Best Lag Window for Spectrum Estimation of Law Order MA Process

In this article, we investigate spectrum estimation of law order moving average (MA) process. (e main tool is the lag window which is one of the important components of the consistent form to estimate spectral density function (SDF).We show, based on a computer simulation, that the Blackman window is the best lag window to estimate the SDF of MA(1) and MA(2) at the most values of parameters βi and series sizes n, except for a special case when β � − 1 and n≥ 40 in MA(1). In addition, the Hanning–Poisson window appears as the best to estimate the SDF ofMA(2) when β1 � β2 � − 0.5 and n≥ 40.


Introduction
A set x t of numerical data (observations) made sequentially in time t is called time series [1]. ere are some important processes of a time series: autoregressive, moving average, and autoregressive-moving average processes.
Spectral analysis can be defined as a process that assigns power versus frequency. One of the time series analysis techniques is spectral analysis. e object of spectral analysis is to estimate and study the spectrum of the time series processes for the phenomena of physics and engineering [2]. e spectrum estimation methods can be classified into parametric and nonparametric methods [3]. e consistent estimate of spectral density function f(ω) is the most important nonparametric spectral analysis method, which depends on lag window λ T (v) and truncation point T [4].
Window functions are used in the estimation of power spectra and bispectra in order to ensure the consistency of the periodogram and the Fourier-type bispectrum estimation methods. A three-dimensional optimum bias lag window is introduced in the estimation of the 4th-order cumulant spectrum, also called trispectrum, which is estimated from the threedimensional Fourier transform of the 4th order cumulants [5].
Zhongsheng et al. [6] suggested that using windows is one important way to improve bispectrum estimation and also an appropriate window function can be used to reduce variance and suppress noise, but it was noticed that sidelodes in a spectrum of window functions can be ended up in spectrum leak. us, one urgent problem which needed to be solved for the application of bispectrum was how to find one appropriate window. He combined a new lag window with Hanning-Poisson window without sidelodes, which is used for nonparametric bispectrum estimation instead of rectangle window. When the spatial location area increases becoming extremely large, it is very difficult [7], or not possible, to evaluate the covariance matrix determined by the set of location distance even for gridded stationary Gaussian process. To alleviate the numerical challenges, he did construct a nonparametric estimator called periodogram of spatial version to represent the sample property in the frequency domain because periodogram requires less computational operation by fast Fourier transform algorithm. Under some regularity conditions on the process, he investigated the asymptotic unbiasedness property of the periodogram as estimator of the spectral density function and achieved the convergence rate. e basic concepts given in Sections 2-5 present white noise, moving average process of order q and their properties, spectral density function (SDF) on general and SDF of MA(q), and the consistent estimate of SDF, and some important lag windows are reviewed. Section 6 presents a simulation for comparison between the SDF and the consistent estimate of SDF.

White Noise
A purely random process ϵ t , t ∈ Z + is called white noise (Gaussian noise) if it consists of a sequence of uncorrelated independent identically distributed (i.i.d) random variables [3], with mean μ ϵ � 0, variance var(ϵ t ) � σ 2 ϵ , and the autocovariance function In addition, the autocorrelation function ρ v is

Moving Average Process
A stochastic process x t , t ∈ Z + is called moving average process of order q and denoted by MA(q). is is given by where ϵ t is the white noise with mean zero and covariance σ 2 ϵ and β i . β i � 0, i > q is the coefficient of the process. e statistical properties of MA(q) is such that ϵ t be the uncorrelated random process. As a result, the autocovariance function R(v) cuts off after a point x t , t > q, and that implies cov(x t , x t+v ), |v| > q and R(v) as e autocorrelation function Note that μ x , σ 2 x , R(v), and ρ v are constants, the finite does not depend on time t for any finite order q. us, the moving average process MA(q) of finite order q is a stationary process [2]. As a special case, MA(1), And μ x � 0, σ 2 x � var(x t ) � σ 2 ϵ (β 2 0 + β 2 1 ), and the autocovariance and autocorrelation functions are given by So, MA (2) is defined as And, the expected value μ x � E(β 0 ϵ t + β 1 ϵ t− 1 + β 2 ϵ t− 2 ) � 0, and the variance is It is clear that σ 2 x � R(v � 0), and the autocovariance and autocorrelation functions are given by

