Testing the equality of several independent stationary and non-stationary time series models

Alexandria Engineering Journal (2021) 60, 1767–1775

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

Testing the equality of several independent stationary and non-stationary time series models with fractional Brownian motion errors Mohammad Reza Mahmoudi b, Dumitru Baleanu c,d, Sultan Noman Qasem e,f, Amirhosein Mosavi g,h,*, Shahab S. Band a,i,* a

Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam Department of Statistics, Faculty of Science, Fasa University, Fasa, Fars, Iran c Department of Mathematics, Faculty of Art and Sciences, Cankaya University, Balgat, 06530 Ankara, Turkey d Institute of Space Sciences, Magurele-Bucharest, Romania e Computer Science Department, College of Computer and information Sciences, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia f Computer Science Department, Faculty of Applied Science, Taiz University, Taiz, Yemen g Environmental Quality, Atmospheric Science and Climate Change Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam h Faculty of Environment and Labour Safety, Ton Duc Thang University, Ho Chi Minh City, Vietnam i Future Technology Research Center, College of Future, National Yunlin University of Science and Technology 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan, ROC b

Received 5 February 2020; revised 20 September 2020; accepted 14 November 2020 Available online 28 November 2020

KEYWORDS Fractional Brownian motion; Friedman test; Repeated measures; Simulation; Simultaneous inference; Test of hypothesis; Time series; RDI

Abstract This work is devoted to apply the parametric and nonparametric techniques to construct test of hypothesis about the equality of the probabilistic behaviors of several time series models with fractional Brownian motion errors fitted on several independent datasets. The accuracy and power of the introduced method are studied using the simulated and real datasets. The results indicate that the introduced approach is more powerful than other alternative approaches, in non-stationary cases. Ó 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

* Corresponding authors.

E-mail addresses: mahmoudi.m.r@fasau.ac.ir (M.R. Mahmoudi), dumitru@cankaya.edu.tr (D. Baleanu), SNMohammed@imamu.edu. sa (S.N. Qasem), amirhosein.mosavi@tdtu.edu.vn (A. Mosavi), shamshirbands@yuntech.edu.tw (S. S. Band).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

1. Introduction For many years, various data analysis techniques are applied to model the natural phenomena such as biology, climatology, economic, electronic, finance, hydrology, management and many. These techniques can usually divide in three categories:

https://doi.org/10.1016/j.aej.2020.11.025 1110-0168 Ó 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1768

M.R. Mahmoudi et al.

artificial intelligence techniques such as deep learning and machine learning [1–5], mathematical techniques such as optimization and numerical analysis [6–11], and statistical techniques such as time series analysis and regression analysis [12–21]. In applications, the scientists may be interested in comparing the parameters of the populations or comparing and classifying the models can be fitted on one population or some populations; for example, the comparison of the parameters of two independent populations such as proportions [22], means [23], variances [24], skewnesses [25], correlations [26], the comparison of two independent regression models [27– 29] and the comparison of several independent regression models [30–32].

When researchers face with the time series analysis on some populations, perhaps they are interested in investigating that the differences between the stochastic mechanisms of the observed time series are significant or not. Really, they may be interested in making comparisons between the probabilistic behaviors of several time series models fitted on several independent datasets. There are many studies on comparison, classification, discrimination, and clustering of two or several stationary time series models [33–48], cyclostationary (periodically correlated) time series models [49,50], almost cyclostationary time series models [51,52], and non-stationary time series models [53–57].

It can be observed that all of the related works have focused on the processes with white noise errors or the comparison of only two processes. Therefore, the comparison, classification, discrimination, and clustering of two or several stationary or non-stationary processes with fractional Brownian motion (fBm) errors were lacked. In this research, for the first time, we are interested in investigating that the differences between the stochastic mechanisms of the several observed time series with fBm errors are significant or not. Really, we are interested in making comparisons between the probabilistic behaviors of several time series models with fBm errors fitted on several independent datasets.

tion of H0 means that at least two of k time series models have significant difference. The acceptance of H0 is also means that there are not significant differences between the time series models. To test H0 the following steps should be ran: Step (1): For each dataset, the best time series model is estimated by

iÞ ðiÞ bðiÞ ¼ fbi Xðt 1 X ; Xt 2 ; ; i ¼ 1; ; k: ð2Þ t These k models can have same or different formulas.

Step (2): For each dataset, the predicted values of Xt is estimated based on all k models by

iÞ ð iÞ bði;jÞ ¼ fbj Xðt 1 X ; Xt 2 ; ; i; j ¼ 1; ; k: t bði;jÞ is the predicted value of dataset XðiÞ In other words, X t t based on best time series model for dataset Xðt jÞ .

