Asymptotic Mean Square Research Articles

Stability of stochastic difference equations with multiplicative and additive fading noise

In this paper, the effects of multiplicative and additive fading stochastic perturbations on the solutions of stochastic difference equations are studied, in which the strength of fading stochastic perturbations is square summable. Different from the existing research, this paper mainly studies the influence of two kinds of stochastic perturbations on the stability of the system, and applies the Lyapunov functional method, the asymptotic mean square trivial sufficiency conditions are obtained for the solutions of stochastic difference equations which based on the influence of multiplicative and additive fading noises. At the same time, the conditions of stochastic perturbation fading rate are obtained. Finally, the above conditions are verified by numerical simulation examples, and the results of numerical simulation show that: (1) the additive perturbations have less influence on the system than the multiplicative perturbations; (2) the fading stochastic perturbations can make the system reach a stable state in a faster time.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ > 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

Asynchronous Control of 2-D Markov Jump Roesser Systems With Nonideal Transition Probabilities.

This article intends to study the asynchronous control problem for 2-D Markov jump systems (MJSs) with nonideal transition probabilities (TPs) under the Roesser model. Two practical considerations motivate the current work. First, considering that the system mode cannot always be observed accurately, a hidden Markov model (HMM) is adopted to describe the relationship between the mismatched modes. Second, considering that the TPs information related to the Markov process and the observation process is difficult to obtain, the nonideal TPs (unknown or uncertain) are simultaneously considered on the two processes. Under the considerations, several new sufficient conditions are developed for concerned closed-loop 2-D MJSs with nonideal TPs, by which the asymptotic mean square stability is ensured with an H∞ performance index. A nonconservative separation strategy is utilized to decouple the system mode TPs and the observation TPs to facilitate the analysis of nonideal TPs. An unified LMI-based condition is finally developed for the concerned closed-loop 2-D MJSs with/without nonideal TPs, showing more satisfactory conservatism than that in the literature. In the end, we present two examples to validate the superiority of the proposed design method.

Asymptotical Mean-Square Stabilization of Networked Control Systems Operating Over Multipath Routing Networks

Asymptotical Mean-Square Stabilization of Networked Control Systems Operating Over Multipath Routing Networks

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

A brief review on stability investigations of numerical methods for systems of stochastic differential equations

<abstract><p>A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.</p></abstract>

Restricted Liu estimator under stochastic linear restrictions in generalized linear models: theory and applications

It is a fact that the problem of multicollinearity adversely effects the estimation of parameters not only in normal linear models but also in generalized linear models (GLMs). In this study, we propose a new estimator by imposing the stochastic restrictions on the parameters and combining the Liu estimator in GLMs to address the problem of multicollinearity. The new estimator is referred to as the stochastic restricted Liu estimator. The novel estimator’s first order approximated form (FOA) is presented to compare it to its competitors using the asymptotic mean square error criterion. Furthermore, two empirical applications using a response variable from the Poisson and Gamma distributions are studied to evaluate the performance of estimators. In addition, a simulation study is conducted by considering a response variable from Binomial distribution for comparisons. Based on simulation study and empirical applications, it is found that the performance of the proposed estimator is better than ML, Liu, and mixed estimators for the appropriate selected biasing parameter.

Channel Estimation for LEO Satellite Massive MIMO OFDM Communications

In this paper, we investigate the massive multiple-input multiple-output orthogonal frequency division multiplexing channel estimation for low-earth-orbit satellite communication systems. First, we use the angle-delay domain channel to characterize the space-frequency domain channel. Then, we show that the asymptotic minimum mean square error (MMSE) of the channel estimation can be minimized if the array response vectors of the user terminals (UTs) that use the same pilot are orthogonal. Inspired by this, we design an efficient graph-based pilot allocation strategy to enhance the channel estimation performance. In addition, we devise a novel two-stage channel estimation (TSCE) approach, in which the received signals at the satellite are manipulated with per-subcarrier space domain processing followed by per-user frequency domain processing. Moreover, the space domain processing of each UT is shown to be identical for all the subcarriers, and an asymptotically optimal vector for the per-subcarrier space domain linear processing is derived. The frequency domain processing can be efficiently implemented by means of the fast Toeplitz system solver. Simulation results show that the proposed TSCE approach can achieve a near performance to the MMSE estimation with much lower complexity.

