Asymptotics for Voronoi tessellations on random samples

AbstractLet V(X1,…,Xn) denote the total edge length of the Voronoi tessellation on random variables X1,…,Xn. If X1,X2,… are independent and have a common continuous density f(x) on the unit square which is bounded away from 0 and ∞ then it is shown thatlimn→∞V(X1,…,Xn)n1/2=2∫[0,1]2(f(x))1/2dxc.c.,where c.c. denotes complete convergence

Localization processes for functional data analysis

We propose an alternative to k-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies stabilization criteria, we establish pointwise and global approximation results in function spaces when the number of data curves is large. We exploit this feature to develop the asymptotic theory when a finite number of curves is observed at time-points given by an i.i.d. sample whose cardinality increases up to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For such problems our methods are competitive with and sometimes superior to benchmark predictions in the field. The R package localFDA provides routines for computing the localization processes and the estimators proposed in this article

Observed photodetachment in parallel electric and magnetic fields

We investigate photodetachment from negative ions in a homogeneous 1.0-T magnetic field and a parallel electric field of approximately 10 V/cm. A theoretical model for detachment in combined fields is presented. Calculations show that a field of 10 V/cm or more should considerably diminish the Landau structure in the detachment cross section. The ions are produced and stored in a Penning ion trap and illuminated by a single-mode dye laser. We present preliminary results for detachment from S- showing qualitative agreement with the model. Future directions of the work are also discussed.Comment: Nine pages, five figures, minor revisions showing final publicatio

A note on limit theorems for perturbed empirical processes

AbstractLet Xi, i⩾ 1, be a sequence of i.i.d.Rk-valued random variables with common distribution P. Let Hnn⩾1, be a sequence of distribution functions (d.f.) such that HnH0, where H0 is the d.f. of the unit mass at zero. The perturbed empirical d.f. is defined by F̃n(x):=n−1Σi⩽nx Hn(x−Xi);P̃n denotes the associated perturbed empirical probability measure. Strong laws of large numbers and weak invariance principles are obtained for the perturbed empirical processes (P̃n−P)(f), f ∈F, where F denotes a class of functions on Rk. The results extend and generalize those of Winter and Yamato and have applications to non-parametric density estimation

Cross-entropy optimization for independent process analysis

We treat the problem of searching for hidden multi-dimensional independent auto-regressive processes. First, we transform the problem to Independent Subspace Analysis (ISA). Our main contribution concerns ISA. We show that under certain conditions, ISA is equivalent to a combinatorial optimization problem. For the solution of this optimization we apply the cross-entropy method. Numerical simulations indicate that the cross-entropy method can provide considerable improvements over other state-of-the-art methods