smooth.basis: Construct a functional data object by smoothing data using a...
In fda: Functional Data Analysis
smooth.basis R Documentation
Construct a functional data object by smoothing data using a roughness
penalty
Description
Discrete observations on one or more curves and for one more more
variables are fit with a set of smooth curves, each defined by an
expansion in terms of user-selected basis functions. The fitting
criterion is weighted least squares, and smoothness can be defined in
terms of a roughness penalty that is specified in a variety of ways.
Data smoothing requires at a bare minimum three elements: (1) a set of
observed noisy values, (b) a set of argument values associated with these
data, and (c) a specification of the basis function system used to
define the curves. Typical basis functions systems are splines for
nonperiodic curves, and fourier series for periodic curves.
Optionally, a set covariates may be also used to take account of
various non-smooth contributions to the data. Smoothing without
covariates is often called nonparametric regression, and with
covariates is termed semiparametric regression.
Usage
smooth.basis(argvals=1:n, y, fdParobj, wtvec=NULL, fdnames=NULL,
covariates=NULL, method="chol", dfscale=1, returnMatrix=FALSE)
Arguments
argvals
a set of argument values corresponding to the observations in array
y. In most applications these values will be common to all curves
and all variables, and therefore be defined as a vector or as a matrix
with a single column. But it is possible that these argument values
will vary from one curve to another, and in this case argvals will
be input as a matrix with rows corresponding to observation points and
columns corresponding to curves. Argument values can even vary from one
variable to another, in which case they are input as an array with
dimensions corresponding to observation points, curves and variables,
respectively. Note, however, that the number of observation points per
curve and per variable may NOT vary. If it does, then curves and variables
must be smoothed individually rather than by a single call to this function.
The default value for argvals are the integers 1 to n, where
n is the size of the first dimension in argument y.
y
an set of values of curves at discrete sampling points or
argument values. If the set is supplied as a matrix object, the rows must
correspond to argument values and columns to replications, and it will be
assumed that there is only one variable per observation. If
y is a three-dimensional array, the first dimension
corresponds to argument values, the second to replications, and the
third to variables within replications. If y is a vector,
only one replicate and variable are assumed. If the data
come from a single replication but multiple vectors, such as data
on coordinates for a single space curve, then be sure to coerce
the data into an array object by using the as.array function
with one as the central dimension length.
fdParobj
a functional parameter object, a functional data object or a
functional basis object. In the simplest case, fdParobj may
be a functional basis object with class "basisfd" defined
previously by one of the "create" functions, and in this case, no
roughness penalty is used. No roughness penalty is also the
consequence of supplying a functional data object with class "fd",
in which case the basis system used for smoothing is that defining
this object. In these two simple cases, smooth.basis is
essentially the same as function Data2fd, and this type of
elementary smoothing is often called "regression smoothing."
However, if the object is a functional parameter object with class
"fdPar", then the linear differential operator object and the
smoothing parameter in this object define the roughness penalty. For
further details on how the roughness penalty is defined, see the help
file for "fdPar". In general, better results can be obtained using a
good roughness penalty than can be obtained by merely varying the
number of basis functions in the expansion.
wtvec
typically a vector of length n that is the length of argvals
containing weights for the values to be smoothed, However, it may also
be a symmetric matrix of order n. If wtvec is a vector,
all values must be positive, and if it is a symmetric matrix, this must
be positive definite. Defaults to all weights equal to 1.
fdnames
a list of length 3 containing character vectors of names for the
following:
- args
-
name for each observation or point in time at which data are
collected for each 'rep', unit or subject.
- reps
-
name for each 'rep', unit or subject.
- fun
-
name for each 'fun' or (response) variable measured repeatedly
(per 'args') for each 'rep'.
covariates
The observed values in y are assumed to be primarily
determined by the height of the curve being estimates, but from time
to time certain values can also be influenced by other known
variables. For example, multi-year sets of climate variables may be
also determined by the presence of absence of an El Nino event, or a
volcanic eruption. One or more of these covariates can be supplied
as an n by p matrix, where p is the number of
such covariates. When such covariates are available, the smoothing
is called "semi-parametric." Matrices or arrays of regression
coefficients are then estimated that define the impacts of each of
these covariates for each curve and each variable.
method
by default the function uses the usual textbook equations for computing
the coefficients of the basis function expansions. But, as in regression
analysis, a price is paid in terms of rounding error for such
computations since they involved cross-products of basis function
values. Optionally, if method is set equal to the string "qr",
the computation uses an algorithm based on the qr-decomposition which
is more accurate, but will require substantially more computing time
when n is large, meaning more than 500 or so. The default
is "chol", referring the Choleski decomposition of a symmetric positive
definite matrix.
dfscale
the generalized cross-validation or "gcv" criterion that is often used
to determine the size of the smoothing parameter involves the
subtraction of an measue of degrees of freedom from n. Chong
Gu has argued that multiplying this degrees of freedom measure by
a constant slightly greater than 1, such as 1.2, can produce better
decisions about the level of smoothing to be used. The default value
is, however, 1.0.
returnMatrix
logical: If TRUE, a two-dimensional is returned using a
special class from the Matrix package.
Details
A roughness penalty is a quantitative measure of the roughness of a
curve that is designed to fit the data. For this function, this penalty
consists of the product of two parts. The first is an approximate integral
over the argument range of the square of a derivative of the curve. A
typical choice of derivative order is 2, whose square is often called
the local curvature of the function. Since a rough function has high
curvature over most of the function's range, the integrated square of
of the second derivative quantifies the total curvature of the function,
and hence its roughness. The second factor is a positive constant called
the bandwidth of smoothing parameter, and given the variable name
lambda here.
In more sophisticated uses of smooth.basis, a derivative may be
replaced by a linear combination of two or more order of derivative,
with the coefficients of this combination themselves possibly varying
over the argument range. Such a structure is called a "linear
differential operator", and a clever choice of operator can result
in much improved smoothing.
