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

Single machine scheduling problems with uncertain parameters and the OWA criterion

In this paper a class of single machine scheduling problems is discussed. It is assumed that job parameters, such as processing times, due dates, or weights are uncertain and their values are specified in the form of a discrete scenario set. The Ordered Weighted Averaging (OWA) aggregation operator is used to choose an optimal schedule. The OWA operator generalizes traditional criteria in decision making under uncertainty, such as the maximum, average, median or Hurwicz criterion. It also allows us to extend the robust approach to scheduling by taking into account various attitudes of decision makers towards the risk. In this paper a general framework for solving single machine scheduling problems with the OWA criterion is proposed and some positive and negative computational results for two basic single machine scheduling problems are provided

Scattering by Interstellar Dust Grains. II. X-Rays

Scattering and absorption of X-rays by interstellar dust is calculated for a model consisting of carbonaceous grains and amorphous silicate grains. The calculations employ realistic dielectric functions with structure near X-ray absorption edges, with resulting features in absorption, scattering, and extinction. Differential scattering cross sections are calculated for energies between 0.3 and 10 keV. The median scattering angle is given as a function of energy, and simple but accurate approximations are found for the X-ray scattering properties of the dust mixture, as well as for the angular distribution of the scattered X-ray halo for dust with simple spatial distributions. Observational estimates of the X-ray scattering optical depth are compared to model predictions. Observations of X-ray halos to test interstellar dust grain models are best carried out using extragalactic point sources.Comment: ApJ, accepted. 27 pages, 12 figures. Much of this material was previously presented in astro-ph/0304060v1,v2,v3 but has been separated into the present article following recommendation by the refere

Analytic Moment-based Gaussian Process Filtering

We propose an analytic moment-based filter for nonlinear stochastic dynamic systems modeled by Gaussian processes. Exact expressions for the expected value and the covariance matrix are provided for both the prediction step and the filter step, where an additional Gaussian assumption is exploited in the latter case. Our filter does not require further approximations. In particular, it avoids finite-sample approximations. We compare the filter to a variety of Gaussian filters, that is, the EKF, the UKF, and the recent GP-UKF proposed by Ko et al. (2007). copyright 2009