Outlier Detection And Estimation In NonLinear Time Series

Abstract. The problem of identifying the time location and estimating the amplitude of outliers in nonlinear time series is addressed. A model‐based method is proposed for detecting the presence of additive or innovational outliers when the series is generated by a general nonlinear model. We use this method for identifying and estimating outliers in bilinear, self‐exciting threshold autoregressive and exponential autoregressive models. A simulation study is performed to test the proposed procedures and comparing them with the methods based on linear models and linear interpolators. Finally, our results are applied for detecting outliers in the Canadian lynx trappings and in the sunspot numbers data.     

Testing the Null of Co‐integration in the Presence of Variance Breaks

Abstract. We show that changes in the innovation covariance matrix of a vector of series can generate spurious rejections of the null hypothesis of co‐integration when applying standard residual‐based co‐integration tests. A bootstrap solution to the inference problem is suggested which is shown to perform well in practice, redressing the size problems associated with the standard test but not losing power relative to the standard test under the alternative.     

A Goodness‐of‐Fit Test for Integer‐Valued Autoregressive Processes

For autoregressive count data time series, a goodness‐of‐fit test based on the empirical joint probability generating function is considered. The underlying process is contained in a general class of Markovian models satisfying a drift condition. Asymptotic theory for the test statistic is provided, including a functional central limit theorem for the non‐parametric estimation of the stationary distribution and a parametric bootstrap method. Connections between the new approach and existing tests for count data time series based on moment estimators appear in limiting scenarios. Finally, the test is applied to a real data set.     

Improvement of the quasi‐likelihood ratio test in ARMA models: some results for bootstrap methods

Abstract. Quasi‐likelihood ratio tests for autoregressive moving‐average (ARMA) models are examined. The ARMA models are stationary and invertible with white‐noise terms that are not restricted to be normally distributed. The white‐noise terms are instead subject to the weaker assumption that they are independently and identically distributed with an unspecified distribution. Bootstrap methods are used to improve control of the finite sample significance levels. The bootstrap is used in two ways: first, to approximate a Bartlett‐type correction; and second, to estimate the p‐value of the observed test statistic. Some simulation evidence is provided. The bootstrap p‐value test emerges as the best performer in terms of controlling significance levels.     

On Bayesian analysis of nonlinear continuous‐time autoregression models

Abstract. This article introduces a method for performing fully Bayesian inference for nonlinear conditional autoregressive continuous‐time models, based on a finite skeleton of observations. Our approach uses Markov chain Monte Carlo and involves imputing data from times at which observations are not made. It uses a reparameterization technique for the missing data, and because of the non‐Markovian nature of the models, it is necessary to adopt an overlapping blocks scheme for sequentially updating segments of missing data. We illustrate the methodology using both simulated data and a data set from the S & P 500 index.     

Nonlinear ARMA models with functional MA coefficients

Abstract. In the present article, we propose and study a new class of nonlinear autoregressive moving‐average (ARMA) models, in which each moving‐average (MA) coefficient is enlarged to an arbitrary univariate function. We first provide a sufficient condition for the existence of the stationary solution and further discuss the moment structure. We investigate the estimation method to the proposed models. The global estimates of parameters and local linear estimates of functional coefficients are obtained by using a back‐fitting algorithm. For testing whether the functional coefficients are some specified parametric forms, a bootstrap test approach is provided. The proposed models are illustrated by both simulated and real data examples.     

Predictive modeling of pharmaceutical processes with missing and noisy data

Lack of knowledge of the first principles, that describe the behavior of processed particulate mixtures, has created significant attention to data‐driven models for characterizing the performance of pharmaceutical processes—which are often treated as black—box operations. Uncertainty contained in the experimental data sets, however, can decrease the quality of the produced predictive models. In this work, the effect of missing and noisy data on the predictive capability of surrogate modeling methodologies such as Kriging and Response Surface Method (RSM) is evaluated. The key areas that affect the final error of prediction and the computational efficiency of the algorithm were found to be: (a) the method used to assign initial estimate values to the missing elements and (b) the iterative procedure used to further improve these initial estimates. The proposed approach includes the combination of the most appropriate initialization technique and the Expectation Maximization Principal Component Analysis algorithm to impute missing elements and minimize noise. Comparative analysis of the use of different initial imputation techniques such as mean, matching procedure, and a Kriging‐based approach proves that the two former used approaches give more accurate, “warm‐start” estimates of the missing data points that can significantly reduce computational time requirements. Experimental data from two case studies of different unit operations of the pharmaceutical powder tablet production process (feeding and mixing) are used as examples to illustrate the performance of the proposed methodology. Results show that by introducing an extra imputation step, the pseudo complete data sets created, produce very accurate predictive responses, whereas discarding incomplete observations leads to loss of valuable information and distortion of the predictive response. Results are also given for different percentages of missing data and different missing patterns. © 2010 American Institute of Chemical Engineers AIChE J, 2010