带变遗忘因子的自适应子空间预测控制器设计

针对实际工业过程具有非线性、时变和多变量的特点,提出一种数据驱动的带有变遗忘因子的自适应子空间预测控制方法。该方法将在线子空间辨识与模型预测控制相结合,同时利用期望输出值与实际输出值的误差实现变遗忘因子的自适应更新,并根据当前变遗忘因子构造了过去与将来的Hankel矩阵,从而实现了预测模型的在线更新,提高了控制器对非线性时变特征的辨识灵敏度和适应能力。最后,利用该控制器对四容水箱对象进行仿真研究,验证了算法的有效性。     

NON‐STATIONARITY AND QUASI‐MAXIMUM LIKELIHOOD ESTIMATION ON A DOUBLE AUTOREGRESSIVE MODEL

This article first studies the non‐stationarity of the first‐order double AR model, which is defined by the random recurrence equation , where γ₀ > 0, α₀ ≥ 0, and {η_t}is a sequence of i.i.d. symmetric random variables. It is shown that the double AR(1) model is explosive under the condition . Based on this, it is shown that the quasi‐maximum likelihood estimator of (φ₀,α₀) is consistent and asymptotically normal so that the unit root problem does not exist in the double AR(1) model. Simulation studies are carried out to assess the performance of the quasi‐maximum likelihood estimator in finite samples.     

A superharmonic prior for the autoregressive process of the second‐order

Abstract. The Bayesian estimation of the spectral density of the AR(2) process is considered. We propose a superharmonic prior on the model as a non‐informative prior rather than the Jeffreys prior. Theoretically, the Bayesian spectral density estimator based on it dominates asymptotically the one based on the Jeffreys prior under the Kullback–Leibler divergence. In the present article, an explicit form of a superharmonic prior for the AR(2) process is presented and compared with the Jeffreys prior in computer simulation.     

On the three-step non-Gaussian quasi-maximum likelihood estimation of heavy-tailed double autoregressive models

This note considers a three-step non-Gaussian quasi-maximum likelihood estimation (TS-NGQMLE) of the double autoregressive model with its asymptotics, which improves efficiency of the GQMLE and circumvents inconsistency of the NGQMLE when the innovation is heavy-tailed. Under mild conditions, the estimator not only can achieve consistency and asymptotic normality regardless of density misspecification of the innovation, but also outperforms the existing estimators, such as the GQMLE and the (weighted) least absolute deviation estimator, when the innovation is indeed heavy-tailed.     

Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004) , we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.     

Analysis of accumulated rounding errors in autoregressive processes

In considering the rounding impact of an autoregressive (AR) process, there are two different models available to be considered. The first assumes that the dynamic system follows an underlying AR model and only the observations are rounded up to a certain precision. The second assumes that the updated observation is a rounded version of an autoregression on previous rounded observations. This article considers the second model and examines behaviour of rounding impacts to the statistical inferences. The conditional maximum‐likelihood estimates for the model are proposed and their asymptotic properties are established, including strong consistency and asymptotic normality. Furthermore, both the classical AR model and the ordinary rounded AR model are no longer reliable when dealing with accumulated rounding errors. The three models are also applied to fit the Ocean Wave data. It turns out that the estimates under distinct models are significantly different. Based on our findings, we strongly recommend that models for dealing with rounded data should be in accordance with the actions of rounding errors.     

Parameter estimation in models with hidden variables : An application to a biotech process

Biological processes are often characterised by significant nonlinearities, noisy measurements and hidden process variables. The dynamic behaviour of such processes can be represented by stochastic differential equations obtained from physical laws. We propose a Bayesian algorithm for parameter estimation in stochastic nonlinear biological processes with unmeasured (or hidden) variables. The proposed algorithm, involves drawing random samples iteratively from a posterior density functions of the parameters and the hidden variables. A Bayesian sampling techniques is used to approximate these posterior density functions. Both Metropolis–Hastings algorithm and Gibbs sampling are used for sample generation. The algorithm is extended to handle multiple data sets and missing observations. The algorithm is applied to an experimental data set collected from an algal bioreactor system. © 2011 Canadian Society for Chemical Engineering     

Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation

Abstract. We provide simulation and theoretical results concerning the finite‐sample theory of quasi‐maximum‐likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)–ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t‐tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean‐squared error of the maximum‐likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)‐ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1–10] invariance properties for the biases in these situations.     

Comparing the bias and misspecification in ARFIMA models

We investigate the bias in both the short-term and long-term parameters for a range of autoregressive fractional integrated moving-average (ARFIMA) models using both semi-parametric and maximum likelihood (ML) estimation methods. The results suggest that, provided the correct model is estimated, the ML method outperforms the semi-parametric methods in terms of the bias and smaller mean square errors in both the long-term and short-term parameter estimates. These biases often cause model selection criteria to select an incorrect ARFIMA specification. Taking account of the potential misspecification the biases associated with the ML procedure tend to increase, although it continues to have a smaller worst-case bias than either of the semi-parametric procedures.     

Assessing the precision of model predictions and other functions of model parameters

Models fitted to data are used extensively in chemical engineering for a variety of purposes, including simulation, design and control. In any of these contexts it is important to assess the uncertainties in the estimated parameters and in any functions of these parameters, including predictions from the fitted model. Profiling is a likelihood ratio approach to estimating uncertainties in parameters and functions of parameters. A comparison is made between the optimization and reparameterization approaches to determining likelihood intervals for functions of parameters. The merits and limitations of generalized profiling are discussed in relation to the linearization approach commonly used in engineering. The benefits of generalized profiling are illustrated with two examples. A geometric interpretation of profiling is used to elucidate its value, and cases are identified for which the numerical algorithm fails. An alternative approach is suggested for these cases.     

An algorithm for the exact likelihood of a stationary vector autoregressive-moving average model

The so-called innovations form of the likelihood function implied by a stationary vector autoregressive-moving average model is considered without directly using a state–space representation. Specifically, it is shown in detail how to compute the exact likelihood by an adaptation to the multivariate case of the innovations algorithm of Ansley (1979 ) for univariate models. Comparisons with other existing methods are also provided, showing that the algorithm described here is computationally more efficient than the fastest methods currently available in many cases of practical interest.     

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.     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

The restricted likelihood ratio test at the boundary in autoregressive series

Abstract.  The restricted likelihood ratio test, RLRT, for the autoregressive coefficient in autoregressive models has recently been shown to be second-order pivotal when the autoregressive coefficient is in the interior of the parameter space and so is very well approximated by the     distribution. In this article, the non-standard asymptotic distribution of the RLRT for the unit root boundary value is obtained and is found to be almost identical to that of the     in the right tail. Together, these two results imply that the     distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution.     

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.     

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

Second order asymptotic efficlency in terms of the asymptotic variance of seqrential estimation procedures in the presence of nrisance parameters

In the presence of nuisance parameters, the Bhattacharyya type bound for the asymptotic variance of estimation procedures is obtained. It is shown that the modified maximum likelihood (ML) estimation procedures together with any stopping rule does not attain the bound. Further it is shown that the modified ML estimation procedure with the appropriate stopping rule is second order asymptotically efficient in some class of estimation procedures in the sense that it attains the lower bound for the asymptotic variance in the class.