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According to the importance of assessing the presence of time delay between the occurrence of various hydrological and meteorological phenomena, the study aim is to introduce a new method (with high ability and non-sensitivity to the abnormality of datasets and the existence of outliers) for determining the time delay in mentioned data series. In this research, a new measure to detect the time delay between two stationary time series (Non-Parametric Cross-Correlation Function or NCCF, called Spearman's CCF or SCCF) is introduced, which has very low sensitivity to abnormality of data series and also the existence of outliers in the data series. The numerical studies verify the ability of the proposed measure. In standard uniform and exponential (with mean 1) time series, at 100% of numerical analyses and in standard Gaussian time series at more than 60% of numerical studies, the ability of SCCF was more than the CCF. The applicability of the proposed measure in practice was also studied using the Reconnaissance Drought Index (RDI) data series of 20 stations over Iran during 1967–2019 in 1, 3, and 12-month time scales. The results of the practical study also proved the appropriate performance of the proposed model in all time scales.

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  • A modified version of the cross-correlation function was presented.

  • Output of this research is a new measure to detect the time delay between two stationary time series.

  • In this research, data series of 20 stations with various climate conditions was used.

  • The results are usable in better understanding the behavior of climatic parameters (especially drought).

Graphical Abstract

Graphical Abstract
Graphical Abstract
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Graphical Abstract
Graphical Abstract
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cross-correlation, drought, non-parametric, stationary, time-delay, time series
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The existence of a relationship between different climatic variables and various hydrological phenomena in terms of temporal and spatial is a proven issue worldwide (Abeysingha et al. 2020; Aksoy et al. 2021; Liang et al. 2021; Liu et al. 2021; Mokarram & Zarei 2021; Salimi et al. 2021; Zarei & Mahmoudi 2021; Zarei et al. 2021a; Lotfirad et al. 2022; Radmanesh et al. 2022; Sun et al. 2022). Generally, this relationship is used by researchers in various studies such as assessing and determining the degree of dependence of different variables, examining changes in one variable under the influence of other variables, evaluating the temporal and spatial changes in variables, examining and determining the delay time of a variable in different regions, etc. (Chen et al. 2018; Zarei & Moghimi 2019; Zarei et al. 2020; Bahrami et al. 2021; Fang et al. 2021; Gumus et al. 2021; Han et al. 2021). Therefore, it can be concluded that the detection of relationships between variables has a vital role in meteorological, hydrological, environmental, and so on, studies.

There are several parametric and non-parametric techniques introduced for detecting the relationships such as Pearson's correlation coefficient (Ablat et al. 2019; Li et al. 2020), Spearman's correlation coefficient (SCCF; Konapala et al. 2020; Tai et al. 2020), Kendall's correlation coefficient (Peña-Gallardo et al. 2019; Mallick et al. 2021), the Sen's slope (Atif et al. 2018; Myronidis et al. 2018), the cross-correlation function (CCF; Seo et al. 2019; Dong et al. 2020), and so on. Aryal & Zhu (2021) assessed the spatiotemporal structure of drought using the standardized precipitation evapotranspiration index (SPEI) and the principal component analysis over the continental United States during 1950–2015 in 12 and 24-month time scales. The results indicated that the areas with severe drought conditions are mostly located in the Northwest, South, upper Midwest, and East regions. Zarei et al. (2021b) used the CCF to assess the susceptibility of winter wheat, barley, and rapeseed to drought using the meteorological data series of 10 stations during 1968–2017 over Iran. They showed that the rapeseed had been the most susceptibility to drought occurrence. This study indicated that the maximum CC between the drought and annual yield of mentioned crops was less than 0.5 in more than 80% of investigated stations (without time lag). Rahmani & Fattahi (2021) used the multifractal cross-correlation to evaluate the sensitivity of meteorological and hydrological drought to temperature and rainfall. Based on the results of this research, the effect of precipitation fluctuations on droughts was more than the temperature fluctuations. Roustaei et al. (2021) assessed the time delay between wind erosion and drought in the Southern regions of Iran using the CCF function. They revealed that the maximum CC between the drought and wind erosion was equal to −0.22 (without time lag). Many researchers have tried to use parametric and non-parametric techniques to detect relationships between climatic and hydrologic variables worldwide (e.g., Baik et al. 2021; Kumar et al. 2021; Lashkari et al. 2021; Lohpaisankrit & Techamahasaranont 2021; Piri & Mobaraki 2021; Tuan & Canh 2021; Zarei et al. 2021c; Zhang et al. 2021).

Pearson's correlation coefficient is sensitive to the abnormality of datasets and the existence of outliers. Moreover, Pearson's correlation coefficient, Spearman's correlation coefficient, and Kendall's correlation coefficient are more applicable in facing independent observations. For assessing the relationship of two time series, the cross-correlation function (CCF, in abbreviation) is suggested. The CCF is somewhat sensitive to the abnormality of datasets and the existence of outliers (similar to Pearson's correlation). In other words, for abnormal populations or for populations with outliers data, the CCF may not work well. To solve this issue, in this research, we define a non-parametric CCF (NCCF, in abbreviation), called Spearman's CCF (SCCF, in abbreviation). The ability of the SCCF to detect a time-delay correlation between two stationary time series is studied. For this purpose, numerous datasets from two stationary time series are produced and analyzed. The ability of SCCF in practice is also investigated by a real example. For this purpose, the annual reconnaissance drought index (RDI) values from three Iranian synoptic are considered and analyzed.

In other words, due to the vital role of determining the delay time of the hydrological phenomena occurrence (such as drought, rain, and flood) in managing and reducing the negative impacts of the phenomena mentioned above, in this study, a new non-parametric statistical test (NCCF) was presented for determining the delay time in hydrological phenomena (It is the most critical aspect of research novelty). Then, its ability was assessed based on the simulated and real data.

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In this section, first, the parametric and non-parametric correlation methods will be explained. Then, the CCF model and its application in determining the time delay in hydrological phenomena will be described. Then, the SCCF model, as a new model for determining the time delay in hydrological phenomena, will be explained. Finally, using simulated data and actual data (based on the RDI drought index in different time scales), the ability of the SCCF model will be investigated.

Parametric correlation

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Assume is a sample of size n from bivariate population ⁠. Pearson's correlation coefficient, which is a parametric tool, is more powerful in facing the bivariate normal datasets and is defined as follows (Equation (1)):
(1)
where and are the values of the X and Y variables, and and are the averages of the X and Y variables in the sample.

Non-parametric correlation

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For abnormal datasets or datasets with outliers, the non-parametric tools such as Spearman's correlation coefficient, Kendall's correlation coefficient, and Sen's slope are more robust. To compute SCCF, at first, and are separately sorted from smallest to largest. Then, Spearman's correlation is defined as the Pearson's correlation coefficient of (Equations (2)–(4)).
(2)
where
(3)
and
(4)
Therefore, Spearman's correlation coefficient is defined as follows (Equation (5)):
(5)

Kendall's correlation coefficient is also computed as follows:

The pairs are called concordant, if and have the same sign (both + or both −). Otherwise, are named discordant. If we show the number of concordant and discordant pairs with A and B, respectively, Kendall's correlation coefficient is calculated as (Equations (6) and (7)):
(6)
In other words, we have:
(7)
where is sign function.

The Sen's slope trend is defined as follows:

First, a set of linear slopes is calculated as (Equation (8)):
(8)
for Sen's slope is then calculated as the median from all slopes (Equation (9)):
(9)

Cross-correlation function

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For assessing the relationship of two time series and ⁠, the cross-correlation function (CCF, in abbreviation) is suggested. For two continuous functions f and g, the CCF in lag is represented as (Equation (10)):
(10)
Such that is the complex conjugate of the function For two discrete functions f and g, the CCF in lag is also represented as (Equation (11)):
(11)

As we can observe, the CCF is the rate of the similarity (correlation) of the function f in (t) and the function g in ⁠.

If and are two weak stationary processes, CCF in lag is represented as (Equations (12) and (13)):
(12)
such that
(13)
denotes the cross-covariance function (CVF, in abbreviation) of and in lag (⁠⁠), and are the means of and ⁠, and and are respectively the standard deviations of and ⁠.
Usually, the is unknown, and therefore, it should be estimated by using sample realizations of and If and is a path of and respectively, the can be estimated by the sample CCF as (Equations (14) and (15):
(14)
such that
(15)
is the sample CVF of and in lag ⁠, and are the means of the two realizations, and and are respectively the standard deviations of the two realizations.

CCF and sample CCF are important tools to measure the rate of time-delay correlation (that is from −1 to +1) between two time series.

Non-parametric cross-correlation function

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The CCF, similar to Pearson's correlation, is somewhat sensitive to the normality assumption. In other words, for abnormal populations or populations with outliers, CCF may not work well. To solve this issue, we define a non-parametric CCF (NCCF, in abbreviation), called Spearman's CCF (SCCF, in abbreviation).

If and are two stationary time series, SCCF in lag is defined as following (Equation (16)):

Consider the pairs
(16)
Sort and ⁠, separately, from smallest to largest. The SCCF in lag is defined as the Pearson's correlation coefficient of (Equations (17)–(19)):
(17)
where
(18)
and
(19)
Therefore, SCCF is defined as following (Equation (20)):
(20)

Power test of SCCF

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Power test of SCCF based on the simulation

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In this section, the ability of the SCCF to detect a time-delay correlation between two stationary time series is studied. For this purpose, numerous datasets from two stationary time series and with time delay are produced and analyzed.

The simulation procedure is as follows:

Step 1: For fixed ⁠, a path of size n from is produced. To investigate the impacts of signal class and data length on the applied mathematical techniques functionality over the data, different distributions such as standard normal, standard uniform, Gaussian, non-Gaussian, Brownian motion and fractional Brownian motion, and exponential with different lengths are considered (Fattahi et al. 2011; Mitková 2019; Han et al. 2022).

Step 2: For fixed time-delay ⁠, a path of size from is produced by (Equation (21))
(21)
where is a random sample of size from standard normal and ⁠.
Step 3: For the time series and ⁠, is estimated by CCF and SCCF, as follows (Equations (22) and (23)):
(22)
(23)

In other words, and are the values of that have the maximum values of and respectively.

Step 4: Steps 1 to 3 are repeated 1,000 times.

Step 5: The power of these approaches are evaluated by (Equations (24) and (25)):
(24)
and
(25)
where and are the counts of the cases that and ⁠, respectively.

Step 5: Calculating the mean absolute error (MAE) and root mean square error (RMSE) indices for CCF and SCCF methods.

Power test of SCCF based on the real data

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In this section, the ability of SCCF in practice is investigated by an actual example. For this purpose, calculated RDI in 1, 3, and 12-month time scales were used. For investigating the stationarity of the datasets, the augmented Dickey–Fuller (ADF) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests were applied. These tests verify that the computed RDI in all three time scales were stationary.

