rwnorm2: The bivariate Wrapped Normal distribution

Description

The bivariate Wrapped Normal distribution

Usage

rwnorm2(n, kappa1 = 1, kappa2 = 1, kappa3 = 0, mu1 = 0, mu2 = 0, ...)
dwnorm2(
  x,
  kappa1 = 1,
  kappa2 = 1,
  kappa3 = 0,
  mu1 = 0,
  mu2 = 0,
  int.displ,
  log = FALSE
)

Value

dwnorm2 gives the density and rwnorm2 generates random deviates.

Arguments

n: number of observations. Ignored if at least one of the other parameters have length k > 1, in which case, all the parameters are recycled to length k to produce k random variates.
kappa1, kappa2, kappa3: vectors of concentration parameters; kappa1, kappa2 > 0, and kappa3^2 < kappa1*kappa2.
mu1, mu2: vectors of mean parameters.
...: additional arguments passed to rmvnorm from package mvtnorm
x: bivariate vector or a two-column matrix with each row being a bivariate vector of angles (in radians) where the densities are to be evaluated.
int.displ: integer displacement. If int.displ = M, then each infinite sum in the density is approximated by a finite sum over 2*M + 1 elements. (See Details.) The allowed values are 1, 2, 3, 4 and 5. Default is 3.
log: logical. Should the log density be returned instead?

Details

The bivariate wrapped normal density at the point $x = (x_1, x_2)$ is given by, $$f(x) = \sqrt((\kappa_1 \kappa_2 - (\kappa_3)^2)) / (2\pi) \sum \exp(-1/2 * (\kappa_1 (T_1)^2 + \kappa_2 (T_2)^2 + 2 \kappa_3 (T_1) (T_2)) )$$ where $$T_1 = T_1(x, \mu, \omega) = (x_1 - \mu_1(2\pi\omega_1))$$ $$T_2 = T_2(x, \mu, \omega) = (x_2 - \mu_1(2\pi\omega_2))$$ the sum extends over all pairs of integers $\omega = (\omega_1, \omega_2)$, and is approximated by a sum over $(\omega_1, \omega_2)$ in $\{-M, -M+1, ..., M-1, M \}^2$ if int.displ = $M$.

Note that above density is essentially the "wrapped" version of a bivariate normal density with mean $$\mu = (\mu_1, \mu_2)$$ and dispersion matrix $\Sigma = \Delta^{-1}$, where

	$\kappa_1$	$\kappa_3$
$\Delta =$
	$\kappa_3$	$\kappa_2$.

Examples

Run this code

kappa1 <- c(1, 2, 3)
kappa2 <- c(1, 6, 5)
kappa3 <- c(0, 1, 2)
mu1 <- c(1, 2, 5)
mu2 <- c(0, 1, 3)
x <- diag(2, 2)
n <- 10

# when x is a bivariate vector and parameters are all scalars,
# dwnorm2 returns single density
dwnorm2(x[1, ], kappa1[1], kappa2[1], kappa3[1], mu1[1], mu2[1])

# when x is a two column matrix and parameters are all scalars,
# dmvsin returns a vector of densities calculated at the rows of
# x with the same parameters
dwnorm2(x, kappa1[1], kappa2[1], kappa3[1], mu1[1], mu2[1])

# if x is a bivariate vector and at least one of the parameters is
# a vector, all parameters are recycled to the same length, and
# dwnorm2 returns a vector of with ith element being the density
# evaluated at x with parameter values kappa1[i], kappa2[i],
# kappa3[i], mu1[i] and mu2[i]
dwnorm2(x[1, ], kappa1, kappa2, kappa3, mu1, mu2)

# if x is a two column matrix and at least one of the parameters is
# a vector, rows of x and the parameters are recycled to the same
# length, and dwnorm2 returns a vector of with ith element being the
# density evaluated at ith row of x with parameter values kappa1[i],
# kappa2[i], # kappa3[i], mu1[i] and mu2[i]
dwnorm2(x, kappa1, kappa2, kappa3, mu1, mu2)

# when parameters are all scalars, number of observations generated
# by rwnorm2 is n
rwnorm2(n, kappa1[1], kappa2[1], kappa3[1], mu1[1], mu2[1])

# when at least one of the parameters is a vector, all parameters are
# recycled to the same length, n is ignored, and the number of
# observations generated by rwnorm2 is the same as the length of the
# recycled vectors
rwnorm2(n, kappa1, kappa2, kappa3, mu1, mu2)

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	\(\kappa_1\)	\( \)	\(\kappa_3\)
\(\Delta =\)	\( \)	\( \)	\( \)
	\(\kappa_3\)	\( \)	\(\kappa_2\).