Goodness of fit test for almost cyclostationary processes

Abstract

This paper is devoted to present a goodness of fit test for discrete-time almost cyclostationary models. The main strategy is based on the spectral support estimation and the application of multiple testing. The results of applying the presented method on simulated and real datasets confirm that the introduced approach acts well in view of power study.

Introduction

Stationarity assumption that is an essential assumption in time series modeling is not satisfied for many datasets, specially for datasets with periodic rhythm. In these situations, cyclostationary (CS) and almost cyclostationary (ACS) processes are good choices to model the rhythmic phenomena. The ACS class of non-stationary time series contains stationary and CS processes. The mean and auto-covariance functions of ACS are almost periodic. Against seasonal time series, the periodicity of ACS processes can not be removed by using differencing operators. The spectra of these processes are supported on lines that are parallel to the main diagonal, $T_j\left(x\right)=x\pm {\alpha }_j$, $j=1,2,\dots$, in spectral square $[0,2\pi )\times [0,2\pi )$. The theories and applications of ACS time series have been discussed and reviewed by many researchers such as the references [5], [6], [7], [9], [10], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [25], [26], [27], [28], [29], [33].

Many references considered spectral techniques for stationary, CS, ACS, and non-stationary processes. Brillinger [2] studied the consistency of the frequency-smoothed periodogram for the power spectrum of stationary processes. The references [4], [5], [6], [7], [12], [13], [14], [30], [31], [32] addressed the problem of spectral correlation density estimation for CS and ACS processes based on time or frequency smoothing the cyclic periodogram and consistent estimators of the spectral correlation density function were provided under some conditions and used these estimators to detect and characterize non-stationary processes. Lii and Rosenblatt [23], [24] provided consistent estimators of the spectral correlation density function for processes whose Loeve bi-frequency spectrum has the support on known lines with not necessarily unit slopes.

Definition 1 Almost Periodic Function [3]

A function $f(t):Z\to R$ is said to be almost periodic in $t\in Z$ if for any $\epsilon >0$, there exists an integer $L_{\epsilon }>0$ such that among any $L_{\epsilon }>0$ consecutive integers there is an integer $p_{\epsilon }>0$ such that:

$$su{p}_{t\in \mathrm{Z}}|f(t+p_{\epsilon })-f(t)|<\epsilon .$$

Definition 2 ACS Process [25]

A second order process $\{X_t:t\in Z\}$ is called ACS if the process has almost periodic mean, $\mu \left(t\right)=E\left(X_t\right)$, and autocovariance, $B(t,\tau )=cov(X_t,X_{t+\tau })$, at t, for every $\tau \in Z$.

As [25], the following assumptions have been considered in this work:

(A1) $\{X_t:t\in Z\}$ is a zero-mean and real-valued time series.

(A2) $X_t$ is an ACS process.

By these assumptions, the autocovariance function $B(t,\tau )$ can be represented by:

$$B(t,\tau )\sim \sum _{\omega \in W_t}a(\omega ,\tau )e^{i\omega t},$$

where

$$a(\omega ,\tau )=\underset{n\to \infty }{\lim }\left(\frac{1}{n}\sum _{j=1}^{n}B(j,\tau )e^{-i\omega t}\right),$$

and for fixed τ. Also as Corduneanu [3] and Hurd [12] indicated, the set $W_{\tau }=\{\omega \in [0,2\pi ):a(\omega ,\tau )\ne 0\}$ is a countable set of frequencies.

(A3) $W={\cup }_{\tau \in Z}{W}_{\tau }$, is a finite set and the spectra of $X_t$ is supported on parallel lines to the main diagonal, $T_j\left(x\right)=x\pm {\alpha }_j$, $j=1,2,\dots$, in spectral square $[0,2\pi )\times [0,2\pi )$. Thus we have:

$$B(t,\tau )=\sum _{\omega \in W}a(\omega ,\tau )e^{i\omega t},$$

and the spectral measure of $X_t$, will be supported on the set:

$$S=\bigcup _{\omega \in W}\{(\nu ,\gamma )\in {[0,2\pi )}^2:\gamma =\nu -\omega \}.$$

Furthermore, the coefficients $a(\omega ,\tau )$ are calculated by:

$$a(\omega ,\tau )=\underset{0}{\overset{2\pi }{\int }}{e}^{i\xi \tau }{r}_{\omega }\left(d\xi \right).$$

The measure $r_{\omega }$ can be identified with the restriction of the spectral measure of the process to the line $\gamma =\nu -\omega$, modulo 2π, where $\omega \in W$.