Study region, data collection, and data evaluation

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In this research, the climatic data series of 20 stations (including Babolsar, Bandar Abbas, Bandar Lengeh, Birjand, Chabahar, Fasa, Qazvin, Iran Shahr, Mashhad, Oroomieh, Ramsar, Semnan, Shiraz, Tabass, Tabriz, Tehran, Torbat Hydarieh, Zabol, Zahedan, and Zanjan stations) with proper spatial distribution, adequacy of the length of the data series, appropriate climate diversity from 1967 to 2019 over Iran were used (Figure 1). Before using data series in all stations, the missing values of the data series were estimated using the Fuzzy regression method (Sadatinejad et al. 2011). The length adequacy of the data series (time duration) was evaluated using the Mockus equation (Zarei & Mahmoudi 2021). The Run-test method was used for assessing the homogeneity of the data series (Mahdavi 2002). Then at all stations, the RDI index in different time scales, including monthly (1-month), seasonal (3-month), and annual (12-month), were calculated. According to the purpose and hypothesis of the research (mentioned in the previous sections), in selecting the stations, it was tried to select and use stations with abnormal RDI data series in 1, 3, and 12-month time scales. Some geographical and meteorological characteristics of chosen stations are presented in Table 1. It should be noted that the required meteorological data were obtained from the Iran Meteorological Organization (www.irimo.ir), and the R 3.6 and SPSS 20 software were used to perform the analysis.
Table 1

Geographical location of selected stations and their meteorological characteristics

StationLatitude (N)Longitude (E)Elevation (m a.s.l)Rainfall (mm/year)Annual average temperature (°C)PET (mm/day)
Babolsar 36.720 52.653 − 21.00 917.05 17.35 3.13 
Bandar Abbas 27.214 56.373 9.80 173.95 26.98 6.58 
Bandar Lengeh 26.528 54.828 22.70 134.13 26.86 6.78 
Birjand 32.891 59.283 1491.00 160.68 16.49 5.70 
Chabahar 25.281 60.651 8.00 113.20 26.33 5.60 
Fasa 28.899 53.719 1268.00 285.55 19.43 5.10 
Qazvin 36.319 50.020 1279.10 319.47 14.07 4.19 
Iran Shahr 27.229 60.718 591.10 109.82 26.95 6.47 
Mashhad 36.236 59.631 999.20 255.32 14.80 4.52 
Oroomieh 37.659 45.055 1328.00 330.65 11.36 3.72 
Ramsar 36.904 50.683 − 20.00 1232.50 16.29 2.81 
Semnan 35.588 53.421 1127.00 139.86 18.29 6.56 
Shiraz 29.561 52.603 1488.00 319.11 18.08 5.29 
Tabass 33.603 56.951 711.00 82.10 22.16 5.15 
Tabriz 38.122 46.242 1361.00 274.55 12.88 4.69 
Tehran 35.693 51.309 1191.00 236.82 17.79 5.17 
Torbat Hydarieh 35.332 59.206 1451.00 261.22 14.40 5.68 
Zabol 31.089 61.543 489.20 54.58 22.41 9.24 
Zahedan 29.472 60.900 1370.00 78.44 18.72 6.31 
Zanjan 36.660 48.522 1659.40 303.57 11.26 2.68 
StationLatitude (N)Longitude (E)Elevation (m a.s.l)Rainfall (mm/year)Annual average temperature (°C)PET (mm/day)
Babolsar 36.720 52.653 − 21.00 917.05 17.35 3.13 
Bandar Abbas 27.214 56.373 9.80 173.95 26.98 6.58 
Bandar Lengeh 26.528 54.828 22.70 134.13 26.86 6.78 
Birjand 32.891 59.283 1491.00 160.68 16.49 5.70 
Chabahar 25.281 60.651 8.00 113.20 26.33 5.60 
Fasa 28.899 53.719 1268.00 285.55 19.43 5.10 
Qazvin 36.319 50.020 1279.10 319.47 14.07 4.19 
Iran Shahr 27.229 60.718 591.10 109.82 26.95 6.47 
Mashhad 36.236 59.631 999.20 255.32 14.80 4.52 
Oroomieh 37.659 45.055 1328.00 330.65 11.36 3.72 
Ramsar 36.904 50.683 − 20.00 1232.50 16.29 2.81 
Semnan 35.588 53.421 1127.00 139.86 18.29 6.56 
Shiraz 29.561 52.603 1488.00 319.11 18.08 5.29 
Tabass 33.603 56.951 711.00 82.10 22.16 5.15 
Tabriz 38.122 46.242 1361.00 274.55 12.88 4.69 
Tehran 35.693 51.309 1191.00 236.82 17.79 5.17 
Torbat Hydarieh 35.332 59.206 1451.00 261.22 14.40 5.68 
Zabol 31.089 61.543 489.20 54.58 22.41 9.24 
Zahedan 29.472 60.900 1370.00 78.44 18.72 6.31 
Zanjan 36.660 48.522 1659.40 303.57 11.26 2.68 

Note: PET is potential evapotranspiration (calculated based on FAO Penman-Monteith).

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Figure 1
Study region, selected stations, and their spatial distribution.
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Study region, selected stations, and their spatial distribution.

Figure 1
Study region, selected stations, and their spatial distribution.
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Study region, selected stations, and their spatial distribution.

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The RDI calculation

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The RDI index was introduced by Tsakiris (2004) and is based on the rainfall and potential evapotranspiration (PET) variables. This index was calculated using the following equations (Tsakiris et al. 2007; Equations (26) and (27):
(26)
(27)
where and are the arithmetic mean and the standard deviation of respectively, is equal to ln (⁠⁠),⁠, and are respectively the amount of rainfall and PET in month of the year during the whole N year of study (hydrological year starts from October in Iran). It is suggested to see Lashkari et al. (2021), Mohammed & Yimam (2021), and El-Tantawi et al. (2021) for more details about the RDI. Some advantages of RDI including: the RDI is a multiscalar index that calculates the drought severity in different time scales. This index is based on P and PET variables and is sensitive to changes in environmental conditions. A flowchart of the research is presented in Figure 2.
Figure 2
Flowchart of the research.
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Flowchart of the research.

Figure 2
Flowchart of the research.
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Flowchart of the research.

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Power test of SCCF based on the simulation

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The computed values of ⁠, for different settings of parameter, are presented in Tables 2,3–4. The results show that the values of and are close to each other for Gaussian (normal) time series. Based on Table 2, in n = 100, 200, 500, and 1,000 at 64, 60, 64, and 60% of lags the was more than the ⁠, respectively. Presented results in Tables 3 and 4 indicated that at 100% of numerical studies for uniform and exponential time series, the is larger than ⁠. In other words, the is more robust than the ⁠, in detecting time delay between two non-Gaussian time series. Tables 5 and 6 indicate that almost at 100% of numerical studies for Brownian motion time series and fractional Brownian motion time series, the is larger than ⁠. In other words, the is more robust than the ⁠, in detecting time delay. The results of the MAE and RMSE also showed that the is more robust than the ⁠, in detecting time delay.

Table 2

The power of and for standard Gaussian time series

MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.989 0.991 0.998 0.994 0.972 0.983 0.986 0.981 0.014 0.013 0.017 0.014 
1 0.983 0.994 0.985 0.982 0.996 0.971 0.987 0.982 0.012 0.018 0.013 0.020 
2 0.963 0.960 0.963 0.987 0.983 0.971 0.984 0.996 0.027 0.022 0.029 0.026 
3 0.986 0.960 0.991 0.962 1.000 0.993 0.982 0.994 0.010 0.023 0.009 0.028 
4 0.952 0.991 0.986 0.979 0.998 0.988 0.986 0.982 0.020 0.015 0.018 0.016 
0.3 0 0.957 0.974 0.969 0.991 0.982 0.980 0.993 0.998 0.025 0.014 0.020 0.017 
1 0.979 0.972 0.963 0.986 0.990 0.977 0.988 0.997 0.020 0.017 0.016 0.019 
2 0.982 0.973 0.982 0.999 0.989 0.990 0.992 0.996 0.014 0.011 0.010 0.015 
3 0.986 0.955 0.988 0.985 0.992 0.979 0.982 0.993 0.013 0.022 0.010 0.026 
4 0.993 0.971 0.962 0.996 0.970 0.976 0.997 0.991 0.020 0.017 0.017 0.019 
0.5 0 0.950 0.954 0.961 0.992 0.978 0.977 0.985 0.983 0.032 0.024 0.024 0.027 
1 0.968 0.959 0.974 0.968 0.986 0.980 0.987 0.997 0.021 0.024 0.016 0.028 
2 0.998 0.970 0.975 0.999 0.989 0.978 0.993 0.980 0.011 0.018 0.010 0.021 
3 0.961 0.996 0.974 0.972 0.997 0.986 0.999 0.990 0.017 0.014 0.017 0.017 
4 0.997 0.987 0.984 0.983 0.988 0.988 0.994 0.985 0.009 0.014 0.007 0.014 
0.7 0 0.994 0.993 0.980 0.999 0.988 0.999 0.998 0.994 0.010 0.004 0.009 0.005 
1 0.961 0.956 0.994 0.972 0.973 0.982 0.986 0.995 0.022 0.024 0.018 0.028 
2 0.977 0.951 0.979 0.977 0.998 0.986 0.991 0.981 0.014 0.026 0.011 0.030 
3 0.992 0.973 0.992 0.986 0.993 0.981 0.991 0.988 0.008 0.018 0.006 0.019 
4 0.990 0.985 0.975 0.972 0.994 0.992 0.988 0.980 0.013 0.018 0.011 0.019 
0.9 0 0.971 0.970 0.991 0.977 0.970 0.979 0.995 0.989 0.018 0.021 0.015 0.022 
1 0.959 0.970 0.968 0.962 0.991 0.999 0.995 0.984 0.022 0.021 0.019 0.026 
2 0.958 0.957 0.989 0.979 0.997 0.977 0.985 0.982 0.018 0.026 0.016 0.028 
3 0.977 0.995 0.963 0.977 0.995 0.991 0.984 0.984 0.020 0.013 0.017 0.015 
4 0.954 0.972 0.964 0.999 0.977 0.978 0.993 0.999 0.028 0.013 0.022 0.018 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.989 0.991 0.998 0.994 0.972 0.983 0.986 0.981 0.014 0.013 0.017 0.014 
1 0.983 0.994 0.985 0.982 0.996 0.971 0.987 0.982 0.012 0.018 0.013 0.020 
2 0.963 0.960 0.963 0.987 0.983 0.971 0.984 0.996 0.027 0.022 0.029 0.026 
3 0.986 0.960 0.991 0.962 1.000 0.993 0.982 0.994 0.010 0.023 0.009 0.028 
4 0.952 0.991 0.986 0.979 0.998 0.988 0.986 0.982 0.020 0.015 0.018 0.016 
0.3 0 0.957 0.974 0.969 0.991 0.982 0.980 0.993 0.998 0.025 0.014 0.020 0.017 
1 0.979 0.972 0.963 0.986 0.990 0.977 0.988 0.997 0.020 0.017 0.016 0.019 
2 0.982 0.973 0.982 0.999 0.989 0.990 0.992 0.996 0.014 0.011 0.010 0.015 
3 0.986 0.955 0.988 0.985 0.992 0.979 0.982 0.993 0.013 0.022 0.010 0.026 
4 0.993 0.971 0.962 0.996 0.970 0.976 0.997 0.991 0.020 0.017 0.017 0.019 
0.5 0 0.950 0.954 0.961 0.992 0.978 0.977 0.985 0.983 0.032 0.024 0.024 0.027 
1 0.968 0.959 0.974 0.968 0.986 0.980 0.987 0.997 0.021 0.024 0.016 0.028 
2 0.998 0.970 0.975 0.999 0.989 0.978 0.993 0.980 0.011 0.018 0.010 0.021 
3 0.961 0.996 0.974 0.972 0.997 0.986 0.999 0.990 0.017 0.014 0.017 0.017 
4 0.997 0.987 0.984 0.983 0.988 0.988 0.994 0.985 0.009 0.014 0.007 0.014 
0.7 0 0.994 0.993 0.980 0.999 0.988 0.999 0.998 0.994 0.010 0.004 0.009 0.005 
1 0.961 0.956 0.994 0.972 0.973 0.982 0.986 0.995 0.022 0.024 0.018 0.028 
2 0.977 0.951 0.979 0.977 0.998 0.986 0.991 0.981 0.014 0.026 0.011 0.030 
3 0.992 0.973 0.992 0.986 0.993 0.981 0.991 0.988 0.008 0.018 0.006 0.019 
4 0.990 0.985 0.975 0.972 0.994 0.992 0.988 0.980 0.013 0.018 0.011 0.019 
0.9 0 0.971 0.970 0.991 0.977 0.970 0.979 0.995 0.989 0.018 0.021 0.015 0.022 
1 0.959 0.970 0.968 0.962 0.991 0.999 0.995 0.984 0.022 0.021 0.019 0.026 
2 0.958 0.957 0.989 0.979 0.997 0.977 0.985 0.982 0.018 0.026 0.016 0.028 
3 0.977 0.995 0.963 0.977 0.995 0.991 0.984 0.984 0.020 0.013 0.017 0.015 
4 0.954 0.972 0.964 0.999 0.977 0.978 0.993 0.999 0.028 0.013 0.022 0.018 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
View Large
Table 3

