Harmonizable processes form a wide class of nonstationary processes, which admit a convenient Fourier analysis and have spectral distributions characterized by correlated components. They have been proven to be useful in many fields of application, e.g., in communication, seismology, EEG data analysis, etc. In this paper, we introduce a parametric form for harmonizable processes, namely Harmonizable Vector AutoRegressive and Moving Average models (HVARMA). In the same spirit as standard VARMA models, they are derived as a unique solution of a difference equation based on a properly defined concept of harmonizable noise. We exhibit their spectral characteristics and derive results for least-squares parameter estimation in a fundamental case. We notably obtain, in some particular cases of the explosive regime, unusual asymptotic laws. Most importantly, we provide an effective way to generate realizations from those novel processes. Our modeling choice for harmonizable noise induces second-order stationary dependencies among the components of the spectral distribution of the harmonizable time series. We choose to model them using a periodic stationary VARMA process, resulting in the so-called HVARMA- model. We characterize its spectral properties and illustrate its ability to capture a vast range of nonstationary behaviors through examples of realizations using various models.

In this article, we show the mean square consistency for a generalized subsampling estimator based on the aggregation of the mean, median, and trimmed mean of some subsampling estimators for general non-stationary time series. Consistency requires standard assumptions, including the existence of moments and alpha-mixing conditions. We apply our results to the Fourier coefficients of the autocovariance function of periodically correlated time series. Furthermore, as in the i.i.d. case, we show that the generalized subsampling estimator satisfies Bernstein inequality and concentrates at an improved rate (under the condition of no or small bias) compared with the original estimator. Finally, we illustrate our results with some simulation data examples.

Spectrally correlated processes are harmonizable processes with spectral measures concentrated on a countable union of curves. Our study focuses on the spectral analysis of spectrally correlated processes, particularly in scenarios where these support curves take the form of lines with possible non-unit slopes. This subclass of spectrally correlated processes finds practical application, for example, in the problem of locating moving sources like aircraft, rockets, or other hostile jamming emitters emitting communication signals. Our research is focused on a frequency-smoothed periodogram along the support line as the spectral density function estimator. We derive its asymptotic distribution and based on that, we discuss the asymptotic properties of the introduced coherence estimator. Additionally, we formulate a consistent subsampling technique tailored for spectral analysis of spectrally correlated processes. We construct subsampling-based confidence intervals for spectral characteristics. Finally, we provide a numerical example illustrating a model relevant to applications in acoustics and communications.

Large-dimensional Markov chains appear in many models and many applications. In this paper, we introduce the Multidimensional Approximative Regenerative Block Bootstrap (MARBB), a bootstrap algorithm designed for high-dimensional Markov chains. We focus, in this paper, on a vector autoregressive (VAR(1)) process with a low-rank structure. We first use a reduction algorithm that transforms the original high-dimensional time series into a lower-dimensional Markov chain. Once the chain is reduced, we leverage the regenerative properties of Harris recurrent Markov chains within a general state space, using the Nummelin splitting technique to extend existing results from the one-dimensional settings to the multidimensional case. This approach enables the identification of approximate regeneration times of the lower-dimensional Markov chain, which in turn leads to the splitting of the original high-dimensional Markov chain into approximate regenerative blocks. These blocks are then used to bootstrap relevant statistics based on regenerative blocks. Finally, we give the MARBB consistency results, and we apply our algorithm to simulation data.

This paper introduces a new semiparametric model for continuous nonstationary processes with irregular cyclicities. The proposed model consists of a superposition of cosines with a nonstationary phase shift process and a stationary amplitude process. The asymptotic properties of the first- and second-order moments of the considered process are investigated. Estimators of the asymptotic mean and autocovariance functions are introduced. The performance of the autocovariance function estimator is examined in a simulation study. Finally, the model is applied to a real biomedical dataset.