The rougnhness penalty is added to the weighted error sum of squares
and the composite is minimized, usually in conjunction with a
high dimensional basis expansion such as a spline function defined by
placing a knot at every observation point. Consequently, the
smoothing parameter controls the relative emphasis placed on fitting
the data versus smoothness; when large, the fitted curve is more smooth,
but the data fit worse, and when small, the fitted curve is more rough,
but the data fit much better. Typically smoothing parameter lambda
is manipulated on a logarithmic scale by, for example, defining it
as a power of 10.
A good compromise lambda value can be difficult to
define, and minimizing the generalized cross-validation or "gcv"
criterion that is output by smooth.basis is a popular strategy
for making this choice, although by no means foolproof. One may also
explore lambda values for a few log units up and down from this
minimizing value to see what the smoothing function and its derivatives
look like. The function plotfit.fd was designed for this purpose.
The size of common logarithm of the minimizing value of lambda
can vary widely, and spline functions depends critically on the typical
spacing between knots. While there is typically a "natural" measurement
scale for the argument, such as time in milliseconds, seconds, and so
forth, it is better from a computational perspective to choose an
argument scaling that gives knot spacings not too different from one.
An alternative to using smooth.basis is to first represent
the data in a basis system with reasonably high resolution using
Data2fd, and then smooth the resulting functional data object
using function smooth.fd.
In warning and error messages, you may see reference to functions
smooth.basis1, smooth.basis2, and smooth.basis3. These
functions are defined within smooth.basis, and are not
normally to be called by users.
The "qr" algorithm option defined by the "method" parameter will
not normally be needed, but if a warning of a near singularity in
the coefficient calculations appears, this choice may be a cure.
Value
an object of class fdSmooth, which is a named list of length 8
with the following components:
fd
a functional data object containing a smooth of the data.
df
a degrees of freedom measure of the smooth
gcv
the value of the generalized cross-validation or GCV criterion.
Ifthere are multiple curves, this is a vector of values, one per
curve. If the smooth is multivariate, the result is a matrix of gcv
values, with columns corresponding to variables.
gcv <- n*SSE/((n-df)^2)
beta
the regression coefficients associated with covariate variables.
These are vector, matrix or array objects depending on whether
there is a single curve, multiple curves or multiple curves and
variables, respectively.
SSE
the error sums of squares. SSE is a vector or a matrix of the
same size as GCV.
penmat
the penalty matrix.
y2cMap
the matrix mapping the data to the coefficients.
argvals, y
input arguments
References
Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009),
Functional data analysis with R and Matlab, Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2005),
Functional Data Analysis, 2nd ed., Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2002),
Applied Functional Data Analysis, Springer, New York.
See Also
lambda2df,
lambda2gcv,
df2lambda,
plot.fd,
project.basis,
smooth.fd,
smooth.monotone,
smooth.pos,
smooth.basisPar
Data2fd,
Examples
oldpar <- par(no.readonly=TRUE)
##
######## Simulated data example 1: a simple regression smooth ########
##
# Warning: In this and all simulated data examples, your results
# probably won't be the same as we saw when we ran the example because
# random numbers depend on the seed value in effect at the time of the
# analysis.
#
# Set up 51 observation points equally spaced between 0 and 1
n <- 51
argvals <- seq(0,1,len=n)
# The true curve values are sine function values with period 1/2
x <- sin(4*pi*argvals)
# Add independent Gaussian errors with std. dev. 0.2 to the true values
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# When we ran this code, we got these values of y (rounded to two
# decimals):
y <- c(0.27, 0.05, 0.58, 0.91, 1.07, 0.98, 0.54, 0.94, 1.13, 0.64,
0.64, 0.60, 0.24, 0.15, -0.20, -0.63, -0.40, -1.22, -1.11, -0.76,
-1.11, -0.69, -0.54, -0.50, -0.35, -0.15, 0.27, 0.35, 0.65, 0.75,
0.75, 0.91, 1.04, 1.04, 1.04, 0.46, 0.30, -0.01, -0.19, -0.42,
-0.63, -0.78, -1.01, -1.08, -0.91, -0.92, -0.72, -0.84, -0.38, -0.23,
0.02)
# Set up a B-spline basis system of order 4 (piecewise cubic) and with
# knots at 0, 0.1, ..., 0.9 and 1.0, and plot the basis functions
nbasis <- 13
basisobj <- create.bspline.basis(c(0,1),nbasis)
plot(basisobj)
# Smooth the data, outputting only the functional data object for the
# fitted curve. Note that in this simple case we can supply the basis
# object as the "fdParobj" parameter
ys <- smooth.basis(argvals=argvals, y=y, basisobj)
Ys <- smooth.basis(argvals=argvals, y=y, basisobj,
returnMatrix=TRUE)
# Ys[[7]] <- Ys$y2cMap is sparse; everything else is the same
all.equal(ys[-7], Ys[-7])
xfd <- ys$fd
Xfd <- Ys$fd
# Plot the curve along with the data
plotfit.fd(y, argvals, xfd)