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

A new class of Poisson Ridge-type estimator

The Poisson Regression Model (PRM) is one of the benchmark models when analyzing the count data. The Maximum Likelihood Estimator (MLE) is used to estimate the model parameters in PRMs. However, the MLE may suffer from various drawbacks that arise due to the existence of multicollinearity problems. Many estimators have been proposed as alternatives to each other to alleviate the multicollinearity problem in PRM, such as Poisson Ridge Estimator (PRE), Poisson Liu Estimator (PLE), Poisson Liu-type Estimator (PLTE), and Improvement Liu-Type Estimator (ILTE). In this study, we define a new general class of estimators which is based on the PRE as an alternative to other existing biased estimators in the PRMs. The superiority of the proposed biased estimator over the other existing biased estimators is given under the asymptotic matrix mean square error sense. Furthermore, two separate Monte Carlo simulation studies are implemented to compare the performances of the proposed biased estimators. Finally, the performances of all considered biased estimators are shown in real data.

Waveform design for higher-resolution localization with MIMO radar

To improve the accuracy and resolution of angle estimation with multiple signal classification (MUSIC) algorithm, we study the design of low peak-to-average-power-ratio waveforms for multiple-input-multiple-output (MIMO) radar. Assuming that the prior knowledge about the target directions and amplitudes is available (e.g., by transmitting initial probing waveforms), we formulate a waveform design problem based on minimizing the asymptotic mean square error of MUSIC. An alternating direction method of multipliers is developed to tackle the encountered nonconvex optimization problem. Compared with the competing waveforms, the waveforms designed by the proposed algorithm exhibit higher resolution and lower estimation error.

A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity

<abstract><p>This paper is concerned with proposing a generalized Liu-type estimator (GLTE) to address the multicollinearity problem of explanatory variable of the linear part in the logistic partially linear regression model. Using the profile likelihood method, we propose the GLTE as a general class of Liu-type estimator, which includes the profile likelihood estimator, the ridge estimator, the Liu estimator and the Liu-type estimator as special cases. The conditional superiority of the proposed GLTE over the other estimators is derived under the asymptotic mean square error matrix (MSEM) criterion. Moreover, the optimal choices of biasing parameters and function of biasing parameter are given. Numerical simulations demonstrate that the proposed GLTE performs better than the existing estimators. An application on a set of real data arising from the study of Indian Liver Patient is shown for illustrating our theoretical results.</p></abstract>

A new improved Liu-type estimator for Poisson regression models

The Poisson Regression Model (PRM) is commonly used in applied sciences such as economics and the social sciences when analyzing the count data. The maximum likelihood method is the well-known estimation technique to estimate the parameters in PRM. However, when the explanatory variables are highly intercorrelated, unstable parameter estimates can be obtained. Therefore, biased estimators are widely used to alleviate the undesirable effects of these problems. In this study, a new improved Liu-type estimator is proposed as an alternative to the other proposed biased estimators. The superiority of the new proposed estimator over the existing biased estimators is given under the asymptotic matrix mean square error criterion. Furthermore, Monte Carlo simulation studies are executed to compare the performances of the proposed biased estimators. Finally, the obtained results are illustrated in real data. Based on the set of experimental conditions which are investigated, the proposed biased estimator outperforms the other biased estimators.

Improving bridge estimators via f-GAN

Bridge sampling is a powerful Monte Carlo method for estimating ratios of normalizing constants. Various methods have been introduced to improve its efficiency. These methods aim to increase the overlap between the densities by applying appropriate transformations to them without changing their normalizing constants. In this paper, we first give a new estimator of the asymptotic relative mean square error (RMSE) of the optimal Bridge estimator by equivalently estimating an f-divergence between the two densities. We then utilize this framework and propose f-GAN-Bridge estimator (f-GB) based on a bijective transformation that maps one density to the other and minimizes the asymptotic RMSE of the optimal Bridge estimator with respect to the densities. This transformation is chosen by minimizing a specific f-divergence between the densities. We show f-GB is optimal in the sense that within any given set of candidate transformations, the f-GB estimator can asymptotically achieve an RMSE lower than or equal to that achieved by Bridge estimators based on any other transformed densities. Numerical experiments show that f-GB outperforms existing methods in simulated and real-world examples. In addition, we discuss how Bridge estimators naturally arise from the problem of f-divergence estimation.

