Moments Of Variables Research Articles

Exploration of dual-attention mechanism-based deep learning for multi-step-ahead flood probabilistic forecasting

In order to improve the flood forecasting accuracy and reflect the forecast uncertainty information in the Three Gorges Reservoir (TGR) interval-basin in China, this study integrates the feature and temporal dual-attention (DA) mechanism and recursive encoder-decoder (RED) structure into the long short-term memory (LSTM) neural network to develop a DA-LSTM-RED model. The feature attention acts on the input variables of the encoder, and the temporal attention mechanism acts on the hidden layer states extracted by the LSTM neural network during encoding process, prompting the proposed model to extract critical input information among different types and moments of input variables to improve the multi-step-ahead flood forecasting accuracy. Second, the copula-based Hydrological Uncertainty Processor (copula-HUP) is used to quantify the forecast uncertainty of the proposed model meanwhile creating multi-step-ahead flood probabilistic forecasts. Combining the long-term 6 h hydrologic data series of the Xiangjiaba-TGR interval-basin and the forecasted precipitation from the European Centre for Medium-Range Weather Forecasts (ECMWF), the effectiveness of the proposed model, the effect of forecast precipitation on multi-step-ahead flood forecasting, and the effect of different copula functions on the probabilistic forecast of copula-HUP are investigated, respectively. The results show that the DA-LSTM-RED model can effectively improve the forecasting accuracy for long forecast horizons (3-7d) compared to the LSTM-RED model, and the average absolute error metrics are reduced by 10%-17%. Meanwhile, the proposed model can identify input variables with a high correlation with the target output variables, which improves the interpretability of deep learning to a certain extent. The Student copula-HUP has the lowest RB and CRPS metrics than the Frank and Gaussian copula-HUP, which can better quantify the DA-LSTM-RED model's forecast uncertainty. Therefore, combining the proposed model with the Student copula-HUP can effectively reduce the forecasting uncertainty, enhance the forecasting reliability and accuracy for future horizons, and provide reliable risk information for the TGR flood control scheduling decision.

New fractal–fractional parametric inequalities with applications

In the present paper, we first establish a general parameterized identity for local fractional twice differentiable functions involving extended fractal–fractional integral operators. Thus by employing generalized convexity on differentiable mappings along with Yang’s Power-mean, Hölder’s and improved fractal integral inequalities lead us to develop variety of new fractal–fractional parameterized inequalities. Several examples are provided with graphical illustrations to prove the validity of new results. We give error analysis of improved bounds numerically and also by 2D, 3D graphical representations. Finally, we show that our main results recapture fractal variants of trapezoid, midpoint, Simpson and Bullen-type inequalities. Some related applications to the fractal means, moment of random variables and wave equations are given as well.

THE USE OF MODEL DISTRIBUTIONS IN THE ANALYSIS OF QUEUING SYSTEMS WITH CORRELATED ARRIVALS

The article considers the task of determining the waiting time for a request in a queue in a general-type single-channel queueing system when time intervals between incoming requests are correlated. For solving the task, it is proposed to carry out decorrelation of specified time intervals using the discrete cosine transform. Generally, to analyze arbitrary distributions included in the expression for a decorrelated sequence, one can suggest an approach based on the approximation of unknown densities by a model distribution. The calculation of cumulants and moments of random variables is very simple compared to the analytical methods. Based on the found final set of cumulants, a model probability density can always be reproduced with a certain error. Generally, to calculate the average waiting time of a request in a queue, it is necessary to determine the correlation properties and one-dimensional probability density of time intervals between incoming requests, synthesize a two-dimensional probability density of a sequence of given intervals that has a measured correlation function, and, further, calculate joint moments and cumulants for correlated values, on which their model density is built.

