A new multimodal and asymmetric bivariate circular distribution

Abstract

Multimodal and asymmetric bivariate circular data arise in several different disciplines and fitting appropriate distribution plays an important role in the analysis of such data. In this paper, we propose a new bivariate circular distribution which can be used to model both asymmetric and multimodal bivariate circular data simultaneously. In fact the proposed density covers unimodality as well as multimodality, symmetry as well as asymmetry of circular bivariate data. A number of properties of the proposed density are presented. A Bayesian approach with MCMC scheme is employed for statistical inference. Three real datasets and a simulation study are provided to illustrate the performance of the proposed model in comparison with alternative models such as finite mixture Cosine model.

Appendix

Proof of Proposition 2.2

The proof is easily obtained since

$$\begin{aligned} C( \delta ,\kappa _{1},\kappa _{2},\kappa _{3},\kappa _{4}) =C\left( \frac{\pi }{4},\kappa _{1},\kappa _{2}\right) C( \delta ,\kappa _{3},\kappa _{4}) \end{aligned}$$

where $C\left( \frac{\pi }{4},\kappa _{1},\kappa _{2}\right)$ and $C(\delta ,\kappa _{3},\kappa _{4})$ are normalizing constants of AGvM and GvM, respectively. Gatto and Jammalamadaka (2007) maintained

$$\begin{aligned} C( \delta ,\kappa _{3},\kappa _{4}) =I_{0}(\kappa _{3}) I_{0}(\kappa _{4}) +\sum _{l=1}^{\infty }I_{2l}(\kappa _{3}) I_{l}(\kappa _{4}) \cos (2l\delta ) . \end{aligned}$$

Also $C\left( \frac{\pi }{4},\kappa _{1},\kappa _{2}\right)$ can be obtained by choosing $\delta =\frac{\pi }{4}$ in Eq. (4).

Table 4 Critical points of MABvM when $\theta _{2}=0$

Table 5 Critical points of MABvM when $\theta _{2}=\pi$

Table 6 Critical points of MABvM when $\theta _{2}=$arctan$( R_{3},R_{4})$ with condition $\kappa _{1}< 2|\kappa _{2}|$ and $\kappa _{3}< 4\kappa _{4}$

Table 7 Critical points of MABvM when $\theta _{2}=$arctan$(-R_{3},R_{4})$ with condition $\kappa _{1}< 2|\kappa _{2}|$ and $\kappa _{3}< 4\kappa _{4}$

Fig. 7

The posterior plots of $\varphi$ in simulation study with real value $\varphi =0$ when $n=30$. a History plot, b Gelman–Rubin diagnostic, c Kernel density plot, d autoregressive plot

Proof of Proposition 2.4

We can derive the conditions for number of modes of the MABvM density when $\mu =\mu _{1}=\mu _{2}=0$ and $\varphi \ne 0$, without loss of generality. To locate the critical values, we take partial derivatives of $\log ( f( \theta _{1},\theta _{2}) )$ in Eq. (3) w.r.t $\theta _{1}$ and $\theta _{2}$. Let $g=\log ( f( \theta _{1},\theta _{2}) ) =\log C+\kappa _{1}\cos ( \theta _{1}-\varphi \theta _{2}) +\kappa _{2}\sin ( 2\theta _{1}-2\varphi \theta _{2}) +\kappa _{3}\cos ( \theta _{2}) +\kappa _{4}\cos ( 2\theta _{2})$. Then

$$\begin{aligned} G_{\theta _{1}}=\frac{\partial g}{\partial \theta _{1}}=\kappa _{1}\sin (\varphi \theta _{2}-\theta _{1})+2\kappa _{2}\cos (2\varphi \theta _{2}-2\theta _{1}) \end{aligned}$$

and

$$\begin{aligned} G_{\theta _{2}}=\frac{\partial g}{\partial \theta _{2}}=-G_{\theta _{1}}+\kappa _{3}\sin (-\theta _{2})+2\kappa _{4}\sin (-2\theta _{2}) \end{aligned}$$

