Estimation of the mean of stationary and nonstationary Ornstein-Uhlenbeck processes and sheets

We consider the problem of estimating an unknown parameter m in case one observes in an interval (rectangle) stationary and nonstationary Ornstein-Uhlenbeck processes (sheets), which are shifted by m times a known deterministic function on the interval (rectangle). It turns out that the maximum likelihood estimator (MLE) has a normal distribution and, for instance, in case of the sheet this MLE is a weighted linear combination of the values at the vertices, integrals on the edges, and the integral on the whole rectangle of the weighted observed process. We do not use partial stochastic differential equations; we apply direct discrete time approach instead. To make the transition from the discrete time to the continuous time, a tool is developed, which might be of independent interest.     

On the large deviations approximation for the stationary distribution of skip-free regulated queueing networks

Large deviations papers like that of Ignatyuk et al. [Russ. Math. Surv. 49 (1994) 41–99] have shown that asymptotically, the stationary distribution of homogeneous regulated networks is of the form

with the coefficient being different in various “boundary influence domains” and also depending on some of these domains on n. In this paper, we focus on the case of constant exponents and on a subclass of networks we call “strongly skip-free” (which includes all Jackson and all two-dimensional skip-free networks). We conjecture that an asymptotic exponent is constant iff it corresponds to a large deviations escape path which progresses gradually (from the origin to the interior) through boundary facets whose dimension always increases by one. Solving the corresponding large deviations problem for our subclass of networks leads to a family of “local large deviation systems” (LLDSs) (for the constant exponents), which are expressed entirely in terms of the cumulant generating function of the network. In this paper, we show that at least for “strongly skip-free” Markovian networks with independent transition processes, the LLDS is closely related to some “local boundary equilibrium systems” (LESs) obtained by retaining from the equilibrium equations only those valid in neighborhoods of the boundary.

Since asymptotic results require typically only that the cumulant generating function is well-defined over an appropriate domain, it is natural to conjecture that these LLDSs will provide the asymptotic constant exponents regardless of any distributional assumptions made on the network.

Finally, we outline a practical recipe for combining the local approximations to produce a global large deviations approximation , with the coefficients K_j determined numerically.     

Simulation of Lévy-driven Ornstein-Uhlenbeck processes with given marginal distribution

A simulation procedure for obtaining discretely observed values of Ornstein-Uhlenbeck processes with given (self-decomposable) marginal distribution is provided. The method proposed, based on inversion of the characteristic function, completely circumvents the problems encountered when trying to reproduce small jumps of Lévy processes. Error bounds for the proposed procedure are provided and its performance is numerically assessed.     

A random-projection based test of Gaussianity for stationary processes

Gaussianity tests have being widely studied in the literature. Regarding the study of Gaussianity tests for stationary processes, these only verify the Gaussianity of a marginal at a fixed finite order, generally order one. Therefore, they do not reject stationary non-Gaussian processes with the one-dimensional Gaussian marginal. Thus, a consistent test is proposed for Gaussianity of stationary processes when a finite sample path of the process is observed. Using random projections, decision rules are applied to the whole distribution of the process and not only on its marginal distribution at a fixed order, as in previous tests. The main idea is to test the Gaussianity of the one-dimensional marginal distribution of some random linear transformations of the process. Note that testing the one-dimensional marginal distribution can be done with previous tests of Gaussianity for stationary processes. It is shown by both theoretical and empirical studies that the proposed test procedure has good properties for a wide range of alternatives.     

A digital twin-based approach for the management of geometrical deviations during assembly processes

The recent transformation in the aeronautical industry gives new prospects in the field of product geometry assurance. These include, in particular the creation of sophisticated virtual models, or digital twins, which can reflect the as-built geometry of physical products and optimize the assembly operations consequently. One of the current obstacles to the implementation of such digital twins is linked to the difficult transition from a conceptual model to a usable virtual representation. In this article, we present the hybrid representation of a product which is capable of integrating the different states of the components at each step of the assembly process. We propose a method to update the virtual representation of already assembled components, in order to include the position and orientation deviations of their surfaces. The B-Rep model of each component is updated from data acquired during the assembly of the product. The various steps of this update, and its associated tools are discussed in the article. Based on the knowledge of the as-built component geometry, the geometry of the yet-to-be-assembled components is adapted so that the final product complies with the functional requirements. To this end, we also discuss a formalism to model the product's functional information and to translate it at a geometrical level thanks to an assembly skeleton.     

An improved method for bounding stationary measures of finite Markov processes

A new method to compute bounds on stationary results of finite Markov processes in discrete or continuous time is introduced. The method extends previously published approaches using polyhedra of eigenvectors for stochastic matrices with a known lower and upper bound of their elements. Known techniques compute bounds for the elements of the stationary vector with respect to the lower bounds of the matrix elements and another set of bounds with respect to the upper bounds of matrix elements. The resulting bounds are usually not sharp, if lower and upper bounds for the elements are known. The new approach combines lower and upper bounds resulting in sharp bounds which are often much tighter than bounds computed using only one bounding value for the matrix elements.     

一类可数Markov控制过程的最优平稳策略

研究了一类具有可数状态空间的Markov控制过程在无限水平平均代价准则下的最优平稳策略问题.对此类过程,引入了折扣Poisson方程,运用无穷小矩阵和性能势的基本性质,导出了平均代价模型在紧致行动集上的最优性方程,并证明了其解的一个存在性定理.     