Modeling the presumed joint probability density function of conditioning variables in stratified turbulent flames

Experimental data from the Cambridge-Sandia burner are used to calculate the conditional average of temperature and the mass fractions of H2O, CO, and CO2 in the context of the Uniform Conditional State (UCS) model. The performance of various presumed joint PDF models are examined for this burner working with different swirl and stratification levels. All of the presented PDF models are based on a few statistical moments of the conditioning variables which are extracted from the data at each measuring point. Primarily, only mixture fraction and reaction progress variable are employed as the conditioning variables. The β function is employed for modeling the marginal PDF of mixture fraction in all cases. For the progress variable, the β function and Modified Laminar Flamelet (MLF)-PDF are tested for modeling the marginal PDF. It is shown that even though the MLF-PDF results in more accurate predictions overall for most of the cases, its superiority with respect to the β-PDF is relatively small in the stratified flames. The performance of the product, Plackett, and Gaussian copulas are also investigated for employing the cross-correlation of mixture fraction and progress variable in building their joint PDF. It is found that the Gaussian copula shows a superior performance over the other two. Next, normalized total enthalpy is employed as a third conditioning variable, and different PDF models are examined for the three-condition version of UCS. It is found that even though adding the third conditioning variable can improve the predictions by considering the effect of heat transfer on the chemistry manifold, such improvements depend on having a proper 3D PDF model for the conditioning variables. In the vicinity of the flame brush, the models employed for the 3D PDFs represent an inferior performance compared to the two-condition version with the use of a Gaussian copula due to complications in the marginal PDF of normalized total enthalpy in these regions. To benefit from considering the impact of heat transfer on chemistry while also avoiding complexities in modeling the 3D joint PDF near the reaction zone, a switch between the two- and three-conditional models which is triggered by the variance of progress variable is proposed. It is shown that the predictions obtained by the use of this switch are almost always superior to all other PDF models investigated in this study.

Risk aggregation with FGM copulas

We offer a new perspective on risk aggregation with FGM copulas. Along the way, we discover new results and revisit existing ones, providing simpler formulas than one can find in the existing literature. This paper builds on two novel representations of FGM copulas based on symmetric multivariate Bernoulli distributions and order statistics. First, we detail families of multivariate distributions with closed-form solutions for the cumulative distribution function or moments of the aggregate random variables. We provide methods to compute the cumulative distribution function of aggregate rvs when the marginals are discrete, then order aggregate random variables under the convex order. Finally, we discuss risk-sharing and capital allocation, providing numerical examples for each.

Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network

We discuss the analysis of a piezoelectric energy harvester for random mechanical vibrations, and we assess the performance improvement guaranteed by interposing a matching network between the transducer and the electrical load, in terms of average output power and power efficiency. The mathematical model describing the harvester is a system of stochastic differential equations, where both cases of linear and nonlinear devices are considered. In the linear case, the power delivered to the load is increased by a factor of about 20 with respect to the direct connection, with a similar increase in the conversion efficiency. In the nonlinear case, we use a moment closure technique to calculate the first- and second-order moments of the electro-mechanical variables in the weak noise limit. Moment calculation is used to determine the optimal values of the matching network components that maximize the performance. In the strong noise limit, the state equations are integrated numerically to determine the same performance metrics. Our analysis shows that a properly designed matching network improves the performance by a significant amount, especially at low noise intensity.

Short-term Wind Speed Prediction Based on Improved Auto Encoder

In order to improve the wind power consumption level of the new power system with new energy as the main body, it is necessary to accurately predict the wind speed. The key is to refine the dynamic trend of the wind power system and the potential physical structure in the wind speed sequence. Firstly, on the basis of the library Koopman dynamics theory and the encoder structures, a physically constrained spatio-temporal neural network is built, which generates the linear evolution moment of the nonlinear variables of the wind farm. Secondly, the system trend is approximated by the linear evolution matrix, and the forward and backward dynamics are fully considered in the prediction process. Then, the bidirectional correlation prediction mechanism and the cost function adapted to different objects are set to reduce the requirements for the reversibility and stability of the prediction sequence. Meanwhile, the hidden vector of feature space is visualized to show the feature interval dependency in the system. Finally, the effectiveness of the proposed method is verified by the wind speed measurement data of Prince Mountain in Beipiao. The results show that the proposed method has high prediction accuracy, strong generalization ability, and high interpretability for strong random and strong fluctuation wind speed series.

