Set Of Independent Random Variables Research Articles

Outlier‐resistant state estimation for complex networks with random false data injection attacks under encoding–decoding mechanism

SummaryThis article focuses on the outlier‐resistant state estimation problem for discrete time‐varying complex networks (TVCNs) affected by random false data injection attacks (FDIAs) under an encoding–decoding mechanism (EDM). From the perspective of information security, a uniform‐quantization‐based EDM is employed to encrypt the transmitted data. During the data transmission process, a set of independent random variables governed by Bernoulli distribution is introduced to characterize the occurrence of random FDIAs. For the purpose of alleviating the passive impact of potential measurement outliers, a saturation structure is adopted during the estimator design. The gain matrix is given by minimizing the upper bound of estimation error covariance. According to the stochastic analysis method, it is shown that the state estimation error is bounded exponentially in mean‐square sense by providing new sufficient condition. It should be noted that we make the first attempt to develop new outlier‐resistant state estimation method with performance evolution criterion in the time‐varying perspective for TVCNs with random FDIAs under EDM. Finally, a simulation example with comparative experiment is presented to illustrate the effectiveness of the newly presented outlier‐resistant estimation algorithm.

Central limit theorem for the number of records in Fα-scheme

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.

Distributed Fault-Tolerant State Estimation for a Class of Nonlinear Systems Over Sensor Networks With Sensor Faults and Random Link Failures

This article is concerned with the problem of distributed fault-tolerant state estimation for sector-bounded nonlinear systems over sensor networks. The advantage of the distributed fault-tolerant state estimation algorithm is that the estimator can have the given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance in the simultaneous presence of sensor faults and communication link failures. Based on the fault-diagnosis technique, the sensor fault of each sensor node is detected and estimated by the designed fault-diagnosis unit. The temporary communication link failures are modeled as a set of independent random variables obeying the Bernoulli distribution. Subsequently, the active fault-tolerant estimation scheme is established and several sufficient conditions are obtained, which guarantee the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance of estimation error system and the detectability of sensor faults. Finally, an illustrative example is provided to verify the effectiveness of the proposed method.

Stochastic isogeometric buckling analysis of composite shell considering multiple uncertainties

An efficient stochastic isogeometric analysis (SIGA) framework considering multiple uncertainties is proposed for the stochastic buckling analysis of composite shells. The framework develops a seamless pipeline enabling uncertainty quantification and uncertainty analysis with the same building blocks of Non-Uniform Rational B-Splines (NURBS). The Gaussian random fields with spatial variability of random parameters are represented by Karhunen-Loève expansion, which is a linear combination of a set of independent random variables and orthogonal functions. The orthogonal functions and corresponding eigenvalues are obtained by solving the Fredholm integral equation of the second kind by the Galerkin isogeometric method. This method exploits the regularity of NURBS basis functions delivering globally smooth eigensolutions and the total mean squared error of the random field is minimized simultaneously. In addition, perturbation technique is developed to solve the stochastic linear buckling equation and predict the second-moment statistics of the buckling load and the probability of failure of structures. The accuracy of the proposed method is assessed by independent Monte Carlo simulation (MCS) based on isogeometric analysis (IGA). Moreover, from the point of view of convergence and computational time, it is evident that the proposed SIGA framework can effectively obtain probabilistic characteristics compared with stochastic finite element method (SFEM).

Stabilization of uncertain networked control systems with actuator saturation and probabilistic cyberattacks

AbstractThis paper investigates the stabilization of uncertain networked control systems (NCSs) with actuator saturation and cyberattacks. The cyberattacks are governed by a set of independent random variables satisfying Bernoulli distribution. To relieve the network bandwidth load effectively, an event‐triggered communication strategy is proposed. By employing Lyapunov stability theory and stochastic analysis techniques, a stability criterion is obtained for the system with actuator saturation and cyberattacks. Moreover, the desired controller gain is derived by solving some matrix inequalities. Finally, the validity and applicability of the criteria are verified through numerical examples.

Distributed Ellipsoidal Intersection Fusion Estimation for Multi-Sensor Complex Systems.

