNA, combines smaller units to create larger ones and this process, which is highly combinatorial, leads to an incredibly large number of possible choices. Individuality and uniqueness are therefore a consequence of this combinatorial explosion. However, although this is a promising point of departure, Reference McMenaminMcMenamin (2002) does not explain why combinatorics should necessarily lead to linguistic individuality and uniqueness. The fact that choices are available does not imply that all of them are selected nor that all individuals must make consistently different choices. Reference McMenaminMcMenamin (2002) only assumes that this is the case: individuals make repeated choices that, in turn, form their own styles, which can be studied using methods from the discipline of stylistics.

McMenamin’s stylistics approach was strongly criticised by Reference ChaskiChaski (2001), who instead argued that the reason why individuals are linguistically unique is rooted in their personalised mental grammar. As a consequence, reliable authorship analysis can only be carried out by looking at the syntactic structures produced by individuals. This syntactic (sometimes also called cognitive) theory of authorship analysis, however, is arguably not substantiated by either theoretical or empirical work. Firstly, Reference ChaskiChaski (2001) does not explain why mental syntactic representations should be different across individuals. Secondly, more than twenty years of research in computational authorship analysis have demonstrated that it is indeed entirely possible to carry out successful authorship analysis work without using any a priori notion of syntactic structure.

In contrast, although Reference ChaskiChaski’s (2001) approach has been sometimes labelled as ‘cognitive’, the explanations put forward by Reference McMenaminMcMenamin (2002) are much more consistent with the cognitive usage-based approach and with the state of the art of psychological understanding of how language works than Reference ChaskiChaski’s (2001). As explained by Reference Nini and GrantNini and Grant (2013), the differences in syntactic structures detected by Reference ChaskiChaski (2001) could still be explained by each individual’s unique history of linguistic usage events as proposed by Reference McMenaminMcMenamin (2002).

The origin of individuality within the stylistic approach is addressed by Reference TurellTurell’s (2010) concept of idiolectal style. This theory explains that individuality is not the result of different repertoires but of different selections of items within the space of possibilities offered by a language. This notion of idiolectal style is consistent with Reference Nini and GrantNini and Grant’s (2013) argument based on Halliday’s Systemic Functional Linguistics, where an idiolect is theorised to be formed by the habitual choices in the various linguistic systems of a language. Reference Nini and GrantNini and Grant (2013) thus agree with Reference TurellTurell (2010) and add that the reason for this individuality lies in the uniqueness of the sociolinguistic history of an individual.

The most recent theoretical advancement linking individuality in language to sociolinguistics is put forward by Reference Grant and MacLeodGrant and MacLeod’s (2020) Resource-Constraint Theory of Identity. Following Reference JohnstoneJohnstone (1996), they maintain that identity is not a property of the individual but of the interaction between the individual and the situation. An identity is not a static attribute of a person but an action or a performance that cannot be found in isolation. Thus, a person’s language is not just the product of their sociolinguistic profile but is actively constituted by the choices they make in a situation. This performance of identity is, however, not unconstrained because it is limited by the linguistic resources belonging to an individual’s repertoire, which in turn depend on their sociolinguistic history. This repertoire is mutable and changes over time, following the individual’s language change.

Another theoretical perspective is provided by Reference CoulthardCoulthard (2004, p. 431), who proposed that ‘every native speaker has their own distinct and individual version of the language they speak and write’ and that ‘this idiolect will manifest itself through distinctive and idiosyncratic choices’. Similarly to Reference McMenaminMcMenamin (2002), therefore, Reference CoulthardCoulthard (2004) explains that the locus of individuality is choice and, more specifically, the combination of choices. He proposes that idiolectal co-selection leads to a high level of idiosyncrasies that resemble a linguistic fingerprint, even though he discourages this analogy. A famous example of the power of idiolectal co-selection is the analysis of the Unabomber’s writings, where a list of twelve linguistic items (e.g. at any rate, clearly, presumably, thereabouts) uniquely connected the Unabomber’s manifesto to an article written by Ted Kaczynski (Reference Coulthard, Johnson and WrightCoulthard et al., 2017). In other words, then, Reference CoulthardCoulthard (2004) connects individuality and uniqueness to the combinatorics of language, although, unlike Reference McMenaminMcMenamin (2002), he does not connect this phenomenon to style. Similarly to Reference McMenaminMcMenamin (2002), however, Coulthard does not provide an explanation as to why individuals should differ in their co-selections.

Empirical validation for Reference CoulthardCoulthard’s (2004, Reference Coulthard2013) proposal was found in the analysis of lexical n-grams in the Enron corpus, a large corpus of emails collected after the collapse of the American company Enron. Reference WrightWright (2013) looked at a sub-sample of four members of this community and discovered that some authors can be extremely distinctive in their combination of choices of greetings and farewells when compared to a similar population of 126 other authors. Then, Reference WrightWright (2017) demonstrated how something seemingly common, such as the word sequences ‘a clean and redlined version’ or ‘please review and let’s discuss’, is unique to a particular author. Even though other similar authors express the same meaning repeatedly in the corpus, they choose formulations that are similar but not identical to those. Using a subset of twelve demographically similar authors, Reference WrightWright (2017) showed that it is possible to use the presence/absence of word n-grams to attribute even relatively short emails in a cohort of 175 candidates. Reference WrightWright (2017) connects these findings to the notions of cognitive linguistics introduced above and he therefore proposes an explanation for Reference CoulthardCoulthard’s (2004) idiolectal co-selection phenomenon: individuality emerges from consistent and repeated usage of a certain sequence of words to realise a speech act, a process leading to this sequence becoming entrenched for that person.

Reference CoulthardCoulthard’s (2004) and Reference McMenaminMcMenamin’s (2002) ideas can therefore be connected to usage-based cognitive linguistic frameworks described in the previous section. Most of or all the cognitive linguistic frameworks indirectly predict that linguistic individuality exists and that its effect on language should be notable. Because the central argument of these frameworks is that the linguistic system is formed through use and experience, individuals must be highly heterogenous and, indeed, unique in the version of their language that they know. For example, Reference LangackerLangacker (1987, p. 62) suggested that, ‘no two speakers control precisely the same set of units […] even if their speech patterns appear to be identical’ and that ‘the set of units mastered by all members of a speech community might be a rather small proportion of the units constituting the linguistic ability of any given speaker’, a position supported by Reference SchmidSchmid (2015, p. 16), who explains how this is a consequence of entrenchment, which is a highly personal phenomenon.

Even though none of the studies reviewed so far explicitly set out to test the predictions of cognitive linguistics, cumulatively there is enough work that constitutes indirect evidence that there is a substantial amount of individuality in language, especially when considering research on authorship analysis.

Another research strand that is important to consider for individuality is the cognitive linguistic research on language acquisition. It is well established that there are substantial and pervasive individual differences both in the acquisition of our first language and in adulthood and that these individual differences depend on both environmental and cognitive factors, such as individual differences in working memory, executive functions, and statistical learning (Reference Kidd, Donnelly and ChristiansenKidd et al., 2018; Reference Kidd, Bidgood, Donnelly, Durrant, Peter, Rowland, Rowland, Theakston, Ambridge and TwomeyKidd et al., 2020). Of particular note in the context of adults’ grammar differences is the work of Reference DąbrowskaDąbrowska (2012, Reference Dąbrowska, Dąbrowska and Divjak2015, Reference Dąbrowska2018), who presents evidence that native speakers of a language do not necessarily acquire all its core grammatical constructions or reach the same conclusions about how to use them. Because individuals possess different combinations of cognitive attributes, even with identical exposure the grammar they arrive at could be quite different. For example, Reference Dąbrowska, Dąbrowska and DivjakDąbrowska (2015) reports how native speakers of English who are not well educated do not know how to process grammatical constructions that are more frequent in written registers that they are less familiar with, such as sentences with parasitic gaps, quantifiers, or even the passive voice. Reference Dąbrowska, Dąbrowska and DivjakDąbrowska’s (2015) conclusion, in line with cognitive linguistic proposals, is that even though it appears as if all members of a community share the same grammar, this is not really the case.

In addition to experiments, cognitive linguistic research using corpora has also shown that linguistic individuality is strong. De Reference SmetSmet (2016) was among the first studies to show how to use corpora to test hypotheses about individual authors’ cognition and, in recent times, this approach has grown considerably, especially in historical linguistics (e.g. Reference Schmid and MantlikSchmid and Mantlik, 2015). Research in this area is growing fast, with studies such as Reference Petré and Van de VeldePetré and Van de Velde (2018), Reference Anthonissen and PetréAnthonissen and Petré (2019), and Reference Fonteyn and NiniFonteyn and Nini (2020) all finding that individual differences are substantial and cannot be simply reduced to noise.

In parallel, more evidence of a high degree of individuality in language is also found using synchronic data. A study by Reference MollinMollin (2009) on the language of former British Prime Minister Tony Blair uncovered evidence that collocations can be highly idiosyncratic. Similarly, Reference BarlowBarlow (2013) found that White House press secretaries can be very distinctive in their language, despite the similarities in their linguistic tasks. Another more recent example is Reference Schmid, Würschinger, Fischer and KüchenhoffSchmid et al. (2021), who uncovered strong individual variation in the way the construction [that’s ADJ] is used, both in terms of the adjective selected and in its pragmatic function. Evidence of chunking being idiosyncratic therefore leading to different and unique repertoires was found by Reference Vetchinnikova, Filppula, Klemola, Mauranen and VetchinnikovaVetchinnikova (2017) after discovering that there exist more fixed sequences in corpora of single individuals than in corpora containing language from multiple individuals, a finding indirectly replicated on a larger scale by Reference Dunn and NiniDunn and Nini (2021).

In sum, it is fair to conclude that individual variation in language is present and is extensive. The large effects found in these studies cannot be attributed to just noise in the data as the findings come from the two complementary perspectives of experimental and corpus studies. For this reason, the development of a Theory of Linguistic Individuality that is based on foundational concepts and assumptions of usage-based cognitive linguistics is licensed and desirable in order to inform both research and practice in authorship analysis. This formal Theory of Linguistic Individuality is the object of the next section.

2.1 Introduction

Firstly, it is important to clarify that the term theory is here intended in its scientific sense of a set of principles and laws based on commonly agreed and verifiable facts that form a coherent explanation of a phenomenon. A theory generates hypotheses which in turn lead to empirically verifiable predictions. If these predictions are correct, then the theory is a good model of reality. If not, the theory must be changed accordingly or, in case its foundations are demolished by the evidence, abandoned altogether in favour of another competing theory.

To the best of my knowledge, there are no formal theories of linguistic individuality or idiolect, let alone theories that are specifically tailored to being applied to authorship analysis. This includes Reference Grant and MacLeodGrant and MacLeod’s (2020) Resource-Constraint Theory of Identity, which is not a theory of linguistic individuality but a theory of how individuals use their repertoire of resources to perform a linguistic identity in a communicative situation. Reference Grant and MacLeodGrant and MacLeod (2020) do not explain how and why individuals possess repertoires that are possibly unique to them at a certain point in time. Instead, the theory that is outlined in this section is focused precisely on this aspect and it can be therefore thought of as complementary to Reference Grant and MacLeodGrant and MacLeod (2020).

As detailed in Section 1.4, several important theoretical points have been made and a body of evidence has been gradually accumulated over the years on linguistic individuality but there has not yet been an attempt to tie all these threads together. The present theory is an attempt to ground all the previous proposals and evidence on what we already know about language processing from the point of view of cognitive science using the foundational notions presented in Section 1.3. The goal of this section and of this Element is therefore to introduce a theory of linguistic individuality that starts from scientifically plausible and established notions of language processing to then build on top of these the foundations for a linguistically and scientifically valid authorship analysis.

2.2 Basic Definitions

2.2.2 Grammar

As explained in Section 1.3, in cognitive linguistics a person’s lexicogrammar (or more simply grammar from now on) is seen as a network of units, similar to the left-hand side set in Figure 2 below. Grammars can therefore be represented mathematically as graphs, which are pairs such as:

$G=\left(V,E\right)$

where both $V$ and $E$ are sets, the first one of nodes and the second one of edges.

