| gamVineCopSelect {gamCopula} | R Documentation |
Sequential pair-copula selection and maximum penalized likelihood estimation of a GAM-Vine model.
Description
This function select the copula family 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.
Usage
gamVineCopSelect(
data,
Matrix,
lin.covs = NULL,
smooth.covs = NULL,
simplified = FALSE,
familyset = NA,
rotations = TRUE,
familycrit = "AIC",
level = 0.05,
trunclevel = NA,
tau = TRUE,
method = "FS",
tol.rel = 0.001,
n.iters = 10,
parallel = FALSE,
verbose = FALSE,
select.once = TRUE,
...
)
Arguments
data |
A matrix or data frame containing the data in [0,1]^d. |
Matrix |
Lower triangular |
lin.covs |
A matrix or data frame containing the parametric (i.e.,
linear) covariates (default: |
smooth.covs |
A matrix or data frame containing the non-parametric
(i.e., smooth) covariates (default: |
simplified |
If |
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!). The vector has to include at least one
pair-copula family that allows for positive and one that allows for negative
dependence. Not listed copula families might be included to better handle
limit cases. If |
rotations |
If |
familycrit |
Character indicating the criterion for bivariate copula
selection. Possible choices: |
level |
Numerical; Passed to |
trunclevel |
Integer; level of truncation. |
tau |
|
method |
|
tol.rel |
Relative tolerance for |
n.iters |
Maximal number of iterations for
|
parallel |
|
verbose |
|
select.once |
if |
... |
Value
gamVineCopSelect returns a gamVine-class object.
See Also
gamVineSeqFit,gamVineStructureSelect,
gamVine-class, gamVineSimulate and
gamBiCopFit.
Examples
require(mgcv)
set.seed(0)
## Simulation parameters
# Sample size
n <- 1e3
# Copula families
familyset <- c(1:2, 301:304, 401:404)
# Define a 4-dimensional R-vine tree structure matrix
d <- 4
Matrix <- c(2, 3, 4, 1, 0, 3, 4, 1, 0, 0, 4, 1, 0, 0, 0, 1)
Matrix <- matrix(Matrix, d, d)
nnames <- paste("X", 1:d, sep = "")
## A function factory
eta0 <- 1
calib.surf <- list(
calib.quad <- function(t, Ti = 0, Tf = 1, b = 8) {
Tm <- (Tf - Ti) / 2
a <- -(b / 3) * (Tf^2 - 3 * Tf * Tm + 3 * Tm^2)
return(a + b * (t - Tm)^2)
},
calib.sin <- function(t, Ti = 0, Tf = 1, b = 1, f = 1) {
a <- b * (1 - 2 * Tf * pi / (f * Tf * pi +
cos(2 * f * pi * (Tf - Ti))
- cos(2 * f * pi * Ti)))
return((a + b) / 2 + (b - a) * sin(2 * f * pi * (t - Ti)) / 2)
},
calib.exp <- function(t, Ti = 0, Tf = 1, b = 2, s = Tf / 8) {
Tm <- (Tf - Ti) / 2
a <- (b * s * sqrt(2 * pi) / Tf) * (pnorm(0, Tm, s) - pnorm(Tf, Tm, s))
return(a + b * exp(-(t - Tm)^2 / (2 * s^2)))
}
)
## Create the model
# Define gam-vine model list
count <- 1
model <- vector(mode = "list", length = d * (d - 1) / 2)
sel <- seq(d, d^2 - d, by = d)
# First tree
for (i in 1:(d - 1)) {
# Select a copula family
family <- sample(familyset, 1)
model[[count]]$family <- family
# Use the canonical link and a randomly generated parameter
if (is.element(family, c(1, 2))) {
model[[count]]$par <- tanh(rnorm(1) / 2)
if (family == 2) {
model[[count]]$par2 <- 2 + exp(rnorm(1))
}
} else {
if (is.element(family, c(401:404))) {
rr <- rnorm(1)
model[[count]]$par <- sign(rr) * (1 + abs(rr))
} else {
model[[count]]$par <- rnorm(1)
}
model[[count]]$par2 <- 0
}
count <- count + 1
}
# A dummy dataset
data <- data.frame(u1 = runif(1e2), u2 = runif(1e2), matrix(runif(1e2 * d), 1e2, d))
# Trees 2 to (d-1)
for (j in 2:(d - 1)) {
for (i in 1:(d - j)) {
# Select a copula family
family <- sample(familyset, 1)
# Select the conditiong set and create a model formula
cond <- nnames[sort(Matrix[(d - j + 2):d, i])]
tmpform <- paste("~", paste(paste("s(", cond, ", k=10, bs='cr')",
sep = ""
), collapse = " + "))
l <- length(cond)
temp <- sample(3, l, replace = TRUE)
# Spline approximation of the true function
m <- 1e2
x <- matrix(seq(0, 1, length.out = m), nrow = m, ncol = 1)
if (l != 1) {
tmp.fct <- paste("function(x){eta0+",
paste(sapply(1:l, function(x)
paste("calib.surf[[", temp[x], "]](x[", x, "])",
sep = ""
)), collapse = "+"), "}",
sep = ""
)
tmp.fct <- eval(parse(text = tmp.fct))
x <- eval(parse(text = paste0("expand.grid(",
paste0(rep("x", l), collapse = ","), ")",
collapse = ""
)))
y <- apply(x, 1, tmp.fct)
} else {
tmp.fct <- function(x) eta0 + calib.surf[[temp]](x)
colnames(x) <- cond
y <- tmp.fct(x)
}
# Estimate the gam model
form <- as.formula(paste0("y", tmpform))
dd <- data.frame(y, x)
names(dd) <- c("y", cond)
b <- gam(form, data = dd)
# plot(x[,1],(y-fitted(b))/y)
# Create a dummy gamBiCop object
tmp <- gamBiCopFit(data = data, formula = form, family = 1, n.iters = 1)$res
# Update the copula family and the model coefficients
attr(tmp, "model")$coefficients <- coefficients(b)
attr(tmp, "model")$smooth <- b$smooth
attr(tmp, "family") <- family
if (family == 2) {
attr(tmp, "par2") <- 2 + exp(rnorm(1))
}
model[[count]] <- tmp
count <- count + 1
}
}
# Create the gamVineCopula object
GVC <- gamVine(Matrix = Matrix, model = model, names = nnames)
print(GVC)
## Not run:
## Simulate and fit the model
sim <- gamVineSimulate(n, GVC)
fitGVC <- gamVineSeqFit(sim, GVC, verbose = TRUE)
fitGVC2 <- gamVineCopSelect(sim, Matrix, verbose = TRUE)
## Plot the results
par(mfrow = c(3, 4))
plot(GVC, ylim = c(-2.5, 2.5))
plot(fitGVC, ylim = c(-2.5, 2.5))
plot(fitGVC2, ylim = c(-2.5, 2.5))
## End(Not run)