Now, we have k predicted values datasets, for each dataset Xðt iÞ ; i ¼ 1; ; k:

bði;1Þ , . . ., Step (3): For each dataset Xðt iÞ ; i ¼ 1; ; k , if X t bði;kÞ are normal, the parametric tests such as repeated X t bði;1Þ , . . ., X bði;kÞ are not normal, the measures [58], and if X t t nonparametric methods such as Friedman test [59] are applied ði;1Þ

ði;1Þ

bði;kÞ , . . ., X bði;kÞ . b ; ; X b ; ; X by using the couples X t;1

Suppose BH ðtÞ is a fBm with Hurst index H 2 ð0; 1Þ, given by Z t 1 1

ðt sÞH 2 dBðsÞ; BH ðtÞ ¼ C H þ 12 0

where B is Brownian motion process, C is gamma function, and integration is with respect to the white noise measure dBðsÞ.BH ðtÞ and BH ðsÞ are two processes with means of zero and following auto-covariance function:

1 2H cðs; tÞ :¼ CovðBH ðsÞ; BH ðtÞÞ ¼ jtj þ jsj2H jt sj2H : 2 The case H ¼ is called standard Brownian motion or Wiener process. The cases H < 12 and H > 12 show the negative and positive autocorrelation between increments, respectively.

For k given independent datasets, suppose k time series models (stationary or non-stationary) are given by

ðiÞ ð iÞ ðiÞ ð1Þ Xðt iÞ ¼ fi Xt 1 ; Xt 2 ; þ BH ðtÞ; i ¼ 1; ; k: 1 , 2

where BH ðtÞ; i ¼ 1; ; k; are independent fBm errors.

t;1

t;ni

t;ni

Nonparametric methods are commonly applied when the observed dataset do not follow the parametric assumptions.

Friedman test that can be applied as nonparametric alternative for the repeated measures, compare dependent datasets that are correspond to the same subjects.

Step (4): For each dataset, the P-value of repeated measures test (or, Friedman test) is computed.

ð1Þ

ð1Þ

hypothesis test H0 : f1 Xt 1 ; Xt 2 ; ¼ ¼

ðkÞ ðkÞ fk Xt 1 ; Xt 2 ; is can be represented by following multiple

2. Method

ðiÞ

In practice, the aim is to test the null hypothesis ð1Þ ð1Þ ðkÞ ðkÞ H0 : f1 Xt 1 ; Xt 2 ; ¼ ¼ fk Xt 1 ; Xt 2 ; . The rejec-

The

tests

0 ð iÞ ðiÞ ð jÞ ðjÞ H0 : fi Xt 1 ; Xt 2 ; ¼ fj Xt 1 ; Xt 2 ; for i; j ¼ 1; ; k: Different multiple testing techniques such as the Bonferroni correction, locally false discovery rate (LFDR) and qvalue strategies [60–64] can be used to perform this test.

Step (5): The multiple tests procedure is applied to testH0 : 3. Simulation study To study the accuracy of introduced procedure, numerous data sets (stationary and non-stationary) are simulated and analyzed. For different sample sizes of ðn1 ; n2 Þ ¼ fð100; 50Þ; ð150; 75Þ; ð200; 100Þ; ð500; 250Þ; ð1000; 500Þ; ð2000; 1000Þg; the estimated size (b a ) and power ð b p Þ of the introduced approach are computed by T ; 10000

Testing the equality of several independent stationary and non-stationary time series models such that T is equal to the number of runs that H0 is rejected.

The simulations are accomplished after 1000 runs and using the R 3.5.3 software. Also the performance of introduced procedure is compared with the methods that were given by [53– 55]. The outlines of their strategies are as following:

1769

PXðtkÞ xj NPXðkÞ xp ¼ : t d XðiÞ Var t

(iii) Compute

3.1. Mahmoudi et al. [55] (i) For each dataset, find the best ARMAð p; qÞ model that minimizes the Akaikee Information Corrected Criterion (AICC) or Bayesian Information Criterion (BIC) (ii) If ARMAðp1 ; q1 Þ and ARMAðp2 ; q2 Þ are respectively best models to fit first and second time series, then assume p ¼ maxfp1 ; p2 g, q ¼ maxfq1 ; q2 g and fit ARMAð p; qÞ model to each time series. (iii) Let Vi as asymptotic covariance matrix given by "

# 1 0 0 E Uðt iÞ VtðiÞ E Uðt iÞ UtðiÞ ; Vi ¼ r

0 0 E Vðt iÞ UtðiÞ E Vðt iÞ VtðiÞ

0

0 ð iÞ ð iÞ where Uðt iÞ ¼ Uðt iÞ ; ; Utþ1 p ; Vðt iÞ ¼ Vðt iÞ ; ; Vtþ1 q and

XLNP;i ¼

m

1 X log NPXðiÞ xp ; t m p¼1

and S2LNP;i ¼

m

2

1 X log NPXðiÞ xp XLNP;i ; t m p¼1

where m is the number of ordinates for periodogram of the series with shorter size. (iv) Compute the test statistic

2

fUt g, fVt g are the autoregressive processes,

DNP

/ðBÞUt ¼ Zt ;

Pm log NPXð2Þ xp p¼1 log NPXðt 1Þ xp t r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼

2 2 SLNP;1 þ SLNP;2 =m 1 m

The test statistic DNP asymptotically follows a standard normal distribution. Consequently, for a given size a, H0 is rejected if jDNP j > Z1 a=2 :

and hðBÞVt ¼ Zt : [65].

