Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.Comment: Full version of a paper to appear at SoCG 201

Deterministic Sampling and Range Counting in Geometric Data Streams

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Developmental Effects of Nicotine Exposure in Drosophila Melanogaster

Approximately 12%-20% of pregnant women smoke at some point during pregnancy, and 10% of pregnant women are reported to have smoked during the last 3 months of pregnancy. Smoking during pregnancy leads to developmental health risks for the fetus and child, including increased mortality, low birth weight, and developmental delays. The direct molecular targets of nicotine are nicotinic acetylcholine receptors (nAChRs) due to the similarities in structure between nicotine and acetylcholine. However, in many cases, it remains unclear what molecular events downstream of nAChRs lead to the deleterious effects of nicotine on development. We have established Drosophila melanogaster as a genetic model system to study the developmental effects of nicotine. So far, we have established that nicotine reduces survival and increases development time in a dose-responsive manner. In addition, we have evidence that developmental nicotine exposure may reduce adult body weight, and that ethanol and nicotine act in a non-additive fashion to reduce survival. Finally, we show that nicotine exposure does not appear to affect brain size in developing larvae. Our results show that the effects of nicotine on fly development are similar to those seen in mammals, and establish Drosophila as a model organism for the study of the deleterious effects of nicotine on development