This paper investigates the problem of distributed ellipsoidal intersection (DEI) fusion estimation for linear time-varying multi-sensor complex systems with unknown input disturbances and measurement data transmission delays. For the problem with external unknown input disturbance signals, a non-informative prior distribution is used to model the problem. A set of independent random variables obeying Bernoulli distribution is also used to describe the situation of measurement data transmission delay caused by network channel congestion, and appropriate buffer areas are added at the link nodes to retrieve the delayed transmission data values. For multi-sensor systems with complex situations, a minimum mean square error (MMSE) local estimator is designed in a Bayesian framework based on the maximum a posteriori (MAP) estimation criterion. In order to deal with the unknown correlations among the local estimators and to select the fusion estimator with lower computational complexity, the fusion estimator is designed using ellipsoidal intersection (EI) fusion technique, and the consistency of the estimator is demonstrated. In this paper, the difference between DEI fusion and distributed covariance intersection (DCI) fusion and centralized fusion estimation is analyzed by a numerical example, and the superiority of the DEI fusion method is demonstrated.

Stochastic monotonicity of dependent variables given their sum

Given a finite set of independent random variables, assume one can observe their sum, and denote with s its value. Efron in 1965, and Lehmann in 1966, described conditions on the involved variables such that each of them stochastically increases in the value s, i.e., such that the expected value of any non-decreasing function of the variable increases as s increases. In this paper, we investigate conditions such that this stochastic monotonicity property is satisfied when the assumption of independence is removed. Comparisons in the stronger likelihood ratio order are considered as well.

Formation of sets of independent components of a multidimensional random variable based on a nonparametric pattern recognition algorithm

The possibility of circumventing the problem of decomposition of the range of values of random variables when testing various hypotheses is considered. A brief review of the literature on this problem is given. A method for forming sets of independent components of a multidimensional random variable is proposed, based on hypotheses testing about the independence of paired combinations of components of a multidimensional random variable. The method uses a two-dimensional non-parametric algorithm for pattern recognition of the kernel type, corresponding to the criterion of maximum likelihood. In contrast to the traditional method based on the application of the Pearson criterion, the proposed approach avoids the problem of decomposing the range of values of random variables into multidimensional intervals. The results of computational experiments performed according to the method of forming sets of independent random variables are presented. Using the information obtained, an information graph is constructed, the vertices of which correspond to the components of a multidimensional random variable, and the edges determine their independence. Then the vertices of the complete subgraphs correspond to groups of independent components of a random variable. The obtained results form the basis for the synthesis of a multi-level nonparametric large volume data processing system, each level of which corresponds to a specific set of independent random variables.

Estimates for the variational distance between two sets of independent random variables

Получены явные нижние и верхние оценки расстояний по вариации между распределениями наборов $(X_1,\ldots,X_n)$ и $(Y_1,\ldots,Y_n)$ независимых, но не обязательно одинаково распределенных (внутри каждого набора) случайных величин в терминах расстояний по вариации $\rho_k$ между распределениями отдельных компонент $X_k$ и $Y_k$, $k=1,\ldots,n$. Оценки для случая однородных выборок были опубликованы в этом журнале в 2018 г. Качественно полученные в настоящей работе оценки согласуются с оценками для однородных случаев.

ОЦІНКА ПОТУЖНОСТІ КРИТЕРІЇВ ТРЕНДУ

An approach to the selection and comparison of the criteria that are used in the analysis of time series of registration parameters of the technical camp of power and power plants based on gas turbine engines is proposed. The approach is based on established important characteristics of trending criteria, namely the power of such criteria, which are considered as criteria for distinguishing complex hypotheses. For analysis, we propose a statistical model for generating data in the form of a combination of deterministic trends and random components. A deterministic component is considered as a linear approximation of its expansion in a Taylor series. This assumption is justified by the need to show a trend in the shortest period of time at which the trend component allows a linear approximation. A random compound is taken as a sample of a general population of independent random variables that have a normal distribution. For analysis, the most common trend criteria were selected: Student's criterion for equality of means; Fisher dispersion ratio criterion; correlation criterion and its varieties. The supporting hypothesis has the form of belonging of a time series to a sample from the general set of independent random variables, and an alternative is belonging to a sample with a linear trend. Trend statistics of the relevant criteria generated on a moving or sectional disjoint analysis window of a given dimension. The trend development parameter was selected as the ratio of the trend growth during the analysis to the standard deviation of the random component. For the considered trend criteria, the obtained dependences of their power on the trend development parameter and the probability of an error of the first kind (erroneous alarm), as well as the operational characteristics of the criteria. The analysis was performed by the methods of analytical estimates and statistical modeling. It has been established that in the case of an alternative, the statistics of the correlation criterion and the Fisher criterion are quickly normalized, and student statistics do not change their type. A comparison of trending power criteria with equal values of the probability of an error of the first kind allows us to establish the advantage of the Student criterion, and the correlation criterion has the worst performance. Obtaining indicators of the power of trend criteria are important for applied applications since it allows you to establish the probability of the second kind of error (skipping a trend).

