High-dimensional Processing Research Articles

Processes of flats induced by higher dimensional processes III

Stationary processes of k-flats in $$\mathbb{E}$$ d can be thought of as point processes on the Grassmannian $$\mathcal{L}$$ k d of k-dimensional subspaces of $$\mathbb{E}$$ d . If such a process is sampled by a (d−k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d−1 and j=0,...,2[r/2]−1, where r is the rank of the manifold $$\mathcal{L}$$ k d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.

A Visualization Technique for Studying the Iterative Estimation of Mixture Densities

Abstract This article focuses on recent work that analyzes the expectation maximization (EM) evolution of mixtures-based estimators. The goal of this research is the development of effective visualization techniques to portray the mixture model parameters as they change in time. This is an inherently high-dimensional process. Techniques are presented that portray the time evolution of univariate, bivariate, and trivariate finite and adaptive mixtures estimators. Adaptive mixtures is a recently developed variable bandwidth kernel estimator where each of the kernels is not constrained to reside at a sample location. The future role of these techniques in developing new versions of the adaptive mixtures procedure is also discussed.

Maximum Likelihood Estimation of Parameters under a Spatial Sampling Scheme

We study in detail asymptotic properties of maximum likelihood estimators of parameters when observations are taken from a two-dimensional Gaussian random field with a multiplicative Ornstein-Uhlenbeck covariance function. Under the complete lattice sampling plan, it is shown that the maximum likelihood estimators are strongly consistent and asymptotically normal. The asymptotic normality here is normalized by the fourth root of the sample size and is obtained through higher order expansions of the likelihood score equations. Extensions of these results to higher-dimensional processes are also obtained, showing that the convergence rate becomes better as the dimension gets higher.

Intelligent Real-Time Control of a Multifingered Robot Gripper by Learning Incremental Actions

Learning control systems are expected to have several advantages over conventional approaches when dealing with complex, high-dimensional processes. One example is the task of controlling grasping operations of a multifingered, mul-tijoined robot gripper, which has been designed and implemented at our robotics lab (the Darmstadt-Hand). The Advanced Gripper Control with Learning Algorithms -AGRICOLA- presented in this paper is able to maintain a stable grasp even if disturbances are applied. Also it works for objects of different sizes for which the grasping has not been learned. Compared to the conventional stiffness approach the performance of the learning system is equal but the design is much easier, since less knowledge about the gripper-hardware has to be taken into account. The main part of the learning control loop is an associative memory storing the grasping behaviour as determined by the choice of an objective function.

Intelligent real-time control of a multifingered robot gripper by learning incremental actions

Learning control systems are expected to have several advantages over conventional approaches when dealing with complex, high-dimensional processes. One example is the task of controlling grasping operations of a multifingered, mul-tijoined robot gripper, which has been designed and implemented at our robotics lab (the Darmstadt-Hand). The Advanced Gripper Control with Learning Algorithms-AGRICOLA- presented in this paper is able to maintain a stable grasp even if disturbances are applied. Also it works for objects of different sizes for which the grasping has not been learned. Compared to the conventional stiffness approach the performance of the learning system is equal but the design is much easier, since less knowledge about the gripper-hardware has to be taken into account. The main part of the learning control loop is an associative memory storing the grasping behaviour as determined by the choice of an objective function.

Processes of flats induced by higher dimensional processes

We discuss a problem which arises in a number of different contexts. The motivation we use is largely derived from an approach through stochastic geometry. A Poisson process of k-flats in d-dimensional Euclidean space gives rise, by means of intersections, to point processes on ( d − k)-flats. The question under investigation is whether or not these induced processes can be used to recover the original. The equivalent cases k = 1 and k = d − 1 have already been discussed and found to yield a positive answer. In this work we show that in all other cases the answer is in the negative. In addition we consider higher dimensional induced processes.

Numerical errors in impurity transport calculations through the boundary of finite-area cells, and a new practical model to reduce such errors

A numerical error which arises during calculation of impurity transport through the boundary is studied. The error increases when rapid change in an impurity profile is expected within the cell near the boundary. The criterion for the cell size to avoid the numerical error is shown, which indicates that a cell with less than 10 −4 nm width is necessary to reduce the error in diffusion simulation of an oxide/silicon structure. A new practical model for impurity flux through the boundary is proposed. Since the model does not require any additional cells at the boundaries, it is easy to implement the model in any higher-dimensional process simulator. The model is applied to the simulations of doped-material source diffusion and also to the boron profile calculation of the channel region in MOS devices, and is found effective to reduce the error due to the finite area of the cell size.

Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

Consider a Poisson point process of density 1 in R d, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e 0, · ··, e n, in Rd and a variety of distances, we obtain the asymptotic behavior of Nd n , the number of points which have e 0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-l p distances related to correlation coefficients.

Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

Consider a Poisson point process of density 1 in Rd, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e0, · ··, en, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn, the number of points which have e0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-lp distances related to correlation coefficients.

Non-causality due to omitted variables

Generally the causal structure of a subprocess of a multivariate stochastic process does not allow conclusions concerning the causal structure of the higher dimensional process. It is well- known that Granger-causality in a bivariate system may be due to an omitted variable. It is also known that non-causality in a bivariate system may theoretically result from neglected variables. Using Canadian income, money and interest rate it is demonstrated that this actually occurs in practice.