Example 1:. Assume the AR(1) model given by b1 þ V b 2. (iv) Compute the matrix R by R ¼ V (v) Compute the test statistic

ðiÞ

ð iÞ

ð iÞ

Xðt iÞ /1 Xt 1 ¼ BH ðtÞ; i ¼ 1; 2: ð1Þ

0

T0 ¼ nðbÞ R 1 ðbÞ; ðiÞ 0 0 ðiÞ ð iÞ b ; ; / b ðiÞ ; b b b ðiÞ ; b h ;

; h ¼ / h ðiÞ ; i ¼ 1; 2; where b bi ¼ / 1 p 1 q are

the

maximum

likelihood estimators of

0 b1 b b2. bi ¼ /ðiÞ ; hðiÞ ; i ¼ 1; 2; and b ¼ b Under H0, T0 is asymptotically distributed as a chi square distribution with ðp þ qÞ degrees of freedom, v2 ðp þ qÞ. Consequently, for a given size a, H0 is rejected ifT0 > v2a ðp þ qÞ: 3.2. Caiado et al. [53–54] n

o ðiÞ (i) For time series X t , compute the periodogram points by

2 ni

X

1 ðiÞ itxj Xt e PXðiÞ xj ¼ ð2pni Þ ; t

t¼1 where xj ¼ 2pj=ni , for j ¼ 1; ; mi ; with mi ¼ ½ni =2 , the largest integer ni =2, and xj is a frequency in ½ p; p . (ii) Compute normalized periodogram points by

For first and second models, assume /1 ¼ 0:5 and 2 f0:1; 0:3; 0:5; 0:7; 0:9g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g:

ð2Þ /1

Example 2:. Assume the MA(1) model given by ðiÞ ðiÞ

ð iÞ

Xðt iÞ ¼ eðt iÞ þ h1 et 1 ; BH ðtÞ; i ¼ 1; 2: ð1Þ

For first and second models, assume h1 ¼ 0:5 and 2 f0:1; 0:3; 0:5; 0:7; 0:9g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g:

ð2Þ h1

Example 3:. Assume the ARMA(1,1) model given by ðiÞ

ð iÞ

ðiÞ ðiÞ

ðiÞ

Xðt iÞ /1 Xt 1 ¼ eðt iÞ þ h1 et 1 ; BH ðtÞ; i ¼ 1; 2: ð1Þ

ð1Þ

For first and second models, assume /1 ¼ 0:2, h1 ¼ 0:5 ð2Þ ð2Þ and /1 ¼ 0:2, h1 2 f 0:9; 0:7; 0:5; 0:3; 0:1g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g: Example 4:. Assume the PAR(1) model ðiÞ

ð iÞ

Xðt iÞ ¼ /ðt iÞ Xt 1 þ BH ðtÞ; i ¼ 1; 2; where

1770

M.R. Mahmoudi et al.

/ðt iÞ ¼ 0:6 þ /ðiÞ cosð2pt=TÞ ð1Þ

For first and second models, assume /1 ¼ 0:5 and 2 f0:1; 0:3; 0:5; 0:7; 0:9g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g:

ð2Þ /1

Example 5:. Assume the PMA(1) model ðiÞ

ðiÞ

Xt ¼ BH ðtÞ þ hðt iÞ BH ðt 1Þ; i ¼ 1; 2; where hðt iÞ

¼ 1 þ hðiÞ cosð2pt=TÞ ð1Þ

For first and second models, assume h1 ¼ 0:5 and ð2Þ h1 2 f0:1; 0:3; 0:5; 0:7; 0:9g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g: Example 6:. Assume the PARMA(1,1) model ð iÞ

ð iÞ

ð iÞ

Xðt iÞ /ðt iÞ Xt 1 ¼ BH ðtÞ þ hðt iÞ BH ðt 1Þi ¼ 1; 2; where /ðt iÞ ¼ 0:6 þ /ðiÞ cosð2pt=TÞ and hðt iÞ ¼ 1 þ hðiÞ cosð2pt=TÞ. ð1Þ

