New Bivariate Wrapped Distributions

Models for univariate directional data are well established. An excellent review of known models can be found in Mardia and Jupp (2000). However, models for bivariate or multivariate directional data have been limited. A common method for constructing models for directional data is wrapping. We are aware of only the bivariate wrapped normal and bivariate wrapped Cauchy distributions being used as models for bivariate directional data. The bivariate wrapped Cauchy distribution has been used for face analysis (Waine, 2001) among others. The bivariate wrapped normal distribution has been used to model online handwriting recognition (Bahlmann, 2006) among others. The aim of this paper is to introduce a range of bivariate wrapped distributions and related measures. Let f(x, y) denote a joint pdf of a random vector (X, Y) in (−∞,+∞) × (−∞,+∞) with characteristic function ψ(s, t). Then a bivariate wrapped distribution can be defined by the pdf:


Introduction
Models for univariate directional data are well established. An excellent review of known models can be found in Mardia and Jupp (2000). However, models for bivariate or multivariate directional data have been limited. A common method for constructing models for directional data is wrapping. We are aware of only the bivariate wrapped normal and bivariate wrapped Cauchy distributions being used as models for bivariate directional data. The bivariate wrapped Cauchy distribution has been used for face analysis (Waine, 2001) among others. The bivariate wrapped normal distribution has been used to model online handwriting recognition (Bahlmann, 2006) among others.
The aim of this paper is to introduce a range of bivariate wrapped distributions and related measures. Let f(x, y) denote a joint pdf of a random vector (X, Y) in (−∞,+∞) × (−∞,+∞) with characteristic function ψ(s, t). Then a bivariate wrapped distribution can be defined by the pdf: Denote the vector of means and let: where, Re denotes the real part and Let: 1,1 1,2 1 1 1:2,1:2 1:2,3:4 3:4,3:4 3:4,1:2 2,1 2,2 where, S a:b,c:d refers to the submatrix of S with rows a to b and columns c to d.
Because of space restrictions, we present details for only two of the fourteen distributions. The variation of the five correlation coefficients for these two bivariate wrapped distributions is illustrated graphically in section 3. The details for other distributions can be obtained from the corresponding author.
The many distributions introduced could encourage further applications of models for directional data. They could also encourage further models being developed for directional data. A future work is to extend the results in this study to the multivariate case.

The Collection
The two distributions discussed here are the bivariate wrapped Laplace and bivariate wrapped normal distributions.
Bivariate wrapped Laplace distribution with: ( )  For illustration, we have considered only the two distributions discussed in section 2. But the variation of the five correlation coefficients with respect to the parameters of the other distributions was similar to those variations reported in Fig. 1 and 2.