Periodic Correlation Research Articles

Robust coherent and incoherent statistics for detection of hidden periodicity in models with non-Gaussian additive noise

We address the issue of detecting hidden periodicity when the signal exhibits periodic correlation, but is additionally affected by non-Gaussian noise with unknown characteristics. This scenario is common in various applications. The conventional approach for identifying periodically correlated (PC) behavior involves the frequency domain-based analysis. In our investigation, we also employ such an approach; however, we use a robust version of the discrete Fourier transform incorporating the Huber function-based M-estimation, unlike the classical algorithm. Building upon this approach, we propose robust coherent and incoherent statistics originally designed to identify hidden periodicity in pure PC models. The novelty of this paper lies in introducing robust coherent and incoherent statistics through the application of the robust discrete Fourier transform in classical algorithms and proposing a new technique for period estimation based on the proposed methodology. We explore two types of PC models and two types of additive noise, resulting in PC signals disturbed by non-Gaussian additive noise. Detecting hidden periodicity in such cases proves to be significantly more challenging than in classical scenarios. Through Monte Carlo simulations, we demonstrate the effectiveness of the proposed robust approaches and their superiority over classical. To further substantiate our findings, we analyze three datasets in which hidden periodicity had previously been confirmed in the literature. Among them, two datasets correspond to the condition monitoring area, being a main motivation of our research.

New estimation method for periodic autoregressive time series of order 1 with additive noise

The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.

On the spectral coherence between two periodically correlated processes

We introduce a general class of multivariate periodically correlated processes and their corresponding time‐domain and spectral‐domain characterizations. A spectral coherence based on the Hilbert–Schmidt inner product of the Fourier transforms is introduced to measure the dependence of two periodically correlated (PC) processes. An estimator for the spectral coherence is introduced and studied. A hypothesis on the presence of significant dependence is formulated and the corresponding testing procedure established. Numerical illustrations on the performance of the spectral coherence and its estimator are given using simulated and real PC time series.

Semi-Lévy driven continuous-time GARCH process

Continuous-time GARCH (COGARCH) processes are one of the influential and successful models in financial data analysis. In contrast to such stationary process, in this paper we study a class of COGARCH processes driven by semi-Lévy process (SL-COGARCH) that has periodically correlated (PC) increments. Under sufficient conditions the strictly periodically stationarity of the state and volatility processes are shown. We verify that the increments with constant length of the SL-COGARCH process constitute a discrete-time PC process. To justify this property, we use simulations of the SL-COGARCH(1,3) process and evaluate its increments. Then we provide the sample spectral coherence test to show the PC behavior of this discrete-time process. We apply the SL-COGARCH(2,2) process to the Dow Jones Industrial Average indices and show that this model provides better prediction of the squared log returns in compare to the retrieved Lévy driven COGARCH method.

Investigating the ability of periodically correlated (PC) time series models to forecast the climate index

Considering the importance of climate change and its effects, especially in recent decades, the forecast of future climate conditions can be useful in managing and planning to reduce its impacts. The hypothesis of this study is that in periodic data series, such as seasonal (3-month) and monthly data, the periodically correlated time series models (PC) have a more ability to predict data series. Therefore, in this research using climatic data series of 18 synoptic stations during 1967–2017 over Iran (with different climate conditions and suitable spatial distribution), initially, the climate conditions based on united nation environmental program (UNEP) aridity index (UAI) in seasonal time scale were assessed, and then, using PC models including Periodic Autoregressive Moving Average (PARMA), Periodic Moving Average (PMA) and Periodic Autoregressive (PAR) the UAI from 2018 to 2030 were predicted and finally, for increasing the applicability of the results of the research the trend of changes in UAI data series on observed data (during 1967–2017) and observed and forecasted data (during 1967–2030) were assessed and compared. The results showed calculated UAI at all stations were periodical (significantly at 0.05% level) and among different PC time series models such as PARMA, PMA and PAR, the PAR model with order 22 [PAR (22)] was the best time series model that fitted in all data series at all stations. The R-squared between the observed and the simulated [based on PAR (22) model] UAI from 1967 to 2017 at all stations were more than 0.610 (significantly at 0.01% level) and the R-squared between the observed and the predicted [based on PAR (22) model] UAI from 2013 to 2017 for validating fitted models at all stations were more than 0.659 (significantly at 0.01% level). Trend assessment of climate conditions showed the climate conditions will be dryer at 94.44% (17 out of 18) of stations (only at Gorgan, the climate conditions will be more humid).

Testing the difference between spectral densities of two independent periodically correlated (cyclostationary) time series models

In this work, the asymptotic distribution for the discrete Fourier transform of periodically correlated (PC) processes is applied to test the equality of two PC time series. Then the performance of the proposed method is investigated through the Monte Carlo simulations.

On infinite dimensional periodically correlated random fields: Spectrum and evolutionary spectra

Infinite dimensional periodically correlated (PC) random fields are studied in spectral domain. A spectral characterization is given and harmonizability is established. The covariance operator is characterized where it is observed that an infinite dimensional PC field is a two-dimensional Fourier transform of a spectral random measure. Also, an evolutionary spectral representation and a space-dependent spectral density are given.