ð1Þ

For first and second models, assume /1 ¼ 0:2, h1 ¼ 0:5 ð2Þ ð2Þ and /1 ¼ 0:2, h1 2 f 0:9; 0:7; 0:5; 0:3; 0:1g, respectively. Also, we consider the Hurst parameter H 2 f0:25; 0:75g: Tables 1–6 indicate the values of b a (Row 7: Proposed Method, Row 8: Mahmoudi et al., Row 9: Caiado et al.) and b p for Examples 1–6, respectively. It can be observed that the values of b a are very close to 0.05, and consequently three methods control the nominal level (a ¼ 0:05), especially asðn1 ; n2 Þ is increased. For stationary models (Examples 1– 3), the power of the proposed method is close to the power of the method given by [55], and is significantly more than the power of the method given by [53,54]. Consequently, for stationary models, the proposed method acts as well as the method given by [55], and acts better than the method given by [53,54]. For non-stationary models (Examples 4–6), the pro-

Table 1

posed method has an excellent ability to discriminate the null and alternative hypotheses for non-stationary models and its power is significantly more than the powers of the alternative methods. Consequently, for non-stationary models, the proposed method acts better than the method alternative methods.

It seems that the proposed technique has many advantages.

First, its power is acceptable, specially for non-stationary models. Second, the proposed technique is not too time consuming.

Moreover, this approach can be used to compare two or several stationary or non-stationary time series models with fBm noises.

4. Real data Now, the behavior of the introduced procedure in real world applications is studied by a real example. The dataset contains the reconnaissance drought index (RDI) values from two Iranian synoptic stations (Iranshahr and Zahedan) from 1980 to 2010.

Figs. 1 and 2 indicate the sample auto-correlation and partial auto-correlation functions (ACF and PACF, respectively) for RDI time series of Iranshahr and Zahedan synoptic sta

tions Xðt 1Þ andXðt 2Þ ; respectively . As can be seen, AR(1) model can be e good model to these datasets.

The results of fitting the best model based on minimum corrected Akaike information criterion (AICC) are shown in Table 7. As can be seen the AR(1) model with coefficients 0.6239 and 0.3869 are the best models that can describe the RDI time series of Iranshahr and Zahedan synoptic stations, respectively.

We are interested in determining whether the RDI time series of two synoptic stations have the same stochastic mechanism or not. Table 8 summarized the results of the introduced technique. As it can be seen, the result of Kolmogorov-Smirnov normality test indicates that all bði;jÞ ; i; j ¼ 1; 2; follow normal distributions. Therefore to comX t pare the two models for each dataset, the repeated measures

Power study for AR(1) model. ðn1 ; n2 Þ

ð2Þ /1

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.802 0.784 0.178 0.501 0.486 0.118 0.054 0.068 0.074 0.498 0.464 0.234 0.823 0.814 0.718

0.904 0.898 0.227 0.678 0.643 0.178 0.053 0.064 0.071 0.689 0.656 0.285 0.965 0.943 0.854

0.999 0.999 0.304 0.801 0.785 0.167 0.052 0.059 0.068 0.843 0.812 0.387 1.000 1.000 0.954

1.000 1.000 0.718 0.988 0.964 0.378 0.051 0.056 0.059 0.999 0.996 0.785 1.000 1.000 0.994

1.000 1.000 0.934 1.000 0.998 0.632 0.049 0.052 0.058 1.000 1.000 0.957 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 0.876 0.050 0.052 0.055 1.000 1.000 1.000 1.000 1.000 1.000

0.3

0.5

0.7

0.9

Testing the equality of several independent stationary and non-stationary time series models Table 2

1771

Power study for MA(1) model. ðn1 ; n2 Þ

ð2Þ h1

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.701 0.695 0.186 0.423 0.387 0.143 0.066 0.065 0.073 0.432 0.430 0.222 0.823 0.798 0.743

0.843 0.834 0.216 0.589 0.578 0.187 0.058 0.063 0.068 0.634 0.625 0.295 0.953 0.932 0.798

0.999 0.985 0.326 0.765 0.734 0.198 0.052 0.059 0.059 0.796 0.786 0.402 1.000 0.995 0.906

1.000 1.000 0.698 0.896 0.873 0.390 0.050 0.058 0.058 0.923 0.909 0.800 1.000 0.998 0.986

1.000 1.000 0.898 0.934 0.921 0.628 0.049 0.058 0.058 1.000 0.998 0.949 1.000 0.999 0.994

1.000 1.000 0.966 0.986 0.943 0.899 0.049 0.057 0.057 1.000 0.999 0.996 1.000 1.000 1.000

0.3

0.5

0.7

0.9

Table 3

Power study for ARMA(1,1) model. ðn1 ; n2 Þ

ð2Þ h1

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.882 0.731 0.205 0.688 0.521 0.112 0.053 0.055 0.054 0.778 0.574 0.098 0.914 0.887 0.812

0.976 0.904 0.294 0.829 0.643 0.134 0.051 0.053 0.053 0.897 0.701 0.102 0.997 0.986 0.943

0.999 0.996 0.364 0.997 0.722 0.138 0.050 0.054 0.053 0.997 0.889 0.116 1.000 1.000 0.978

