fdaconcur (version 0.1.3)

ConcReg_Lag: 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)

Value

A list of the following:

beta0

a vector of length(workGridY) representing the fitted \(\beta_0(t)\)

beta

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

workGridY

a vetor representing the working grid for the response.

Lag

a vector representing the lags for all predictors

Arguments

Y

a list which contains functional responses in the form of a list LY and the time points LT at which they are observed (i.e., list(Ly = LY,Lt = LT)).

X

a list of lists which contains the observed functional predictors list Lxj and the time points list Ltj at which they are observed. It needs to be of the form list(list(Ly = Lx1,Lt = Lxt1),list(Ly = Lx2,Lt = Lxt2),...).

Lag

a length length(X) vector denoting the lags for all predictors.

optnsY

a list of options control parameters for the response specified by list(name=value). See `Details' in fdapace::FPCA.

optnsX

a list of options control parameters for the predictors specified by list(name=value). See `Details' in fdapace::FPCA.

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.

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)