Calculate the dependence function \(A\) for the multivariate logistic and multivariate asymmetric logistic models; plot the estimated function in the trivariate case.
amvevd(x = rep(1/d,d), dep, asy, model = c("log", "alog"), d = 3, 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)A vector of length d or a matrix with d
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.
\(A(1/d,\dots,1/d)\) is returned by default since it is often
a useful summary of dependence.
The dependence parameter(s). For the logistic model,
should be a single value. For the asymmetric logistic model,
should be a vector of length \(2^d-d-1\), or a single value,
in which case the value is used for each of the \(2^d-d-1\)
parameters (see rmvevd).
The asymmetry parameters for the asymmetric logistic
model. Should be a list with \(2^d-1\) vector elements
containing the asymmetry parameters for each separate component
(see rmvevd and Examples).
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 in
rmvevd.
The dimension; an integer greater than or equal to two.
The trivariate case d = 3 is the default.
Logical; if TRUE, and the dimension d is
three (the default dimension), the dependence function of a
trivariate model is plotted. For plotting in the bivariate case,
use abvevd. If FALSE (the default), the
following arguments are ignored.
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.
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.
For plotting, the function is evaluated at grid^2
points.
The minimum value for which colours are plotted. By defualt \(\code{lower} = 1/3\) as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.
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.The argument alters the way in
which the function is plotted; it does not change the function
definition.
A character vector of length three, in which case the
ith margin is labelled using the ith component,
or NULL, in which case no labels are given. The actual
location of the margins, and hence the labels, is defined by
ord.
A numerical value giving the amount by which the
labels should be scaled relative to the default. Ignored
if lab is NULL.
A numeric vector of values. If plotting, the smallest evaluated function value is returned invisibly.
Let \(z = (z_1,\dots,z_d)\) and \(w = (w_1,\dots,w_d)\). Any multivariate extreme value distribution can be written as $$G(z) = \exp\left\{- \left\{\sum\nolimits_{j=1}^{d} y_j \right\} A\left(\frac{y_1}{\sum\nolimits_{j=1}^{d} y_j}, \dots, \frac{y_d}{\sum\nolimits_{j=1}^{d} y_j}\right)\right\}$$ for some function \(A\) defined on the simplex \(S_d = \{w \in R^d_+ : \sum\nolimits_{j=1}^{d} w_j = 1\}\), where $$y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}$$ for \(1+s_i(z_i-a_i)/b_i > 0\) and \(i = 1,\dots,d\), and where the (generalized extreme value) marginal parameters are given by \((a_i,b_i,s_i)\), \(b_i > 0\). If \(s_i = 0\) then \(y_i\) is defined by continuity.
\(A\) is called (by some authors) the dependence function. It follows that \(A(w) = 1\) when \(w\) is one of the \(d\) vertices of \(S_d\), and that \(A\) is a convex function with \(\max(w_1,\dots,w_d) \leq A(w)\leq 1\) for all \(w\) in \(S_d\). The lower and upper limits of \(A\) are obtained under complete dependence and mutual independence respectively. \(A\) does not depend on the marginal parameters.
# NOT RUN {
amvevd(dep = 0.5, model = "log")
s3pts <- matrix(rexp(30), nrow = 10, ncol = 3)
s3pts <- s3pts/rowSums(s3pts)
amvevd(s3pts, dep = 0.5, model = "log")
# }
# NOT RUN {
amvevd(dep = 0.05, model = "log", plot = TRUE, blty = 1)
# }
# NOT RUN {
amvevd(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))
amvevd(s3pts, dep = 0.15, asy = asy, model = "alog")
amvevd(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7)
# }
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