The power of and for standard uniform time series

MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.964 0.942 0.984 0.929 0.994 0.928 0.982 0.942 0.019 0.065 0.022 0.065 
1 0.964 0.935 0.988 0.923 0.984 0.936 0.992 0.939 0.018 0.067 0.021 0.067 
2 0.987 0.930 0.974 0.914 0.997 0.938 0.995 0.936 0.012 0.071 0.015 0.071 
3 0.991 0.947 0.977 0.916 0.975 0.929 0.983 0.943 0.019 0.066 0.014 0.067 
4 0.965 0.927 0.980 0.927 0.979 0.943 0.996 0.936 0.020 0.067 0.016 0.067 
0.3 0 0.993 0.918 0.967 0.916 0.980 0.920 0.990 0.936 0.018 0.078 0.014 0.078 
1 0.960 0.910 0.978 0.942 0.984 0.930 0.987 0.941 0.023 0.069 0.018 0.070 
2 0.959 0.917 0.967 0.912 0.970 0.945 0.995 0.933 0.027 0.073 0.021 0.074 
3 0.972 0.915 0.974 0.915 0.995 0.923 0.989 0.944 0.018 0.076 0.014 0.077 
4 0.989 0.908 0.961 0.936 0.978 0.945 0.993 0.939 0.020 0.068 0.016 0.069 
0.5 0 0.968 0.925 0.964 0.923 0.977 0.930 0.988 0.930 0.026 0.073 0.019 0.073 
1 0.996 0.941 0.992 0.927 0.990 0.924 0.992 0.949 0.008 0.065 0.006 0.066 
2 0.986 0.924 0.963 0.930 0.992 0.930 0.982 0.931 0.019 0.071 0.016 0.071 
3 0.973 0.915 0.998 0.910 0.998 0.924 0.997 0.944 0.009 0.077 0.010 0.078 
4 0.998 0.917 0.982 0.939 0.997 0.950 0.994 0.948 0.007 0.062 0.007 0.063 
0.7 0 0.996 0.909 1.000 0.932 0.989 0.930 0.985 0.945 0.008 0.071 0.007 0.072 
1 0.987 0.920 0.977 0.930 0.987 0.934 1.000 0.937 0.012 0.070 0.010 0.070 
2 0.964 0.914 0.995 0.929 0.988 0.940 0.986 0.944 0.017 0.068 0.014 0.069 
3 0.950 0.938 0.961 0.925 0.981 0.947 0.997 0.941 0.028 0.062 0.023 0.063 
4 0.987 0.940 0.986 0.936 0.989 0.935 0.992 0.949 0.012 0.060 0.008 0.060 
0.9 0 0.994 0.938 0.969 0.914 0.975 0.934 0.989 0.936 0.018 0.070 0.015 0.070 
1 0.988 0.907 0.961 0.940 0.988 0.935 0.988 0.933 0.019 0.071 0.016 0.072 
2 0.951 0.921 0.979 0.913 0.974 0.944 0.990 0.933 0.027 0.072 0.021 0.073 
3 0.952 0.930 0.967 0.924 0.974 0.927 0.981 0.947 0.032 0.068 0.024 0.069 
4 0.951 0.939 0.991 0.929 0.971 0.948 0.999 0.934 0.022 0.063 0.020 0.063 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.964 0.942 0.984 0.929 0.994 0.928 0.982 0.942 0.019 0.065 0.022 0.065 
1 0.964 0.935 0.988 0.923 0.984 0.936 0.992 0.939 0.018 0.067 0.021 0.067 
2 0.987 0.930 0.974 0.914 0.997 0.938 0.995 0.936 0.012 0.071 0.015 0.071 
3 0.991 0.947 0.977 0.916 0.975 0.929 0.983 0.943 0.019 0.066 0.014 0.067 
4 0.965 0.927 0.980 0.927 0.979 0.943 0.996 0.936 0.020 0.067 0.016 0.067 
0.3 0 0.993 0.918 0.967 0.916 0.980 0.920 0.990 0.936 0.018 0.078 0.014 0.078 
1 0.960 0.910 0.978 0.942 0.984 0.930 0.987 0.941 0.023 0.069 0.018 0.070 
2 0.959 0.917 0.967 0.912 0.970 0.945 0.995 0.933 0.027 0.073 0.021 0.074 
3 0.972 0.915 0.974 0.915 0.995 0.923 0.989 0.944 0.018 0.076 0.014 0.077 
4 0.989 0.908 0.961 0.936 0.978 0.945 0.993 0.939 0.020 0.068 0.016 0.069 
0.5 0 0.968 0.925 0.964 0.923 0.977 0.930 0.988 0.930 0.026 0.073 0.019 0.073 
1 0.996 0.941 0.992 0.927 0.990 0.924 0.992 0.949 0.008 0.065 0.006 0.066 
2 0.986 0.924 0.963 0.930 0.992 0.930 0.982 0.931 0.019 0.071 0.016 0.071 
3 0.973 0.915 0.998 0.910 0.998 0.924 0.997 0.944 0.009 0.077 0.010 0.078 
4 0.998 0.917 0.982 0.939 0.997 0.950 0.994 0.948 0.007 0.062 0.007 0.063 
0.7 0 0.996 0.909 1.000 0.932 0.989 0.930 0.985 0.945 0.008 0.071 0.007 0.072 
1 0.987 0.920 0.977 0.930 0.987 0.934 1.000 0.937 0.012 0.070 0.010 0.070 
2 0.964 0.914 0.995 0.929 0.988 0.940 0.986 0.944 0.017 0.068 0.014 0.069 
3 0.950 0.938 0.961 0.925 0.981 0.947 0.997 0.941 0.028 0.062 0.023 0.063 
4 0.987 0.940 0.986 0.936 0.989 0.935 0.992 0.949 0.012 0.060 0.008 0.060 
0.9 0 0.994 0.938 0.969 0.914 0.975 0.934 0.989 0.936 0.018 0.070 0.015 0.070 
1 0.988 0.907 0.961 0.940 0.988 0.935 0.988 0.933 0.019 0.071 0.016 0.072 
2 0.951 0.921 0.979 0.913 0.974 0.944 0.990 0.933 0.027 0.072 0.021 0.073 
3 0.952 0.930 0.967 0.924 0.974 0.927 0.981 0.947 0.032 0.068 0.024 0.069 
4 0.951 0.939 0.991 0.929 0.971 0.948 0.999 0.934 0.022 0.063 0.020 0.063 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
View Large
Table 4

The power of and for exponential (with mean 1) time series

MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.993 0.921 0.984 0.931 0.987 0.921 0.994 0.934 0.011 0.073 0.011 0.073 
1 0.975 0.949 0.967 0.930 0.989 0.925 0.985 0.943 0.021 0.063 0.023 0.064 
2 0.970 0.937 0.981 0.931 0.987 0.946 0.981 0.948 0.020 0.060 0.021 0.060 
3 0.960 0.935 0.998 0.937 0.990 0.924 0.998 0.945 0.014 0.065 0.015 0.065 
4 0.965 0.930 0.986 0.935 0.990 0.930 0.985 0.949 0.019 0.064 0.015 0.064 
0.3 0 0.990 0.920 0.969 0.938 0.982 0.932 0.992 0.937 0.017 0.068 0.013 0.069 
1 0.974 0.945 0.972 0.941 0.976 0.948 0.993 0.947 0.021 0.055 0.016 0.055 
2 0.969 0.935 0.973 0.935 0.987 0.933 0.988 0.949 0.021 0.062 0.016 0.062 
3 0.965 0.948 0.986 0.926 0.987 0.936 0.982 0.947 0.020 0.061 0.015 0.061 
4 0.971 0.900 0.970 0.945 0.995 0.945 0.985 0.946 0.020 0.066 0.016 0.069 
0.5 0 0.965 0.949 0.974 0.917 0.987 0.921 0.982 0.950 0.023 0.066 0.017 0.068 
1 0.976 0.928 0.981 0.937 0.972 0.934 0.994 0.931 0.019 0.067 0.015 0.068 
2 0.969 0.901 0.972 0.923 0.997 0.946 0.987 0.949 0.019 0.070 0.016 0.073 
3 0.957 0.933 0.961 0.940 0.991 0.934 0.990 0.942 0.025 0.063 0.021 0.063 
4 0.984 0.947 0.972 0.937 0.981 0.930 0.996 0.940 0.017 0.062 0.013 0.062 
0.7 0 0.980 0.911 0.990 0.918 0.996 0.948 0.980 0.946 0.014 0.069 0.011 0.071 
1 0.953 0.906 0.973 0.912 0.978 0.932 0.994 0.934 0.026 0.079 0.021 0.080 
2 0.994 0.914 0.986 0.934 0.979 0.948 0.990 0.931 0.013 0.068 0.010 0.069 
3 0.992 0.908 0.964 0.932 0.983 0.927 0.981 0.948 0.020 0.071 0.016 0.073 
4 0.971 0.919 0.983 0.950 0.994 0.929 0.984 0.949 0.017 0.063 0.013 0.065 
0.9 0 0.966 0.939 0.995 0.942 0.992 0.932 0.995 0.950 0.013 0.059 0.013 0.060 
1 0.957 0.934 0.985 0.941 0.999 0.921 0.981 0.936 0.020 0.067 0.017 0.067 
2 0.972 0.919 0.997 0.918 0.981 0.949 0.985 0.946 0.016 0.067 0.013 0.069 
3 0.977 0.925 0.988 0.941 0.978 0.929 0.987 0.947 0.018 0.065 0.013 0.065 
4 0.999 0.933 0.991 0.922 0.980 0.927 0.982 0.934 0.012 0.071 0.010 0.071 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.993 0.921 0.984 0.931 0.987 0.921 0.994 0.934 0.011 0.073 0.011 0.073 
1 0.975 0.949 0.967 0.930 0.989 0.925 0.985 0.943 0.021 0.063 0.023 0.064 
2 0.970 0.937 0.981 0.931 0.987 0.946 0.981 0.948 0.020 0.060 0.021 0.060 
3 0.960 0.935 0.998 0.937 0.990 0.924 0.998 0.945 0.014 0.065 0.015 0.065 
4 0.965 0.930 0.986 0.935 0.990 0.930 0.985 0.949 0.019 0.064 0.015 0.064 
0.3 0 0.990 0.920 0.969 0.938 0.982 0.932 0.992 0.937 0.017 0.068 0.013 0.069 
1 0.974 0.945 0.972 0.941 0.976 0.948 0.993 0.947 0.021 0.055 0.016 0.055 
2 0.969 0.935 0.973 0.935 0.987 0.933 0.988 0.949 0.021 0.062 0.016 0.062 
3 0.965 0.948 0.986 0.926 0.987 0.936 0.982 0.947 0.020 0.061 0.015 0.061 
4 0.971 0.900 0.970 0.945 0.995 0.945 0.985 0.946 0.020 0.066 0.016 0.069 
0.5 0 0.965 0.949 0.974 0.917 0.987 0.921 0.982 0.950 0.023 0.066 0.017 0.068 
1 0.976 0.928 0.981 0.937 0.972 0.934 0.994 0.931 0.019 0.067 0.015 0.068 
2 0.969 0.901 0.972 0.923 0.997 0.946 0.987 0.949 0.019 0.070 0.016 0.073 
3 0.957 0.933 0.961 0.940 0.991 0.934 0.990 0.942 0.025 0.063 0.021 0.063 
4 0.984 0.947 0.972 0.937 0.981 0.930 0.996 0.940 0.017 0.062 0.013 0.062 
0.7 0 0.980 0.911 0.990 0.918 0.996 0.948 0.980 0.946 0.014 0.069 0.011 0.071 
1 0.953 0.906 0.973 0.912 0.978 0.932 0.994 0.934 0.026 0.079 0.021 0.080 
2 0.994 0.914 0.986 0.934 0.979 0.948 0.990 0.931 0.013 0.068 0.010 0.069 
3 0.992 0.908 0.964 0.932 0.983 0.927 0.981 0.948 0.020 0.071 0.016 0.073 
4 0.971 0.919 0.983 0.950 0.994 0.929 0.984 0.949 0.017 0.063 0.013 0.065 
0.9 0 0.966 0.939 0.995 0.942 0.992 0.932 0.995 0.950 0.013 0.059 0.013 0.060 
1 0.957 0.934 0.985 0.941 0.999 0.921 0.981 0.936 0.020 0.067 0.017 0.067 
2 0.972 0.919 0.997 0.918 0.981 0.949 0.985 0.946 0.016 0.067 0.013 0.069 
3 0.977 0.925 0.988 0.941 0.978 0.929 0.987 0.947 0.018 0.065 0.013 0.065 
4 0.999 0.933 0.991 0.922 0.980 0.927 0.982 0.934 0.012 0.071 0.010 0.071 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
View Large
Table 5