In this work, the model is simplified by assuming that the grammar of an individual is just the set of nodes, $V$ . In other words, as shown in Figure 2, we can remove the links that connect the units and pretend that the grammar of a person is just an unstructured repository. The set representation is a simplification where we maintain the composition of the elements of the grammar but we remove the connection across these elements for the purpose of simplicity.

This view of the grammar of a person as a finite set of units that they can produce automatically is not unrealistic because, although we can and we do create novel utterances all the time, the set of units we use to make them is dependent on our repertoire, which depends on our memory, which in turn is finite. If we can imagine the grammar of a person as a set, then we can use the notations and tools from another area of discrete mathematics, set theory. Thus, the example set in Figure 2 of the grammar $G$ of an individual $A$ , which we call ${\text{G}}^A$ , would be:

${\text{G}}^A=\left\{\left[ontheotherhand,\right],\left[the\text{N}of\right],\left[\text{P}\text{N}\right],\left[hope\right],\left[\text{A},\right],\dots \right\}$

where each element of the set is separated by a comma and each unit is contained in square brackets.

In reality, this way of representing grammar as a set of units is somewhat comparable to the traditional representation of grammar as a list of rules. Chunks can indeed be equated to rules because, for example, the word-punctuation unit [and,] could be described as the rule of using a comma after the coordinator and. Any complex unit is, in essence, a rule for how to order or combine sub-units over time (see Section 1.3.2 and Reference Christiansen and ChaterChristiansen and Chater, 2016).

Figure 2 Graphical representation of a mental grammar as a set (right) compared with a graph representation (left).

This definition is, however, incomplete because it does not take entrenchment into account. As Reference LangackerLangacker (1987, p. 59) suggests, it is not realistic to assume that there is a hard boundary of entrenchment that makes something a unit for an individual. Instead, it is better to think of units as standing on a continuum of various levels of entrenchment. Something that an author uses more often is more entrenched for them than something that the author almost never uses, even though they are still able to understand it. This is probably what leads to some strings or sentences being units for one person but novel structures for another person, even though this second person can still understand what is being communicated.

Therefore, if we want to represent the grammar of a person as a set, a more realistic model is one where this set is fuzzy. In mathematics, a fuzzy set is a set where each element has a probability of belonging. Rather than, for example, $A=\left\{a,b,c,\dots \right\}$ , a fuzzy set looks like $A=\left\{\left(a,0.3\right),\left(b,0.7\right),\left(c,1\right),\dots \right\}$ , where each element is enclosed in round brackets containing both the element itself and its probability of belonging to the set. We can think of this probability of belonging as the mathematical formalisation of the concept of entrenchment and we can call it $\epsilon$ .

This model of grammar is pictured in Figure 3, where the dashed lines become more solid as entrenchment increases from a theoretical 0, meaning the person has never used a particular structure, which can therefore only be a unit for another person, to an ideal 1, meaning that the unit is maximally entrenched and the speaker produces it completely automatically.

Figure 3 Graphical representation of a mental grammar with different degrees of entrenchment, from a theoretical 1 to a theoretical 0.

As done before, however, although unit membership is continuous, for the sake of this work we can say that any unit with entrenchment above a certain threshold that we call $\pi$ is an element of the grammar of an individual, thus a unit that they can automatically produce. Thus, the grammar of an individual $A$ , ${\text{G}}^A$ , is a set of units obtained by taking only those units for which $\epsilon \ge \pi$ . Because this threshold makes the membership discrete again (either a unit belongs to a grammar or not), then for simplicity we can treat ${\text{G}}^A$ as a normal, non-fuzzy set.

This definition of grammar is still not complete, however, because an individual’s grammar is not a static entity but one that changes over time, even in adulthood. Speakers constantly learn how to use new units and also lose units, depending on their usage and, consequently, the level of entrenchment. Therefore, time is an important and essential component to be added to the definition of a grammar, the final definition of which is Definition 3.

Definition 3. A grammar $G$ of a person $A$ at a time $t$ , or $G_t^A$ , is the set of units with entrenchment $\epsilon \ge \pi$ for person $A$ at time $t$ .

The inclusion of time in the theory is also fundamental for authorship analysis. The fact that we can, at least theoretically, only observe a snapshot of a speaker’s mental grammar at a particular time implies that authorship analysis cannot be equated to fingerprint or DNA identification because these two biological characteristics of a person’s identity do not change over time.

2.2.4 Language and Variety

Because languages constantly evolve, it only makes sense to talk about languages at particular points in time. We therefore define a language as in Definition 5.

Definition 5. A language or variety $L$ including $N$ speakers at a particular point in time $t$ , or $L_t$ , is a fuzzy set of units $\left[u\right]$ with its membership function ${\mu }_L$ being the proportion of the $N$ grammars $G$ for which $\left[u\right]\in G$ .

In simple words, a unit is deemed to be part of a language or variety to the extent that a number of its speakers have it in their grammar (with $\epsilon >0$ ). For example, if we say that in the community for language/variety $L$ half of them have $\left[u\right]$ in their $G_t$ , then ${\mu }_L\left(\left[u\right]\right)=0.5$ . For English we can reasonably argue that $\left[the\right]$ is a unit that all speakers share. Thus, ${\mu }_{English}\left(\left[the\right]\right)=1$ because if we imagine the overlap of all the grammars of its speakers, then this is equal to the number of speakers. In contrast, ${\mu }_{English}\left(\left[jg\right]\right)=0$ , because this is not a sequence of characters/sounds of English. Then, of course, there are many units that are units for only some people, for example ${\mu }_{English}\left(\left[nowt\right]\right)=0.02$ or ${\mu }_{English}\left(\left[Ithen\right]\right)=0.01$ , where the numbers are just made-up examples that ideally would correspond to, respectively, the proportion of speakers of a certain dialect or a certain register. Therefore, dialects, sociolects, registers, or any variety can also be defined following Definition 5, the only change being how its speakers are defined.

An important consequence of defining a language or variety in this way is that units for just one person are part of the language/variety. However, this definition of a language is actually realistic for usage-based frameworks. As explained by Reference SchmidSchmid’s (2015) entrenchment-and-conventionalisation model and as also conceptualised by the view of language as a complex adaptive system, the individual’s mind and the community are linked inextricably and what is a unit only held in the mind of one person today can become a unit that belongs to an entire community tomorrow.

We can imagine two different individuals’ grammars embedded in a language $L$ at a time $t$ like in Figure 4.

Figure 4 Graphical representation of the grammars of two individuals contextualised within a language at a particular time $t$ .

This graphical representation shows how the theory outlined here predicts that there will be units that are shared by individuals and units that only certain individuals possess, as explained in the relevant section above. There will also be many units that none of the two individuals $A$ and $B$ considered in Figure 4 possess and that therefore are outside of both their grammars. However, these units must exist in the mental grammar of other individuals that are not represented in Figure 4 for convenience. In other words, any grammar at any point in time is a subset of the language at that point in time, $G_t\subset L_t$ .

2.3 The Principle of Linguistic Individuality

After establishing the definitions above, it is now possible to use the vocabulary of the theory to express its most fundamental principle, the Principle of Linguistic Individuality:

This principle states explicitly what several theories of language assume: no two individuals share the same version of the grammar of the language they speak and write at any one moment. This also implies that it is unlikely for two individuals to have the exact same idiolect at the end of their lives. That no two individuals are the same or, at least, that it is very unlikely to meet two identical linguistic individuals has been the most fundamental working assumption adopted by studies and casework on authorship analysis. However, as explained in Section 1.4, a substantial body of indirect evidence has been accrued such that it makes sense to consider this assumption a basic principle.

The main reason why each individual’s grammar must be unique is that something that counts as a unit for one individual might not necessarily count as a unit for another individual. Although there is enough evidence to support this claim, what is far less known is the extent to which individuals share units. A weak form of the principle would mean that differences between individuals are minimal and just enough for them to be identified through the language they use, as demonstrated by research in authorship analysis. In contrast, a strong version of the principle could imply that our conception that all speakers of a language share most of its grammar is an illusion given by the fact that, although each individual’s grammar is unique, these grammars are mutually understandable.

2.3.1 The Linguistic Fingerprint

The Principle of Linguistic Individuality therefore comes very close to arguing for a linguistic fingerprint. However, because an individual’s grammar is constantly shaped by usage, the Principle of Linguistic Individuality makes it clear that this metaphor is only valid if we consider a specific time window, unlike actual fingerprints. The theory, however, does not specify exactly the length of this time window $t$ . For the present work, it is only important that this insight is included in the theory, pending future research on the stability of an individual’s mental grammars over time. Because we know that authorship analysis is possible, we do not expect this window to be very short, as analyses of historical data also suggest (Reference FonteynFonteyn, 2021).

Another objection to the notion of linguistic fingerprints apart from change over time has been their incommensurability. For example, Reference CoulthardCoulthard (2004, p. 432) argued that:

the concept of the linguistic fingerprint is an unhelpful, if not actually misleading metaphor […] because it leads us to imagine the creation of massive databanks consisting of representative linguistic samples (or summary analyses) of millions of idiolects […] such an enterprise is, and for the foreseeable future will continue to be, impractical if not impossible [..] any linguistic sample, even a very large one, provides only very partial information about its creator’s idiolect.

Similarly, Reference Grant and MacLeodGrant and MacLeod (2018, p. 83) wrote that ‘it is clear that even vast quantities of linguistic data from many individuals could not fully substantiate Coulthard’s strong assertion of linguistic uniqueness’.

Although the assumption of the impossibility of capturing an idiolect is plausible as an a priori position, this limitation can be challenged by this theory if we consider what we know about the mathematical laws of language. For example, the relationship between word tokens observed and new word types discovered is very well known in linguistics and goes by the name of Herdan’s law (Reference HerdanHerdan, 1960). The law can be described in the following manner:

$V=N^{\alpha }$

where $V$ is the number of types in a corpus and $N$ is the number of tokens in the same corpus and $\alpha$ is an exponent that depends on the data set but that is a positive number smaller than 1. This law is actually also known as Heaps’ law because Heaps demonstrated that it applies to many other phenomena involving a relationship between tokens and types (Reference HeapsHeaps, 1978). Assuming $\alpha =0.7$ , the equation can be graphed as in Figure 5.

Figure 5 Herdan–Heaps’ law with $\alpha =0.7$ .

A consequence of this law is that there are diminishing returns in how many types are discovered the more tokens are seen. For example, let’s assume a scenario where we want to study the vocabulary of a person and we start collecting data from them in batches of 1,000 words. With $\alpha =0.7$ , then after 1,000 tokens are collected we expect to find 126 types that are repeated several times, meaning that 13% of the 1,000 tokens is the vocabulary we discovered. Adding another 1,000 tokens means that we discover about 78 new types, bringing the number of total types to 204, which is, however, only 10% of the tokens. Thus, collecting more tokens leads to a decreasing number of new types being discovered proportionally speaking. In other words, the more tokens we add the lower the probability that we encounter new types. Once we are in the range of millions of tokens, the fraction of them that are different token types is only 1.6% and adding another 1,000 does not really make any difference. Although all this concerns the lexicon, in principle this relationship should apply to grammar as well, if we conceive of a grammar as a repository of units.

Herdan–Heaps’ law does not have an upper boundary, meaning that no matter how many tokens one observes, the number of types will keep growing infinitely. This has been empirically demonstrated on very large corpora of language where increasing the number of tokens always leads to an increasing number of types. This result is however dependent on the fact that a token is defined as a sequence of characters in between spaces, which therefore includes typos or sequences such as haha that can be extended. In reality, there is a point of inflection after which the core vocabulary is exhausted and this core vocabulary is in the order of thousands or tens of thousands of units (Reference Gerlach and AltmannGerlach and Altmann, 2013). The expectation is that the same should be true for a $G_t$ , where a set of units constitutes the core and this number is not so large as to become impractical.

