Theoretical Study of Some Angle Parameter Trigonometric Copulas

Copulas are important probabilistic tools to model and interpret the correlations of measures involved in real or experimental phenomena. The versatility of these phenomena implies the need for diverse copulas. In this article, we describe and investigate theoretically new two-dimensional copulas based on trigonometric functions modulated by a tuning angle parameter. The independence copula is, thus, extended in an original manner. Conceptually, the proposed trigonometric copulas are ideal for modeling correlations into periodic, circular, or seasonal phenomena. We examine their qualities, such as various symmetry properties, quadrant dependence properties, possible Archimedean nature, copula ordering, tail dependences, diverse correlations (medial, Spearman, and Kendall), and two-dimensional distribution generation. The proposed copulas are fleshed out in terms of data generation and inference. The theoretical findings are supplemented by some graphical and numerical work. The main results are proved using two-dimensional inequality techniques that can be used for other copula purposes.


Introduction
Multidimensional functions called copulas are important in modeling multivariate random variables and understanding their dependence structures. They find their origin and prime property in the Sklar theorem (see [1]). Mathematically, for any integer n, a n-dimensional copula can be defined as a cumulative distribution function defined on [0, 1] n with standard uniform marginal distributions. For the purposes of this article, the definition of a two-dimensional copula in the absolutely continuous case is given below. This definition has a certain flexibility, but the main critical point remains the validity of the two-increasing property. On the practical side, copulas are crucial for estimating and interpreting the correlations of measures involved in real or experimental phenomena. Due to the plurality of possible phenomena of interest, a large number of copulas with various attributes are necessary. This has motivated researchers over the years to introduce valuable copulas, and apply them in concrete, real-life scenarios. The Frank, Gumbel-Hougaard, Ali-Mikhail-Haq, Joe, Farlie-Gumbel-Morgenstern, Clayton, Plackett, Raftery, elliptical, Fréchet, Galambos, and Marshall-Olkin are some of the classical copulas. Their definitions are based on motivated transformations of power-polynomial-exponential-logarithmic functions. They have specific qualities that make them of great interest in the context of random dependence modeling. Further details on these copulas can be found in the books of [1][2][3][4].
Recently, a lot of attention has been paid to copulas defined (partially or not) with trigonometric functions. The inclusion of trigonometric functions in this context confers on the copula some oscillating features that are appropriate to model the correlations into phenomena of periodic, circular, or seasonal nature. In particular, they are ideal for analyzing correlations involved in movement data, circular data, and environmental data. The theory of the classical trigonometric copulas can be found in [5][6][7][8][9][10][11]. For practice, we refer to [12][13][14][15][16]. Furthermore, the R package named Cylcop, recently developed by [17], gives the trigonometric (and circular) copulas a new dimension of applicability.
In this article, we present and study trigonometric copulas depending on a tuning angle parameter. They are derived from the very general family of copulas (not especially trigonometric) introduced in [18], but with the importance of the angle parameter in mind. To be more specific, the proposed copulas are defined by the following form: where f (x) is a simple one-dimensional function involving a trigonometric function, and θ is the angle parameter that only modulates this trigonometric function. Despite the potential for modeling correlations into periodic, circular, or seasonal phenomena, this area of research appears to have received little attention; none of the references [5][6][7][8][9][10][11] consider such an angle parameter approach. Thus, we describe the most intuitive of such angle parameter copulas, and study them on the theoretical side with a maximum of details. In particular, we emphasize the optimal set of values for θ such that the corresponding copula remains valid. We determine the expressions of the corresponding copula density, survival copula, and survival copula density. Some mixed trigonometric copulas are also given. Then, we examine a maximum of theoretical properties of the proposed angle parameter copulas, including symmetry properties, quadrant dependence properties, copula ordering, various expansions, tail dependences, medial correlation, Spearman correlation, Kendall correlation, and two-dimensional distribution generation. Data generation and inference from the proposed copulas are sketched. The theory is illustrated by means of graphics. The proofs of the main results are based on some two-dimensional inequality techniques that can be of independent interest.
The following is the outline for the rest of the article: A cosine angle parameter copula is presented in Section 2, along with its main theoretical properties. In Section 3, an analogue sine copula is introduced and studied. Section 4 contains the conclusions and perspectives.

Cosine Angle Parameter Copula
This section is devoted to a simple cosine copula with an angle parameter.

