Predict method of a Generalized Additive model for the copula parameter or Kendall's tau

Predict method of a Generalized Additive model for the copula parameter or Kendall's tau

gamBiCopPredict(object, newdata = NULL, target = "calib", alpha = 0,
  type = "link")

Arguments

object	gamBiCop-class object.
newdata	(Same as in predict.gam from the mgcv package) A matrix or data frame containing the values of the model covariates at which predictions are required. If this is not provided then predictions corresponding to the original data are returned. If newdata is provided then it should contain all the variables needed for prediction: a warning is generated if not.
target	Either 'calib', 'par' or 'tau' or a combination of those. 'calib' (default) corresponds to the calibration function, 'par' to the copula parameter and 'tau' to Kendall's tau.
alpha	In (0,1) to return the corresponding confidence interval.
type	(Similar as in predict.gam from the mgcv package, only active for type = 'calib'). When this has the value 'link' (default), the calibration function is returned. When type = 'terms' each component of the linear predictor is returned separately (possibly with standard errors): this includes parametric model components, followed by each smooth component, but excludes any offset and any intercept. When type = 'lpmatrix' then a matrix is returned which yields the values of the linear predictor (minus any offset) when post-multiplied by the parameter vector (in this case alpha is ignored).

Value

If target = 'calib', then a list with 1 item calib. If target = 'par', target = 'tau' or target = c('par', 'tau'), then a list with 2, 2 or 3 items, namely calib and par, tau and par, or calib, tau and par.

If alpha is in (0,1), then a additional items of the list are calib.CI as well as e.g. par.CI and/or tau.CI depending on the value of target.

Otherwise, if type = 'lpmatrix' (only active for type = 'calib'), then a matrix is returned which will give a vector of linear predictor values (minus any offset) at the supplied covariate values, when applied to the model coefficient vector (similar as predict.gam from the mgcv).

See also

gamBiCop and gamBiCopFit.

Examples

require(copula)
set.seed(0)

## Simulation parameters (sample size, correlation between covariates,
## Clayton copula family)
n <- 5e2
rho <- 0.5
fam <- 1

## A calibration surface depending on three variables
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)))
  }
)

## 3-dimensional matrix X of covariates
covariates.distr <- mvdc(normalCopula(rho, dim = 3),
  c("unif"), list(list(min = 0, max = 1)),
  marginsIdentical = TRUE
)
X <- rMvdc(n, covariates.distr)
colnames(X) <- paste("x", 1:3, sep = "")

## U in [0,1]x[0,1] with copula parameter depending on X
U <- condBiCopSim(fam, function(x1, x2, x3) {
  eta0 + sum(mapply(function(f, x)
    f(x), calib.surf, c(x1, x2, x3)))
}, X[, 1:3], par2 = 6, return.par = TRUE)

## Merge U and X
data <- data.frame(U$data, X)
names(data) <- c(paste("u", 1:2, sep = ""), paste("x", 1:3, sep = ""))

## Model fit with penalized cubic splines (via min GCV)
basis <- c(3, 10, 10)
formula <- ~ s(x1, k = basis[1], bs = "cr") +
  s(x2, k = basis[2], bs = "cr") +
  s(x3, k = basis[3], bs = "cr")
system.time(fit <- gamBiCopFit(data, formula, fam))
#>    user  system elapsed 
#>   0.155   0.000   0.156 

## Extract the gamBiCop objects and show various methods
(res <- fit$res)
#> Gaussian copula with tau(z) = (exp(z)-1)/(exp(z)+1) where 
#> z ~ s(x1, k = basis[1], bs = "cr") + s(x2, k = basis[2], bs = "cr") + 
#>     s(x3, k = basis[3], bs = "cr")
EDF(res)
#> [1] 1.000000 1.999178 7.577543 5.665301
pred <- gamBiCopPredict(fit$res, X, target = c("calib", "par", "tau"))