Uncorrelated Random Variables Research Articles

State estimation for 2-D uncertain systems with redundant channels and deception attacks: A set-membership method

This paper considers the estimation issue of two-dimensional uncertain systems with redundant channels and randomly occurring deception attacks. The redundant channel transmission is applied to improve transmission reliability, with channel packet losses characterized by a set of mutually uncorrelated random variables that follow the Bernoulli distribution. To simulate the random deception attacks, a Bernoulli distribution-based sequence is employed. The purpose of this study is to build a set-membership estimator for the considered system with redundant channels in the insecure network environment that limits the state estimate error to a predetermined ellipsoidal area. By employing the set-membership method with the mathematical induction technique, sufficient criteria are derived to guarantee the expected estimated performance. In addition, an optimization algorithm is further provided to ensure locally optimal estimation performance. Finally, a numerical simulation of the suggested set-membership estimation technique is presented to demonstrate its usefulness.

A spectral surrogate model for stochastic simulators computed from trajectory samples

Stochastic simulators are non-deterministic computer models which provide a different response each time they are run, even when the input parameters are held at fixed values. They arise when additional sources of uncertainty are affecting the computer model, which are not explicitly modeled as input parameters. The uncertainty analysis of stochastic simulators requires their repeated evaluation for different values of the input variables, as well as for different realizations of the underlying latent stochasticity. The computational cost of such analyses can be considerable, which motivates the construction of surrogate models that can approximate the original model and its stochastic response, but can be evaluated at much lower cost.We propose a surrogate model for stochastic simulators based on spectral expansions. Considering a certain class of stochastic simulators that can be repeatedly evaluated for the same underlying random event, we view the simulator as a random field indexed by the input parameter space. For a fixed realization of the latent stochasticity, the response of the simulator is a deterministic function, called trajectory. Based on samples from several such trajectories, we approximate the latter by sparse polynomial chaos expansion and compute analytically an extended Karhunen–Loève expansion (KLE) to reduce its dimensionality. The uncorrelated but dependent random variables of the KLE are modeled by advanced statistical techniques such as parametric inference, vine copula modeling, and kernel density estimation. The resulting surrogate model approximates the marginals and the covariance function, and allows to obtain new realizations at low computational cost. We observe that in our numerical examples, the first mode of the KLE is by far the most important, and investigate this phenomenon and its implications.

Data-Driven Probabilistic Methodology for Aircraft Conflict Detection Under Wind Uncertainty

Assuming the availability of a reliable aircraft trajectory planner, this paper presents a probabilistic methodology to detect conflicts between aircraft in the cruise phase of flight in the presence of wind velocity forecasting uncertainty. This uncertainty is quantified by ensemble weather forecasts, the members of which are regarded as realizations of correlated random processes and used to derive the eastward and northward components of the wind velocity. First, the Karhunen-Loève expansion is used to obtain a series expansion of the components of the wind velocity in terms of a set of uncorrelated random variables and deterministic coefficients. Then, the uncertainty generated by these uncorrelated random variables in the outputs of the aircraft trajectory planner is quantified using the arbitrary polynomial chaos technique. Finally, the probability density function of the great circle distance between each pair of aircraft is derived from the polynomial expansions using a Gaussian kernel density estimator and used to estimate the probability of conflict. The arbitrary polynomial chaos technique allows the effects of uncertainty in complex nonlinear dynamical systems, such as those underlying aircraft trajectory planners, to be quantified with high computational efficiency, only requiring the existence of a finite number of statistical moments of the random variables of the Karhunen-Loève expansion while avoiding any assumptions on their probability distributions. To demonstrate the effectiveness of the proposed conflict detection method, numerical experiments are conducted via an optimal control-based aircraft trajectory planner for a given wind velocity forecast represented by an ensemble prediction system.

Correlated Sample-Based Prior in Bayesian Inversion Framework for Microwave Tomography

When using the statistical inversion framework in microwave tomography (MWT), generally, the real and imaginary parts of the unknown dielectric constant are treated as uncorrelated and independent random variables. Thereby, in the maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> estimates, the two recovered variables may show different structural changes inside the imaging domain. In this work, a correlated sample-based prior model is presented to incorporate the correlation of the real part with the imaginary part of the dielectric constant in the statistical inversion framework. The method is used to estimate the inhomogeneous moisture distribution (as dielectric constant) in a large cross section of polymer foam. The targeted application of MWT is in industrial drying to derive intelligent control methods based on tomographic inputs for selective heating purposes. One of the features of the proposed method shows how to integrate lab-based dielectric characterization, often available in MWT application cases, in the prior modeling. The method is validated with numerical and experimental MWT data for the considered moisture distributions.

