GENERALIZED INFORMATION THEORY AND DECOMPOSABILITY OF SYSTEMS

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

Two experiments used post-class formation within-class relational assessment test performances to evaluate whether participants demonstrated preference for certain members of an equivalence class based on the type of relation that existed between class members. In Experiment 1, two 5-node 7-member equivalence classes, consisting entirely of nonsense syllables, were established using the simultaneous protocol. Only 1 of the 6 participants in Experiment 1 formed classes. After class formation, the effects of the different relations between stimuli were evaluated using within-class relational assessment tests, and the 1 participant showed an absolute preference for transitive over equivalence relations, and for baseline over symmetrical relations. Experiment 2 was identical to Experiment 1, except that one of the nonsense syllable stimuli in each class was replaced by a pictorial stimulus. Under these conditions, classes were formed by 5 of 13 participants. During the relational assessment tests, the 5 participants who formed classes demonstrated almost exclusive preferences for transitive relations over equivalence relations and for trained baseline relations over symmetrical relations. Thus, this research demonstrates that the members of equivalence classes are differentially related to each other based on relational type.

Similarity relations are reflexive, symmetric, and transitive fuzzy relations. They help to make approximate inferences, replacing the notion of equality. Similarity-based unification has been quite intensively investigated, as a core computational method for approximate reasoning and declarative programming. In this paper we consider solving constraints over several similarity relations, instead of a single one. Multiple similarities pose challenges to constraint solving, since we can not rely on the transitivity property anymore. Existing methods for unification with fuzzy proximity relations (reflexive, symmetric, non-transitive relations) do not provide a solution that would adequately reflect particularities of dealing with multiple similarities. To address this problem, we develop a constraint solving algorithm for multiple similarity relations, prove its termination, soundness, and completeness properties, and discuss applications.

Rough set theory and vague set theory are powerful tools for managing uncertain, incomplete and imprecise information. This paper extends the rough vague set model based on equivalence relations and the rough fuzzy set model based on fuzzy relations to vague sets. We mainly focus on the lower and upper approximation operators of vague sets based on vague relations, and investigate the basic properties of approximation operators on vague sets. Specially, we give some essential characterizations of the lower and upper approximation operators generated by reflexive, symmetric, and transitive vague relations. Finally, we structure a parameterized roughness measure of vague sets and similarity measure methods between two rough vague sets, and obtain some properties of the roughness measure and similarity measures. We also give some valuable counterexamples and point out some false properties of the roughness measure in the paper of Wang et al.

Quantities governed by randomness correspond to functions on the probability space called random variables. The value taken by a random variable is subject to chance, and the associated likelihoods are described by a function called the distribution function. Two important classes of random variables are discussed, namely discrete variables and continuous variables. The law of averages, known also as the law of large numbers, states that the proportion of successes in a long run of independent trials converges to the probability of success in any one trial. This result provides a mathematical basis for a philosophical view of probability based on repeated experimentation. Worked examples involving random variables and their distributions are included, and the chapter terminates with sections on random vectors and on Monte Carlo simulation.

Artificial bee colony (ABC) algorithm is widely known for its distinguished exploration ability. However, its exploitation ability is relatively poor. To solve the problem, we propose a novel combinatorial search strategy, whose guided vector can be freely switched between a random vector and the global best vector. It can help improve the exploitation ability of ABC. At the same time, a random vector is beneficial to regulate the enhanced exploitation ability. In addition, both of them can pass information on to a current vector instead of only perturbing a current vector itself. The two guided vectors are chosen with a probability depending on the ratio of fitness information of a current vector to that of the global-best vector. Thus, one of the two guided vectors can be adaptively selected to direct the search. In addition, a mechanism of frequency of perturbation is employed to enhance the scale of information sharing between a current vector and a guided vector for each onlooker bee. Moreover, a modified greedy selection mechanism is designed to choose a child vector inspired by simulated annealing. Furthermore, the search strategy of multiple scouts is also employed in the last stage. Based on all these modifications, an improved ABC (IABC) is proposed. Finally, a few experiments are carried on 58 benchmark problems, including CEC2014 benchmark problems. The computational results exhibit the merit of IABC.

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed, and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.

Testing for mutual independence among several random vectors is a challenging problem, and in recent years, it has gained significant attention in statistics and machine learning literature. Most of the existing tests of independence deal with only two random vectors, and they do not have straightforward generalizations for testing mutual independence among more than two random vectors of arbitrary dimensions. On the other hand, there are various tests for mutual independence among several random variables, but these univariate tests do not have natural multivariate extensions. In this article, we propose two general recipes, one based on inter-point distances and the other based on linear projections, for multivariate extensions of these univariate tests. Under appropriate regularity conditions, these resulting tests turn out to be consistent whenever we have consistency for the corresponding univariate tests. We carry out extensive numerical studies to compare the empirical performance of these proposed methods with the state-of-the-art methods.