Definition and Graphics
The following proposition presents the considered cosine copula.
Proof. Let us prove the main points defining an absolutely continuous two-dimensional copula, as recalled in Definition 1.
For the purposes of this study, we call the copula C * (x, y) in Equation (1) as the cos-copula. It has the feature of having a tuning angle parameter θ that modulates its correlation features. For θ = 0, we clearly have C * (x, y) = xy; the cos-copula is reduced to the independence copula. Furthermore, we can remark that the cos-copula is still valid for θ ∈ [−π/2, 0] because of the evenness of the cosine function. We can, thus, define the cos-copula with θ ∈ [−π/2, π/2] without loss of generality.
To end the presentation, Figure 1 represents the two-dimensional plot of the cos-copula for selected values of θ. From Figure 1, we see that the parameter θ skews the triangular shape of the cos-copula. The next parts are devoted to the functions related to the cos-copula, as well as its main properties.

Related Functions
The density associated with the cos-copula is the function c * : [0, 1] 2 → [0, +∞) given by It is worth noting that c * (0, 0) = cos(θ) which can be negative for some values of θ. In particular, C * (x, y) is not a copula for θ ∈ (π/2, π]. In this sense, if we restrict our attention to the interval [0, π], [0, π/2] is the optimal set of values for θ. As usual, the copula density can be involved in various moment-type measures, and estimation methods (see, for instance, Refs. [13,17]). Thus, we see how the parameter θ affects the overall shape of the cos-copula density. As a last important function, the survival cos-copula is the function It defines a valid copula, which is also a new trigonometric copula with an angle parameter represented by θ. Figure 3 represents the two-dimensional plot of the survival cos-copula for selected values of θ. The density associated with the survival cos-copula is given by To have an idea of its shapes, Figure 4 shows the two-dimensional plot of the survival cos-copula density for selected values of θ.  Because any convex linear combination of copulas is a copula (see [1]), the cos-copula can be used in a variety of new mixed copulas. For instance, we can consider the following mixed cos-copulas: • Mixed copula 1: For any angle parameters θ 1 ∈ [0, π/2] and θ 2 ∈ [0, π/2] and λ ∈ [0, 1], by setting C * (x, y) = C * (x, y; θ), a possible mixed copula is given as • Mixed copula 2: A second example is They are also new angle parameter trigonometric copulas by construction.

Properties
We now list the important properties of the cos-copula C * (x, y) as specified in Equation (1). • As already mentioned before: -For θ = 0, it is clear that C * (x, y) = xy. Therefore, the cos-copula is reduced to the independence copula.

-
If we restrict our attention to the interval [0, π], the set [0, π/2] is the optimal set of values for θ for validating C * (x, y) as a copula.

-
In terms of power series, by using the cosine series expansion and binomial formula, we get In particular, upon differentiation with respect to x and y on the interior of the domain of convergence, one has This expansion can be used in a variety of mathematical applications, such as determining various moment-type measurements.
• By arbitrary taking θ = π/2, we notice that As a result, the cos-copula is not Archimedean (see [1]). In other words, there is no generator function ψ : The cos-copula is not radially symmetric since there clearly exists (x, y) such that As any copula, the Fréchet-Hoeffding bounds can be expressed as follows: For any Thanks to the Kober inequality, the following inequality holds: 1] refers to the Farlie-Gumbel-Morgenstern (FGM) copula (see [1]). Since −(2/π)θ ∈ [−1, 1], the cos-copula and FGM copula are involved in a complete copula ordering. • For any θ ∈ [0, π/2], the two following results are obtained: Hence, the cos-copula has no tail dependence (see [1]). • The medial correlation of the cos-copula is defined by It is clearly a decreasing and negative function with respect to θ for θ ∈ [0, π/2], with M = 0 for θ = 0 and M = −0.07612047 for θ = π/2. Figure 5 represents the medial correlation for θ ∈ [0, 1.5]. Thus, the cos-copula has a weak medial correlation with M ∈ [−0.076, 0]. • A useful dependence measure based on copula is the Spearman rho (see [1]). The Spearman rho of the cos-copula, as an example of copula, is defined by Based on well-known mathematical techniques, the following proposition provides a mathematical expression for this measure.

Proposition 2.
The Spearman rho of the cos-copula can be expressed as where Ci(x) denotes the cosine integral defined by Ci(x) = −  Proof.

•
For θ = 0, we have C * (x, y) = xy and, by Equation (7), it is immediate that ρ = 0. • For θ ∈ (0, π/2], based on Equation (7), only the integral term requires attention. By using a step-by-step integration and some properties of the special functions Ci(x) and Si(x), we obtain The desired result follows immediately.
The proof of Proposition 2 ends.