The power of and for Brownian motion time series

MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.965 0.928 0.978 0.932 0.995 0.934 0.991 0.934 0.018 0.068 0.021 0.068 
1 0.972 0.951 0.973 0.950 0.981 0.956 0.981 0.956 0.023 0.047 0.024 0.047 
2 0.970 0.931 0.978 0.938 0.987 0.947 0.992 0.951 0.018 0.058 0.020 0.059 
3 0.976 0.936 0.978 0.938 0.989 0.944 0.989 0.950 0.017 0.058 0.013 0.058 
4 0.978 0.943 0.986 0.908 0.989 0.930 0.992 0.946 0.014 0.068 0.010 0.070 
0.3 0 0.965 0.917 0.985 0.930 0.994 0.938 0.997 0.947 0.015 0.067 0.014 0.068 
1 0.965 0.907 0.970 0.943 0.997 0.949 0.998 0.949 0.018 0.063 0.016 0.065 
2 0.968 0.903 0.968 0.929 0.972 0.949 0.986 0.961 0.027 0.065 0.019 0.068 
3 0.967 0.903 0.976 0.904 0.976 0.912 0.980 0.963 0.025 0.080 0.018 0.083 
4 0.963 0.925 0.978 0.937 0.982 0.941 0.998 0.955 0.020 0.061 0.017 0.061 
0.5 0 0.950 0.934 0.963 0.942 0.978 0.953 0.984 0.961 0.031 0.053 0.024 0.054 
1 0.967 0.948 0.980 0.957 0.985 0.964 0.991 0.969 0.019 0.041 0.015 0.041 
2 0.963 0.925 0.974 0.943 0.981 0.944 0.986 0.956 0.024 0.058 0.018 0.059 
3 0.962 0.909 0.975 0.918 0.985 0.935 0.992 0.955 0.022 0.071 0.017 0.073 
4 0.990 0.918 0.995 0.956 0.995 0.964 0.998 0.969 0.006 0.048 0.004 0.052 
0.7 0 0.978 0.912 0.987 0.939 0.991 0.941 0.994 0.965 0.013 0.061 0.010 0.064 
1 0.974 0.901 0.980 0.926 0.988 0.939 0.995 0.960 0.016 0.069 0.012 0.072 
2 0.957 0.919 0.978 0.933 0.991 0.958 0.997 0.961 0.019 0.057 0.017 0.060 
3 0.931 0.955 0.961 0.963 0.989 0.964 0.998 0.972 0.030 0.037 0.028 0.037 
4 0.968 0.952 0.973 0.955 0.973 0.955 0.992 0.962 0.024 0.044 0.018 0.044 
0.9 0 0.966 0.919 0.979 0.925 0.984 0.932 0.995 0.942 0.019 0.071 0.015 0.071 
1 0.973 0.904 0.976 0.940 0.989 0.947 0.992 0.958 0.018 0.063 0.014 0.066 
2 0.968 0.932 0.972 0.934 0.981 0.944 0.995 0.963 0.021 0.057 0.017 0.058 
3 0.952 0.938 0.971 0.943 0.990 0.954 0.997 0.959 0.023 0.052 0.020 0.052 
4 0.965 0.929 0.974 0.933 0.986 0.965 0.995 0.967 0.020 0.052 0.016 0.054 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
MAE
RMSE
100
200
500
1,000
SCCFCCFSCCFCCF
0.1 0 0.965 0.928 0.978 0.932 0.995 0.934 0.991 0.934 0.018 0.068 0.021 0.068 
1 0.972 0.951 0.973 0.950 0.981 0.956 0.981 0.956 0.023 0.047 0.024 0.047 
2 0.970 0.931 0.978 0.938 0.987 0.947 0.992 0.951 0.018 0.058 0.020 0.059 
3 0.976 0.936 0.978 0.938 0.989 0.944 0.989 0.950 0.017 0.058 0.013 0.058 
4 0.978 0.943 0.986 0.908 0.989 0.930 0.992 0.946 0.014 0.068 0.010 0.070 
0.3 0 0.965 0.917 0.985 0.930 0.994 0.938 0.997 0.947 0.015 0.067 0.014 0.068 
1 0.965 0.907 0.970 0.943 0.997 0.949 0.998 0.949 0.018 0.063 0.016 0.065 
2 0.968 0.903 0.968 0.929 0.972 0.949 0.986 0.961 0.027 0.065 0.019 0.068 
3 0.967 0.903 0.976 0.904 0.976 0.912 0.980 0.963 0.025 0.080 0.018 0.083 
4 0.963 0.925 0.978 0.937 0.982 0.941 0.998 0.955 0.020 0.061 0.017 0.061 
0.5 0 0.950 0.934 0.963 0.942 0.978 0.953 0.984 0.961 0.031 0.053 0.024 0.054 
1 0.967 0.948 0.980 0.957 0.985 0.964 0.991 0.969 0.019 0.041 0.015 0.041 
2 0.963 0.925 0.974 0.943 0.981 0.944 0.986 0.956 0.024 0.058 0.018 0.059 
3 0.962 0.909 0.975 0.918 0.985 0.935 0.992 0.955 0.022 0.071 0.017 0.073 
4 0.990 0.918 0.995 0.956 0.995 0.964 0.998 0.969 0.006 0.048 0.004 0.052 
0.7 0 0.978 0.912 0.987 0.939 0.991 0.941 0.994 0.965 0.013 0.061 0.010 0.064 
1 0.974 0.901 0.980 0.926 0.988 0.939 0.995 0.960 0.016 0.069 0.012 0.072 
2 0.957 0.919 0.978 0.933 0.991 0.958 0.997 0.961 0.019 0.057 0.017 0.060 
3 0.931 0.955 0.961 0.963 0.989 0.964 0.998 0.972 0.030 0.037 0.028 0.037 
4 0.968 0.952 0.973 0.955 0.973 0.955 0.992 0.962 0.024 0.044 0.018 0.044 
0.9 0 0.966 0.919 0.979 0.925 0.984 0.932 0.995 0.942 0.019 0.071 0.015 0.071 
1 0.973 0.904 0.976 0.940 0.989 0.947 0.992 0.958 0.018 0.063 0.014 0.066 
2 0.968 0.932 0.972 0.934 0.981 0.944 0.995 0.963 0.021 0.057 0.017 0.058 
3 0.952 0.938 0.971 0.943 0.990 0.954 0.997 0.959 0.023 0.052 0.020 0.052 
4 0.965 0.929 0.974 0.933 0.986 0.965 0.995 0.967 0.020 0.052 0.016 0.054 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
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Table 6

