face

Description

Fast Covariance Estimation for Sparse Functional Data

Details

Author(s)

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu

Maintainer: Cai Li <cai.li.stats@gmail.com>

References

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

Extraction of correlation and mean from a face.sparse object

Description

Extraction of correlation and mean from a face.sparse object

Usage

cor.face(object,argvals.new,option="raw")

Arguments

Value

Author(s)

Luo Xiao <lxiao5@ncsu.edu>

References

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

Examples

# See the examples for "face.sparse".

Internal functions for face package

Description

Internal function.

Fast covariance estimation for sparse functional data

Description

The function is to estimate the mean and covariance function from a cluster of functions/longitudinal observations.

Usage

face.sparse(data, newdata = NULL,
             center = TRUE, argvals.new = NULL,
             knots = 7, 
             p = 3, m = 2, lambda = NULL, lambda_mean = NULL,
             search.length = 14, 
             lower = -3, upper = 10, lower2 = -3, upper2 = 5,
             calculate.scores = FALSE,pve=0.99,two_step=FALSE)

Arguments

Details

This is a generalized version of bivariate P-splines (Eilers and Marx, 2003) for covariance smoothing of sparse functional or longitudinal data. It uses tensor product B-spline basis functions and employes a differencing penalty on the assosciated parameter matrix. The only smoothing parameter in the method is selected by leave-one-subject-out cross validation and is implemented with a fast algorithm.

There are two steps for estimation. During the first step, the objective function to minimize is the penalized least squares on empirical estimates of covariance function. During the second step, the covariance between the empirical estimates (depending on the estimates of covariance function) are accounted and thus a generalized penalized least squares are minimized.

If center is TRUE, then a population mean will be calculated and is smoothed by univariate P-spline smoothing:pspline (Eilers and Marx, 1996). This univariate smoothing uses leave-one-subject-out cross validation to select the smoothing parameter.

The knots are "equally-spaced", the differencing penalty in Eilers and Marx (2003) is used.

If the functional data are observed at the same grid for each function/curve and can be organized into a data matrix, then fpca.face in the package refund should instead be used. fpca.face allows a small percentage (less than 30 percent) of missing data in the data matrix.

Value

Author(s)

Luo Xiao <lxiao5@ncsu.edu> and Cai Li <cli9@ncsu.edu>

References

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

Paul Eilers and Brian Marx, Multivariate calibration with temperature interaction using two-dimensional penalized signal regression, Chemometrics and Intelligent Laboratory Systems 66 (2003), 159-174.

Paul Eilers and Brian Marx, Flexible smoothing with B-splines and penalties, Statist. Sci., 11, 89-121, 1996.

Simon N. Wood, P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data, Stat. Comput., doi: 10.1007/s11222-016-9666-x.

See Also

fpca.face and fpca.sc in refund

Examples

## Not run: 
##########################
#### CD4 data example
##########################

require(refund)
data(cd4)
n <- nrow(cd4)
Tt <- ncol(cd4)

id <- rep(1:n,each=Tt)
t <- rep(-18:42,times=n)
y <- as.vector(t(cd4))
sel <- which(is.na(y))


## organize data and apply FACEs
data <- data.frame(y=log(y[-sel]),
argvals = t[-sel],
subj = id[-sel])
data <- data[data$y>4.5,]
fit_face <- face.sparse(data,argvals.new=(-20:40))

## set calculate.scores to TRUE if want to get scores
fit_face <- face.sparse(data,argvals.new=(-20:40),calculate.scores=TRUE)
scores <- fit_face$rand_eff$scores

data.h <- data
tnew <- fit_face$argvals.new

## scatter plots
Xlab <- "Months since seroconversion"
Ylab <- "log (CD4 count)"
par(mfrow=c(1,1),mar = c(4.5,4.5,3,2))
id <- data.h$subj
uid <- unique(id)
plot(data.h$argvals,data.h$y,
type = "n", ylim = c(4.5,8),
xlab = Xlab, ylab = Ylab,
cex.lab = 1.25,cex.axis=1.25,cex.main = 1.25)

for(i in 1:10){
seq <- which(id==uid[i])
lines(data.h$argvals[seq],data.h$y[seq],lty=1,col="gray",lwd=1,type="l")
#points(data.h$argvals[seq],data.h$y[seq],col=1,lty=1,pch=1)
}

Sample <- seq(10,50,by=10)
for(i in Sample){
seq <- which(id==uid[i])
lines(data.h$argvals[seq],data.h$y[seq],lty=1,col="black",lwd=1,type="l")
}
lines(tnew,fit_face$mu.new,lwd=2,lty=2,col="red")

## plots of variance/correlation functions

Cov <- fit_face$Chat.new
Cov_diag <- diag(Cov)
Cor <- fit_face$Cor.new

par(mfrow=c(1,2),mar=c(4.5,4.1,3,4.5))


plot(tnew,Cov_diag,type="l",
xlab = Xlab, ylab="",main= "CD4: variance function",
#ylim = c(0.8,1.5),
cex.axis=1.25,cex.lab=1.25,cex.main=1.25,lwd=2)

require(fields)
image.plot(tnew,tnew,Cor,
xlab=Xlab, ylab = Xlab,
main = "CD4: correlation function",
cex.axis=1.25,cex.lab=1.25,cex.main=1.25,
axis.args = list(at = c(0,0.2,0.4,0.6,0.8,1.0)),
legend.shrink=0.75,legend.line=-1.5)


