Abstract
This paper studies circular correlations for the bivariate von Mises sine and cosine distributions. These are two simple and appealing models for bivariate angular data with five parameters each that have interpretations connected to those in the ordinary bivariate normal model. However, the variability and association of the angle pairs cannot be easily deduced from the model parameters unlike the bivariate normal. Thus to compute such summary measures, tools from circular statistics are needed. We derive analytic expressions and study the properties of the Jammalamadaka–Sarma and Fisher–Lee circular correlation coefficients for the von Mises sine and cosine models. Likelihoodbased inference of these coefficients from sample data is then presented. The correlation coefficients are illustrated with numerical and visual examples, and the maximum likelihood estimators are assessed on simulated and real data, with comparisons to their nonparametric counterparts. Implementations of these computations for practical use are provided in our R package BAMBI.
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Notes
In the literature the JS and FL correlation coefficients have been denoted by \(\rho _c\) and \(\rho _T\) respectively, following the authors’ notations. We however, shall use \(\rho _{\text {FL}}\) and \(\rho _{\text {JS}}\) in this paper for clarity.
Circular variance is defined for an angular variable \(\Theta \) as \(\text {var}(\Theta ) = 1E(\cos (\Theta ))\) (see, e.g., Jammalamadaka and Sengupta (2001)). Expressions for \(\text {var}(\Theta )\) and \(\text {var}(\Phi )\) for the von Mises sine distribution were first provided in Singh et al. (2002).
Options for computing jackknife and bootstrapbased standard error estimates are provided in our R package BAMBI.
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Appendix A: Technical results required in the proofs of Theorem 2.1 and 2.2
Appendix A: Technical results required in the proofs of Theorem 2.1 and 2.2
Proposition A.0.1
Let \((\Theta , \Phi ) \sim {{\,\mathrm{{vM}_{s}}\,}}(\kappa _1, \kappa _2, \kappa _3, 0, 0)\). Then

(i)
\(E\left( \sin \Theta \sin \Phi \right) = \frac{1}{C_s} \frac{\partial C_s}{\partial \kappa _3}\).

(ii)
\({{\,\mathrm{{sgn}}\,}}( E(\sin \Phi \sin \Theta )) = {{\,\mathrm{{sgn}}\,}}(\kappa _3)\).

(iii)
\(E\left( \cos \Theta \cos \Phi \right) = \frac{1}{C_s} \frac{\partial ^2 C_s}{\partial \kappa _1 \partial \kappa _2}\).

(iv)
\(E\left( \cos \Theta \right) = \frac{1}{C_s} \frac{\partial C_s}{\partial \kappa _1}\), and \(E\left( \cos \Phi \right) = \frac{1}{C_s} \frac{\partial C_s}{\partial \kappa _2}\).

(v)
\(E\left( \cos ^2 \Theta \right) = \frac{1}{C_s} \frac{\partial ^2 C_s}{\partial \kappa _1^2}\), and \(E\left( \cos ^2 \Phi \right) = \frac{1}{C_s} \frac{\partial ^2 C_s}{\partial \kappa _2^2}\).

(vi)
\(E(\sin \Phi \cos \Theta ) = E(\sin \Theta \cos \Phi ) = 0\).

(vii)
\(E(\sin \Theta \cos \Theta ) = E(\sin \Phi \cos \Phi ) = 0\).
Proof
Because the integrand in (A1) is smooth and has continuous first and second order partial derivatives with respect to the parameters \((\kappa _1, \kappa _2, \kappa _3)\), and the limits of the integral are finite and constant (free of the parameters), partial differentiation with respect to the parameters, and the integration can be done in interchangeable orders (Leibniz’s rule).

(1)
Differentiating both sides of (A1) partially with respect to \(\kappa _3\), and then applying Leibniz’s rule, we get
$$\begin{aligned} \frac{\partial C_s}{\partial \kappa _3}&= \int _{\pi }^{\pi } \int _{\pi }^{\pi } \sin \theta \sin \phi \, \exp \left( \kappa _1 \cos \theta + \kappa _2 \cos \phi + \kappa _3 \sin \theta \sin \phi \right) \, d\theta \, d\phi \\&= C_s E\left( \sin \Theta \sin \Phi \right) . \end{aligned}$$ 
(2)
Let \(g(\lambda ) = \frac{\partial C_c}{\partial \lambda }\). Since \(C_s > 0\), following part (i), it is enough to show that \({{\,\mathrm{{sgn}}\,}}(g(\lambda )) = {{\,\mathrm{{sgn}}\,}}(\lambda )\). From the infinite series representation (2.7) we get
$$\begin{aligned} g(\lambda ) = 8 \pi ^2 \sum _{m=1}^{\infty } m \left( {\begin{array}{c}2m\\ m\end{array}}\right) \frac{\lambda ^{2m1}}{(4\kappa _1 \kappa _2)^m} I_{m}(\kappa _1) I_{m}(\kappa _2) \lesseqgtr 0 \end{aligned}$$according as \(\lambda \lesseqgtr 0\). This completes the proof.

