Simultaneous Confidence Bands for Mean Functions

Description

Statistical methods for estimating and inferring the mean of functional data. The methods include simultaneous confidence bands, local polynomial fitting, bandwidth selection by plug-in and cross-validation, goodness-of-fit tests for parametric models, equality tests for two-sample problems, and plotting functions.

Details

Author(s)

David Degras <ddegrasv@gmail.com>

References

Azzalini, A. and Bowman, A. (1993). On the use of nonparametric regression for checking linear relationships. Journal of the Royal Statistical Society. Series B 55, 549-557.

Benhenni, K. and Degras, D. (2014). Local polynomial estimation of the mean function and its derivatives based on functional data and regular designs. ESAIM: Probability and Statistics 18, 881-899.

Degras, D. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica 21, 1735-1765.

Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society. Series B 53, 233-243.

Leave-One-Curve-out Cross-Validation Score

Description

Compute the cross-validation score of Rice and Silverman (1991) for the local polynomial estimation of a mean function.

Usage

cv.score(bandwidth, x, y, degree = 1, gridsize = length(x))

Arguments

Details

The cross-validation score is obtained by leaving in turn each curve out and computing the prediction error of the local polynomial smoother based on all other curves. For a bandwith value h, this score is

CV(h) = \frac{1}{np} \sum_{i=1}^n \sum_{j=1}^p \left( Y_{ij} - \hat{\mu}^{-(i)}(x_j;h) \right)^2,

where Y_{ij} is the measurement of the i-th curve at location x_j for i=1,\ldots,n and j=1,\ldots,p, and \hat{\mu}^{-(i)}(x_j;h) is the local polynomial estimator with bandwidth h based on all curves except the i-th.

If the x values are not equally spaced, the data are first smoothed and evaluated on a grid of length gridsize spanning the range of x. The smoothed data are then interpolated back to x.

Value

the cross-validation score.

References

Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society. Series B (Methodological), 53, 233–243.

See Also

cv.select

Examples

## Artificial example 
x <- seq(0, 1, len = 100)
mu <- x + .2 * sin(2 * pi * x)
y <- matrix(mu + rnorm(2000, sd = .25), 20, 100, byrow = TRUE)
h <- c(.005, .01, .02, .05, .1, .15)
cv <- numeric()
for (i in 1:length(h)) cv[i] <- cv.score(h[i], x, y, 1)
plot(h, cv, type = "l")

## Plasma citrate data
## Compare cross-validation scores and bandwidths  
## for local linear and local quadratic smoothing
## Not run: 
data(plasma)
time <- 8:21
h1 <- seq(.5, 1.3, .05)
h2 <- seq(.75, 2, .05)
cv1 <- sapply(h1, cv.score, x = time, y = plasma, degree = 1)
cv2 <- sapply(h2, cv.score, x = time, y = plasma, degree = 2)
plot(h1, cv1, type = "l", xlim = range(c(h1,h2)), ylim = range(c(cv1, cv2)), 
  xlab = "Bandwidth (hour)", ylab = "CV score", 
  main = "Cross validation for local polynomial estimation")
lines(h2, cv2, col = 2)
legend("topleft", legend = c("Linear", "Quadratic"), lty = 1, 
  col = 1:2, cex = .9)

## Note: using local linear (resp. quadratic) smoothing 
## with a bandwidth smaller than .5 (resp. .75) can result 
## in non-definiteness or numerical instability of the estimator. 

## End(Not run)

Cross-Validation Bandwidth Selection for Local Polynomial Estimation

Description

Select the cross-validation bandwidth described in Rice and Silverman (1991) for the local polynomial estimation of a mean function based on functional data.

Usage

 cv.select(x, y, degree = 1, interval = NULL, gridsize = length(x), ...) 

Arguments

Details

The cross-validation score is obtained by leaving in turn each curve out and computing the prediction error of the local polynomial smoother based on all other curves. For a bandwith value h, this score is

CV(h) = \frac{1}{np} \sum_{i=1}^n \sum_{j=1}^p \left( Y_{ij} - \hat{\mu}^{-(i)}(x_j;h) \right)^2,

where Y_{ij} is the measurement of the i-th curve at location x_j for i=1,\ldots,n and j=1,\ldots,p, and \hat{\mu}^{-(i)}(x_j;h) is the local polynomial estimator with bandwidth h based on all curves except the i-th.