Remark 2. Based on Equations
Of course, this result is of independent interest to our copula study. It is, however, fascinating to see the versatility in the nature of the functions involved, i.e., logarithmic, special integral function, cosine function, and Euler-Mascheroni constant.

•
In complement of the Spearman rho, we can present the Kendall tau of the cos-copula. It is defined by The closed form expression for τ is unmanageable due to the complexity of the product function C * (x, y)c * (x, y). We can, however, state that it is a decreasing function with respect to θ for θ ∈ [0, π/2], with τ = 0 for θ = 0 and τ = −0.06641903 for θ = π/2. Figure 7 represents the Kendall tau for θ ∈ [0, 1.5]. The small values of τ confirm the fact that the cos-copula is ideal to model weak negative correlations.

•
The cos-copula opens some interesting perspectives in distribution theory and modeling. The most immediate of these perspectives is the creation of simple and new two-dimensional distributions with cumulative distribution functions of the following form: H(x, y) = C * (F(x), G(y)), so where F(x) and G(x) denote two cumulative distribution functions. This gives twodimensional trigonometric distributions, which seem slightly underexplored in the literature. By considering the exponential distribution as the parent, we may define the cos-two-exponential distribution by the following cumulative distribution: and H(x, y; α, β) = 0 for (x, y) ∈ [0, +∞) 2 . To understand the importance of the trigonometric distributions in theory and practice for the uni-dimensional case, we may refer to [22,23].

Data Generation and Inference
The most straightforward method of generating random data (or values) from a distribution defined by a copula is what might be termed the inverse conditional method. For any positive integer n, with the aim of generating n data from the cos-copula, this method may be described as follows:

1.
Generate n data (s 1 , t 1 ), . . . , (s n , t n ) from a random vector (S, T), where S and T are independent random variables with the uniform distribution over (0, 1).
Then (s 1 , u 1 ), . . . , (s n , u n ) are n data generated from the cos-copula defined with the chosen θ.
Other methods exist (see [24]). Such simulated data can be used for computational tests, or estimation purposes.
On the other hand, in a data analysis scenario, the angle parameter θ is generally unknown. Its evaluation is, thus, of interest for precise data fitting, or at least to know if θ is close to 0 corresponding to the independent case, or close to π/2, corresponding to the most highly correlated case. For this evaluation, it can be estimated from n data (x 1 , y 1 ), . . . , (x n , y n ), that are susceptible to coming from the cos-copula distribution, by the maximum likelihood method; θ is thus estimated by provided that it is unique. This estimation method ensures satisfying qualities to θ, such as underlying consistency and asymptotic normality, which are the basis for the construction of confidence intervals and statistical tests (see, for instance, Ref. [13,17]). At this point, concrete applications of the aforementioned techniques to real-world datasets remain a possibility.

Sine Angle Parameter Copula
This section completes the findings of the previous section; it is devoted to a simple sine copula with an angle parameter. This copula can be viewed as a parametric generalization of one copula introduced in [18]. Indeed, in ( [18], Example 9), it is proved that the function C : [0, 1] 2 → [0, 1] defined by is a valid copula. Among the questions that arise are: • Can we replace the angle-value π/4 with a tuning parameter and, if so, what is its "optimal values set"? • What are the related functions of such a copula? • What are its theoretical properties?
The answers to these questions are given below.

Definition and Graphics
The following proposition presents the considered angle parameter sine copula.
Proof. Let us prove the main points defining an absolutely continuous two-dimensional copula, as recalled in Definition 1.

