Minimizing the Size of the Uncertainty Regions for Centers of Moving Entities

In this paper, we study the problems of computing the 1-center, centroid, and 1-median of objects moving with bounded speed in Euclidean space. We can acquire the exact location of only a constant number of objects (usually one) per unit time, but for every other object, its set of potential locations, called the object's uncertainty region, grows subject only to the speed limit. As a result, the center of the objects may be at several possible locations, called the center's uncertainty region. For each of these center problems, we design query strategies to minimize the size of the center's uncertainty region and compare its performance to an optimal query strategy that knows the trajectories of the objects, but must still query to reduce their uncertainty. For the static case of the 1-center problem in R^1, we show an algorithm that queries four objects per unit time and is 1-competitive against the optimal algorithm with one query per unit time. For the general case of the 1-center problem in R^1, the centroid problem in R^d, and the 1-median problem in R^1, we prove that the Round-robin scheduling algorithm is the best possible competitive algorithm. For the center of mass problem in R^d, we provide an O(log n)-competitive algorithm. In addition, for the general case of the 1-center problem in R^d (d >= 2), we argue that no algorithm can guarantee a bounded competitive ratio against the optimal algorithm.Comment: 22 pages, 3 figures, accepted to LATIN 202

High resolution satellite imagery orientation accuracy assessment by leave-one-out method: accuracy index selection and accuracy uncertainty

The Leave-one-out cross-validation (LOOCV) was recently applied to the evaluation of High Resolution Satellite Imagery orientation accuracy and it has proven to be an effective method alternative with respect to the most common Hold-out-validation (HOV), in which ground points are split into two sets, Ground Control Points used for the orientation model estimation and Check Points used for the model accuracy assessment. On the contrary, the LOOCV applied to HRSI implies the iterative application of the orientationmodel using all the known ground points as GCPs except one, different in each iteration, used as a CP. In every iteration the residual between imagery derived coordinates with respect to CP coordinates (prediction error of the model on CP coordinates) is calculated; the overall spatial accuracy achievable from the oriented image may be estimated by computing the usual RMSE or, better, a robust accuracy index like the mAD (median Absolute Deviation) of prediction errors on all the iterations. In this way it is possible to overcome some drawbacks of the HOV: LOOCVis a reliable and robustmethod, not dependent on a particular set of CPs and on possible outliers, and it allows us to use each known ground point both as a GCP and as a CP, capitalising all the available ground information. This is a crucial problem in current situations, when the number of GCPs to be collected must be reduced as much as possible for obvious budget problems. The fundamentalmatter to deal with was to assess howwell LOOCVindexes (mADand RMSE) are able to represent the overall accuracy, that is howmuch they are stable and close to the corresponding HOV RMSE assumed as reference. Anyway, in the first tests the indexes comparison was performed in a qualitative way, neglecting their uncertainty. In this work the analysis has been refined on the basis of Monte Carlo simulations, starting from the actual accuracy of ground points and images coordinates, estimating the desired accuracy indexes (e.g. mAD and RMSE) in several trials, computing their uncertainty (standard deviation) and accounting for them in the comparison. Tests were performed on a QuickBird Basic image implementing an ad hoc procedure within the SISAR software developed by the Geodesy and Geomatics Team at the Sapienza University of Rome. The LOOCV method with accuracy evaluated by mAD seemed promising and useful for practical case

Robust regression with imprecise data

We consider the problem of regression analysis with imprecise data. By imprecise data we mean imprecise observations of precise quantities in the form of sets of values. In this paper, we explore a recently introduced likelihood-based approach to regression with such data. The approach is very general, since it covers all kinds of imprecise data (i.e. not only intervals) and it is not restricted to linear regression. Its result consists of a set of functions, reflecting the entire uncertainty of the regression problem. Here we study in particular a robust special case of the likelihood-based imprecise regression, which can be interpreted as a generalization of the method of least median of squares. Moreover, we apply it to data from a social survey, and compare it with other approaches to regression with imprecise data. It turns out that the likelihood-based approach is the most generally applicable one and is the only approach accounting for multiple sources of uncertainty at the same time

Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)

In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structure on the covariance or reducing the number of input vectors.Comment: 10 pages - paper accepted at ICML 201