Stability of Euler Methods for Fuzzy Differential Equation

The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ.

Spectral criteria to stability and observability of mean-field stochastic periodic systems

This paper addresses stability and observability of discrete-time mean-field stochastic systems with periodic coefficients and multiplicative noise. By constructing a monodromy operator associated to the periodic coefficients of the considered dynamics, spectral criterion and Lyapunov-type criterion are presented for asymptotic mean square stability and weak stability, respectively. Based on the Lyapunov criterion, a necessary and sufficient condition is supplied for regional stabilizability. Furthermore, Popov–Belevitch–Hautus (PBH) criterion is obtained for observability of mean-field stochastic periodic systems. By the proposed spectral criterion of stability/observability, the intrinsic relationships are clarified for stability/observability among deterministic periodic systems, stochastic periodic systems and mean-field stochastic periodic systems. Finally, as an application of PBH criterion, a Barbashin–Krasovskii stability theorem is established to indicate the connection between observability and stability of the considered systems.

Optimal subsampling for large‐sample quantile regression with massive data

AbstractTo balance the explosive growth of data volume and limited budgets for computational resources, one of the popular methods is downscaling the data volume by subsampling a subdataset that inherits the relevant property of the full data. As an alternative to the mean regression model, the quantile regression model has been studied extensively when the data are independent and the data scale is medium. This article focuses on quantile regression with massive data where the sample size n (greater than in general) is extraordinarily large but the dimension d (smaller than 20 in general) is small. We first formulate the general subsampling procedure and establish the asymptotic property of the resultant estimator. Then, with the help of optimality criteria in experimental design, we derive two subsampling probabilities that are optimal in the sense of smallest asymptotic mean square error. Since the optimal subsampling probabilities depend on the full data estimator, we develop a two‐step optimal subsampling algorithm and study the consistency and asymptotic normality of the resultant estimator. The empirical performance of the optimal subsampling algorithm is evaluated with synthetic and real datasets.

Decentralized observer‐based controller design for large‐scale systems with quantized measurements and actuator faults

AbstractThis paper is focused on designing observer‐based decentralized memory feedback controller for ensuring the asymptotic mean square stability of the large‐scale systems with minimum H∞ performance index. Precisely, the unknown interconnection between each subsystems of a large‐scale system is assumed to satisfy quadratic bounds, and measured output is quantized by a logarithmic quantizer. Also, the signals are transmitted through the actuator component wherein the occurrence of fault is indispensable. Thus, the impact of faults in actuator is considered in control design to tolerate the fault effects and also for ensuring robust performance. A state space representation of the system is formulated to reconstruct the unmeasurable states via the available informations of input/output dynamics. Based on the designed observer, a decentralized memory feedback controller is developed. Specifically, in terms of linear matrix inequalities, the stability conditions are derived and which are sufficient to guarantee the desired result. At last, simulations are carried out for two numerical examples to validate the potential of the theoretical result.

Stability analysis of delayed Markovian jump systems with delay switching and state signals and applications

AbstractThis article addresses the asymptotical mean square stability problem of continuous‐time delayed Markovian jump systems (MJSs). Different from the existing delayed MJS models, time‐varying delays are contained in not only system state but also switching signal and could lead to great difficulties. An auxiliary system approach is developed to handle the difficulties, in which the traditionally delayed state and a new delayed exponential matrix are both involved. Based on the established model, sufficient stability conditions with solvable forms are given by using a Lyapunov function approach and some novel techniques. More extensions about an MJS under sampled switching and state signals are further considered, and so does the stabilization problem. A numerical example is offered to verify the effectiveness and superiority of the methods proposed in this study.

Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces

• In this article we have studied the asymptotic stability and mean square stability of fractional-order (1,2] stochastic differential equations (SDEs). • We have considered the family of SDEs with variable delay in state. • For obtaining main results we used Banach fixed point theorem and imposed Lipschitz condition on nonlinearity. • Results are advanced and weighted enough as a contribution to stability theory of fractional SDEs. In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2 . We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory.