Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013.

Long-duration animal tracking in difficult lighting conditions.

High-throughput analysis of animal behavior requires software to analyze videos. Such software typically depends on the experiments' being performed in good lighting conditions, but this ideal is difficult or impossible to achieve for certain classes of experiments. Here, we describe techniques that allow long-duration positional tracking in difficult lighting conditions with strong shadows or recurring "on"/"off" changes in lighting. The latter condition will likely become increasingly common, e.g., for Drosophila due to the advent of red-shifted channel rhodopsins. The techniques enabled tracking with good accuracy in three types of experiments with difficult lighting conditions in our lab. Our technique handling shadows relies on single-animal tracking and on shadows' and flies' being accurately distinguishable by distance to the center of the arena (or a similar geometric rule); the other techniques should be broadly applicable. We implemented the techniques as extensions of the widely-used tracking software Ctrax; however, they are relatively simple, not specific to Drosophila, and could be added to other trackers as well.

The Rational Adolescent: Strategic Information Processing during Decision Making Revealed by Eye Tracking.

Adolescence is often viewed as a time of irrational, risky decision-making - despite adolescents' competence in other cognitive domains. In this study, we examined the strategies used by adolescents (N=30) and young adults (N=47) to resolve complex, multi-outcome economic gambles. Compared to adults, adolescents were more likely to make conservative, loss-minimizing choices consistent with economic models. Eye-tracking data showed that prior to decisions, adolescents acquired more information in a more thorough manner; that is, they engaged in a more analytic processing strategy indicative of trade-offs between decision variables. In contrast, young adults' decisions were more consistent with heuristics that simplified the decision problem, at the expense of analytic precision. Collectively, these results demonstrate a counter-intuitive developmental transition in economic decision making: adolescents' decisions are more consistent with rational-choice models, while young adults more readily engage task-appropriate heuristics.