Class Of Nonstationary Processes Research Articles

Estimation of nonstationary nonparametric regression model with multiplicative structure

Summary This paper presents a multiplicative nonstationary nonparametric regression model which allows for a broad class of nonstationary processes. We propose a three-step estimation procedure to uncover the conditional mean function and establish uniform convergence rates and asymptotic normality of our estimators. The new model can also be seen as a dimension-reduction technique for a general two-dimensional time-varying nonparametric regression model, which is especially useful in small samples and for estimating explicitly multiplicative structural models. We consider two applications: estimating a pricing equation for the US aggregate economy to model consumption growth and estimating the shape of the monthly risk premium for S&P 500 Index data.

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of non stationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.

Generalizations of Cyclostationarity: A New Paradigm for Signal Processing for Mobile Communications, Radar, and Sonar

The relative motion between transmitter and receiver introduces nonstationarity to stationary signals and modifies the nonstationarity of nonstationary signals. Several Doppler channel models of interest transform almost-cyclostationary (ACS) processes, which are an appropriate nonstationary model for modulated signals adopted in communications, radar, sonar, and telemetry, into generalized ACS (GACS) processes or spectrally correlated (SC) processes. Both classes of processes include the ACS processes as special cases. A review of these new classes of nonstationary processes is presented. The problems of probabilistic characterization, statistical function estimation, and determining the effects of time-sampling on these models are addressed. The impact of these models on signal detection and source location problems is discussed.

Clustering High-Dimensional Time Series Based on Parallelism

This article considers the problem of clustering high-dimensional time series based on trend parallelism. The underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors, a special class of nonstationary processes. For each group where the parallelism holds, I semiparametrically estimate its representative trend function and vertical shifts of group members, and establish their central limit theorems. An information criterion, consisting of in-group similarities and number of groups, is then proposed for the purpose of clustering. I prove its theoretical consistency and propose a splitting-coalescence algorithm to reduce the computational burden in practice. The method is illustrated by both simulation and a real-data example.

Asymptotics of partial sums of linear processes with changing memory parameter*

We study the limit distribution of partial sums of nonstationary truncated linear process {Xt, t = 1,…, n} with long memory and changing memory parameter dt,n ∈ (0,∞). Two classes of linear processes are investigated, namely, (i) the class of FARIMA-type truncated moving averages with time-varying fractional integration parameter and (ii) the class of time-varying fractionally integrated processes introduced in [A. Philippe, D. Surgailis, and M.-C. Viano, Invariance principle for a class of nonstationary processes with long memory, C. R. Acad. Sci., Paris, Ser. I, 342:269–274, 2006; A. Philippe, D. Surgailis, and M.-C. Viano, Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl., 52:651–673, 2008]. The cases of fast-changing memory parameter (dt,n = dt does not depend on n) and slowly changing memory parameter (dt,n = d(t/n) for some function d(τ), τ ∈ [0, 1]) are discussed. In the case of fast-changing memory, the limit partial sums process is a type II fractional Brownian motion (fBm) with the Hurst parameter equal to the global maximum of (dt) for class (i) processes and to the mean value of (dt) for class (ii) processes. In the case of slowly changing memory, the limit of partial sums for both classes (i) and (ii) is degenerate and “localized” at the global maximum of the memory function d(∙); however, a nondegenerate limit of the partial sums process is shown to exist when time is suitably rescaled in the vicinity of the maximum point.

On the Causality between Multiple Locally Stationary Processes

When one would like to describe the relations between multivariate time series, the concepts of dependence and causality are of importance. These concepts also appear to be useful when one is describing the properties of an engineering or econometric model. Although the measures of dependence and causality under stationary assumption are well established, empirical studies show that these measures are not constant in time. Recently one of the most important classes of nonstationary processes has been formulated in a rigorous asymptotic framework by Dahlhaus in (1996), (1997), and (2000), called locally stationary processes. Locally stationary processes have time-varying spectral densities whose spectral structures smoothly change in time. Here, we generalize measures of linear dependence and causality to multiple locally stationary processes. We give the measures of linear dependence, linear causality from one series to the other, and instantaneous linear feedback, at time t and frequency λ.

Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance

The random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena. Random walk is included in unit root process which is a class of nonstationary processes. Due to its nonstationarity, the least squares estimator (LSE) of random walk does not satisfy asymptotic normality. However, it is well known that the sequence of partial sum processes of random walk weakly converges to standard Brownian motion. This result is so-called functional central limit theorem (FCLT). We can derive the limiting distribution of LSE of unit root process from the FCLT result. The FCLT result has been extended to unit root process with locally stationary process (LSP) innovation. This model includes different two types of nonstationarity. Since the LSP innovation has time-varying spectral structure, it is suitable for describing the empirical financial time series data. Here we will derive the limiting distributions of LSE of unit root, near unit root and general integrated processes with LSP innovation. Testing problem between unit root and near unit root will be also discussed. Furthermore, we will suggest two kind of extensions for LSE, which include various famous estimators as special cases.

Testing the Null Hypothesis of Nonstationary Long Memory Against the Alternative Hypothesis of a Nonlinear Ergodic Model

Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. This paper provides a formal method of testing for nonstationary long memory against the alternative of a particular form of nonlinear ergodic processes; namely, exponential smooth transition autoregressive processes. In this regard, the current paper provides a significant generalization to existing unit root tests by allowing the null hypothesis to encompass a much larger class of nonstationary processes. The asymptotic theory associated with the proposed Wald statistic is derived, and Monte Carlo simulation results confirm that the Wald statistics have reasonably correct size and good power in small samples. In an application to real interest rates and the Yen real exchange rates, we find that the tests are able to distinguish between these competing processes in most cases, supporting the long-run Purchasing Power Parity (PPP) and Fisher hypotheses. But, there are a few cases in which long memory and nonlinear ergodic processes display similar characteristics and are thus confused with each other in small samples.

A Spectral Representation of a Class of Nonstationary Processes

In this paper, an extension of the weakly harmonizable class of processes is considered. This class, termed almost periodic contractive harmonizable, is based upon the natural contractive operator associated with harmonizable processes. A spectral representation of these processes is obtained. A relation between the almost periodic contractive harmonizable and the oscillatory harmonizable classes is considered. The paper concludes with a series representation for the almost periodic contractive harmonizable class.

A self-normalized confidence interval for the mean of a class of nonstationary processes

We construct an asymptotic confidence interval for the mean of a class of nonstationary processes with constant mean and time-varying variances. Due to the large number of unknown parameters, traditional approaches based on consistent estimation of the limiting variance of sample mean through moving block or non-overlapping block methods are not applicable. Under a block-wise asymptotically equal cumulative variance assumption, we propose a self-normalized confidence interval that is robust against the nonstationarity and dependence structure of the data. We also apply the same idea to construct an asymptotic confidence interval for the mean difference of nonstationary processes with piecewise constant means. The proposed methods are illustrated through simulations and an application to global temperature series.

Nonparametric regression estimation in a null recurrent time series

We derive an asymptotic theory of nonparametric estimation for a time series regression model Z t = f( X t )+ W t , where { X t } and { Z t } are observed nonstationary processes, and { W t } is an unobserved stationary process. The class of nonstationary processes allowed for { X t } is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process { W t } is assumed to be linear and stationary.

Asymptotic properties of an empirical K-function for inhomogeneous spatial point processes

The K-function is one of the most commonly used summary statistics. It plays the role for spatial point processes that the covariance function or the variogram plays for continuous observation. The asymptotic properties of the nonparametric estimation of K-function in stationary spatial point processes have been studied. However, in practice, stationary is often not a reasonable assumption. In this article, we investigate the asymptotic behaviour of the nonparametric estimation of K-function for a class of nonstationary processes.

Nonhomogeneous fractional integration and multifractional processes

Extending the recent work of Philippe et al. [A. Philippe, D. Surgailis, M.-C. Viano, Invariance principle for a class of non stationary processes with long memory, C. R. Acad. Sci. Paris, Ser. 1. 342 (2006) 269–274; A. Philippe, D. Surgailis, M.-C. Viano, Time varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. (2007) (in press)] on time-varying fractionally integrated operators and processes with discrete argument, we introduce nonhomogeneous generalizations I α ( ⋅ ) and D α ( ⋅ ) of the Liouville fractional integral and derivative operators, respectively, where α ( u ) , u ∈ R , is a general function taking values in ( 0 , 1 ) and satisfying some regularity conditions. The proof of D α ( ⋅ ) I α ( ⋅ ) f = f relies on a surprising integral identity. We also discuss properties of multifractional generalizations of fractional Brownian motion defined as white noise integrals X t = ∫ 0 t ( I α ( ⋅ ) B ̇ ) ( s ) d s and Y t = ∫ 0 t ( D − α ( ⋅ ) B ̇ ) ( s ) d s.

