On processes with hyperbolically decaying autocorrelations

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (?∞, 0), which enjoys non‐summable partial autocorrelations and autocorrelations decaying as rapidly as ρ_n ? n^?1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).     

ASYMPTOTIC PROPERTIES OF THE SAMPLE AUTOCORRELATIONS AND PARTIAL AUTOCORRELATIONS OF A MULTIPLICATIVE ARIMA PROCESS

Abstract. We shall investigate the asymptotic behaviour of the sample autocorrelations and partial autocorrelations of a multiplicative ARIMA process and derive their limiting distributions. Some simulations are presented to illustrate the results obtained.     

ON WEIGHTED PORTMANTEAU TESTS FOR TIME‐SERIES GOODNESS‐OF‐FIT

Recent work in the literature has shown weighted variants of the classic portmanteau test for time series can be more powerful in many situations. In this article, we study the asymptotic distribution of weighted sums of the squared residual autocorrelations where both the sample size n and maximum lag of the statistic m grow large. Several weighting schemes are introduced, including a data‐adaptive statistic in which the weights are determined by a function of the sample partial autocorrelations. These statistics can provide more power than other portmanteau tests found in the literature and are much less sensitive to the choice of the maximum correlation lag. The efficacy of the proposed methods is further demonstrated through an analysis of Australian red wine sales.     

Testing for serial dependence in time series models of counts

Abstract. In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first-order and second-order autoregressive as well as first-order moving average processes.     

DIAGNOSTIC CHECKING OF PERIODIC AUTOREGRESSION MODELS WITH APPLICATION

Abstract. An overview of model building with periodic autoregression (PAR) models is given emphasizing the three stages of model development:identification, estimation and diagnostic checking. New results on the distribution of residual autocorrelations and suitable diagnostic checks are derived. The validity of these checks is demonstrated by simulation. The methodology discussed is illustrated with an application. It is pointed out that the PAR approach to model development offers some important advantages over the more general approach using periodic autoregressive moving-average models.     

INVERSE AUTOCOVARIANCES AND A MEASURE OF LINEAR DETERMINISM FOR A STATIONARY PROCESS

Abstract. The inverse autocorrelations and autocovariances of a stationary stochastic process are generally used in the identification of ARMA models and linear systems, but they are also useful for studying linear relations inside the process as a whole.
Using inverse autocovariances, for any stationary process an 'inverse process' may be defined which may be considered a minimum variance linear filter, and turns out to be the best linear two-sided interpolator for one unknown value.
Basing on these results an index of linear determinism is introduced to measure to what degree a stationary process satisfies a linear deterministic constraint. The behaviour of the index for ARMA processes is finally examined.     

The Effect of the Estimation on Goodness‐of‐Fit Tests in Time Series Models

Abstract. We analyze, by simulation, the finite‐sample properties of goodness‐of‐fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving‐average time‐series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite‐sample estimation efficiencies and residual regeneration methods.     

Computation and Characterization of Autocorrelations and Partial Autocorrelations in Periodic ARMA Models

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving-average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.     

The restricted likelihood ratio test for autoregressive processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (?1, 1] × (?1, 1)^p?1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.     

Synthesis and properties of acrylic‐based superabsorbent

Acrylic‐based superabsorbent polymers with partial‐neutralized acrylic acid were synthesized by inverse suspension polymerization. The effects of reaction parameters, including the concentration of crosslinking agent and initiator, the neutralization degree of monomer, the monomer concentration, and phase ratio of a system, on the water absorption have been investigated. Furthermore, to improve the properties of salt‐resistance and antielectrolyte for the polymer, acrylamide monomer containing nonionic and hydrophilic groups was introduced into the system, which can suppress the salt effect and homoion effect. The water absorption Q^5/3 is proportional to the reciprocal of ionic strength, 1/S, which follows Flory's elasticity gel theory. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 92: 619–624, 2004     

Selection of oxygen permeation models for different mixed ionic‐electronic conducting membranes

Permeation data of several mixed ionic‐electronic conducting (MIEC) membranes were analyzed by two oxygen permeation models (i.e., Zhu's model and Xu–Thomson's model), respectively, to find a concise method to guide the choice of permeation models. We found that Zhu's model can well fit the permeation data of perovskite‐type membranes, like Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3‐δ (BSCF) and BaCe_0.05Fe_0.95O_3‐δ (BCF), and dual‐phase membranes, like 75 wt % Ce_0.85Sm_0.15O_1.925–25 wt % Sm_0.6Sr_0.4Al_0.3Fe_0.7O_3‐δ (SDC‐SSAF), whose oxygen vacancy concentrations are almost independent of the oxygen partial pressure at elevated temperatures. However, Zhu's model was not appropriate for membranes whose oxygen vacancy concentration changed obviously with oxygen partial pressure at elevated temperatures, such as La_0.6Sr_0.4Co_0.2Fe_0.8O_3‐δ (LSCF) and La_0.7Sr_0.3CoO_3‐δ (LSC). On the contrary, Xu–Thomson's model can fit the data of LSCF and LSC well, but it is inapplicable for BSCF, BCF, and SDC‐SSAF. Therefore, the dependence of oxygen vacancy concentration on oxygen partial pressure was suggested as an index for the selection of the permeation models. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4043–4053, 2017     

