CD4 cell counts

Description

A dataset containing the logarithm of CD4 cell counts for 190 patients with AIDS from June 1997 to January 2002. The data come from a human immunodeficiency virus (HIV) study by Wohl et al. (2005) and can be obtained from Cao et al. (2015).

Usage

CD4

Format

A data frame with 741 rows and 3 variables:

PATIENT

Patient ID.

CD4OBS

Logarithm of CD4 cell counts.

CD4DATE

Day of measurement.

References

David A. Wohl, Donglin Zeng, Paul Stewart, Nicolas Glomb, Timothy Alcorn, Suzanne Jones, Jean Handy, Susan Fiscus, Adriana Weinberg, Deepthiman Gowda, and Charles van der Horst (2005). "Cytomegalovirus viremia, mortality, and end-organ disease among patients with aids receiving potent antiretroviral therapies." Journal of Acquired Immune Deficiency Syndromes, 38(5):538-544.

Hongyuan Cao, Donglin Zeng, and Jason P. Fine (2015). "Regression analysis of sparse asynchronous longitudinal data." Journal of The Royal Statistical Society Series B-statistical Methodology, 77(4):755-776.

Least square estimates of functional principal component scores

Description

Least square estimates (LSE) of functional principal component scores.

Usage

FPCscoreLSE(Lt, Ly, kern, bw, FPC_dis, RegGrid, more = FALSE)

Arguments

Value

If more = FALSE, a n by nK matrix containing the estimates of the FPC scores is returned, where n is the sample size. If more = TRUE, a list containing the following components is returned:

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 3:5, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
basis <- fda::create.bspline.basis(interval, nbasis = 13, norder = 4,
                              breaks = seq(0, 10, length.out = 11))
Klist <- KFPCA(DataNew$Lt, DataNew$Ly, interval, nK = 2, bw = 1,
               nRegGrid = 51, fdParobj = basis)
# Just an example to explain the use of FPCscoreLSE().
# One can obtain FPC scores estimates for KFPCA method
# by KFPCA() directly. Note that FPCscoreLSE() can also be used
# to estimate FPC scores for methods except KFPCA.
scoreKFPCA <- FPCscoreLSE(DataNew$Lt, DataNew$Ly, kern = "epan",
                          bw = Klist$bwmean, FPC_dis = Klist$FPC_dis,
                          RegGrid = seq(interval[1], interval[2], length.out = 51))

Generate functional/longitudinal data via KL expansion

Description

Generate functional/longitudinal data via Karhunen–Loève expansion.

Usage

GenDataKL(n, interval, sparse, regular, meanfun, score, eigfun, sd)

Arguments

Value

A list containing the following components:

Examples

n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))

Bandwidth selection through GCV for one-dimension cases

Description

Bandwidth selection through generalized cross-validation (GCV) for one-dimension cases.

Usage

GetGCVbw1D(Lt, Ly, kern, dataType = "Sparse")

Arguments

Value

A scalar denoting the optimal bandwidth.

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
# Optimal bandwidth for mean function estimate
bwOpt <- GetGCVbw1D(DataNew$Lt, DataNew$Ly, kern = "epan")

Bandwidth selection through GCV for two-dimension cases

Description

Bandwidth selection through generalized cross-validation (GCV) for two-dimension cases.

Usage

GetGCVbw2D(tPairs, yin, Lt, kern, ObsGrid, RegGrid, dataType = "Sparse")

Arguments

Value

A scalar denoting the optimal bandwidth.

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
# Optimal bandwidth for the estimate of
# E{X(s)X(t)} = cov(X(s), X(t)) + mu(s) * mu(t)
xin2D <- NULL
yin2D <- NULL
for(i in 1:n){
  xin2D <- rbind(xin2D, t(utils::combn(DataNew$Lt[[i]], 2)))
  yin2D <- rbind(yin2D, t(utils::combn(DataNew$Ly[[i]], 2)))
}
tPairs <- xin2D
yin <- yin2D[,1] * yin2D[, 2]
bwOpt <- GetGCVbw2D(tPairs = tPairs, yin = yin, Lt = DataNew$Lt,
                    kern = "epan", ObsGrid = sort(unique(unlist(DataNew$Lt))),
                    RegGrid = seq(interval[1], interval[2], length.out = 51))

Kendall Functional Principal Component Analysis (KFPCA) for sparse design

Description

KFPCA for non-Gaussian functional data with sparse design or longitudinal data.

Usage

KFPCA(
  Lt,
  Ly,
  interval,
  dataType = "Sparse",
  nK,
  kern = "epan",
  bw,
  kernK = "epan",
  bwK = "GCV",
  kernmean = "epan",
  bwmean = "GCV",
  nRegGrid,
  fdParobj,
  more = TRUE
)

Arguments

Value

A list containing the following components:

References

Rou Zhong, Shishi Liu, Haocheng Li, Jingxiao Zhang (2021). "Robust Functional Principal Component Analysis for Non-Gaussian Longitudinal Data." Journal of Multivariate Analysis, https://doi.org/10.1016/j.jmva.2021.104864.