If the x values are not equally spaced, the data are first smoothed and evaluated on a grid of length gridsize spanning the range of x. The smoothed data are then interpolated back to x.

cv.select uses the standard R function optimize to optimize cv.score. If the argument interval is not specified, the lower bound of the search interval is by default (x_2-x_1)/2 if degree < 2 and x_2-x_1 if degree >= 2. The default value of the upper bound is (\max(x)-\min(x))/2. These values guarantee in most cases that the local polynomial estimator is well defined. It is often useful to plot the function to be optimized for a range of argument values (grid search) before applying a numerical optimizer. In this way, the search interval can be narrowed down and the optimizer is more likely to find a global solution.

Value

a bandwidth that minimizes the cross-validation score.

References

Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society. Series B (Methodological), 53, 233–243.

See Also

cv.score, plugin.select

Examples

## Not run: 
## Plasma citrate data
## Compare cross-validation scores and bandwidths  
## for local linear and local quadratic smoothing

data(plasma)
time <- 8:21   				
## Local linear smoothing						
cv.select(time, plasma, 1)	# local solution h = 3.76, S(h) = 463.08			
cv.select(time, plasma, 1, interval = c(.5, 1))	# global solution = .75, S(h) = 439.54

## Local quadratic smoothing						
cv.select(time, plasma, 2)	# global solution h = 1.15, S(h) = 432.75			
cv.select(time, plasma, 2, interval = c(1, 1.5))	# same

## End(Not run)

Phoneme data

Description

Log-periodograms of five phonemes ("sh" as in "she", "dcl" as in "dark", "iy" as the vowel in "she", "aa" as the vowel in "dark", and "ao" as the first vowel in "water") spoken by a sample of ~50 male speakers. For each phoneme, 400 log-periodograms are observed on a uniform grid of 150 frequencies.

Usage

data(phoneme)

Format

A data frame with 2000 rows and 151 columns. Each row contains a discretized log-periodogram (150 first columns) followed by a phoneme indicator (last column) coded as follows:

Source

This dataset was created by the STAPH group in Toulouse, France.
The larger original dataset can be found at
https://hastie.su.domains/ElemStatLearn/.

Examples

## Not run: 
data(phoneme)
freq <- 1:150
classes <- phoneme[,151]
phoneme <- phoneme[,-151]
classnames <- c("sh", "iy", "dcl", "aa", "ao")

## Local linear fit to the mean log-periodogram for each phoneme 
llfit <- mapply(locpoly, y = by(phoneme, classes, colMeans), 
          MoreArgs = list(x = freq, bandwidth = 2, degree = 1, 
          gridsize = length(freq)))
llfit.y <- matrix(unlist(llfit["y",]), 150, 5)  
matplot(freq, llfit.y, type = "l", lty = 1, xlab = "Frequency (scaled)", 
	ylab = "Log-intensity", 
	main = "Local linear estimation\nof the population mean log-periodogram")
legend("topright", legend = classnames, col = 1:5, lty = 1)

## End(Not run)

Plasma citrate data

Description

Plasma citrate concentrations measured on 10 human subjects on the same day. The measurements on an individual were taken each hour from 8:00AM to 9:00PM. A possible statistical analysis is to estimate the population mean plasma citrate concentration as a function of time of day.

Usage

data(plasma)

Format

A matrix with 10 rows (corresponding to subjects) and 14 columns (corresponding to hours).

Source

Andersen, A. H., Jensen, E. B. and Schou, G. (1981). Two-way analysis of variance with correlated errors. International Statistical Review 49, 153–157.

Hart, T. D. and Wehrly, T. E. (1986). Kernel regression estimation using repeated measurements data. Journal of the American Statistical Association 81, 1080–1088.

Examples

## Not run: 
data(plasma)
matplot(x = 8:21, y = t(plasma), cex = .75, type = "l", col = 1, lty = 1,
  lwd = .5, xlab = "Time of day (hour)", ylab = "Concentration", 
  main = "Plasma citrate data")
lines(8:21, colMeans(plasma), col = 2, lwd = 1.5)
legend("topright", col = 2, lty = 1, lwd = 1.5, legend = "Mean")
## End(Not run)

Plot a SCBand Object

Description

plot method for class "SCBand".

