atvpar {evd} | R Documentation

## Parametric Dependence Functions of Trivariate Extreme Value Models

### Description

Calculate or plot the dependence function A for the trivariate logistic and trivariate asymmetric logistic models.

### Usage

    atvpar(x = rep(1/3,3), dep, asy, model = c("log", "alog"), plot =
        FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50,
        lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1)

### Arguments

`x` | A vector of length three or a matrix with three columns, in which case the dependence function is evaluated across the rows (ignored if plot is `TRUE`). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. A(1/3,1/3,1/3) is returned by default since it is often a useful summary of dependence.
`dep` | The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length four, or a single value, in which case the value is used for each of the four parameters (see `rmvevd`).
`asy` | The asymmetry parameters for the asymmetric logistic model. Should be a list with seven vector elements; three of length one, three of length two and one of length three, containing the asymmetry parameters for each separate component (see `rmvevd` and Examples).
`model` | The specified model; a character string. Must be either `"log"` (the default) or `"alog"` (or any unique partial match), for the logistic and asymmetric logistic models respectively. The definition of each model is given (for general dimensions) in `rmvevd`.
`plot` | Logical; if `TRUE` the function is plotted. The minimum (evaluated) value is returned invisibly. If `FALSE` (the default), the following arguments are ignored.
`col` | A list of colours (see `image`). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between `lower` and one.
`blty` | The border line type, for the border that surrounds the triangular image. By default `blty` is zero, so no border is plotted. Plotting a border leads to (by default) an increase in `grid` (and hence computation time), to ensure that the image fits within it.
`grid` | For plotting, the function is evaluated at `grid^2` points.
`lower` | The minimum value for which colours are plotted. By defualt `lower` = 1/3 as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.
`ord` | A vector of length three, which should be a permutation of the set {1,2,3}. The points (1,0,0), (0,1,0) and (0,0,1) (the vertices of the simplex) are depicted clockwise from the top in the order defined by `ord`.
`lab` | A character vector of length three, in which case the `i`th margin is labelled using the `i`th component, or `NULL`, in which case no labels are given. The actual location of the margins, and hence the labels, is defined by `ord`.
`lcex` | A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if `lab` is `NULL`.

### Details

Let z = (z1,z2,z3) and w = (w1,w2,w3). Any trivariate extreme value distribution can be written as

G(z) = exp{-(y1+y2+y3) A[y1/(y1+y2+y3),y2/(y1+y2+y3), y3/(y1+y2+y3)]}

for some function A defined on the simplex S_3 = {w: w1 + w2 + w3 = 1}, where

yi = {1+si(zi-ai)/bi}^(-1/si)

for 1+si(zi-ai)/bi > 0 and i = 1,2,3, and where the (generalized extreme value) marginal parameters are given by (ai,bi,si), bi > 0. If si = 0 then yi is defined by continuity.

A is called (by some authors) the dependence function. It follows that A(1,0,0) = A(0,1,0) = A(0,0,1) = 1, and that A is a convex function with max(w1,w2,w3) <= A(w) <= 1 for all w in S_3. The lower and upper limits of A are obtained under complete dependence and mutual independence respectively. A does not depend on the marginal parameters.

### Value

`atvpar` calculates or plots the dependence function for the trivariate logistic and trivariate asymmetric logistic models, at specified parameter values.

### See Also

`atvnonpar`, `abvpar`, `rmvevd`, `image`

### Examples

    atvpar(dep = 0.5, model = "log")
    s3pts <- matrix(rexp(30), nrow = 10, ncol = 3)
    s3pts <- s3pts/rowSums(s3pts)
    atvpar(s3pts, dep = 0.5, model = "log")
    ## Not run: atvpar(dep = 0.05, model = "log", plot = TRUE, blty = 1)
    atvpar(dep = 0.95, model = "log", plot = TRUE, lower = 0.94)

    asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
    atvpar(s3pts, dep = 0.15, asy = asy, model = "alog")
    atvpar(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7)

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