Challenges to studying Zipf’s law of abbreviation in birdsong

Assessing the evidence for ZLA in birdsong presents several challenges that we do not encounter when studying ZLA in humans. First, relative to the number of words in human languages, the number of note types used by most bird populations is small. A small number of note types makes it more difficult to detect a significant concordance between note type frequency and duration [13,19]. If the number of note types is very small, as in the calls of African penguins [40], then even a perfect concordance between the frequency and duration of note types may provide only weak evidence for ZLA. No amount of additional study can resolve this problem. Thus, we may never be able to say with confidence that ZLA exists in some populations. Instead, researchers interested in ZLA in birdsong may need to assess large numbers of populations and draw conclusions based on the full body of evidence.

A second challenge stems from the fact that different birds in the same population can have very different note type repertoires [47]. In humans, individuals in a population that shares a language are likely to use similar sets of words with similar frequencies. Thus, it may be reasonable to study ZLA at the population level. Researchers can select representative multiauthor texts and assess the concordance between the length and frequency of use of each word using simple rank correlations [13]. In contrast, in many bird species, individuals in the same population use different and sometimes non-overlapping sets of note types [47]. This makes it difficult to adequately sample the use of note types in those populations. The problem is compounded by the fact that, in at least some species, song durations themselves appear to be constrained: birds that use longer note types sing fewer notes in each song [48,49]. In such species, if birds that sing shorter note types are at least as common as birds that sing longer note types, then we might see patterns consistent with ZLA at the population level even if no individual bird uses short note types more frequently than it uses long ones. However, such a pattern would not provide evidence for the principle of least effort proposed to underlie ZLA.

Because the principle of least effort suggests that individuals should use shorter types (i.e., words or notes) more frequently than longer ones, we might wish to look for ZLA at the level of individuals rather than populations. That is, if we choose a random individual from a population, are we likely to find that this individual uses shorter note types more frequently than longer ones? However, this question is made difficult by the fact that songs produced by individual birds in the same population may not be independent. In many species, songs are highly stereotyped and birds learn their songs from others [47,50,51]. If we find that two birds have note use consistent with ZLA, then the pattern may have arisen independently in each bird, or it may have arisen only once and both birds may have learned it from the same source. The second case is weaker evidence for ZLA. Thus, any attempt to study ZLA at the level of individuals must adequately account for the potential non-independence of individuals’ songs.

Finally, perhaps the biggest challenge to studying ZLA in birdsong arises from the inherent difficulty of classifying notes. In human languages, especially in the written form, we can usually agree on whether two units represent the same word or different words [13,15,52]. In birdsong, determining whether two notes belong to the same note type is less straightforward. Notes are usually assigned to types by expert inspection of spectrograms [47,53] or sometimes by computational clustering [42,54–56]. Both methods are highly repeatable [47,56]. However, high repeatability does not ensure that the assigned note types match the intent of the birds that produced those notes. Different birds may produce notes that are very similar but are nonetheless objectively distinguishable among individuals (e.g., because they have slightly different peak frequencies or durations [47]). Should we assign these notes to the same or different types? Similarly, individual birds may produce objectively distinguishable versions of similar notes at different points in their song. In general, we cannot know whether the bird intends to produce slightly different notes, or whether it is attempting to produce the same note each time but its performance is constrained by the position of the note in the song. It is not clear that we can resolve this problem empirically. We could ask whether listening birds can distinguish between notes, but the ability of other birds to distinguish between notes does not necessarily indicate the intent of the producer. By analogy, I might attempt to imitate a word pronounced by my colleague, but listeners may still be able to distinguish my attempt from theirs. In some study populations (e.g., [47]), we may know which birds learned their songs from which others, and we may be able to use analogies in song structure to infer that similar-sounding notes are attempts to produce the same note type. However, for most populations, we do not know which birds learned from which others, and inferring the intent behind individual notes may be fundamentally beyond our grasp. In cases where it is difficult to decide whether notes should be assigned to the same or different types, it is likely that the ambiguous notes will have similar characteristics, and therefore the decision to split or merge types may have little effect on the durations of the note types. However, the decision to split or merge types will necessarily affect the frequencies with which those types appear, and so may affect our inferences about ZLA.

An especially difficult problem arises if classification errors are not independent of note durations. For example, we might be more likely to split longer note types, or to merge shorter note types, simply because longer note types give us more opportunity to identify potential differences among notes. If this is true, then our classification system will overestimate the frequencies of short note types and underestimate the frequencies of long note types in the population repertoire. This would produce a pattern consistent with ZLA not because of how birds use note types but rather because of how we perceive the note types they use.