Knowing that such a critical threshold exists, then the empirical question is: how much text does one need to observe to approximate the core grammar of a person, or at very least the core grammar for one register? Although we will not know the exact amount until we have some empirical verification with real data, mathematically and theoretically speaking this task is not as impossible as previously theorised. If Herdan–Heaps’ law applies to $G_t$ , then, although this is infinitely productive, it can still be described, exactly like the lexicon of a language.

2.4 The Statistical Approximation Hypothesis

Before moving on to the next two sections showing how the theory can be implemented in practice, this section establishes a link between the present theory and past work in authorship analysis.

As explained above, there is no good reason to explain why individuals manifest certain preferences for functors which make up the key profiles. For example, saying that an author uses the 12.3% of the times they write a word is meaningless because, firstly, this item does not necessarily constitute a choice in itself and, secondly, it cannot just be due to preference. However, linguistic units as defined in this theory can and indeed often are themselves made up of functors such as the and therefore a connection between this theory and the frequency methods can be established. This connection is formulated as the Statistical Approximation Hypothesis:

The hypothesis is essentially stating that frequency of functors identifies authors because it implicitly contains information about their $G_t$ . For example, consider these two toy grammars from two hypothetical individuals, $A$ and $B$ , sampled from texts they produced within the same time window:Footnote ⁴

$G_t^A=\left\{\left[\text{V N}onthe\text{N}\right],\left[ratherthan\right],\left[give\text{N }to\text{ N}\right],\left[half\text{N}\right],\left[help\text{V}\right],\left[tryto\text{V}\right],\left[sortof\right]\right\}$

$G_t^B=\left\{\left[\text{V N}with\text{N}\right],\left[insteadof\right],\left[give\text{N N}\right],\left[halfof\text{N}\right],\left[helpto\text{V}\right],\left[tryand\text{V}\right],\left[kindof\right]\right\}$

If we then analyse two texts written by these two individuals, $T_t^A$ and $T_t^B$ , where each unit appears only once, we can compile the matrix of word counts in Table 3.

Table 3 Sample of word token count matrix from toy texts written by two individuals.

	on	the	rather	than	give	to	half	help	try	sort	of	with	instead	and	kind
$T_t^A$	1	1	1	1	1	2	1	1	1	1	1	0	0	0	0
$T_t^B$	0	0	0	0	1	1	1	1	1	0	3	1	1	1	1

This matrix of word token counts traditionally used for authorship analysis, especially when only considering function words or other functors, could therefore be a statistical representation of the units in the author’s grammar. This hypothesis effectively explains why frequency of word tokens or short character n-grams work: each unit contains a different configuration of function words and morphemes and, if people’s inventories of units are different, then this will also result in different profiles of frequencies of function words and morphemes because the frequency with which a word occurs reflects an individual’s selection of units.

This hypothesis makes verifiable predictions. If the hypothesis is correct, then methods that better reflect this Theory of Linguistic Individuality should perform at least equally well or outperform methods based on frequencies, which are instead just approximations to the first. The rest of this Element will address this prediction and show how the theory can be translated into a methodology to analyse and compare texts for the purposes of authorship analysis.

3.1 Grammar Induction

Theoretically, there are many ways to extract a grammatical representation from a text or corpus, with varying degrees of cognitive realism. The best function, in particular for the purposes of authorship analysis, can only be found through many empirical studies.

The most basic and yet cognitively plausible way to extract a grammar from a corpus is to use n-grams. These representations are more cognitively realistic than simple words (see Section 1.3.2). In computational linguistics, the first pioneering language models were indeed created using n-grams before eventually becoming more sophisticated through the use of neural networks, making them more effective but less amenable to interpretation (Reference Jurafsky and MartinJurafsky and Martin, 2009). Training a neural network on a corpus could be a way to arrive at a ${\hat{G}}_t$ too but for the purposes of forensic linguistics the lack of transparency is an issue.

Although somewhat acceptable, n-grams are generally unsatisfactory as a representation of grammar because they do not capture sophisticated abstractions or units that are not sequential. For example, word n-grams fail to recognise word forms that belong to the same lemma. Although this could be fixed by using a part-of-speech tagger, it is clearly a less than ideal solution. In addition, any n-gram requires the decision as to what $n$ to use. The ideal grammar induction function should contain as few arbitrary decisions as possible by including knowledge from cognitive science to guide the decisions.

An example of a middle ground between just n-grams and more complex representations is Reference DunnDunn’s (2017) algorithm to automatically extract a construction grammar given a corpus. This algorithm considers various levels of abstraction at the same time, thus returning units that mix words, lemmas, parts of speech, and semantic tags so long as they meet a collocation statistic threshold.

Despite promising advances, at this point in time there are no ready solutions to this problem and, even if there were, it would still be useful to establish a benchmark using the simplest possible method. In this work, therefore, the grammatical induction function is simply the extraction of n-grams. A grammar induction function based on n-grams needs at the very least three parameters to be set: (1) the level of abstraction of the n-gram; (2) the n; and (3) the value of frequency to promote a token to unit.

Four levels of abstraction are considered: character n-grams, word n-grams, part-of-speech (POS) n-grams, and frame n-grams. The last type is a novel way to extract n-grams at a medium level of abstraction, similar to Reference Wible and TsaoWible and Tsao’s (2010) hybrid n-grams, Sinclair’s collocational frameworks (Reference SinclairRenouf and Sinclair, 1991), or the patterns of Pattern Grammar (Reference Hunston, Francis and FrancisHunston and Francis, 2000). These frame n-grams are created by first tagging the data using a POS tagger and then keeping only the tags for nouns, personal pronouns, and verbs, which are the most likely categories to contain content information, and keeping only the word forms for other parts of speech. Table 4 shows examples of 3-grams of these four types using the same short phrase.

As demonstrated in this table, punctuation marks are always retained as if they were a word token.

Due to the explorational nature of the present work, it is useful to include as many values of n as possible. For this reason, the lengths considered are: $2\le n\le 9$ for character n-grams, $1\le n\le 8$ for word n-grams, and $1\le n\le 10$ for both POS n-grams and frame n-grams.

Finally, a minimum frequency could be set in order to decide what n-grams to include in the ${\hat{G}}_t$ and what to exclude as potential noise. For instance, an n-gram like ‘golden_bands_,’ could occur only once in a text and one way to proceed would be to exclude it because there is not enough evidence that this is a unit for the individual who produced it. Thus, one could set the minimum frequency to > 1 to avoid this potential confounding factor. For the experiments illustrated below, however, no minimum frequency was chosen. Any n-gram found even only once was included in the final set representing the ${\hat{G}}_t$ .

3.2 Measures of Similarity

Once a grammar induction function has been applied to a text, a set of units is produced, which is what we call ${\hat{G}}_t$ . As explained in Section 2.2.2, although the best representation for a grammar is as a network, for simplicity a set of units is used instead. The question of comparing grammars therefore boils down to how to measure the similarity between sets. From a purely mathematical perspective, there are many ways to compare sets. To find the right way, one can select the best methods either theoretically or empirically via testing, which is the approach taken in this work.

In its most basic sense, measuring the amount of similarity between two sets of items could simply mean quantifying the overlap, as in Figure 4. That overlap, $G^A\cap G^B$ , contains the following units:

$G^A{\cap }^{\text{}}{G}^B=\left\{\left[dog\right],\left[hope\right], \left[the\text{N}of\right]\right\}$

An alternative mathematical representation of the two overlapping sets embedded in a language $L_t$ in Figure 4 can be obtained by using binary vectors. Similarly to the numeric vectors explained in Section 1.2.1, a binary vector is a vector that can contain only 0 (a feature is absent) or 1 (a feature is present), as in Table 5.

Table 5 Representation of Figure 4 using binary vectors.

Units	$G_t^A$	$G_t^B$	$G_t^A{\cap }^{\text{}}{G}_t^B$
[arguably]	0	0	0
[technician]	0	0	0
[N V N P N]	0	0	0
[although]	1	0	0
[-ness]	1	0	0
[respectfully]	1	0	0
[N V P]	1	0	0
[dog]	1	1	1
[hope]	1	1	1
[the N of]	1	1	1
[However,]	0	0	0
[can V A V]	0	1	0

Table 5 shows more clearly how the two grammars can also have in common the absence of certain units which, however, exist in $L_t$ . An important methodological decision is therefore whether common absences should be counted or not.

Even if we decided to only focus on the common presences, just counting the items in the overlap is not the only way or the best way possible to quantify similarity. In fact, the most common way of measuring the similarity of binary vectors of this kind is the Jaccard coefficient, initially introduced as a way to measure similarities in ecology in Reference JaccardJaccard (1912). This coefficient is calculated using the following formula, where the vertical bars indicate the cardinality of the set; that is, the number of items it contains:

$J_{\left(G_t^A,G_t^B\right)}=\frac{\left|G_t^A{\cap }^{\text{}}{G}_t^B\right|}{\left|G_t^A{\cup }^{\text{}}{G}_t^B\right|}\text{.}$

In other words, the Jaccard coefficient is the ratio between the overlap and the union of two sets. For the example in Table 5, the overlap would be the third column, where a 1 marks the elements that appear in both sets. The union, instead, would be the sum of all the units that appear in either one of the grammars, so the sum of all the 1s. The Jaccard coefficient for the example in Table 5 is therefore $\frac{3}{8}=0.375$ . The Jaccard coefficient is one of the most common ways to measure similarities between two binary vectors because when dividing the overlap by the union one is effectively putting both sets on to the same playing field: when using the Jaccard coefficient the elements that are present in the universe outside of the two sets being compared (in our case $L_t$ ), are not considered.

The Jaccard coefficient has already been used in forensic linguistics. Originally introduced in the field by Reference Grant and CoulthardGrant (2010) as a way to measure similarity between text messages, it was later adopted for various other studies in forensic authorship analysis, such as Reference WrightWright (2013, Reference Wright2017) and Reference NiniNini (2018). The Jaccard coefficient was initially adopted by Reference Grant and CoulthardGrant (2010) precisely because it does not count common absences. The argument is that this is a desirable property when dealing with text messages because these texts are short and therefore the absence of a feature does not automatically mean that that feature is absent in the grammar of the author.

Although Reference Grant and CoulthardGrant’s (2010) assumption is plausible, the question of including or excluding common absences in binary vector comparison is not at all straightforward and it has been widely discussed, for example in biology and ecology (Reference Hayek and HeyerHayek, 1994; Reference WarrensWarrens, 2008, p. 5). To resolve this issue, the approach taken in this work is to carry out a survey of all possible binary coefficients, like Jaccard, and then test empirically which one is the most suitable. For this reason, a set of 22 coefficients is considered. Before presenting the list of coefficients, however, it is necessary to introduce the terminology used to describe them.

Let us imagine two sets such as the ones in Figure 6, which is a more general abstract representation of the linguistic example shown in Figure 4.

Figure 6 Example of intersection between two sets, $A$ and $B$ , in a universe of features $U$ .

Figure 6 shows the quantities that are commonly considered when calculating binary coefficients in both graphical representation and set notation. The figure shows a universe of features $U$ with two sets, $A$ and $B$ , and their intersection. Each quadrant of the features represents a quantity that can be calculated from this setting and that is indicated by a letter in small capitals:

• The number of elements in the intersection, $\left|A\cap B\right|$ , is commonly indicated with $a$ .

• The letters $b$ and $c$ respectively indicate the number of elements that only belong to one of the two sets and that are not shared with the other.

• The letter $d$ indicates everything else in the universe of features that is not in either of the two sets. This can be indicated as in Figure 6 as $\left|\overline{A\cup B}\right|$ , or alternatively, as the number of elements in the universe minus the union of $A$ and $B$ , thus $\left|U\right|-\left|A\cup B\right|$ .

• Finally, the letter $p$ , not in the picture, usually indicates the sum of all the quantities above, so $p=a+b+c+d$ .

