ConcReg_Lag {fdaconcur}

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

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.

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

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)