1.000 1.000 0.703 0.999 0.828 0.304 0.049 0.052 0.052 1.000 0.996 0.264 1.000 1.000 0.989

1.000 1.000 0.934 1.000 1.000 0.498 0.049 0.051 0.048 1.000 1.000 0.424 1.000 1.000 0.998

1.000 1.000 0.988 1.000 1.000 0.862 0.048 0.050 0.052 1.000 1.000 0.803 1.000 1.000 0.999

0.3

0.5

0.7

0.9

Table 4

Power study for PAR(1) model. ðn1 ; n2 Þ

/ð2Þ

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.784 0.178 0.149 0.486 0.118 0.109 0.068 0.074 0.067 0.464 0.234 0.212 0.814 0.718 0.578

0.898 0.227 0.198 0.643 0.178 0.189 0.064 0.071 0.064 0.656 0.285 0.278 0.943 0.854 0.691

0.999 0.304 0.278 0.785 0.197 0.197 0.059 0.068 0.065 0.812 0.387 0.376 1.000 0.954 0.782

1.000 0.718 0.597 0.964 0.378 0.312 0.056 0.059 0.062 0.996 0.785 0.695 1.000 0.994 0.821

1.000 0.934 0.798 0.998 0.632 0.526 0.052 0.058 0.060 1.000 0.957 0.832 1.000 1.000 0.898

1.000 1.000 0.876 1.000 0.876 0.712 0.052 0.055 0.057 1.000 1.000 0.901 1.000 1.000 0.912

0.3

0.5

0.7

0.9

1772

M.R. Mahmoudi et al.

Table 5

Power study for PMA(1) model. ðn1 ; n2 Þ

h

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.678 0.186 0.145 0.423 0.143 0.165 0.066 0.073 0.065 0.432 0.222 0.193 0.823 0.743 0.543

0.823 0.216 0.215 0.589 0.187 0.186 0.058 0.068 0.064 0.634 0.295 0.269 0.953 0.798 0.598

0.966 0.326 0.298 0.765 0.198 0.211 0.052 0.059 0.062 0.796 0.402 0.370 1.000 0.906 0.765

0.999 0.698 0.523 0.896 0.390 0.387 0.050 0.058 0.056 0.923 0.800 0.692 1.000 0.986 0.843

1.000 0.898 0.741 0.934 0.628 0.565 0.049 0.058 0.055 1.000 0.949 0.774 1.000 0.994 0.875

1.000 0.966 0.879 0.986 0.899 0.781 0.049 0.057 0.054 1.000 0.996 0.865 1.000 1.000 0.912

ð2Þ

0.3

0.5

0.7

0.9

Table 6

Power study for PARMA(1,1) model. ðn1 ; n2 Þ

h

Method

ð100; 50Þ

ð150; 75Þ

ð200; 100Þ

ð500; 250Þ

(1000, 500)

(2000,1000)

0.1

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

Proposed Mahmoudietal: Caiado et al.

0.731 0.205 0.189 0.521 0.112 0.132 0.055 0.054 0.058 0.574 0.098 0.113 0.887 0.124 0.176

0.904 0.294 0.301 0.643 0.134 0.165 0.053 0.053 0.057 0.701 0.102 0.123 0.986 0.143 0.212

0.996 0.364 0.365 0.722 0.138 0.156 0.054 0.053 0.056 0.889 0.116 0.126 1.000 0.306 0.353

1.000 0.703 0.657 0.828 0.304 0.289 0.052 0.052 0.056 0.996 0.264 0.176 1.000 0.698 0.563

1.000 0.934 0.876 1.000 0.498 0.387 0.051 0.048 0.053 1.000 0.424 0.324 1.000 0.902 0.710

1.000 0.988 0.914 1.000 0.862 0.756 0.050 0.052 0.053 1.000 0.803 0.597 1.000 0.984 0.734

ð2Þ

0.3

0.5

0.7

0.9

Sample ACF

1.00 .80

.80

.60

.60

.40

.40

.20

.20

.00

.00

-.20

-.20

-.40

-.40

-.60

-.60

-.80

-.80

-1.00

0

5

10

15

Fig. 1

20

25

Sample PACF

1.00

30

35

40

-1.00

0

5

10

15

20

25

Plot of sample ACF/PACF for RDI time series of Iranshahr synoptic station.

30

35

40

Testing the equality of several independent stationary and non-stationary time series models Sample ACF

1.00

.80

.60

.60

.40

.40

.20

.20

.00

.00

-.20

-.20

-.40

-.40

-.60

-.60

-.80

-.80

-1.00

0

5

10

15

Fig. 2

Table 7

20

25

Sample PACF

1.00

.80

30

35

40

-1.00

1773

0

5

10

15

20

25

30

35

40

Plot of sample ACF/PACF for RDI time series of Zahedan synoptic station.

Best model for RDI time series of Iranshahr and Zahedan synoptic stations based on minimum AICC.