This notation is far more convenient because, for example, we can now rewrite the Jaccard coefficient as:

$J_{\left(G_t^A,G_t^B\right)}=\frac{a}{a+b+c}\text{.}$

This way of representing similarity coefficients for binary data is more economical and will be implemented throughout the rest of the work.

The coefficients explored in this work were taken from the comprehensive survey of 51 binary coefficients carried out by Reference Todeschini, Consonni, Xiang, Holliday, Buscema and WillettTodeschini et al. (2012). The analysis of their correlations found that 28 out of 51 were either perfectly or highly correlated with other coefficients in terms of the relative ranking that they produce. Because the use of these coefficients in this work is mostly about producing a rank, highly correlated coefficients can be safely excluded from further consideration. The analyses in the present work are therefore carried out using the coefficients listed in Table 6.

Table 6 List of coefficients considered in the present work. The table lists the name of the coefficient, the formula, the key references in which it was proposed, and its class.

Name	Formula	References	Class
Simple Matching	$\frac{a+d}{p}$	Reference Sokal and MichenerSokal and Michener (1958)	symmetric
Jaccard	$\frac{a}{a+b+c}$	Reference JaccardJaccard (1912)	asymmetric
Russell–Rao	$\frac{a}{p}$	Reference Russell and RaoRussell and Rao (1940)	asymmetric
Simpson	$\frac{a}{\text{min}\left(\left(a+b\right),\left(a+c\right)\right)}$	Reference SimpsonSimpson (1943)	asymmetric
Braun-Blanquet	$\frac{a}{\max \left(\left(a+b\right),\left(a+c\right)\right)}$	Reference Braun-BlanquetBraun-Blanquet (1932)	asymmetric
Ochiai	$\frac{a}{\sqrt{\left(a+b\right)\left(a+c\right)}}$	Reference Driver and KroeberDriver and Kroeber (1932) Reference OchiaiOchiai (1957)	asymmetric
Kulczynski	$\frac{1}{2}\left(\frac{a}{a+b}+\frac{a}{a+c}\right)$	Reference KulczynskiKulczynski (1927)	asymmetric
Mountford	$\frac{2a}{ab+ac+2bc}$	Reference Mountford and MurphyMountford (1962)	asymmetric
Rogot–Goldberg	$\frac{a}{2a+b+c}+\frac{d}{2d+b+c}$	Reference Rogot and GoldbergRogot and Goldberg (1966)	symmetric
Hawkins–Dotson	$\frac{1}{2}\left(\frac{a}{a+b+c}+\frac{d}{d+b+c}\right)$	Reference Hawkins, Dotson, Ramp and SembHawkins and Dotson (1968)	symmetric
Yule	$\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}$	Reference YuleYule (1900, Reference Yule1912)	correlation
Cole	$\frac{ad-bc}{\left(a+c\right)\left(c+d\right)}$	Reference ColeCole (1949)	correlation
Goodman–Kruskal	$\frac{2\text{min}\left(\mathrm{a,}d\right)-b-c}{2\text{min}\left(\mathrm{a,}d\right)+b+c}$	Reference Goodman and KruskalGoodman and Kruskal (1954)	symmetric
Sokal–Sneath	$\frac{a}{\sqrt{\left(a+b\right)\left(a+c\right)}}\frac{d}{\sqrt{\left(d+b\right)\left(d+c\right)}}$	Reference Sokal and SneathSokal and Sneath (1963)	symmetric
Phi	$\frac{ad-bc}{\sqrt{\left(a+b\right)\left(a+c\right)\left(c+d\right)\left(b+d\right)}}$	Reference Pearson and HeronPearson and Heron (1913)	correlation
Sorgenfrei	$\frac{a^2}{\left(a+b\right)\left(a+c\right)}$	Reference SorgenfreiSorgenfrei (1958)	asymmetric
Cohen	$\frac{2\left(ad-bc\right)}{\left(a+b\right)\left(b+d\right)+\left(a+c\right)\left(c+d\right)}$	Reference CohenCohen (1960)	correlation
Consonni–Todeschini 3	$\frac{\ln \left(1+a\right)}{\ln \left(1+p\right)}$	Reference Consonni and TodeschiniConsonni and Todeschini (2012)	asymmetric
Consonni–Todeschini 5	$\frac{\ln \left(1+ad\right)-\ln \left(1+bc\right)}{\ln \left(1+\frac{p^2}{4}\right)}$	Reference Consonni and TodeschiniConsonni and Todeschini (2012)	symmetric

The list also contains information about the class of the coefficient.Footnote ⁵ The main difference is whether a coefficient contains the quantity $d$ or not, meaning whether the coefficient counts common absences, or negative matches, or only common presences, or positive matches. Reference Todeschini, Consonni, Xiang, Holliday, Buscema and WillettTodeschini et al. (2012) classify as symmetric those coefficients that equally weight positive and negative matches, while the asymmetric coefficients are those that only consider positive matches. All the coefficients are similarity (or distance) coefficients, like Delta. Their value ranges from 1, if the two sets are identical, to 0, if the two sets are completely different. The only exceptions are the ones in the correlation class, which instead output a value between -1 and 1 representing the level of correlation between the two binary vectors considered. These correlation coefficients can also be mathematically transformed so that their output is bounded between 0 and 1.

Table 6 shows the variety of possibilities that are available to quantify the similarity between two sets, which demonstrates in principle that it is not immediately obvious how one would choose the way to proceed to measure similarity of binary vectors for authorship analysis. This question is addressed empirically in the rest of this section, where the coefficients above are compared to each other and to Delta.

3.3 A Reinterpretation of the N-gram Tracing Method

Besides the Jaccard coefficient as implemented by Reference Johnson and WrightJohnson and Wright (2014) and Reference WrightWright (2017), another authorship analysis method that uses a binary coefficient to compare sets is the n-gram tracing method introduced by Reference Grieve, Clarke and ChiangGrieve et al. (2019), which is also quite similar to Reference CoulthardCoulthard’s (2013) method.

The n-gram tracing algorithm is the following:

1. 1. Take the disputed text ( $Q$ ) and break it down into n-grams (e.g. of characters or words) of a length n.

2. 2. For each text belonging to the candidate authors:

   1. a. Take the candidate author’s text and break it down into the same n-gram type and length as done for $Q$ so as to create a set of n-grams for each candidate’s text.

   2. b. Take the union of all the candidate’s sets of n-grams.

3. 3. Attribute the $Q$ text to the candidate whose n-gram set shares the highest percentage of n-gram types with $Q$ .

The method can be represented using the following formula:

$S_{\left(Q,A\right)}=\frac{\left|Q\cap \cup T_i^A\right|}{\left|Q\right|}\text{.}$

where $Q$ is the set of n-grams in the $Q$ text, $T_i^A$ is the set of n-grams extracted from the text $T_i$ of the set of texts written by author $A$ , $A=\left\{T_1,T_2,T_3,\dots T_n\right\}$ , and thus ${\cup }^{\text{}}{T}_i^A$ is the union of all these sets of n-grams. The resulting $S_{\left(Q,A\right)}$ is the similarity coefficient that represents how much of $Q$ is in $A$ . Because it is assumed that in all cases $\left|Q\right|<<\left|{\cup }^{\text{}}{T}_i^A\right|$ , which is a realistic assumption in forensic contexts, then the formula above is essentially the same as the one for the overlap or Simpson’s coefficient (see Table 6).

The formula above can be easily adapted to the theory by substituting $Q$ with ${\hat{G}}_t^Q$ , the grammar extracted from the disputed texts, and $\cup T_i^A$ with ${\hat{G}}_t^A$ , the grammar extracted from the texts collected for the candidate. Thus,

$S_{\left(Q,A\right)}=\frac{\left|{\hat{G}}_t^Q{\cap }^{\text{}}{\hat{G}}_t^A\right|}{\left|{\hat{G}}_t^Q\right|}\text{.}$

For example, let us assume we have one disputed text, $Q$ , and we break it down into the set of its constituting n-grams, ${\hat{G}}_t^Q=\left\{\left[respectfully\right],\left[can\text{ V A V}\right],\left[although\right],\left[dog\right]\right\}$ . The number of items in this set is $\left|{\hat{G}}_t^Q\right|=4$ . Looking back at Table 5, we can now calculate $S_{\left(Q,A\right)}$ for candidates $A$ and B.

$S_{\left(Q,A\right)}=\frac{\left|{\hat{G}}_t^Q{\cap }^{\text{}}{\hat{G}}_t^A\right|}{\left|{\hat{G}}_t^Q\right|}=\frac{3}{4}=0.75$

$S_{\left(Q,B\right)}=\frac{\left|{\hat{G}}_t^Q{\cap }^{\text{}}{\hat{G}}_t^B\right|}{\left|{\hat{G}}_t^Q\right|}=\frac{2}{4}=0.5$

Because the score for $A$ is higher, then $A$ is more likely to be the author of $Q$ than $B$ .

The example above shows the comparison of two grammars represented as sets of units using n-gram tracing, or the Simpson coefficient. The three experiments in this section are fundamentally based on the same procedure but use the array of coefficients seen in Table 6 to compare the performance to the benchmark of Delta.

3.4 Experiment 1: The refcor Corpus

The first computational experiment reported in this work is a comparison between Delta and the method described in Section 3.3 based on the present theory. The aim of this experiment is twofold: (1) to test whether this method works as well as or better than Delta, and thus whether there is support for the Statistical Approximation Hypothesis; (2) to test which ones among the binary coefficients and the features are better for identification to substantiate the validity of the theoretical claims made above. Providing the answers to these research questions can inform the pursuit of an application of the present theory to authorship analysis.

This first experiment uses the English section of the refcor corpus of literary works (Reference Jannidis, Pielström, Schöch and Vitt.Jannidis et al., 2015; Reference Evert, Proisl and JannidisEvert et al., 2017).Footnote ⁶ This section of the corpus contains texts collected from Project Gutenberg – more precisely, 75 novels from 25 authors with each author contributing 3 novels. Although texts of this kind are not particularly forensically realistic, novels from famous authors have the advantage of containing a lot of data, which is a requirement to experiment with various parameters.

As a benchmark to evaluate performance, the binary coefficients listed in Table 6 are compared to the best Delta coefficient types tested by Reference Evert, Proisl and JannidisEvert et al. (2017): Cosine Delta and Cosine Delta with Evert’s modification, from now on called Evert’s Delta. In contrast to Reference Evert, Proisl and JannidisEvert et al. (2017), however, the present experiment does not measure performance using a clustering technique but by using the k-nearest neighbours algorithm with k = 1, as most commonly done for analyses using Delta. In simple words, this means that the candidates are ranked from most similar to most dissimilar to the questioned text and the most similar candidate is assigned as the author of the questioned text. The algorithm adopted for this experiment was the following:

1. (1) For each $Q$ length: 50, 100, 300, 500, 1000, 2000, 5000.

2. (2) For each K length: 2,000, 5,000, 10,000, 20,000, 50,000, 100,000, 130,000.

3. (3) For each text in the refcor corpus:

   1. (a) the text was removed and labelled as the Q text;

   2. (b) a random sample of the Q texts of size equal to one of the lengths in (1) was taken;

   3. (c) the remaining texts in the corpus were combined by author to form one large sample for each author;

   4. (d) a random sample was taken from each author’s corpus equal to one of the lengths in (2);

   5. (e) the Q random sample and the authors’ samples were recombined into one data set;

   6. (f) this data set was broken down into n-grams as shown in Table 4;

   7. (g) the coefficients in Table 6 were calculated. In addition, Cosine Delta and Evert’s Delta were also calculated to be used as benchmarks. For these, only the 3,000 most frequent features in the corpus were considered (Reference Evert, Proisl and JannidisEvert et al., 2017);

   8. (h) the rank of the correct author was determined. If there was a tie, then the max method was used, meaning that both authors received the lowest rank (e.g. if author A and author B are both tied in first position, they both receive the second position). This is the most conservative way to calculate a rank so that if the correct author is tied with someone else this is still considered a failed test.