Quantifying statistical interdependence, part III: N > 2 point processes

Stochastic event synchrony (SES) is a recently proposed family of similarity measures. First, "events" are extracted from the given signals; next, one tries to align events across the different time series. The better the alignment, the more similar the N time series are considered to be. The similarity measures quantify the reliability of the events (the fraction of "nonaligned" events) and the timing precision. So far, SES has been developed for pairs of one-dimensional (Part I) and multidimensional (Part II) point processes. In this letter (Part III), SES is extended from pairs of signals to N > 2 signals. The alignment and SES parameters are again determined through statistical inference, more specifically, by alternating two steps: (1) estimating the SES parameters from a given alignment and (2), with the resulting estimates, refining the alignment. The SES parameters are computed by maximum a posteriori (MAP) estimation (step 1), in analogy to the pairwise case. The alignment (step?2) is solved by linear integer programming. In order to test the robustness and reliability of the proposed N-variate SES method, it is first applied to synthetic data. We show that N-variate SES results in more reliable estimates than bivariate SES. Next N-variate SES is applied to two problems in neuroscience: to quantify the firing reliability of Morris-Lecar neurons and to detect anomalies in EEG synchrony of patients with mild cognitive impairment. Those problems were also considered in Parts I and II, respectively. In both cases, the N-variate SES approach yields a more detailed analysis.     

On the exact asymptotics for the stationary sojourn time distribution in a tandem of queues with light-tailed service times

We study the asymptotics of the stationary sojourn time Z of a “typical customer” in a tandem of single-server queues. It is shown that in a certain “intermediate” region of light-tailed service time distributions, Z may take a large value mostly due to a large value of a single service time of one of the customers. Arguments used in the paper also allow us to obtain an elementary proof of the logarithmic asymptotics for the tail distribution of the stationary sojourn time in the whole class of light-tailed distributions.     

Exact Algorithms for <Emphasis Type="Italic">L</Emphasis>(2,1)-Labeling of Graphs

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O ^*((k+1) ⁿ ) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006 ⁿ ). Furthermore we show that dynamic programming can be used to establish an O(3.8730 ⁿ ) algorithm to compute an optimal L(2,1)-labeling.     

Force calculation for orbital-dependent potentials with FP-(L)APW + lo basis sets

Within the linearized augmented plane-wave method for electronic structure calculations, a force expression was derived for such exchange-correlation energy functionals that lead to orbital-dependent potentials (e.g., LDA+U or hybrid methods). The forces were implemented into the WIEN2k code and were tested on systems containing strongly correlated d and f electrons. The results show that the expression leads to accurate atomic forces.     

量子纠错码[[7,1,4]]_p(p>3)存在性的图论构造方法

利用由Schingemann和Werner两人提出的构造量子纠错码的图论方法,证明了量子纠错码[[7,1,4]]p(p>3)的存在性。     

Exact and efficient top-<Emphasis Type="Italic">K</Emphasis> inference for multi-target prediction by querying separable linear relational models

Many complex multi-target prediction problems that concern large target spaces are characterised by a need for efficient prediction strategies that avoid the computation of predictions for all targets explicitly. Examples of such problems emerge in several subfields of machine learning, such as collaborative filtering, multi-label classification, dyadic prediction and biological network inference. In this article we analyse efficient and exact algorithms for computing the top-K predictions in the above problem settings, using a general class of models that we refer to as separable linear relational models. We show how to use those inference algorithms, which are modifications of well-known information retrieval methods, in a variety of machine learning settings. Furthermore, we study the possibility of scoring items incompletely, while still retaining an exact top-K retrieval. Experimental results in several application domains reveal that the so-called threshold algorithm is very scalable, performing often many orders of magnitude more efficiently than the naive approach.     

The law of large numbers for fuzzy processes and the estimation problem

We study how the law of large numbers can be extended to the case of fuzzy numbers. We work with L-fuzzy sets, where L is a CL_∞ commutative monoid with an operation 1. We define the 1-independence of fuzzy numbers. We find, when 1 is the operation Λ of the lattice, that the arithmetical mean of Λ-independent fuzzy numbers does not converge. We think that this case could be a limit one; we show, for particular cases of the 1-operation, that generally convergence holds. We build estimators for fuzzy models; we particularly consider the “minimum fuzziness estimator.” We study the convergence of these estimators and compare them by way of the fuzzy subsets they induce. We lastly study the case where we have dependent fuzzy numbers and particularly the case of linearly filtered “1-autoindependent” fuzzy processes.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

Exact analytical solutions for some popular benchmark problems in topology optimization III: <Emphasis Type="Italic">L</Emphasis>-shaped domains

In this paper exact, analytical solutions are derived for another highly popular benchmark problem, namely, L-shaped domains having a horizontal line support and one or several point loads. The optimal topologies are obtained in the context of Michell structures, i.e., least-weight, stress, or compliance-controlled trusses with a single load condition.     

Fast, prime factor, discrete Fourier transform algorithms over GF(2) for 8 ? m ? 10

In this paper it is shown that Winograd’s algorithm for computing convolutions and a fast, prime factor, discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2^m − 1 points over GF(2^m), for 8 ? m ? 10. These new transform techniques can be used to decode Reed-Solomon (RS) codes of block length 2^m − 1. The complexity of this new transform algorithm is reduced substantially from more conventional methods. A computer simulation verifies these new results.