Cyclocopula Technique to Study the Relationship Between Two Cyclostationary Time Series with Fractional Brownian Motion Errors

Detection of the relationship between two time series is so important in environmental and hydrological studies. Several parametric and non-parametric approaches can be applied to detect relationships. These techniques are usually sensitive to stationarity assumptions. In this research, a new copula-based method is introduced to detect the relationship between two cylostationary time series with fractional Brownian motion (fBm) errors. The numerical studies verify the performance of the introduced approach.


Introduction
The ways of modeling dependency between two time series has always been one of the main focuses in practice. The choice of the better model depends on the dependency structure of obligators and is crucial part of the modeling. The applied methods in previous studies include Pearson's correlation coefficient [1][2][3][4][5][6], Spearman's correlation coefficient [7][8][9][10], Kendall's correlation coefficient [11][12][13][14], Sen's slope [15][16][17], cross-correlation function [18][19][20] and copula [21][22][23][24][25][26]. When we face with the relationship of two stationary time series, crosscorrelation function and copula are suggested. Cross-correlation function, as same as Pearson's correlation, is somewhat sensitive to abnormality of datasets and existence of outliers. In other words, for abnormal populations or when we face with outliers, cross-correlation function may not work well. An efficient way to model dependency is to use new modeling mechanism of Copula Theory which helps understand the correlation beyond linearity. In contrast, Copula is a function which transfers the multivariate distribution function to its marginal distribution function by quantiles. It is a great tool for modeling dependency where correlation follows the random distribution. Copula technique is most efficient for stationary time series and may not One of the main advantages of copulas is allowing to identify tail dependence across the multiple distributions. There are several Copulas that can be selected. The choice of most appropriate Copula has been an important issue. In practice, independent and perfectly correlated variables are naturally inapplicable for copulas. Due to its familiarity, Gaussian Copula [30] was the most famous one among others, however it failed to capture asymmetry, non-linearity and heavy tail dependency.
Therefore, alternative copulas have been used to model joint dependences. Several papers have been implied Copula families in modeling joint default dependencies in hydrology. Gumbel copula [31] is able to capture right tail dependence which is a particularly explored aspect of default dependency. Clayton copula [32] is defined to have left tail dependence. Student's copula [33] captures symmetric tail dependence with equally right and left tail dependence while Gaussian and Frank copulas [34] are defined as symmetric dependence without any tail dependence.

Cyclostationary Time Series
Stationarity is an important condition in classical time series analysis. But because of existence of cyclic rhythm in many practical situations such as hydrology and climatology, the stationarity assumption is not satisfied. As a suitable alternative, cyclostationary (CS) time series [35][36] are employed to describe the cyclic rhythms of rhythmic processes.
A time series , is CS with cycle T (CS-T), if T is the minimum natural number so that The cyclic correlation can be detected by employing the coherent statistics [36]. These statistics were defined by Since the spectral domain of CS-T time series is supported on the lines ( ) = ± 2 ( − 1) , = 1, … , , therefore, we expect the coherent statistics plot of a CS process emphasis these lines.
Because of cyclic rhythms in CS processes, using usual copula regression to compute the relationship of two CS-T time series is somewhat wrong. To solve this issue, in this research, a new approach is employed to evaluate relationship of two CS-T time series.

Copula for Cyclostationary Time Series
As previously discussed, the copula technique is most efficient for stationary time series and may not work well for non-stationary time series such as cyclostationary time series. To solve this issue, in this research, we introduce a copula-based regression technique. (ii) Let , and as the distribution function of the members of ( ) , ( ) , and ( , ) ( ) , respectively. Define the copula of ( ) and ( ) by and estimate the copula of ( ) and ( ) by ̂i =̂( i , i ), i = 1, … , T.
(iv) Combine the regression equations to next uses such as prediction or goodness of fit tests. where lcm refers to smallest common multiple of 1 and 2 .

Simulation
In this section, the ability of the proposed method to detect relationship between two CS time series is studied. For this purpose, numerous datasets from two CS time series and are produced and analyzed.
The simulation procedure is as following: Step In this study, we apply five different copula families, namely Gaussian, Student's t copula and three Archimedean copulas; Clayton, Gumbel and Frank. We capture symmetric dependence without tail dependence with Gaussian and Frank copulas, symmetric dependence with upper and lower tail dependence with t copula, left (lower) tail dependence with Clayton copula, right (upper) tail dependence with Gumbel copula.

Gaussian Copula
Gaussian (normal) copula, as the name implies, assumes joint distribution follows bivariate standard normal distribution. Gaussian copula is the most commonly applied copula in practice due to its convenient properties. The bivariate Gaussian copula is presented by: where is the joint bivariate cumulative standard normal distribution with ∈ [−1,1], as the correlation of the bivariate normal distribution and Φ −1 is the inverse of a univariate standard normal distribution.

Copula
The bivariate Student's copula is represented by: where , is the standardized bivariate Student's distribution with covariance ∈ [−1,1] and degree of freedom. −1 ( ) indicates the inverse of Student's cumulative distribution function. The main advantage of Student's copula over Gaussian copula is assuming a non-zero tail dependence even if correlation is zero.

Clayton copula
To estimate the lower tail dependency, the Clayton copula, is mostly suggested. The bivariate Clayton copula is represented by: is the copula. The parameter is related to Kendall's tau rank correlation as following:

Gumbel copula
To capture weak lower tail dependence and strong upper tail dependence, the Gumbel copula is developed. The bivariate Gumbel copula is given by: where ≥ 1 is the copula parameter. The Gumbel copula represents just independent and positive dependence.
The parameter is related to Kendall's tau rank correlation as following:

Frank copula
The bivariate Frank copula is given by: where ≠ 0 is the copula parameter. Unlike the Gumbel and Clayton copulas, Frank copula allows both negative and positive dependence in data. The parameter is related to Kendall's tau rank correlation as following: Step 3: For each copula, the regression analysis is applied to estimate the equation of ( ) based on ( ) , ̂= 0, + 0, , = 1, … , , = , + , + ( − 1) .
Step 7: For each parameter setting, the means of , and of all 1000 runs and five copulas are computed.
We consider first order periodic autoregressive with fBm error (PARFBM(1)) time series.
Assume the process

Conclusion
The analysis of hydrological and climatological datasets often requires the detection of relationships between different variables. When we face with the relationship of two time series, cross-correlation function and copula models are suggested. Cross-correlation function is somewhat sensitive to abnormality of datasets and existence of outliers. An efficient way to model dependency is to use new modeling mechanism of Copula Theory which helps understand the correlation beyond linearity. Copula technique is most efficient for stationary time series and may not work well for non-stationary time series such as cyclostationary time series. To solve this issue, in this research, we introduced a copula-based regression technique. The ability of the proposed approach to detect relationship between two cyclostationary time series with fractional Brownian motion errors was studied. For this purpose, numerous datasets from two cyclostationary time series with fractional Brownian motion errors were produced and analyzed.
The results indicated that the values of goodness of fit indices were close to one, and consequently, the proposed method was robust to detect relationship between two cyclostationary time series with fractional Brownian motion errors.