GENERALIZED LEAST SQUARES ESTIMATION OF ARMA MODELS

Abstract. Two multistage methods for estimating scalar ARMA models are investigated. Both estimate innovations using an autoregression; these are used to obtain initial ARMA parameter estimates by regression and finally the initial estimates are refined by generalized least squares or nonlinear optimization to achieve efficiency. We provide a proof of the generalized least squares procedure.     

Model-based control of a molten carbonate fuel cell (MCFC) process

To improve availability and performance of fuel cells, the operating temperature of molten carbonate fuel cells (MCFC) stack should be strictly maintained within a specified operation range, and an efficient control technique should be employed to meet this objective. While most modern control strategies are based on process models, many existing models of MCFC are not ready to be applied in synthesis and operation of control systems. In this study, we developed an auto-regressive moving average (ARMA) model and machine learning methods of least squares support vector machine (LS-SVM), artificial neural network (ANN) and partial least squares (PLS) for the MCFC system based on input-output operating data. The ARMA model showed the best tracking performance. A model predictive control method for the operation of MCFC system was developed based on the proposed ARMA model. The control performance of the proposed MPC methods was compared with that of conventional controllers using numerical simulations performed on various process models including an MCFC process. Numerical results show that ARMA model based control provides improved control performance compared to other control methods.     

AN APPLICATION OF THE SCHUR-COHN ALGORITHM TO TIME SERIES ANALYSIS

Abstract. Standard least squares analysis of autoregressive moving-average (ARMA) processes with errors-in-variables entails the construction of a new set of parameters which are functions of the original ARMA parameters, and requires that derivatives of these new parameters of order three or less with respect to the ARMA parameters exist and be bounded. The boundedness of these derivatives in turn depends critically on the nonsingularity of a matrix B which is a function of the ARMA parameters via the new parameters in the model. A particular version of the classical Schur–Cohn algorithm enables us to establish this nonsingularity.     

ASYMPTOTIC INFERENCE FOR NON-INVERTIBLE MOVING-AVERAGE TIME SERIES

Abstract. This paper is concerned with statistical inference of nonstationary and non-invertible autoregressive moving-average (ARMA) processes. It makes use of the fact that a derived process of an ARMA( p, q ) model follows an AR( q ) model with an autoregressive (AR) operator equivalent to the moving-average (MA) part of the original ARMA model. Asymptotic distributions of least squares estimates of MA parameters based on a constructed derived process are obtained as corresponding analogs of a nonstationary AR process. Extensions to the nearly non-invertible models are considered and the limiting distributions are obtained as functionals of stochastic integrals of Brownian motions and Ornstein-Uhlenbeck processes. For application, a two-stage procedure is proposed for testing unit roots in the MA polynomial. Examples are given to illustrate the application.     

A comparative study of models for molten carbonate fuel cell (MCFC) processes

The necessity of this work arose from the need for identification of a comprehensive plant model that can be used in the model-based control of the MCFC plant. Various models for molten carbonate fuel cell (MCFC) processes are presented and evaluated in this paper. Both a rigorous model based on mass and energy balances and implicit models based on operation data were investigated and analyzed. In particular, auto-regressive moving average (ARMA) model, least-squares support vector machine (LSSVM) model, artificial neural network (ANN) model and partial least squares (PLS) model for a MCFC system were developed based on input-output operating data. Among these models, the ARMA model showed the best agreement with plant operation data.     

Asymptotic laws of successive least squares estimates for seasonal arima models and application

Abstract. In view of detecting the stochastic non-stationarity in time series, successive Yule–Walker estimates are considered for general seasonal ARIMA models and their asymptotic laws are obtained. This extends results known on least squares estimates for stable–unstable ARMA. Furthermore, these asymptotic laws are then compared with analogous results obtained under some additive seasonal model that corresponds to the case of deterministic seasonal behaviour. These results, combined with a simulation study, reveal that successive autoregressions provide a very useful tool both for identifying seasonal ARIMA processes and for distinguishing between stochastic and deterministic seasonal behaviours.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Portmanteau tests for ARMA models with infinite variance

Abstract. Autoregressive and moving‐average (ARMA) models with stable Paretian errors are some of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking of fitted models has not been addressed. In this article, we develop portmanteau tests for checking the randomness of a time series with infinite variance and for ARMA diagnostic checking when the innovations have infinite variance. It is assumed that least squares or an asymptotically equivalent estimation method, such as Gaussian maximum likelihood, is used. It is also assumed that the distribution of the innovations is identically and independently distributed (i.i.d.) stable Paretian. It is seen via simulation that the proposed portmanteau tests do not converge well to the corresponding limiting distributions for practical series length so a Monte Carlo test is suggested. Simulation experiments show that the proposed Monte Carlo test procedure works effectively. Two illustrative applications to actual data are provided to demonstrate that an incorrect conclusion may result if the usual portmanteau test based on the finite variance assumption is used.     

