Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

Second-order properties of locally stationary processes

Abstract.  In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties.     

ON SIMULATION OF A GAUSSIAN STATIONARY PROCESS

In this paper we study the convergence of a simulated Gaussian stationary process to the actual process. First we propose a method of simulation of a Gaussian stationary process for a given spectral density function. We prove that the sample functions of this simulated process converge to those of the actual process uniformly on any finite time interval and obtain the convergence rate. We also discuss this method and compare it with the most often used simulation method first proposed by Rice. Our simulation results show that the Rice method simulates just as well as the method studied in this paper.     

Simulation of Real Discrete Time Gaussian Multivariate Stationary Processes with Given Spectral Densities

In this article we establish a simulation procedure to generate values for a real discrete time multivariate stationary process, based on a factor of spectral density matrix. We prove the convergence of the simulator, at each time epoch, to the actual process, and provide the corresponding rate of convergence. We merely assume that the spectral density matrix is continuous and of bounded variation. By using the positive root factor, we provide an extended version for the Sun and Chaika ( 1997 ) simulator, for real univariate stationary processes.     

THE DETECTION OF A SINGLE ADDITIVE OUTLIER OF UNKNOWN POSITION

Abstract. Kudo (On the testing of outlying observations. Sankhya 17 (1956), 67–73) has derived an optimal invariant detector of a single additive outlier of unknown position in the context of an underlying Gaussian process consisting of independent and identically distributed random variables. We show how this author's arguments can be extended to derive an invariant detector of an additive outlier of unknown position for an underlying zero-mean Gaussian stochastic process. This invariant detector depends on the parameters of this process; its properties are analysed further for the particular case of an underlying zero-mean Gaussian AR( p ) process. It provides an upper bound on the performance of any invariant detector based solely on the data and it may be 'bootstrapped' to provide an invariant detector based solely on the data. A plausibility argument is presented in favour of the proposition that the bootstrapped detector is nearly optimal for sufficiently large data length n. The truth of this proposition has been confirmed by simulation results for zero-mean Gaussian AR(1) and AR(2) processes (for certain sets of possible outlier positions). The bootstrapped detector is shown to be closely related to the detector based on the approximate likelihood ratio criteria of Fox (Outliers in time series. J. Roy. Statist. Soc. Ser. B 34 (1972), 350–63) and the leave-one-out diagnostic of Bruce and Martin (Leave- k -out diagnostics in time series. J. Roy. Statist. Soc. Ser B 51 (1989), 363–424). It is also shown how the case of an underlying Gaussian process with arbitrary mean can be reduced to the case of an underlying zero-mean Gaussian process.     

Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat ( J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi ( J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m -vector processes. It is shown that, for Gaussian vector AR( p ) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤ k ≤ p . Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA( q ) processes.     

Testing for Stationarity in Multivariate Locally Stationary Processes

In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap‐assisted test based on a Kolmogorov–Smirnov‐type statistic, which tracks the deviation of the time‐varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non‐stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite‐sample properties of the new method and contains a comparison with existing approaches.     

Nonparametric Tests of Change-Points with Tapered Data

In this work we build two families of nonparametric tests using tapered data for the off-line detection of change-points in the spectral characteristics of a stationary Gaussian process. This is done using the Kolmogorov–Smirnov statistics based on integrated tapered periodograms. Convergence is obtained under the null hypothesis by means of a double indexed (frequency– time) process together with some extensions of Dirichlet and Fejer kernels. Consistency is proved using these statistics under the alternative. Then, using numerical simulations, we observe that the use of tapered data significantly improves the properties of the test, especially in the case of small samples.     

PREDICTION ERROR OF MULTIVARIATE TIME SERIES WITH MIS-SPECIFIED MODELS

We consider the situation in which an incorrectly specified autoregressive moving-average model is used to predict future values of a stationary multivariate time series. The use of an incorrect model for prediction results in an increase in mean-square prediction error over that of the optimal predictor, and an expression for this increase is first given for fixed values of the parameters in the incorrect model. For the case in which the incorrect model is an autoregression, we also take into account parameter estimation error by first deriving the asymptotic distribution and limiting moment properties of the least-squares estimator of the parameters in the mis-specified model. An asymptotic approximation to the increase in mean-square prediction error is then obtained. Numerical examples are provided to demonstrate the accuracy of the asymptotic approximation in finite samples. Our results are consistent with those obtained in the univariate case, indicating that fitted autoregressions of high order can yield substantially sub-optimal forecasts.     

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.     

Reaction steady states and stationary states in modelling semi-batch enzyme hydrolysis

Dimensionless material balance equations describing an uninhibited enzyme hydrolysis process in a semi-batch reactor (i.e. fed-batch reactor) are formulated; numerical solution of these equations provided concentration profiles of the enzyme-substrate complex by using published kinetic parameters. The unrestricted values obtained are compared with estimates based separately on the reaction steady state and stationary state assumptions. Results are discussed in terms of the enzyme/substrate inventory used and it is found that the reaction steady state is a satisfactory approximation only when this ratio is sufficiently small. The stationary state may be a better approximation at other values, particularly when enzyme is added to substrate or when an empty tank is being filled. Reaction yields from semi-batch and batch operations are compared. Processing takes longer in the semi-batch operations and complete conversions are only practical in this mode when enzyme is added to substrate.     