Remark

In the rest of the paper, all the equalities of frequencies are modulo 2π.

(A4) $r_0$ is an absolute continuous measure with respect to the Lebesgue measure.

Under this assumption and the assumption that $\sum _{\tau =-\infty }^{\infty }|a(\omega ,\tau )|<\infty$, Dehay and Hurd [4] showed for any $\omega \in W$, exists a spectral density function

$$f_{\omega }\left(\nu \right)=\frac{1}{2\pi }\sum _{\tau =-\infty }^{\infty }a(\omega ,\tau )e^{-i\nu \tau }.$$

Consequently, an ACS process with support on a finite number of cyclic frequencies is represented by:

$$X_t=\underset{0}{\overset{2\pi }{\int }}{e}^{-itx}\zeta \left(dx\right),t\in \mathrm{Z},$$

where ζ is a random spectral measure on $[0,2\pi )$ such that:

$$E\left(\zeta \left(d\theta \right)\overline{\zeta \left(d{\theta }^{\prime }\right)}\right)=0,(\theta ,{\theta }^{\prime })\notin S.$$

As Mahmoudi et al. [25] indicated, the spectral distribution and the density matrices of ζ, are defined by:

$$\mathit{F}\left(d\lambda \right)={\left[F_{k,j}\left(d\lambda \right)\right]}_{j,k=1,\dots ,m},$$

and

$$\mathit{f}\left(\lambda \right)=\frac{d\mathit{F}}{d\lambda }={\left[f_{k,j}\left(\lambda \right)\right]}_{j,k=1,\dots ,m},$$

respectively, where

$$F_{k,j}\left(d\lambda \right)=E\left(\zeta (d\lambda +{\alpha }_k)\overline{\zeta (d\lambda +{\alpha }_j)}\right),k,j=1,\dots ,m,$$

and $f_{k,j}$ is spectral density correspond to $F_{k,j}$.

Definition 3 Discrete Fourier Transform (DFT)

Let $X_0,\dots ,X_{N-1}$, are a sample of size N from ACS process $\{X_t:t\in Z\}$. The DFT of the finite sequence $X_0,\dots ,X_{N-1}$, is defined by:

$$d_X\left(\lambda \right)=N^{-1/2}\sum _{t=0}^{N-1}{X}_t{e}^{-it\lambda },\lambda \in [0,2\pi ).$$

Definition 4 Periodogram

Assume that we have a sample $X_0,\dots ,X_{N-1}$, from ACS process $\{X_t:t\in Z\}$. The periodogram of the finite sequence $X_0,\dots ,X_{N-1}$, is defined by:

$$I_X\left(\lambda \right)={\left|d_X\left(\lambda \right)\right|}^2,\lambda \in [0,2\pi ).$$

The distribution of the DFT and the periodogram of ACS processes have been widely studied by [17], [18], [20], [25]. Lii and Rosenblatt [23], [24] considered the spectral estimation of non-stationary but harmonizable processes including CS and ACS processes. For a single realization of the process with spectral supports on lines, they proposed periodogram-like and consistent estimators. Then they presented a constructive method to determine the lines of support of spectra under appropriate conditions.

The purpose of this work is to introduce a goodness of fit test about ACS processes. The motivation of this work is based on this fact that the ACS time series have spectral measure support on parallel lines to the main diagonal in the spectral square. The main idea is estimating the support of spectra and applying multiple testing. First, the spectral support of observed dataset is estimated by using the periodogram's asymptotic distribution. Then multiple testing strategies are applied to check the appropriateness of ACS model. In Section 2, based on the periodogram's asymptotic distribution, a goodness of fit test for ACS time series is presented. Then Monte Carlo simulations and real data analysis are applied to evaluate the performance of the presented approach.