The cycle above was repeated five times to mitigate the effect of chance, which means that for each text in refcor five random samples for each sample size were extracted and the results discussed below are the average values of these five repetitions.

Figure 7 is a visual representation as a heatmap of the performance of the coefficients and of the features. Each cell of the matrix represents the median rank of the correct author for that combination of coefficient and feature. Darker shades indicate lower ranks, meaning that the darker the cell is, the better the performance of that combination. To aid interpretation, both rows and columns were sorted by rank so that the combinations of features and coefficients close to the bottom left corner are the ones that best perform overall.

Figure 7 Heatmap visualising the matrix with each coefficient as rows and each feature as columns. The cells are the median rank for each combination of coefficient and feature. The order of the rows and columns is given by the rank.

Several facts about the coefficients are evident from this graph. Firstly, coefficients at the top of the matrix such as Mountford, Russell–Rao, or Simple Matching do not perform well, no matter what feature is used. In contrast, the best coefficients are Cole, Consonni-Todeschini 5, Yule, Kulczynski, and Phi. The Simpson coefficient, which is essentially n-gram tracing, performs well but not as well as these other coefficients. Three of these coefficients contain information about negative matches because they are either symmetrical or correlational coefficients, which is very surprising given the small sizes of Q considered. The Delta coefficients do not perform as well as one would expect given previous studies but it is evident that Cosine Delta is a valuable coefficient if used with the right features, character 4-grams, 3-grams, or 2-grams, or POS 1-grams or 2-grams. Another interesting point to make is that Jaccard does not perform very well overall. As for the features, the evidence points to the best ones being long character n-grams, in particular n = 8 to n = 9, followed by word n-grams, n = 2 to n = 3, then frames with n = 2 to n = 3, and finally POS with n = 4 to n = 5.

Figure 8 shows the relationship between the accuracy of the identification, in the form of median rank, and the feature types. The value on the vertical axis is the median rank achieved by each feature (only considering the best coefficients: Cole, Consonni-Todeschini 5, Kulczynski, Phi, and Yule) while the horizontal axis shows the increasing length of the sample for the author, while each section of the graph represents the values for a size of Q.

Figure 8 Chart showing the median rank of the correct author depending on the length of the Q text and the length of the authors’ samples for the best feature of each type.

The figure suggests that even though the correct author might not be always the most similar one to Q, it is very often at least in the top half of the ranking, out of 25 possible positions.

The figure also makes very clear that the best feature is overwhelmingly character 9-grams, almost always followed by word 2-grams, and then frame 3-grams and POS 5-grams. Despite their lower performance, the last two are still valid features, especially when the length of the texts becomes larger and the gap between the feature types becomes small. This pattern means that the feature type becomes less important when a lot of comparison data is available. For the strictest test of a Q text of 50 tokens, the results suggest that it is possible to narrow down the correct author within the top three using character 9-grams only when 130,000 token known samples are available. However, even for texts as short as 50 tokens, it is still possible with any feature type to narrow down the search to the top half of a list of 25 authors. Performance then improves as Q and K samples become longer, as expected, until reaching the median rank of 1 for character 9-grams when the Q text is 300 tokens and the known samples are 130,000 tokens. When the Q text is 2,000 or 5,000 tokens, then known samples of 20,000 tokens are sufficient to obtain a correct attribution.

Figure 9 shows the performance of the coefficients. This figure should be read in the same way as Figure 8, the difference being that the dots represent the top-performing coefficients plus Cosine Delta and Simpson (n-gram tracing), which are added for comparison.

Figure 9 Graph showing the median rank of the correct author achieved as a function of the length of the candidate authors’ samples and the size of the Q text for some of the top performing coefficients (Cole, Kulczynski,) plus Cosine Delta and Simpson.

The most evident pattern is the gap between Cosine Delta and the other coefficients when short Q texts are considered, with Cosine Delta being considerably outperformed. The only setting in which Cosine Delta is comparable to the binary coefficients is when the Q text is 5,000 words and this is compatible with previous results (Reference Proisl, Evert, Jannidis, Schöch, Konle and PielströmProisl et al., 2018). In contrast, the remaining binary coefficients are all pretty much equal in performance. The only other pattern to note is that Cole seems to be even better than others when Q is very short. It seems therefore that Cole should be considered the best coefficient overall because it performs well for both short and long Q texts. It is crucial to note that Cole is a correlational coefficient that includes information about common absences and the fact that such a coefficient work very well even for short Q texts means that ignoring common absences leads to a loss of information. In conclusion, the results confirm that the binary coefficients work as well if not better than Delta and that the best coefficients are the ones that count negative matches.

3.4.2 Maximising Known Sample Length

The maximum known sample limit of 130,000 was chosen as this is the amount of data available for the author with the least amount of data. However, considering the above finding that an increase of the sample sizes leads to better performance, it would be interesting to find out what happens when much larger samples are considered, such as for the twelve authors in the corpus that have at least 300,000 tokens of data.

The same experiment was therefore carried out but this time keeping the known sample length constant at 300,000 tokens and varying only the length of Q and the features and coefficients. The only drawback of this follow-up experiment is the number of candidate authors is reduced by half and it is therefore more likely that the correct author is identified by chance.

The results are presented in Figure 10 and Figure 11, which respectively show the percentage of times that the correct author is found to be the most similar to Q or within the top three most similar authors to Q.

Figure 10 Percentage of times that the correct author is the most similar to Q depending on the size of Q for 300,000 token known samples and for four features, character 9-grams, frame 4-grams, POS 5-grams and word 2-grams. Each graph belongs to one of the top-performing coefficients (Cole, Consonni-Todeschini 5, Kulczynski, Phi, Simpson, Yule, and Cosine Delta for reference).

Figure 11 Percentage of times that the correct author is within the top three most similar authors to Q depending on the size of Q for 300,000 token known samples and for four features, character 9-grams, frame 4-grams, POS 5-grams and word 2-grams. Each graph belongs to one of the top-performing coefficients (Cole, Consonni-Todeschini 5, Kulczynski, Phi, Simpson, Yule, and Cosine Delta for reference).

All results are greater than the chance baseline (8% and 25%). The figures show that, for Q texts as short as 50 tokens, more than 40% of the time the correct author is the top one out of twelve and in more than 60% of the time it is one of the top three. This performance increases gradually until it reaches close to 100% for Q texts with 5,000 tokens. Comparing the results of the binary coefficients to the performance of Cosine Delta, the latter only improves with the increase in Q size and for short Qs it remains around or just above baseline chance.

The results remain unchanged in terms of which feature is best (character 9-grams and word 2-grams) and show that, at this size, the coefficients that still perform well are only Cole, Kulczynski, and Simpson.

Although these results could suggest that adding more data to the known samples improves the results, the much smaller pool of authors could also be responsible for this improvement. For this reason, Figure 12 plots the performance for these same 12 authors using 150,000 token samples vs 300,000 token samples, only using the Simpson coefficient.

Figure 12 Accuracy of identification for Simpson coefficient for the four features (character 9-grams, word 2-grams, frames 4-grams, and POS 5-grams), different lengths of Q, and two known sample sizes, 150,000 tokens or 300,000 tokens.

The graph shows that in most cases adding 150,000 tokens does not really justify the improvement obtained. This conclusion seems to be in line with what has been proposed in Section 2.3.1 regarding the effect of the Herdan–Heaps’ law on the collection of an individual’s full grammar.

3.5 Experiment 2: The c50 Corpus

The first experiment and its follow-ups have been carried out on a corpus that is realistic for literary applications of authorship analysis but unusual for forensic linguistics. In addition, because the English refcor is made up of texts that are not synchronic, a possible objection could be that the findings do not generalise to modern texts. Apart from this limitation, there is also a problem of replicability. Outside of some natural sciences, it is unlikely that a single experiment can find incontrovertible evidence of a widely generalisable phenomenon and replications are therefore necessary.

For this reason, Experiment 2 is designed as a replication experiment involving modern data and it was carried out only after the results of the previous experiments were obtained. Thus, this study can be seen as a pre-registered study, with the previous results providing the hypotheses to confirm. The goal of this second experiment is to verify whether the same conclusions that were obtained in the refcor corpus are also found in a completely different and unrelated data set including modern data.

The corpus chosen for this experiment is the c50 corpus (Reference Houvardas, Stamatatos, Euzenat and DomingueHouvardas and Stamatatos, 2006), which contains 50 authors, each having 100 documents, 50 for training and 50 for test, sampled from the corporate and industrial news category of the Reuters Corpus, Volume 1 (Reference Lewis, Yang, Rose and LiLewis et al., 2004). All texts in this corpus are therefore sampled from compatible registers because most of the aspects of the situation of production are compatible. The reason why this corpus was chosen is twofold: firstly, it contains many possible candidates and a substantial amount of data for each author, which is necessary to test the most extreme parameters of known sample size; secondly, the c50 corpus has been used by several independent authorship analysis studies and it is therefore possible to compare the present results to those.

3.5.3 Results

The results in c50 confirm the conclusions reached using refcor, as demonstrated by the heatmap in Figure 13, a replication of Figure 7.

Figure 13 Heatmap visualising the matrix with each coefficient as rows and each feature as columns for the c50 corpus. The cells are the median rank for each combination of coefficient and feature. Rows and columns are ordered by rank.

This figure shows the same coefficients and features being the best for both corpora, with just slight variations. Some binary coefficients are again better than Cosine or Evert’s Delta. For c50, it seems that it is harder to detect which coefficient is the best since many of them perform equally well and are able to detect the correct author within the top ten most similar author to the Q text. Regarding features, again short word n-grams and long character n-grams are the best ones, followed by frames and then POS. It is hard to understand which features are the best specifically but the important point is that the overall pattern is confirmed. The results demonstrate that the tendencies discovered in refcor generalise to an unseen modern data set.

To compare these results to previous studies, Table 8 shows the top ten configurations of coefficients and features with the highest percentages of accuracy, plus the best results for frames and POS n-grams for comparison.

Table 8 Top ten performing feature and coefficient configurations in terms of percentage of times that the correct author is top or within the top three. In addition, the best-performing configuration with the frames and POS n-gram types are displayed for comparison.

Rank	Feature	Coefficient	% top 1			Rank	Feature	Coefficient	% top 3
1	words 3	Cole	73.2			1	words 3	Phi	91.36
2	words 3	Phi	72.96			2	words 3	Cole	91.32
3	words 3	Kulczynski	72.24			3	words 3	Kulczynski	90.8
4	characters 9	Phi	72			4	characters 9	Phi	90.72
5	characters 8	Phi	71.88			5	characters 8	Phi	90.64
6	words 3	Simpson	71.44			6	words 2	Phi	90.64
7	characters 7	Phi	70.92			7	words 3	Simpson	90.32
8	words 2	Phi	70.72			8	characters 9	Cole	89.84
9	words 2	Cole	70.28			9	characters 7	Phi	89.76
10	characters 9	Cole	70.2			10	words 2	Cole	89.6
…	…	…	…			…	…	…	…
46	frames 4	Cole	59.2			50	frames 4	Cole	79.24
…	…	…	…			…	…	…	…
90	POS 5	Phi	48			88	POS 5	Phi	68.16

These results are very surprising because the accuracy levels of this experiment are more or less the same as the ones obtained using machine learning and are comparable if not equal to the performance of very complex and computationally demanding neural network approaches.

A small difference between the two studies is that, compared to refcor, for c50 it seems that even rather long word/punctuation n-grams are useful. This could be because there is a lot of formulaic language in the corpus. A manual inspection largely confirms this hypothesis as there are several word/punctuation 7-grams shared by authors between their training sample and the test sample, even though there are no duplicate texts in the corpus. Table 9 shows a random selection of these 7-grams.

Table 9 Random selection of word/punctuation 7-grams only found in common between an author’s training texts and the same author’s test texts.

the internet society, which helps develop

sato, an analyst at ubs securities

at the start of this year.

