A Study of Mexican Free-Tailed Bat Chirp Syllables: Bayesian Functional Mixed Models for Nonstationary Acoustic Time Series

Abstract

We describe a new approach to analyze chirp syllables of free-tailed bats from two regions of Texas in which they are predominant: Austin and College Station. Our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps. The data are analyzed by modeling spectrograms of the chirps as responses in a Bayesian functional mixed model. Given the variable chirp lengths, we compute the spectrograms on a relative time scale interpretable as the relative chirp position, using a variable window overlap based on chirp length. We use two-dimensional wavelet transforms to capture correlation within the spectrogram in our modeling and obtain adaptive regularization of the estimates and inference for the regions-specific spectrograms. Our model includes random effect spectrograms at the bat level to account for correlation among chirps from the same bat and to assess relative variability in chirp spectrograms within and between bats. The modeling of spectrograms using functional mixed models is a general approach for the analysis of replicated nonstationary time series, such as our acoustical signals, to relate aspects of the signals to various predictors, while accounting for between-signal structure. This can be done on raw spectrograms when all signals are of the same length and can be done using spectrograms defined on a relative time scale for signals of variable length in settings where the idea of defining correspondence across signals based on relative position is sensible. Supplementary materials for this article are available online.

KEY WORDS:

• Acoustic data
• Bat syllable
• Bayesian analysis
• Functional data analysis
• Functional mixed model
• Isomorphic transformation
• Nonstationary time series
• Spectrogram
• Variable overlap

Acknowledgments

Martinez was supported by a post-doctoral training grant from the National Cancer Institute (R25T-CA90301). Morris was supported by a grant from the National Cancer Institute (R01-CA107304). Carroll's research was supported by a grant from the National Cancer Institute (R37-CA05730) and by Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST). We thank Richard C. Herrick for implementing the functional mixed model into the WFMM software used to obtain the results presented in this work. Additionally, this work was supported by the Statistical and Applied Mathematical Sciences Institute (SAMSI) Program on the Analysis of Object Data. We also thank the editor, associate editor, and reviewers for their insightful comments that helped improve this article.