Usage

## S3 method for class 'SCBand'
plot(x , y = NULL, xlim = NULL, ylim = NULL, main = NULL, xlab = NULL, 
     ylab = NULL, col = NULL, cex = NULL, pch = NULL, lty = NULL, lwd = NULL, 
     legend = TRUE, where = NULL, text = NULL, legend.cex = NULL, horiz = TRUE, 
     bty = "n", ...)

Arguments

Details

The argument y can be used to plot subsets of the y data. If non null, this argument has priority over the component x$y for plotting.

The graphical parameters col, cex, pch, lty, and lwd apply to the following components to be plotted: data, parametric estimate, nonparametric estimate(s), normal simultaneous confidence bands (SCB), and bootstrap SCB. More precisely, cex and pch must be specified as vectors of length equal to the number of y data sets to be plotted (0, 1, or 2); lty and lwd must specified as numeric vectors of length equal to the total number of estimates and SCB components; col applies to all components and should be specified accordingly. If necessary, graphical parameters are recycled to match the required length.

By default a legend is plotted horizontally at the bottom of the graph.

Examples

## Not run: 
## Plasma citrate data 
time <- 8:21
data(plasma)
h <- cv.select(time, plasma, degree = 1, interval = c(.5, 1))
scbplasma <- scb.mean(time, plasma, bandwidth = h, scbtype = "both",
                      gridsize = 100)
plot(scbplasma, cex = .2, legend.cex = .85, xlab = "Time of day", 
     ylab = "Concentration", main = "Plasma citrate data")

## End(Not run)

Pseudo-Likelihood Ratio Test for Models of the Mean Function

Description

Implement the Pseudo-Likelihood Ratio Test (PLRT) of Azzalini and Bowman (1993) with functional data. The test is used to assess whether a mean function belongs to a given finite-dimensional function space.

Usage

 plrt.model(x, y, model, verbose = FALSE) 

Arguments

Value

the p value of the PLRT.

References

Azzalini, A. and Bowman, A. (1993). On the use of nonparametric regression for checking linear relationships. Journal of the Royal Statistical Society. Series B 55, 549-557.

See Also

scb.model

Examples

## Example: Gaussian process with mean = linear function + bump 
## and Onstein-Uhlenbeck covariance. The bump is high in the y   
## direction and narrow in the x direction. The SCB and PLRT 
## tests are compared.

# The departure from linearity in the mean function is strong  
# in the supremum norm (SCB test) but mild in the euclidean norm 
# (PLRT). With either n = 20 or n = 100 curves, the SCB test 
# strongly rejects the incorrect linear model for the mean 
# function while the PLRT retains it. 

p    <- 100    # number of observation points
x 	  <- seq(0, 1, len = p)
mu <- -1 + 1.5 * x + 0.2 * dnorm(x, .6, .02) 
plot(x, mu, type = "l")
R 	  <- (.25)^2 * exp(20 * log(.9) * abs(outer(x,x,"-"))) # covariance 
eigR <- eigen(R, symmetric = TRUE)  	
simR <- eigR$vectors %*% diag(sqrt(eigR$values)) 	 

n  <- 20
set.seed(100)
y  <- mu + simR %*% matrix(rnorm(n*p), p, n) 		
y  <- t(y) 	
points(x, colMeans(y))									
h  <- cv.select(x, y, 1)	
scb.model(x, y, 1, bandwidth = h)		 # p value: <1e-16
plrt.model(x, y, 1, verbose = TRUE)	# p value: .442
n  <- 100
y  <- mu + simR %*% matrix(rnorm(n*p), p, n) 		
y  <- t(y) 	
h  <- cv.select(x, y, 1)	
scb.model(x, y, 1, bandwidth = h)		 # p value: <1e-16
plrt.model(x, y, 1, verbose = TRUE)	# p value: .456

Plug-in Bandwidth Selection for Local Polynomial Estimation

Description

Select the plug-in bandwidth described in Benhenni and Degras (2011) for the local polynomial estimation of a mean function and its first derivative based on functional data.

Usage

plugin.select(x, y, drv = 0L, degree = drv+1L, gridsize = length(x), ...) 

Arguments

Details

The plug-in method should not be used with small data sets, since it is based on asymptotic considerations and requires reasonably accurate estimates of derivatives of the mean and covariance functions. Both the number of observed curves and observation points should be moderate to large. The plug-in bandwidth is designed to minimize the asymptotic mean integrated squared estimation error

AMISE(h) = \int (\mu(t) - \hat{\mu}(t; h))^2 dt,

where \mu(x) is the mean function and \hat{\mu}(t;h) is a local polynomial estimator with kernel bandwidth h. The expression of the plug-in bandwidth can be found in Benhenni and Degras (2011).