, ‘ said mms international analyst katherine

estimates from first call. Reporting dates

july, when the british colony of

software that controls the basic functions of

Although it could be argued that topic is having an impact here, this is actually not correct. The longer an n-gram is, the less likely is its re-occurrence due to topic. These are therefore original encodings of meaning that the author repeats in similar circumstances. For example, there are six different authors that use the word July with colony within 5 words to the left or the right or there are three authors that use the word controls or control with software within 5 words to left or right. None of these authors, however, uses the exact same 7-gram. The objection to this argument is that certain words contained in the 7-grams are topical and are indeed found only in texts written by a certain author, for example SATO or Internet Society. These problems of the c50 corpus are not easy to address because they are intrinsic to the data set. However, the present experiment can be directly compared to all other studies that worked on this same corpus and that could therefore exploit the formulaic nature of this register.

The only results for which it is guaranteed that topic information is ignored are frame and POS n-grams. Although their position in the ranking in Table 8 is low, it is important to notice that these results are way better than chance and that they are not too far off from the state-of-the-art performance of machine learning techniques. When using frame 4-grams with Cole the correct author is first about 60% of the time (and within the top three 79% of the time), while using POS 5-grams with Phi the correct author is top almost half of the time (and within the top three 70% of the time). Table 10 presents a selection of random n-grams of these two types that are found across training and test texts for only one author. These n-grams, which largely have nothing to do with topic, can be considered either real grammatical units belonging to the grammar of the author or very close proxies to these units.

Table 10 List of random examples of frame 4-grams and POS 5-grams that are found in at least two texts, one training and one test, belonging to only one author.

Frame 4-grams	POS 5-grams
[NN NNS VBP excessive]	[, VBP JJ CD NNS]
[domestic NNS who VBP]	[VB NNP NNS TO IN]
[due mainly to lower]	[DT NN VBD A CD]
[NNS that previously VBD]	[POS CD NNS VBD NN]
[VBZ VBN the chinese]	[, WP NNS VBP VBG]
[NNS VBD by three]	[NN IN NNP RB NN]
[that what VBZ up]	[DT NNP VBD CD NNP]
[in an antitrust NN]	[DT NNP TO DT NN]
[VBD almost one NN]	[WDT VBZ NNP NNP TO]
[NN during NNP and]	[IN JJR NN CC JJ]

In conclusion, it is fair to say that the results observed for the refcor corpus are not just idiosyncratic to the Gutenberg novels because the findings generalise to an unseen modern data set. The fact that these c50 texts are more formulaic means that the details of the results are slightly different but the general patterns hold. Most importantly, these results suggest that there is no substantial loss in performance using a method based on this theory compared to machine learning or neural network approaches that use frequency information. This finding is worthy of further scrutiny and can be considered evidence in favour of the Statistical Approximation Hypothesis.

3.6 Experiment 3: The Likelihood Ratio Framework

To bring this theory to fruition for real-life forensic linguistic casework, the likelihood ratio approach, which is the standard way in which forensic sciences present their results, should be adopted.

The role of the forensic scientist in court is to help the trier of fact’s determination and the likelihood ratio represents the forensic scientist’s contribution which the trier of fact, typically the jury, uses to arrive at their determination. During a trial, jurors hear a number of witnesses and at the end they express their judgement on whether the defendant is or is not guilty. We can call this final expression of judgement using Bayesian statistics terminology the posterior odds. The way the jury reaches this conclusion is by adjusting these odds as they hear more about the case and the evidence. Each bit of evidence moves the final posterior odds in favour of the prosecution or defence hypothesis according to its strength. This probability adjustment is the contribution of each piece of evidence and, mathematically speaking, is measured by the likelihood ratio. The forensic expert is not concerned about the initial (or prior) odds or the posterior odds: their job is just to present the evidence relevant to their expertise so that they can help the jury to adapt the odds of a guilty verdict accordingly.

Although this approach is well developed and used extensively in other forensic sciences, forensic authorship analysis is lagging behind. However, recent important work on the adoption of the likelihood ratio framework in the context of authorship analysis adopting a distance-based approach has been carried out by Reference IshiharaIshihara (2021a) using Cosine Delta. As Reference Carne and IshiharaCarne and Ishihara (2021) argue, a feature-based method is theoretically and empirically superior to a distance-based method. The former implies the calculation of a likelihood ratio for each feature (e.g. word or n-gram) and then their fusion in a final likelihood ratio. The latter, instead, is based on one coefficient that summarises this variation. Despite the theoretical superiority, they conclude that the use of distance measures can often be more practical and easier to compute, a claim empirically confirmed by Reference IshiharaIshihara (2021b).

To calculate likelihood ratios starting from a distance coefficient such as Cosine Delta or the binary coefficients examined in the present work, the following steps should be followed:

1. (1) A comparison data set of similar background data to the test data is compiled.

2. (2) The coefficient is calculated for all possible combinations of same author and different author in the background data.

3. (3) The two distributions of the coefficient for both same-author and different-author comparisons are extracted from the background data.

4. (4) The coefficient is calculated for all possible combinations of same author and different author in the test data.

5. (5) For each pair of either same-author or different-author in the test data, the likelihood ratio is calculated as the odds of obtaining that value in the case of same author vs different author for the background data.

6. (6) The resulting likelihood ratio expresses the odds of observing a certain value of the coefficient in the hypothesis that this coefficient is the result of a same-author comparison vs the hypothesis that this coefficient is the result of a different-author comparison.

This series of steps can be applied to any distance/similarity coefficient and the next section demonstrates this.

3.6.2 Experiment 3.1: c50 Background Data

In this first experiment, the training half of the c50 is used to extract the distributions of the Simpson coefficient in the case in which the coefficient measures a different-author pair vs the case in which the coefficient measures a same-author pair. These distributions can be seen in Figure 14.

Figure 14 Density graphs showing the distribution of the Simpson coefficient for the four features considered (word 2-grams, character 9-grams, frame 4-grams, and POS 5-grams) for same-author comparisons vs different-author comparisons in the ‘known’ half of the c50 corpus.

The density graphs clearly show how same-author comparisons result in higher values of the Simpson coefficient in all four feature types. Although the peaks for same-author comparisons vary slightly, from eyeballing the graph it would be fair to conclude that if a comparison results in a Simpson coefficient larger than 0.5 it is very likely that this is because the candidate author is the author of the Q text. These odds can be expressed through a likelihood ratio.

This operation of calibration can be performed in various ways but, following Reference IshiharaIshihara (2021a), the logistic regression method was applied here, as implemented by van Reference LeeuwenLeeuwen (2015). In addition, rather than producing likelihood ratios, we adopt log-likelihood ratios, which have the advantage of expressing the ratio with a positive number if the evidence supports the same-author hypothesis or a negative number if the evidence supports the different-author hypothesis.

To assess the usefulness of these log-likelihood ratios we need the second half of the c50 corpus. The background data is needed to generate a logistic regression model that transforms Simpson coefficients into log-likelihood ratios. The performance of this model is measured on the unseen half of the c50 based on the extent to which these log-likelihood ratios make correct predictions.

The performance of a likelihood ratio is assessed by using a coefficient called $C_{llr}$ . If the value of this coefficient is above 1, then this means that the information returned by the likelihood ratios is not useful. Any value below 1 instead means that there is useful information in the likelihood ratios and the closer the value of $C_{llr}$ is to 0, the better. $C_{llr}$ is made up of two components: $C_{llr}=C_{llr}^{min}+C_{llr}^{cal}$ . The former measures the performance in terms of discrimination while the latter in terms of calibration. Table 11 shows the values of these performance metrics for this first test.

Table 11 Table indicating the values of $C_{llr}$ and $C_{llr}^{min}$ for each feature, all tested on c50 with the first half of c50 as background data.

	$C_{llr}$	$C_{llr}^{min}$
word 2-grams	0.30	0.29
character 9-grams	0.28	0.27
frame 4-grams	0.49	0.48
POS 5-grams	0.56	0.55

All the $C_{llr}$ are indeed lower than 1, meaning that they provide useful information. The most effective features are word 2-grams and character 9-grams, as expected, even though we see that frames and POS do indeed contain useful information. The results can also be visualised in the form of Tippet plots, another standard way to report results about likelihood ratios.

Figure 15 shows the Tippet plot for word 2-grams only as an example. In this plot, the value of the log-likelihood ratio is on the horizontal axis and the proportion of cases that score as much or greater on the vertical axis, with the two lines representing, on the right, same-author comparisons and, on the left, different-author comparisons. For example, we observe that almost 90% of same-author comparisons have a log-likelihood ratio above 0, which is only the case in less than 10% of different-author comparisons. Log-likelihood ratios equal to or greater than 2 are only obtained by same-author comparisons and never by different-author comparisons. The same distribution, in reverse, is seen for values lower than -2. We can therefore conclude that these results are well calibrated for this data set.

Figure 15 Tippet plots demonstrating the performance of the log-likelihood ratio for word 2-grams using the first half of c50 as background data.

3.6.3 Experiment 3.2: refcor Background Data

The intent of the second experiment is instead to verify the generalisability of the findings of the first. The same procedure as above is repeated, with the exception that the background data is not the first half of c50 but the refcor corpus. In order to make this computational experiment fair, only one configuration of Q and known sample length was selected: 500 tokens for Q length and 20,000 tokens for the known samples. In this way, the sizes are roughly the same across the two data sets. Apart from the similarity of size, however, the two data sets are rather different from each other, since refcor is mostly constituted by nineteenth-century novels.

Firstly, let’s examine the distributions of the Simpson coefficient for each feature type in the refcor.

Figure 16 Density graphs showing the distribution of the Simpson coefficient for the four features considered (word 2-grams, character 9-grams, frame 4-grams, and POS 5-grams) for same-author comparisons vs different-author comparisons in the refcor corpus.

Figure 16 shows how the distributions in refcor are less skewed and look as if they are almost normally distributed, in contrast to c50. Although the difference is less prominent, in refcor we see again how same-author comparisons obtain scores for Simpson that are higher than different-author comparisons. Even though the shapes of the two distributions are different, it is possible to say that if a Simpson coefficient is greater than 0.4 then it is more likely that the comparison is a same-author comparison. In other words, a similarity of that magnitude is only likely if the Q text has been written by the same author from whom we extracted the known sample. The test to determine if this is a correct guess is again the evaluation of how useful the likelihood ratios are in measuring these odds.

The performance results are presented in Table 12. Except for character 9-grams, all other features have values of $C_{llr}$ lower than 1. These scores are larger than in Experiment 1, meaning that the performance is poorer, which is not surprising. The only exception is word 2-grams, where actually the $C_{llr}=0.43$ is not at all incompatible with the values found in Experiment 1.

Table 12 Table indicating the values of $C_{llr}$ and $C_{llr}^{min}$ for each feature, all tested on c50 with the refcor corpus used as background data.

	$C_{llr}$	$C_{llr}^{min}$
word 2-grams	0.43	0.29
character 9-grams	1.04	0.27
frame 4-grams	0.92	0.48
POS 5-grams	0.90	0.55

The fact that there is a detectable signal even when the background data is so different is a surprising finding. The inconsistency between $C_{llr}$ and $C_{llr}^{min}$ suggests that this poor performance is due to lack of calibration, not lack of discrimination. As before, the Tippet plot in Figure 17 shows what these performance metrics mean in the case of word 2-grams. This Tippet plot is quite similar to the one in Figure 15 obtained when c50 is used as background data

Figure 17 Tippet plots demonstrating the performance of the log-likelihood ratios for word 2-grams using refcor as background data.

4.1 The Cognitive Reality of n-grams and the Individuality Equilibrium

In much of recent computational research in authorship analysis, word n-grams, POS n-grams, and long character n-grams have largely been ignored. When their frequency is used, word or part-of-speech n-grams simply do not work as well as single words or short character n-grams. Reference Antonia, Craig and ElliottAntonia et al. (2014), who considered the frequency of n-grams precisely because of their cognitive realism, also concluded that single words are indeed more effective than n-grams.

