The appropriate use of Zipf's law in animal communication studies

A Zipf plot (or statistic) is a log–log plot of the frequency of occurrence of signalling units (letters, words, phonemes, etc.) against their rank order (1st, 2nd, 3rd). Zipf’s law emerges for almost all languages’ letters and words as an approximate slope of 1 in this log–log plot, a result George Zipf (1949) stated was due to the ‘principle of least effort’ in communication systems, representing a ‘balance’ between the repetition desired by the listener, and the diversity desired by the transmitter. There have been many applications (some correct and some not) of this plot in animal communication studies (reviewed in McCowan et al. 1999) with a recent critique being that of McCowan et al.’s (1999) work by Suzuki et al. (2004), to which we herein reply. The purpose of our reply is to clarify the inferences made in McCowan et al. (1999) as well as the correct use of Zipf’s law in animal communication studies. There is much, on a one-by-one basis, that could be addressed from Suzuki et al. (2004); however, for the sake of brevity, we chose to paint this reply with a broader brush. We summarize the important points just below, which we then address in some additional detail in the main text. (1) Suzuki et al. (2004) claim that McCowan et al. (1999) attributed linguistic properties to Zipf’s law and used Zipf’s law as a language detector. McCowan et al. (1999) ‘never’ attributed linguistic properties to Zipf’s law; on the contrary, we outlined its correct application and the limitations of using this statistic in our paper (which we quote below). It appears to us that the conflict over the Zipf statistic having linguistic value (or semantic content, i.e. meaning) is really between Suzuki et al. (2004) and Cancho & Solé (2003) because Suzuki et al. (2004) state that ‘.Zipf’s law is not an appropriate route to conclude anything about the linguistic nature or potential capacity for communication transfer’ (page 11), while Cancho & Solé (2003) state ‘Our finding strongly suggests that Zipf’s law is a hallmark of symbolic reference and not a meaningless feature’ (page 788). Again, Suzuki et al. (2004) state ‘Zipf’s law is not even a necessary condition for a data sequence to have semantic content.’ (page 14), while Cancho & Solé state, ‘Our results strongly suggest that Zipf’s law is required by symbolic systems’ (page 791). McCowan et al. (1999) only apply the Zipf statistic as an ‘indicator of potential structure’ in the distribution of signals, and then only in a differential sense with changes in the Zipf slopes being indicative of changes in the structural distribution of a signalling system repertoire (and then only at the repertoire level). Because Hailman et al. (1986) did attribute linguistic properties to bird calls using a Mandelbrot fit to Zipf’s law, this paper also may be drawing conclusions contrary to those of Suzuki et al. (2004). (2) Suzuki et al. (2004, page 14) imply that Zipf slopes cannot be used even in a differential sense when they state that it cannot be used ‘as a comparison of two communication schemes’. While we are not certain what constitute ‘two communication schemes’ for Suzuki et al. (2004), we do apply it as an indicator of changes in the distribution of bottlenose dolphin, Tursiops truncatus, signals with age in McCowan et al. (1999) and the distribution of squirrel monkey, Saimiri sciureus, signals with age in McCowan et al. (2002). These results have also been confirmed by a bootstrapping of both infant (!1 month old) and adult signal data sets (see Fig. 1). These bootstrap analyses show that the differences in Zipf slopes Correspondence: B. McCowan, Population Health and Reproduction, School of Veterinary Medicine, University of California, Davis, 18830 Road 112, Tulare, CA 93274, U.S.A. (email: bmccowan@vmtrc. ucdavis.edu). L. R. Doyle is at the SETI Institute, 2035 Landings Drive, Mountain View, CA 94043, U.S.A. J. M. Jenkins is at the SETI Institute, MS 245-3, NASA Ames Research Center, Moffett Field, CA 94035, U.S.A. S. F. Hanser is at the Ecology Graduate Group, University of California, One Shields Avenue, Davis, CA 95616, U.S.A.