rvmsin {BAMBI} | R Documentation |

## The bivariate von Mises sine model

### Description

The bivariate von Mises sine model

### Usage

```
rvmsin(
n,
kappa1 = 1,
kappa2 = 1,
kappa3 = 0,
mu1 = 0,
mu2 = 0,
method = "naive"
)
dvmsin(x, kappa1 = 1, kappa2 = 1, kappa3 = 0, mu1 = 0, mu2 = 0, log = FALSE)
```

### 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; |

`mu1` , `mu2` |
vectors of mean parameters. |

`method` |
Rejection sampling method to be used. Available choices are |

`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. |

`log` |
logical. Should the log density be returned instead? |

### Details

The bivariate von Mises sine model density at the point `x = (x_1, x_2)`

is given by

`f(x) = C_s (\kappa_1, \kappa_2, \kappa_3) \exp(\kappa_1 \cos(T_1) + \kappa_2 \cos(T_2) + \kappa_3 \sin(T_1) \sin(T_2))`

where

`T_1 = x_1 - \mu_1; T_2 = x_2 - \mu_2`

and `C_s (\kappa_1, \kappa_2, \kappa_3)`

denotes the normalizing constant for the sine model.

Two different rejection sampling methods are implemented for random generation. If `method = "vmprop"`

, then first the y-marginal
is drawn from the associated marginal density, and then x is generated from the conditional distributio of x given y. The marginal generation of
y is implemented in a rejection sampling scheme with proposal being either von Mises (if the target marginal density is unimodal), or a mixture of
von Mises (if bimodal), with optimally chosen concentration. This the method suggested in Mardia et al. (2007). On the other hand, when
`method = "naive"`

(default) a (naive) bivariate rejection sampling scheme with (bivariate) uniform propsoal is used.

Note that although method = `"vmprop"`

may provide better efficiency when the density is highly concentrated, it does have
an (often substantial) overhead due to the optimziation step required to find a reasonable proposal concentration parameter.
This can compensate the efficiency gains of this method, especially when `n`

is not large.

### Value

`dvmsin`

gives the density and `rvmsin`

generates random deviates.

### Examples

```
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,
# dvmsin returns single density
dvmsin(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
dvmsin(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
# dvmsin 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]
dvmsin(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 dvmsin 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]
dvmsin(x[1, ], kappa1, kappa2, kappa3, mu1, mu2)
# when parameters are all scalars, number of observations generated
# by rvmsin is n
rvmsin(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 rvmsin is the same as the length of the
# recycled vectors
rvmsin(n, kappa1, kappa2, kappa3, mu1, mu2)
```

*BAMBI*version 2.3.5 Index]