Nonparametric estimation of the spectral density of amplitude-modulated time series with missing observations

Consider a real-valued and second-order stationary time series with mean zero. The aim is to estimate its spectral density. A minimax solution of this problem is known when either the time series is observed directly, or some observations are missed according to an independent Bernoulli process, or for some special cases when the time series is multiplied by an amplitude-modulating time series with known distribution. It is shown that if a time series of interest, a Bernoulli time series defining missing mechanism, and an amplitude-modulating time series are mutually independent, then the shape of spectral density of an underlying time series of interest can be estimated with the minimax rate known for the case of direct observations. Furthermore, in some special cases the spectral density can be estimated with the minimax rate known for directly observed time series of interest.     

Stability and boundedness of solutions of Stieltjes Differential Equations

Dynamical equations on time scales are formulated by means of Stieltjes differential equations, which, depending on the time integrator, include ordinary differential equations and difference equations as well as mixtures of both. Explicit conditions for the boundedness and stability of solutions are presented here for linear and nonlinear Stieltjes differential equations. In addition, the continuous dependence of solutions on the time integrator is established by means of a Gronwall-like inequality for equations with different time integrators.     

Adaptive weak approximation of stochastic differential equations

Adaptive time‐stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading‐order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps. © 2001 John Wiley & Sons, Inc.     

A Stochastic Analysis of Power Doubling Time for a Subcritical System

The power doubling time for a subcritical system is identified as a stochastic first-passage time problem. Using stochastic point kinetics equations, relations for the mean doubling time and the standard deviation in doubling time are derived. It is shown that the power doubling time for a subcritical system is independent of whether the system is fast or thermal, weakly depends on source strength, and is approximately proportional to the inverse of the reactivity for small negative values of the reactivity.     

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.     

Distribution of the time of first arrival of a Wiener process at a given boundary

An exact expression is derived for the characteristic function of the distribution of the first arrival time of a Wiener process at a given boundary, which is fixed in time. An approximate expression is obtained for the same characteristic function assuming that the given boundary is a function of time. The properties of the distribution of the first arrival time are analyzed.Translated from Statisticheskie Metody, pp. 31–42, 1980.     

分数阶微分中立型系统的有限时间镇定性及有界性

讨论了一类分数阶微分中立型系统的有限时间状态反馈镇定性及有界性问题.利用Gronwall 不等式, 获得了使得系统有限时间状态反馈镇定及有限时间有界的充分性条件.     

Novel method of identifying time series based on network graphs

In this article, we propose a novel method for transforming a time series into a complex network graph. The proposed algorithm is based on the spatial distribution of a time series. The characteristics of geometric parameters of a network represent the dynamic characteristics of a time series. Our algorithm transforms, respectively, a constant series into a fully connected graph, periodic time series into a regular graph, linear divergent time series into a tree, and chaotic time series into an approximately power law distribution network graph. We find that when the dimension of reconstructed phase space increases, the corresponding graph for a random time series quickly turns into a completely unconnected graph, while that for a chaotic time series maintains a certain level of connectivity. The characteristics of the generated network, including the total edges, the degree distribution, and the clustering coefficient, reflect the characteristics of the time series, including diverging speed, level of certainty, and level of randomness. This observation allows a chaotic time series to be easily identified from a random time series. The method may be useful for analysis of complex nonlinear systems such as chaos and random systems, by perceiving the differences in the outcomes of the systems—the time series—in the identification of the systemic levels of certainty or randomness. © 2011 Wiley Periodicals, Inc. Complexity, 2011     

复杂系统中混沌排斥子的动力学特性分析及应用研究

研究了由一类复杂系统排斥子所生成的时间序列的分形特征、分维值,利用相空间重构理论对排斥子所生成的混沌时序数据进行了重构．研究了时序数据的零均值处理、傅立叶滤波对预测结果的影响,研究了预测样本值的选取对预测的相对误差、预测长度影响等相关问题．结果表明:该模型对于这类排斥子所生成的时序数据建模和预测都具有实用性,且混沌排斥子样本数据的零均值处理对预测结果有一定的量的改变,但对排斥子样本数据进行Fourier滤波处理会降低预测的精度,这对于复杂系统排斥子的研究有着较为重要的理论和实际意义．     

Instability in a delay-PDE model of earthquake occurrence

Summary Ductile deformation prior to brittle fracture in rocks causes fracture to take place with a time delay after the critical stress for fracture is reached, in the presence of an increasing load stress. We discuss the stability of a stochastic model of interactive earthquake occurrence under the influence of time delays resulting from the ductile process. A threshold for oscillatory behavior is found for large enough coupling and discrete time delays. The system is stable if the time delays are spread over a broad time interval, even for large coupling.     