Value

the plug-in bandwidth.

References

Benhenni, K. and Degras, D. (2011). Local polynomial estimation of the average growth curve with functional data. http://arxiv.org/abs/1107.4058

See Also

cv.select

Examples

## Not run: 
## Phoneme data
data(phoneme)
classes <- phoneme[,151]
phoneme <- phoneme[,-151]
freq    <- 1:150
plugin.bandwidth  <- numeric(5) 
cv.bandwidth  <- numeric(5)  # compare with cross-validation
for (i in 1:5) {
  plugin.bandwidth[i] <- plugin.select(x = freq, y = phoneme[classes == i, ],
                          drv = 0, degree = 1) 
  cv.bandwidth[i]     <- cv.select(x = freq, y = phoneme[classes == i, ], 
                          degree = 1)
}

round(cbind(plugin.bandwidth, cv.bandwidth), 4)


## End(Not run)

Print a SCBand Object

Description

print method for class "SCBand".

Usage

## S3 method for class 'SCBand'
print(x,...)

Arguments

Details

The function print.SCBand concisely displays the information of an object of class "SCBand". More precisely it shows the data range, bandwidth used in local polynomial estimation, and key information on SCB and statistical tests.

See Also

plot.SCBand, summary.SCBand

Examples

## Not run: 
# Plasma citrate data
data(plasma)
time <- 8:21
h <- cv.select(time, plasma, 1)
scbplasma <- scb.mean(time, plasma, bandwidth = h, scbtype = "both", gridsize = 100)
scbplasma

## End(Not run)

Compare Two Mean Functions

Description

This two-sample test builds simultaneous confidence bands (SCB) for the difference between two population mean functions and retains the equality assumption if the null function is contained in the bands. Equivalently, SCB are built around one of the local linear estimates (the one for say, population 1), and the equality hypothesis is accepted if the other estimate (the one for population 2) lies within the bands.

Usage

scb.equal(x, y, bandwidth, level = .05, degree = 1, 
	scbtype = c("normal","bootstrap","both","no"), gridsize = NULL, 
	keep.y = TRUE, nrep = 2e4, nboot = 1e4, parallel = c("no","multicore","snow"), 
	ncpus = getOption("boot.ncpus",1L), cl = NULL)

Arguments

Value

A list object of class "SCBand". Depending on the function used to create the object (scb.mean, scb.model, or scb.equal), some of its components are set to NULL. For scb.mean, the object has components:

Depending on the value of scbtype, some or all of the fields pnorm, qnorm, normscb, nrep, pboot, qboot, normboot and nboot may be NULL.

References

Degras, D. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica, 21, 1735–1765.

See Also

scb.mean, scb.model

Examples

## Not run: 
# Phoneme data: compare the mean log-periodograms 
# for phonemes "aa" as the vowel in "dark" and "ao" 
# as the first vowel in "water"
data(phoneme)
n <- nrow(phoneme)
N <- ncol(phoneme) 
classes <- split(1:n,phoneme[,N])
names(classes) <- c("sh", "iy", "dcl", "aa", "ao")
freq    <- 1:150
compare.aa.ao <- scb.equal(freq, list(phoneme[classes$aa,-N], 
  phoneme[classes$ao,-N]), bandwidth = c(.75, .75), scbtype = "both", nboot = 2e3)
summary(compare.aa.ao)

## End(Not run)

Build Simultaneous Confidence Bands for Mean Functions

Description

Fit a local linear estimator and build simultaneous confidence bands (SCB) for the mean of functional data.

Usage

scb.mean(x, y, bandwidth, level = .95, degree = 1, 
	scbtype = c("normal","bootstrap","both","no"), gridsize = length(x), 
	keep.y = TRUE, nrep = 2e4, nboot = 5e3, parallel = c("no", "multicore", "snow"), 
	ncpus = getOption("boot.ncpus",1L), cl = NULL)

Arguments

Details

The local polynomial fitting uses a standard normal kernel and is implemented via the locpoly function. Bootstrap SCB are implemented with the boot function and typically require more computation time than normal SCB.

Value

An object of class "SCBand". To accommodate the different functions creating objects of this class (scb.mean, scb.model, and scb.equal), some components of the object are set to NULL. The component list is:

Depending on the value of scbtype, some of the fields qnorm, normscb, nrep, qboot, normboot and nboot may be NULL.

