Stationary Autoregressive Models via a Bayesian Nonparametric Approach

Abstract. An approach to constructing strictly stationary AR(1)‐type models with arbitrary stationary distributions and a flexible dependence structure is introduced. Bayesian nonparametric predictive density functions, based on single observations, are used to construct the one‐step ahead predictive density. This is a natural and highly flexible way to model a one‐step predictive/transition density.     

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.     

Computation and Characterization of Autocorrelations and Partial Autocorrelations in Periodic ARMA Models

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving-average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.     

A Generalised Fractional Differencing Bootstrap for Long Memory Processes

A bootstrap methodology suitable for use with stationary and non‐stationary fractionally integrated time series is further developed in this article. The resampling algorithm involves estimating the degree of fractional integration, applying the fractional differencing operator, resampling the resulting approximation to the underlying short memory series and, finally, cumulating to obtain a resample of the original fractionally integrated process. This approach extends existing methods in the literature by allowing for general bootstrap schemes including blockwise bootstraps. Furthermore, we show that it can also be validly used for non‐stationary fractionally integrated processes. We establish asymptotic validity results for the general method and provide simulation evidence which highlights a number of favourable aspects of its finite sample performance, relative to other commonly used bootstrap methods.     

A Note on the Behaviour of Nonparametric Density and Spectral Density Estimators at Zero Points of their Support

The asymptotic behaviour of nonparametric estimators of the stationary density and of the spectral density function of a stationary process have been studied in some detail in the last 50–60years. Nevertheless, less is known about the behaviour of these estimators when the target function happens to vanish at the point of interest. In the article at hand, we fill this gap and show that asymptotic normality still holds true but with super‐efficient and different rates of convergence for the density and for the spectral density estimators that are affected also by the dependence structure of the process.     

Moving Fourier Analysis for Locally Stationary Processes with the Bootstrap in View

We introduce a moving Fourier transformation for locally stationary time series, which captures the time‐varying spectral density in a similar manner as the classical Fourier transform does for stationary time series. In particular, the resulting Fourier coefficients as well as moving local periodograms are shown to be (almost all) asymptotically uncorrelated. The moving local periodogram is obtained by thinning the local periodogram to avoid multiple information present at different but close points in time. We obtain consistent estimators for the local spectral density at each point in time by smoothing the moving local periodogram. Furthermore, the moving Fourier coefficients, respectively periodograms, are well suited to adapt stationary frequency domain bootstrap methods to the locally stationary case. For the wild time frequency toggle bootstrap, it is shown that the corresponding bootstrap covariance of a global locally stationary bootstrap samples captures the time‐varying covariance structure of the underlying locally stationary time series correctly. Furthermore, this bootstrap in addition to adaptations of other frequency domain bootstrap methods is used in a simulation study to obtain uniform confidence bands for the time‐varying autocorrelation at lag 1. Finally, this methodology is applied to a wind data set.     

New Algorithm for Constructing Models of the Chemical Reaction Kinetics in the Quasistationary and Quasiequilibrium Approximations

An analysis is made regarding the possibility of using the concepts of routes, overall equations, and stationary rates in constructing kinetic models simultaneously in the quasiequilibrium and quasistationary approximations. A general algorithm is proposed to construct routs, formulate overall equations, and derive expressions for stationary rates. Features of the algorithm are revealed, which are related to some quantitative characteristics of the reaction mechanism.     

ASYMPTOTIC EFFICIENCY OF THE SAMPLE COVARIANCES IN A GAUSSIAN STATIONARY PROCESS

Abstract. This paper deals with the asymptotic efficiency of the sample autocovariances of a Gaussian stationary process. The asymptotic variance of the sample autocovariances and the Cramer–Rao bound are expressed as the integrals of the spectral density and its derivative. We say that the sample autocovariances are asymptotically efficient if the asymptotic variance and the Cramer–Rao bound are identical. In terms of the spectral density we give a necessary and sufficient condition that they are asymptotically efficient. This condition is easy to check for various spectra.     

ESTIMATION AND INTERPOLATION OF MISSING VALUES OF A STATIONARY TIME SERIES

Abstract. A complete solution of the important problem of estimating (interpolating) the missing values of a stationary time series is obtained by decomposing it into a prediction plus regression problem. This makes it possible to estimate the missing values by finding the multistep-ahead predictors and using the existing computer packages for time series analysis. Such a solution is vital for the E step of the EM algorithm, and it is shown how this algorithm can be used to develop a simultaneous procedure for estimating the parameters and missing values of a time series.     

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.     

Dissolution of a Polydisperse Ensemble of Particles in a Flow Apparatus

A mathematical model of continuous dissolution of salts in a liquid solution based on the crystal size distribution density function is considered. It is assumed that crystals have a spherical shape and the particle dissolution rate is a power-law function of the particle radius. A stationary solution is obtained, and its stability is analyzed. Features of the solution are noted for a particular case—batch process. For a nonstationary solution, the evolution equation for the undersaturation of the liquid solution is derived. An approximate analytical method for solving this equation is proposed; the solution found by this algorithm is quite close to that obtained by numerical methods. The concept of dissolution efficiency coefficient in continuous dissolution is introduced; analytical expressions for this coefficient are derived.     

Frequency domain generalized empirical likelihood method

This paper is concerned with a version of empirical likelihood method for spectral restrictions, which handles stationary time series data via the frequency domain approach. The asymptotic properties of frequency domain generalized empirical likelihood are studied for either strictly stationary processes with vanishing cumulant spectral density function of order 4 or linear processes generated by iid innovations with possibly non‐zero fourth order cumulant. Several statistics for testing parametric restrictions, over‐identified spectral restrictions, and additional spectral restrictions are shown to have the limiting chi‐squared distributions. Some numerical results are presented to investigate the finite sample performance of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

COMPUTATION OF CANONICAL CORRELATION BETWEEN PAST AND FUTURE OF A TIME SERIES

Abstract. The canonical correlation between the past and future of a stationary time series is shown to be the limit of the canonical correlation between the infinite past and finite future, and computation of the latter is reduced to an eigenvalue problem invovling finite matrices. This provides a convenient finite-dimensional algorithm for computing canonical correlations and components of a time series. An upper bound is conjectured for the largest canonical correlation.     

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.     

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.     

LEAVE-K-OUT DIAGNOSTICS IN STATE-SPACE MODELS

Abstract. The paper derives an algorithm for computing leave- k -out diagnostics for the detection of patches of outliers for stationary and nonstationary state-space models with regression effects. The algorithm is based on a reverse run of the Kalman filter on the smoothing errors and is both efficient and easy to implement. The US index of industrial production for textiles is used to illustrate the application of the algorithm.     

Robust empirical likelihood for time series

This article introduces a robust frequency domain empirical likelihood inference procedure for the parametric component in the spectral densities of stationary processes. We construct the empirical likelihood function by using a new spectral estimating function to achieve robustness against contamination in the spectral density. Simulation studies demonstrate the good performance of the proposed robust frequency domain empirical likelihood method, which produces more accurate confidence regions than the ordinary empirical likelihood counterpart.     

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 new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.     

面向水下环境的图像对比度增强算法研究

在采集水下图像时,由于水面对光的反射和散射会导致水下图像对比度下降。针对这一情况,提出了一种基于离散平稳小波变换的算法,对图像进行去噪和对比度增强的处理。实验表明：新算法能够有效地增强水下图像的对比度,同时还能够很好地抑制图像中的白噪声。