## prediction of several subjects

par(mfrow=c(2,2),mar=c(4.5,4.5,3,2))
Sample <- c(30,40,50,60)
for(i in 1:4){
sel <- which(id==uid[Sample[i]])
dati <- data.h[sel,]

seq <- -20:40
k <- length(seq)
dati_pred <- data.frame(y = rep(NA,nrow(dati) + k ),
argvals = c(rep(NA,nrow(dati)),seq),
subj=rep(dati$subj[1],nrow(dati) + k )
)

dati_pred[1:nrow(dati),] <- dati
yhat2 <- predict(fit_face,dati_pred)

data3 <- dati
Ylim <- range(c(data3$y,yhat2$y.pred))

plot(data3$argvals,data3$y,xlab=Xlab,ylab=Ylab, main = paste("Male ",i,sep=""),
ylim = c(4,8.5),
cex.lab=1.25,cex.axis = 1.25,cex.main = 1.25,pch=1,xlim=c(-20,40))

Ord <- nrow(dati) + 1:k
lines(dati_pred$argvals[Ord],yhat2$y.pred[Ord],col="red",lwd=2)
lines(dati_pred$argvals[Ord],
yhat2$y.pred[Ord] - 1.96*yhat2$se.pred[Ord], col="red",lwd=1,lty=2)
lines(dati_pred$argvals[Ord],
yhat2$y.pred[Ord] + 1.96*yhat2$se.pred[Ord], col="red",lwd=1,lty=2)

lines(tnew,fit_face$mu.new,lty=3,col="black",lwd=2)
legend("bottomleft",c("mean","prediction"),lty=c(3,1),col=1:2,lwd=2,bty="n")
}

## End(Not run)

Subject-specific curve prediction from a face.sparse fit

Description

Predict subject-specific curves based on a fit from "face.sparse".

Usage

## S3 method for class 'face.sparse'
predict(object, newdata,...)

Arguments

Details

This function makes prediction based on observed data for each subject. So for each subject, it requires at least one observed data. For the time points prediction is desired but no observation is available, just make the corresponding data$y as NA.

Value

Author(s)

Luo Xiao <lxiao5@ncsu.edu>

References

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

Examples

#See the examples for "face.sparse".

Mean prediction from a P-spline smoothing fit

Description

Predict mean values based on a fit from "pspline".

Usage

## S3 method for class 'pspline.face'
predict(object, argvals.new,...)

Arguments

Value

Predicted means at argvals.new.

Author(s)

Luo Xiao <lxiao5@ncsu.edu>

References

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

Examples

#See the examples for "pspline".

Univariate P-spline smoothing

Description

Univariate P-spline smoothing with the smoothing parameter selected by leave-one-subject-out cross validation.

Usage

pspline(data, argvals.new = NULL, knots = 35,
            p = 3, m = 2, lambda = NULL,
            search.length = 100,
            lower = -20, upper = 20)

Arguments

Details

The function is an implementation of the P-spline smoothing in Eilers and Marx (1996). P-splines uses B-splines as basis functions and employs a differencing penalty on the coefficients. Leave-one-subject-out cross validation is used for selecting the smoothing parameter and a fast algorithm is implemented.

Value

Author(s)

Luo Xiao <lxiao5@ncsu.edu>

References

Paul Eilers and Brian Marx, Flexible smoothing with B-splines and penalties, Statist. Sci., 11, 89-121, 1996.

Luo Xiao, Cai Li, William Checkley and Ciprian Crainiceanu, Fast covariance estimation for sparse functional data, Stat. Comput., doi: 10.1007/s11222-017-9744-8.

See Also

gam in mgcv.

Examples

## Not run: 
## cd4 data
require(refund)
data(cd4)
n <- nrow(cd4)
T <- ncol(cd4)

id <- rep(1:n,each=T)
t <- rep(-18:42,times=n)
y <- as.vector(t(cd4))
sel <- which(is.na(y))

## organize data
data <- data.frame(y=log(y[-sel]),
argvals = t[-sel],
subj = id[-sel])
data <- data[data$y>4.5,]

## smooth
fit <- pspline(data)

## plot
plot(data$argvals,fit$mu.new,type="p")

## prediction
pred <- predict(fit,quantile(data$argvals,c(0.2,0.6)))
pred

## End(Not run)

Knots selection for P-spline smoothing

Description

Construct knots from either quantiles of observed time points or equally-spaced time points.

Usage

select.knots(t,knots=10,p=3,option="equally-spaced")

Arguments

Details

The number of knots in the output will be knot plus 2 times p; and the B-spline basis matrix constructed from this vector of knots with degrees p will be knots plus p.

Value

A vector of knots

Author(s)

Luo Xiao <lxiao5@ncsu.edu>

Examples

t <- rnorm(100)
knots <- select.knots(t)