The power of and for fractional Brownian motion time series

100
200
500
1,000
MAE
RMSE
SCCFCCFSCCFCCF
0.1 0 0.965 0.929 0.974 0.933 0.986 0.965 0.995 0.967 0.020 0.052 0.023 0.054 
1 0.974 0.915 0.986 0.924 0.990 0.941 0.993 0.956 0.014 0.066 0.016 0.068 
2 0.966 0.939 0.968 0.943 0.995 0.946 0.997 0.955 0.019 0.054 0.024 0.055 
3 0.969 0.939 0.973 0.945 0.978 0.954 0.997 0.962 0.021 0.050 0.017 0.051 
4 0.972 0.927 0.980 0.933 0.985 0.949 0.994 0.959 0.017 0.058 0.013 0.059 
0.3 0 0.968 0.935 0.975 0.937 0.982 0.951 0.989 0.959 0.022 0.055 0.016 0.055 
1 0.973 0.936 0.981 0.947 0.991 0.956 0.992 0.961 0.016 0.050 0.012 0.051 
2 0.966 0.937 0.980 0.942 0.989 0.953 0.999 0.965 0.017 0.051 0.014 0.052 
3 0.968 0.936 0.979 0.943 0.983 0.958 0.993 0.969 0.019 0.049 0.015 0.050 
4 0.960 0.935 0.982 0.936 0.987 0.961 0.992 0.969 0.020 0.050 0.016 0.052 
0.5 0 0.976 0.925 0.985 0.938 0.991 0.956 0.993 0.966 0.014 0.054 0.011 0.056 
1 0.970 0.929 0.983 0.934 0.990 0.941 0.990 0.956 0.017 0.060 0.013 0.061 
2 0.958 0.915 0.971 0.935 0.989 0.943 0.997 0.959 0.021 0.062 0.018 0.064 
3 0.963 0.919 0.973 0.934 0.987 0.941 0.995 0.964 0.021 0.061 0.017 0.063 
4 0.965 0.925 0.969 0.931 0.987 0.945 0.995 0.958 0.021 0.060 0.017 0.062 
0.7 0 0.970 0.933 0.982 0.942 0.993 0.952 0.998 0.964 0.014 0.052 0.013 0.054 
1 0.966 0.915 0.987 0.926 0.992 0.939 0.994 0.942 0.015 0.070 0.013 0.070 
2 0.965 0.929 0.968 0.940 0.985 0.948 0.998 0.955 0.021 0.057 0.018 0.058 
3 0.976 0.930 0.977 0.945 0.989 0.957 0.994 0.961 0.016 0.052 0.013 0.053 
4 0.982 0.922 0.986 0.936 0.989 0.951 0.998 0.963 0.011 0.057 0.009 0.059 
0.9 0 0.976 0.923 0.984 0.936 0.989 0.957 0.999 0.960 0.013 0.056 0.011 0.058 
1 0.964 0.932 0.981 0.936 0.982 0.955 0.992 0.961 0.020 0.054 0.016 0.055 
2 0.977 0.930 0.984 0.938 0.987 0.959 1.000 0.966 0.013 0.052 0.011 0.054 
3 0.969 0.928 0.982 0.939 0.992 0.953 0.995 0.965 0.016 0.054 0.013 0.056 
4 0.972 0.926 0.990 0.945 0.991 0.963 0.993 0.968 0.014 0.050 0.011 0.052 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
100
200
500
1,000
MAE
RMSE
SCCFCCFSCCFCCF
0.1 0 0.965 0.929 0.974 0.933 0.986 0.965 0.995 0.967 0.020 0.052 0.023 0.054 
1 0.974 0.915 0.986 0.924 0.990 0.941 0.993 0.956 0.014 0.066 0.016 0.068 
2 0.966 0.939 0.968 0.943 0.995 0.946 0.997 0.955 0.019 0.054 0.024 0.055 
3 0.969 0.939 0.973 0.945 0.978 0.954 0.997 0.962 0.021 0.050 0.017 0.051 
4 0.972 0.927 0.980 0.933 0.985 0.949 0.994 0.959 0.017 0.058 0.013 0.059 
0.3 0 0.968 0.935 0.975 0.937 0.982 0.951 0.989 0.959 0.022 0.055 0.016 0.055 
1 0.973 0.936 0.981 0.947 0.991 0.956 0.992 0.961 0.016 0.050 0.012 0.051 
2 0.966 0.937 0.980 0.942 0.989 0.953 0.999 0.965 0.017 0.051 0.014 0.052 
3 0.968 0.936 0.979 0.943 0.983 0.958 0.993 0.969 0.019 0.049 0.015 0.050 
4 0.960 0.935 0.982 0.936 0.987 0.961 0.992 0.969 0.020 0.050 0.016 0.052 
0.5 0 0.976 0.925 0.985 0.938 0.991 0.956 0.993 0.966 0.014 0.054 0.011 0.056 
1 0.970 0.929 0.983 0.934 0.990 0.941 0.990 0.956 0.017 0.060 0.013 0.061 
2 0.958 0.915 0.971 0.935 0.989 0.943 0.997 0.959 0.021 0.062 0.018 0.064 
3 0.963 0.919 0.973 0.934 0.987 0.941 0.995 0.964 0.021 0.061 0.017 0.063 
4 0.965 0.925 0.969 0.931 0.987 0.945 0.995 0.958 0.021 0.060 0.017 0.062 
0.7 0 0.970 0.933 0.982 0.942 0.993 0.952 0.998 0.964 0.014 0.052 0.013 0.054 
1 0.966 0.915 0.987 0.926 0.992 0.939 0.994 0.942 0.015 0.070 0.013 0.070 
2 0.965 0.929 0.968 0.940 0.985 0.948 0.998 0.955 0.021 0.057 0.018 0.058 
3 0.976 0.930 0.977 0.945 0.989 0.957 0.994 0.961 0.016 0.052 0.013 0.053 
4 0.982 0.922 0.986 0.936 0.989 0.951 0.998 0.963 0.011 0.057 0.009 0.059 
0.9 0 0.976 0.923 0.984 0.936 0.989 0.957 0.999 0.960 0.013 0.056 0.011 0.058 
1 0.964 0.932 0.981 0.936 0.982 0.955 0.992 0.961 0.020 0.054 0.016 0.055 
2 0.977 0.930 0.984 0.938 0.987 0.959 1.000 0.966 0.013 0.052 0.011 0.054 
3 0.969 0.928 0.982 0.939 0.992 0.953 0.995 0.965 0.016 0.054 0.013 0.056 
4 0.972 0.926 0.990 0.945 0.991 0.963 0.993 0.968 0.014 0.050 0.011 0.052 
MAE 0.025 0.027 0.022 0.017 0.013 0.017 0.010 0.011     
RMSE 0.042 0.046 0.030 0.020 0.011 0.016 0.006 0.008     
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Ability assessment of SCCF using the real data

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In this section, the ability of SCCF in practice is investigated by an actual example. For this purpose, the calculated data of RDI in 1, 3, and 12-month time scales were used. The ADF and KPSS tests were used to investigate the stationarity of the datasets. The results of the ADF and KPSS tests verify that the calculated RDI in all time scales were stationary.

Data collection and evaluation

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In this research, the humidity data in 1971 at Tabass were estimated using the Fuzzy regression. Based on the results, the time duration of the data series (Based on the Mockus equation) was significantly adequate at all stations (at a 0.99 significant level). The results of the Run-test indicated that all data series at all stations were homogeneous.

Calculated RDI

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Calculated RDI in selected stations at different time scales showed that at all stations and all time scales, the normal class of drought severity (with RDI value between 0.99 and −0.99) had the most occurrence frequency (Figures 3,4–5). Based on the results, the most severe drought occurred in Babolsar, Bandar Abbas, Bandar Lengeh, Birjand, Chabahar, Fasa, Qazvin, Iran Shahr, Mashhad, Oroomieh, Ramsar, Semnan, Shiraz, Tabass, Tabriz, Tehran, Torbat Hydarieh, Zabol, Zahedan, and Zanjan stations in 2010, 2010, 2001, 2018, 2010, 2010, 2017, 2008, 2000, 2010, 2016, 2016, 2008, 2008, 2001, 1999, 1973, 2014, 2017, and 1967 years, respectively. The main reason for the occurrence of severe droughts in the years mentioned above is a significant decrease in the amount of rainfall. This issue has been confirmed in Zarei et al. (2022) research.
Figure 3
Calculated monthly (1-month) RDI in some of the selected stations.
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Calculated monthly (1-month) RDI in some of the selected stations.

Figure 3
Calculated monthly (1-month) RDI in some of the selected stations.
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Calculated monthly (1-month) RDI in some of the selected stations.

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Figure 4
Calculated seasonal (3-month) RDI in some of the selected stations.
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Calculated seasonal (3-month) RDI in some of the selected stations.

Figure 4
Calculated seasonal (3-month) RDI in some of the selected stations.
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Calculated seasonal (3-month) RDI in some of the selected stations.

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Figure 5
Calculated annually (12-month) RDI in some of the selected stations.
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Calculated annually (12-month) RDI in some of the selected stations.

Figure 5
Calculated annually (12-month) RDI in some of the selected stations.
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Calculated annually (12-month) RDI in some of the selected stations.

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Normality test of RDI data series

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In this study, the Kolmogorov–Smirnov test was used to assess the normality of the calculated RDI in 1, 3, and 12-month time scales. Based on the results of the normality test, the calculated RDI at all stations and all chosen time scales were non-normal at the 0.05 significant level (Table 7). Table 7 shows that in all stations and all time scales, the calculated significance levels for the Kolmogorov–Smirnov were less than 0.05.

Table 7

Results of the normality test of calculated RDI in 1, 3, and 12-month time scales (based on Kolmogorov–Smirnov test)

Station12-month (Annual)3-month (Seasonal)1-month (monthly)
Significant level
Babolsar 0.0000 0.0000 0.0000 
Bandar Abbas 0.0000 0.0000 0.0000 
Bandar Lengeh 0.0031 0.0000 0.0000 
Birjand 0.0000 0.0000 0.0000 
Chabahar 0.0000 0.0000 0.0000 
Fasa 0.0000 0.0000 0.0000 
Qazvin 0.0030 0.0000 0.0000 
Iran Shahr 0.0000 0.0000 0.0000 
Mashhad 0.0173 0.0000 0.0000 
Oroomieh 0.0105 0.0000 0.0000 
Ramsar 0.0024 0.0000 0.0000 
Semnan 0.0000 0.0000 0.0000 
Shiraz 0.0000 0.0000 0.0000 
Tabass 0.0002 0.0000 0.0000 
Tabriz 0.0014 0.0000 0.0000 
Tehran 0.0001 0.0000 0.0000 
Torbat Hydarieh 0.0000 0.0000 0.0000 
Zabol 0.0001 0.0000 0.0000 
Zahedan 0.0036 0.0000 0.0000 
Zanjan 0.0000 0.0000 0.0000 
Station12-month (Annual)3-month (Seasonal)1-month (monthly)
Significant level
Babolsar 0.0000 0.0000 0.0000 
Bandar Abbas 0.0000 0.0000 0.0000 
Bandar Lengeh 0.0031 0.0000 0.0000 
Birjand 0.0000 0.0000 0.0000 
Chabahar 0.0000 0.0000 0.0000 
Fasa 0.0000 0.0000 0.0000 
Qazvin 0.0030 0.0000 0.0000 
Iran Shahr 0.0000 0.0000 0.0000 
Mashhad 0.0173 0.0000 0.0000 
Oroomieh 0.0105 0.0000 0.0000 
Ramsar 0.0024 0.0000 0.0000 
Semnan 0.0000 0.0000 0.0000 
Shiraz 0.0000 0.0000 0.0000 
Tabass 0.0002 0.0000 0.0000 
Tabriz 0.0014 0.0000 0.0000 
Tehran 0.0001 0.0000 0.0000 
Torbat Hydarieh 0.0000 0.0000 0.0000 
Zabol 0.0001 0.0000 0.0000 
Zahedan 0.0036 0.0000 0.0000 
Zanjan 0.0000 0.0000 0.0000 

Note: Calculated RDI data series at all station and all time scales were non-normal.

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SCCF ability assessment

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To assess the ability of SCCF using the actual RDI data in 1, 3, and 12-month time scales at first, 15 groups of meteorological stations were defined; 10 out of 15 groups included two stations with short spatial distance, and 5 out of 15 groups included two stations with long spatial distance. Groups with short spatial distance, including group 1 including Fasa and Shiraz stations, group 2 including Zabol and Zahedan stations, group 3 including Bandar Abass and Bandar Lenge stations, group 4 including Oroomieh and Tabriz stations, group 5 including Ramsar and Babolsar stations, group 6 including Birjand and Tabass stations, group 7 including Torbat Hydarieh and Mashhad stations, group 8 including Iran Shahr and Chabahar stations, group 9 including Qazvin and Zanjan stations, and group 10 including Semnan, and Tehran stations and groups with long spatial distance including group 11 including Fasa and Tabriz stations, group 12 including Qazvin and Zahedan stations, group 13 including Bandar Abass and Mashhad stations, group 14 including Oroomieh, and Shiraz stations, and group 15 including Chabahar and Semnan stations. Then the SCCF between the RDI values of two stations in each group at different time scales in 21 lags, including 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, −1, −2, −3, −4, −5, −6, −7, −8, −9, and −10 were estimated. It is expected that, if the SCCF had good ability, the result of SCCF between stations with short spatial distance does not reveal delay time in drought occurrence. In stations with long spatial distances, this result should be reversed.

