Performance and Robustness Analysis of Co-Prime and Nested Sampling

Coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than the Nyquist rate, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This thesis studies the performance of coprime and nested samplers in spatial and temporal domains, when these assumptions are violated. In spatial domain, the robustness of these samplers is studied by considering arrays with perturbed sensor locations (with unknown perturbations). Simplified expressions for the Fisher Information matrix for perturbed coprime and nested arrays are derived, which explicitly highlight the role of co-array. It is shown that even in presence of perturbations, it is possible to resolve $O(M^2)$ under appropriate conditions on the size of the grid. The assumption of small perturbations leads to a novel ``bi-affine" model in terms of source powers and perturbations. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers. This thesis also studies the robustness of coprime sampling to finite number of samples and sampling jitter, by analyzing their effects on the quality of the estimated autocorrelation sequence. A variety of bounds on the error introduced by such non ideal sampling schemes are computed by considering a statistical model for the perturbation. They indicate that coprime sampling leads to stable estimation of the autocorrelation sequence, in presence of small perturbations. Under appropriate assumptions on the distribution of WSS signals, sharp bounds on the estimation error are established which indicate that the error decays exponentially with the number of samples. The theoretical claims are supported by extensive numerical experiments.

ANALYSIS OF MACROINVERTEBRATE COMMUNITIES IN SEASONAL WETLANDS THROUGH TIME, ACROSS SPACE, AND USING SPECIES TRAITS

Restoration of natural wetlands may be informed by macroinvertebrate community composition. Macroinvertebrate communities of wetlands are influenced by environmental characteristics such as vegetation, soil, hydrology, land use, and isolation. This dissertation explores multiple approaches to the assessment of wetland macroinvertebrate community composition, and demonstrates how these approaches can provide complementary insights into the community ecology of aquatic macroinvertebrates. Specifically, this work focuses on macroinvertebrates of Delmarva Bays, isolated seasonal wetlands found on Maryland’s eastern shore. A comparison of macroinvertebrate community change over a nine years in a restored wetland complex indicated that the macroinvertebrate community of a rehabilitated wetlands more rapidly approximated the community of a reference site than did a newly created wetland. The recovery of a natural macroinvertebrate community in the rehabilitated wetland indicated that wetland rehabilitation should be prioritized over wetland creation and long-term monitoring may be needed to evaluate restoration success. This study also indicated that characteristics of wetland vegetation reflected community composition. The connection between wetland vegetation and macroinvertebrate community composition led to a regional assessment of predaceous diving beetle (Coleoptera: Dytiscidae) community composition in 20 seasonal wetlands, half with and half without sphagnum moss (Sphagnum spp.). Species-level identifications indicated that wetlands with sphagnum support unique and diverse assemblages of beetles. These patterns suggest that sphagnum wetlands provide habitat that supports biodiversity on the Delmarva Peninsula. To compare traits of co-occurring beetles, mandible morphology and temporal and spatial variation were measured between three species of predaceous diving beetles. Based on mandible architecture, all species may consume similarly sized prey, but prey characteristics likely differ in terms of piercing force required for successful capture and consumption. Therefore, different assemblages of aquatic beetles may have different effects on macroinvertebrate community structure. Integrating community-level and species-level data strengthens the association between individual organisms and their ecological role. Effective restoration of imperiled wetlands benefits from this integration, as it informs the management practices that both preserve biodiversity and promote ecosystem services.