# Compute the root-mean-squared-error (RMSE) of the fit relative to the
# truth
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
print(RMSE) # We obtained 0.069
# RMSE <- 0.069 seems good relative to the standard error of 0.2.
# Range through numbers of basis functions from 4 to 12 to see if we
# can do better. We want the best RMSE, but we also want the smallest
# number of basis functions, which in this case is the degrees of
# freedom for error (df). Small df implies a stable estimate.
# Note: 4 basis functions is as small as we can use without changing the
# order of the spline. Also display the gcv statistic to see what it
# likes.
for (nbasis in 4:12) {
basisobj <- create.bspline.basis(c(0,1),nbasis)
ys <- smooth.basis(argvals, y, basisobj)
xfd <- ys$fd
gcv <- ys$gcv
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
# progress report:
# cat(paste(nbasis,round(RMSE,3),round(gcv,3),"\n"))
}
# We got RMSE <- 0.062 for 10 basis functions as optimal, but gcv liked
# almost the same thing, namely 9 basis functions. Both RMSE and gcv
# agreed emphatically that 7 or fewer basis functions was not enough.
# Unlike RMSE, however, gcv does not depend on knowing the truth.
# Plot the result for 10 basis functions along with "*" at the true
# values
nbasis <- 10
basisobj <- create.bspline.basis(c(0,1),10)
xfd <- smooth.basis(argvals, y, basisobj)$fd
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
# Homework:
# Repeat all this with various values of sigerr and various values of n
##
####### Simulated data example 2: a roughness-penalized smooth ########
##
# A roughness penalty approach is more flexible, allowing continuous
# control over smoothness and degrees of freedom, can be adapted to
# known features in the curve, and will generally provide better RMSE
# for given degrees of freedom.
# It does require a bit more effort, though.
# First, we define a little display function for showing how
# df, gcv and RMSE depend on the log10 smoothing parameter
plotGCVRMSE.fd <- function(lamlow, lamhi, lamdel, x, argvals, y,
fdParobj, wtvec=NULL, fdnames=NULL, covariates=NULL) {
loglamvec <- seq(lamlow, lamhi, lamdel)
loglamout <- matrix(0,length(loglamvec),4)
m <- 0
for (loglambda in loglamvec) {
m <- m + 1
loglamout[m,1] <- loglambda
fdParobj$lambda <- 10^(loglambda)
smoothlist <- smooth.basis(argvals, y, fdParobj, wtvec=wtvec,
fdnames=fdnames, covariates=covariates)
xfd <- smoothlist$fd # the curve smoothing the data
loglamout[m,2] <- smoothlist$df
# degrees of freedom in the smoothing curve
loglamout[m,3] <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
loglamout[m,4] <- mean(smoothlist$gcv) # the mean of the N gcv values
}
cat("log10 lambda, deg. freedom, RMSE, gcv\n")
for (i in 1:m) {
cat(format(round(loglamout[i,],3)))
cat("\n")
}
par(mfrow=c(3,1))
plot(loglamvec, loglamout[,2], type="b")
title("Degrees of freedom")
plot(loglamvec, loglamout[,3], type="b")
title("RMSE")
plot(loglamvec, loglamout[,4], type="b")
title("Mean gcv")
return(loglamout)
}
# Use the data that you used in Example 1, or run the following code:
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
sigerr <- 0.2
err <- matrix(rnorm(x),n,1)*sigerr
y <- x + err
# We now set up a basis system with a knot at every data point.
# The number of basis functions will equal the number of interior knots
# plus the order, which in this case is still 4.
# 49 interior knots + order 4 <- 53 basis functions
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Note that there are more basis functions than observed values. A
# basis like this is called "super-saturated." We have to use a
# positive smoothing parameter for it to work. Set up an object of
# class "fdPar" that penalizes the total squared second derivative,
# using a smoothing parameter that is set here to 10^(-4.5).
lambda <- 10^(-4.5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
cat(round(c(df,RMSE,gcv),3),"\n")
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
# Repeat these analyses for a range of log10(lambda) values by running
# the function plotGCVRMSE that we defined above.
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# When we ran this example, the optimal RMSE was 0.073, and was achieved
# for log10(lambda) <- -4.25 or lambda <- 0.000056. At this level of
# smoothing, the degrees of freedom index was 10.6, a value close to
# the 10 degrees of freedom that we saw for regression smoothing. The
# RMSE value is slightly higher than the regression analysis result, as
# is the degrees of freedom associated with the optimal value.
# Roughness penalty will, as we will see later, do better than
# regression smoothing; but with slightly red faces we say, "That's
# life with random data!" The gcv statistic agreed with RMSE on the
# optimal smoothing level, which is great because it does not need to
# know the true values. Note that gcv is emphatic about when there is
# too much smoothing, but rather vague about when we have
# under-smoothed the data.
# Homework:
# Compute average results taken across 100 sets of random data for each
# level of smoothing parameter lambda, and for each number of basis
# functions for regression smoothing.
##
## Simulated data example 3:
## a roughness-penalized smooth of a sample of curves
##
n <- 51 # number of observations per curve
N <- 100 # number of curves
argvals <- seq(0,1,len=n)
# The true curve values are linear combinations of fourier function
# values.
# Set up the fourier basis
nfourierbasis <- 13
fourierbasis <- create.fourier.basis(c(0,1),nfourierbasis)
fourierbasismat <- eval.basis(argvals,fourierbasis)
# Set up some random coefficients, with declining contributions from
# higher order basis functions
basiswt <- matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N)
xcoef <- matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt
xfd <- fd(xcoef, fourierbasis)
x <- eval.fd(argvals, xfd)
# Add independent Gaussian noise to the data with std. dev. 0.2
sigerr <- 0.2
y <- x + matrix(rnorm(c(x)),n,N)*sigerr
# Set up spline basis system
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Set up roughness penalty with smoothing parameter 10^(-5)
lambda <- 10^(-5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
# Display the results, note that a gcv value is returned for EACH curve,
# and therefore that a mean gcv result is reported
cat(round(c(df,RMSE,mean(gcv)),3),"\n")
# the fits are plotted interactively by plotfit.fd ... click to advance
# plot
plotfit.fd(y, argvals, xfd)
