Efficient estimation and particle filter for max‐stable processes

Extreme values are often correlated over time, for example, in a financial time series, and these values carry various risks. Max‐stable processes such as maxima of moving maxima (M3) processes have been recently considered in the literature to describe time‐dependent dynamics, which have been difficult to estimate. This article first proposes a feasible and efficient Bayesian estimation method for nonlinear and non‐Gaussian state space models based on these processes and describes a Markov chain Monte Carlo algorithm where the sampling efficiency is improved by the normal mixture sampler. Furthermore, a unique particle filter that adapts to extreme observations is proposed and shown to be highly accurate in comparison with other well‐known filters. Our proposed algorithms were applied to daily minima of high‐frequency stock return data, and a model comparison was conducted using marginal likelihoods to investigate the time‐dependent dynamics in extreme stock returns for financial risk management.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

Nickel‐Catalyzed Suzuki–Miyaura Coupling of a Tertiary Iodocyclopropane with Wide Boronic Acid Substrate Scope: Coupling Reaction Outcome Depends on Radical Species Stability

We describe a nickel‐catalyzed Suzuki–Miyaura arylation of a tertiary iodocyclopropane with arylboronic acids; this is an efficient and convergent strategy for providing various enantioenriched arylcyclopropanes with a quaternary stereogenic center. This is the first metal‐catalyzed coupling between a tertiary alkyl electrophile and a wide range of aromatics, including heteroaromatics. We found that the outcome of the Ni‐catalyzed coupling with halides as electrophiles was dependent on the stability of the radical species formed during the reaction. The use of tert‐butyl alcohol (t‐BuOH) as the reaction solvent was very effective, because of its stability under the radical‐generating reaction conditions.

     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.     

On backward periodic autoregressive processes

It has been shown that there is a one-to-one correspondence between a multivariate autoregress ive (AR) process and a scalar periodic AR process. So we can analyze periodic processes by the theory of multivariate AR processes and vice versa.
Multivariate AR processes have been widely studied and the Levinson–Whittle–Wiggins–Robinson algorithm is well known for obtaining the predictor coeffic ient matrices. In addition, the circular Levinson algorithm has been derived for obtaining the coefficients of periodic AR processes.
In this paper, we construct backward periodic AR processes from the auxiliary coefficients used in this algorithm. A numerical example is also presented and the statistical properties of the estimated coefficients of backward periodic AR processes based on a sample of finite size are derived.     

On residual empirical processes of GARCH‐SM models: application to conditional symmetry tests

Abstract. Considering the generalized autoregressive conditionally heteroskedastic with stochastic mean (GARCH‐SM) model, we establish in this article the consistency and the weak representation of a functional of its residual empirical process. Based on this result, a symmetry test for GARCH‐SM model is developed. Simulations are given to show the asymptotic behaviour and normality of the test statistic.     

MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES

We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale.     

Time Series Extended Finite‐State Machine‐Based Relevance Vector Machine Multi‐Fault Prediction

Fault prediction means to detect faults that can occur in the future. While most studies focus on predicting one fault at a time, multi‐fault prediction is more practical for industrial processes as multiple faults can cause much more damage than a single one. A time series extended finite‐state machine (TS‐EFSM)‐based relevance vector machine (RVM) approach is proposed for multi‐fault prediction. Time lags and correlation coefficients between the process variables and process states are determined. Then, a variable and a state dependence diagram based on the correlation coefficients is established with the EFSM. Furthermore, the RVM is applied to identify parameters for the sake of better prediction accuracy and shorter testing times. With the prediction parameters, faults can be predicted using the aforementioned TS‐EFSM state transitions.     

Contemporaneous aggregation of GARCH processes

Abstract. In this article, the effect of contemporaneous aggregation of heterogeneous generalized autoregressive conditionally heteroskedastic (GARCH) processes, as the cross‐sectional size diverges to infinity is studied. We analyse both cases of cross‐sectionally dependent and independent individual processes. The limit aggregate does not belong to the class of GARCH processes. Dynamic conditional heteroskedasticity is only preserved when the individual processes are sufficiently cross‐correlated, although long memory for the limit aggregate volatility is not attainable. We also explore more general forms of cross‐sectional dependence and various types of aggregation schemes.     

A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

We approach the problem of non‐parametric estimation for autoregressive Markov switching processes. In this context, the Nadaraya–Watson‐type regression functions estimator is interpreted as a solution of a local weighted least‐square problem, which does not admit a closed‐form solution in the case of hidden Markov switching. We introduce a non‐parametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte Carlo step and estimates the regression function via a Robbins–Monro step. We prove that non‐parametric autoregressive models with Markov switching are identifiable when the hidden Markov process has a finite state space. Consistency of the estimator is proved using the strong α‐mixing property of the model. Finally, we present some simulations illustrating the performances of our non‐parametric estimation procedure.     

