# Bootstrap pointwise confidence intervals for the coefficient functions in functional concurrent regression for densely observed data. ```r GetCI_Dense(dat, tGrid, level = 0.95, R = 10, bw, kernel_type) ``` ## Arguments - `dat`: A list of input functional/scalar covariates. Each field corresponds to a functional (a matrix) or scalar (a vector) variable. The last entry is assumed to be the functional response if no entry is names `'Y'`. If a field corresponds to a functional variable, it should be an $n$-by-$m$ matrix, where each row holds the observations for one subject on the common grid `tGrid`. If a field corresponds to a scalar covariate, it should be a vector of length $n$. - `tGrid`: A vector of length $m$ with the input time points. - `level`: A number taking values in [0,1] determing the confidence level. Default: 0.95. - `R`: An integer holding the number of bootstrap replicates. Default: 999. - `bw`: Scalar holding the bandwidth. - `kernel_type`: Character holding the kernel type (see `Lwls1D` for supported kernels). ## Returns 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 intercept function 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 ```r set.seed(1) n <- 50 nGridIn <- 101 tGrid <- seq(0, 1, length.out=nGridIn) # Functional data support muX1 <- tGrid * 2 # mean function for X_1 sigma <- 1 beta0 <- 0 beta <- rbind(cos(tGrid), 1.5 + sin(tGrid)) Z <- MASS::mvrnorm(n, rep(0, 2), diag(2)) X_1 <- Z[, 1, drop=FALSE] %*% matrix(1, 1, nGridIn) + matrix(muX1, n, nGridIn, byrow=TRUE) epsilon <- rnorm(n, sd=sigma) Y <- t(sapply(seq_len(n), function(i) { beta0 + beta[1,] * X_1[i, ] + beta[2,] * Z[i, 2] + epsilon[i] })) dat <- list(X1=X_1, Z1=Z[, 2], Y=Y) res <- ptFCReg(tGrid = tGrid, dat = dat) smres <- smPtFCRegCoef(res, bw = 2.5 / (nGridIn-1), kernel_type = 'epan') res_CI = GetCI_Dense(dat, tGrid, level = 0.95, R = 10, bw = 2.5 / (nGridIn-1), kernel_type = 'epan') beta1 = res_CI$CI_beta[[1]] ##extracting CI for beta1 beta1a = beta1$CI_beta1.lower beta1b = beta1$CI_beta1.upper true_beta = beta[1,] ##extracting true coef beta1 in the simulation setting est_beta = smres$beta[1,] ## ##extracting estimated coef beta1 from ###fitting the concurrent regression model plot(beta1$CIgrid, beta1a, type= 'l', ylim = c(0,2)) ##plot of lower CI for beta1 lines(beta1$CIgrid, beta1b) ##plot of lower CI for beta1 lines(beta1$CIgrid, true_beta, col ='red') ##plot of true coef beta1 in the simulation setting lines(beta1$CIgrid, est_beta, col ='blue') ##plot of estimated coef beta1 from fitting the concurrent regression model ```

GetCI_Dense function