# Repeat these results for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# Our results were:
# log10 lambda, deg. freedom, RMSE, GCV
# -6.000 30.385 0.140 0.071
# -5.750 26.750 0.131 0.066
# -5.500 23.451 0.123 0.062
# -5.250 20.519 0.116 0.059
# -5.000 17.943 0.109 0.056
# -4.750 15.694 0.104 0.054
# -4.500 13.738 0.101 0.053
# -4.250 12.038 0.102 0.054
# -4.000 10.564 0.108 0.055
# -3.750 9.286 0.120 0.059
# -3.500 8.177 0.139 0.065
# -3.250 7.217 0.164 0.075
# -3.000 6.385 0.196 0.088
# RMSE and gcv both indicate an optimal smoothing level of
# log10(lambda) <- -4.5 corresponding to 13.7 df. When we repeated the
# analysis using regression smoothing with 14 basis functions, we
# obtained RMSE <- 0.110, about 10 percent larger than the value of
# 0.101 in the roughness penalty result. Smooth the data, outputting a
# list containing various quantities
# Homework:
# Sine functions have a curvature that doesn't vary a great deal over
# the range the curve. Devise some test functions with sharp local
# curvature, such as Gaussian densities with standard deviations that
# are small relative to the range of the observations. Compare
# regression and roughness penalty smoothing in these situations.
if(!CRAN()){
##
####### Simulated data example 4: a roughness-penalized smooth ########
## with correlated error
##
# These three examples make GCV look pretty good as a basis for
# selecting the smoothing parameter lambda. BUT GCV is based an
# assumption of independent errors, and in reality, functional data
# often have autocorrelated errors, with an autocorrelation that is
# usually positive among neighboring observations. Positively
# correlated random values tend to exhibit slowly varying values that
# have long runs on one side or the other of their baseline, and
# therefore can look like trend in the data that needs to be reflected
# in the smooth curve. This code sets up the error correlation matrix
# for first-order autoregressive errors, or AR(1).
rho <- 0.9
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
Rerr <- matrix(0,n,n)
for (i in 1:n) {
for (j in 1:n) Rerr[i,j] <- rho^abs(i-j)
}
# Compute the Choleski factor of the correlation matrix
Lerr <- chol(Rerr)
# set up some data
# Generate auto-correlated errors by multipling independent errors by
# the transpose of the Choleski factor
sigerr <- 0.2
err <- as.vector(crossprod(Lerr,matrix(rnorm(x),n,1))*sigerr)
# See the long-run errors that are genrated
plot(argvals, err)
y <- x + err
# Our values of y were:
y <- c(-0.03, 0.36, 0.59, 0.71, 0.97, 1.2, 1.1, 0.96, 0.79, 0.68,
0.56, 0.25, 0.01,-0.43,-0.69, -1, -1.09,-1.29,-1.16,-1.09,
-0.93, -0.9,-0.78,-0.47, -0.3,-0.01, 0.29, 0.47, 0.77, 0.85,
0.87, 0.97, 0.9, 0.72, 0.48, 0.25,-0.17,-0.39,-0.47,-0.71,
-1.07,-1.28,-1.33,-1.39,-1.45, -1.3,-1.25,-1.04,-0.82,-0.55, -0.2)
# Retaining the above data, now set up a basis system with a knot at
# every data point: the number of basis functions will equal the
# number of interior knots plus the order, which in this case is still
# 4.
# 19 interior knots + order 4 <- 23 basis functions
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
fdParobj <- fdPar(fdobj)
# Smooth these results for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# Our results without weighting were:
# -6.000 30.385 0.261 0.004
# -5.750 26.750 0.260 0.005
# -5.500 23.451 0.259 0.005
# -5.250 20.519 0.258 0.005
# -5.000 17.943 0.256 0.005
# -4.750 15.694 0.255 0.006
# -4.500 13.738 0.252 0.006
# -4.250 12.038 0.249 0.007
# -4.000 10.564 0.246 0.010
# -3.750 9.286 0.244 0.015
# -3.500 8.177 0.248 0.028
# -3.250 7.217 0.267 0.055
# -3.000 6.385 0.310 0.102
# Now GCV still is firm on the fact that log10(lambda) over -4 is
# over-smoothing, but is quite unhelpful about what constitutes
# undersmoothing. In real data applications you will have to make the
# final call. Now set up a weight matrix equal to the inverse of the
# correlation matrix
wtmat <- solve(Rerr)
# Smooth these results for a range of log10(lambda) values using the
# weight matrix
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj,
wtvec=wtmat)