Analytic Nonstationary Processes

In 1952, Cramer introduced a class of nonstationary processes. This broad class of processes contains the important harmonizable and stationary classes of processes. The Cramer class can have additional structure imposed upon it through Cesaro summability considerations. This paper obtains conditions for the analyticity of the sample paths of a class of these processes. A sampling theorem is presented as a natural application.

Nonparametric estimation in a nonlinear cointegration type model

We derive an asymptotic theory of nonparametric estimation for a time series regression model Zt=f(Xt)+Wt, where {Xt} and {Zt} are observed nonstationary processes and {Wt} is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for {Xt} is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that {Wt} is a Markov chain satisfying some mixing conditions. The finite-sample properties of f̂(x) are studied by means of simulation experiments.

Stationarizing two classes of nonstationary processes by wavelet

The main purpose of this paper is to discuss the stationarity of two classes of nonstationary processes after wavelet transformations. Owing to the fact that the wavelet transformation possesses localization and implicit differencing property, the authors show that after wavelet transformation, the fractionally differenced process and the harmonizable periodically correlated process may be changed into stationary processes.

Invariance principle for a class of non stationary processes with long memory

We prove a functional central limit theorem for the partial sums of a class of time varying processes with long memory. To cite this article: A. Philippe et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Time-frequency analysis— [formula omitted]-stationary processes

Methods such as wavelets, short term Fourier transforms and time deformation [Gray and Zhang, 1988. On a class of nonstationary processes. J. Time Ser. Anal. 9(2), 133–154 and Gray, Vijverberg and Woodward, 2005. Nonstationary data analysis by time deformation. Commun. Statist. A, to appear] have been developed to analyze the time-frequency properties of a process where frequency changes with time. When the frequencies of a process are either monotonically increasing or monotonically decreasing with time, a rather general time deformation approach is to apply an appropriate Box–Cox transformation to the time axis for the given signal in order to obtain a new stationary data set. This data set can then be analyzed by standard methods. Processes which are transformed to a stationary process after Box–Cox transformation of the time scale are called G ( λ ) -stationary processes, where λ is the corresponding parameter of the Box–Cox transformation. In this paper it is shown that the standard concept of stationarity can be viewed as stationarity under a linear transformation of the index set, while transforming time (time deformation) by a non-linear monotonic transformation introduces the concept of stationarity on a non-linear index set. Thus the notion of stationarity is broadened considerably to allow stationarity on different scales. The method is illustrated with analysis of both simulated and real data. Finally, it is shown that such processes can be transformed to stationarity by sampling properly. The software for performing the analysis discussed in this paper can be downloaded from the website http://faculty.smu.edu/hgray/research.htm.

Cyclostationary Random Vibration of a Ship Propeller

A special class of nonstationary processes with periodically varying statistics, called cyclostationary (CS), is investigated. This process is encountered in many engineering problems involving rotating machinery, such as turbines, propellers, helicopter rotors, and diesel engines. The objective of this paper is to show that a CS model of the flow-induced excitation on a propeller can describe the physics of the problem more accurately than a traditional stationary model. The mean values of the hydrodynamic forces are calculated using the vortex panel method and the vortex theory of propellers. Considering the randomness in the axial and the tangential components of the wake velocity, we calculate the covariance matrix of the forces. This analysis shows that the hydrodynamic forces acting on the propeller are CS processes. Then we calculate the standard deviation (root mean square [RMS]) of the blade response. We show that the CS model predicts the timewise variation of the statistics of the excitation and the response (e.g., the RMS), including their peaks. A traditional stationary model cannot provide this information because it assumes constant statistics. Finally, a parametric analysis is performed to demonstrate the effects of the correlation structure of the velocity field behind a ship hull on the RMS of the blade deflection.

Some Results on the Wavelet Packet Decomposition of Nonstationary Processes

Wavelet/wavelet packet decomposition has become a very useful tool in describing nonstationary processes. Important examples of nonstationary processes encountered in practice are cyclostationary processes or almost-cyclostationary processes. In this paper, we study the statistical properties of the wavelet packet decomposition of a large class of nonstationary processes, including in particular cyclostationary and almost-cyclostationary processes. We first investigate in a general framework, the existence and some properties of the cumulants of wavelet packet coefficients. We then study more precisely the almost-cyclostationary case, and determine the asymptotic distributions of wavelet packet coefficients. Finally, we particularize some of our results in the cyclostationary case before providing some illustrative simulations.