Overview of the current study

With these challenges in mind, Lewis and colleagues [41] developed a novel method for assessing ZLA in bird populations. This method studies ZLA at the individual level and is appropriate for systems where the repertoires of individual birds may be different and where frequencies of use may be learned and thus non-independent among birds. Here, we introduce the R [57] package ZLAvian (https://CRAN.R-project.org/package=ZLAvian) to implement Lewis and colleagues’ [41] method. We apply ZLAvian to assess evidence for ZLA in 11 populations from 7 species of songbirds with songs archived in Bird-DB, an open access repository of annotated birdsong [58]. We synthesise results from the 11 populations as a first pass at quantifying ZLA in birdsong. Our work offers both a computational tool and an empirical foundation for studying ZLA in bird communication.

Box 1. Formalisation of the method proposed by Lewis and colleagues

Let be the number of note types and b be the number of birds in a dataset. Let be the log-transformed duration of instance of note type produced by bird . Let be the number of times bird produced note type , and let be the total number of instances of note type across all birds in the dataset. Then, the mean log-transformed duration of note type as produced by bird is . Let be the concordance (i.e., Kendall’s ) between the s and the s in bird . Then, is a random variable with variance given by

where is the number of note types in the repertoire of bird [59]. The inverse-variance weighted mean concordance in the population is . This is our test statistic for ZLA in the population.

Each differs from the expected log-transformed duration for note type in the population due to variation within and among birds. We obtain , the expected log-transformed duration for note type in the population, by fitting the random effects model

for each note type using the package lme4 in R [60]. Here, the vector holds the log-transformed durations of all instances of note type in the data, and is an indicator matrix where entry if the ^th instance of note type was produced by bird and otherwise. The vector has entries drawn from andthe vector has entries drawn from . Thus, and are the variances in the mean log-transformed durations of note type among and within birds, respectively. Let D be the deviation matrix in which entry describes the deviation of the mean log-transformed duration of note type as produced by bird from the expectation for the population.

We can obtain one value of under the null by permuting the log-transformed note durations among the note types without changing the set of note types that each bird produces. To do this, let be an matrix with each column equal to , where is some permutation of the s. Then, the matrix represents a resampling of the mean log-transformed note type durations under the assumption that the note type duration is independent of the frequency of use, and where each bird’s note type durations differ from the expected value for the population by the same magnitude as in the original data. Let be the concordance between the s and the s in bird . Then, is one possible value of under the null. We obtain a null distribution for by repeated permutations.

Statistical approach

ZLAvian works by computing the concordances between note type duration and frequency of use in individual birds sampled from a population, computing the weighted mean concordance across all birds in the sample as a test statistic, and comparing that test statistic to a null distribution obtained by permuting durations among note types while maintaining the song structure within and among birds. The analysis requires birdsong data in which notes have been assigned to types, the duration of each note has been measured, and each note can be attributed to an individual bird. In its default mode, ZLAvian computes and studies log-transformed note durations rather than working with raw note durations. This is important for the computation of the null distribution. We explain the decision to study log-transformed note durations after we explain how the null distribution is computed.

To obtain a test statistic for ZLA in a sampled population, ZLAvian computes the mean log-transformed duration of each note type as produced by each bird in the sample, and counts the number of times that each bird produced each note type. Then, it computes the concordance (i.e., Kendall’s ) between the mean log-transformed duration and the frequency of use of note types within birds. This results in one value of for each bird. ZLAvian uses these s to compute , the weighted mean value of in the population. In the default mode, ZLAvian weights each value of by its inverse variance [59]. Because is a random variable, weighting by the inverse variance accounts for the fact that the s in birds with larger note type repertoires provide more accurate estimates of the population mean [61]. However, ZLAvian also offers users the option of computing with each bird weighted equally. In either case, the computed serves as a test statistic but also has a useful biological interpretation. Kendall’s , and therefore , ranges from −1 to 1 and is linearly related to the probability of concordance among observations. That is, if we were to randomly select a bird from the study population and then randomly select two note types as produced by that bird, then is the probability that the longer note type would appear more frequently. This makes an intuitive metric for comparing the observed strength of ZLA across populations.

Next, ZLAvian computes a null distribution for . To do this, it first computes the expected log-transformed duration for each note type in the population. It estimates each expected log-transformed duration as the intercept in an intercept-only random effects model of the observed log-transformed durations for that note type, with a random effect for each bird that produced the note type. This method accords more weight to birds that produced the note type more frequently, because we can more accurately estimate the mean note type duration in those birds. ZLAvian then permutes the expected log-transformed durations among the note types at the population level. Thus, if a note type is assigned a particular duration by permutation, it is assigned that same duration in all birds that produced that note type. This permutation results in a set of population mean log-transformed durations for note types that we might see under the null hypothesis that note type durations and frequencies of use are independent, but it maintains the observed distribution of note type frequencies within and among birds in the population. This accounts for the fact that birds may learn note types and frequencies of use from other birds.