• For any
For any (x, y) ∈ [0, 1] 2 , using standard derivation techniques, we have Let us now study the sign of the above function by distinguishing the case θ ∈ [0, 1] and the case θ ∈ [−1, 0).
Case θ ∈ [0, 1]: We can write and Let us prove that A 1 ≥ 0 and A 2 ≥ 0. For A 1 , let us first remark that Therefore, since θ ∈ [0, 1] and cos(a) ∈ [0, 1] for any a ∈ [0, 1], we have The two-increasing property is proved. Case θ ∈ [−1, 0): For this case, we develop a strategy different to the previous case. We can write and Let us prove that B 1 ≥ 0 and B 2 ≥ 0.
As a result, C • (x, y) is a valid two-dimensional copula. This ends the proof of Proposition 3.
For the purposes of this paper, we call the copula C • (x, y) in Equation (9) as the sincopula. Like the cos-copula, it has the feature to have a tuning angle parameter θ. Clearly, for θ = 0, we have C • (x, y) = xy; the sin-copula is reduced to the independence copula, and for θ = π/4, it is reduced to the copula in ( [18], Example 9).
To end the presentation, Figure 8 represents the two-dimensional plot of the sin-copula for selected values of θ. From Figure 8, we see that the parameter θ skews the triangular shape of the sin-copula. The functions of the sin-copula, as well as its key features, are discussed in the following parts.
As a result of Figure 9, we can see how θ influences the overall form of the sin-copula density.
As a last important function, the survival sin-copula is the function It establishes a valid copula, which is also a new trigonometric copula with θ as an angle parameter. Figure 10 represents the two-dimensional plot of the survival sin-copula for selected values of θ. The copula density associated with the survival sin-copula is given by To have an idea of its shapes, Figure 11 shows its two-dimensional plot for selected values of θ. On the other hand, the sin-copula can be used in a variety of new mixed copulas. For instance, we can consider the following mixed sin-copulas: • Mixed copula 1: For any angle parameters θ 1 ∈ [−1, 1] and θ 2 ∈ [−1, 1], and λ ∈ [0, 1], by setting C • (x, y) = C • (x, y; θ), we can consider • Mixed copula 2: Similarly, for any angle parameters θ 1 ∈ [0, π/2] and θ 2 ∈ [−1, 1], and λ ∈ [0, 1], by setting C * (x, y) = C * (x, y; θ) and C • (x, y) = C • (x, y; θ), we can set • Mixed copula 3: Another example is They are also new angle parameter trigonometric copulas by construction.

Properties
The main features of the sin-copula C • (x, y) as described in Equation (9) are now listed.

•
As already mentioned before: -For θ = 0, it is clear that C • (x, y) = xy. Therefore, the sin-copula is reduced to the independence copula.
As a result, we can observe that the sin-copula has inherent analytical complexity.

-
In terms of power series, by using the cosine series expansion and binomial formula, we get In particular, upon differentiation with respect to x and y on the interior of the domain of convergence, one has This expansion can be used to determine various moment-type measurements in a range of mathematical applications.
• By arbitrary taking θ = −1, we notice that As a result, the sin-copula is not Archimedean (see [1]). • The sin-copula is not radially symmetric since there exists (x, y) such that C • (x, y) = C • (x, y).

•
Thanks to the inequality: sin(a) ≤ a and the Jordan inequality: sin(a) ≥ (2/π)a for a ∈ [0, π/2] (see [20]), we have a copula ordering between the sin-copula and FGM copula: -For θ ∈ [−1, 0], the contrary holds: The following relationship between the cos-copula and sin-copula holds: Therefore, the following ordering results are established: -For θ ∈ [−1, 0], since the first sine term in Equation (12) is negative, and the second one is positive, we have -For θ ∈ [0, 1], since the first sine term in Equation (12) is positive, and the second one too, we have C • (x, y) ≥ C * (x, y).
• For any θ ∈ [0, π/2], the two following results are obtained: Hence, the sin-copula has no tail dependence. • The medial correlation of the sin-copula is defined by It is clearly an increasing function with respect to θ for θ ∈ [−1, 1], with M = −0.247404 for θ = −1 and M = 0.247404 for θ = 1. Figure 12 represents the medial correlation for θ ∈ [−1, 1]. The Spearman rho of the sin-copula, as an example of copula, is defined by Using well-known mathematical methods, the following assertion provides a mathematical expression for this measure.

Proposition 4.
The Spearman rho of the sin-copula can be expressed as

Proof.
• For θ = 0, we have C • (x, y) = xy and, by Equation (13), it is immediate that ρ = 0. • For θ ∈ (0, 1], still based on the definition of ρ in Equation (13), we have By using a step-by-step integration, we obtain Immediately, the intended result occurs. • For θ ∈ [−1, 0), thanks to the oddity of the sine function, we can write Since −θ ∈ (0, 1], the expression of ρ can be transposed with −θ instead of θ, with the minus in factor of the overall expression.
The stated proposition is proved.

