On Bartlett's Formula for Non-linear Processes

Bartlett's formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth-order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L ² and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non-negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study     

Estimation of GARCH Models from the Autocorrelations of the Squares of a Process

This paper shows how the parameters of a stable GARCH(1, 1) model can be estimated from the autocorrelations of the squared process. Specifically, the method applies a minimum distance estimator (MDE) to the sample autocorrelations of the squared realization. The asymptotic efficiency of the estimator is calculated from using the first g autocorrelations. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases, can be more efficient than the quasi maximum likelihood estimator (QMLE). Also, the estimated process can better fit the pattern of observed autocorrelations of squared returns than those from models estimated by maximum likelihood estimation (MLE). The estimator is applied to a series of hourly exchange rate returns, which are extremely non Gaussian.     

一类化工过程多变量系统的自适应非线性预测控制

针对化工过程的一类多变量非线性系统,提出了一种自适应非线性预测控制(ANMPC)算法。在采用递归最小二乘法进行预测模型参数在线辨识的基础上,将系统的静态非线性关系用一个反向传播(BP)神经网络稳态模型来表示,通过稳态模型求得的动态增益来进一步校正预测模型的参数。详述了ANMPC控制器设计步骤,通过在一个多变量pH中和过程中的仿真验证了本算法的可行性和有效性。     

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ?_t = σ_tη_t and σ_t = f (σ_t?1, σ_t?2,…, σ_t?p, ?_t?1, ?_t?2,…, ?_t?q), when {?_t} is a process with just barely infinite variance, that is, {?_t} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.     

The Limiting Distribution of a Non-Stationary Integer Valued GARCH(1,1) Process

We consider the integer valued GARCH(1,1) process defined by the two equation system and λ_{n + 1} = ω + αY_n + βλ_n. When α + β < 1 this process has a stationary solution and properties are well understood. In this note we find the limiting distribution of λ_n and Y_n for the case of α + β = 1.     

Nonlinear adaptive switching control for a class of non-affine nonlinear systems

An improved nonlinear adaptive switching control method is presented to relax the assumption on the higher order nonlinear terms of a class of discrete-time non-affine nonlinear systems. The proposed control strategy is composed of a linear adaptive controller, a neural network (NN) based nonlinear adaptive controller and a switching mechanism. An incremental model is derived to represent the considered system and an improved robust adaptive law is chosen to update the parameters of the linear adaptive controller. A new performance criterion of the switching mechanism is designed to select the proper controller. Using this control scheme, all the signals in the system are proved to be bounded. Numerical examples verify the effectiveness of the proposed algorithm.     

Large sample properties of spectral estimators for a class of stationary nonlinear processes

Abstract. We consider the standard spectral estimators based on a sample from a class of strictly stationary nonlinear processes which include, in particular, the bilinear and Volterra processes. It is shown that these estimators, under certain mild regularity conditions are both consistent and asymptotically normal.     

Measuring nonlinear dependence in time‐series,a distance correlation approach

We extend the concept of distance correlation of Szekely et al. (2007) and propose the auto distance correlation function (ADCF) to measure the temporal dependence structure of time‐series. Unlike the classic measures of correlations such as the autocorrelation function, the proposed measure is zero if and only if the measured time‐series components are independent. In this article, we propose and theoretically verify a subsampling methodology for the inference of sample ADCF for dependent data. Our methodology provides a useful tool for exploring nonlinear dependence structures in time‐series.     

A cyber-secure control-detector architecture for nonlinear processes

This work presents a detector-integrated two-tier control architecture capable of identifying the presence of various types of cyber-attacks, and ensuring closed-loop system stability upon detection of the cyber-attacks. Working with a general class of nonlinear systems, an upper-tier Lyapunov-based Model Predictive Controller (LMPC), using networked sensor measurements to improve closed-loop performance, is coupled with lower-tier cyber-secure explicit feedback controllers to drive a nonlinear multivariable process to its steady state. Although the networked sensor measurements may be vulnerable to cyber-attacks, the two-tier control architecture ensures that the process will stay immune to destabilizing malicious cyber-attacks. Data-based attack detectors are developed using sensor measurements via machine-learning methods, namely artificial neural networks (ANN), under nominal and noisy operating conditions, and applied online to a simulated reactor-reactor-separator process. Simulation results demonstrate the effectiveness of these detection algorithms in detecting and distinguishing between multiple classes of intelligent cyber-attacks. Upon successful detection of cyber-attacks, the two-tier control architecture allows convenient reconfiguration of the control system to stabilize the process to its operating steady state.     

Optimal periodic forcing of nonlinear chemical processes for performance improvements

Unsteady-state periodic operations can improve the optimal steady-state performance of nonlinear chemical processes. To examine if the optimal periodic operation is proper and to obtain the optimal forcing functions subject to various control and state constraints it is suggested in this paper to convert the problems into a form which is suitable for constrained nonlinear programming. The adopted numerical optimization method is based on employing the control parametrization technique and is thus capable of dealing with the problem of multiple input forcings and obtaining optimal forcing functions and/or parameters while subject to general constraints. Besides, it provides information about to what extent the process performance can be improved by adopting the optimal periodic control.     