The results of the SCCF of the calculated RDI data series in groups 1–10, including stations with short spatial distance, indicated that in 1, 3, and 12-month time scales at all groups, the maximum values of SCCF were estimated in lag 0 (Tables 8,9–10). In other words, in all stations close to each other, the drought occurrence is simultaneous and has no time delay. According to the results of the SCCF of the RDI data series in groups 11–15, including stations with long spatial distances, in 1, 3, and 12-month time scales, at all groups, the occurrence of drought at one station versus another station has time delay (Tables 11,12–13). This result proves the good ability of the SCCF model. Based on the results, in the monthly time scale, the maximum absolute value of SCCF in groups 11, 12, 13, 14, and 15 occurred in lags 6, 6, −6, 1, and 1, respectively. In the seasonal time scale, the maximum absolute value of SCCF in groups 11, 12, 13, 14, and 15 occurred in lags −4, 2, 4, 4, and 4, respectively. In annual time scale, the maximum absolute value of SCCF in groups 11, 12, 13, 14, and 15 occurred in lags 4, −4, −10, −4, and 4, respectively (Figure 6). It seems the main reason for the greater ability of the SCCF model compared to the CCF model is that the SCCF model is a hybrid model (a hybrid of the Spearman and SSF models). Roustaei et al. (2021) assessed the time delay between wind erosion and drought in the Southern regions of Iran using the CCF function. They showed that the maximum CC between drought and wind erosion was equal to −0.22 (without time lag). While the calculated correlation values by the SCCF model in this research were much higher.
Table 8

Spearman cross-correlations (SCCF) between calculated annually (12-month) RDI values in groups with stations with low spatial distance

SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 0.0005 0.1049 − 0.0202 0.2995 − 0.0403 0.1082 0.1234 − 0.1101 0.1185 − 0.1329 
−9 − 0.0307 0.0443 − 0.0065 0.2463 0.1416 − 0.1320 − 0.0497 − 0.2020 − 0.0040 − 0.0858 
−8 − 0.1168 0.2533 − 0.0339 0.3018 − 0.1088 − 0.0138 0.0601 − 0.0543 0.0160 − 0.0364 
−7 0.0803 0.1000 0.0288 0.1830 − 0.0510 0.0815 0.1339 0.1147 − 0.0714 − 0.0528 
−6 0.0245 0.1347 0.0004 0.1420 0.0950 0.0186 0.2456 − 0.0553 0.0411 − 0.0900 
−5 0.0035 0.0029 0.0869 0.2460 − 0.0505 0.1119 0.2367 0.0023 − 0.1538 0.1058 
−4 0.0702 0.0819 0.0649 0.1850 0.1297 − 0.0557 0.0347 0.1344 − 0.2147 0.0852 
−3 0.0121 − 0.1011 0.1224 0.1245 − 0.0456 0.1338 0.1384 0.0419 − 0.0503 0.1009 
−2 0.0951 0.0476 0.2386 0.1611 − 0.1913 0.0019 0.0581 0.3827 − 0.2104 0.1566 
−1 − 0.1325 0.1336 0.0397 0.3220 0.0918 0.2396 0.1591 0.1876 − 0.0391 − 0.1055 
0 0.4297 0.5673 0.8729 0.7821 0.3120 0.6616 0.6968 0.3965 0.5961 0.6890 
1 − 0.1149 0.1047 0.0931 0.2272 0.1486 0.0794 0.1338 − 0.0817 − 0.0275 − 0.1269 
2 − 0.0568 0.1947 0.1702 0.2074 0.0631 0.1151 0.1174 0.0856 0.0948 0.1294 
3 − 0.0810 − 0.0547 0.1493 − 0.0521 − 0.0767 0.1301 − 0.1865 − 0.0649 0.0127 0.2378 
4 − 0.0623 − 0.0093 0.0224 0.0975 0.1113 0.1562 0.1326 − 0.1634 − 0.0374 0.0781 
5 − 0.3050 − 0.0793 0.0945 0.2404 0.2806 0.1541 0.1126 − 0.1559 − 0.0812 0.3509 
6 − 0.0113 0.0185 0.0130 − 0.0191 0.0144 0.0058 0.1522 − 0.1215 − 0.1526 0.0177 
7 − 0.1435 − 0.1232 − 0.0400 0.1418 − 0.1067 0.1551 0.2771 − 0.2987 − 0.0980 0.1085 
8 0.0107 − 0.1440 0.0628 0.0439 0.1732 − 0.1410 − 0.1135 − 0.1076 − 0.1564 0.1201 
9 − 0.1781 − 0.1499 − 0.0630 0.1401 0.0102 0.2065 − 0.0261 − 0.0937 0.0450 − 0.1033 
10 0.1346 0.0691 − 0.0632 0.2079 0.2402 0.0120 0.0588 − 0.1474 0.0529 0.0882 
SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 0.0005 0.1049 − 0.0202 0.2995 − 0.0403 0.1082 0.1234 − 0.1101 0.1185 − 0.1329 
−9 − 0.0307 0.0443 − 0.0065 0.2463 0.1416 − 0.1320 − 0.0497 − 0.2020 − 0.0040 − 0.0858 
−8 − 0.1168 0.2533 − 0.0339 0.3018 − 0.1088 − 0.0138 0.0601 − 0.0543 0.0160 − 0.0364 
−7 0.0803 0.1000 0.0288 0.1830 − 0.0510 0.0815 0.1339 0.1147 − 0.0714 − 0.0528 
−6 0.0245 0.1347 0.0004 0.1420 0.0950 0.0186 0.2456 − 0.0553 0.0411 − 0.0900 
−5 0.0035 0.0029 0.0869 0.2460 − 0.0505 0.1119 0.2367 0.0023 − 0.1538 0.1058 
−4 0.0702 0.0819 0.0649 0.1850 0.1297 − 0.0557 0.0347 0.1344 − 0.2147 0.0852 
−3 0.0121 − 0.1011 0.1224 0.1245 − 0.0456 0.1338 0.1384 0.0419 − 0.0503 0.1009 
−2 0.0951 0.0476 0.2386 0.1611 − 0.1913 0.0019 0.0581 0.3827 − 0.2104 0.1566 
−1 − 0.1325 0.1336 0.0397 0.3220 0.0918 0.2396 0.1591 0.1876 − 0.0391 − 0.1055 
0 0.4297 0.5673 0.8729 0.7821 0.3120 0.6616 0.6968 0.3965 0.5961 0.6890 
1 − 0.1149 0.1047 0.0931 0.2272 0.1486 0.0794 0.1338 − 0.0817 − 0.0275 − 0.1269 
2 − 0.0568 0.1947 0.1702 0.2074 0.0631 0.1151 0.1174 0.0856 0.0948 0.1294 
3 − 0.0810 − 0.0547 0.1493 − 0.0521 − 0.0767 0.1301 − 0.1865 − 0.0649 0.0127 0.2378 
4 − 0.0623 − 0.0093 0.0224 0.0975 0.1113 0.1562 0.1326 − 0.1634 − 0.0374 0.0781 
5 − 0.3050 − 0.0793 0.0945 0.2404 0.2806 0.1541 0.1126 − 0.1559 − 0.0812 0.3509 
6 − 0.0113 0.0185 0.0130 − 0.0191 0.0144 0.0058 0.1522 − 0.1215 − 0.1526 0.0177 
7 − 0.1435 − 0.1232 − 0.0400 0.1418 − 0.1067 0.1551 0.2771 − 0.2987 − 0.0980 0.1085 
8 0.0107 − 0.1440 0.0628 0.0439 0.1732 − 0.1410 − 0.1135 − 0.1076 − 0.1564 0.1201 
9 − 0.1781 − 0.1499 − 0.0630 0.1401 0.0102 0.2065 − 0.0261 − 0.0937 0.0450 − 0.1033 
10 0.1346 0.0691 − 0.0632 0.2079 0.2402 0.0120 0.0588 − 0.1474 0.0529 0.0882 

Note: Each group includes two adjacent stations (with low spatial distance). Group1 including Fasa and Shiraz stations, group 2 including Zabol and Zahedan stations, group 3 including Bandar Abass and Bandar Lenge stations, group 4 including Oroomieh and Tabriz stations, group 5 including Ramsar and Babolsar stations, group 6 including Birjand and Tabass stations, group 7 including Torbat Hydarieh and Mashhad stations, group 8 including Iran Shahr and Chabahar stations, group 9 including Qazvin and Zanjan stations, and group 10 including Semnan and Tehran stations.

Bold characters are the maximum values of SCCF.

View Large
Table 9

Spearman cross-correlations (SCCF) between calculated seasonal (3-month) RDI values in groups with stations with low spatial distance

SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 − 0.7360 − 0.5862 − 0.5520 − 0.5898 − 0.7800 − 0.7278 − 0.7676 − 0.1937 − 0.7275 − 0.7269 
−9 0.0428 − 0.0306 0.0245 − 0.0367 0.0203 0.1093 0.0059 0.0540 0.0208 0.0140 
−8 0.7475 0.6389 0.6299 0.7129 0.7749 0.7444 0.7919 0.3915 0.7744 0.7251 
−7 0.0064 0.1026 0.0165 − 0.0076 − 0.0143 − 0.0658 0.0204 − 0.1182 − 0.0625 − 0.0186 
−6 − 0.7424 − 0.6635 − 0.5639 − 0.5768 − 0.7684 − 0.7906 − 0.8059 − 0.1550 − 0.7396 − 0.7229 
−5 0.0285 − 0.0414 0.0254 − 0.0286 0.0017 0.1055 − 0.0289 0.0354 0.0064 − 0.0017 
−4 0.7411 0.5975 0.5693 0.7520 0.8005 0.7605 0.8108 0.4160 0.7891 0.7336 
−3 − 0.0202 0.0714 − 0.0103 0.0170 − 0.0143 − 0.0637 0.0263 − 0.0786 − 0.0672 − 0.0190 
−2 − 0.7467 − 0.6046 − 0.5533 − 0.6234 − 0.8023 − 0.7444 − 0.8016 − 0.2052 − 0.7405 − 0.7378 
−1 0.0221 − 0.0688 0.0162 − 0.0216 0.0090 0.1165 − 0.0066 0.0448 0.0006 0.0341 
0 0.9505 0.8668 0.8959 0.9163 0.9015 0.9076 0.9490 0.5419 0.9252 0.8977 
1 − 0.0236 0.0889 − 0.0238 − 0.0179 − 0.0304 − 0.0817 0.0262 − 0.1146 − 0.0557 − 0.0177 
2 − 0.7374 − 0.5869 − 0.5790 − 0.6108 − 0.7950 − 0.7444 − 0.7889 − 0.2621 − 0.7396 − 0.7409 
3 0.0480 − 0.0567 0.0439 − 0.0411 0.0408 0.0882 − 0.0097 − 0.0610 0.0025 0.0382 
4 0.7322 0.6167 0.5979 0.7450 0.8178 0.7583 0.8053 0.4519 0.7819 0.7545 
5 − 0.0014 0.0824 − 0.0004 − 0.0233 − 0.0298 − 0.0695 0.0090 − 0.1389 − 0.0708 − 0.0322 
6 − 0.7539 − 0.6606 − 0.6218 − 0.5967 − 0.7882 − 0.7896 − 0.8002 − 0.2217 − 0.7376 − 0.7245 
7 0.0217 − 0.0571 − 0.0059 − 0.0395 0.0311 0.1028 − 0.0026 0.0019 0.0165 0.0491 
8 0.7188 0.6764 0.6377 0.7331 0.7883 0.7473 0.7849 0.3762 0.7847 0.7572 
9 − 0.0218 0.0764 − 0.0244 − 0.0027 − 0.0280 − 0.0881 0.0427 − 0.1812 − 0.0440 − 0.0159 
10 − 0.7299 − 0.5909 − 0.5938 − 0.6077 − 0.7710 − 0.7414 − 0.7600 − 0.2742 − 0.7053 − 0.7073 
SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 − 0.7360 − 0.5862 − 0.5520 − 0.5898 − 0.7800 − 0.7278 − 0.7676 − 0.1937 − 0.7275 − 0.7269 
−9 0.0428 − 0.0306 0.0245 − 0.0367 0.0203 0.1093 0.0059 0.0540 0.0208 0.0140 
−8 0.7475 0.6389 0.6299 0.7129 0.7749 0.7444 0.7919 0.3915 0.7744 0.7251 
−7 0.0064 0.1026 0.0165 − 0.0076 − 0.0143 − 0.0658 0.0204 − 0.1182 − 0.0625 − 0.0186 
−6 − 0.7424 − 0.6635 − 0.5639 − 0.5768 − 0.7684 − 0.7906 − 0.8059 − 0.1550 − 0.7396 − 0.7229 
−5 0.0285 − 0.0414 0.0254 − 0.0286 0.0017 0.1055 − 0.0289 0.0354 0.0064 − 0.0017 
−4 0.7411 0.5975 0.5693 0.7520 0.8005 0.7605 0.8108 0.4160 0.7891 0.7336 
−3 − 0.0202 0.0714 − 0.0103 0.0170 − 0.0143 − 0.0637 0.0263 − 0.0786 − 0.0672 − 0.0190 
−2 − 0.7467 − 0.6046 − 0.5533 − 0.6234 − 0.8023 − 0.7444 − 0.8016 − 0.2052 − 0.7405 − 0.7378 
−1 0.0221 − 0.0688 0.0162 − 0.0216 0.0090 0.1165 − 0.0066 0.0448 0.0006 0.0341 
0 0.9505 0.8668 0.8959 0.9163 0.9015 0.9076 0.9490 0.5419 0.9252 0.8977 
1 − 0.0236 0.0889 − 0.0238 − 0.0179 − 0.0304 − 0.0817 0.0262 − 0.1146 − 0.0557 − 0.0177 
2 − 0.7374 − 0.5869 − 0.5790 − 0.6108 − 0.7950 − 0.7444 − 0.7889 − 0.2621 − 0.7396 − 0.7409 
3 0.0480 − 0.0567 0.0439 − 0.0411 0.0408 0.0882 − 0.0097 − 0.0610 0.0025 0.0382 
4 0.7322 0.6167 0.5979 0.7450 0.8178 0.7583 0.8053 0.4519 0.7819 0.7545 
5 − 0.0014 0.0824 − 0.0004 − 0.0233 − 0.0298 − 0.0695 0.0090 − 0.1389 − 0.0708 − 0.0322 
6 − 0.7539 − 0.6606 − 0.6218 − 0.5967 − 0.7882 − 0.7896 − 0.8002 − 0.2217 − 0.7376 − 0.7245 
7 0.0217 − 0.0571 − 0.0059 − 0.0395 0.0311 0.1028 − 0.0026 0.0019 0.0165 0.0491 
8 0.7188 0.6764 0.6377 0.7331 0.7883 0.7473 0.7849 0.3762 0.7847 0.7572 
9 − 0.0218 0.0764 − 0.0244 − 0.0027 − 0.0280 − 0.0881 0.0427 − 0.1812 − 0.0440 − 0.0159 
10 − 0.7299 − 0.5909 − 0.5938 − 0.6077 − 0.7710 − 0.7414 − 0.7600 − 0.2742 − 0.7053 − 0.7073 