Synoptic Station

Best Model

Model’s Formula

Noise0 sVariance

AICC

Iranshahr

AR(1)

¼ 0:6239Xt 1

0.6107

77.6102

Zahedan

AR(1)

bð1Þ X t bð1Þ X t

¼ 0:3869Xt 1

0.8503

87.5374

Table 8

Comparison of two AR(1) models in each station.

Kolmogorov-Smirnov Normality Test

Comparison

Bonferroni Correction Test

Synoptic Station

Data

Kolmogorov-Smirnov

pvalue

Suitable Test

pvalue

pvalue

Iranshahr

bð1;1Þ X t

0:709

0.697

Repeated Measures

0.978

1.000

0:709

0.697

0:694

0.721

Repeated Measures

0.822

1.000

0:694

0.721

bð1;2Þ X t Zahedan

bð2;1Þ X t bð2;2Þ X t

test should be applied. The results of the repeated measures test determined that for each dataset two AR(1) didn’t significantly differ. Also, the results of multiple Bonferroni correction test showed that the RDI time series of two synoptic stations have the same stochastic mechanism.

5. Conclusion Comparison, classification, discrimination, and clustering of the models fitted on several independent observed time series are one of important issues in time series analysis. In this work, we tried to solve this important problem for several stationary or non-stationary processes with fractional Brownian motion. At first, all of time series models were fitted on datasets. Then, for observed datasets, the predicted values of

all models were estimated based on fitted models formulas.

Finally, the multiple testing techniques were applied to compare predicted values. The ability of the proposed technique was studied using simulation. The results indicated that the proposed approach controlled the nominal size of the test.

Moreover, the introduced approach could nicely discriminate different fitted models, specially for non-stationary models. The introduced approach had many advantages.

The method was more powerful than other alternatives.

The method was not too time consuming. The method could be used to compare several stationary or non-stationary time series. The applicability of the proposed approach in real world applications was also studied by investigating the fitted models on RDI values from two Iranian synoptic stations.

1774 Declaration of Competing Interest The authors declared that there is no conflict of interest.

References [1] Jalali, S. M. J., Karimi, M., Khosravi, A., & Nahavandi, S. (2019, October). An efficient neuroevolution approach for heart disease detection. In 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC) (pp.

3771-3776). IEEE. [2] Jalali, S. M. J., Khosravi, A., Alizadehsani, R., Salaken, S. M., Kebria, P. M., Puri, R., & Nahavandi, S. (2019). Parsimonious Evolutionary-based Model Development for Detecting Artery Disease. In ICIT (pp. 800-805).

[3] S.M.J. Jalali, S. Ahmadian, A. Khosravi, S. Mirjalili, M.R. Mahmoudi, S. Nahavandi, Neuroevolution-based Autonomous Robot Navigation: A Comparative Study, Cognit. Syst. Res. (2020).

[4] S.J. Mousavirad, G. Schaefer, S.M.J. Jalali, I. Korovin, July). A benchmark of recent population-based metaheuristic algorithms for multi-layer neural network training, Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion (2020) 1402–1408.

[5] S.M.J. Jalali, S. Ahmadian, P.M. Kebria, A. Khosravi, C.P. Lim, S. Nahavandi, Evolving artificial neural networks using butterfly optimization algorithm for data classification, Springer, Cham, 2019, pp. 596–607.

[6] M.H. Heydari, Z. Avazzadeh, M.R. Mahmoudi, Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion, Chaos, Solitons Fractals 124 (2019) 105–124.

[7] R. Roohi, M.H. Heydari, M. Aslami, M.R. Mahmoudi, A comprehensive numerical study of space-time fractional bioheat equation using fractional-order Legendre functions, European Phys. J. Plus 133 (2018) 412.

[8] A. Babaei, H. Jafari, S. Banihashemi, A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise, Symmetry 12 (6) (2020) 904.

[9] A. Babaei, H. Jafari, S. Banihashemi, Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J. Comput. Appl. Math. 112908 (2020).

[10] A. Babaei, H. Jafari, M. Ahmadi, A fractional order HIV/ AIDS model based on the effect of screening of unaware infectives, Mathematical Methods Appl. Sci. 42 (7) (2019) 2334–2343.

[11] A. Babaei, B.P. Moghaddam, S. Banihashemi, J.A.T. Machado, Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations, Commun.

Nonlinear Sci. Numer. Simul. 82 (2020) 104985.

[12] M. Maleki, M.R. Mahmoudi, D. Wraith, K.H. Pho, Time series modelling to forecast the confirmed and recovered cases of COVID-19, Travel Med. Infect. Dis. 101742 (2020).

[13] M. Bahrami, M.J. Amiri, M.R. Mahmoudi, S. Koochaki, Modeling caffeine adsorption by multi-walled carbon nanotubes using multiple polynomial regression with interaction effects, J. Water Health 15 (4) (2017) 526–535.

