Time‐series clustering via quasi U‐statistics

The problem of time‐series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U‐statistics and subgroup decomposition tests. The decomposition may be applied to any concave time‐series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non‐identically distributed groups of time‐series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non‐stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.     

NORMING RATES AND LIMIT THEORY FOR SOME TIME‐VARYING COEFFICIENT AUTOREGRESSIONS

A time‐varying autoregression is considered with a similarity‐based coefficient and possible drift. It is shown that the random‐walk model has a natural interpretation as the leading term in a small‐sigma expansion of a similarity model with an exponential similarity function as its AR coefficient. Consistency of the quasi‐maximum likelihood estimator of the parameters in this model is established, the behaviours of the score and Hessian functions are analysed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian function standardization. A large family of unit root models with stationary and explosive alternatives is characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied, and a large‐sample limit theory provided, which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by sustained stochastic departures from unity. The findings provide a composite case for time‐varying coefficient dynamic modelling. Some simulations and a brief empirical application to data on international Exchange Traded Funds are included. Copyright © 2014 Wiley Publishing Ltd     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

Statistical analysis of a spatio‐temporal model with location‐dependent parameters and a test for spatial stationarity

Abstract. In this article, we define a spatio‐temporal model with location‐dependent parameters to describe temporal variation and spatial nonstationarity. We consider the prediction of observations at unknown locations using known neighbouring observations. Further, we propose a local least squares‐based method to estimate the parameters at unobserved locations. The sampling properties of these estimators are investigated. We also develop a statistical test for spatial stationarity. To derive the asymptotic results, we show that the spatially nonstationary process can be locally approximated by a spatially stationary process. We illustrate the methods of estimation with some simulations.     

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.     

Sub‐stage PCA modelling and monitoring method for uneven‐length batch processes

For the characteristic of multistage of batch processes, a new PCA‐based sub‐stage division algorithm is proposed. This algorithm is based on the fact that production transition can be detected by analysing the loading matrixes and principal component matrixes, which reveal objectively evolvement of underlying process behaviours. Sub‐stage PCA models for each stage are built, and then extended to monitor the batch processes with uneven‐length durations by choosing the right sub‐stage model according to the principle of minimum similarity distance of principal component matrixes. With the proposed method, multi‐stage batch processes with durations of uneven‐length can be monitored effectively.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

A Bayesian framework for real‐time identification of locally weighted partial least squares

Just‐in‐time (JIT) learning methods are widely used in dealing with nonlinear and multimode behavior of industrial processes. The locally weighted partial least squares (LW‐PLS) method is among the most commonly used JIT methods. The performance of LW‐PLS model depends on parameters of the similarity function as well as the structure and parameters of the local PLS model. However, the regular LW‐PLS algorithm assumes that the parameters of the similarity function and structure of the local PLS model are known and do not fully utilize available knowledge to estimate the model parameters. A Bayesian framework is proposed to provide a systematic way for real‐time parameterization of the similarity function, selection of the local PLS model structure, and estimation of the corresponding model parameters. By applying the Bayes' theorem, the proposed framework incorporates the prior knowledge into the identification process and takes into account the different contribution of measurement noises. Furthermore, Bayesian model structure selection can automatically deal with the model complexity problem to avoid the overfitting issue. The advantages of this new approach are highlighted through two case studies based on the real‐world near infrared data. © 2014 American Institute of Chemical Engineers AIChE J, 61: 518–529, 2015     

Analysis of a pH‐Based Potentiometric Biosensor Using the Homotopy Perturbation Method

A mathematical model of steady‐state and non‐steady‐state responses of a pH‐based potentiometric biosensor immobilizing organophosphorus hydrolase was developed. The model is based on non‐stationary diffusion equations containing a nonlinear term related to the Michaelis‐Menten kinetics of an enzymatic reaction. An analytical expression for the substrate concentration was obtained for all values of parameter a (Thiele modulus) using the homotopy perturbation method. From this result, the concentrations of the deprotonation products of an organophosphodiester (P_hH, ZH and AH) were obtained. Our analytical results were compared with available simulation results. A satisfactory agreement with the simulation data is noted.     

Asymptotic results for Fourier‐PARMA time series

Periodically stationary times series are useful to model physical systems whose mean behavior and covariance structure varies with the season. The Periodic Auto‐Regressive Moving Average (PARMA) process provides a powerful tool for modelling periodically stationary series. Since the process is non‐stationary, the innovations algorithm is useful to obtain parameter estimates. Fitting a PARMA model to high‐resolution data, such as weekly or daily time series, is problematic because of the large number of parameters. To obtain a more parsimonious model, the discrete Fourier transform (DFT) can be used to represent the model parameters. This article proves asymptotic results for the DFT coefficients, which allow identification of the statistically significant frequencies to be included in the PARMA model.     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.     

A Wavelet Characterization of Continuous‐Time Periodically Correlated Processes with Application to Simulation

We introduce a wavelet characterization of continuous‐time periodically correlated processes based on a linear combination of infinite‐dimensional stationary processes. The finite version of this linear combination converges to the main process. The first‐order and second‐order estimators based on the wavelets are presented. Under a simple and easy algorithm, the periodically correlated process is simulated for a given autocovariance function. The proposed algorithm has two main advantages: first, it is fast, and second, it is distribution free. We indicate through four examples that the simulated data are periodically correlated with the desired period.     