# Our results with weighting were:
# -6.000 46.347 0.263 0.005
# -5.750 43.656 0.262 0.005
# -5.500 40.042 0.261 0.005
# -5.250 35.667 0.259 0.005
# -5.000 30.892 0.256 0.005
# -4.750 26.126 0.251 0.006
# -4.500 21.691 0.245 0.008
# -4.250 17.776 0.237 0.012
# -4.000 14.449 0.229 0.023
# -3.750 11.703 0.231 0.045
# -3.500 9.488 0.257 0.088
# -3.250 7.731 0.316 0.161
# -3.000 6.356 0.397 0.260
# GCV is still not clear about what the right smoothing level is.
# But, comparing the two results, we see an optimal RMSE without
# smoothing of 0.244 at log10(lambda) <- -3.75, and with smoothing 0.229
# at log10(lambda) <- -4. Weighting improved the RMSE. At
# log10(lambda) <- -4 the improvement is 9 percent.
# Smooth the data with and without the weight matrix at log10(lambda) =
# -4
fdParobj <- fdPar(fdobj, 2, 10^(-4))
smoothlistnowt <- smooth.basis(argvals, y, fdParobj)
fdobjnowt <- smoothlistnowt$fd # the curve smoothing the data
df <- smoothlistnowt$df # the degrees of freedom in the smoothing curve
GCV <- smoothlistnowt$gcv # the value of the GCV statistic
RMSE <- sqrt(mean((eval.fd(argvals, fdobjnowt) - x)^2))
cat(round(c(df,RMSE,GCV),3),"\n")
smoothlistwt <- smooth.basis(argvals, y, fdParobj, wtvec=wtmat)
fdobjwt <- smoothlistwt$fd # the curve smoothing the data
df <- smoothlistwt$df # the degrees of freedom in the smoothing curve
GCV <- smoothlistwt$gcv # the value of the GCV statistic
RMSE <- sqrt(mean((eval.fd(argvals, fdobjwt) - x)^2))
cat(round(c(df,RMSE,GCV),3),"\n")
par(mfrow=c(2,1))
plotfit.fd(y, argvals, fdobjnowt)
plotfit.fd(y, argvals, fdobjwt)
par(mfrow=c(1,1))
plot(fdobjnowt)
lines(fdobjwt,lty=2)
points(argvals, y)
# Homework:
# Repeat these analyses with various values of rho, perhaps using
# multiple curves to get more stable indications of relative
# performance. Be sure to include some negative rho's.
##
######## Simulated data example 5: derivative estimation ########
##
# Functional data analyses often involve estimating derivatives. Here
# we repeat the analyses of Example 2, but this time focussing our
# attention on the estimation of the first and second derivative.
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
Dx <- 4*pi*cos(4*pi*argvals)
D2x <- -(4*pi)^2*sin(4*pi*argvals)
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# We now use order 6 splines so that we can control the curvature of
# the second derivative, which therefore involves penalizing the
# derivative of order four.
norder <- 6
nbasis <- n + norder - 2
basisobj <- create.bspline.basis(c(0,1),nbasis,norder)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Note that there are more basis functions than observed values. A
# basis like this is called "super-saturated." We have to use a
# positive smoothing parameter for it to work. Set up an object of
# class "fdPar" that penalizes the total squared fourth derivative. The
# smoothing parameter that is set here to 10^(-10), because the squared
# fourth derivative is a much larger number than the squared second
# derivative.
lambda <- 10^(-10)
fdParobj <- fdPar(fdobj, Lfdobj=4, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
Dxhat <- eval.fd(argvals, xfd, Lfd=1)
D2xhat <- eval.fd(argvals, xfd, Lfd=2)
RMSED <- sqrt(mean((Dxhat - Dx )^2))
RMSED2 <- sqrt(mean((D2xhat - D2x)^2))
cat(round(c(df,RMSED,RMSED2,gcv),3),"\n")
# Four plots of the results row-wise: data fit, first derivative fit,
# second derivative fit, second vs. first derivative fit
# (phase-plane plot)
par(mfrow=c(2,2))
plotfit.fd(y, argvals, xfd)
plot(argvals, Dxhat, type="p", pch="o")
points(argvals, Dx, pch="*")
title("first derivative approximation")
plot(argvals, D2xhat, type="p", pch="o")
points(argvals, D2x, pch="*")
title("second derivative approximation")
plot(Dxhat, D2xhat, type="p", pch="o")
points(Dx, D2x, pch="*")
title("second against first derivative")
# This illustrates an inevitable problem with spline basis functions;
# because they are not periodic, they fail to capture derivative
# information well at the ends of the interval. The true phase-plane
# plot is an ellipse, but the phase-plane plot of the estimated
# derivatives here is only a rough approximtion, and breaks down at the
# left boundary.
# Homework:
# Repeat these results with smaller standard errors.
# Repeat these results, but this time use a fourier basis with no
# roughness penalty, and find the number of basis functions that gives
# the best result. The right answer to this question is, of course, 3,
# if we retain the constant term, even though it is here not needed.
# Compare the smoothing parameter preferred by RMSE for a derivative to
# that preferred by the RMSE for the function itself, and to that
# preferred by gcv.
## Simulated data example 6:
## a better roughness penalty for derivative estimation
##
# We want to see if we can improve the spline fit.
# We know from elementary calculus as well as the code above that
# (4*pi)^2 sine(2*p*x) <- -D2 sine(2*p*x), so that
# Lx <- D2x + (4*pi)^2 x is zero for a sine or a cosine curve.
# We now penalize roughness using this "smart" roughness penalty
# Here we set up a linear differential operator object that defines
# this penalty
constbasis <- create.constant.basis(c(0,1))
xcoef.fd <- fd((4*pi)^2, constbasis)
Dxcoef.fd <- fd(0, constbasis)
bwtlist <- vector("list", 2)
bwtlist[[1]] <- xcoef.fd
bwtlist[[2]] <- Dxcoef.fd
Lfdobj <- Lfd(nderiv=2, bwtlist=bwtlist)
# Now we use a much larger value of lambda to reflect our confidence
# in power of calculus to solve problems!
lambda <- 10^(0)
fdParobj <- fdPar(fdobj, Lfdobj=Lfdobj, lambda=lambda)
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
Dxhat <- eval.fd(argvals, xfd, Lfd=1)
D2xhat <- eval.fd(argvals, xfd, Lfd=2)
RMSED <- sqrt(mean((Dxhat - Dx )^2))
RMSED2 <- sqrt(mean((D2xhat - D2x)^2))
cat(round(c(df,RMSED,RMSED2,gcv),3),"\n")
# Four plots of the results row-wise: data fit, first derivative fit,
# second derivative fit, second versus first derivative fit
# (phase-plane plot)
par(mfrow=c(2,2))
plotfit.fd(y, argvals, xfd)
plot(argvals, Dxhat, type="p", pch="o")
points(argvals, Dx, pch="*")
title("first derivative approximation")
plot(argvals, D2xhat, type="p", pch="o")
points(argvals, D2x, pch="*")
title("second derivative approximation")
plot(Dxhat, D2xhat, type="p", pch="o")
points(Dx, D2x, pch="*")
title("second against first derivative")