[14] A.R. Zarei, M.R. Mahmoudi, Evaluation of changes in RDIst index effected by different Potential Evapotranspiration calculation methods, Water Resour Manag 31 (15) (2017) 4981–4999.

[15] A.R. Abbasi, M.R. Mahmoudi, Z. Avazzadeh, Diagnosis and clustering of power transformer winding fault types by crosscorrelation and clustering analysis of FRA results, IET Gener.

Transm. Distrib. 12 (19) (2018) 4301–4309.

M.R. Mahmoudi et al.

[16] M.R. Mahmoudi, M. Maleki, A New Method to Detect Periodically Correlated Structure, Comput. Statistics 32 (4) (2017) 1569–1581.

[17] A.R. Nematollahi, A.R. Soltani, M.R. Mahmoudi, Periodically Correlated Modeling by Means of the Periodograms Asymptotic Distributions, Stat. Pap. 58 (4) (2017) 1267–1278.

[18] M.R. Mahmoudi, M.H. Heydari, Z. Avazzadeh, On the Asymptotic Distribution for the Periodograms of Almost Periodically Correlated (Cyclostationary) Processes, Digital Signal Process. 81 (2018) 186–197.

[19] M.R. Mahmoudi, M.H. Heydari, Z. Avazzadeh, K.H. Pho, Goodness of fit test for almost cyclostationary processes, Digital Signal Process. 96 (2020) 102597.

[20] M.R. Mahmoudi, D. Baleanu, B.A. Tuan, K.H. Pho, A novel method to detect almost cyclostationary structure, Alexandria Engineering Journal 59 (4) (2020) 2339–2346.

[21] M.R. Mahmoudi, A.R. Nematollahi, A.R. Soltani, On the detection and estimation of the simple harmonizable processes, Iranian Journal of Science and Technology (Sciences) 39 (2) (2015) 239–242.

[22] H. Haghbin, M.R. Mahmoudi, Z. Shishebor, Large Sample Inference on the Ratio of Two Independent Binomial Proportions, J. Math. Ext. 5 (1) (2011) 87–95.

[23] M.R. Mahmoudi, J. Behboodian, M. Maleki, Inference on the Ratio of Means in Two Independent Populations, J. Stat.

Theory Appl. 16 (3) (2017) 366–374.

[24] M.R. Mahmoudi, M. Mahmoodi, Inferrence on the Ratio of Variances of Two Independent Populations, J. Math. Ext. 7 (2) (2014) 83–91.

[25] M.R. Mahmoudi, R. Nasirzadeh, M. Mohammadi, On the Ratio of Two Independent Skewnesses, Commun. Stat-Theor.

Methods 48 (7) (2019) 1721–1727.

[26] M.R. Mahmoudi, M. Mahmoodi, Inferrence on the Ratio of Correlations of Two Independent Populations, J. Math. Ext. 7 (4) (2014) 71–82.

[27] M.R. Mahmoudi, M. Mahmoudi, E. Nahavandi, Testing the Difference between Two Independent Regression Models, Commun. Stat. Theory Methods 45 (21) (2016) 6284–6289.

[28] M.R. Mahmoudi, M. Maleki, A. Pak, Testing the Equality of Two Independent Regression Models, Commun. Stat. Theory Methods 47 (12) (2018) 2919–2926.

[29] M.R. Mahmoudi, On Comparing Two Dependent Linear and Nonlinear Regression Models, J. Test. Eval. 47 (1) (2018) 449– 458.

[30] P. Ji-jun, M.R. Mahmoudi, D. Baleanu, M. Maleki, On Comparing and Classifying Several Independent Linear and Non-Linear Regression Models with Symmetric Errors, Symmetry 11 (6) (2019) 820.

[31] M.R. Mahmoudi, M.H. Heydari, K.H. Pho, Fuzzy clustering to classify several regression models with fractional Brownian motion errors, Alexandria Engineering Journal 59 (4) (2020) 2811–2818.

[32] M.R. Mahmoudi, M. Mahmoudi, A. Pak, On comparing, classifying and clustering several dependent regression models, J. Stat. Comput. Simul. 89 (12) (2019) 2280–2292.

[33] A.M. Alonso, J.R. Berrendero, A. Hernandez, A. Justel, Time series clustering based on forecast densities, Comput. Stat. Data Anal. 51 (2006) 762–776.

[34] H. Dette, T. Hildebrandt, A note on testing hypothesis for stationary processes in the frequency domain, J. Multivariate Anal. 104 (2011) 101–114.

[35] H. Dette, T. Kinsvater, M. Vetter, Testing nonparametric hypotheses for stationary processes by estimating minimal distances, J. Time Ser. Anal. 32 (2010) 447–461.

[36] H. Dette, E. Paparoditis, Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities, Journal of the Royal Statistical Society B 71 (2009) 831–8571.