In nature, individual birds may produce the same note types in slightly different ways, and therefore the mean duration of each note type as produced by each bird will differ from the mean duration of that note type in the population. As a result, the same note types may have different rank order durations in different birds. How each bird produces each note type may be learned from other birds, and durations may not be independent among the note types a bird produces. For example, a bird that produces a longer version of one note type may be more likely to produce longer versions of some other note types. We want to account for differences in note type durations among birds under maximally conservative assumptions about how note type versions are learned. To do this, ZLAvian computes the deviation of each bird’s mean log-transformed note type duration from the population mean for each note type that the bird produced. This results in a deviation matrix with one entry per note type per bird. With equal probability, ZLAvian either adds the deviation matrix to or subtracts the deviation matrix from the permuted population mean log-transformed durations. This allows the rank order of note type durations to differ among birds, but maintains the structure of deviations in note type durations within and among birds. The result is a matrix of permuted and adjusted note type durations that might have been produced by birds in the sample if note type durations and frequencies of use were independent.

Finally, ZLAvian computes , the analog of in the permuted and adjusted data. The null distribution for is the set of s computed for every possible permutation of the mean log-transformed note type durations with the positive or negative deviation structures added. In most cases, the set of possible permutations is too large to compute all possible values of . Therefore, ZLAvian estimates the null distribution from a randomly chosen subset of the possible permutations. The p-value for the hypothesis that ZLA manifests in the study population is the proportion of the null distribution in which is equal to or smaller than Lewis and colleagues [41] developed this method to study the concordance between note type durations and frequencies of use in birdsong, but the method is transferable to other taxa and to other measures of production effort (e.g., bandwidth, concavity, excursion [42]).

The log transformation of note durations is important when the observed variability among birds in note type durations is reassigned to the note types after the population mean durations have been permuted. In many bird populations, the variability among individuals in the durations of particular note types scales with the population mean durations of those note types [48] (see also S1 Appendix). This is what we would expect if, for example, there are more opportunities for errors in duration to accumulate in longer note types. Furthermore, the differences among the durations of shorter note types tend to be smaller than the differences among the durations of longer note types – that is, the distributions of population mean note type durations are often right-skewed. If we were to permute the population mean raw durations among the note types and then reassign the original raw deviances to the permuted note type durations, we would often assign large deviances from long note types to note types that received short durations by permutation. As a result, the rank orders of note types with longer durations in the observed data would vary more among individuals in the permuted data than they did in the observed data, and the rank orders of note types with shorter durations in the observed data would vary less among individuals in the permuted data than they did in the observed data. This would change the relative importance of long and short note types in generating concordances at the population level. Log transformation reduces or removes the relationship between the mean and the interindividual variability of note type durations, and eliminates the right skew in the distribution of note type durations. Thus, we can permute log-transformed durations among note types without biasing interindividual variability in the rank orders of note types towards note types with longer observed durations. Some authors have advocated studying median rather than mean durations to assess ZLA in animal communication [20], but log transformation makes the distributions of durations approximately symmetrical, so means and medians are similar. Nonetheless, ZLAvian offers users the option to study ZLA using medians rather than means, and in the supplementary information we show that the qualitative results are similar in the populations we studied (S1 Table). ZLAvian also offers users the option to study raw rather than log-transformed measures of production effort. We do not advocate this when using durations to study ZLA, but it may be appropriate for users studying measures of production effort for which interindividual variability does not scale with the mean.

The method proposed by Lewis and colleagues [41] assesses evidence for ZLA within individuals while accounting for non-independence among individuals due to song learning or stereotypy. However, it does not correct for flaws in the classification of notes into types. Such flaws result in errors in the data, and in general statistical methods cannot correct such errors. However, we can assess how different kinds of note type misclassifications will affect our inferences (S2 Appendix). If we incorrectly merge note types (i.e., assign notes to the same type when they should belong to different types), then we will overestimate the variance of the null distribution and our inferences will be conservative. If we incorrectly split note types, then we will underestimate the variance of the null distribution and our inferences will be anticonservative. Attempts to assess ZLA in birdsong must be interpreted in light of these potential biases.