FILTERING OF SIGNALS WITH UNKNOWN INTERFERENCE POWER AND RANDOM JUMPING PERTURBATION

A linear stochastic dynamic system under the influence of random disturbances and interference is considered. The perturbation is a sequence of uncorrelated random variables with a distribution in the range [–1, 1]. This sequence is modeled by a random binary signal with values 1 and 0 and is described by a Markov chain with known probabilities of transitions from one state to another. The modulated signal is fed to the input of the linear control unit. The output signal of the control object is measured with an error, which is a sequence of uncorrelated random variables with an unknown distribution in the range [–1, 1].The problem under consideration differs from the optimal linear filtration problems based on the application of the Kalman filter and its modifications. Its novelty consists in the following: 1) the input signal is a random jumping process – uncorrelated noise modulated by a random binary signal; 2) the variances of random processes – the input signal and the interference characterizing their power are unknown. А posteriori mathematical expectation and the a posteriori variance of the filtering error are determined by the methods of Bayesian estimation and the theory of systems with a random jump structure. The optimal estimation algorithm is described by a system of recurrent equations. It consists of five interconnected blocks: 1) a meter of mathematical expectation and dispersion of an additive mixture of an output signal with interference; 2) an indicator of a random structure; 3) a classifier of a random structure; 4) a dispersiometer; 5) a filter.

Law of large numbers for the sequence of uncorrelated fuzzy random variables

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy random variables, weak and strong laws of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

Bone mineral density modeling via random field: Normality, stationarity, sex and age dependence

Background and ObjectiveCapturing the population variability of bone properties is of paramount importance to biomedical engineering. The aim of the present paper is to describe variability and correlations in bone mineral density with a spatial random field inferred from routine computed tomography data. MethodsRandom fields were simulated by transforming pairwise uncorrelated Gaussian random variables into correlated variables through the spectral decomposition of an age-detrended correlation matrix. The validity of the random field model was demonstrated in the spatiotemporal analysis of bone mineral density. The similarity between the computed tomography samples and those generated via random fields was analyzed with the energy distance metric. ResultsThe random field of bone mineral density was found to be approximately Gaussian/slightly left-skewed/strongly right-skewed at various locations. However, average bone density could be simulated well with the proposed Gaussian random field for which the energy distance, i.e., a measure that quantifies discrepancies between two distribution functions, is convergent with respect to the number of correlation eigenpairs. ConclusionsThe proposed random field model allows the enhancement of computational biomechanical models with variability in bone mineral density, which could increase the usability of the model and provides a step forward in in-silico medicine.

Non-fragile asynchronous state estimation for Markovian switching CVNs with partly accessible mode detection: The discrete-time case

This article is devoted to the non-fragile asynchronous state estimation problem for Markovian switching complex-valued networks subject to randomly occurring nonlinearities (RONs) and external stochastic disturbances. Two mutually uncorrelated random variables with known statistical properties are introduced to depict the existing RONs. Considering the case that the information concerning to the system modes cannot be completely acquired by the state estimator to be designed, the asynchronous phenomenon is considered via a hidden Markov model with only partial mode detection probabilities being accessible. By resorting to the intensive stochastic analysis as well as an improved complex-valued reciprocal convex inequality, some mode-dependent criteria are provided which ascertain that the estimation error system is asymptotically mean square stable. In addition, the estimator gains desired can be appropriately designed by resorting to feasible solutions of a set of complex matrix inequalities. One numerical example is also provided to demonstrate effectiveness of the estimation scheme proposed.

Extreme statistics of the excitations in the random transverse Ising chain

In random quantum magnets, like the random transverse Ising chain, low energy excitations are localized in rare regions and there are only weak correlations between them. It is an open question whether these correlations are relevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and---for shorter chains---by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Considering the finite size corrections, the two numerical methods give very similar results, but these differ from the correction term for uncorrelated random variables, indicating that the weak correlations between low-energy excitations in random quantum magnets are relevant.

