Date: 2021-10-01

# Completion of Functional Fragments

#### 1 October 2021

## Overview

We utilize the package ``fdapace`` (Gajardo et al. 2021) to showcase the completion of simulated functional fragments when one has available a densely observed sample of curves (Carroll et al. 2020).

    ### Simulate functional fragments

    set.seed(1)
    # set parameters for simulation
    n <- 50 # number of sample trajectories
    m <- 30 # number of observation per curve
    grid <- seq(0, 1, len = m) # set up regular grid
    lambda.cos <- 3^(-(2*(1:25)-1)) # eigenvalues
    lambda.sin <- 3^(-(2*(1:25)))
    mpct <- 0.25 # proportion of domain missing
    densegr <- ((1:100) - .5)/100 # dense regular grid to generate the data
    gr <- densegr[(100/m)*(1:m)] # observation grid
    sigma2 <- 0.1 # variance for noise term

    # generate discretely observed complete functional data
    x.full <- matrix(0, n, length(densegr))
    for (j in 1:length(lambda.cos)) {
      f <- sqrt(lambda.cos[j])*sqrt(2)*cos(2*pi*(2*j - 1)*densegr)
      x.full <- x.full + rnorm(n, 0, 1)%*%t(f)
    }
    for (j in 1:length(lambda.sin)) {
      f <- sqrt(lambda.sin[j])*sqrt(2)*sin(2*pi*(2*j)*densegr)
      x.full <- x.full + rnorm(n, 0, 1)%*%t(f)
    }

    # subset the original dense true curve to the k-equidistance sampling scheme;
    x <- x.full[, densegr %in% gr]

    # generate observation periods by deleting a random subinterval from domain,
    # with 20% of data is complete
    # matrix of indicators for inclusion/exclusion of observation
    x.obs <- matrix(1, n, length(grid))
    centers <- c(0.25, 0.5, 0.75)
    x.obs[1, ] <- grid <= 0.75
    for (i in floor(n*0.2):n) {
      cen <- runif(1)
      x.obs[i, (cen - mpct/2 < grid)&(grid < cen + mpct/2)] <- FALSE
      # missing interval = (cen-u,cen+u)
    }
    # remove missing periods
    x[!x.obs] <- NA

    # distort by error term
    for(i in 1:dim(x)[1]){
      for(j in 1:dim(x)[2]){
        x[i,j] <- x[i,j] + rnorm(n = 1, sd = sqrt(sigma2))
      }
    }

    # visualize the functional fragments
    matplot(gr, t(x[c(1, 9:15), ]), type="l", lty=c(1, 1, 2, 2, 2, 2, 2, 2),
            xlab="", ylab="")
    title(sub = 'Figure 1: Error-contaminated observations from functional fragments.',
          line = 2)

    ### Use FPCA to complete the missing fragments

    # convert x matrix into 3 vectors
    id <- c()
    time <- c()
    y <- c()
    for(i in 1:n){
      id <- append(id, rep(i, sum(x.obs[i, ])))
      time <- append(time, grid[x.obs[i, ] == 1])
      y <- append(y, x[i, x.obs[i,] == 1])

    }
    L3 <- MakeFPCAInputs(IDs = id, tVec = time, y)
    optns <- list(dataType = 'Sparse', usergrid = TRUE, methodXi = 'CE')
    fit <- FPCA(L3$Ly, L3$Lt, optns = optns)
    # completed trajectory for the 1st test sample
    PACE.pred <- as.vector(predict(fit, newLy = L3$Ly[1], newLt = L3$Lt[1],
                                   K = fit$selectK, xiMethod = "CE")$predCurves)

    # Visualize and compare the observations, true trajectory and completed trajectory
    # scatterplot of original data with missing parts
    matplot(gr, x[1, ], pch = c(16), col = 1, xlab = "", ylab = "", ylim = c(-2, 2))
    lines(grid, PACE.pred, lty = 1, col = 2, xlab = "", ylab = "")# completed functional data
    lines(densegr, x.full[1, ], col = 3)# discretely observed complete functional data
    legend("topleft", legend = c("discretedly observed data", "completed trajectory",
                                 "true trajectory"),
           bty = "n", lty = c(NA, 1, 1), pch = c(16, NA, NA), col = 1:3)
    title(sub = 'Figure 2: PACE completion for functional fragments with 25% of
          subinterval missing.', line = 3)

## References

Carroll, Cody, Satarupa Bhattacharjee, Yaqing Chen, Paromita Dubey, Jianing Fan, Alvaro Gajardo, Xiner Zhou, Hans-Georg Müller, and Jane-Ling Wang. 2020. “Time Dynamics of COVID-19.” Scientific Reports 10 (1): 1–14.
Gajardo, Alvaro, Satarupa Bhattacharjee, Cody Carroll, Yaqing Chen, Xiongtao Dai, Jianing Fan, Pantelis Z. Hadjipantelis, et al. 2021. fdapace: Functional Data Analysis and Empirical Dynamics. https://CRAN.R-project.org/package=fdapace.