Asymptotic Mean Square Research Articles

Some r – k class proportional hazard regression models in the presence of collinearity: an evidence from Indian infant mortality

Proportional hazard regression (PHR) model is used to analyze the time to event data in terms of a set of explanatory variables. The estimation and interpretation of the model parameters are unstable, when there is collinearity between explanatory variables. In order to improve the estimation of proportional hazard model with continuous covariates, the r-k class proportional hazard estimator is proposed, which combines the ridge proportional hazard regression (ridge PHR) and principal component proportional hazard regression (PCPHR). The comparisons of the r-k class PHR, ridge PHR, and PCPHR estimators to the maximum likelihood (ML) according to the asymptotic scalar mean square error (MSE) criterion are done. Simulation study is done to evaluate its performance. Furthermore, the proposed method is applied to assess the infant mortality in India.

Modified-mean-shift-based noisy label detection for hyperspectral image classification

Labeling mistakes occur in the many real training sets of hyperspectral image (HSI) classification, due to mistakes in the collection of labeled training data samples phase. Label noise is an essential problem in classification, with many possible negative consequences. In this paper, a new method is proposed to detect and delete noisy label samples and therefore remove the effect of errors in the classification process. For this purpose, first, a modified mean shift (MMS) method originated from minimum Bayesian risk is proposed and used to improve the separability of the mislabeled samples from training sets. Then denoised training samples are given to the classification algorithms such as SVM, KNN, MLR, and KOMP to classify the HSI data sets. Then, the effect of several types of loss functions is investigated for the proposed MMS method. Also, the analysis of bounded differences inequality, mean square error (MSE), asymptotic mean square error (AMSE), and asymptotic normality are obtained for our method. The performance of the mentioned compared methods greatly improves at high levels of noise, when the MMS method are combined to them. The obtained experimental results show that the proposed MMS method improved the performance of these classification methods for real HSI data sets.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Efficient estimation methods for non-Gaussian regression models in continuous time

In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Levy noises in the case when the unknown functions belong to Sobolev ellipses. Using the Pinsker’s approach, we provide a sharp lower bound for the normalized asymptotic mean square accuracy. However, the main result obtained by Pinsker for the Gaussian white noise model is not correct without additional conditions for the ellipse coefficients. We find such constructive sufficient conditions under which we develop efficient estimation methods. We show that the obtained conditions hold for the ellipse coefficients of an exponential form. For exponential coefficients, the sharp lower bound is calculated in explicit form. Finally, we apply this result to signals number detection problems in multi-pass connection channels and we obtain an almost parametric convergence rate that is natural for this case, which significantly improves the rate with respect to power-form coefficients.

HMM-based quantized dissipative control for 2-D Markov jump systems

In this paper, the dissipative quantized control problem is addressed for Markov jump two-dimensional systems based on Roesser model, in which both asynchronous phenomenon and signal quantization between system modes and controller modes are taken into consideration simultaneously. Moreover, the hidden Markov model (HMM) is adopted to tackle such an asynchronous phenomenon. The principal goal is to devise a state feedback controller, which guarantees that the established closed-loop system achieves asymptotic mean square stability as well as satisfies a prescribed extended dissipative property. Drawing support from Lyapunov function approach and inequality technique, some less conservative criteria ensuring the implementability of the desired controller are derived. Ultimately, the availability and practicability of the developed results are certified through a simulation example.

