| Description |
We study robust estimators for uncertain points and sketching of lines, trajectories, and other shapes. For locationally uncertain points, 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 and other robust estimators of an uncertain point set. More generally, for robust estimators, we also give a result about the robustness of composite estimators: Under mild conditions on the individual estimators, the breakdown point of the composite estimator is the product of the breakdown points of the individual estimators. Another contribution of this work is a sketched representation based on a set of landmarks for geometric objects. Using this representation, we develop a new class of distances for objects including lines, hyperplanes, and trajectories. These distances easily and interpretably map objects to a Euclidean space, are simple to compute, and perform well in data analysis tasks. For trajectories, they match and in some cases significantly outperform all state-of-the-art other metrics, can effortlessly be used in k-means clustering, and fast approximate nearest neighbor algorithms, which greatly improves the efficiency of trajectory similarity search. Under reasonable and often simple conditions, these distances are metrics. We also show how to use sensitivity sampling to approximate such landmarkbased distances, bound the required size of the sketched vector, and give an algorithm to recover a trajectory from its vectorized representation. |