Accuracy of multivalue methods during change of time‐step size: Adams‐Moulton

Multivalue methods are a class of time‐stepping procedures for numerical solution of differential equations that progress to a new time level using the approximate solution for the function of interest and its derivatives at a single time level. The methods differ from multistep procedures, which make use of solutions to the differential equation at multiple time levels to advance to the new time level. Multistep methods are difficult to employ when a change in time‐step is desired, because the standard formulas (e.g., Adams‐Moulton or Gear) must be modified to accommodate the change. Multivalue methods seem to possess the desirable feature that the time‐step may be changed arbitrarily as one proceeds, since the solution proceeds from a single time level. However, in practice, changes in the time‐step introduce lower order errors or alter the coefficient in the truncation error term. Here, the multivalue Adams‐Moulton method is presented based on a general interpolation procedure. Modifications required to retain the high‐order accuracy of these methods during a change in time‐step are developed. Additionally, a formula for the unknown initial derivatives is presented. Finally, two examples are provided to illustrate the potential merit of the modification to the standard multivalue methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partials Differential Eq 16: 312–326, 2000     

Finite‐time synchronization of memristive neural networks with time‐varying delays via two control methods

In this paper, on the basis of the Lyapunov stability theory and finite‐time stability lemma, the finite‐time synchronization problem for memristive neural networks with time‐varying delays is studied by two control methods. First, the discontinuous state‐feedback control rule containing integral part for square sum of the synchronization error and the discontinuous adaptive control rule are designed for realizing synchronization of drive‐response memristive neural networks in finite time, respectively. Then, by using some important inequalities and defining suitable Lyapunov functions, some algebraic sufficient criteria guaranteeing finite‐time synchronization are deduced for drive‐response memristive neural networks in finite time. Furthermore, we give the estimation of the upper bounds of the settling time of finite‐time synchronization. Lastly, the effectiveness of the obtained sufficient criteria guaranteeing finite‐time synchronization is validated by simulation.     

An eigenvalue approach to the limiting behavior of time series aggregates

Many time series variables such as rainfall, industrial production, and sales exist only in some aggregated forms. To see the implication of time series aggregation it is important to know the limiting behavior of the time series aggregates. From the relationship of autocovariances between the underlying time series variable and its aggregates, we show that the limiting behavior of time series aggregates is closely related to the eigenvalues and the eigenvectors of the aggregation operator. Specifically, the vector of admissible autocorrelations of the limiting model for the time series aggregates is the eigenvector associated with the largest eigenvalue of the aggregation transformation. This provides an interesting and simple method for deriving the limiting model for time series aggregates. Systematic sampling of time series can be treated similarly. The method is illustrated with an empirical example.     

Almost sure convergence of continuous time branching processes

We prove necessary and sufficient conditions for the transience of the non-zero states in a non-homogeneous, continuous time Markov branching process. The result is obtained by passing from results about the discrete time skeleton of the continuous time chain to the continuous time chain itself. An alternative proof of a result for continuous time Markov branching processes in random environments is then given, showing that earlier moment conditions were not necessary.     

Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case*

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.     

一类非线性抛物系统解的爆破现象

研究了具有依赖于时间的系数的非线性抛物方程解的爆破现象.对已知数据项进行一定的假设并设置一些辅助函数,应用微分不等式技术,得到了方程的解发生爆破的条件.当爆破发生时,分别推导了方程在二维区域和三维区域上解的爆破时间的下界.     

医院统计需求预测方法研究

本文从医院管理实际出发，研究和探索了时间顺序预测，因果预测和预报性预测这三种分别适用于短期，中期和长期的科学预测方法。它在我国医院管理中具有实际的指导意义。     

Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics

The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.     

无限批量调度中最小化加权完工时间和问题的一个线性时间近似方案

本文考虑n个工件的无限批量机器调度问题．一台机器可以同时加工B≥n个工件．每个工件具有一个正权因子、一个释放时间和一个加工时间．一个批次的加工时间是该批次所包含所有工件的加工时间的最大者．在同一批次中加工的工件有相同的完工时间，即它们的共同开始时间加上该批次的加工时间．对于最小化加权完工时间和问题，本文给出了第一个多项式时间近似方案(PTAS)．对任意给定精度，该算法的运行时间为线性的．     

Commutativity of first‐order discrete‐time linear time‐varying systems

After introducing the concept of commutativity for continuous‐time linear time‐varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete‐time linear time‐varying systems, the problem is formulated for first‐order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first‐order discrete‐time linear time‐varying systems are derived, and their advantageous use in digital system design is illustrated, which are the main objectives of the paper. The results are verified by examples which include an application in amplitude modulation for digital telecommunication.