Multi-Party Dynamic State Estimation That Preserves Data and Model Privacy

In this paper we focus on the dynamic state estimation which harnesses a vast amount of sensing data harvested by multiple parties and recognize that in many applications, to improve collaborations between parties, the estimation procedure must be designed with the awareness of protecting participants’ data and model privacy, where the latter refers to the privacy of key parameters of observation models. We develop a state estimation paradigm for the scenario where multiple parties with data and model privacy concerns are involved. Multiple parties monitor a physical dynamic process by deploying their own sensor networks and update the state estimate according to the average state estimate of all the parties calculated by a cloud server and security module. The paradigm taps additively homomorphic encryption which enables the cloud server and security module to jointly fuse parties’ data while preserving the data privacy. Meanwhile, all the parties collaboratively develop a stable (or optimal) fusion rule without divulging sensitive model information. For the proposed filtering paradigm, we analyze the stabilization and the optimality. First, to stabilize the multi-party state estimator while preserving observation model privacy, two stabilization design methods are proposed. For special scenarios, the parties directly design their estimator gains by the matrix norm relaxation. For general scenarios, after transforming the original design problem into a convex semi-definite programming problem, the parties collaboratively derive suitable estimator gains based on the alternating direction method of multipliers (ADMM). Second, an optimal collaborative gain design method with model privacy guarantees is provided, which results in the asymptotic minimum mean square error (MMSE) state estimation. Finally, numerical examples are presented to illustrate our design and theoretical findings.

Which bridge estimator is the best for variable selection?

We study the problem of variable selection for linear models under the high-dimensional asymptotic setting, where the number of observations $n$ grows at the same rate as the number of predictors $p$. We consider two-stage variable selection techniques (TVS) in which the first stage uses bridge estimators to obtain an estimate of the regression coefficients, and the second stage simply thresholds this estimate to select the “important” predictors. The asymptotic false discovery proportion ($\operatorname{AFDP}$) and true positive proportion (ATPP) of these TVS are evaluated. We prove that for a fixed ATPP, in order to obtain a smaller $\operatorname{AFDP}$, one should pick a bridge estimator with smaller asymptotic mean square error in the first stage of TVS. Based on such principled discovery, we present a sharp comparison of different TVS, via an in-depth investigation of the estimation properties of bridge estimators. Rather than “orderwise” error bounds with loose constants, our analysis focuses on precise error characterization. Various interesting signal-to-noise ratio and sparsity settings are studied. Our results offer new and thorough insights into high-dimensional variable selection. For instance, we prove that a TVS with Ridge in its first stage outperforms TVS with other bridge estimators in large noise settings; two-stage LASSO becomes inferior when the signal is rare and weak. As a by-product, we show that two-stage methods outperform some standard variable selection techniques, such as $\operatorname{LASSO}$ and Sure Independence Screening, under certain conditions.

Phytoplankton-zooplankton interaction under environmental stochasticity: Survival, extinction and stability

Deterministic mathematical models are frequently used to represent the predator-prey interaction for a deeper understanding of population dynamics. This approach, however, always has some limitations as it does not capture the environmental noise or demographic stochasticity which are important features of any natural system. In this paper, we study a minimal deterministic model of phytoplankton-zooplankton (prey and predator, respectively) interaction and compare its dynamics with its stochastic version formulated by two different stochastic perturbation techniques. In the first method, two parameters of the deterministic system are replaced by its average value plus an error term. In the second method, the stochastic perturbation is considered proportional to the distance of state variables from their deterministic equilibrium value. We analyze both the stochastic models and explore their dynamics. In particular, we determine sufficient conditions for extinction probability, stochastic persistence of populations, and the existence of stationary distribution of the first stochastic system. For the other system, we determine sufficient conditions for the asymptotic mean square stability by defining a suitable Lyapunov function. Different analytical results are illustrated numerically and interpreted biologically. The stochastic behaviours of the system are compared with its deterministic counterpart. A case study has also been done considering the 24 months data of phytoplankton-zooplankton interaction in the Lake Trasimeno, Italy.

Sparse Doubly-Selective Channel Estimation Techniques for OSTBC MIMO-OFDM Systems: A Hierarchical Bayesian Kalman Filter Based Approach

