Analysis of convergence for initial condition estimation of coupled map lattices based on symbolic dynamics

A novel approach to the inverse problem of diffusively coupled map lattices is systematically investigated by utilizing the symbolic vector dynamics. The relationship between the performance of initial condition estimation and the structural feature of dynamical system is proved theoretically. It is found that any point in a spatiotemporal coupled system is not necessary to converge to its initial value with respect to sufficient backward iteration, which is directly relevant to the coupling strength and local mapping function. When the convergence is met, the error bound in estimating the initial condition is proposed in a noiseless environment, which is determined by the dimension of attractors and metric entropy of the system. Simulation results further confirm the theoretic analysis, and prove that the presented method provides the important theory and experimental results for better analysing and characterizing the spatiotemporal complex behaviours in an actual system.

Estimation and Prediction of the Condition of the Vehicle Engine Based on the Correlation Dimension

This paper applies the fractal dimension as a characteristic to describe the engine抯 operating condition and its developmental trend. A correlation dimension is one of the quantities that are usually used to characterize a strange attractor. With the operation of the phase space reconstruction, respective correlation dimensions of a series of vibration signals obtained under different conditions are calculated to find the intrinsic relationship between the indicator and the operating condition. The experiment result shows that the correlation dimension is sensitive to the condition evolution and convenient for the identification of abnormal operational states. In advanced prognostic algorithm based on the BP neural network is then applied on the correlation dimensions to predict the short-term running conditions in order to avoid severe faults and realize in-time maintenance. Experimental results are presented to illustrate the proposed methodology.

A method of estimating initial conditions of coupled map lattices based on time-varying symbolic dynamics

A novel computationally efficient algorithm in terms of the time-varying symbolic dynamic method is proposed to estimate the unknown initial conditions of coupled map lattices （CMLs）. The presented method combines symbolic dynamics with time-varying control parameters to develop a time-varying scheme for estimating the initial condition of multi-dimensional spatiotemporal chaotic signals. The performances of the presented time-varying estimator in both noiseless and noisy environments are analysed and compared with the common time-invariant estimator. Simulations are carried out and the obtained results show that the proposed method provides an efficient estimation of the initial condition of each lattice in the coupled system. The algorithm cannot yield an asymptotically unbiased estimation due to the effect of the coupling term, but the estimation with the time-varying algorithm is closer to the Cramer-Rao lower bound （CRLB） than that with the time-invariant estimation method, especially at high signal-to-noise ratios （SNRs）.

Parameter Estimation for the NEAR(p) Model

As to the acronym NEAR（p）, it means “New Exponential Autoregressive Process of order p”. The NEAR（p） model is defined by

A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics

Based on symbolic dynamics, a novel computationally efficient algorithm is proposed to estimate the unknown initial vectors of globally coupled map lattices （CMLs）. It is proved that not all inverse chaotic mapping functions are satisfied for contraction mapping. It is found that the values in phase space do not always converge on their initial values with respect to sufficient backward iteration of the symbolic vectors in terms of global convergence or divergence （CD）. Both CD property and the coupling strength are directly related to the mapping function of the existing CML. Furthermore, the CD properties of Logistic, Bernoulli, and Tent chaotic mapping functions are investigated and compared. Various simulation results and the performances of the initial vector estimation with different signal-to- noise ratios （SNRs） are also provided to confirm the proposed algorithm. Finally, based on the spatiotemporal chaotic characteristics of the CML, the conditions of estimating the initial vectors usiug symbolic dynamics are discussed. The presented method provides both theoretical and experimental results for better understanding and characterizing the behaviours of spatiotemporal chaotic systems.

ON THE SENSITIVITY OF THE LDU FACTORIZATION

In this paper,we give the sensitivity analyses by two approaches for L,D,U in factorization A=LDU of general perturbations in A which sufficiently small in norm. By the matrix vector equation approach,we derive the sharp condition number for L,D and U factors .By the matrix equation approach we derive corresponding condition estimates. When A is a symmetric matrix,the corresponding results can be obtained for LDL T factorization.

Asymptotic Properties of Estimators for Ornstein-Uhlenbeck Processes with Small Symmetricα-Stable Motions

The asymptotic behaviors for estimators of the drift parameters in the Ornstein-Uhlenbeck process driven by small symmetricα-stable motion are studied in this paper.Based on the discrete observations,the conditional least squares estimators(CLSEs)of all the parameters involved in the Ornstein–Uhlenbeck process are proposed.We establish the consistency and the asymptotic distributions of our estimators asεgoes to 0 and n goes to∞simultaneously.

A Constrained Interval-Valued Linear Regression Model:A New Heteroscedasticity Estimation Method

Linear regression models for interval-valued data have been widely studied.Most literatures are to split an interval into two real numbers,i.e.,the left-and right-endpoints or the center and radius of this interval,and fit two separate real-valued or two dimension linear regression models.This paper is focused on the bias-corrected and heteroscedasticity-adjusted modeling by imposing order constraint to the endpoints of the response interval and weighted linear least squares with estimated covariance matrix,based on a generalized linear model for interval-valued data.A three step estimation method is proposed.Theoretical conclusions and numerical evaluations show that the proposed estimator has higher efficiency than previous estimators.

On the Asymptotic Behavior of Nearest Neighbor Estimator of Conditional Median

This paper deals with the estimation in nonparametrio regression model.Sincethe conditional mean is sensitive to the tail behavior of the conditional distributionof the model,instead conditional median is considered.For estimation of theconditional median,the sequence of the nearest neighbor estimators is shown to beasymptotio normal and consistent.

EXPLICIT ERROR ESTIMATES FOR MIXED AND NONCONFORMING FINITE ELEMENTS

In this paper, we study the explicit expressions of the constants in the error estimates of the lowest order mixed and nonconforming finite element methods. We start with an explicit relation between the error constant of the lowest order Raviart-Thomas interpolation error and the geometric characters of the triangle. This gives an explicit error constant of the lowest order mixed finite element method. Furthermore, similar results can be ex- tended to the nonconforming P1 scheme based on its close connection with the lowest order Raviart-Thomas method. Meanwhile, such explicit a priori error estimates can be used as computable error bounds, which are also consistent with the maximal angle condition for the optimal error estimates of mixed and nonconforming finite element methods.