# Sieve estimation: B-spline based estimation procedure for time-varying additive models. The VCAM function can be used to perform function-to-scalar regression. ```r VCAM(Lt, Ly, X, optnAdd = list(), optnVc = list()) ``` ## Arguments - `Lt`: An **n**-dimensional list of **N_i**-dimensional vectors whose elements consist of longitudinal time points for each **i**-th subject. - `Ly`: An **n**-dimensional list of **N_i**-dimensional vectors whose elements consist of longitudinal response observations of each **i**-subject corresponding to **Lt**. - `X`: An **n** by **d** matrix whose row vectors consist of covariate vector of additive components for each subject. - `optnAdd`: A list of options controls B-spline parameters for additive components, specified by list(name=value). See 'Details'. - `optnVc`: A list of options controls B-spline parameters for varying-coefficient components, specified by list(name=value). See 'Details'. ## Returns A list containing the following fields: - **Lt**: The same with input given by **Lt** - **LyHat**: Fitted values corresponding to **Ly** - **phiEst**: An **N** by **d** matrix whose column vectors consist of estimates for each additive component function evaluated at **gridX**. - **beta0Est**: An **M**-dimensional vector for overall mean function estimates evaluated at **gridT**. - **betaEst**: An **M** by **d** matrix whose column vectors consist of estimates for each varying-coefficient components evaluated at **gridT**. - **gridX**: The same with input given by **optnAdd$grid** - **gridT**: The same with input given by **optnVc$grid** ## Details `VCAM` provides a simple algorithm based on B-spline basis to estimate its nonparametric additive and varying-coefficient components. Available control options for **optnAdd** are - **nKnot**: A **d**-dimensional vector which designates the number of knots for each additive component function estimation (default=10). - **order**: A **d**-dimensional vector which designates the order of B-spline basis for each additive component function estimation (default=3). - **grid**: A **N** by **d** matrix whose column vector consist of evaluation grid points for each component function estimation. and control options for **optnVc** are - **nKnot**: A **(d+1)**-dimensional vector which designates the number of knots for overall mean function and each varying-coefficient component function estimation (default=10). - **order**: A **(d+1)**-dimensional vector which designates the order of B-spline basis for overall mean function and each varying-coefficient component function estimation (default=3). - **grid**: A **M** by **(d+1)** matrix whose column vectors consist of evaluation grid points for overall mean function and each varying-coefficient component function estimation. ## Examples ```r library(MASS) set.seed(100) n <- 100 d <- 2 Lt <- list() Ly <- list() m <- rep(0,2) S <- matrix(c(1,0.5,0.5,1),nrow=2,ncol=2) X <- pnorm(mvrnorm(n,m,S)) beta0 <- function(t) 1.5*sin(3*pi*(t+0.5)) beta1 <- function(t) 3*(1-t)^2 beta2 <- function(t) 4*t^3 phi1 <- function(x) sin(2*pi*x) phi2 <- function(x) 4*x^3-1 for (i in 1:n) { Ni <- sample(10:20,1) Lt[[i]] <- sort(runif(Ni,0,1)) Ly[[i]] <- beta0(Lt[[i]]) + beta1(Lt[[i]])*phi1(X[i,1]) + beta2(Lt[[i]])*phi2(X[i,2]) + rnorm(Ni,0,0.1) } vcam <- VCAM(Lt,Ly,X) op <- par(no.readonly = TRUE) par(mfrow=c(1,1)) plot(unlist(vcam$LyHat),unlist(Ly),xlab='observed Y',ylab='fitted Y') abline(coef=c(0,1),col=8) par(mfrow=c(1,2)) plot(vcam$gridX[,1],vcam$phiEst[,1],type='l',ylim=c(-1,1),xlab='x1',ylab='phi1') points(vcam$gridX[,1],phi1(vcam$gridX[,1]),col=2,type='l',lty=2,lwd=2) legend('topright',c('true','est'),lwd=2,lty=c(1,2),col=c(1,2)) plot(vcam$gridX[,2],vcam$phiEst[,2],type='l',ylim=c(-1,3),xlab='x2',ylab='phi2') points(vcam$gridX[,2],phi2(vcam$gridX[,2]),col=2,type='l',lty=2,lwd=2) legend('topleft',c('true','est'),lwd=2,lty=c(1,2),col=c(1,2)) par(mfrow=c(1,3)) plot(vcam$gridT,vcam$beta0Est,type='l',xlab='t',ylab='beta0') points(vcam$gridT,beta0(vcam$gridT),col=2,type='l',lty=2,lwd=2) legend('topright',c('true','est'),lwd=2,lty=c(1,2),col=c(1,2)) plot(vcam$gridT,vcam$betaEst[,1],type='l',xlab='t',ylab='beta1') points(vcam$gridT,beta1(vcam$gridT),col=2,type='l',lty=2,lwd=2) legend('topright',c('true','est'),lwd=2,lty=c(1,2),col=c(1,2)) plot(vcam$gridT,vcam$betaEst[,2],type='l',xlab='t',ylab='beta2') points(vcam$gridT,beta2(vcam$gridT),col=2,type='l',lty=2,lwd=2) legend('topright',c('true','est'),lwd=2,lty=c(1,2),col=c(1,2)) par(op) ``` ## References Zhang, X. and Wang, J.-L. (2015), "Varying-coefficient additive modelsfor functional data", Biometrika, Vol.102, No.1, p.15-32.