Gaussian Semiparametric Estimation of Non-stationary Time Series

Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non-stationary log-periodogram regression, Forthcoming J. Economet. , 1997) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long-range-dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat . 23 (1995), 1630–61) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d ∈ (−½, 1) and asymptotically normal for d ∈ (−½,¾) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric efficiency of this estimate for stationary sequences also extends to the non-stationary framework.     

Simulation of Real‐Valued Discrete‐Time Periodically Correlated Gaussian Processes with Prescribed Spectral Density Matrices

Abstract. In this article, we provide a spectral characterization for a real‐valued discrete‐time periodically correlated process, and then proceed on to establish a simulation procedure to simulate such a Gaussian process for a given spectral density. We also prove that the simulated process, at each time index, converges to the actual process in the mean square.     

Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Chebyshev配置点谱方法直接求解离散坐标辐射传递方程

针对三维长方形炉内具有吸收-发射介质的辐射换热,基于Chebyshev配置点谱方法和Schur分解开发了直接求解辐射离散坐标方程的求解器。针对离散后所得到的三维矩阵方程,分别用两种方法进行求解,一种是用张量积将三维转变成二维然后直接用Schur分解求解;另一种是自行开发三维Schur分解直接求解。数值实验表明,在相同的输入参数下,新求解器具有很好的精度,尤其相比于标准离散坐标法,新求解器能节省大量计算时间。特别是基于三维Schur分解的直接求解器,在相同的输入参数下,计算时间只有标准离散坐标法的10%～1%。     

A Data‐Driven Gaussian Process Regression Model for Two‐Chamber Microbial Fuel Cells

Rapidly and accurately modeling of microbial fuel cells (MFCs) plays an important role not only in thorough understanding of the effects of operating conditions on system performance, but also in the successful implementation of real‐time maximization of power output. Although the first principle electrochemical model has better generalization performance, it is often time‐consuming for model construction and is hard to real‐time application. In this study, a nonparametric Gaussian process regression (GPR) model is used to capture the nonlinear relationship between operating conditions and output voltage in the MFCs. A simple online learning strategy is proposed to recursively update the hyper‐parameters of the GPR model. The applicability and effectiveness of the proposed method is validated by both the simulation and experimental datasets from the acetate and the glucose and glutamic acid two‐chamber MFCs. The results illustrate that the online GPR model provides a promising method for capturing the complex nonlinearity phenomenon in MFCs, which can be greatly helpful for further real‐time optimization of MFCs.     

Bootstrap-assisted Goodness-of-fit Tests in the Frequency Domain

The validity of a stationary time series model may be measured by the goodness of fit of the spectral distribution function. Anderson (Technical Report 27, 1991; Technical Report 309, 1995; Stanford University) has worked out the closed-form characteristic functions for the Cramer–von Mises criterion for general linear processes, under the condition that all values of the parameters are specified. The asymptotic approach is not easily implemented and usually requires a case by case analysis. In this paper we propose a bootstrap goodness-of-fit test in the frequency domain. By properly resampling the residuals, we can consistently estimate the p values for many weakly dependent semiparametric models with unspecified parameter values. This is the content of the main theorem that we try to explain. A group of simulations is conducted, showing consistent significance level and good power. The special tests are applied to the lynx data and reveal structure unexplained by the AR(1) model fitted by Tong ( J. R. Stat. Soc. A 140 (1977), 432–36). A possible generalization with application to financial data analysis is also discussed.     

TESTING CHANGE-POINTS IN THE EXPLOSIVE GAUSSIAN AUTOREGRESSIVE PROCESSES

Abstract. A functional limit theorem with a particular function class and topology is derived for non-ergodic type time series. This limit theorem allows us to study the asymptotic law of the associated likelihood ratio test (LRT) statistic for testing the presence of a change in the covariance parameter in the explosive Gaussian autoregressive model. We show that the level of the LRT cannot be approximated without introducing appropriate normalization. The limit law of a particular weighted likelihood ratio test is examined through a simulation study and is compared with the well-known Kolmogorov distribution obtained in the stationary case; we conclude that for practical applications when the root is really close to unity one can use the same thresholds as in the stationary case. This procedure is applied to the study of three real time series known to be non-stationary.     

CLASSIFICATION OF TEXTURES USING SECOND-ORDER SPECTRA

Abstract. We consider the classification of textures as realizations of stationary random fields using non-parametric estimates of their second-order spectra. The random fields in each class are assumed to be stationary with the same spectrum, which we estimate from a finite sample by smoothing its periodogram. The classification rule can be interpreted as maximizing a mean square convergent approximation to the averaged log-likelihood if the random fields are Gaussian, and in general as minimizing the discrepancy of the periodogram from the spectrum of the class. The limiting behaviour of the probability of misclassification as the sample size tends to infinity is studied under certain cumulant conditions. The classification rule is illustrated with real texture data.     

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.