Measuring association and dependence between random vectors

A random vector ${\bf X}$ is weakly stable if and only if for all $a,b \in {\mathbb R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where ${\bf X}'$ is an independent copy of ${\bf X}$ and $\Theta$ is independent of ${\bf X}$. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195--213]) to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta,$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define a weak generalized convolution of measures defined by the formula ${\cal L}(Q_1) \otimes_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\bf X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this ...

Modifiers connect L-fuzzy sets to topological spaces

Ground Penetrating Radar (GPR) is a geophysical method to detect the structure and characteristics of underground burial targets by using high frequency electromagnetic wave. However, for traditional pulse GPR systems, There is still room of progress in stability and signal-to-noise (SNR) of reflections. In this work, in order to reconstruct the 3-D structure of complex objects in underground space, we apply stepped frequency continuous wave (SFCW) technology to GPR system and design a reconfigurable multistate GPR (RM-GPR). In addition, we expand back projection imaging algorithm into 3-D space and improve the computational efficiency of the algorithm by limiting the single imaging area and simplifying the self convolution process. The unique aspect of the RM-GPR system is that they can reconstruct the real 3-D contour of the targets in the underground complex space and help people to have an intuitive understanding of unknown targets in underground space.

Since its first demonstration in 1995, terahertz time‐domain imaging has attracted an increasingly growing interest for its ability to reveal spectral fingerprints of materials and probe changes in refractive index and absorption, as well as detect the inner structure of complex objects via time‐of‐flight measurements. Practically, however, its widespread use has been hampered by the very long acquisition time typically required to spatially raster‐scan the object, and for each spatial point, record the field in time via a delay line. Here, this fundamental bottleneck is addressed by implementing a scanless single‐pixel imaging scheme, which sets the path for an unprecedented reduction of both system complexity and acquisition time. By properly exploiting natural wave diffraction, time‐to‐space encoding applied to terahertz point detection allows for an almost instantaneous capture of the terahertz waveforms, while multidimensional images are reconstructed via a computational approach. The scheme is a promising solution for the development of next‐generation fast and compact terahertz imagers perfectly suitable for high‐repetition‐rate laser sources.

Input modeling is the selection of a probability distribution to capture the uncertainty in the input environment of a stochastic system. Example applications of input modeling include the representation of the randomness in the time to failure for a machining process, the time between arrivals of calls to a call center, and the demand received for a product of an inventory system. Building simulations of stochastic systems requires the development of input models that adequately represent the uncertainty in such random variables. Since there are an abundance of probability distributions that can be used for this purpose, a natural question to ask is how to identify the probability distribution that best represents the particular situation under study. For example, is the exponential distribution a reasonable choice to represent the time to failure for a machining process, or is it better to use an empirical distribution function obtained from the historical time-to-failure data? Recognizing the fact that there is no true input model waiting to be found, the goal of stochastic input modeling is to obtain an approximation that captures the key characteristics of the system inputs. The development of a good input model requires the collection of as much information as possible about the relevant randomness in the system as well as the historical data consisting of the past realizations of the random variables of interest. In the presence of a data set, the input model can be identified by fitting a probability distribution to the historical data. However, it may be difficult and/or costly to collect data for the stochastic system under study; it can also be impossible to properly collect any data at all such as when the proposed system does not exist. In the absence of historical data, any relevant information (e.g., expert opinion and the conventional bounds suggested by the underlying physical situation) can be used for input modeling. This article addresses the key issues that arise in stochastic input modeling both in the presence and in the absence of historical data. The first step in input modeling is to identify the sources of randomness in the input environment of the system under study. Many stochastic systems contain multiple sources of uncertainty, e.g., the completion time of an item on a particular machine, the potential breakdown of the machine, and the percentage of defective items produced by the machine might be among the sources of uncertainty in a manufacturing setting. Throughout, the random vector X = (X1, X2, …, X K )′ is used to represent the collection of K different inputs of a stochastic system, where X k is the random variable denoting the kth system input. The K components of this random vector might also be correlated with each other. Therefore, the stochastic properties of the random inputs X k , k = 1, 2, …, K, are captured in the joint probability

Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the present general setting. The new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other. Although the setting is quite general, the result are already of interest (and new) for random vectors in $\mathbb{R}^n$. In that special case, quantitative bounds are provided for the stability of the equality conditions in the entropy power inequality; a reverse entropy power inequality for log-concave random vectors is proved; an information-theoretic analog of the Rogers-Shephard inequality for convex bodies is established; and it is observed that some of these results lead to new inequalities for the determinants of positive-definite matrices. Moreover, by considering the multiplicative subgroups of the complex plane, one obtains new inequalities for the differential entropies of products and ratios of nonzero, complex-valued random variables.