Two Universality Properties Associated with the Monkey Model of Zipf's Law

The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: (1) the power law exponent converges strongly to $-1$ as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on $[0,1]$; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem for the logarithm of sample spacings from Shao and Hahn, and the second property follows from Anscombe's central limit theorem for a random number of i.i.d. random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.Comment: 14 pages, 3 figure

Calculation of Precise Constants in a Probability Model of Zipf's Law Generation and Asymptotics of Sums of Multinomial Coefficients

© 2017 Vladimir Bochkarev and Eduard Lerner.Let ω0,ω1,⋯,ωn be a full set of outcomes (symbols) and let positive pi, i=0,⋯,n, be their probabilities (∑i=0npi=1). Let us treat ω0 as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the nonincreasing order of their probabilities. Let pr be the probability of the rth word in this list. We prove that if at least one of the ratios log pi/log pj, i,j ∈ {1,⋯,n}, is irrational, then the limit limr→∞pr/r-1/γ exists and differs from zero; here γ is the root of the equation ∑i=1n piγ=1. The limit constant can be expressed (rather easily) in terms of the entropy of the distribution (p1γ,⋯,pnγ)

Linguistic Laws and Compression in a Comparative Perspective: A Conceptual Review and Phylogenetic Test in Mammals

Over the last several decades, the application of “Linguistic Laws” - statistical regularities underlying the structure of language- to studying human languages has exploded. These ideas, adopted from Information Theory, and quantitative linguistics, have been useful in helping to understand the evolution of the underlying structures of communicative systems. Moreover, since the publication of a seminal article in 2010, the field has taken a comparative approach to assess the degree of similarities and differences underlying the organisation of communication systems across the natural world. In this thesis, I begin by surveying the state of the field as it pertains to the study of linguistic laws and compression in nonhuman animal communication systems. I subsequently identify a number of theoretical and methodological gaps in the current literature and suggest ways in which these might be rectified to strengthen conclusions in future and enable the pursuit of novel theoretical questions. In the second chapter, I undertake a phylogenetically controlled analysis, which aims to demonstrate the extent of conformity to Zipf’s Law of Abbreviation in mammalian vocal repertoires. I test each individual repertoire, and then examine the entire collection of repertoires together. I find mixed evidence of conformity to the Law of Abbreviation, and conclude with some implications of this work, and future directions in which it might be extended

Measuring acoustic complexity in continuously varying signals: how complex is a wolf howl?

Communicative complexity is a key behavioural and ecological indicator in the study of animal cognition. Much attention has been given to measures such as repertoire size and syntactic structure in both bird and mammal vocalizations, as large repertoires and complex call combinations may give an indication of the cognitive abilities both of the sender and receiver. However, many animals communicate using a continuous vocal signal that does not easily lend itself to be described by concepts such as ‘repertoire’. For example, dolphin whistles and wolf howls both have complex patterns of frequency modulation, so that no two howls or whistles are quite the same. Is there a sense in which some of these vocalizations may be more ‘complex’ than others? Can we arrive at a quantitative metric for complexity in a continuously varying signal? Such a metric would allow us to extend familiar analyses of communicative complexity to those species where vocal behaviour is not restricted to sequences of stereotyped syllables. We present four measures of complexity in continuous signals (Wiener Entropy, Autocorrelation, Inflection Point Count, and Parsons Entropy), and examine their relevance using example data from members of the genus Canis. We show that each metric can lead to different conclusions regarding which howls could be considered complex or not. Ultimately, complexity is poorly defined and researchers must compare metrics to ensure that they reflect the properties for which the hypothesis is being tested.AK is supported by a Herchel Smith postdoctoral fellowship at the University of Cambridge. Part of this work was carried out while AK was a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville

Measuring acoustic complexity in continuously varying signals: how complex is a wolf howl?

Communicative complexity is a key behavioural and ecological indicator in the study of animal cognition. Much attention has been given to measures such as repertoire size and syntactic structure in both bird and mammal vocalizations, as large repertoires and complex call combinations may give an indication of the cognitive abilities both of the sender and receiver. However, many animals communicate using a continuous vocal signal that does not easily lend itself to be described by concepts such as ‘repertoire’. For example, dolphin whistles and wolf howls both have complex patterns of frequency modulation, so that no two howls or whistles are quite the same. Is there a sense in which some of these vocalizations may be more ‘complex’ than others? Can we arrive at a quantitative metric for complexity in a continuously varying signal? Such a metric would allow us to extend familiar analyses of communicative complexity to those species where vocal behaviour is not restricted to sequences of stereotyped syllables. We present four measures of complexity in continuous signals (Wiener Entropy, Autocorrelation, Inflection Point Count, and Parsons Entropy), and examine their relevance using example data from members of the genus Canis. We show that each metric can lead to different conclusions regarding which howls could be considered complex or not. Ultimately, complexity is poorly defined and researchers must compare metrics to ensure that they reflect the properties for which the hypothesis is being tested.AK is supported by a Herchel Smith postdoctoral fellowship at the University of Cambridge. Part of this work was carried out while AK was a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville

The Phylogeny and Function of Vocal Complexity in Geladas

The complexity of vocal communication varies widely across taxa – from humans who can create an infinite repertoire of sound combinations to some non-human species that produce only a few discrete sounds. A growing body of research is aimed at understanding the origins of ‘vocal complexity’. And yet, we still understand little about the evolutionary processes that led to, and the selective advantages of engaging in, complex vocal behaviors. I contribute to this body of research by examining the phylogeny and function of vocal complexity in wild geladas (Theropithecus gelada), a primate known for its capacity to combine a suite of discrete sound types into varied sequences. First, I investigate the phylogeny of vocal complexity by comparing gelada vocal communication with that of their close baboon relatives and with humans. Comparisons of vocal repertoires reveal that geladas – specifically the males – produce a suite of unique or ‘derived’ call types that results in a more diversified vocal repertoire than baboons. Also, comparisons of acoustic properties reveal that geladas produce vocalizations with greater spectro-temporal modulation, a feature shared with human speech, than baboons. Additionally, I show that the same organizational principle – Menzerath’s law – underpins the structure of gelada vocal sequences (i.e., combinations of derived and homologous call types) and human sentences. Second, I investigate the function of vocal complexity by examining the perception of male complex vocal sequences (i.e., those with more derived call types), the contexts in which they are produced, and how their production differs across individuals. A playback experiment shows that female geladas perceive ‘complex’ and ‘simple’ vocal sequences as being different. Then, two observational studies show that male production of complex vocal sequences mediates their affiliative interactions with females, both during neutral periods and periods of uncertainty (e.g., following conflicts). Finally, I find evidence that vocal complexity can act as a signal of male ‘quality’, in that more dominant males exhibit higher levels of vocal complexity than their subordinate counterparts. Collectively, the work presented in this dissertation presents an integrative investigation of the ultimate origins of complex communication systems, and in the process, it highlights the critical importance of approaching the study of complexity from several scientific perspectives.PHDPsychologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138479/1/gustison_1.pd