Application of ZLAvian to birdsong archived in Bird-DB

We downloaded 660 annotations of birdsong representing seven bird species (California thrasher, Toxostoma redivivum; redthroat, Pyrrholaemus brunneus; black-headed grosbeak, Pheucticus melanocephalus; sage thrasher, Oreoscoptes montanus; Cassin’s vireo, Vireo cassinii; western tanager, Piranga ludoviciana; gray shrikethrush, Colluricincla harmonica) from the open access repository Bird-DB [58] on 18 April 2022. Annotations on Bird-DB can include one or more songs, but all songs in the same annotation are from the same bird. The phrases in each annotation are classified to types, and the starting and ending times for each phrase are reported. We used the reported starting and ending times to compute the phrase durations. The recordings represented in Bird-DB were collected on different days and at different locations, and we assumed that each annotation represents a different bird. Most phrases in Bird-DB are monosyllabic, and thus correspond to individual notes. A small number of phrases consist of short sequences of notes that typically appear together. Such polysyllabic phrases are often analysed as single units, and we follow this convention in the body of this paper. Our results are qualitatively similar if we divide the polysyllabic phrases into individual notes and use notes rather than phrases as the primary unit of analysis (S3 Appendix).

We cleaned the annotations downloaded from Bird-DB prior to analysis. Phrase types in Bird-DB are identified by two- or three-letter strings. We excluded any phrase with a type identifier that includes non-alphabetic characters, or that comprises fewer than two or more than three characters. These are likely to be data entry errors, and we cannot confidently assign these phrases to types. We also excluded annotations that include only one repeated monosyllabic phrase type. These annotations may represent alarm calls, and alarm calls may adhere to different rules than other calls or songs. Assessing concordance within annotations requires at least two phrase types, so no information about concordance was lost by excluding annotations that consisted of only one phrase type.

In some cases, songs from the same species on Bird-DB were annotated under different classification systems. We cannot analyse these songs together, because we do not know which phrases in one classification system correspond to which phrases in the other. If we treat phrases from different classification systems as different when they are in fact the same, we will overestimate the number of phrases in the species’ repertoire and underestimate the variances of null distributions, and the p-values we obtain when testing for ZLA in that species will be anticonservative (S2 Appendix). Therefore, when songs from the same species were annotated using different classification systems, we treated annotations classified by each system as different populations. Tests conducted on multiple populations from the same species cannot be regarded as independent, because populations may share phrases and song structures. Nonetheless, tests of different populations that produce similar results can provide corroborating evidence for or against the presence of ZLA in that species.

For each population represented in Bird-DB, we report the number of annotations studied, the total number of phrase types across all annotations, the mean number of phrases and phrase types that appear in each annotation, the mean Shannon diversity of phrase types in annotations, the concordance between phrase type duration and frequency of use at the population level, and the mean concordance between phrase type duration and frequency of use by individuals in the population (i.e., ). For population-level and individual-level concordances, we report the one-tailed p-values testing whether each concordance is more negative than we would expect under the null hypothesis that the note type duration and the frequency of use are unrelated. Thus, small p-values indicate patterns strongly consistent with ZLA and large p-values (i.e., close to 1) indicate patterns strongly contrary to ZLA. Finally, for each population we report the threshold value of that would be necessary to infer statistically significant () support for ZLA at the individual level. We call this the detection threshold. All reported measures were computed by ZLAvian.

Because phrase type repertoires in birdsong can be small, biologically plausible negative concordances between phrase durations and frequencies of use may not be statistically significant in individual bird populations, and researchers may need to synthesise across many populations to understand ZLA in birdsong. To do this, we fit an intercept-only model of in the populations we studied, with a random effect of species to account for correlation among the concordances observed in populations of the same species. This analysis asks whether we would expect the concordance between phrase duration and frequency of use in a randomly selected bird species to be negative, even if concordances are not significantly negative when bird populations are studied individually.

If birdsong adheres to ZLA, we would like to know whether the strength of ZLA in birdsong is different from that in human languages. Following [13], we measured the length in characters and the frequency of use of words in 462 translations of the Universal Declaration of Human Rights downloaded from https://unicode.org/udhr/index.html on 21 May 2023. We computed the concordance between the length and the frequency of use of words in each translation. We compared the mean concordances we observed in our seven bird species to those we found in written human languages using a t-test with a Welch correction for unequal variance. This approach discards information because it averages among populations within each bird species. Therefore, we also fit a Bayesian hierarchical model of in birdsong and written human language, with a random effect of species in the birdsong data, and allowing unequal variance in the birdsong and human language data. We computed the probability of direction for the difference in the mean in birdsong and written human language to confirm the inference from our t-test [62].

To understand how large phrase or note type repertoires in birdsong need to be for researchers to detect statistically significant evidence for ZLA in a population, we regressed the repertoire sizes in the populations we studied on the detection thresholds we achieved for each population, allowing random intercepts for each species. If researchers know the strength of ZLA they hope to detect in a population, this analysis allows them to estimate how large a phrase repertoire they would need for that strength of ZLA to be statistically significant.