fdaconcur: Concurrent Regression and History Index Models for Functional Data

Description

This package provides tools for functional concurrent regression and history index models.

Details

fdaconcur for Functional Concurrent Regression and History Index Models

Provides an implementation of concurrent or varying coefficient regression methods for functional data. The implementations are done for both dense and sparsely observed functional data. Pointwise confidence bands can be constructed for each case. Further, the influence of past predictor values are modeled by a smooth history index function, while the effects on the response are described by smooth varying coefficient functions, which are very useful in analyzing real data such as COVID data.

References: Yao, F., Müller, H.G., Wang, J.L. (2005) <doi: 10.1214/009053605000000660>. Sentürk, D., Müller, H.G. (2010) <doi: 10.1198/jasa.2010.tm09228>.

PACE is based on the idea that observed functional data are generated by a sample of underlying (but usually not fully observed) random trajectories that are realizations of a stochastic process. It does not rely on pre-smoothing of trajectories, which is problematic if functional data are sparsely sampled.

Maintainer: Su I Iao siao@ucdavis.edu

Author(s)

Satarupa Bhattacharjee Yaqing Chen Changbo Zhu Han Chen Yidong Zhou Álvaro Gajardo Poorbita Kundu Hang Zhou

Hans-Georg Müller hgmueller@ucdavis.edu

See Also

Useful links:

• https://github.com/functionaldata/tFDAconcur

• Report bugs at https://github.com/functionaldata/tFDAconcur/issues

Functional Concurrent Regression with Lag Model

Description

Functional concurrent regression model with lag for dense functional responses and dense functional predictors.

Usage

ConcReg_Lag(Y, X, Lag = NULL, optnsY = NULL, optnsX = NULL)

Arguments

Details

The functional concurrent regression model with lag is defined as

E[Y(t)|X_1(t), \cdots, X_p(t)] = \beta_0(t) + \sum_{j=1}^p\beta_j(t)X_j(t-Lag[j])

For more details we refer to Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264.

Value

A list of the following:

References

Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264. Yao, F., Müller, H.G., Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Annals of Statistics 33, 2873–2903. Hall, P., Horowitz, J.L. (2007). Methodology and convergence rates for functional linear regression. The Annals of Statistics, 35(1), 70–91.

Examples

phi1 <- function(t) sin(pi*t / 5) / sqrt(5)
phi2 <- function(t) cos(pi*t / 5) / sqrt(5)
lambdaX <- c(10, 5)
n <- 50
N <- 101
Xi <- matrix(rnorm(2*n), nrow = n, ncol = 2)
denseLt <- list()
denseLy <- list()
t0 <- seq(0, 15, length.out = N)
for (i in 1:n) {
  denseLt[[i]] <- t0
  denseLy[[i]] <- lambdaX[1]*Xi[i, 1]*phi1(t0) + lambdaX[2]*Xi[i, 2]*phi2(t0)
}
denseX0 <- list(Ly = denseLy,Lt = denseLt)

Lag <- c(3)
t0_out <- t0[t0>=max(Lag)]
beta_1 <- function(t) 5*sin(pi*t/10)
beta_0 <- function(t) t^2/2
### functional response Y(t), t in t0_out ### 
denseLt <- list()
denseLy <- list()
for (i in 1:n) {
  denseLt[[i]] <- t0_out
  denseLy[[i]] <- beta_0(t0_out) +  
    denseX0$Ly[[i]][1:length(t0_out)]* beta_1(t0_out) + 
    rnorm(length(t0_out), 0, 0.1) 
}
denseY <- list(Ly = denseLy, Lt = denseLt)
model = ConcReg_Lag(Y=denseY, X=list(X1 = denseX0), Lag=Lag)

print(model$workGridY) # workGrid for Y between 6 to 15
plot(beta_1(model$workGridY)) # workGrid for X1 between 3 to 12 
plot(beta_0(model$workGridY))

plot(model$beta[1,])
plot(model$beta0)

Functional Concurrent Regression using 2D smoothing

Description

Functional concurrent regression with dense or sparse functional data for scalar or functional dependent variables. Note: function-to-scalar regression can also be handled using the VCAM function in fdapace.

Usage

ConcurReg(
  vars,
  outGrid,
  userBwMu = NULL,
  userBwCov = NULL,
  kern = "gauss",
  measurementError = FALSE,
  diag1D = "all",
  useGAM = FALSE,
  returnCov = TRUE
)

Arguments

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:

References

• Yao, F., Müller, H.G., Wang, J.L. "Functional Linear Regression Analysis for Longitudinal Data." Annals of Statistics 33, (2005): 2873-2903.(Dense data)

• Sentürk, D., Müller, H.G. "Functional varying coefficient models for longitudinal data." J. American Statistical Association, 10, (2010): 1256–1264.