Using NSGA‐II and TOPSIS Methods for Interior Ballistic Optimization Based on One‐Dimensional Two‐Phase Flow Model

In order to meet the design demands of new gun systems or new types of projectiles, the interior ballistic charge design seems especially important. In this paper, a one‐dimensional two‐phase flow model is presented. The model describes the transient combustion of granular propellants in a gun, and pressure waves are considered as an objective. This study adopts a hybrid method to solve the problem. In the first stage, the non‐dominated sorting genetic algorithm (NSGA‐II) with “a “filter” is employed to approximate a set of Pareto‐optimal solutions. In the subsequent stage, a multi‐attribute decision‐making (MADM) approach is adopted to rank these solutions from the best to the worst. The ranking of Pareto‐optimal solutions is based on the technique ordered preference by similarity to ideal solution (TOPSIS) method. In TOPSIS method each objective needs a corresponding weight coefficient, and a practical problem is introduced. Both the entropy method and linear analysis method are adopted to get two sets of weights for the objectives, respectively. The two pairs of final, best compromise solutions are compared for satisfying the designer’s aim. For the analysis of the results, a two‐phase flow interior ballistic model for design optimization is established, and the hybrid approach could get a reasonable design scenario.     

Inference for non‐stationary time‐series autoregression

The article considers simultaneous inference for a class of non‐stationary autoregressive models where the model parameters are allowed to vary smoothly over time. Simultaneous confidence tubes with asymptotically correct coverage probabilities are constructed to assess the overall patterns and magnitudes of the parameter functions over time. Simulation studies are conducted, and a real time‐series dataset is analyzed to demonstrate the usefulness of the proposed methodology.     

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.     

Subsampling inference for the mean of heavy‐tailed long‐memory time series

In this article, we revisit a time series model introduced by MCElroy and Politis (2007a) and generalize it in several ways to encompass a wider class of stationary, nonlinear, heavy‐tailed time series with long memory. The joint asymptotic distribution for the sample mean and sample variance under the extended model is derived; the associated convergence rates are found to depend crucially on the tail thickness and long memory parameter. A self‐normalized sample mean that concurrently captures the tail and memory behaviour, is defined. Its asymptotic distribution is approximated by subsampling without the knowledge of tail or/and memory parameters; a result of independent interest regarding subsampling consistency for certain long‐range dependent processes is provided. The subsampling‐based confidence intervals for the process mean are shown to have good empirical coverage rates in a simulation study. The influence of block size on the coverage and the performance of a data‐driven rule for block size selection are assessed. The methodology is further applied to the series of packet‐counts from ethernet traffic traces.     

Tests for Volatility Shifts in Garch Against Long‐Range Dependence

Many empirical findings show that volatility in financial time series exhibits high persistence. Some researchers argue that such persistency is due to volatility shifts in the market, while others believe that this is a natural fluctuation explained by stationary long‐range dependence models. These two approaches confuse many practitioners, and forecasts for future volatility are dramatically different depending on which models to use. In this article, therefore, we consider a statistical testing procedure to distinguish volatility shifts in generalized AR conditional heteroscedasticity (GARCH) model against long‐range dependence. Our testing procedure is based on the residual‐based cumulative sum test, which is designed to correct the size distortion observed for GARCH models. We examine the validity of our method by providing asymptotic distributions of test statistic. Also, Monte Carlo simulations study shows that our proposed method achieves a good size while providing a reasonable power against long‐range dependence. It is also observed that our test is robust to the misspecified GARCH models.     

Asymptotic Properties of a Probabilistic‐Statistical Model of Particle Classification in Hydrocyclones

The peculiarities of practical implementation of a probabilistic‐statistical model for a hydrodynamic stage of particle classification process of liquid‐solid polydisperse systems in cylinder‐conic hydrocyclones of small size have been investigated. Within reasonable assumptions, stationary solutions of the Fokker‐Planck‐Kolmogorov kinetic equation were obtained for the considered separation process. In order to describe changes in characteristics of suspension separation in hydrocyclones it was proposed to use stationary distributions, which parameters depend not only on hydraulic and dynamic features of flows inside an apparatus, but also are determined by relative magnitudes of the impact of particle classification and centrifugal forces in comparison with the intensity of random perturbations.     

Subgrid scale scalar variance in high‐Schmidt‐number turbulence

The subgrid scale (SGS) variance for a high‐Schmidt‐number passive scalar of Sc >> 1 is measured using a high‐resolution planar laser‐induced fluorescence technique in a grid‐generated turbulent liquid flow, and the values of the model coefficients in the scale‐similarity model and the scalar‐gradient model used for estimating the SGS scalar variance are experimentally evaluated. The results show that for both models, the measured values are much larger than the well‐known values obtained in the previous studies done for non‐high‐Sc scalars of Sc ? 1. Similarly, the measured value of the model coefficient in the scalar‐gradient model tends to be larger than the value estimated by the dynamic procedure. The increases in the measured values of the model coefficients for the high‐Sc scalar can be explained by the presence of the viscous‐convective range showing a nearly (?1)‐slope in the high‐wavenumber range of the power spectrum of concentration fluctuation. © 2011 American Institute of Chemical Engineers AIChE J, 2012