Structure selection and estimation of a GAM-Vine model.

This function select the structure and estimates the parameter(s) of a Generalized Additive model (GAM) Vine model, where GAMs for individual edges are specified either for the copula parameter or Kendall's tau. It solves the maximum penalized likelihood estimation for the copula families supported in this package by reformulating each Newton-Raphson iteration as a generalized ridge regression, which is solved using the mgcv package.

gamVineStructureSelect(udata, lin.covs = NULL, smooth.covs = NULL,
  simplified = TRUE, type = 0, familyset = NA, rotations = TRUE,
  familycrit = "AIC", treecrit = "tau", level = 0.05,
  trunclevel = NA, tau = TRUE, method = "FS", tol.rel = 0.001,
  n.iters = 10, parallel = FALSE, verbose = FALSE,
  select.once = TRUE)

Arguments

udata	A matrix or data frame containing the data in [0,1]^d.
lin.covs	A matrix or data frame containing the parametric (i.e., linear) covariates (default: lin.covs = NULL).
smooth.covs	A matrix or data frame containing the non-parametric (i.e., smooth) covariates (default: smooth.covs = NULL).
simplified	If TRUE (default), then a simplified vine is fitted (which is possible only if there are exogenous covariates). If FALSE, then a non-simplified vine is fitted.
type	type = 0 (default) for a R-Vine and type = 1 for a C-Vine.
familyset	An integer vector of pair-copula families to select from (the independence copula MUST NOT be specified in this vector unless one wants to fit an independence vine!). Not listed copula families might be included to better handle limit cases. If familyset = NA (default), selection among all possible families is performed. Coding of pair-copula families: 1 Gaussian, 2 Student t, 5 Frank, 301 Double Clayton type I (standard and rotated 90 degrees), 302 Double Clayton type II (standard and rotated 270 degrees), 303 Double Clayton type III (survival and rotated 90 degrees), 304 Double Clayton type IV (survival and rotated 270 degrees), 401 Double Gumbel type I (standard and rotated 90 degrees), 402 Double Gumbel type II (standard and rotated 270 degrees), 403 Double Gumbel type III (survival and rotated 90 degrees), 404 Double Gumbel type IV (survival and rotated 270 degrees).
rotations	If TRUE, all rotations of the families in familyset are included.
familycrit	Character indicating the criterion for bivariate copula selection. Possible choices: familycrit = 'AIC' (default) or 'BIC', as in BiCopSelect from the VineCopula package.
treecrit	Character indicating how pairs are selected in each tree. treecrit = "tau" uses the maximum spanning tree of the Kendall's tau (i.e., the tree of maximal overall dependence), treecrit = "rho" uses the Spearman's rho.
level	Numerical; Passed to gamBiCopSelect, it is the significance level of the test for removing individual predictors (default: level = 0.05) for each conditional pair-copula.
trunclevel	Integer; level of truncation.
tau	TRUE (default) for a calibration function specified for Kendall's tau or FALSE for a calibration function specified for the Copula parameter.
method	'NR' for Newton-Raphson and 'FS' for Fisher-scoring (default).
tol.rel	Relative tolerance for 'FS'/'NR' algorithm.
n.iters	Maximal number of iterations for 'FS'/'NR' algorithm.
parallel	TRUE (default) for parallel selection of copula family at each edge or FALSE for the sequential version. for the Copula parameter.
verbose	TRUE if informations should be printed during the estimation and FALSE (default) for a silent version. from mgcv.
select.once	if TRUE the GAM structure is only selected once, for the family that appears first in familyset.

Value

gamVineSeqFit returns a gamVine-class object.

See also

gamVineSeqFit,gamVineCopSelect, gamVine-class, gamVineSimulate and gamBiCopSelect.

Examples

require(VineCopula)
set.seed(0)


## An example with a 3-dimensional GAM-Vine

# Sample size
n <- 1e3

# Define a R-vine tree structure matrix
d <- 3
Matrix <- c(2, 3, 1, 0, 3, 1, 0, 0, 1)
Matrix <- matrix(Matrix, d, d)
nnames <- paste("x", 1:d, sep = "")

# Copula families for each edge
fam <- c(301, 401, 1)

# Parameters for the first tree (two unconditional copulas)
par <- c(1, 2)

# Pre-allocate the GAM-Vine model list
count <- 1
model <- vector(mode = "list", length = d * (d - 1) / 2)

# The first tree contains only the two unconditional copulas
for (i in 1:(d - 1)) {
  model[[count]] <- list(family = fam[count], par = par[count], par2 = 0)
  count <- count + 1
}

# The second tree contains a unique conditional copula
# In this first example, we take a linear calibration function (10*x-5)

# Set-up a dummy dataset
tmp <- data.frame(u1 = runif(1e2), u2 = runif(1e2), x1 = runif(1e2))

# Set-up an arbitrary linear model for the calibration function
model[[count]] <- gamBiCopFit(tmp, ~x1, fam[count])$res

# Update the coefficients of the model
attr(model[[count]], "model")$coefficients <- c(-5, 10)

# Define gamVine object
GVC <- gamVine(Matrix = Matrix, model = model, names = nnames)
GVC
#> GAM-Vine matrix: 
#>      [,1] [,2] [,3]
#> [1,]    2    0    0
#> [2,]    3    3    0
#> [3,]    1    1    1
#> 
#>  Where 
#> 1  <->  x1 
#> 2  <->  x2 
#> 3  <->  x3 
#> 
#>  Tree 1: 
#> x2,x1: Clayton type 1 (standard and 90 degrees rotated) 
#> x3,x1: Gumbel type 1 (standard and 90 degrees rotated) 
#> 
#>  Tree 2: 
#> x2,x3|x1 : Gaussian copula with tau(z) = (exp(z)-1)/(exp(z)+1) where 
#> z ~ x1
# NOT RUN {
# Simulate new data
simData <- data.frame(gamVineSimulate(n, GVC))
colnames(simData) <- nnames

# Fit data using sequential estimation assuming true model known
summary(fitGVC <- gamVineSeqFit(simData, GVC))

# Fit data using structure selection and sequential estimation
summary(fitGVC2 <- gamVineStructureSelect(simData, simplified = FALSE))
# }