Spectral Density Function
If x t , t ∈ Z is a discrete stochastic process with autocorrelation function ρ v [3,8], a spectral density function (SDF) f(ω) is defined as a Fourier transform of autocorrelation function ρ v and is given as . e formula is rewritten as Since autocorrelation is an even function [9], it implies Hence, 4. 1. SDF of MA(q). Let x t be the moving average process defined in (3) with autocovariance function R(v) and autocorrelation function ρ v . e spectral density function f(ω) defined in (13) is given as From (7), Hence, As a special case, MA (1), the spectral density function will be And spectral density function of MA (2) is given by

The Consistent Estimate of SDF
Let X t , t ∈ Z be a real-valued, weakly stationary, discrete stochastic process (time series) with zero mean and autocovariance function R v with lag v and autocorrelation function ρ v [3]. e consistent estimate of R v and ρ v are If X t is a stochastic process of size n, then the consistent form to estimate the spectral density function is [2] f(ω) � 1 2π where T is the truncation point 0 ≤ T ≤ n and λ T (v) is the lag window, which weighting the autocorrelation function. e consistent estimate of SDF depends on two important sides, select an appropriate value of a truncation point T and an appropriate lag window λ T (v).
ere are a lot of lag windows suggested by researchers [3,6,[10][11][12]. Table 1 contains the most important of lag windows as shown in previous papers.

Windows Different values of parameter
(28) (7) e criterion used to evaluate the windows performance was the mean square error (MSE) calculated with the following formula: where k and L were defined in (3) and (5), respectively, and f j (ω i ) is the consistent estimate of the SDF formula in (26). Table 6: MSE values of MA(1) spectrum estimation with size (n � 500).  Table 7: MSE values of MA(1) spectrum estimation with size (n � 1000).

Results
(1) For the first order of moving average process MA(1), when we study the different values of parameter β as shown in Table 2 and size n of the process, we get the results given in Tables 4-8. (2) For the second order of moving average process MA (2), when we study the different values of parameters β 1 , β 2 and size of process n, we get the results given in Tables 9-13.

Conclusion
(1) In MA(1) with the different parameters and series sizes, the best lag window which gives the minimum mean square error (MSE) between the SDF f(ω) and the consistent estimate of SDF f(ω), and the results in Tables 4-8 can be summarized in Table 14. (2) e results in Tables 9-13 shows that, in MA (2), with the following different parameters and series sizes, the best lag window which gives the minimum mean 6.317333361 6.316064139 6.319364383 6.332573978 6.345797365 6.349105366 6.347832956 6.345797365 Triangular Table 9: MSE values of MA(2) spectrum estimation with size (n � 10).

Different values of parameters
square error (MSE) between the SDF f(ω) and the consistent estimate of SDF f(ω) is shown in Table 15. (3) In MA(1) with series sizes n < 40 and any value of parameter β, the best lag window which gives the minimum mean square error (MSE) between the SDF f(ω) and the consistent estimate of SDF f(ω) is the Blackman window, as shown in Table 9. (4) In MA(1) with series sizes n ≥ 40 and parameter β � − 1 or belongs to neighborhood − 1 with radius 0.3, the best lag window which gives the minimum mean square error (MSE) between the SDF f(ω) and the consistent estimate of SDF f(ω) is the Hanning-Poisson window, as shown in Table 9. (5) Blackman window is the best window to estimate the SDF for white noise ϵ t , where MA(1) � ϵ t when β � 0, as shown in Table 9. (6) In MA(2) with series sizes n < 40 and any values of parameters β 1 , β 2 , the best lag window which gives the minimum mean square error (MSE) between the  (2) spectrum estimation with size (n � 100).
Data Availability e data included in the article were calculated using the MATLAB software and the method of calculation of the data is given in Section 6.

Conflicts of Interest
e authors declare that they have no conflicts of interest.