Certain Periodically Correlated Multicomponent Locally Stationary Processes

By introducing $X^{ls}(t)$ as a random mixture of two stationary processes where the time dependent random weights have exponentially convex covariance, we show that this process has a multicomponent locally stationary covariance function in Silverman's sense. We also define $X^p(t)$ as a certain continuous time periodically correlated (PC) process where its covariance function is generated by the covariance function of a discrete time through defining some simple random measure on a real line. We also impose a biperiodic correlation for this PC process with $X^{ls}(t)$. The existence of such a random measure is proved. Then by defining $X(t)=X^{ls}(t)+X^p(t)$ as a certain PC multicomponent locally stationary process, the covariance structure and time varying spectral representation of such processes are characterized.

Causal Wiener filter banks for periodically correlated time series

A causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC) time series is developed. By converting a PC time series into a vector-valued wide-sense stationary (WSS) time series, the existing literature on factorization of spectral density matrices may be utilized. However, because PC analytic and equivalent baseband signals are generally complex improper, spectral factorization algorithms must be adapted to the improper case. Based on the factorization of the spectral density matrix for the equivalent WSS vector process, causal synthesis and whitening filters for PC time series can be built. These techniques are exploited to implement a causal cyclic Wiener filter as a multirate filter bank or an equivalent polyphase structure. This filter bank is shown to be an efficient equivalent implementation of a frequency shift (FRESH) filter. Therefore, the results derived in this paper also show how to build a causal FRESH cyclic Wiener filter.

Periodograms asymptotic distributions in periodically correlated processes and multivariate stationary processes: An alternative approach

Discrete time periodically correlated (PC) processes are viewed as the processes with time-dependent spectra. This, together with an auxiliary operator which is defined here is employed to apply classical results on the asymptotic distribution of the periodogram of the univariate white noise (innovations) to derive the asymptotic distributions of the periodograms for the PC processes and also for the multivariate stationary processes. We assume only the continuity and positive definiteness of the spectral densities together with the independence of the innovations.

Characterization of the Spectra of Periodically Correlated Processes

A complete characterization of the spectrum of a locally square integrable periodically correlated (PC) processes is obtained. The result makes use of the author's recent theorem establishing a one to one correspondence between PC processes and a certain class on infinite dimensional stationary processes. In terms of distributions it is proved that the Fourier transform of a positive definite distribution on the plane which is the sum of complex uniformly bounded measures supported on equidistant lines parallel to diagonal is a locally square integrable function.

PERIODIC CORRELATION IN STRATOSPHERIC OZONE DATA

Abstract. A 50‐year time series of monthly stratospheric ozone readings from Arosa, Switzerland, is analyzed. The time series exhibits the properties of a periodically correlated (PC) random sequence with annual periodicities. Spectral properties of PC random sequences are reviewed and a test to detect periodic correlation is presented. An autoregressive moving‐average (ARMA) model with periodically varying coefficients (PARMA) is fitted to the data in two stages. First, a periodic autoregressive model is fitted to the data. This fit yields residuals that are stationary but non‐white. Next, a stationary ARMA model is fitted to the residuals and the two models are combined to produce a larger model for the data. The combined model is shown to be a PARMA model and yields residuals that have the correlation properties of white noise.

Ergodic behavior and estimation for periodically correlated processes

A complex second-order stochastic process { X( t); t ∈ R } is periodically correlated (PC) when its mean function m( t) and covariance R( s, t) are periodic with the period T. In the following note the relationship between the hierarchy of chaos and estimation for the stochastic process is investigated. The results are based on the Correspondence Principle for the PC processes.

Correlation theory of almost periodically correlated processes

A second-order stochastic process X is called almost periodically correlated (PC) in the sense of Gladyshev if its mean function m( t) and covariance R( t + τ, t) are uniformly continuous with respect to t, τ and are almosst periodic functions of t for every τ. We show that the mean uniformly almost periodic processes discussed by Kawata are also almost PC in the sense of Gladyshev. If X is almost PC, then for each fixed τ the function R( t + τ, t) has the Fourier series R(t+τ, t) ∼ ∑ λ ϵ ∇τ a(λ, τ) exp(iλt) and a(λ τ) = lim A→∞ 1 2A ∫ A+C −A+C R(t+τ, t) exp(−iλt) dt exists for every λ and τ, independently of the constant c. Assuming only that a(λ, τ) exists in this sense for every λ and τ, we show a(λ, τ) is a Fourier transform a(λ τ) = ∫ R exp(iyτ) r lambda;(dy) if and only if a(0, τ) is continuous at τ = 0; under this same condition, the set Λ = {λ: a(λ, τ) ≠ 0 for some τ} is countable. We show that a strongly harmonizable process is almost PC if and only if its spectral measure is concentrated on a countable set of diagonal lines S λ = {( γ 1, γ 2): γ 2 = γ 1 − λ}; further, one may identify the spectral measure on the k th line with the measure r λ k appearing in (iii). Finally we observe that almost PC processes are asymptotically stationary and give conditions under which strongly harmonizable almost PC processes may be made stationary by an independent random time shift.