Remark 3.
Similarly to Remark 2, a fascinating series formula, not repertoried to the best of our knowledge, can be proved. Based on Equations (6) and (8), upon integration over [0, 1] 2 , the following formula holds: In complement of the Spearman rho, we can present the Kendall tau of the sin-copula. It is defined by The complexity of the product function C • (x, y)c • (x, y) makes the closed form expression for τ unmanageable. We can, however, show that it is an increasing function with respect to θ for θ ∈ [−1, 1], with τ = 0 for θ = 0, τ = −0.2185653 for θ = −1 and τ = 0.2185653 for θ = 1. Figure where F(x) and G(x) denote two cumulative distribution functions. As a result, probabilistic or statistical modeling in this context becomes more feasible.

Data Generation and Inference
The data generation method described in Section 2.4 can be configured for the sincopula. For any positive integer n, we can generate n data from the sin-copula by proceeding as follows:
Then (s 1 , u 1 ), . . . , (s n , u n ) are n data generated from the sin-copula defined with the chosen value of θ.
In a data analysis scenario, the angle parameter θ is usually unknown. Its estimation is thus useful for exact data fitting, or at the very least for determining if θ is close to 0, corresponding to the independent case, or close to −1 or 1, corresponding to the most highly negatively or positively correlated situations, respectively. For this evaluation, it can be estimated from n data (x 1 , y 1 ), . . . , (x n , y n ), that are susceptible to coming from the sin-copula distribution, by the maximum likelihood method; θ is thus estimated by Concrete applications of the aforementioned methodologies to real-world datasets are thus possible.

Conclusions
We have introduced and studied several trigonometric copulas that have the features to depend on a tuning angle parameter. We have shown that they are quite simple from the mathematical point of view, and possess interesting properties.
For the first copula, called cos-copula, we have demonstrated that: (i) it extends the independence copula; (ii) it has the negative quadrant property; (iii) it is symmetric; (iv) it is not Archimedean; (v) it is not radially symmetric; (vi) a special ordering exists between it and the FGM copula; (vii) it has no tail dependence; (viii) the medial correlation belongs to the interval [−0.0762, 0]; (ix) the Spearman rho belongs to the interval [−0.1, 0]; (x) the Kendall tau belongs to the interval [−0.067, 0]; (xi) it can serve to create a plethora of trigonometric two-dimensional distributions. Thus, the cos-copula is adapted to model weak negative correlations. It is thus not adapted to model moderate or large correlations. The corresponding copula density, survival copula, and survival copula density have been expressed, as well as two mixed copula versions.
For the second copula, called the sin-copula, its properties can be listed as follows: (i) it extends the independence copula; (ii) it has the negative and positive quadrant properties; (iii) it is symmetric; (iv) it is not Archimedean; (v) it is not radially symmetric; (vi) comprehensive ordering exists between it and both the FGM copula and cos-copula; (vii) it has no tail dependence; (viii) the medial correlation belongs to the interval [−0.25, 0.25]; (ix) the Spearman rho belongs to the interval [−0.33, 0.33]; (x) the Kendall tau belongs to the interval [−0.22, 0.22]; (xi) it can also serve to create a plethora of trigonometric twodimensional distributions. Thus, the sin-copula is adapted to model moderate negative or positive correlations. The corresponding copula density, survival copula, and survival copula density have been expressed, as well as three mixed copula versions.
The shapes of the principal functions and their related features have been visually observed using graphics.

Perspectives
Thus, the first elements for the promotion of the proposed copulas are in this article. Perspectives of further research includes the following points: • Following the spirit of some power-extended FGM copulas (see [25]), one can think of considering some extensions of the cos-copula and sin-copula of the forms: respectively, where α, β and c are newly introduced shape parameters. However, the possible values of α, β and c such that C * * (x, y) and C •• (x, y) are valid copulas remain to discover. • The n-dimensional versions of the cos-copula and sin-copula, which can be defined as C * : [0, 1] n → [0, 1] and C • : [0, 1] n → [0, 1], respectively, where respectively, deserve to be investigated for n ≥ 3. Especially, the possible values for θ in this case need to be determined. • The use of the sin-copula and cos-copula for data analysis, following the methodology in [13,15], is an important perspective. Furthermore, the development of an R package is envisageable, inspired by the research design of Cylcop developed by [17]. • Last but not least, other simple angle parameter copulas can be created on the basis of this study. One could think of C : [0, 1] 2 → [0, 1] defined by C (x, y) = xy{1 + tan[θ(1 − x)(1 − y)]}, but the optimal values of θ such that C (x, y) is a valid copula remain unknown. It is proven that θ ∈ [0, π/4] satisfies the required conditions (in an unpublished work), but this set is thought to be less than optimal.
Funding: This research received no external funding.