(3)
The result is obtained by partially differentiating (A1) twice, once with respect \(\kappa _1\) and then with respect to \(\kappa _2\), and then by applying Leibniz’s rule.

(4)
The proof is given in (Singh et al., 2002, Theorem 2(b)).

(5)
The first half is obtained by partially differentiating (A1) twice with respect to \(\kappa _1\), and the second half, with respect to \(\kappa _2\); followed by an application of Leibniz’s rule.

(6)
We shall only prove the first half. The proof of the second half is similar. It follows (see Singh et al. (2002)) that the conditional distribution of \(\Phi \) given \(\Theta = \theta \) is univariate von Mises \({{\,\mathrm{{vM}}\,}}\left( \kappa = a(\theta ), \mu = b(\theta ) \right) \), and the marginal density of \(\Theta \) is given by:
$$\begin{aligned} f_\Theta (\theta ) = \frac{2 \pi I_0(a(\theta ))}{C_s} \exp (\kappa _1 \cos \theta ) \mathbbm {1}_{[\pi , \pi )} (\theta ) \end{aligned}$$where
$$\begin{aligned} a(\theta ) = \left\{ \kappa _2^2 + \kappa _3^2 \sin ^2 \theta \right\} ^{1/2} \text { and } b(\theta ) = \tan ^{1} \left( \frac{\kappa _3}{\kappa _2} \sin \theta \right) . \end{aligned}$$Note that \(f_\Theta \) is symmetric about \((\mu _1 = )\; 0\). Therefore, we have
$$\begin{aligned} E\left( \sin \Phi \cos \Theta \right)&= E \left[ \cos \Theta \, E\left( \sin \Phi \mid \Theta \right) \right] \\&= E \left[ \cos \Theta \, \frac{I_1(a(\Theta ))}{I_0(a(\Theta )) } \, \sin (\beta (\Theta )) \right] \\&= E \left[ \cos \Theta \, \frac{I_1(a(\Theta ))}{I_0(a(\Theta )) } \, \frac{(\kappa _3/\kappa _2) \sin \Theta }{\sqrt{1 + (\kappa _3/\kappa _2)^2 \sin ^2 \Theta }} \right] \\&= 0, \end{aligned}$$where the second equality follows from Proposition A.0.2, and the last from the fact that the associated integral is an odd function.

(7)
These results are immediate consequences of symmetry of the marginal distributions.
\(\square \)
Proposition A.0.2
Let X have a univariate von Mises distribution \({{\,\mathrm{{vM}}\,}}(\kappa , \mu )\). Then \(E(\sin X) = \frac{I_1(\kappa )}{I_0(\kappa )} \sin \mu \).
Proof
Because the density of X is symmetric about \(\mu \), we have,
Also (see, e.g., Abramowitz and Stegun 1964, §9.6.19)),
Solving for \(E(\sin X)\) from (A2) and (A3) yields \(E(\sin X) = \frac{I_1(\kappa )}{I_0(\kappa )} \sin \mu \). \(\square \)
Proposition A.0.3
Let \((\Theta , \Phi ) \sim {{\,\mathrm{{vM}_{c}}\,}}(\kappa _1, \kappa _2, \kappa _3, 0, 0)\). Then

(i)
\(E\left( \cos \Theta \cos \Phi \right) = \frac{1}{C_c} \frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2}\).

(ii)
\(E\left( \sin \Theta \sin \Phi \right) = \frac{1}{C_c} \left\{ \frac{\partial C_c}{\partial \kappa _3}  \frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2} \right\} \).

(iii)
\({{\,\mathrm{{sgn}}\,}}( E(\sin \Phi \sin \Theta )) = {{\,\mathrm{{sgn}}\,}}(\kappa _3)\).

(iv)
\(E\left( \cos \Theta \right) = \frac{1}{C_s} \frac{\partial C_s}{\partial \kappa _1}\), and \(E\left( \cos \Phi \right) = \frac{1}{C_s} \frac{\partial C_s}{\partial \kappa _2}\).

(v)
\(E\left( \cos ^2 \Theta \right) = \frac{1}{C_c} \frac{\partial ^2 C_c}{\partial \kappa _1^2}\), and \(E\left( \cos ^2 \Phi \right) = \frac{1}{C_c} \frac{\partial ^2 C_c}{\partial \kappa _2^2}\).