• Sentürk, D., Nguyen, D.V. "Varying Coefficient Models for Sparse Noise-contaminated Longitudinal Data", Statistica Sinica 21(4), (2011): 1831-1856. (Sparse data)

Examples

# Y(t) = beta_0(t) + beta_1(t) X_1(t) + beta_2(t) Z_2 + epsilon
#
# Settings
set.seed(1)
n <- 30
nGridIn <- 50
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)
 outGrid <- seq(min(T), max(T), by=0.05)
# 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)
set.seed(1)
Ysp <- fdapace::Sparsify(Y, T, sparsity)
vars <- list(X_1=X_1sp, Z_2=Z[, 2], Y=Ysp)
res2 <- ConcurReg(vars, outGrid, userBwMu=c(.5,0,.5), userBwCov=c(.5,0,.5), kern='gauss',
 measurementError=TRUE, diag1D='none', useGAM = FALSE, returnCov=TRUE)

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

Description

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

Usage

GetCI_Dense(dat, tGrid, level = 0.95, R = 10, bw, kernel_type)

Arguments

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 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

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

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

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)

Fitted functional responses from a ptFCReg object.

Description

Fitted functional responses from a ptFCReg object.

Usage

fitted_ptFCReg(object)

Arguments

Value

An n-by-m matrix, where each row corresponds to one subject, and each column corresponds to a time point in object$tGrid.

Examples

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')
fit_res <- stats::fitted(res)
fit_smres <- stats::fitted(smres)

Functional History Index Model

Description

Functional history index model for dense functional responses and dense functional predictors.

Usage

historyIndexDense(Y, X, Lag = NULL, optnsY = NULL, optnsX = NULL)

Arguments

Details

The functional history index model is defined as E[Y(t)|X_1(t), \cdots, X_p(t)] = \beta_0(t) + \sum_{i=1}^p\beta_i(t)\int_0^{\Delta_i}\gamma_i(s)X_i(t-s)ds for t\in[\max_i\{\Delta_i\}, T] with a suitable T>0. Write \alpha_i(t, s)=\beta_i(t)\gamma_i(s). It becomes E[Y(t)|X_1(t), \cdots, X_p(t)] = \beta_0(t) + \sum_{i=1}^p\int_0^{\Delta_i}\alpha_i(t, s)X_i(t-s)ds. For more details we refer to Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264.

Value

A list of the following:

References

Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264. Yao, F., Müller, H.G., Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Annals of Statistics 33, 2873–2903. Hall, P., Horowitz, J.L. (2007). Methodology and convergence rates for functional linear regression. The Annals of Statistics, 35(1), 70–91.

Examples

set.seed(1)
### functional covariate X(t) ###
phi1 <- function(t) sin(pi*t / 5) / sqrt(5)
phi2 <- function(t) cos(pi*t / 5) / sqrt(5)
lambdaX <- c(10, 5)
n <- 150
N <- 101
Xi <- matrix(rnorm(2*n), nrow = n, ncol = 2)
denseLt <- list()
denseLy <- list()
t0 <- seq(0, 15, length.out = N)
for (i in 1:n) {
  denseLt[[i]] <- t0
  denseLy[[i]] <- lambdaX[1]*Xi[i, 1]*phi1(t0) + lambdaX[2]*Xi[i, 2]*phi2(t0)
}
denseX0 <- list(Ly = denseLy, Lt = denseLt)

### generate coefficient function gamma(u), beta(u) ###
Lag <- 5
u0 <- t0[t0<=Lag]
t0_out <- t0[t0>=Lag]
gamma_u <- function(u) sqrt(2/5) * cos(pi * u /5)
beta_1 <- function(t) 5*sin(pi*t/10)
beta_0 <- function(t) t^2/2

### functional response Y(t), t in t0_out ###
denseLt <- list()
denseLy <- list()
for (i in 1:n) {
  denseLt[[i]] <- t0_out
  Xt <- denseX0$Ly[[i]]
  Xtu <- t(sapply((1:N)[t0>=Lag], function(j){
    rev(Xt[(j-length(u0)+1):j])  #history index for X[t-u:t]
  }))
  IntGammaXtu <- apply(Xtu, 1, function(v){
    fdapace::trapzRcpp(u0, gamma_u(u0) * v)
  })
  #append 0 in the first length(u0)-1 element(useless info. in our modeling)
  denseLy[[i]] <- beta_0(t0_out) + IntGammaXtu * beta_1(t0_out) + rnorm(length(t0_out), 0, 0.1)
}
denseY <- list(Ly = denseLy, Lt = denseLt)

### functional predictor X(t) (adjust for t0_out) ###
denseLt <- list()
denseLy <- list()
for (i in 1:n){
  denseLt[[i]] <- t0_out
  denseLy[[i]] <- denseX0$Ly[[i]][t0>=Lag]
}
denseX <- list(Ly = denseLy,
               Lt = denseLt)
fit <- historyIndexDense(Y = denseY, X = list(X = denseX), Lag = Lag)
fit$beta0
fit$alpha[[1]]

Functional History Index Model

Description

Functional history index model for functional responses and functional predictors.

