  With the dramatic growth in the number of application domains that generate
probabilistic, noisy and uncertain data, there has been an increasing interest
in designing algorithms for geometric or combinatorial optimization problems
over such data. In this paper, we initiate the study of constructing
$\epsilon$-kernel coresets for uncertain points. We consider uncertainty in the
existential model where each point's location is fixed but only occurs with a
certain probability, and the locational model where each point has a
probability distribution describing its location. An $\epsilon$-kernel coreset
approximates the width of a point set in any direction. We consider
approximating the expected width (an \expkernel), as well as the probability
distribution on the width (an \probkernel) for any direction. We show that
there exists a set of $O(1/\epsilon^{(d-1)/2})$ deterministic points which
approximate the expected width under the existential and locational models, and
we provide efficient algorithms for constructing such coresets. We show,
however, it is not always possible to find a subset of the original uncertain
points which provides such an approximation. However, if the existential
probability of each point is lower bounded by a constant, an exp-kernel\ (or an
fpow-kernel) is still possible. We also construct an quant-kernel coreset in
linear time. Finally, combining with known techniques, we show a few
applications to approximating the extent of uncertain functions, maintaining
extent measures for stochastic moving points and some shape fitting problems
under uncertainty.