Note: Bold characters are the maximum values of SCCF.

View Large
Table 10

Spearman cross-correlations (SCCF) between calculated monthly (1-month) RDI values in groups with stations with low spatial distance

SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 0.2455 0.2711 0.1482 0.2984 0.3444 0.2795 0.3243 − 0.0039 0.2846 0.2648 
−9 − 0.0538 0.0363 − 0.0351 − 0.0438 0.0422 − 0.0281 − 0.0025 − 0.1307 − 0.0720 − 0.0203 
−8 − 0.3405 − 0.1914 − 0.2239 − 0.2744 − 0.2814 − 0.3596 − 0.3362 − 0.1850 − 0.3819 − 0.3327 
−7 − 0.5113 − 0.3451 − 0.3167 − 0.4641 − 0.5212 − 0.5729 − 0.6202 − 0.1233 − 0.5841 − 0.5384 
−6 − 0.5922 − 0.4408 − 0.3648 − 0.5205 − 0.6452 − 0.6372 − 0.7029 − 0.1153 − 0.6520 − 0.6144 
−5 − 0.5233 − 0.4117 − 0.3365 − 0.4947 − 0.5760 − 0.5525 − 0.6105 − 0.1083 − 0.5694 − 0.5324 
−4 − 0.2904 − 0.2892 − 0.2101 − 0.3397 − 0.3360 − 0.3055 − 0.3594 − 0.1211 − 0.3441 − 0.3173 
−3 − 0.0044 − 0.0807 − 0.0494 − 0.0610 − 0.0358 0.0326 0.0041 − 0.0028 − 0.0120 0.0077 
−2 0.2970 0.2140 0.2024 0.2996 0.2577 0.3687 0.3519 0.1595 0.3395 0.3459 
−1 0.5965 0.4368 0.4275 0.5758 0.5294 0.6599 0.6640 0.3338 0.6497 0.6084 
0 0.9405 0.7992 0.8767 0.8846 0.8016 0.8669 0.9152 0.4048 0.8983 0.8636 
1 0.5642 0.4710 0.4132 0.5697 0.5667 0.5789 0.6487 0.2020 0.6126 0.5763 
2 0.2604 0.2969 0.1866 0.2979 0.3577 0.2971 0.3625 − 0.0122 0.3081 0.2959 
3 − 0.0476 0.0390 − 0.0687 − 0.0288 0.0158 − 0.0533 0.0060 − 0.1788 − 0.0451 − 0.0056 
4 − 0.3178 − 0.1742 − 0.2180 − 0.3092 − 0.2859 − 0.3335 − 0.3357 − 0.2182 − 0.3936 − 0.3454 
5 − 0.5251 − 0.3427 − 0.3261 − 0.4508 − 0.5190 − 0.5679 − 0.5997 − 0.1423 − 0.5915 − 0.5636 
6 − 0.5927 − 0.4490 − 0.3737 − 0.5216 − 0.6382 − 0.6390 − 0.6887 − 0.1275 − 0.6539 − 0.6272 
7 − 0.5166 − 0.3871 − 0.3354 − 0.5066 − 0.5470 − 0.5658 − 0.6158 − 0.1267 − 0.5627 − 0.5230 
8 − 0.3194 − 0.2830 − 0.2047 − 0.3260 − 0.3336 − 0.3129 − 0.3477 − 0.1226 − 0.3368 − 0.3145 
9 0.0017 − 0.0434 0.0181 − 0.0749 − 0.0359 0.0180 − 0.0140 − 0.0715 − 0.0184 − 0.0206 
10 0.3037 0.1686 0.2092 0.2455 0.2809 0.3405 0.3371 0.1136 0.3367 0.3182 
SCCF
LagsGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10
−10 0.2455 0.2711 0.1482 0.2984 0.3444 0.2795 0.3243 − 0.0039 0.2846 0.2648 
−9 − 0.0538 0.0363 − 0.0351 − 0.0438 0.0422 − 0.0281 − 0.0025 − 0.1307 − 0.0720 − 0.0203 
−8 − 0.3405 − 0.1914 − 0.2239 − 0.2744 − 0.2814 − 0.3596 − 0.3362 − 0.1850 − 0.3819 − 0.3327 
−7 − 0.5113 − 0.3451 − 0.3167 − 0.4641 − 0.5212 − 0.5729 − 0.6202 − 0.1233 − 0.5841 − 0.5384 
−6 − 0.5922 − 0.4408 − 0.3648 − 0.5205 − 0.6452 − 0.6372 − 0.7029 − 0.1153 − 0.6520 − 0.6144 
−5 − 0.5233 − 0.4117 − 0.3365 − 0.4947 − 0.5760 − 0.5525 − 0.6105 − 0.1083 − 0.5694 − 0.5324 
−4 − 0.2904 − 0.2892 − 0.2101 − 0.3397 − 0.3360 − 0.3055 − 0.3594 − 0.1211 − 0.3441 − 0.3173 
−3 − 0.0044 − 0.0807 − 0.0494 − 0.0610 − 0.0358 0.0326 0.0041 − 0.0028 − 0.0120 0.0077 
−2 0.2970 0.2140 0.2024 0.2996 0.2577 0.3687 0.3519 0.1595 0.3395 0.3459 
−1 0.5965 0.4368 0.4275 0.5758 0.5294 0.6599 0.6640 0.3338 0.6497 0.6084 
0 0.9405 0.7992 0.8767 0.8846 0.8016 0.8669 0.9152 0.4048 0.8983 0.8636 
1 0.5642 0.4710 0.4132 0.5697 0.5667 0.5789 0.6487 0.2020 0.6126 0.5763 
2 0.2604 0.2969 0.1866 0.2979 0.3577 0.2971 0.3625 − 0.0122 0.3081 0.2959 
3 − 0.0476 0.0390 − 0.0687 − 0.0288 0.0158 − 0.0533 0.0060 − 0.1788 − 0.0451 − 0.0056 
4 − 0.3178 − 0.1742 − 0.2180 − 0.3092 − 0.2859 − 0.3335 − 0.3357 − 0.2182 − 0.3936 − 0.3454 
5 − 0.5251 − 0.3427 − 0.3261 − 0.4508 − 0.5190 − 0.5679 − 0.5997 − 0.1423 − 0.5915 − 0.5636 
6 − 0.5927 − 0.4490 − 0.3737 − 0.5216 − 0.6382 − 0.6390 − 0.6887 − 0.1275 − 0.6539 − 0.6272 
7 − 0.5166 − 0.3871 − 0.3354 − 0.5066 − 0.5470 − 0.5658 − 0.6158 − 0.1267 − 0.5627 − 0.5230 
8 − 0.3194 − 0.2830 − 0.2047 − 0.3260 − 0.3336 − 0.3129 − 0.3477 − 0.1226 − 0.3368 − 0.3145 
9 0.0017 − 0.0434 0.0181 − 0.0749 − 0.0359 0.0180 − 0.0140 − 0.0715 − 0.0184 − 0.0206 
10 0.3037 0.1686 0.2092 0.2455 0.2809 0.3405 0.3371 0.1136 0.3367 0.3182 

Note: Bold characters are the maximum values of SCCF.

View Large
Table 11

Spearman cross-correlations (SCCF) between calculated annually (12-month) RDI values in groups with stations with high spatial distance

SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 − 0.0668 0.0276 0.3513 − 0.0221 0.1147 
−9 − 0.2115 − 0.2205 0.0486 − 0.1589 − 0.0097 
−8 − 0.2520 0.0413 0.1312 − 0.0187 − 0.1709 
−7 − 0.2186 0.0551 0.1971 0.0335 − 0.1040 
−6 − 0.3258 0.0734 0.0479 − 0.0526 − 0.1118 
−5 − 0.2457 − 0.0065 0.3010 − 0.0851 − 0.1078 
−4 − 0.1856 − 0.2342 0.1299 − 0.3455 − 0.1568 
−3 − 0.3329 0.0059 0.1595 − 0.1185 − 0.0834 
−2 − 0.3511 0.0937 0.1868 0.1633 0.0147 
−1 − 0.3277 0.1091 0.0972 0.0164 0.0174 
0 − 0.1373 0.1957 0.3486 0.2586 0.1058 
1 − 0.3702 − 0.1639 0.0564 0.1563 0.1180 
2 − 0.2430 0.1877 0.1703 0.1126 0.1327 
3 − 0.3586 0.1930 0.0422 0.0409 0.0270 
4 − 0.4308 − 0.0142 0.1813 − 0.2206 0.2030 
5 − 0.2828 − 0.0188 0.0194 − 0.0577 0.0093 
6 − 0.2104 0.0440 0.0994 − 0.0033 0.0233 
7 − 0.2756 − 0.1147 0.0860 − 0.0030 0.1517 
8 − 0.1661 0.0094 − 0.1831 0.0772 − 0.0103 
9 − 0.2960 − 0.0145 0.0470 0.1980 − 0.0270 
10 − 0.1275 − 0.0410 0.3056 0.3418 − 0.0342 
SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 − 0.0668 0.0276 0.3513 − 0.0221 0.1147 
−9 − 0.2115 − 0.2205 0.0486 − 0.1589 − 0.0097 
−8 − 0.2520 0.0413 0.1312 − 0.0187 − 0.1709 
−7 − 0.2186 0.0551 0.1971 0.0335 − 0.1040 
−6 − 0.3258 0.0734 0.0479 − 0.0526 − 0.1118 
−5 − 0.2457 − 0.0065 0.3010 − 0.0851 − 0.1078 
−4 − 0.1856 − 0.2342 0.1299 − 0.3455 − 0.1568 
−3 − 0.3329 0.0059 0.1595 − 0.1185 − 0.0834 
−2 − 0.3511 0.0937 0.1868 0.1633 0.0147 
−1 − 0.3277 0.1091 0.0972 0.0164 0.0174 
0 − 0.1373 0.1957 0.3486 0.2586 0.1058 
1 − 0.3702 − 0.1639 0.0564 0.1563 0.1180 
2 − 0.2430 0.1877 0.1703 0.1126 0.1327 
3 − 0.3586 0.1930 0.0422 0.0409 0.0270 
4 − 0.4308 − 0.0142 0.1813 − 0.2206 0.2030 
5 − 0.2828 − 0.0188 0.0194 − 0.0577 0.0093 
6 − 0.2104 0.0440 0.0994 − 0.0033 0.0233 
7 − 0.2756 − 0.1147 0.0860 − 0.0030 0.1517 
8 − 0.1661 0.0094 − 0.1831 0.0772 − 0.0103 
9 − 0.2960 − 0.0145 0.0470 0.1980 − 0.0270 
10 − 0.1275 − 0.0410 0.3056 0.3418 − 0.0342 