ON THE CORRELATION STRUCTURE FOR THE GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTIC PROCESS

Abstract. The correlation structure for the squares from the generalized autoregressive conditional heteroskedastic (GARCH) process is presented. It is shown that the behaviour of the correlations for the squares mimics the usual correlations of an appropriately defined ARMA process, although the admissible regions for the correlations are somewhat more restrictive. Simulation experiments are used to study the applicability of the theoretical results for order identification and diagnostic checking. Finally, an empirical example is given for the IBM stock market price series from Box and Jenkins (1976).     

ALTERNATIVE ESTIMATORS AND UNIT ROOT TESTS FOR THE AUTOREGRESSIVE PROCESS

Abstract. We compare several estimators for the second-order autoregressive process and compare the associated tests for a unit root. Monte Carlo results are reported for the ordinary least squares estimator, the simple symmetric least squares estimator and the weighted symmetric least squares estimator. The weighted symmetric least squares estimator of the autoregressive parameters generally has smaller mean square error than that of the ordinary least squares estimator, particularly when one root is close to one in absolute value. For the second-order model with known zero intercept, the one-sided ordinary least squares test for a unit root is more powerful than the symmetric tests. For the model with an estimated intercept, the one-sided weighted symmetric least squares test is the most powerful test.     

ON THE RELATIONSHIP BETWEEN GENERALIZED LEAST SQUARES AND GAUSSIAN ESTIMATION OF VECTOR ARMA MODELS

Abstract. This paper investigates theoretical aspects of the relationship between the generalized least squares and Gaussian estimation schemes for vector autoregressive moving-average models. The asymptotic convergence of the generalized least squares estimator to the Gaussian estimator is established and an alternative numerical method for implementing the generalized least squares scheme is proposed. Finally, some simulation results are presented to illustrate the theory.     

小波变换-偏最小二乘法用于苯胺、苯酚和苯甲酸的同时测定

将小波变换与偏最小二乘法相结合,对苯酚、苯胺、苯甲酸三组分体系进行同时测定。先对光谱信号进行小波变换,去除噪声后重构信号,再用偏最小二乘法预报样品浓度。结果表明,经小波变换后处理结果回收率明显提高。     

A GENERALIZED FRACTIONALLY INTEGRATED AUTOREGRESSIVE MOVING-AVERAGE PROCESS

Abstract. This paper considers the long memory Gegenbauer autoregressive movingaverage (GARMA) process that generalizes the fractionally integrated ARMA (ARFIMA) process to allow for hyperbolic and sinusoidal decay in autocorrelations. We propose the conditional sum of squares method for estimation (which is asymptotically equivalent to the maximum likelihood estimation) and develop the asymptotic theory. Many results are in sharp contrast to those of the ARFIMA model. Simulations are conducted to assess the performance of the proposed estimators in small sample applications. Two applications to the sunspot data and the US inflation rates based on the wholesale price index are provided.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.     

An evaluation of methods for determining the parameters in the BWR equation of state using volumetric and heat capacity data

The use of linear programming was compared with non-linear least squares techniques for estimating Benedict-Webb-Rubin (BWR) equation of state parameters to correlate the volumetric behavior of a fluid. The techniques were extended to estimate the parameters from the volumetric and heat capacity properties of a pure component. In the non-linear least squares technique a convergence scheme, which is a combination of the Levenberg and Marquardt algorithms, was used. The volumetric behavior of carbon dioxide, methane and their mixtures was best correlated by the parameter sets obtained by the non-linear least squares technique when volume was chosen as the dependent variable. For the simultaneous correlation of volumetric and heat capacity properties of carbon dioxide the non-linear least squares technique was found to be superior to the linear programming technique.     

A least squares approach to model reduction and the design of low-order controllers

State space process models and feedback control laws are simplified by eliminating selected state variables using a modified least squares technique. The revised technique uses random perturbations to excite the high order system and the resulting reduced-order models tend to be more robust than those derived using conventional least squares and deterministic inputs. Low-order controllers were designed for a pilot scale, double effect evaporator using the modified least squares approach. These controllers performed well in both experimental and simulated response tests.     

一种基于时序误差补偿的动态软测量建模方法

针对目前静态软测量建模方法无法反映工业过程动态信息,造成预测模型精度低、鲁棒性差等问题,提出了一种基于最小二乘支持向量机(LS-SVM)和自回归-滑动平均模型(ARMA)的软测量建模方法。首先,建立了基于LS-SVM的软测量模型,利用ARMA模型对预测误差的动态估计,通过增加动态校正环节,实现了对静态模型的动态校正以改善系统动态响应特性。最后将上述方法用于乙烯精馏过程中乙烷浓度的软测量建模,仿真结果表明:与单一使用LSSVM模型相比,该方法具有跟踪性能好、泛化能力强等优点,是一种有效的软测量建模方法。     

A METHODOLOGICAL NOTE ON THE DISAGGREGATION OF TIME SERIES TOTALS

Abstract. This paper sheds new light on a generalized least squares approach for disaggregating a series of time series totals to estimate an underlying unaggregated series. By reinterpreting the generalized least squares problem as a time series prediction problem we can produce considerable computational savings relative to standard least squares approaches. Our reinterpretation gives us insight into the nature of the matrices which need to be inverted to compute the disaggregates.