Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Passive Acoustics: A Multifaceted Tool for Marine Mammal Conservation

Marine mammals exploit the efficiency of sound propagation in the marine environment for essential activities like communication and navigation. For this reason, passive acoustics has particularly high potential for marine mammal studies, especially those aimed at population management and conservation. Despite the rapid realization of this potential through a growing number of studies, much crucial information remains unknown or poorly understood. This research attempts to address two key knowledge gaps, using the well-studied bottlenose dolphin (Tursiops truncatus) as a model species, and underwater acoustic recordings collected on four fixed autonomous sensors deployed at multiple locations in Sarasota Bay, Florida, between September 2012 and August 2013. Underwater noise can hinder dolphin communication. The ability of these animals to overcome this obstacle was examined using recorded noise and dolphin whistles. I found that bottlenose dolphins are able to compensate for increased noise in their environment using a wide range of strategies employed in a singular fashion or in various combinations, depending on the frequency content of the noise, noise source, and time of day. These strategies include modifying whistle frequency characteristics, increasing whistle duration, and increasing whistle redundancy. Recordings were also used to evaluate the performance of six recently developed passive acoustic abundance estimation methods, by comparing their results to the true abundance of animals, obtained via a census conducted within the same area and time period. The methods employed were broadly divided into two categories – those involving direct counts of animals, and those involving counts of cues (signature whistles). The animal-based methods were traditional capture-recapture, spatially explicit capture-recapture (SECR), and an approach that blends the “snapshot” method and mark-recapture distance sampling, referred to here as (SMRDS). The cue-based methods were conventional distance sampling (CDS), an acoustic modeling approach involving the use of the passive sonar equation, and SECR. In the latter approach, detection probability was modelled as a function of sound transmission loss, rather than the Euclidean distance typically used. Of these methods, while SMRDS produced the most accurate estimate, SECR demonstrated the greatest potential for broad applicability to other species and locations, with minimal to no auxiliary data, such as distance from sound source to detector(s), which is often difficult to obtain. This was especially true when this method was compared to traditional capture-recapture results, which greatly underestimated abundance, despite attempts to account for major unmodelled heterogeneity. Furthermore, the incorporation of non-Euclidean distance significantly improved model accuracy. The acoustic modelling approach performed similarly to CDS, but both methods also strongly underestimated abundance. In particular, CDS proved to be inefficient. This approach requires at least 3 sensors for localization at a single point. It was also difficult to obtain accurate distances, and the sample size was greatly reduced by the failure to detect some whistles on all three recorders. As a result, this approach is not recommended for marine mammal abundance estimation when few recorders are available, or in high sound attenuation environments with relatively low sample sizes. It is hoped that these results lead to more informed management decisions, and therefore, more effective species conservation.