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
basis <- fda::create.bspline.basis(interval, nbasis = 13, norder = 4,
                              breaks = seq(0, 10, length.out = 11))
# KFPCA
Klist <- KFPCA(DataNew$Lt, DataNew$Ly, interval, nK = 2, bw = 1,
               nRegGrid = 51, fdParobj = basis)
plot(Klist$FPC_smooth)

Kendall Functional Principal Component Analysis (KFPCA) for dense and regular design

Description

KFPCA for non-Gaussian functional data with dense and regular design.

Usage

KFPCA_reg(Lt, Ly, nGrid, nK, fdParobj)

Arguments

Value

A list containing the following components:

References

Rou Zhong, Shishi Liu, Haocheng Li, Jingxiao Zhang (2021). "Functional principal component analysis estimator for non-Gaussian data." <arXiv: https://arxiv.org/abs/2102.01286>.

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 16 #the first eigenvalue
lambda_2 <- 9 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 51, regular = TRUE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.25))
basis <- fda::create.bspline.basis(interval, nbasis = 13, norder = 4,
                              breaks = seq(0, 10, length.out = 11))
#KFPCA
Klist <- KFPCA_reg(DataNew$Lt, DataNew$Ly, nGrid = 51, nK = 2, fdParobj = basis)
plot(Klist$FPC_list[[1]])
plot(Klist$FPC_list[[2]])

Local linear estimates of mean function

Description

Local linear estimates of mean function.

Usage

MeanEst(Lt, Ly, kern, bw, gridout)

Arguments

Value

A list containing the following components:

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){x}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
# Mean function estimate at all observation time points
bwOpt <- GetGCVbw1D(DataNew$Lt, DataNew$Ly, kern = "epan")
meanest <- MeanEst(DataNew$Lt, DataNew$Ly, kern = "epan", bw = bwOpt,
                   gridout = sort(unique(unlist(DataNew$Lt))))
plot(meanest$Grid, meanest$mean)

Sparse plot

Description

Create sparse plot to see the sparsity of the data.

Usage

SparsePlot(Lt, interval, ...)

Arguments

Details

For the sparse plot, x-axis is the observation time while y-axis represents various subjects.

Value

Create the corresponding sparse plot.

Examples

# Generate data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
# DataNew1 and DataNew2 have different sparsity
DataNew1 <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                      meanfun = function(x){0}, score = score,
                      eigfun = eigfun, sd = sqrt(0.1))
DataNew2 <- GenDataKL(n, interval = interval, sparse = 2:4, regular = FALSE,
                      meanfun = function(x){0}, score = score,
                      eigfun = eigfun, sd = sqrt(0.1))
# Create sparse plots
par(mfrow = c(1, 2))
SparsePlot(DataNew1$Lt, interval = interval)
SparsePlot(DataNew2$Lt, interval = interval)
par(mfrow = c(1, 1))

Kernel Functions

Description

Some common-used kernel functions.

Usage

kernfun(type)

Arguments

Value

The corresponding kernel function.

Examples

x <- seq(-2, 2, 0.01)
par(mfrow = c(2,2))
plot(x, kernfun("epan")(x), type = "l", main = "Epanechnikov")
plot(x, kernfun("unif")(x), type = "l", main = "Uniform")
plot(x, kernfun("quar")(x), type = "l", main = "Quartic")
plot(x, kernfun("gauss")(x), type = "l", main = "Gaussian")
par(mfrow = c(1,1))

Predict FPC scores

Description

Predict FPC scores using least square estimate (LSE) for a new sample.

Usage

## S3 method for class 'KFPCA'
predict(object, newLt, newLy, nK, more = FALSE, ...)

Arguments

Value

If more = FALSE, a n by nK matrix containing the predictions of the FPC scores is returned, where n is the new sample size. If more = TRUE, a list containing the following components is returned:

Examples

# Generate training data
n <- 100
interval <- c(0, 10)
lambda_1 <- 9 #the first eigenvalue
lambda_2 <- 1.5 #the second eigenvalue
eigfun <- list()
eigfun[[1]] <- function(x){cos(pi * x/10)/sqrt(5)}
eigfun[[2]] <- function(x){sin(pi * x/10)/sqrt(5)}
score <- cbind(rnorm(n, 0, sqrt(lambda_1)), rnorm(n, 0, sqrt(lambda_2)))
DataNew <- GenDataKL(n, interval = interval, sparse = 6:8, regular = FALSE,
                     meanfun = function(x){0}, score = score,
                     eigfun = eigfun, sd = sqrt(0.1))
basis <- fda::create.bspline.basis(interval, nbasis = 13, norder = 4,
                              breaks = seq(0, 10, length.out = 11))
Klist <- KFPCA(DataNew$Lt, DataNew$Ly, interval, nK = 2, bw = 1,
               nRegGrid = 51, fdParobj = basis)
# Generate test data
n_test <- 20
score_test <- cbind(rnorm(n_test, 0, sqrt(lambda_1)),
                    rnorm(n_test, 0, sqrt(lambda_2)))
Data_test <- GenDataKL(n_test, interval = interval, sparse = 6:8, regular = FALSE,
                       meanfun = function(x){0}, score = score_test,
                       eigfun = eigfun, sd = sqrt(0.1))
# Prediction
score_pre <- predict(Klist, Data_test$Lt, Data_test$Ly, nK = 2)
plot(score_test[,1], score_pre[,1])