Note: Each group is including two stations with high spatial distance. Group 11 including Fasa and Tabriz stations, group 12 including Qazvin and Zahedan stations, group 13 including Bandar Abass and Mashhad stations, group 14 including Oroomieh and Shiraz stations, and group 15 including Chabahar and Semnan stations.

Bold characters are the maximum values of SCCF (based on the absolute value).

View Large
Table 12

Spearman cross-correlations (SCCF) between calculated seasonal (3-month) RDI values in groups with stations with high spatial distance

SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 − 0.6587 − 0.6653 − 0.6882 − 0.7068 − 0.5130 
−9 0.0231 − 0.1244 − 0.0429 − 0.0047 0.0527 
−8 0.6732 0.6756 0.6798 0.7336 0.4814 
−7 0.0189 0.1608 0.1180 0.0072 − 0.0003 
−6 − 0.6802 − 0.6716 − 0.6912 − 0.7085 − 0.5640 
−5 0.0032 − 0.1588 − 0.1003 0.0024 0.0274 
−4 0.7140 0.6122 0.7765 0.7373 0.4655 
−3 0.0101 0.1682 0.1330 − 0.0059 0.0188 
−2 − 0.7026 − 0.7639 − 0.6776 − 0.6980 − 0.5108 
−1 0.0241 − 0.1581 − 0.0793 − 0.0108 0.0666 
0 0.7464 0.7336 0.7890 0.7350 0.4345 
1 0.0278 0.1641 0.1450 − 0.0002 0.0341 
2 − 0.6801 − 0.7962 − 0.6802 − 0.7203 − 0.4757 
3 − 0.0166 − 0.1551 − 0.0979 − 0.0044 0.0695 
4 0.6847 0.6726 0.7927 0.7607 0.5271 
5 0.0109 0.1535 0.1258 − 0.0199 − 0.0047 
6 − 0.6849 − 0.6901 − 0.6816 − 0.7023 − 0.5107 
7 0.0381 − 0.1251 − 0.1000 − 0.0096 0.0235 
8 0.7002 0.6572 0.6726 0.7299 0.4743 
9 0.0312 0.1656 0.1488 0.0118 0.0487 
10 − 0.6873 − 0.6308 − 0.6594 − 0.6777 − 0.4542 
SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 − 0.6587 − 0.6653 − 0.6882 − 0.7068 − 0.5130 
−9 0.0231 − 0.1244 − 0.0429 − 0.0047 0.0527 
−8 0.6732 0.6756 0.6798 0.7336 0.4814 
−7 0.0189 0.1608 0.1180 0.0072 − 0.0003 
−6 − 0.6802 − 0.6716 − 0.6912 − 0.7085 − 0.5640 
−5 0.0032 − 0.1588 − 0.1003 0.0024 0.0274 
−4 0.7140 0.6122 0.7765 0.7373 0.4655 
−3 0.0101 0.1682 0.1330 − 0.0059 0.0188 
−2 − 0.7026 − 0.7639 − 0.6776 − 0.6980 − 0.5108 
−1 0.0241 − 0.1581 − 0.0793 − 0.0108 0.0666 
0 0.7464 0.7336 0.7890 0.7350 0.4345 
1 0.0278 0.1641 0.1450 − 0.0002 0.0341 
2 − 0.6801 − 0.7962 − 0.6802 − 0.7203 − 0.4757 
3 − 0.0166 − 0.1551 − 0.0979 − 0.0044 0.0695 
4 0.6847 0.6726 0.7927 0.7607 0.5271 
5 0.0109 0.1535 0.1258 − 0.0199 − 0.0047 
6 − 0.6849 − 0.6901 − 0.6816 − 0.7023 − 0.5107 
7 0.0381 − 0.1251 − 0.1000 − 0.0096 0.0235 
8 0.7002 0.6572 0.6726 0.7299 0.4743 
9 0.0312 0.1656 0.1488 0.0118 0.0487 
10 − 0.6873 − 0.6308 − 0.6594 − 0.6777 − 0.4542 

Note: Bold characters are the maximum values of SCCF (based on the absolute value).

View Large
Table 13

Spearman cross-correlations (SCCF) between calculated monthly (1-month) RDI values in groups with stations with high spatial distance

SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 0.2855 0.3676 0.3086 0.3338 0.1834 
−9 0.0483 0.1293 0.0630 − 0.0144 0.0312 
−8 − 0.2492 − 0.1596 − 0.2090 − 0.3292 − 0.1890 
−7 − 0.4985 − 0.4058 − 0.4493 − 0.5704 − 0.3062 
−6 − 0.6202 − 0.5605 − 0.5517 − 0.6471 − 0.3211 
−5 − 0.5633 − 0.5398 − 0.4992 − 0.5186 − 0.2933 
−4 − 0.2945 − 0.3763 − 0.2874 − 0.2696 − 0.1152 
−3 0.0302 − 0.1133 − 0.0595 0.0305 0.0310 
−2 0.3369 0.1722 0.2240 0.3179 0.1757 
−1 0.5515 0.4158 0.4829 0.6399 0.3047 
0 0.5838 0.5048 0.5176 0.6302 0.2887 
1 0.4721 0.5639 0.5049 0.7127 0.3766 
2 0.3088 0.3884 0.3173 0.3254 0.2121 
3 0.0669 0.1400 0.0724 − 0.0050 − 0.0188 
4 − 0.2201 − 0.1891 − 0.1966 − 0.3523 − 0.2005 
5 − 0.4924 − 0.4182 − 0.4110 − 0.5645 − 0.3012 
6 − 0.6273 − 0.5788 − 0.5316 − 0.6339 − 0.3304 
7 − 0.5462 − 0.5453 − 0.4948 − 0.5247 − 0.2603 
8 − 0.2933 − 0.3684 − 0.3058 − 0.2695 − 0.1226 
9 0.0257 − 0.1401 − 0.0550 0.0171 0.0042 
10 0.3038 0.1621 0.2228 0.3100 0.1785 
SCCF
LagsGroup 11Group 12Group 13Group 14Group 15
−10 0.2855 0.3676 0.3086 0.3338 0.1834 
−9 0.0483 0.1293 0.0630 − 0.0144 0.0312 
−8 − 0.2492 − 0.1596 − 0.2090 − 0.3292 − 0.1890 
−7 − 0.4985 − 0.4058 − 0.4493 − 0.5704 − 0.3062 
−6 − 0.6202 − 0.5605 − 0.5517 − 0.6471 − 0.3211 
−5 − 0.5633 − 0.5398 − 0.4992 − 0.5186 − 0.2933 
−4 − 0.2945 − 0.3763 − 0.2874 − 0.2696 − 0.1152 
−3 0.0302 − 0.1133 − 0.0595 0.0305 0.0310 
−2 0.3369 0.1722 0.2240 0.3179 0.1757 
−1 0.5515 0.4158 0.4829 0.6399 0.3047 
0 0.5838 0.5048 0.5176 0.6302 0.2887 
1 0.4721 0.5639 0.5049 0.7127 0.3766 
2 0.3088 0.3884 0.3173 0.3254 0.2121 
3 0.0669 0.1400 0.0724 − 0.0050 − 0.0188 
4 − 0.2201 − 0.1891 − 0.1966 − 0.3523 − 0.2005 
5 − 0.4924 − 0.4182 − 0.4110 − 0.5645 − 0.3012 
6 − 0.6273 − 0.5788 − 0.5316 − 0.6339 − 0.3304 
7 − 0.5462 − 0.5453 − 0.4948 − 0.5247 − 0.2603 
8 − 0.2933 − 0.3684 − 0.3058 − 0.2695 − 0.1226 
9 0.0257 − 0.1401 − 0.0550 0.0171 0.0042 
10 0.3038 0.1621 0.2228 0.3100 0.1785 

Note: Bold characters are the maximum values of SCCF (based on the absolute value).

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Figure 6
The lag number with maximum value (based on the absolute value) of SCCF in calculated 1, 3, and 12-month RDI in groups with stations with high spatial distance.
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The lag number with maximum value (based on the absolute value) of SCCF in calculated 1, 3, and 12-month RDI in groups with stations with high spatial distance.

Figure 6
The lag number with maximum value (based on the absolute value) of SCCF in calculated 1, 3, and 12-month RDI in groups with stations with high spatial distance.
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The lag number with maximum value (based on the absolute value) of SCCF in calculated 1, 3, and 12-month RDI in groups with stations with high spatial distance.

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The analysis of environmental, meteorological, and hydrological datasets often requires the detection of relationships between their variables. Generally, for assessing the relationship between two time series datasets, the CCF is suggested. The CCF is somewhat sensitive to the abnormality of datasets and the existence of outliers. Therefore, for abnormal datasets or for datasets with outliers, the CCF may not work well. To solve this issue, in this research, a non-parametric cross-correlation function was defined (SCCF). The ability of the proposed measure to detect the time-delay correlation between two stationary time series was studied. For this purpose, numerous datasets from two stationary time series were produced and analyzed. The results indicated that the proposed measure was more robust than the comparative measure, in detecting time delay between two non-Gaussian time series. The ability of the proposed measure in practice was also investigated by a real example. For this purpose, the RDI values of 20 synoptic stations in 1, 3, and 12-month time scales were considered and analyzed. The analysis results based on the actual data also proved the ability of the SCCF model. Finally, considering the low power of the CCF and high power of the SCCF to estimate CC in non-normal data series, it is suggested to assess the CC and calculate the time delay in non-normal hydrological data series the CCF replaced with the SCCF. On the other hand, it is recommended that SCCF's ability to determine the time delay in other hydrological variables such as rain and flood be investigated by other researchers.

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The authors would like to thank National Meteorological Organization of Iran for providing the necessary climatic data.

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The participation of M.R.M. and A.R.Z. includes the data collection, analyzing the results, and writing the article.

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The authors confirm that this article is original research and has not been published or presented previously in any journal or conference in any language (in whole or in part).

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Data cannot be made publicly available; readers should contact the corresponding author for details.

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The authors declare there is no conflict.

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This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/).