# The results are nearly perfect in spite of the fact that we are not using
# periodic basis functions. Notice, too, that we have used 2.03
# degrees of freedom, which is close to what we would use for the ideal
# fourier series basis with the constant term dropped.
# Homework:
# These results depended on us knowing the right period, of course.
# The data would certainly allow us to estimate the period 1/2 closely,
# but try various other periods by replacing 1/2 by other values.
# Alternatively, change x by adding a small amount of, say, linear trend.
# How much trend do you have to add to seriously handicap the results?
##
######## Simulated data example 7: Using covariates ########
##
# Now we simulate data that are defined by a sine curve, but where the
# the first 20 observed values are shifted upwards by 0.5, and the
# second shifted downwards by -0.2. The two covariates are indicator
# or dummy variables, and the estimated regression coefficients will
# indicate the shifts as estimated from the data.
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# the n by p matrix of covariate values, p being here 2
p <- 2
zmat <- matrix(0,n,p)
zmat[ 1:11,1] <- 1
zmat[11:20,2] <- 1
# The true values of the regression coefficients
beta0 <- matrix(c(0.5,-0.2),p,1)
y <- y + zmat
# The same basis system and smoothing process as used in Example 2
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
fdParobj <- fdPar(fdobj, 2, lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj, covariates=zmat)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
beta <- smoothlist$beta # the regression coefficients
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
cat(round(c(beta,df,RMSE,gcv),3),"\n")
par(mfrow=c(1,1))
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
print(beta)
# The recovery of the smooth curve is fine, as in Example 2. The
# shift of the first 10 observations was estimated to be 0.62 in our run,
# and the shift of the second 20 was estimated to be -0.42. These
# estimates are based on only 10 observations, and these estimates are
# therefore quite reasonable.
# Repeat these analyses for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj,
covariates=zmat)
# Homework:
# Try an example where the covariate values are themselves are
# generated by a smooth known curve.
##
## Simulated data example 8:
## a roughness-penalized smooth of a sample of curves and
## variable observation points
##
n <- 51 # number of observations per curve
N <- 100 # number of curves
argvals <- matrix(0,n,N)
for (i in 1:N) argvals[,i] <- sort(runif(1:n))
# The true curve values are linear combinations of fourier function
# values.
# Set up the fourier basis
nfourierbasis <- 13
fourierbasis <- create.fourier.basis(c(0,1),nfourierbasis)
# Set up some random coefficients, with declining contributions from
# higher order basis functions
basiswt <- matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N)
xcoef <- matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt
xfd <- fd(xcoef, fourierbasis)
x <- matrix(0,n,N)
for (i in 1:N) x[,i] <- eval.fd(argvals[,i], xfd[i])
# Add independent Gaussian noise to the data with std. dev. 0.2
sigerr <- 0.2
y <- x + matrix(rnorm(c(x)),n,N)*sigerr
# Set up spline basis system
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
# Set up roughness penalty with smoothing parameter 10^(-5)
lambda <- 10^(-5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
#RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
eval.x <- eval.fd(argvals, xfd)
e.xfd <- (eval.x-x)
mean.e2 <- mean(e.xfd^2)
RMSE <- sqrt(mean.e2)
# Display the results, note that a gcv value is returned for EACH
# curve, and therefore that a mean gcv result is reported
cat(round(c(df,RMSE,mean(gcv)),3),"\n")
# Function plotfit.fd is not equipped to handle a matrix of argvals,
# but can always be called within a loop to plot each curve in turn.
# Although a call to function plotGCVRMSE.fd works, the computational
# overhead is substantial, and we omit this here.
##
## Real data example 9. gait
##
# These data involve two variables in addition to multiple curves
gaitnbasis <- 21
gaittime <- (1:20)/21
gaitrange <- c(0,1)
gaitbasis <- create.fourier.basis(gaitrange, gaitnbasis)
gaitfd <- fd(matrix(0,gaitnbasis,1), gaitbasis)
lambda <- 10^(-11.5)
harmaccelLfd <- vec2Lfd(c(0, 0, (2*pi)^2, 0))
gaitfdPar <- fdPar(gaitfd, harmaccelLfd, lambda)
gaitSmooth <- smooth.basis(gaittime, gait, gaitfdPar)
gaitfd <- gaitSmooth$fd
if (!CRAN()){
# by default creates multiple plots, asking for a click between plots
plotfit.fd(gait, gaittime, gaitfd)
}
}
# end of if (!CRAN)
par(oldpar)
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| smooth.basis | R Documentation |
Construct a functional data object by smoothing data using a roughness penalty
Description
Discrete observations on one or more curves and for one more more variables are fit with a set of smooth curves, each defined by an expansion in terms of user-selected basis functions. The fitting criterion is weighted least squares, and smoothness can be defined in terms of a roughness penalty that is specified in a variety of ways.
Data smoothing requires at a bare minimum three elements: (1) a set of observed noisy values, (b) a set of argument values associated with these data, and (c) a specification of the basis function system used to define the curves. Typical basis functions systems are splines for nonperiodic curves, and fourier series for periodic curves.
Optionally, a set covariates may be also used to take account of various non-smooth contributions to the data. Smoothing without covariates is often called nonparametric regression, and with covariates is termed semiparametric regression.