Estimation of coefficient of variation for structural analysis: The correlation interval approach

The paper is focused on the efficient estimation of the coefficient of variation for functions of correlated and uncorrelated random variables. Specifically, the paper deals with time-consuming functions solved by the non-linear finite element method. In this case, the semi-probabilistic methods must reduce the number of simulations as much as possible under several simplifying assumptions while preserving the accuracy of the obtained results. The selected commonly used methods are reviewed with the intent of investigating their theoretical background, assumptions and limitations. It is shown, that Taylor series expansion can be modified for fully correlated random variables, which leads to a significant reduction in the number of simulations independent of the dimension of the stochastic model (the number of input random variables). The concept of the interval estimation of the coefficient of variation using Taylor series expansion is proposed and applied to numerical examples of increasing complexity. It is shown that the obtained results correspond to the theoretical conclusions of the proposed method.

A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

A new probe of Gaussianity and isotropy with application to cosmic microwave background maps

We introduce a new mathematical tool (a direction-dependent probe) to analyze the randomness of purported isotropic Gaussian random fields on the sphere. If the field is isotropic and Gaussian then the probe coefficients for a given direction should be realizations of uncorrelated scalar Gaussian random variables. To study the randomness of a field, we use the autocorrelation of the sequence of probe coefficients (which are just the Fourier coefficients [Formula: see text] if the [Formula: see text]-axis is taken in the probe direction). We introduce a particular function on the sphere (called the AC discrepancy) that accentuates the departure from Gaussianity and isotropy. We apply the probe to assess the full-sky cosmic microwave background (CMB) temperature maps produced by the Planck collaboration (PR2 2015 and PR3 2018), with special attention to the inpainted maps. We find that for some of the maps, there are many directions for which the departures are significant, especially near the galactic plane. We also look briefly at the noninpainted Planck maps, for which the computed AC discrepancy maps have a very different character, with features that are global rather than local.

Fatigue crack-growth predictions for a railway axle under material data variability

This document reports a probabilistic approach for residual fatigue life predictions of a cracked railway axle with allowance for variability of the input data. A NASGRO formulation was used for such a purpose, with its main parameters considered as uncorrelated random variables varying in given ranges. A load spectrum was considered in order to take into account of the actual loading conditions that these components undergo. The presented approach aimed to perform residual fatigue life predictions in terms of both mean value and dispersion around it. The results were also used to drive deterministic approaches in calculating conservative crack-growth assessments.

Angle-of-Arrival Estimation Technique for Fast Beamforming Using Monopulse Signal in the Antenna Array Systems

In this paper we propose an anlge-of-arrival (AoA) estimation technique using the monopulse signal to achieve a reliable link for the ground-to-unmanned aerial vehicle (UAV) or UAV-to-UAV. We first find the exact statistical characteristics of the monopulse signal, which is the random variable as a form of the ratio between the uncorrelated quadratic random variables with non-zero means. Specifically, we derive the probability density function (PDF) and the cumulative distribution function (CDF) of the monopulse signal in antenna array in closed-form. Then using the statistical characteristics of the monopulse signal, we develop the adaptive AoA estimator, which can be applied to fast beamforming with high accuracy. Based on our exact analytical results, we derive the mean squared error (MSE) performance of the proposed method. We show from our simulations that the proposed method achieves better performance with relatively less complexity and overhead than other conventional methods.

Kolmogorov's strong law of large numbers holds for pairwise uncorrelated random variables

Используя подход Н. Этемади (1981 г.) к усиленному закону больших чисел (УЗБЧ) и развитие этого подхода, предпринятое в работе Ш. Чeргe, К. Тандори и В. Тотика (1983 г.), мы приводим более слабые условия, при которых УЗБЧ все еще справедлив, для попарно некоррелированных (а также для "почти некоррелированных") случайных величин. Мы сосредоточиваем внимание, в частности, на случайных величинах, не являющихся одинаково распределенными. Наш подход позволяет получить еще одно, простое доказательство классического УЗБЧ.