Although cognitive realism is not much discussed in modern authorship analysis, the current state of the art seems to suggest a false opposition between scientific plausibility and efficiency. However, if instead of counting how many times a feature occurs the presence or absence of a feature is considered, then n-grams either outperform or perform as well as traditional frequency-based methods. This is a significant finding because it opens the way to authorship analysis methods that are both cognitively plausible, justified theoretically, explainable, and, with more research, also effective and efficient.

In this study, the features that were selected to approximate a linguistic unit are simple n-grams. As explained in the theory, a set is only a rough approximation to a real mental grammar, since the way a mental grammar is modelled in cognitive linguistics is as a network. However, the representation used in this work is more plausible than the traditional model used in authorship analysis of a bag-of-words. Thus, in addition to the theoretical arguments in its favour, the successful results of the studies above point to this set of n-grams model being, at the very least, a good enough approximation to reality despite this simplification. This is especially the case for long character n-grams. Following cognitive linguistics, this theory predicts that the word does not have special unit status in a person’s grammar. The fact that character 9-grams are the best feature generally speaking can be seen as a confirmation of how a usage-based cognitive linguistic theory of language is a valid one for authorship analysis. The use of characters allows the identification of units that are both plausible and distinctive enough for each author but that would not be captured if a traditional grammatical segmentation or tokenisation is adopted.

The computational experiments above also suggest how the fewer atomic elements to form an n-gram there are, the larger is the n you need to achieve good accuracy. These results are in line with what is predicted by the constraints on human working memory predicted by well-established cognitive psychology theories. The fact that the size of each of the best n-gram features is compatible with the predictions made by these cognitive theories is encouraging because it is suggestive that these units could be real mental units. The fact that these units can distinguish authors is consequently explainable by the Principle of Linguistic Individuality.

N-grams range from short sequences that are very common (character 2-grams) to very long n-grams that tend to be unique even for the individual who uses them (word 7-grams). For an authorship analysis method based on binary coefficients, the performance is poor for both of these types of n-gram but the range between these two points will be like a parabola, with a peak signalling the best n-gram length to capture individuality. These n-grams are in the ‘Goldilocks zone’ of the linguistic individuality space: they do not appear in everyone’s grammar so as to be useless, yet they are not so rare and unique as to only be uttered once and never repeated. Instead, they recur just enough in the language of an author to make it identifiable. This sweet spot of individuality is like an equilibrium between absolute pervasiveness and absolute uniqueness, and it varies in a predictable fashion depending on the feature type.

The hypothesis put forward to explain this effect is that when an n-gram is short and common, then it is likely that the n-gram is not a real linguistic unit for an individual but a sub-component of a unit. For example, character 1-grams are unlikely to be units but they are definitely components of units. Similarly, a word 10-gram is unlikely to be a unit and more likely to be a structure containing linguistic units, thus a unique creation that an individual would not and could not store because of working memory constraints. Instead, n-grams such as word 2-grams or character 9-grams or POS 5-grams are the right size to be realistic units because they are convenient enough to be memorised for reuse.

Another tentative conclusion is that the peak of this individuality equilibrium changes with the register. We can propose this conclusion because, for example, the best $n$ for n-grams in c50 is slightly larger than for refcor. A potential explanation could be that the more idiomatic and repetitive a register is, the more likely it is that individuals process it in larger chunks, as this way of processing would be more advantageous in those cases.

All in all, despite their simplicity, it is possible that n-grams are relatively good approximations to real linguistic units. However, new methods based on this theory should also consider more flexible and comprehensive sequences as candidate units instead. The grammar induction function used is in itself a way to test various hypotheses of how the linguistic individual processes language. The function used in this work is quite basic and does not allow for units of various sizes to be extracted. However, better functions would select units of variable sizes, as long as certain requirements, potentially cognitively realistic ones, are met.

4.2 Evidence for the Statistical Approximation Hypothesis

This notion of an individuality equilibrium is very well connected to the Statistical Approximation Hypothesis introduced in Section 2.4. In a nutshell, this hypothesis proposes that frequency-based methods for authorship analysis are essentially a numeric summary of a $G_t$ . If this is true, then what explains our ability to identify an author, even when using frequency methods, is still the Principle of Linguistic Individuality. The only difference is that, when using Delta, instead of measuring $G_t$ directly a statistical imprint of this unique grammar is captured through word or character counts. The prediction made by this hypothesis is therefore that a more direct method of capturing the grammar of an individual should be equally good or superior in performance to the frequency methods.

This is precisely what is seen in the studies reported above. Especially in the results for the c50 corpus, methods introduced in this work based on binary coefficients of n-grams perform equally well if not better than methods using frequency and far more complex statistical procedures, such as machine learning algorithms. In addition, another important pattern uncovered in these studies is the complementarity of the binary coefficients and Delta. For example, when the character n-grams are short then Delta works well and binary coefficients do not. When, instead, the character n-grams are long it is the binary coefficients that work well while Delta fails. The performance of Delta decreases gradually as the performance of the binary coefficients increases. In other words, Delta works very well only when a sub-section of the space of features is considered, typically the one composed of the top most frequent words, which tend to be, precisely, sub-units that constitute larger n-grams, which are instead more likely to be real mental units. This fascinating phenomenon is therefore remarkably suggestive of what should be observed if the hypothesis was correct: the frequency of sub-units stops being useful when considering presence/absence of full units because the information being picked up is the same.

This is only a tentative conclusion, however, since this pattern could also have other explanations. For example, it is possible that frequency methods using character 2-grams and Phi using character 9-grams are complementary because they capture useful but different information about authorship which is not captured by the other. Or it is possible that the Principle of Linguistic Individuality is responsible for only some of the variation in functor frequencies, while other aspects of linguistic individuality, which are not covered by the present theory, contribute to explaining the rest of the variation. The only way to answer these questions is through more empirical work.

4.3 Capturing an Individual Grammar in Full

The (un)feasibility of collecting an entire individual’s grammar was addressed theoretically in Section 2.3.1 in relation to Herdan–Heaps’ law. After the empirical studies above, it is possible to advance this discussion a little further by reflecting on a few key points. Firstly, the empirical explorations revealed that, in contrast to traditional frequency-based methods for which the addition of more features does not lead to improvement in performance, for the authorship analysis methods based on binary coefficients the more features the better. Secondly, although an improvement was noted when reaching 300,000 tokens, at least for the refcor corpus we have evidence that 150,000 tokens are almost as good as 300,000.

The latter result precisely reflects the consequences of the Herdan–Heaps’ law for authorship analysis. After 150,000 tokens, adding the same amount over again to the known sample leads to very little difference in identification performance, which seems to suggest that a critical level corresponding to the coverage of the core grammar has been hit. It would be difficult to imagine that reaching, for example, one million tokens for an author would massively change the results and we know this because of Herdan–Heaps’ law. In sum, at least for refcor, we have a threshold after which we can say that we captured the closest approximation to the full grammar of an individual, at least for one particular register. Although these 150,000 tokens do not exactly contain the entirety of the grammar of a person for a register, they are a good enough approximation for authorship analysis because they might represent the core grammar of the individual.

Another important point emerging from these studies is that a large amount of comparison data can compensate for small disputed texts. The results show how even texts as short as 50 words could be attributed much better than chance as long as hundreds of thousands of words of data are available for comparison. This is a very important finding for authorship analysis and points to the far greater importance of having a large comparison data set than a large questioned data set.

In conclusion, there are two take-home messages for authorship analysis that should be considered for further scrutiny. The first is that, contrary to what has been hypothesised until now, it is actually possible to capture a very close approximation to a person’s grammar. We do not know yet whether this approximation only concerns one single register or if these thresholds are register-specific but we know that it is possible, a fact that should encourage more studies on other types of data. The second message is that, once these known samples are created, then the length of the Q text is not as important.

4.4 The Evaluation of Similarity

The importance of being able to weight the linguistic similarities between two sets of texts has been extensively pointed out by scholars in forensic linguistics. For example, Reference Solan and TiersmaSolan and Tiersma (2005, pp. 163–4) noted that ‘pointing out the similarities between documents without taking into account the likelihood of such overlap, especially by people in the academic community, and without any analysis of the differences between the authors, is not good science’ and that ‘without taking base rates into account, it is impossible to know how much to make of each observed similarity’. Similarly, Reference CoulthardCoulthard (2004, pp. 444–5) wrote: ‘work by many people on a large number of cases has shown that there is no longer any dispute that the occurrence of shared identical items is conclusive evidence that two texts have not been independently created; what remains to be agreed is how few shared identical items are necessary to support a decision’.

Firstly, the present studies demonstrate that pointing out the similarities alone is actually not enough. The most successful binary coefficients, like Phi or Cole, included information about negative matches, or common absences. This was especially true for the cases including large comparison samples, suggesting that once a large amount of data is available, then absences of features are indeed evidence of absence in an individual’s grammar. Thus, in contrast to a priori speculations, empirical evidence demonstrates that absences have a role to play in authorship analysis.

Secondly, the studies also demonstrate that the advancement called for by Reference Solan and TiersmaSolan and Tiersma (2005) and Reference CoulthardCoulthard (2004) must and can be done using the likelihood ratio framework. The likelihood ratio analysis demonstrated in Section 3.6 is precisely a way to evaluate the shared similarities between two samples, one being the Q text and the other the sample of a known author, compared against the baseline of what is similarity by chance in a comparable sample. The likelihood ratio is therefore the way to express numerically and then verbally how much to make of the observed similarity. Much more research must be done in this area to arrive at standards of practice that are widely agreed upon. However, the studies in Section 3.6 and their promising results should pave the way for conducting further work in the near future.

Although the assumption about the values associated with these thresholds of similarity should be that they vary case by case, at least for now the results of Section 3.6 show us that there could be more universality in it than one could in principle assume. The fact that thresholds obtained from nineteenth-century data work for modern texts in another register is at the very least surprising. And yet, to a certain extent, the results can be justified theoretically by the Principle of Linguistic Individuality: if individuality in language is pervasive and considerable and our feeling that we all use the same grammar is an illusion, then it makes sense that two independently produced texts have very low thresholds of similarity compared to two texts produced by the same author. Although this critical threshold will vary depending on the coefficient and the feature type used, its existence can be verified relatively quickly once more evidence is available through replication of this study over many different data sets. In the a priori unlikely case that ‘universal’ thresholds of similarity exist, then one could foresee a future where the forensic linguist can immediately evaluate whether doing a full analysis is advisable or not by simply measuring the similarity, even without calculating a likelihood ratio.

In conclusion, though, whatever future empirical studies will show, it is undeniable that the move from authorship analysis to forensic authorship analysis can only happen with the adoption of the likelihood ratio framework, especially as this is readily applicable as demonstrated in this work.

4.5 An Individual’s Grammar As a Behavioural Biometric

A biometric is a measurement of a physiological or behavioural human feature that meets four properties (Reference Jain, Ross and PrabhakarJain et al., 2004):

1. (1) universality: each individual has the feature;

2. (2) distinctiveness: each individual is different in regard to this feature;

3. (3) permanence: the feature is invariant over time;

4. (4) collectability: the feature can be measured.

We can then define a biometric system as a set of methodologies and tools to use a biometric to identify individuals. Reference Jain, Ross and PrabhakarJain et al. (2004) list various physiological and behavioural biometrics, such as DNA, fingerprints, gait, iris and retina, vein patterns on a hand, voice, and signatures.

The argument that authorship analysis is a biometric system is not new. Reference Narayanan, Paskov and GongNarayanan et al. (2012, p. 4), for example, wrote that ‘stylometric fingerprints can be considered a [sic] extension of behavioral biometrics’. Similarly, Reference Pokhriyal, Tayal, Nwogu and GovindarajuPokhriyal et al. (2017) claimed that the extraction of stylometric markers constitutes a cognitive biometric, even though they do not refer to any research that has to do with writing, linguistics, or cognitive science.

Although this and other studies called for stylometry to be listed as biometric, this is technically and theoretically incorrect because stylometry is not the biometric itself, the feature being measured, but only a biometric system, the tools through which we measure a characteristic. The missing piece that seems to be ignored is: what human biometric characteristic is measured with stylometry? This is not a trivial question because it is not possible to conclude that whatever these ‘stylometric fingerprints’ measure is a biometric without an understanding of what is being measured.