Testing the equality of several independent stationary and non-stationary time series models [37] D.S. Coates, P.J. Diggle, Tests for Comparing Two Estimated Spectral Densities, J. Time Ser. Anal. 7 (1986) 7–20.

[38] P.J. Diggle, N.I. Fisher, Nonparametric Comparison of Cumulative Periodograms, Appl. Stat. 40 (1991) 423–434.

[39] K. Fokianos, A. Savvides, On Comparing Several Spectral Densities, Technometrics 50 (3) (2008) 317–331.

[40] S. Fruhwirth-Schnatter, S. Kaufmann, Model-based clustering of multiple time series”, CEPR Discussion Paper No. 4650 (2004).

[41] C. Jentsch, A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes, J. Time Series Anal. 33 (2012) 177–192.

[42] C. Jentsch, M. Pauly, A note on using periodogram-based distances for comparing spectral densities, Statist. Probab. Lett.

82 (2012) 158–164.

[43] C. Jentsch, M. Pauly, Testing equality of spectral densities using randomization techniques, Bernoulli 21 (2) (2015) 697–739.

[44] Y. Kakizawa, R.H. Shumway, M. Taniguchi, Discrimination and Clustering for Multivariate Time series, J. Am. Stat. Assoc.

93 (1998) 328–340.

[45] E.A. Maharaj, A significance test for classifying ARMA models, J. Stat. Comput. Simul. 54 (1996) 305–331.

[46] E.A. Maharaj, Comparison and classification of stationary multivariate time series, Pattern Recogn. 32 (1999) 1129–1138.

[47] E.A. Maharaj, Clusters of time series, J. Classif. 17 (2000) 297– 314.

[48] D. Piccolo, A distance measure for classifying ARIMA models, J. Time Ser. Anal. 11 (1990) 153–164.

[49] M.R. Mahmoudi, M.H. Heydari, R. Roohi, A new method to compare the spectral densities of two independent periodically correlated time series, Math Comput Simulat 160 (2019) 103– 110.

[50] M.R. Mahmoudi, M.H. Heydari, Z. Avazzadeh, Testing the difference between spectral densities of two independent periodically correlated (cyclostationary) time series models, Commun Stat Theory Methods 48 (9) (2019) 2320–2328.

[51] M.R. Mahmoudi, M. Maleki, K. Borodin, K.H. Pho, D. Baleanu, On comparing and clustering the spectral densities of

[52]

[53]

[54]

[55]

[56]

[57]

[58] [59] [60]

[61] [62] [63] [64]

[65]

1775

several almost cyclostationary processes, Alexandria Engineering Journal 59 (4) (2020) 2555–2565.

R. Zhou, M.R. Mahmoudi, S.N.Q. Mohammed, K.H. Pho, Testing the equality of the spectral densities of several uncorrelated almost cyclostationary processes, Alexandria Engineering Journal 59 (2020) 3545–3550.

J. Caiado, N. Crato, D. Pena, A Periodogram-based Metric for Time Series Classification, Comput. Stat. Data Anal. 50 (2006) 2668–2684.

J. Caiado, N. Crato, D. Pena, Comparison of times series with unequal length in the frequency domain, Comm. Statist.

Simulation Comput. 38 (2009) 527–540.

M.R. Mahmoudi, M. Maleki, A. Pak, Testing the Difference between Two Independent Time Series Models, Iran J. Sci. Technol. A (Sciences) 41 (2017) 665–669.

G.E. Salcedo, R.F. Porto, P.A. Morettin, Comparing nonstationary and irregularly spaced time series, Comput. Stat.

Data Anal. 56 (12) (2012) 3921–3934.

E.A. Maharaj, Comparison of Non-stationary Time Series in the Frequency Domain, Comput. Stat. Data Anal. 40 (2002) 131– 141.

D.C. Montgomery, Design and Analysis of Experiments, Eighth ed., Arizona State University. John Wiley & Sons Inc., 2015.

W.J. Conover, Practical Nonparametric Statistics, 3rd ed., John Wiley, Hoboken, 1980.

Y. Benjamini, Y. Hochberg, Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing, J. R Stat. Soc. Series B Stat. Methodol. 57 (1) (1995) 125–133.

S. Holm, A Simple Sequentially Rejective Multiple Test Procedure, Scand. J. Stat. 6 (2) (1979) 65–70.

J.D. Storey, A Direct Approach to False Discovery Rates, J. R Stat. Soc. Series B Stat. Methodol. 64 (2002) 479–498.

J.D. Storey, R. Tibshirani, Statistical Significance for Genomewide Studies, PNAS 100 (2003) 9440–9445.

K. Strimmer, fdrtool: A Versatile R Package for Estimating Local and Tail Area- based False Discovery Rates, Bioinformatics 24 (2008) 1461–1462.

P.J. Brockwell, R.A. Davis, Time Series: Theory and Methods, Springer-Verlag, Berlin, 1991.