Usage

historyIndexSparse(Y, X, Lag = NULL, optnsY = NULL, optnsX = NULL)

Arguments

Details

The functional history index model is defined as E[Y(t)|X_1(t), \cdots, X_p(t)] = \beta_0(t) + \sum_{j=1}^p\beta_j(t)\int_0^{\Delta_j}\gamma_j(s)X_j(t-s)ds for t\in[\max_j\{\Delta_j\}, T] with a suitable T>0. Write \alpha_j(t, s)=\beta_j(t)\gamma_j(s). It becomes E[Y(t)|X_1(t), \cdots, X_p(t)] = \beta_0(t) + \sum_{j=1}^p\int_0^{\Delta_j}\alpha_j(t, s)X_j(t-s)ds. For more details we refer to Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264.

Value

A list of the following:

References

Şentürk, D. and Müller, H.G., (2010). Functional varying coefficient models for longitudinal data. Journal of the American Statistical Association, 105(491), pp.1256-1264. Yao, F., Müller, H.G., Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Annals of Statistics 33, 2873–2903. Hall, P., Horowitz, J.L. (2007). Methodology and convergence rates for functional linear regression. The Annals of Statistics, 35(1), 70–91.

Examples

set.seed(1)
### functional covariate X(t) ###
phi1 <- function(t) sin(pi*t / 5) / sqrt(5)
phi2 <- function(t) cos(pi*t / 5) / sqrt(5)
lambdaX <- c(10, 5)
n <- 150
N <- 101
Xi <- matrix(rnorm(2*n), nrow = n, ncol = 2)
denseLt <- list()
denseLy <- list()
t0 <- seq(0, 15, length.out = N)
for (i in 1:n) {
  denseLt[[i]] <- t0
  denseLy[[i]] <- lambdaX[1]*Xi[i, 1]*phi1(t0) + lambdaX[2]*Xi[i, 2]*phi2(t0)
}
denseX0 <- list(Ly = denseLy,Lt = denseLt)

### generate coefficient function gamma(u), beta(u) ###
Lag <- 5
u0 <- t0[t0<=Lag]
t0_out <- t0[t0>=Lag]
gamma_u <- function(u) sqrt(2/5) * cos(pi * u /5)
beta_1 <- function(t) 5*sin(pi*t/10)
beta_0 <- function(t) t^2/2

### functional response Y(t), t in t0_out ### 
denseLt <- list()
denseLy <- list()
for (i in 1:n) {
  denseLt[[i]] <- t0_out
  Xt <- denseX0$Ly[[i]]
  Xtu <- t(sapply((1:N)[t0>=Lag], function(j){
    rev(Xt[(j-length(u0)+1):j])  #history index for X[t-u:t]
  }))
  IntGammaXtu <- apply(Xtu, 1, function(v){
    fdapace::trapzRcpp(u0, gamma_u(u0) * v)
  })
  #append 0 in the first length(u0)-1 element(useless info. in our modeling)
  denseLy[[i]] <- beta_0(t0_out) + IntGammaXtu * beta_1(t0_out) + rnorm(length(t0_out), 0, 0.1) 
}
denseY <- list(Ly = denseLy, Lt = denseLt)

### functional predictor X(t) (adjust for t0_out) ###
denseLt <- list()
denseLy <- list()
for (i in 1:n){
  denseLt[[i]] <- t0_out
  denseLy[[i]] <- denseX0$Ly[[i]][t0>=Lag]
}
denseX <- list(Ly = denseLy,
               Lt = denseLt)
               
### sparify the dense data ###
SparseY <- fdapace::Sparsify(samp = t(sapply(denseY$Ly, function(v) v)), 
                    pts = t0_out,
                    sparsity = sample(10:20, n, replace = TRUE)
)
SparseX <- fdapace::Sparsify(samp = t(sapply(denseX$Ly, function(v) v)),
                    pts = t0_out,
                    sparsity = sample(10:20, n, replace = TRUE))

### model fitting for sparse case ###
Fit_result <- historyIndexSparse(Y = SparseY, X = list(X = SparseX), Lag = Lag)
Fit_result$beta0
Fit_result$beta1
Fit_result$gamma

Functional concurrent regression using pointwise multiple linear regression.

Description

Functional concurrent regression using pointwise multiple linear regression.

Usage

ptFCReg(tGrid, dat)

Arguments

Value

A list containing the following fields:

beta0

A vector containing the time-varying intercept evaluated on tGrid.

beta

A matrix for the concurrent regression effects, where rows correspond to different predictors and columns to different time points in tGrid.

tGrid

The input tGrid.

R2

A vector of the time-varying R^2(t), evaluated at t in tGrid.

Ldf

A list holding the input data, each element of which is a data frame holding the data observed at one element of tGrid.

Examples

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)

Smooth the concurrent effects functions in a ptFCReg object using local linear regression. The local linear regression is implemented using the function Lwls1D.

Description

Smooth the concurrent effects functions in a ptFCReg object using local linear regression. The local linear regression is implemented using the function Lwls1D.

Usage

smPtFCRegCoef(object, bw, kernel_type)

Arguments

Value

An object of class ptFCReg, where the fields beta0 and beta hold the smoothed intercept functions and concurrent effects functions, respectively. See ptFCReg for a complete list of the fields.

Examples

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')