rvine_structure {rvinecopulib}

R-vine structure

Description

R-vine structures are compressed representations encoding the tree structure of the vine, i.e. the conditioned/conditioning variables of each edge. The functions ⁠[cvine_structure()]⁠ or ⁠[dvine_structure()]⁠ give a simpler way to construct C-vines (every tree is a star) and D-vines (every tree is a path), respectively (see Examples).

Usage

rvine_structure(order, struct_array = list(), is_natural_order = FALSE)

cvine_structure(order, trunc_lvl = Inf)

dvine_structure(order, trunc_lvl = Inf)

rvine_matrix(matrix)

Arguments

order

a vector of positive integers.

struct_array

a list of vectors of positive integers. The vectors represent rows of the r-rvine structure and the number of elements have to be compatible with the order vector. If empty, the model is 0-truncated.

is_natural_order

whether struct_array is assumed to be provided in natural order already (a structure is in natural order if the anti- diagonal is 1, .., d from bottom left to top right).

trunc_lvl

the truncation level

matrix

an R-vine matrix, see Details.

Details

The R-vine structure is essentially a lower-triangular matrix/triangular array, with a notation that differs from the one in the VineCopula package. An example array is

4 4 4 4
3 3 3
2 2
1

which encodes the following pair-copulas:

tree	edge	pair-copulas
0	0	⁠(1, 4)⁠
	1	⁠(2, 4)⁠
	2	⁠(3, 4)⁠
1	0	⁠(1, 3; 4)⁠
	1	⁠(2, 3; 4)⁠
2	0	⁠(1, 2; 3, 4)⁠

An R-vine structure can be converted to an R-vine matrix using as_rvine_matrix(), which encodes the same model with a square matrix filled with zeros. For instance, the matrix corresponding to the structure above is:

4 4 4 4
3 3 3 0
2 2 0 0
1 0 0 0

Similarly, an R-vine matrix can be converted to an R-vine structure using as_rvine_structure().

Denoting by M[i, j] the array entry in row i and column j (the pair-copula index for edge e in tree t of a d dimensional vine is ⁠(M[d + 1 - e, e], M[t, e]; M[t - 1, e], ..., M[1, e])⁠. Less formally,

1. Start with the counter-diagonal element of column e (first conditioned variable).

2. Jump up to the element in row t (second conditioned variable).

3. Gather all entries further up in column e (conditioning set).

Internally, the diagonal is stored separately from the off-diagonal elements, which are stored as a triangular array. For instance, the off-diagonal elements off the structure above are stored as

4 4 4
3 3
2

for the structure above. The reason is that it allows for parsimonious representations of truncated models. For instance, the 2-truncated model is represented by the same diagonal and the following truncated triangular array:

4 4 4
3 3

A valid R-vine structure or matrix must satisfy several conditions which are checked when rvine_structure(), rvine_matrix(), or some coercion methods (see as_rvine_structure() and ⁠as_rvine_matrix(⁠) are called:

1. It can only contain numbers between 1 and d (and additionally zeros for R-vine matrices).

2. The anti-diagonal must contain the numbers 1, ..., d.

3. The anti-diagonal entry of a column must not be contained in any column further to the right.

4. The entries of a column must be contained in all columns to the left.

5. The proximity condition must hold: For all t = 1, ..., d - 2 and e = 1, ..., d - t there must exist an index j > d, such that ⁠(M[t, e], {M[1, e], ..., M[t - 1, e]})⁠ equals either ⁠(M[d + 1 - j, j], {M[1, j], ..., M[t - 1, j]})⁠ or ⁠(M[t - 1, j], {M[d + 1 - j, j], M[1, j], ..., M[t - 2, j]})⁠.

Condition 5 already implies conditions 2-4, but is more difficult to check by hand.

Value

Either an rvine_structure or an rvine_matrix.

See Also

as_rvine_structure(), as_rvine_matrix(), plot.rvine_structure(), plot.rvine_matrix(), rvine_structure_sim(), rvine_matrix_sim()

Examples

# R-vine structures can be constructed from the order vector and struct_array
rvine_structure(order = 1:4, struct_array = list(
  c(4, 4, 4),
  c(3, 3),
  2
))

# R-vine matrices can be constructed from standard matrices
mat <- matrix(c(4, 3, 2, 1, 4, 3, 2, 0, 4, 3, 0, 0, 4, 0, 0, 0), 4, 4)
rvine_matrix(mat)

# coerce to R-vine structure
str(as_rvine_structure(mat))

# truncate and construct the R-vine matrix
mat[3, 1] <- 0
rvine_matrix(mat)

# or use directly the R-vine structure constructor
rvine_structure(order = 1:4, struct_array = list(
  c(4, 4, 4),
  c(3, 3)
))

# throws an error
mat[3, 1] <- 5
try(rvine_matrix(mat))

# C-vine structure
cvine <- cvine_structure(1:5)
cvine
plot(cvine)

# D-vine structure
dvine <- dvine_structure(c(1, 4, 2, 3, 5))
dvine
plot(dvine)