This paper discusses the problem of choosing the optimal block length for two block bootstrap methods designed for periodically correlated processes. These are the Generalized Seasonal Block Bootstrap and the Extension of Moving Block Bootstrap. Two estimation problems are considered: the overall mean and the seasonal means. In both cases, the optimal block length is obtained by minimizing the mean squared error of the corresponding bootstrap variance estimator and in all cases it is proportional to the cube root of the sample size and should be a multiple of the period length plus one observation to avoid some bias. Finally, the results of the performed simulation are presented, in which optimal blocks lengths are estimated for several periodically correlated time series.

We study the estimation problem of the spectral density function for harmonizable non-stationary processes. More precisely, we consider spectrally correlated processes whose spectral measure has the support contained in the union of unknown lines with possibly non-unit slopes. We propose the frequency-smoothed periodogram along the estimated support line as an estimator of the spectral density function. We show the mean-square consistency of the proposed estimator. Additionally, we discuss the estimation of the support line in a specific model with its applications in locating a moving source. Finally, we present simulations confirming the proven results.

This paper discusses the problem of choosing the optimal block length for two block bootstrap methods designed for periodically correlated processes. These are the Generalized Seasonal Block Bootstrap and the Extension of Moving Block Bootstrap. Two estimation problems are considered: the overall mean and the seasonal means. In both cases, the optimal block length is obtained by minimizing the mean squared error of the corresponding bootstrap variance estimator and in all cases it is proportional to the cube root of the sample size and should be a multiple of the period length plus one observation to avoid some bias. Finally, the results of the performed simulation are presented, in which optimal blocks lengths are estimated for several periodically correlated time series.

Abstract We introduce a new approach for nonparametric spectral density estimation based on the subsampling technique, which we apply to the important class of nonstationary time series. These are almost periodically correlated sequences. In contrary to existing methods, our technique does not require demeaning of the data. On the simulated data examples, we compare our estimator of spectral density function with the classical one. Additionally, we propose a modified estimator, which allows to reduce the leakage effect. Moreover, in the supplementary materials, we provide a simulation study and two real data economic applications. Supplementary materials for this article are available online.

: In this paper, we develop tools for statistical inference on repli- cated realizations of spatiotemporal processes that are locally time-harmonizable. Our method estimates both the rescaled spatial time-varying Loève-spectrum and the spatial time-varying dual-frequency coherence function under realistic modeling assumptions. We construct confidence intervals for these parameters of interest using the Circular Block Bootstrap method and prove its consistency. We illustrate the application of our methodology on a dataset arising from an experiment in neuropsychology. From EEG recordings, our method allows studying the dynamic functional connectiv- ity within the brain associated to visual working memory performance

The aim of this paper is to provide readers with basic concepts and techniques for analysis of strongly periodically correlated fields (SCF) over Z^2 . We show that every SCF over Z^2 can be transformed into a coordinate-wise SCF (Fact 3.1) studied in [13]. The main result of the paper however is a specific decomposition of a strongly periodically correlated field (Theorem 4.1) which was not available for coordinate-wise SCFs. As consequences of the latter we obtain a description and an easy proof of existence of the spectral measures of an SCF (Theorem 5.1) as well as a functional description of an absolutely continuous SCF (Theorem 6.1). Most of the facts are explained in details and proved, with an exception of the proof of Theorem 6.1, which was too long for this publication and is left for a forthcoming paper.

In this paper consistency of the Frequency Domain Bootstrap for differentiable functionals of spectral density function of a linear stationary time series is discussed. The notion of influence function in the time domain on spectral measures is introduced. Moreover, the Frechet and Hadamard differentiability of functionals of spectral measures are defined in the time domain. Sufficient and necessary conditions for consistency of the FDB in the considered problems are provided and the second order correctness is discussed for some functionals. Finally, validity of the FDB for the empirical processes is considered. As an illustration the notions of quantile and range in the time domain are discussed. A simulation study is provided, in which performance of the FDB is analyzed.