(vi)
\(E(\sin \Phi \cos \Theta ) = E(\sin \Theta \cos \Phi ) = 0\).

(vii)
\(E(\sin \Theta \cos \Theta ) = E(\sin \Phi \cos \Phi ) = 0\).
Proof
We have
Using the same arguments as in the von Mises sine case, it follows that partial differentiation with respect to the parameters, and the integration can be done in interchangeable orders (Leibniz’s rule).

(i)
Differentiating both sides of (A4) twice, once with respect \(\kappa _1\) and then with respect to \(\kappa _2\), and then by applying Leibniz’s rule, we get
$$\begin{aligned}&\frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2} \\&\quad = \int _{\pi }^{\pi } \int _{\pi }^{\pi } \sin \theta \sin \phi \, \exp \left( \kappa _1 \cos \theta + \kappa _2 \cos \phi + \kappa _3 \cos (\theta  \phi ) \right) \, d\theta \, d\phi \\&\quad = C_c E \left( \cos \Theta \cos \Phi \right) . \end{aligned}$$ 
(ii)
Differentiating (A4) partially with respect to \(\kappa _3\), and then applying Leibniz’s rule, we get
$$\begin{aligned} \frac{\partial C_c}{\partial \kappa _3}&= \int _{\pi }^{\pi } \int _{\pi }^{\pi } \cos (\theta \phi ) \, \exp \left( \kappa _1 \cos \theta + \kappa _2 \cos \phi + \kappa _3 \cos (\theta  \phi ) \right) \, d\theta \, d\phi \\&= C_c E\cos \left( \Theta  \Phi \right) = C_c E\cos \left( \cos \Theta \cos \Phi + \sin \Theta \sin \Phi \right) . \end{aligned}$$This, together with part (i) yields
$$\begin{aligned} \frac{\partial C_c}{\partial \kappa _3}  \frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2} = C_c \, E (\sin \Theta \sin \Phi ) \end{aligned}$$ 
(iii)
Let \(g(\kappa _3) = \frac{\partial C_c}{\partial \kappa _3}  \frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2}\). Since \(C_c > 0\), following part (i), it is enough to show that \({{\,\mathrm{{sgn}}\,}}(g(\kappa _3)) = {{\,\mathrm{{sgn}}\,}}(\kappa _3)\). Straightforward algebra on the infinite series representations (2.13) and (2.16) of \(\frac{\partial C_c}{\partial \kappa _3}\) and \(\frac{\partial ^2 C_c}{\partial \kappa _1 \partial \kappa _2}\) yields,
$$\begin{aligned}&g(\kappa _3) \nonumber \\&\quad = 2 \pi ^2 \left\{ \sum _{m=1}^\infty I_{m1}(\kappa _1) I_{m1}(\kappa _2) I_m(\kappa _3)  \sum _{m=1}^\infty I_{m1}(\kappa _1) I_{m+1}(\kappa _2) I_m(\kappa _3) \right. \nonumber \\&\qquad \left.  \sum _{m=1}^\infty I_{m+1}(\kappa _1) I_{m1}(\kappa _2) I_m(\kappa _3) + \sum _{m=1}^\infty I_{m+1}(\kappa _1) I_{m+1}(\kappa _2) I_m(\kappa _3) \right\} \nonumber \\&\quad = 2 \pi ^2 \sum _{m=1}^\infty [I_{m1}(\kappa _1)  I_{m+1}(\kappa _1)] [I_{m1}(\kappa _2)  I_{m+1}(\kappa _2)] I_m(\kappa _3) \nonumber \\&\quad = \sum _{m=1}^\infty a_m \, I_m(\kappa _3) \end{aligned}$$(A5)where \(a_m = 2 \pi ^2 [I_{m1}(\kappa _1)  I_{m+1}(\kappa _1)] [I_{m1}(\kappa _2)  I_{m+1}(\kappa _2)]\). Note that \((a_m)_{m \ge 1}\) is a decreasing sequence of positive real numbers since \(I_n(x) > I_{n+1}(x)\) for \(n \ge 1\) and \(x \ge 0\). We consider the cases \(\kappa _3 = 0\), \(\kappa _3 > 0\) and \(\kappa _3 < 0\) separately, and note the sign of \(g(\kappa _3)\) in each case.

(a)
If \(\kappa _3 = 0\), then \(I_m(\kappa _3) = 0\) for all \(m = 1, 2, \ldots \). Consequently, the right hand side of (A5) becomes zero.

