  Integration of heterogeneous sensors is a challenging problem across a range
of applications. Prominent among these are multi-target tracking, where one
must combine observations from different sensor types in a meaningful way to
track multiple targets. Because sensors have differing error models, we seek a
theoretically-justified quantification of the agreement among ensembles of
sensors, both overall for a sensor collection, and also at a fine-grained level
specifying pairwise and multi-way interactions among sensors. We demonstrate
that the theory of mathematical sheaves provides a unified answer to this need,
supporting both quantitative and qualitative data. The theory provides
algorithms to globalize data across the network of deployed sensors, and to
diagnose issues when the data do not globalize cleanly. We demonstrate the
utility of sheaf-based tracking models based on experimental data of a wild
population of black bears in Asheville, North Carolina. A measurement model
involving four sensors deployed among the bears and the team of scientists
charged with tracking their location is deployed. This provides a sheaf-based
integration model which is small enough to fully interpret, but of sufficient
complexity to demonstrate the sheaf's ability to recover a holistic picture of
the locations and behaviors of both individual bears and the bear-human
tracking system. A statistical approach was developed for comparison, a dynamic
linear model which was estimated using a Kalman filter. This approach also
recovered bear and human locations and sensor accuracies. When the observations
are normalized into a common coordinate system, the structure of the dynamic
linear observation model recapitulates the structure of the sheaf model,
demonstrating the canonicity of the sheaf-based approach. But when the
observations are not so normalized, the sheaf model still remains valid.

Data Analysis, Statistics and Probability