The argument proposed in this section is that the present Theory of Linguistic Individuality allows us to rightly call authorship analysis a biometric system measuring a specific behavioural biometric of an individual, their $G_t$ . If we understand linguistic individuality as an emerging property of an individual connected to the development of their unique repository of lexicogrammatical units, then we can also check how and to what extent the criteria for a biometric are met for $G_t$ . The Principle of Linguistic Individuality tells us how $G_t$ meets the first three criteria for a window of time $t$ . Cognitive usage-based linguistics provides evidence that language is a learned behaviour and it also explains why and how individuals differ in the creation of their own personalised mental grammar. The last criterion of collectability has been demonstrated empirically in this work but it is generally speaking work in progress as new and better ways to approximate $G_t$ are developed over time. It is therefore this link to the science of language that enables us to say with confidence that authorship analysis can be classified as a behavioural biometric system.

4.6 What About Style?

The elephant in the room of the present Element is the concept of style. As it stands, the present theory does not include this notion, which is, however, a concept called upon relatively often in research in authorship analysis. How does style fit into this theory?

As explained in Section 1, style is linked to register: a style is a variety, so a constellation of linguistic features, that is associated with a social group, community, or individual when producing a text in a particular communicative situation. This definition of style is compatible with the functional linguistics tradition, such as work done by Reference BiberBiber (1988) and Reference Biber and ConradBiber and Conrad (2009) but also Reference Halliday and MatthiessenHalliday and Matthiessen (2004) and Reference Hasan and WebsterHasan (2009b, Reference Hasan and Webster2009a). This tradition teaches us how style and register are not in an additive relationship: one cannot simply subtract register from a text and get the style. Instead, a style can be seen as a register that belongs to a group. For example, let’s take academic prose. If one collects a large sample of academic texts, all the constellation of linguistic features that they have in common could be called the register of academic texts. However, if one collects academic texts written by undergraduate students, one is observing the way this group interacts with the context, thus getting the register of a social group, which is what we call a style. Similarly, if one collects academic texts written by a group of individuals who have never encountered this type of text then the variety of language they produce is going to be their style of academic text. In reality, a better term for this type of style is code, as proposed by Reference Hasan, Cloran, Butt and WilliamsHasan (1996) and applied to authorship analysis by Reference Nini and GrantNini and Grant (2013).

That the notion of style is insufficient and unrealistic to characterise authorship analysis can be confirmed by carrying out a thought experiment on what constitutes a ‘stylistic choice’, as in Reference McMenaminMcMenamin’s (2002) sense. Let us assume that an individual selects the word get as opposed to obtain because they have never come across the latter; can we call this a stylistic choice? Or, alternatively, can an individual who uses the double-object construction give me it, which is attested in northern dialects of British English, be compared to other individuals who are native speakers of other varieties of English for whom this construction is deemed ungrammatical? Or again, in the example above of undergraduate students writing academic prose, to what extent can their overuse of complex sentences as opposed to complex noun phrases be called a stylistic choice rather than a lack of familiarity with these structures due to lack of exposure? If a choice is not in someone’s repertoire, then it cannot possibly be called a ‘choice’ at all for that person.

Modelling individuality as the result of stylistic choices implies that we all have all options at our disposal and we select the ones we want to use. This conceptualisation of idiolect, which is consistent with Turell’s notion of idiolectal style, would model a person’s idiolect as the personalised sets of choices among several options. However, as demonstrated by the above thought experiments, this understanding of individuality is not necessarily realistic in all situations. Instead, given what we know from cognitive usage-based linguistics studies about language production, it is actually more likely that very often we do not have so much of a choice of what to say because our language production is a reflection of the level of entrenchment of the units that make up our grammar.

For the reasons outlined above, continuing to think about authorship analysis as the comparison of styles is simply untenable linguistically and scientifically. The implications of this conclusion for authorship analysis is that, unlike Reference KestemontKestemont’s (2014) analogy, authorship analysis should not be akin to Morelli’s method of recognising painters from the way they painted ears, noses, leaves, clouds, or other elements that are frequent but not essential. How does style fit in in this theory, then? The answer is in all likelihood that some of the aspects that are typically called ‘style’ in the context of authorship analysis are instead better explained by the Principle of Linguistic Individuality and the theory outlined in this work. This is particularly the case for what Reference McMenaminMcMenamin (2002) calls ‘habits’ that make up our style.

Having said that, though, because of the nature of language production in writing, the present theory might not fully account for all authorial information in a text. For example, it is very common for an author to write a piece of text and then go back and change it. Perhaps the author looks up a word in a dictionary and substitutes it with another word and so on. All these mechanics of revision are not taken into account in the present theory, which instead focuses exclusively on the act of producing language in the moment.

It is therefore possible that something akin to the naive understanding of style, such as the notion of aesthetically acceptable or unacceptable, could still play a part in authorship analysis and, if that is the case, this aspect should be included in any future expansion of this theory. The way this must be done, however, is again by including insight from cognitive science and established scientific knowledge of the act of writing a text. Although there is not much research on cognitive models of linguistic production in writing, the available models suggest that writing consists of two major phases: translation and editing (Reference GalbraithGalbraith, 2009). The former is the process of converting ideas into words and it is likely that the factors explored above in relation to cognitive linguistics explain this process of linguistic production. The editing phase, instead, is a more conscious process that works very differently. This distinction could hold in the future and lead to a differentiation between the selection of entrenched units on the one hand and the choice of style markers on the other. The notion of stylistic choice, which is not valid for the translation phase, could instead apply to style markers that emerge during the editing phase. All in all, more research to disentangle these two types of evidence is needed. For example, it is likely that at least for forensic purposes it is better to focus on translation-generated markers than editing-generated marks, as the latter could be in principle easier to disguise or imitate. This unresolved matter leaves plenty of room for further expansions and enhancements of the present work.

4.7 A New Perspective on Authorship Analysis

The final reflection is on the consequences of this theory and the evidence above for the way authorship analysis is conceived. As previously explained, the present theory suggests that we can abandon the notion of authorship analysis as ‘style detection’. In addition, cognitive linguistics and cognitive science reveal how Reference Mosteller and WallaceMosteller and Wallace’s (1963) implicit model of language production as a word-by-word choice is untenable. Interestingly, as demonstrated empirically in this work, a method of authorship analysis grounded on cognitive theories of language processing instead looks much more similar to the methods commonly employed to tackle plagiarism.

The point being made here is that authorship analysis should be reinterpreted from ‘style analysis’ to a kind of plagiarism detection. This follows logically from this theory. If each individual has a unique repertoire of units, then recognising an author is like detecting self-plagiarism. The empirical component of this work indeed implemented a very simple comparison of common n-grams, which is the classic and most straightforward way to test for plagiarism. The only difference was the size of the n-grams. For evidence of plagiarism, common n-grams longer than the ones that work for authorship analysis are needed. This is because, as empirically shown, once a sequence reaches a certain length then this becomes unique to the author as well, making it therefore a unique fingerprint of the text produced by this person. The mechanics of authorship analysis is not far from that, though, as the present work demonstrated how collections of shorter sequences equally become ‘linguistic fingerprints’ of an author.

For this reason, another important prediction of the present theory is that better success in authorship analysis will be achieved with techniques that are similar to the ones currently employed in plagiarism detection. In this field, brute force common n-grams detection has been superseded by more complex but efficient techniques, such as fingerprinting or alignments algorithms also used in genetics (Reference OakesOakes, 2014). Although these techniques as they stand cannot function for authorship analysis, if this theory holds true it should be possible to tweak them to more efficiently and accurately capture the same kind of authorial signal detected in the present empirical studies. Ideally, the best techniques would be the ones that extend this approach to measure the actual entity being studied – that is, the author’s grammar – such as techniques that extract networks of nested units rather than sequences.

5.1 A Research Agenda for Future Work

There are several lines of enquiry that the present theory leaves open for future studies. It is important to note that these questions existed even before the introduction of the present theory. However, the benefit of having this Theory of Linguistic Individuality is that these questions can now be contextualised and given a proper name. In this way, new knowledge is generated and scaffolded systematically so that progress is made in a precise direction.

What follows is therefore a list of research questions for the future. The answers to these questions would serve to confirm the theory and, if or where this fails, to expand it and modify it. The answers to these questions are of great benefit for both our understanding of linguistic individuality and for the practice in (forensic) authorship analysis.

1. (1) What is the value of $t$ ? This is one of the most important questions because the entire notion of the unique linguistic individual rests on this number. The value of $t$ is essentially the length of the time window during which the grammar of an individual $G_t$ remains constant. Although we do not have data to even make a guess, we know that, at least in most cases, this window is quite large as otherwise we would not be able to make great progress in authorship analysis at all. This question is not going to be easy to answer but there is no doubt that, with laboratory experiments and longitudinal corpus analyses, some approximations can be obtained.

2. (2) How big is a $G_t$ ? Although we do know that the set of units that an individual can use is extremely large, we also know that it is not infinite. If it is not infinite, then there must be a number or at the very least an approximate magnitude of its size that we can estimate. Research of this kind for the lexicon is being carried out and we do have rough estimations of the size of individuals’ lexicons. At the moment there is no research on the size of an individual’s repository of units but there is no reason why this cannot be done in principle. Research on this aspect is marginally important for cognitive linguistics but it would be quite important for forensic linguistics and authorship analysis.

3. (3) How does Herdan–Heaps’ law apply to $G_t$ ? Or, better, what is the relationship between unit types and unit tokens? This research would be important for authorship analysis because, as explained in Section 2.3.1, it would tell us at what point one can say confidently that collecting more data for an individual becomes unnecessary.

4. (4) Is there a critical entrenchment threshold after which an individual is more likely to use a particular unit? Reference LangackerLangacker (1987) argued that this threshold does not exist and that entrenchment is a continuum. Although this is likely to be the case, it is still possible that there is a critical threshold that leads to a sudden higher probability of usage by an individual. If this exists, what is it? This is a very difficult question to answer but answering it would lead us closer to finding methods to extract realistic units. In the present work, the role of entrenchment has been put aside for the sake of simplification, an unrealistic but necessary assumption to carry out initial empirical work. However, the prediction made by this theory is that taking entrenchment into account should improve authorship analysis.

5. (5) Another simplification that was adopted for the present work is the treatment of grammars as sets as opposed to graphs. The obvious next step would therefore be the use of graph theory instead of set theory. This representation of grammar would be more realistic than the one portrayed here but also far more complex. Nonetheless, if the theory is right, authorship analysis methods based on graph theory should outperform the ones based on set theory.

6. (6) As anticipated in Section 4.6, a better integration of this theory with cognitive models of written language production should be pursued. Although the notion of style is not explored in this work, it is possible that it could feature in an expansion of this theory that is better integrated with known cognitive science facts about the process of writing.

7. (7) And, finally, a last but not least important question: is all this valid in other languages? The present work is only based on English and, although the theories and principles that are derived from cognitive linguistics should apply to any language, again this should be confirmed empirically.

5.2 Final Conclusions

In this Element, a Theory of Linguistic Individuality has been delineated with the goal of directing and encouraging future research in authorship analysis towards a direction that is both computationally successful and scientifically plausible. There is one consideration about the practice of authorship analysis that emerges from this work and that should be adopted in the future, especially but not exclusively in forensic authorship analysis: the foundation of the practice of authorship analysis should be based on the science of estimating the similarity between two samples of language. As such, it should rely on our knowledge of how much similarity is to be expected between two samples generated by the same grammar vs two samples generated by two different grammars given the conditions of production of the samples. However, for the field to move towards this goal, it is evident that we need to know more about the way texts are produced. Progress in this direction must advance fuelled by existing widely accepted understanding of human cognition.

5.3 Supplementary Material: Data and Code

The data and the R code used for the experiments can be found here: www.cambridge.org/nini.