R/gamVineCopSelect.R
gamVineCopSelect.RdThis 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.
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)
| 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 |
gamVineCopSelect returns a gamVine-class object.
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)#> GAM-Vine matrix: #> [,1] [,2] [,3] [,4] #> [1,] 2 0 0 0 #> [2,] 3 3 0 0 #> [3,] 4 4 4 0 #> [4,] 1 1 1 1 #> #> Where #> 1 <-> X1 #> 2 <-> X2 #> 3 <-> X3 #> 4 <-> X4 #> #> Tree 1: #> X2,X1: Gumbel type 3 (survival and 90 degrees rotated) #> X3,X1: Gaussian #> X4,X1: Gumbel type 1 (standard and 90 degrees rotated) #> #> Tree 2: #> X2,X4|X1 : Gumbel type 1 (standard and 90 degrees rotated) copula with tau(z) = (exp(z)-1)/(exp(z)+1) where #> z ~ s(X1, k = 10, bs = "cr") #> X3,X4|X1 : Gumbel type 2 (standard and 270 degrees rotated) copula with tau(z) = (exp(z)-1)/(exp(z)+1) where #> z ~ s(X1, k = 10, bs = "cr") #> #> Tree 3: #> X2,X3|X4,X1 : Gumbel type 4 (survival and 270 degrees rotated) copula with tau(z) = (exp(z)-1)/(exp(z)+1) where #> z ~ s(X1, k = 10, bs = "cr") + s(X4, k = 10, bs = "cr")# 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)) # }