(b)
If \(\kappa _3 > 0\), then \(I_m(\kappa _3) > 0\) for all \(m = 1, 2, \ldots \). Therefore, the right hand side of (A5) is a series of positive terms, and hence is positive.

(c)
If \(\kappa _3 < 0\), then \(I_m(\kappa _3) = (1)^m I_m(\vert \kappa _3\vert )\) for \(m = 1, 2, \ldots \), and the right hand side of (A5) is an (absolutely convergent) alternating series
$$\begin{aligned} S = \sum _{m = 1}^\infty (1)^m \, a_m \, I_m(\vert \kappa _3\vert ). \end{aligned}$$Note that
$$\begin{aligned} S&=  \sum _{m = 1}^\infty a_{2m  1} \, I_{2m  1}(\vert \kappa _3\vert ) + \sum _{m = 1}^\infty a_{2m} \, I_{2m}(\vert \kappa _3\vert ) \\&<  \sum _{m = 1}^\infty a_{2m  1} \, I_{2m  1}(\vert \kappa _3\vert ) + \sum _{m = 1}^\infty a_{2m1} \, I_{2m}(\vert \kappa _3\vert ) \\&=  \sum _{m = 1}^\infty a_{2m  1} \, [I_{2m  1}(\vert \kappa _3\vert )  I_{2m}(\vert \kappa _3\vert )] =  S^* \\&< 0. \end{aligned}$$where the inequality in the second line follows from the fact that \((a_m)_{m \ge 1}\) is decreasing and positive, and that in the last line is a consequence of the fact that \(S^*\), being a series of positive terms (since \(I_{2m  1}(\vert \kappa _3\vert ) > I_{2m}(\vert \kappa _3\vert )\) for all \(m \ge 1\)), is positive.

(a)

(iv)
The first part is proved by partially differentiating (A4) with respect to \(\kappa _1\), and the second part, with respect to \(\kappa _2\); followed by an application of Leibniz’s rule.

(v)
The first half is obtained by partially differentiating (A4) twice with respect to \(\kappa _1\), and the second half, with respect to \(\kappa _2\); followed by an application of Leibniz’s rule.

(vi)
We shall only prove the first half. The proof of the second half is similar. It follows from Mardia et al. (2007) that the conditional distribution of \(\Phi \) given \(\Theta = \theta \) is univariate von Mises \({{\,\mathrm{{vM}}\,}}\left( \kappa = \kappa _{13}, \mu = \theta _0 \right) \), and the marginal density of \(\Theta \) is given by:
$$\begin{aligned} g_\Theta (\theta ) = \frac{2 \pi I_0(\kappa _{13}(\theta ))}{C_c} \, \exp (\kappa _2 \cos \theta ) \, \mathbbm {1}_{[\pi , \pi )} (\theta ) \end{aligned}$$where
$$\begin{aligned} \kappa _{13} (\theta ) = \kappa _1^2 + \kappa _3^2 + 2 \kappa _1 \kappa _3 \cos \theta \text { and } \theta _0 = \tan ^{1} \left( \frac{\kappa _3 \sin \theta }{\kappa _1 + \kappa _3 \cos \theta } \right) . \end{aligned}$$Note that \(f_\Theta \) is symmetric about \((\mu _1 = )\; 0\). Therefore, we have
$$\begin{aligned} E\left( \sin \Phi \cos \Theta \right)&= E \left[ \cos \Theta \, E\left( \sin \Phi \mid \Theta \right) \right] \\&= E \left[ \cos \Theta \, \frac{I_1(\kappa _{13}(\Theta ))}{I_0(\kappa _{13}(\Theta )) } \, \sin \tan ^{1} \left( \frac{\kappa _3 \sin \Theta }{\kappa _1 + \kappa _3 \cos \Theta } \right) \right] \\&= E \left[ \cos \Theta \, \frac{I_1(\kappa _{13}(\Theta ))}{I_0(\kappa _{13}(\Theta )) } \, \frac{\left( \frac{\kappa _3 \sin \Theta }{\kappa _1 + \kappa _3 \cos \Theta } \right) }{\sqrt{1 + \left( \frac{\kappa _3 \sin \Theta }{\kappa _1 + \kappa _3 \cos \Theta } \right) ^2}} \right] \\&= 0, \end{aligned}$$where the second equality follows from Proposition A.0.2, and the last from the fact that the associated integral is an odd function.

(vii)
These results are immediate consequences of symmetry of the marginal distributions.
\(\square \)
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Chakraborty, S., Wong, S.W.K. On the circular correlation coefficients for bivariate von Mises distributions on a torus. Stat Papers 64, 643–675 (2023). https://doi.org/10.1007/s00362022013339
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DOI: https://doi.org/10.1007/s00362022013339