On Some Inequalities of Differentiable Uniformly Convex Mapping with Applications

The widely known hermite-hadamard-Fejer type inequalities are so important in the field of mathematical analysis. Many researchers have studied on these inequalities. In this paper, we have obtained several inequalities related to the Hermite-Hadamard inequality for a special class of the functions called uniformly convex functions. We have also presented applications of these obtained inequalities in some error estimates for higher moments of random variables.

Some integral inequalities for Hadamard fractional moments of continuous random variables

Using Hadamard fractional integral operator, some new integral inequalities for the Hadamard fractional moments of continuous random variables are obtained.

New easy to compute formulas of the moments of random variables appearing in the coupon collector problem

New easy to compute formulas of the moments of random variables appearing in the coupon collector problem

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.

Generalized first-order second-moment method for uncertain random structures

<abstract> <p>In this paper, a new reliability assessing method for structures influenced by both aleatory and epistemic uncertainty simultaneously is developed. To handle hybrid types of uncertainties, chance theory is introduced to define a new hybrid reliability index. By mathematical derivation and theorems proofs, the new index is showed to be effective and compatible with hybrid types of uncertainties. Correspondingly, a generalized first-order second-moment (GFOSM) algorithm is established for practical reliability assessment of structures with hybrid uncertainties. Based on the first two moments of basic variables, the GFOSM method can perform fast and effective reliability assessment without large-scale integration operations and can be considered as an extension and expansion of the traditional FOSM method. Two numerical cases further illustrate the effectiveness and practicability of the proposed method from different perspectives.</p> </abstract>

Probabilistic Bounds with Quadratic-Exponential Moments for Quantum Stochastic Systems

This paper is concerned with quadratic-exponential moments (QEMs) for dynamic variables of quantum stochastic systems with position-momentum type canonical commutation relations. The QEMs play an important role for statistical “localisation” of the quantum dynamics in the form of upper bounds on the tail probability distribution for a positive definite quadratic function of the system variables. We employ a randomised representation of the QEMs in terms of the moment-generating function (MGF) of the system variables, which is averaged over its parameters using an auxiliary classical Gaussian random vector. This representation is combined with a family of weighted L2 -norms of the MGF, leading to upper bounds for the QEMs of the system variables. These bounds are demonstrated for open quantum harmonic oscillators with vacuum input fields and non-Gaussian initial states.

An Efficient High-Order-Moment-Based Reliability Method Combining the Maximum Likelihood Point and Cubic Normal Transformation

Moment-based methods can measure the safety degrees of mechanical systems affected by unavoidable uncertainties, utilizing only the statistical moments of random variables for reliability analysis. For the conventional derivation of the first four statistical moments based on the second-order Taylor expansion series evaluated at the most likelihood point (MLP), skewness and kurtosis involve the higher fourth raw moments of random variables and thus are unfavorable for engineering applications. This paper develops new computing formulae for the first four statistical moments which require only the first four central moments of random variables, and the probability distribution of the performance function is approximated using cubic normal transformation. Several numerical examples are given to demonstrate the accuracy of the proposed methods. Comparisons of the two proposed approaches and the maximum entropy method (ME) are also made regarding reliability assessment.

Reliability analysis of compressed timber studs on the buckling criterion

Introduction. Ensuring safety is a priority goal in the design, construction and operation of building and structures. A quantitative assessment of safety can be the failure probability of a structural element. The article presents algorithms for reliability analysis of timber studs by the buckling criterion under central compression force.
 
 Materials and methods. The FOSM (First Order Second Moment) method is a classic approach for solving many reliability analyses tasks in the engineering sector. This method uses the second statistical moments of random variables (mathematical expectation and variance) and the method of statistical linearization with the decomposition of the function into a Taylor series of the first order. However, it is often necessary to deal with nonlinear limit state functions, where the statistical linearization method can lead to incorrect results. In this case, it is necessary to use other algorithms for reliability analysis: the Hasofer – Lind method, SORM (Second Order Reliability Method), etc.
 