In this paper a simulation comparison of the bootstrap confidence intervals for the coefficients of the autocovariance function of a periodically correlated time series is provided. Two bootstrap methods are used: the circular version of the Extension of Moving Block Bootstrap and the circular version of the Generalized Seasonal Block Bootstrap. The bootstrap pointwise and simultaneous confidence intervals for the real and the imaginary parts of the Fourier coefficients of the autocovariance function are constructed. The actual coverage probabilities, the average lengths and the average upper and lower quantiles values are calculated. On the basis of the performed simulation, the choice of the block length that is an integer multiple of the period length is advised when the Moving Block Bootstrap is used. Moreover, two methods for the block length choice designed for stationary data: the Minimum Volatility Method and the approach based on the logarithm of quantile are verified not to be valid for periodic nonstationary case. Finally, a heuristic method of the block length choice is proposed.

In this paper we consider almost cyclostationary processes with jitter effect. We propose a bootstrap approach based on the Moving Block Bootstrap method to construct pointwise and simultaneous confidence intervals for the Fourier coefficients of the autocovariance function of such processes. In the simulation study we show how our results can be used to detect the significant frequencies of the autocovariance function. We compare the behavior of our approach for jitter effects caused by perturbations from two distributions, namely uniform and truncated normal. Moreover, we present a real data application of our methodology.

This research is dedicated to the study of periodic characteristics of periodically correlated time series such as seasonal means, seasonal variances and autocovariance functions. Two bootstrap methods are used: the extension of the usual Moving Block Bootstrap (EMBB) and the Generalised Seasonal Block Bootstrap (GSBB). The first approach is proposed, because the usual Moving Block Bootstrap does not preserve the periodic structure contained in the data and cannot be applied for the considered problems. For the aforementioned periodic characteristics the bootstrap estimators are introduced and consistency of the EMBB in all cases is obtained. Moreover, the GSBB consistency results for seasonal variances and autocovariance function are presented. Additionally, the bootstrap consistency of both considered techniques for smooth functions of the parameters of interest is obtained. Finally, the simultaneous bootstrap confidence intervals are constructed. A simulation study to compare their actual coverage probabilities is provided. A real data example is presented.

In this paper we consider a continuous almost periodically correlated process {X( t), t is an element of R} that is observed at the jump moments of a stationary Poisson point process {N( t), t >= 0}. The processes {X( t), t. R} and {N( t), t >= 0} are assumed to be independent. We define the kernel estimators of the Fourier coefficients of the autocovariance function of X( t) and investigate their asymptotic properties. Moreover, we propose a bootstrap method that provides consistent pointwise and simultaneous confidence intervals for the considered coefficients. Finally, to illustrate our results we provide a simulated data example.

In this paper a subsampling approach for nonstationary time series with a non-zero mean function is proposed. It is applied for periodically and almost periodically processes. Two statistical tests are constructed. An example with real data is presented.

Methods for estimating parameters of periodic autoregressive moving average PARMA systems when the periodic coefficients are represented by Fourier series remain to be employed in practice and still offer problem areas for future research. As pointed out by early writers on the subject, such as Hannan (1955) and Jones and Brelsford (1967), if the periodic variations are smooth within the fundamental period, as might be expected in many physical time series, a substantial reduction in the number of estimated parameters may be realized by setting many of the Fourier coefficients to be zero; this restricts the estimated solutions to a subspace. The identification problem becomes the determination of lags and frequencies with significant amplitudes. While progress has been made in this area, improvements and new methods are needed. In reviewing the development of Fourier‐PARMA methods, we naturally view many of the main advances in PARMA time series analysis under the usual parameterization. Two simulations are presented that demonstrate further potential and open problems associated with the Fourier methods. WIREs Comput Stat 2016, 8:130–149. doi: 10.1002/wics.1380This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

In this paper we review the methods of solving PARMA systems when the periodic coefficients are represented by Fourier series. This method, introduced by Jones and Brelsford (1967), remains a method employed in practice yet still offers problem areas for future research. The primary benefit of the method is that for PARMA systems in which the periodic variations are smooth within the fundamental period, a substantial reduction in the number of estimated parameters may be realized by forcing many of the Fourier coefficients to be zero, hence restricting estimated solutions to a subspace. While reviewing the development of Fourier PARMA methods we naturally view many of the main advances in PARMA time series analysis under the usual parameterization. Two simulations are presented that demonstrate further potential and open problems associated with the methods.