IEEE Workshop on Underwater Acoustic Signal Processing, October 1999
Title: Information Entropy of Humpback Whale Song
Authors: Ryuji Suzuki, John R. Buck, Peter L. Tyack

Humpback whales produce songs in their winter breeding seasons in warm water. A song consists of a series of more than 12 discrete units, where a unit is the shortest sound element separated by silence. The fundamental frequency ranges from 30 to 3000Hz, and the duration of the longest observed vocalization session was over 24 hours. Several papers have analyzed the song structure with manual classification and visual inspection. Payne and McVay (1971) proposed a hierarchical syntax, which is currently most commonly accepted, but which also has caused a linguistic question: do songs really have a hierarchical syntax, or is it a result of human bias? Tape-recorded songs were digitized and individual units detected and extracted using an automatical detector. The individual units were encoded into spectrograms with logarithmic frequency axis, and classified in two layers of self-organizing maps (a neural network), symbolizing the entire song. This process was performed separately for each song. The rest of analysis takes the symbolized song sequences to examine the structure of the songs using information theory.

The entropy of the song generation process was estimated with two parametric models and a non-parametric estimator for comparison, rather than justifying the model selection. We chose independent identical distribution, a.k.a. stationary memoryless model and first order empirical Markov model, the two most popular choices in animal communication studies, and the sliding window match length (SWML) estimator (Kontoyiannis et. al. 1998), derived from Wyner-Ziv result (Wyner and Ziv 1989). This SWML estimator is universal, asymptotically unbiased, robust and fast converging. Its operation somewhat resembles LZ’77 universal compression scheme (Ziv and Lempel 1977).

Non-parametric estimates were so low that the redundancy of the messaes were at least 0.75. This indicates that production of the songs are governed by structural or syntactic constraints. Also, the estimates from parametric models were significantly higher than that from the non-parametric estimator, indicating that both memoryless and first order Markov models are too simple to model the syntax of humpback whale songs.

The empirical distribution of individual units were observed to be locally but not globally stationary. This implies that empirical Markov models of any order can not reasonably model the song structure. The autocorrelation of symbolized song sequences exhibited repetitive structure with two different periods, approximately 8 and 200. Such a phenomenon can be simply described by assuming a hierarchy in syntax. Otherwise, without hierarchy, the model has to manage a huge number of parameters. This result is consistent with the Payne and McVay (1971) syntax.

Back to Publications Page