 Results. The numerical example of reliability analysis for compressed timber studs on the buckling criterion is considered. To illustrate the problem of the research, the dimensions of the cross-section of the studs and the strength of the wood are taken as random variables, and the load is considered a deterministic value. In the following, an algorithm for reliability analysis is described for a timber studs under random load. The classical FOSM approach showed an overestimated reliability index by 35 % compared to the results of the Monte Carlo numerical experiment and the analytical solution using the Hasofer – Lind algorithm.
 
 Conclusions. The use of the traditional FOSM method for reliability analysis of wooden studs according to the buckling criterion can lead to overestimated reliability index estimates. It is unacceptable from the point of view of the structural safety. The calculation and analysis of the reliability of timber studs should be carried out on the basis of the Hasofer – Lind algorithm, the SORM method or other optimization methods.

Motzkin path decompositions of functionals in noncommutative probability

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional [Formula: see text] are naturally described in terms of Motzkin paths identified with reduced Motzkin words [Formula: see text]. Using this fact, we demonstrate that the mixed moments of order [Formula: see text] of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to [Formula: see text] with summation extending over [Formula: see text], the set of reduced Motzkin paths of length [Formula: see text]. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice [Formula: see text] into sublattices [Formula: see text] of partitions which are monotonically adapted to colors in [Formula: see text]. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.

An electroelastic Kirchhoff rod theory incorporating free space electric energy

This work presents a geometrically exact Kirchhoff-like electroelastic rod theory wherein the contribution of free space energy is also factored in. In addition to the usual mechanical variables such as the rod’s centerline and cross-section orientation, three electric potential parameters are also introduced to account for the variation in electric potential within the rod’s cross-section as well as along the rod length. The free space energy is included through an electric flux-like variable acting on the lateral surface of the rod. The governing equations of this rod are derived through balance of forces and moments (arising from both mechanical and electric sources) for mechanical variables, weighted integration of the three-dimensional electric displacement equation in the rod’s cross-section for electric potential parameters and through the asymptotic expansion of a boundary integral equation for the electric flux variable. We also obtain the expressions of internal contact moment and conjugates of electric potential parameters in terms of the rod’s aforementioned mechanical and electric variables. A specific choice of the three-dimensional electroelastic stored energy density is further taken catering to piezoelectric materials and dielectric elastomers and the rod’s constitutive equations are rederived which exhibit interesting interplay between mechanical and electric variables. The presented work can help improve the design of soft electroelastic rod-like actuators and sensors.

Stochastic analysis of structures under limited observations using kernel density estimation and arbitrary polynomial chaos expansion

Over the past decades there has been considerable interest among the scientific community in modeling random system inputs from limited observations and propagating the system responses. In this paper, we develop a novel method for the reasonable modeling of system inputs and efficient propagation of response. Our method firstly constructs the random model of system inputs from observations by developing a novel kernel density estimator (KDE) for Karhunen–Loeve (KL) variables. By further implementing the arbitrary polynomial chaos (aPC) formulation on dependent non-Gaussian KL variables, the associated aPC-based response propagation is then developed. In our method, the developed random model can accurately represent the input parameters from limited observations as the developed KDE of KL variables can incorporate the inherent relation between marginals of input parameters and KL variables, while empowering the input model to preserve the second-order correlations. Furthermore, since the statistical moments of KL variables can be analytically evaluated, the aPC formulation for achieving optimal convergence of system responses can be accurately determined. In addition, the system response can be accurately propagated in an efficient way by developing an aPC-based regression method using space-filling design, in which the conditional distributions in Rosenblatt transformation of aPC variables can be efficiently determined without evaluations of multi-dimensional integrations. In this way, the current work provides an effective framework for the reasonable stochastic modeling​ and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented to highlight the effectiveness of the developed method.

This is the moment for probabilistic loops

We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.

Preconditioners for robust optimal control problems under uncertainty

AbstractThe discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle‐point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriously challenging. In this manuscript, we study efficient preconditioners for all‐at‐once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle‐point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi‐iterative method are proposed to solve this reduced system. We consider a random elliptic partial differential equation whose diffusion coefficient is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates of the dependence of the spectrum of the preconditioned system matrix on the statistical properties of the random field and on the discretization of the probability space. Such estimates involve either the first or second moment of the random variables and , where is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.