References

Degras, D. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica, 21, 1735–1765.

See Also

scb.equal, scb.model

Examples

## Not run: 
## Plasma citrate data
data(plasma)
time <- 8:21
h <- cv.select(time, plasma, 1, c(.5, 1))
scbplasma <- scb.mean(time, plasma, bandwidth = h, scbtype = "both", gridsize = 100)
scbplasma
plot(scbplasma, cex = .2, legend.cex = .85, xlab = "Time", ylab = "Concentration", 
  main = "Plasma citrate data")

## End(Not run)

Goodness-Of-Fit of a Model for the Mean Function

Description

This is the goodness-of-fit test for parametric models of the mean function described in Degras (2011). The candidate model must be a finite-dimensional function space (curvilinear regression). The test is based on the sup-norm distance between a smoothed parametric estimate and a local linear estimate. Graphically, the candidate model is retained whenever one of the estimates lies within the SCB built around the other.

Usage

scb.model(x, y, model, bandwidth, level = .05, degree = 1, 
	scbtype = c("normal","bootstrap","both","no"), gridsize = length(x), 
	keep.y = TRUE, nrep = 2e4, nboot = 5e3, parallel = c("no", "multicore", "snow"), 
	ncpus = getOption("boot.ncpus",1L), cl = NULL)

Arguments

Value

An object of class "SCBand". To accommodate the different functions creating objects of this class (scb.mean, scb.model, and scb.equal), some components of the object are set to NULL. The component list is:

Depending on the value of scbtype, some or all of the fields pnorm, qnorm, normscb, nrep, pboot, qboot, normboot and nboot may be NULL.

References

Degras, D. (2011). Simultaneous confidence bands for nonparametric regression with functional data.
Statistica Sinica, 21, 1735–1765.

See Also

scb.equal, scb.mean

Examples

## Example from Degras (2011)
## Gaussian process with polynomial mean function 
## and Ornstein-Uhlenbeck covariance function
## The SCB and PLRT tests are compared

set.seed(100)
p    <- 100  	# number of observation points
x 	  <- seq(0, 1, len = p)
mu	  <- 10 * x^3 - 15 * x^4 + 6 * x^5	# mean 
R 	  <- (.25)^2 * exp(20 * log(.9) * abs(outer(x,x,"-"))) # covariance 
eigR <- eigen(R, symmetric = TRUE)  	
simR <- eigR$vectors %*% diag(sqrt(eigR$values)) 	 

# Candidate model for mu: polynomial of degree <= 3
# This model, although incorrect, closely approximates mu.
# With n = 50 curves, the SCB and PLRT incorrectly retain the model.
# With n = 70 curves, both tests reject it. 
n <- 50  
y <- mu + simR %*% matrix(rnorm(n*p), p, n) 	# simulate data  
y <- t(y) 	# arrange the trajectories in rows
h <- cv.select(x, y, 1)	
scb.model(x, y, 3, bandwidth = h)		  # p value: .652
plrt.model(x, y, 3, verbose = TRUE)	# p value: .450 
n <- 70  	
y <- mu + simR %*% matrix(rnorm(n*p), p, n) 		
y <- t(y) 										
h <- cv.select(x, y, 1)	
scb.model(x, y, 3, bandwidth = h)		  # p value: .004
plrt.model(x, y, 3, verbose = TRUE)	# p value: .001

# Correct model: polynomials of degree <= 5
scb.model(x, y, 5, bandwidth = h)  	# p value: .696
plrt.model(x, y, 5, verbose = TRUE)	# p value: .628

Summarize a SCBand Object

Description

summary method for class "SCBand"

Usage

## S3 method for class 'SCBand'
summary(object, ...)

Arguments

Details

The function summary.SCBand displays all fields of a SCBand object at the exception of x, y, par, nonpar, normscb, and bootscb which are potentially big. It provides information on the function call, data, local polynomial fit, SCB, and statistical tests.

See Also

plot.SCBand, print.SCBand

Examples

## Not run: 
## Plasma citrate data 
data(plasma)
time <- 8:21
h <- cv.select(time, plasma, 1)
scbplasma <- scb.mean(time, plasma, bandwidth = h, scbtype = "both", gridsize = 100)
summary(scbplasma)

## End(Not run)