Usage
smooth.basis(argvals=1:n, y, fdParobj, wtvec=NULL, fdnames=NULL,
covariates=NULL, method="chol", dfscale=1, returnMatrix=FALSE)
Arguments
argvals |
a set of argument values corresponding to the observations in array
|
y |
an set of values of curves at discrete sampling points or
argument values. If the set is supplied as a matrix object, the rows must
correspond to argument values and columns to replications, and it will be
assumed that there is only one variable per observation. If
|
fdParobj |
a functional parameter object, a functional data object or a
functional basis object. In the simplest case, |
wtvec |
typically a vector of length |
fdnames |
a list of length 3 containing character vectors of names for the following:
|
covariates |
The observed values in |
method |
by default the function uses the usual textbook equations for computing
the coefficients of the basis function expansions. But, as in regression
analysis, a price is paid in terms of rounding error for such
computations since they involved cross-products of basis function
values. Optionally, if |
dfscale |
the generalized cross-validation or "gcv" criterion that is often used
to determine the size of the smoothing parameter involves the
subtraction of an measue of degrees of freedom from |
returnMatrix |
logical: If TRUE, a two-dimensional is returned using a special class from the Matrix package. |
Details
A roughness penalty is a quantitative measure of the roughness of a
curve that is designed to fit the data. For this function, this penalty
consists of the product of two parts. The first is an approximate integral
over the argument range of the square of a derivative of the curve. A
typical choice of derivative order is 2, whose square is often called
the local curvature of the function. Since a rough function has high
curvature over most of the function's range, the integrated square of
of the second derivative quantifies the total curvature of the function,
and hence its roughness. The second factor is a positive constant called
the bandwidth of smoothing parameter, and given the variable name
lambda here.
In more sophisticated uses of smooth.basis, a derivative may be
replaced by a linear combination of two or more order of derivative,
with the coefficients of this combination themselves possibly varying
over the argument range. Such a structure is called a "linear
differential operator", and a clever choice of operator can result
in much improved smoothing.
The rougnhness penalty is added to the weighted error sum of squares
and the composite is minimized, usually in conjunction with a
high dimensional basis expansion such as a spline function defined by
placing a knot at every observation point. Consequently, the
smoothing parameter controls the relative emphasis placed on fitting
the data versus smoothness; when large, the fitted curve is more smooth,
but the data fit worse, and when small, the fitted curve is more rough,
but the data fit much better. Typically smoothing parameter lambda
is manipulated on a logarithmic scale by, for example, defining it
as a power of 10.
A good compromise lambda value can be difficult to
define, and minimizing the generalized cross-validation or "gcv"
criterion that is output by smooth.basis is a popular strategy
for making this choice, although by no means foolproof. One may also
explore lambda values for a few log units up and down from this
minimizing value to see what the smoothing function and its derivatives
look like. The function plotfit.fd was designed for this purpose.
The size of common logarithm of the minimizing value of lambda
can vary widely, and spline functions depends critically on the typical
spacing between knots. While there is typically a "natural" measurement
scale for the argument, such as time in milliseconds, seconds, and so
forth, it is better from a computational perspective to choose an
argument scaling that gives knot spacings not too different from one.
An alternative to using smooth.basis is to first represent
the data in a basis system with reasonably high resolution using
Data2fd, and then smooth the resulting functional data object
using function smooth.fd.
In warning and error messages, you may see reference to functions
smooth.basis1, smooth.basis2, and smooth.basis3. These
functions are defined within smooth.basis, and are not
normally to be called by users.
The "qr" algorithm option defined by the "method" parameter will not normally be needed, but if a warning of a near singularity in the coefficient calculations appears, this choice may be a cure.
Value
an object of class fdSmooth, which is a named list of length 8
with the following components:
fd |
a functional data object containing a smooth of the data. |
df |
a degrees of freedom measure of the smooth |
gcv |
the value of the generalized cross-validation or GCV criterion. Ifthere are multiple curves, this is a vector of values, one per curve. If the smooth is multivariate, the result is a matrix of gcv values, with columns corresponding to variables.
|
beta |
the regression coefficients associated with covariate variables. These are vector, matrix or array objects depending on whether there is a single curve, multiple curves or multiple curves and variables, respectively. |
SSE |
the error sums of squares. SSE is a vector or a matrix of the same size as GCV. |
penmat |
the penalty matrix. |
y2cMap |
the matrix mapping the data to the coefficients. |
argvals, y |
input arguments |
References
Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.
See Also
lambda2df,
lambda2gcv,
df2lambda,
plot.fd,
project.basis,
smooth.fd,
smooth.monotone,
smooth.pos,
smooth.basisPar
Data2fd,
Examples
oldpar <- par(no.readonly=TRUE)
##
######## Simulated data example 1: a simple regression smooth ########
##
# Warning: In this and all simulated data examples, your results
# probably won't be the same as we saw when we ran the example because
# random numbers depend on the seed value in effect at the time of the
# analysis.
#
# Set up 51 observation points equally spaced between 0 and 1
n <- 51
argvals <- seq(0,1,len=n)
# The true curve values are sine function values with period 1/2
x <- sin(4*pi*argvals)
# Add independent Gaussian errors with std. dev. 0.2 to the true values
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# When we ran this code, we got these values of y (rounded to two
# decimals):
y <- c(0.27, 0.05, 0.58, 0.91, 1.07, 0.98, 0.54, 0.94, 1.13, 0.64,
0.64, 0.60, 0.24, 0.15, -0.20, -0.63, -0.40, -1.22, -1.11, -0.76,
-1.11, -0.69, -0.54, -0.50, -0.35, -0.15, 0.27, 0.35, 0.65, 0.75,
0.75, 0.91, 1.04, 1.04, 1.04, 0.46, 0.30, -0.01, -0.19, -0.42,
-0.63, -0.78, -1.01, -1.08, -0.91, -0.92, -0.72, -0.84, -0.38, -0.23,
0.02)
# Set up a B-spline basis system of order 4 (piecewise cubic) and with
# knots at 0, 0.1, ..., 0.9 and 1.0, and plot the basis functions
nbasis <- 13
basisobj <- create.bspline.basis(c(0,1),nbasis)
plot(basisobj)
# Smooth the data, outputting only the functional data object for the
# fitted curve. Note that in this simple case we can supply the basis
# object as the "fdParobj" parameter
ys <- smooth.basis(argvals=argvals, y=y, basisobj)
Ys <- smooth.basis(argvals=argvals, y=y, basisobj,
returnMatrix=TRUE)
# Ys[[7]] <- Ys$y2cMap is sparse; everything else is the same
all.equal(ys[-7], Ys[-7])
xfd <- ys$fd
Xfd <- Ys$fd
# Plot the curve along with the data
plotfit.fd(y, argvals, xfd)