Protocol-Based Tobit Kalman Filter Under Integral Measurements and Probabilistic Sensor Failures

This paper is concerned with the Tobit Kalman filtering problem for a class of discrete time-varying systems subject to censored observations, integral measurements and probabilistic sensor failures under the Round-Robin protocol (RRP). The censored observations are characterized by the Tobit observation model, the integral measurements are described as functions of system states over a certain time interval required for data acquisition, and the sensor failures are governed by a set of uncorrelated random variables. The RRP is employed to decide the transmission sequence of sensors in order to alleviate undesirable data collisions. By resorting to the augmentation technique and the orthogonality projection principle, a protocol-based Tobit Kalman filter (TKF) is developed with the coexistence of integral measurements and sensor failures that lead to a couple of augmentation-induced terms. Moreover, the performance of the proposed filter is analyzed through examining the statistical property of the error covariance of the state estimation. Further analysis shows the existence of self-propagating upper and lower bounds on the estimation error covariance. A case study on ballistic roll rate estimation is presented to illustrate the efficacy of the developed filter.

Kolmogorov's Strong Law of Large Numbers Holds for Pairwise Uncorrelated Random Variables

Using the approach of Etemadi for the strong law of large numbers [Z. Wahrsch. Verw. Gebiete, 55 (1981), pp. 119--122] and its elaboration by Csorgo, Tandori, and Totik [Acta Math. Hungar., 42 (198...

Territorial behaviour of buzzards versus random matrix spacing distributions

A deeper understanding of the processes underlying the distribution of animals in space is crucial for both basic and applied ecology. The Common buzzard (Buteo buteo) is a highly aggressive, territorial bird of prey that interacts strongly with its intra- and interspecific competitors. We propose and use random matrix theory to quantify the strength and range of repulsion as a function of the buzzard population density, thus providing a novel approach to model density dependence. As an indicator of territorial behaviour, we perform a large-scale analysis of the distribution of buzzard nests in an area of 300 square kilometres around the Teutoburger Wald, Germany, as gathered over a period of 20 years. The nearest and next-to-nearest neighbour spacing distribution between nests is compared to the two-dimensional Poisson distribution, originating from uncorrelated random variables, to the complex eigenvalues of random matrices, which are strongly correlated, and to a two-dimensional Coulomb gas interpolating between these two. A one-parameter fit to a time-moving average reveals a significant increase of repulsion between neighbouring nests, as a function of the observed increase in absolute population density over the monitored period of time, thereby proving an unexpected yet simple model for density-dependent spacing of predator territories. A similar effect is obtained for next-to-nearest neighbours, albeit with weaker repulsion, indicating a short-range interaction. Our results show that random matrix theory might be useful in the context of population ecology.

Uncertainty Quantification of Waveguide Dispersion Using Sparse Grid Stochastic Testing

To quantify waveguide dispersion uncertainty, we propose a stochastic analysis technique based on the polynomial chaos method, where Smolyak sparse grids are used as sample spaces for the recently developed nonintrusive stochastic testing (ST) technique. This strategy allows for a smaller set of candidate collocation nodes in the ST algorithm compared with the computationally expensive tensor grid approach. The method is tailored to the analysis of waveguide structures with a statistically spatially varying dielectric-property profile, which is then captured by a discrete set of uncorrelated random variables by means of Karhunen–Loeve transforms. In this way, the propagation properties of realistic stochastic waveguides can be predicted, which is of major importance in the electronic design process. The presented method, which is constructed around a second-order full-wave deterministic finite element solver, is illustrated through the uncertainty quantification of two waveguide dispersion problems and is validated using commercial electromagnetic field solvers and the Monte Carlo method.

Distributed filtering for a class of discrete-time systems over wireless sensor networks

Abstract This paper addresses the distributed filter design problem for a class of dynamic systems over wireless sensor networks. The missing measurements and the correlation among state noises and measurement noises are considered, where a set of mutually uncorrelated random variables is employed to describe the missing phenomena. Firstly, the construction of the designed filter is proposed and the prediction of the state at each node is given. Then, the filtering error covariance is presented and the filter parameters are determined to minimize the trace of such a covariance, where the network topology data are used to simplified the singular matrix. Subsequently, the relationship between the filter performance and missing probability of the measurement is discussed. Finally, a numerical simulation is presented to illustrate the effectiveness and capability of the proposed distributed filters.