Distributed economic model predictive control for operational safety of nonlinear processes

Achieving operational safety of chemical processes while operating them in an economically‐optimal manner is a matter of great importance. Our recent work integrated process safety with process control by incorporating safety‐based constraints within model predictive control (MPC) design; however, the safety‐based MPC was developed with a centralized architecture, with the result that computation time limitations within a sampling period may reduce the effectiveness of such a controller design for promoting process safety. To address this potential practical limitation of the safety‐based control design, in this work, we propose the integration of a distributed model predictive control architecture with Lyapunov‐based economic model predictive control (LEMPC) formulated with safety‐based constraints. We consider both iterative and sequential distributed control architectures, and the partitioning of inputs between the various optimization problems in the distributed structure based on their impact on process operational safety. Moreover, sufficient conditions that ensure feasibility and closed‐loop stability of the iterative and sequential safety distributed LEMPC designs are given. A comparison between the proposed safety distributed EMPC controllers and the safety centralized EMPC is demonstrated via a chemical process example. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3404–3418, 2017     

Partial autocorrelation parameterization for subset autoregression

Abstract. A new version of the partial autocorrelation plot and a new family of subset autoregressive models are introduced. A comprehensive approach to model identification, estimation and diagnostic checking is developed for these models. These models are better suited to efficient model building of high‐order autoregressions with long time series. Several illustrative examples are given.     

Machine learning-based distributed model predictive control of nonlinear processes

This work explores the design of distributed model predictive control (DMPC) systems for nonlinear processes using machine learning models to predict nonlinear dynamic behavior. Specifically, sequential and iterative DMPC systems are designed and analyzed with respect to closed-loop stability and performance properties. Extensive open-loop data within a desired operating region are used to develop long short-term memory (LSTM) recurrent neural network models with a sufficiently small modeling error from the actual nonlinear process model. Subsequently, these LSTM models are utilized in Lyapunov-based DMPC to achieve efficient real-time computation time while ensuring closed-loop state boundedness and convergence to the origin. Using a nonlinear chemical process network example, the simulation results demonstrate the improved computational efficiency when the process is operated under sequential and iterative DMPCs while the closed-loop performance is very close to the one of a centralized MPC system.     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

Efficient recursive kernel canonical variate analysis for monitoring nonlinear time‐varying processes

Kernel canonical variate analysis (KCVA) cannot be adopted for monitoring nonlinear time‐varying processes because of changes in variance, mean, and correlation between variables. Efficient recursive kernel canonical variate analysis (ERKCVA) is thus proposed to monitor the nonlinear time‐varying processes. In a high‐dimensional feature space, the covariance matrix can be updated recursively by the exponentially weighted moving average approach. The first‐order perturbation theory is introduced to obtain the recursive singular value decomposition of the Hankel matrix, which can significantly reduce the computational cost of the proposed method. Prediction errors and state variables are non‐Gaussian; thus, upper control limits can be derived from the estimated probability density function by kernel density estimation. The proposed method is demonstrated by simulating a continuous stirred tank reactor. Simulation results indicate that ERKCVA could efficiently capture the predefined normal and natural changes in nonlinear time‐varying processes. In addition, ERKCVA can also identify 4 types of sensor faults.     

Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes

We demonstrate that the fast and exact Davies–Harte algorithm is valid for simulating a certain class of stationary Gaussian processes – those with a negative autocovariance sequence for all non-zero lags. The result applies to well known classes of long memory processes: Gaussian fractionally differenced (FD) processes, fractional Gaussian noise (fGn) and the nonstationary fractional Brownian Motion (fBm).     

The Use of Aggregate Time Series in Testing for Gaussianity

Many time series encountered in practice are non-Gaussian. Because of the process of data collection or the practice or researchers, time series used in analysis and modelling are frequently temporal aggregates. In this paper, we study the effects of the use of aggregate time series on testing for Gaussianity. We analyse how the test statistic is affected by aggregation and how that affects the power of the test. The results show that the use of aggregate time series induces Gaussianity and that the degree of inducement increases with the order of aggregation. In fact, the use of aggregate time series reduces the power of the test, although the effect is not significant for low orders of aggregation.     

On the spectral density of a class of chaotic time series

The purpose of this paper is to show explicitly the spectral density function of the stationary stochastic process determined by a certain class of two-dimensional maps F_α defined below (α is a parameter in (0, 1)), the random variable φ(x, y) = x and the invariant probability described below. We first define the transformation T_α: [0, 1]←[0, 1] given by T _α(x) = {x/α if 0 ≤x < α and (α(x?α)/1 ?α) if α≤x≤ 1 where α∈ (0, 1) is a constant. The map T_α describes a model for a particle (or the probability of a certain kind of element in a given population) that moves around, in discrete time, in the interval [0, 1]. The results presented here can be stated either for T_α or for F_α but we prefer the latter. The results for T_α can be obtained from the more general setting described by F_α. The map F_α is defined from K = ([0, 1]× (0, α)) ∨ ([0, α]×[α, 1]) ?;R;² to itself and is given by F_α(x, y) = (T_α(x), G_α(x, y)) for (x, y) ∈K, where G _α(x, y) = {αy if 0 ≤x < α and α + ((1 ?α)/α)y if α≤x < 1. The spectral density function of the stationary process with probability ν (invariant for F_α and absolutely continuous with respect to the Lebesgue measure) Z_t = X_t + ξ_t = φ{F^t_α(X₀, Y₀)} + ξ_t for t∈Z where (X₀, Y₀) ∈R² and ξ_t}_t∈Z is a white noise process, is given explicitly (Theorem 1) by f _Z (λ) = f_X(λ) + (σ²_ξ/2π) = (1/2πvar(X_t))[γ{exp(iλ)}?C(0)] + (σ²/2π) for all λ∈[0, 2π), where var(X_t) = (α²?α + 1)(α²? 5α + 5){12(2 ?α)²}^?1, γ is given by Equation (2.10) of Proposition 5 and C(0) = (1 + α²?α³){3(2 ?α)}^?1. We also estimate the parameter α based on a time series.     

Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.