GetCI_Sparse {fdaconcur}

Bootstrap pointwise confidence intervals for the coefficient functions in functional concurrent regression for sparsely observed data.

Description

Bootstrap pointwise confidence intervals for the coefficient functions in functional concurrent regression for sparsely observed data.

Usage

GetCI_Sparse(
  vars,
  outGrid = NULL,
  level = 0.95,
  R = 999,
  userBwMu = NULL,
  userBwCov = NULL,
  kern = "gauss",
  measurementError = FALSE,
  diag1D = "all",
  useGAM = FALSE
)

Arguments

vars

A list of input functional/scalar covariates. Each field corresponds to a functional (a list) or scalar (a vector) covariate. The last entry is assumed to be the response if no entry is named 'Y'. If a field corresponds to a functional covariate, it should have two fields: 'Lt', a list of time points, and 'Ly', a list of functional values.

outGrid

A vector with the output time points, which need to be within 5% and 95% of the range of functional covariates. If NULL, outGrid will be generated from 5% to 95% of the range of functional covariates with 51 grids for fitting. Default: NULL

level

A number taking values in [0,1] determining the confidence level. Default: 0.95.

R

An integer holding the number of bootstrap replicates. Default: 999.

userBwMu

A scalar/vector bandwidth used for smoothing the mean function. Each entry in the vector represents the bandwidth used for the corresponding covariate in vars. For the scalar covariates, you can input 0 as a placeholder. If you only input a scalar, the function will use the same bandwidth to smooth all mean functions. — a scalar/vector of positive numeric - default: NULL — if no scalar/vector value is provided, the bandwidth value for the smoothed mean function is chosen using 'GCV';

userBwCov

A scalar/vector bandwidth used for smoothing the auto or cross-covariances. If you use 1D smoothing for the diagonal line of the covariance (diag1D="all"), only one scalar input is needed. If you use 2D smoothing for the covariance (diag1D="none"), a vector of bandwidth is required. Each entry in the vector represents the bandwidth used for the corresponding covariate in vars. For the scalar covariates, you can input 0 as a placeholder. — a scalar/vector of positive numeric - default: NULL — if no scalar/vector is provided, the bandwidth value for the smoothed cross-covariance function is chosen using 'GCV';

kern

Smoothing kernel choice, common for mu and covariance; "rect", "gauss", "epan", "gausvar", "quar" (default: "gauss")

measurementError

Assume measurement error in the data; logical - default: FALSE. If TRUE the diagonal raw covariance will be removed when smoothing.

diag1D

A string specifying whether to use 1D smoothing for the diagonal line of the covariance. 'none': don't use 1D smoothing; 'all': use 1D for both auto- and cross-covariances. (default : 'all')

useGAM

Use GAM smoothing instead of local linear smoothing (semi-parametric option); logical - default: FALSE.

Details

If measurement error is assumed, the diagonal elements of the raw covariance will be removed. This could result in highly unstable estimate if the design is very sparse, or strong seasonality presents. WARNING! For very sparse functional data, setting measurementError = TRUE is not recommended.

Value

A list containing the following fields:

CI_beta0

CI for the intercept function — A data frame holding three variables: CIgrid — the time grid where the CIs are evaluated, CI_beta0.lower and CI_beta0.upper — the lower and upper bounds of the CIs for the intercept function on CIgrid.

CI_beta

A list containing CIs for the slope functions — the length of the list is same as the number of covariates. Each list contains the following fields: A data frame holding three variables: CIgrid — the time grid where the CIs are evaluated, CI_beta_j.lower and CI_beta_j.upper — the lower and upper bounds of the CIs for the coefficient functions on CIgrid for j = 1,2,\dots.

CI_R2

CI the time-varying R^2(t) — A data frame holding three variables: CIgrid — the time grid where the CIs are evaluated, CI_R2.lower and CI_R2.upper — the lower and upper bounds of the CIs for the time-varying R^2(t) on CIgrid.

level

The confidence level of the CIs.

Examples

set.seed(1)
n <- 30
nGridIn <- 100
sparsity <- 5:10 # Sparse data sparsity
T <- round(seq(0, 1, length.out=nGridIn), 4) # Functional data support
bw <- 0.1
outGrid <- round(seq(min(T), 1, by=0.05), 2)
# Simulate functional data
mu <- T * 2 # mean function for X_1
sigma <- 1

beta_0 <- 0
beta <- rbind(cos(T), 1.5 + sin(T))
beta_2 <- 1

Z <- MASS::mvrnorm(n, rep(0, 2), diag(2))
X_1 <- Z[, 1, drop=FALSE] %*% matrix(1, 1, nGridIn) + matrix(mu, n, nGridIn, byrow=TRUE)
epsilon <- rnorm(n, sd=sigma)
Y <- matrix(NA, n, nGridIn)
for (i in seq_len(n)) {
  Y[i, ] <- beta_0 + beta[1,] * X_1[i, ] + beta[2,] * Z[i, 2] + epsilon[i]
}

# Sparsify functional data
set.seed(1)
X_1sp <- fdapace::Sparsify(X_1, T, sparsity)
Ysp <- fdapace::Sparsify(Y, T, sparsity)
vars <- list(X_1=X_1sp, Z_2=Z[, 2], Y=Ysp)
res <-  GetCI_Sparse(vars, outGrid[-c(1,21)], level = 0.95, R = 2,
                     userBwMu = c(.1,.1,.1), userBwCov = c(.1,.1,.1),
                     kern='gauss', measurementError=TRUE, diag1D='none',
                     useGAM = FALSE)