Hierarchical Bayesian Kalman filter (HBKF) based schemes are conceived for doubly-selective sparse channel estimation in orthogonal space-time block coded (OSTBC) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) wireless systems. Initially, a pilot based multiple measurement vector (MMV) model is formulated for estimating the OSTBC MIMO-OFDM channel. This is followed by the development of a low-complexity, online pilot-based HBKF (P-HBKF) scheme for tracking the sparse time-varying frequency-selective channel. The salient advantages of the proposed P-HBKF technique are that it requires significantly lower number of pilot subcarriers, while also exploiting the inherent sparsity of the wireless channel. Subsequently, data detection is also incorporated in the proposed framework, leading to the development of a procedure for joint sparse doubly-selective channel estimation and symbol detection. Recursive Bayesian Cramér-Rao bounds and closed form expressions are also obtained for the asymptotic mean square error (MSE) based on the solution of the Riccati equation for the KF for benchmarking the performance. Simulation results are presented for validating the theoretical bounds and for comparing the performance of the proposed and existing techniques.

Стiйкiсть розв’язку стохастичних динамiчних систем випадкової структури iз зовнiшнiми збуреннями, пуассоновими перемиканнями i всiєю передiсторiєю

The Ito-Skorokhod stochastic dynamical system with external random disturbances, with Markov switching and all prehistory is considered in this paper. Basic definitions of stability of a strong solution for such systems are introduced and sufficient conditions for asymptotic stability in probability in general and the asymptotic mean square stability in general are received.

Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method

This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results.

Mean Square Stability of Impulsive Stochastic Delayed Reaction-Diffusion Equations via Comparison Principle with Razumikhin Method

This paper investigates the mean square stability problem for impulsive stochastic delayed reaction-diffusion equations. By employing stochastic analysis theory, impulsive differential inequality technique and Razumikhin method, comparison principle for impulsive stochastic delayed reaction-diffusion equations is firstly established. Then, by using the comparison principle, some sufficient conditions are derived to ensure the mean square stability, mean square uniform stability, mean square asymptotic stability and mean square exponential stability of such systems. Finally, an example is provided to show the effectiveness of the proposed results.

Boundary control of stochastic reaction‐diffusion systems with Markovian switching

SummaryThis article focuses on the boundary control of stochastic Markovian reaction‐diffusion systems (SMRDSs). Both the cases of completely known and partially unknown transition probabilities are taken into account. By using the Lyapunov functional method, a sufficient condition is obtained under the designed boundary controllers to guarantee the asymptotic mean square stability for SMRDSs with completely known transition probabilities. For the case of partially unknown transition probabilities, we introduce free‐connection weighting matrices to handle the boundary control problem. When external disturbance enters the system, a sufficient criterion of H‐infinity boundary control is developed. Furthermore, robust stabilization is investigated for parametric uncertain SMRDSs in both cases. Two examples are presented to demonstrate the efficiency of the proposed approaches.

The Improved Value-at-Risk for Heteroscedastic Processes and Their Coverage Probability

A risk measure commonly used in financial risk management, namely, Value-at-Risk (VaR), is studied. In particular, we find a VaR forecast for heteroscedastic processes such that its (conditional) coverage probability is close to the nominal. To do so, we pay attention to the effect of estimator variability such as asymptotic bias and mean square error. Numerical analysis is carried out to illustrate this calculation for the Autoregressive Conditional Heteroscedastic (ARCH) model, an observable volatility type model. In comparison, we find VaR for the latent volatility model i.e., the Stochastic Volatility Autoregressive (SVAR) model. It is found that the effect of estimator variability is significant to obtain VaR forecast with better coverage. In addition, we may only be able to assess unconditional coverage probability for VaR forecast of the SVAR model. This is due to the fact that the volatility process of the model is unobservable.

Asynchronous control for two‐dimensional hidden Markovian jump systems with partly known mode observation conditional probabilities

SummaryThis study focuses on the asynchronous control problem for two‐dimensional discrete‐time hidden Markovian jump systems where the mode observation conditional probability matrix is partly known. Considering the original system modes are invisible, the observed modes emitted from an observer serve as an alternative for stability analysis and controller design where a mode observation conditional probability matrix is constructed to characterize the emission between system modes and observed modes. Specially, only partly known information of the mode observation conditional probability matrix is accessible. With the introduction of the free‐connection weighting matrices, the asymptotic mean square stability criterion is firstly derived based on Lyapunov method. This introduction provides a further degree of relaxation and less conservatism is therefore achieved. Secondly, we present synthesis conditions for asynchronous state feedback controller design given in terms of a set of interconnected linear matrix inequalities. Moreover, cluster concept based on the partitions of observed modes is adopted which helps to decrease the number of controllers and simplify the design complexity. A numerical example, regarding the cases with and without clustering of the observed modes, is presented to illustrate the effectiveness of the proposed method.

Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

A Variable Regularized Recursive Subspace Model Identification Algorithm With Extended Instrumental Variable and Variable Forgetting Factor

This paper proposes a new smoothly clipped absolute deviation (SCAD) regularized recursive subspace model identification algorithm with square root(SR) extended instrumental variable (EIV) and locally optimal variable forgetting factor (LOFF). The proposed algorithm is based on a new local polynomial modeling (LPM)-based variable FF EIV projection approximation subspace tracking (PAST) and multivariate recursive least squares algorithms for estimating respectively the subspace of the extended observability matrix, and the input matrix and feedthrough matrix. The SREIV-PAST algorithm offers improved resilience to additive noise over the PAST algorithm and it is implemented using the more stable square-root form. The asymptotic mean square error of the corresponding LPM-based model is derived and minimized at each time instant to obtain the proposed LOFF for improving the convergence speed and steady state error. A recursive bi-iteration singular value decomposition (SVD) algorithm is also proposed for recursive computation of the pseudo-inverse of the state transition matrix and its eigenvalues. This facilitates online estimation of its model order using classical model selection criteria. Moreover, a new criterion based on the percentage of explained variance is also proposed with improved performance. The SCAD regularization is proposed for automatic model selection of the input and feedthrough matrices, as it is asymptotically unbiased. Efficient techniques for incorporating SCAD and estimating other required quantities are also developed. The proposed algorithms are evaluated using computer simulations under stationary and nonstationary environments and a real dataset on wing flutter data. Results show that the proposed algorithms offer improved convergence speed and steady state performance over the conventional algorithms and it also provides an online estimate of the system model order.

Computational Implementation and Asymptotic Statistical Performance Analysis of Range Frequency Autocorrelation Function for Radar High-Speed Target Detection

The range frequency autocorrelation function (RFAF) based algorithm is proposed for radar target detection and motion parameter estimation in our previous work. In the RFAF-based method, the symmetric autocorrelation function (SAF) is constructed with respect to the range frequency, and three dimensional (slow time, range frequency, and shift frequency) energy accumulation can be completed coherently. In this article, as a further study of the RFAF-based method, the fast implementation and detailed asymptotic statistical performance analyses of RFAF are studied. First, on the basis of the frequency circular convolution theorem, the fast implementation is proposed. Then, to evaluate the anti-noise performance and estimation accuracy, the output signal to noise ratio (SNR) and asymptotic mean square errors (AMSEs) of estimated parameters are derived in closed forms. Theoretical analyses and numerical simulation results reveal that the RFAF-based method outperforms several state-of-the-art algorithms in terms of anti-noise performance and parameter estimation accuracy, and its computational efficiency is high.

$\mathcal H_{2}$ Performance Analysis and Applications of 2-D Hidden Bernoulli Jump System

The asymptotic mean square stability and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance of the 2-D hidden Bernoulli jump system is analyzed in this paper. This system, essentially, has a bi-jumping property, in other words, its jumps depend on two jumping parameters, one is the Bernoulli process, the other is dependent on the Bernoulli process via a conditional probability matrix, they together determine which subsystem is active and are coined as hidden Bernoulli model. By means of Lyapunov function method, a sufficient condition is derived, which reveals that the system is asymptotically mean square stable and has a certain H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance if a set of matrix inequalities are satisfied. Furthermore, the issues of asynchronous control and filtering are addressed for 2-D Bernoulli jump system based on the hidden Bernoulli model and the derived result. Finally, numerical simulations indicate that these proposed theories and methods are reliable and efficient.

Dynamical thermalization in time-dependent billiards.

Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.