ON THE ASYMPTOTIC DISTRIBUTION OF THE GENERALIZED PARTIAL AUTOCORRELATION FUNCTION IN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. It has been conjectured and illustrated that the estimate of the generalized partial autocorrelation function (GPAC), which has been used for the identification of autoregressive moving-average (ARMA) models, has a thick-tailed asymptotic distribution. The purpose of this paper is to investigate the asymptotic behaviour of the GPAC in detail. It will be shown that the GPAC can be represented as a ratio of two functions, known as the θ function and the Λ function, each of which itself has a useful pattern for ARMA model identification. We shall show the consistencies of the extended Yule-Walker estimates of the three functions and present their asymptotic distributions.     

The sampling properties of conditional independence graphs for I(1) structural VAR models

Abstract. Structural vector autoregressions allow dependence among contemporaneous variables. If such models have a recursive structure, the relationships among the variables can be represented by directed acyclic graphs. The identification of these relationships for stationary series may be enabled by the examination of the conditional independence graph constructed from sample partial autocorrelations of the observed series. In this article, we extend this approach to the case when the series follows an I(1) vector autoregression. For such a model, estimated regression coefficients may have non‐standard asymptotic distributions and in small samples this affects the distribution of sample partial autocorrelations. We show that, nevertheless, in large samples, exactly the same inference procedures may be applied as in the stationary case.     

High‐temperature structural stability of ceria‐based inverse opals

The use of ceria‐based inverse opals as a catalyst system for the thermochemical production of fuels from sunlight offers the potential of improved fuel production kinetics over materials with random porosity. Quantitative methods for characterizing ordered porosity are lacking, thus limiting the ability to predict the lifetime of ordered structures at elevated temperatures. In the present work, Fourier transform image analysis was used to determine the effect of composition and temperature on ordered porosity for a series of CeO₂‐ZrO₂ inverse opals having pore sizes ranging from 300 nm to 1 μm. An order parameter, γ, derived from the image analysis, was applied to scanning electron microscopy images and used to determine the degree of order in the inverse opal. The thermal stability studies indicate that loss of ordered porosity is highly dependent on temperature and that gas cycling effects have a minor effect on periodicity. A minimum Zr content of 20 at.% is necessary to retain periodicity for annealing up to 1000°C with pore diameters larger than 1 μm. These results show that CeO₂‐ZrO₂ inverse opals can be used at higher temperatures than previously thought for efficient thermochemical hydrogen production without loss of the benefits associated with ordered porosity.     

Study nonlinear vibration of cross‐ply laminated plates using scale models

This article describes the establishment of necessary similarity conditions for geometrically nonlinear vibrations of thin cross‐ply laminated plates. The Von‐karman's strain‐displacement relations have been employed to model structural nonlinearity of the system. The Galerkin procedure is used to reduce the nonlinear partial differential equations to a nonlinearly second‐order ordinary differential equation. An analytical investigation based on the direct use of the governing equations of the systems is undertaken to derive the necessary scaling laws and similarity conditions. In this study, a set of scaling laws are introduced that can predict the nonlinear free vibration frequency of the prototypes by projecting the frequency of the model, accurately. The effects of distorted models with relaxations in the number of plies, stacking sequence, and different vibration amplitudes are studied. The results presented herein indicate that models with different relaxations can predict the nonlinear vibration frequency of prototypes with good accuracy. POLYM. COMPOS., 35:752–758, 2014. © 2013 Society of Plastics Engineers     

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.     

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Bartlett's formula for a general class of nonlinear processes

Abstract.  A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.     

Compressible and incompressible 1‐D linear wave propagation assessment in fast fluidized beds

We propose two equivalent 1‐D perturbation models for a fast fluidized bed considering compressibility effects. The first model is the explicit summation of incompressible and compressible terms. All compressible terms appear multiplied by the inverse squared gas sound propagation speed, s. In the second model, a lumped waving structure is presented, with just one term corresponding to each wave hierarchy. In both proposed models, the incompressible part was retrieved in the limit s → ∞. The Liu's linear stability analysis, based on Whitham's, was extended to estimate the compressibility contribution. Stability conditions on the propagation speeds and a criterion on the wave number were developed. This method was applied to two physical systems whose solid properties differ widely. It was shown that the effect of the fluid compressibility is at least as important as the effect of the solid compressibility modulus. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

Shifted Legendre function approximation of differential equations; application to crystallization processes

An effective new method solves either the stretched functional ordinary differential equation describing a crystallization process using breakage models, or the partial differential equations for a population balance which represent the transient crystallization process. The population balance density function is first expanded into a series of shifted Legendre polynomial functions. The partial differential equation (or the ordinary differential equation) is first transformed into a series of ordinary differential equations (or of algebraic equations) for the expansion coefficients which are solved by numerical computation. The computational time is greatly reduced through a recursive algorithm for the integration of the triple product of the shifted Legendre polynomial functions. Illustrative examples are given and the results are compared with data available in the literature. The proposed method is powerful, accurate and more precise than previously documented ones.