Diffusion Studies of Glucose and Sucrose in Chitosan Membranes and Beads for Enzymatic Production Processes

The diffusion of glucose and sucrose was investigated in membrane and bead experiments. Concentration‐dependent diffusion coefficients of pure glucose and sucrose were determined in precipitated chitosan membranes of varying thickness using diffusion cell experiments. Contrary to fructose, the resulting diffusion coefficients of glucose and sucrose did not reach their free diffusion coefficients at infinite dilution suggesting additional interactions between chitosan and these two sugars. Counter‐diffusion in bead experiments showed a good agreement between predicted and measured data allowing the inclusion of the diffusion data in the simulation of the proposed production process for laminaribiose. In conclusion, the encapsulation in chitosan presented a good trade‐off between increased mass transfer resistance as evaluated by the Thiele modulus and improved thermal stability and antibacterial activity.     

A light‐tailed conditionally heteroscedastic model with applications to river flows

Abstract. A conditionally heteroscedastic model, different from the more commonly used autoregressive moving average–generalized autoregressive conditionally heteroscedastic (ARMA‐GARCH) processes, is established and analysed here. The time‐dependent variance of innovations passing through an ARMA filter is conditioned on the lagged values of the generated process, rather than on the lagged innovations, and is defined to be asymptotically proportional to those past values. Designed this way, the model incorporates certain feedback from the modelled process, the innovation is no longer of GARCH type, and all moments of the modelled process are finite provided the same is true for the generating noise. The article gives the condition of stationarity, and proves consistency and asymptotic normality of the Gaussian quasi‐maximum likelihood estimator of the variance parameters, even though the estimated parameters of the linear filter contain an error. An analysis of six diurnal water discharge series observed along Rivers Danube and Tisza in Hungary demonstrates the usefulness of such a model. The effect of lagged river discharge turns out to be highly significant on the variance of innovations, and nonparametric estimation approves its approximate linearity. Simulations from the new model preserve well the probability distribution, the high quantiles, the tail behaviour and the high‐level clustering of the original series, further justifying model choice.     

Tests for the Equality of Two Processes' Spectral Densities with Unequal Lengths Using Wavelet Methods

Testing procedures for assessing whether two stationary and independent linear processes with unequal lengths have the same spectral densities or same auto‐covariance functions are investigated. New test statistics are proposed based on the difference of the two wavelet‐based estimates of the two spectral densities. The asymptotic normal distributions of the empirical wavelet coefficients are derived based on Bartlett type approximation of a quadratic form with dependent variables by the corresponding quadratic form with independent and identically distributed (i.i.d.) random variables. The limit distributions of the proposed test statistics are derived from those asymptotic results, and they asymptotically follow known chi‐square distributions. The advantage of those new procedures is that those test statistics are constructed very simply and can be used for two time series with arbitrary lengths. The performance of those new tests is compared with some recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed tests are very comparable to the current tests.     

Some Notes on Mutual Information Between Past and Future

Abstract. We present some new results on the mutual information between past and future for Gaussian stationary sequences. We provide several formulae to calculate this quantity. As a by‐product, we establish the so‐called reflectrum identity that links partial autocorrelation coefficients and cepstrum coefficients. So as to obtain these results, we provide an account of several regularity conditions for Gaussian stationary processes in terms of properties of the associated Toeplitz and Hankel operators. We discuss conditions under which the mutual information is finite. These results lead us to an interesting perspective towards the definition of long‐memory processes. Our result implies that zeros on the unit circle can cause mutual information to be infinite. Examples include fractional autoregressive integrated moving average (ARIMA) models. In addition, we consider a finite sample from a Gaussian stationary sequence. In the expansion of the determinant of its covariance matrix, the Toeplitz matrix, the first and second term are, entropy and mutual information respectively. A form of approximation to the likelihood using entropy and mutual information is presented.     

Long‐Term Stability,Regeneration and Recycling of Imidazolium‐based Ionic Liquids

Ionic liquids (ILs) are discussed in many current research papers extensively in terms of their potential use in the chemical industry, as process aids and novel materials. The long‐term stability of the IL is for industrial applications as important as to know which species arise during the degradation due to thermal, mechanical, chemical or electrochemical stress. The investigation of the long‐term stability of two selected ILs over several months under process‐like conditions is presented with a subsequent analysis by LC‐MS to identify the resulting decomposition products. Knowledge about the occurring species and their analytical quantification are basis for the selection of appropriate processes for the separation of the decomposition products and the development of recycling processes for ILs. Particularly melt crystallization processes are suitable for separating structurally similar decomposition products that typically occur in the IL degradation.     

Stability of nonlinear AR‐GARCH models

Abstract. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.     

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