Assembling Components with Probabilistic Lead Times

This paper examines a production model in which N different types of components are used to assemble a finished product. The components are acquired from various suppliers in lots received at or before the beginning of the assembly process. The lead time between placing and receiving each of the N orders is assumed to be a random variable having a known probability density function. Based on recent results regarding the probability distribution and the expectation of the maximum of a set of independent random variables, the mathematical model describing this production/inventory situation is developed. The mathematical model is then used to derive a closed form formula for the optimal solution that minimizes the total production/inventory cost function. Moreover, the reorder point is shown to depend on the probability distribution and expected value of the random variable representing the maximum among the N lead times. The case in which each of the N lead times follows a Weibull distribution is investigated, and a numerical example is given to illustrate this case.

A variance‐constrained approach to event‐triggered distributed extended Kalman filtering with multiple fading measurements

SummaryIn this paper, a distributed extended Kalman filtering problem is studied for discrete‐time nonlinear systems with multiple fading measurements. To alleviate the network communication burden, the event‐triggered communication scheme is employed in both sensor‐to‐estimator channel and estimator‐to‐estimator channel. As such, the data transmission is executed only when the predefined event occurs. In addition, a set of independent random variables with known statistical properties is defined to represent the phenomenon of multiple fading measurements. The variance‐constrained approach is adopted to derive an upper bound for the estimation error covariance in consideration of the event‐triggered mechanism and truncated error by linearization. The filter gain for each node is then designed to minimize such an upper bound by recursively solving two Raccati‐like difference equations. By virtue of the stochastic stability theory, a sufficient condition is provided to guarantee the boundedness of the estimation error. Finally, a simulation example is presented to illustrate the feasibility and effectiveness of the proposed filtering algorithm.

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

ABSTRACTLet X1, …, Xn be independent random variables with Xi ∼ EWG(α, β, λi, pi), i = 1, …, n, and Y1, …, Yn be another set of independent random variables with Yi ∼ EWG(α, β, γi, qi), i = 1, …, n. The results established here are developed in two directions. First, under conditions p1 = ⋅⋅⋅ = pn = q1 = ⋅⋅⋅ = qn = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ1 = ⋅⋅⋅ = λn = γ1 = ⋅⋅⋅ = γn = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p1, …, pn) and (q1, …, qn). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.

Rank constrained distribution and moment computations

Consider a set of independent random variables with specified distributions or a set of multivariate normal random variables with a product correlation structure. This paper shows how the distributions and moments of these random variables can be calculated conditional on a specified ranking of their values. This can be useful when the ordering of the variables can be determined without observing the actual values of the variables, as in ranked set sampling, for example. Thus, prior information on the distributions and moments from their individual specified distributions can be updated to provide improved posterior information using the known ranking. While these calculations ostensibly involve high dimensional integral expressions, it is shown how the previously developed general recursive integration methodology can be applied to this problem so that they can be evaluated in a straightforward manner as a series of one-dimensional or two-dimensional integral calculations. Furthermore, the proposed methodology possesses a self-correction mechanism in the computation that prevents any serious growth of the errors. Examples illustrate how different kinds of ranking information affect the distributions, expectations, variances, and covariances of the variables, and how they can be employed to solve a decision making problem.