There are sixteen critical points for $G_{\theta _{1}}=G_{\theta _{2}}=0$. Let $D_{\theta _{1},\theta _{2}}=G_{\theta _{1}, \theta _{1}}G_{\theta _{2}, \theta _{2}}-G_{\theta _{1}, \theta _{2}}^{2}$,

• (i) if $D_{\theta _{1}, \theta _{2}}>0$ and $G_{\theta _{1},\theta _{1}}<0$ then $(\theta _{1},\theta _{2})$ is a mode.

• (ii) if $D_{\theta _{1}, \theta _{2}}>0$ and $G_{\theta _{1},\theta _{1}}>0$ then $(\theta _{1},\theta _{2})$ is an anti-mode.

• (iii) if $D_{\theta _{1}, \theta _{2}}<0$ then $(\theta _{1},\theta _{2})$ is a saddle point.

We can find the number of modes after simplification of underlying algebraic equations. The critical points are given in Tables 4, 5, 6 and 7 where $R_{1}=(4\kappa _{2})^{-2}\left( -\kappa _{1}^{2}+16\kappa _{2}^{2}+\sqrt{\kappa _{1}^{4}+32\kappa _{1}^{2}\kappa _{2}^{2}}\right)$, $R_{2}=(4\kappa _{2})^{-2}\left( -2\kappa _{1}^{2}+32\kappa _{2}^{2}-2\sqrt{\kappa _{1}^{4}+32\kappa _{1}^{2}\kappa _{2}^{2}}\right)$, $R_{3}=\frac{\sqrt{-\kappa _{3}^{2}+16\kappa _{4}^{2}}}{4\kappa _{4} }$, and $R_{4}=-\frac{\kappa _{3}}{4\kappa _{4}}$.

In these tables $\arctan (x,y)$ gives the arc tangent of $\frac{y}{x}$. Table 4 illustrates that there is no restrictions on unimodality when $\theta _{1}=$ arctan$\left( \frac{2\kappa _{2}}{\kappa _{1}}\left( \frac{1}{2} R_{1}-1\right) ,\sqrt{R_{1}}\right)$ and $\theta _{2}=0$. Checking all of the critical points shows that if $\kappa _{1}< 2\vert \kappa _{2}\vert$ or $\kappa _{3}< 4\kappa _{4}$ then MABvM has two modes. Also if $\kappa _{1}< 2\vert \kappa _{2}\vert$ and $\kappa _{3}< 4\kappa _{4}$ then MABvM has four modes.

Posterior plots in simulation study when $n=30$ and $\varphi =0$ Figure 7 shows the history, Gelman–Rubin diagnostic, kernel density, and autocorrelation plots from the simulation study. We define two sequences from different starting points. The Gelman–Rubin diagnostic shows that the behavior of the sequence of chains is the same. Therefore, the variance within the chains is the same as the variance across the chains. Also, the autocorrelation plot reveals there is low correlation between successive samples and the history plot moves up and down around the mode of the distribution. Thus the samples will reach a stationary distribution.

3-D plots of protein data and the fitted models The difference between the proposed models and the alternative models is an advantage of our study, since we have defined a different model to analysis bivariate circular data. An additional figure and comment on the difference are now provided as follows. Accordingly, the MABvM model seems to provide a better fit for the ruggedness of the histogram (i.e. the frequencies that should not be ignored). The 3-D histogram of protein data and the kernel density plots of the competitive models are given in Fig. 8.

Fig. 8

The 3-D histogram of protein data and kernel density plots of the fitted models. a Histogram plot, b MABvM model, c EMABvM model, d Shieh and Johnson model, e Kato model, f bivariate Cosine model, g bivariate Sine model, h mixture of bivariate Cosine model, i mixture of bivariate Sine model