historyIndexDense {fdaconcur}

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

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 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:

beta0

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

alpha

a list of matrices with the i-th element representing the fitted \alpha_i(t, s)

yHat

an n by length(workGridY) matrix of fitted Y_i(t)'s.

workGridY

a vetor representing the working grid for the response.

workGridLag

a list of vectors with the i-th element representing the working grid in [0, \Delta_i].

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]]