Search for anomalies in the decay of radioactive Mn-54

Recent papers have reported that 54Mn, which decays by electron capture (a weak nuclear interaction) with half-life ∼312 days, is influenced by solar activity. Should this actually occur, new physics would be needed to explain it. This paper reports results of an analysis of 54Mn activity measured over a time interval of ∼3.6 half-lives. If standard nuclear physics applies, the logarithmic residuals of 54Mn activities should form a stationary set of independent random variables whose statistics are determined solely by a constant decay rate λ and initial mean count μ. Analysis of the time-variation, autocorrelation, and power spectra of the 54Mn logarithmic residuals agrees exquisitely with standard nuclear physics. Computer-simulated activities exhibiting periodic decay of amplitude show that anomalies would be detectable by these statistical tests for values of α as low as ∼1 part in 104. This limit is about 10 times lower than reported deviations from exponential decay.

FastHiC: a fast and accurate algorithm to detect long-range chromosomal interactions from Hi-C data.

How chromatin folds in three-dimensional (3D) space is closely related to transcription regulation. As powerful tools to study such 3D chromatin conformation, the recently developed Hi-C technologies enable a genome-wide measurement of pair-wise chromatin interaction. However, methods for the detection of biologically meaningful chromatin interactions, i.e. peak calling, from Hi-C data, are still under development. In our previous work, we have developed a novel hidden Markov random field (HMRF) based Bayesian method, which through explicitly modeling the non-negligible spatial dependency among adjacent pairs of loci manifesting in high resolution Hi-C data, achieves substantially improved robustness and enhanced statistical power in peak calling. Superior to peak callers that ignore spatial dependency both methodologically and in performance, our previous Bayesian framework suffers from heavy computational costs due to intensive computation incurred by modeling the correlated peak status of neighboring loci pairs and the inference of hidden dependency structure. In this work, we have developed FastHiC, a novel approach based on simulated field approximation, which approximates the joint distribution of the hidden peak status by a set of independent random variables, leading to more tractable computation. Performance comparisons in real data analysis showed that FastHiC not only speeds up our original Bayesian method by more than five times, bus also achieves higher peak calling accuracy. FastHiC is freely accessible at:http://www.unc.edu/∼yunmli/FastHiC/ CONTACTS: : yunli@med.unc.edu or ming.hu@nyumc.org Supplementary data are available at Bioinformatics online.

Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables

ABSTRACTIn this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let be independent random variables with Further, let be another set of independent random variables with First, when and a matrix with different parameters changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when and , we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.

Can You Play a Fair Game of Craps with a Loaded Pair of Dice?

The parts-to-totals map sends the distributions of a set of independent random variables on a finite set of probability spaces to the total distribution of their sum on the product space. We study, in special cases modeled by dice, the geometry of the extension to complex pseudoprobabilities of this map, arithmetic questions about the existence of real points in certain fibers, and, when these exist, of strict points, having all coordinates in the unit interval.

A frequentist view on cycle-transitivity of reciprocal relations

We establish a connection between two transitivity frameworks: the transitivity of fuzzy relations based on a commutative quasi-copula and the cycle-transitivity of reciprocal relations w.r.t. the dual quasi-copula as an upper bound function. Loosely speaking, it turns out that the latter can be characterized by imposing a lower bound on the relative frequency with which the former is fulfilled, when applied to reciprocal relations. We provide two compelling cases: the 4/6 theorem, expressing that the winning probability relation of a set of independent random variables is at least 66.66% product-transitive, and the 5/6 theorem, expressing that the mutual rank probability relation associated with a given poset is at least 83.33% product-transitive. Moreover, these lower bounds turn out be rather conservative, illustrating that, from a frequentist point of view, transitivity is abundant.

On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables

Negatively associated (NA) random variables are a more general class of random variables which include a set of independent random variables and have been applied to many practical fields. In this paper, the complete moment convergence of weighted sums for arrays of row-wise NA random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of row-wise NA random variables are established. Moreover, under the weaker conditions, we extend the results of Baek et al. [J. Korean Stat. Soc. 37 (2008), pp. 73–80] and Sung [Abstr. Appl. Anal. 2011 (2011)]. As an application, the complete moment convergence of moving average processes based on an NA random sequence is obtained, which improves the result of Li and Zhang [Stat. Probab. Lett. 70 (2004), pp. 191–197 ].