# Compute the root-mean-squared-error (RMSE) of the fit relative to the
# truth
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
print(RMSE) # We obtained 0.069
# RMSE <- 0.069 seems good relative to the standard error of 0.2.
# Range through numbers of basis functions from 4 to 12 to see if we
# can do better. We want the best RMSE, but we also want the smallest
# number of basis functions, which in this case is the degrees of
# freedom for error (df). Small df implies a stable estimate.
# Note: 4 basis functions is as small as we can use without changing the
# order of the spline. Also display the gcv statistic to see what it
# likes.
for (nbasis in 4:12) {
basisobj <- create.bspline.basis(c(0,1),nbasis)
ys <- smooth.basis(argvals, y, basisobj)
xfd <- ys$fd
gcv <- ys$gcv
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
# progress report:
# cat(paste(nbasis,round(RMSE,3),round(gcv,3),"\n"))
}
# We got RMSE <- 0.062 for 10 basis functions as optimal, but gcv liked
# almost the same thing, namely 9 basis functions. Both RMSE and gcv
# agreed emphatically that 7 or fewer basis functions was not enough.
# Unlike RMSE, however, gcv does not depend on knowing the truth.
# Plot the result for 10 basis functions along with "*" at the true
# values
nbasis <- 10
basisobj <- create.bspline.basis(c(0,1),10)
xfd <- smooth.basis(argvals, y, basisobj)$fd
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
# Homework:
# Repeat all this with various values of sigerr and various values of n
##
####### Simulated data example 2: a roughness-penalized smooth ########
##
# A roughness penalty approach is more flexible, allowing continuous
# control over smoothness and degrees of freedom, can be adapted to
# known features in the curve, and will generally provide better RMSE
# for given degrees of freedom.
# It does require a bit more effort, though.
# First, we define a little display function for showing how
# df, gcv and RMSE depend on the log10 smoothing parameter
plotGCVRMSE.fd <- function(lamlow, lamhi, lamdel, x, argvals, y,
fdParobj, wtvec=NULL, fdnames=NULL, covariates=NULL) {
loglamvec <- seq(lamlow, lamhi, lamdel)
loglamout <- matrix(0,length(loglamvec),4)
m <- 0
for (loglambda in loglamvec) {
m <- m + 1
loglamout[m,1] <- loglambda
fdParobj$lambda <- 10^(loglambda)
smoothlist <- smooth.basis(argvals, y, fdParobj, wtvec=wtvec,
fdnames=fdnames, covariates=covariates)
xfd <- smoothlist$fd # the curve smoothing the data
loglamout[m,2] <- smoothlist$df
# degrees of freedom in the smoothing curve
loglamout[m,3] <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
loglamout[m,4] <- mean(smoothlist$gcv) # the mean of the N gcv values
}
cat("log10 lambda, deg. freedom, RMSE, gcv\n")
for (i in 1:m) {
cat(format(round(loglamout[i,],3)))
cat("\n")
}
par(mfrow=c(3,1))
plot(loglamvec, loglamout[,2], type="b")
title("Degrees of freedom")
plot(loglamvec, loglamout[,3], type="b")
title("RMSE")
plot(loglamvec, loglamout[,4], type="b")
title("Mean gcv")
return(loglamout)
}
# Use the data that you used in Example 1, or run the following code:
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
sigerr <- 0.2
err <- matrix(rnorm(x),n,1)*sigerr
y <- x + err
# We now set up a basis system with a knot at every data point.
# The number of basis functions will equal the number of interior knots
# plus the order, which in this case is still 4.
# 49 interior knots + order 4 <- 53 basis functions
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Note that there are more basis functions than observed values. A
# basis like this is called "super-saturated." We have to use a
# positive smoothing parameter for it to work. Set up an object of
# class "fdPar" that penalizes the total squared second derivative,
# using a smoothing parameter that is set here to 10^(-4.5).
lambda <- 10^(-4.5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
cat(round(c(df,RMSE,gcv),3),"\n")
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
# Repeat these analyses for a range of log10(lambda) values by running
# the function plotGCVRMSE that we defined above.
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# When we ran this example, the optimal RMSE was 0.073, and was achieved
# for log10(lambda) <- -4.25 or lambda <- 0.000056. At this level of
# smoothing, the degrees of freedom index was 10.6, a value close to
# the 10 degrees of freedom that we saw for regression smoothing. The
# RMSE value is slightly higher than the regression analysis result, as
# is the degrees of freedom associated with the optimal value.
# Roughness penalty will, as we will see later, do better than
# regression smoothing; but with slightly red faces we say, "That's
# life with random data!" The gcv statistic agreed with RMSE on the
# optimal smoothing level, which is great because it does not need to
# know the true values. Note that gcv is emphatic about when there is
# too much smoothing, but rather vague about when we have
# under-smoothed the data.
# Homework:
# Compute average results taken across 100 sets of random data for each
# level of smoothing parameter lambda, and for each number of basis
# functions for regression smoothing.
##
## Simulated data example 3:
## a roughness-penalized smooth of a sample of curves
##
n <- 51 # number of observations per curve
N <- 100 # number of curves
argvals <- seq(0,1,len=n)
# The true curve values are linear combinations of fourier function
# values.
# Set up the fourier basis
nfourierbasis <- 13
fourierbasis <- create.fourier.basis(c(0,1),nfourierbasis)
fourierbasismat <- eval.basis(argvals,fourierbasis)
# Set up some random coefficients, with declining contributions from
# higher order basis functions
basiswt <- matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N)
xcoef <- matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt
xfd <- fd(xcoef, fourierbasis)
x <- eval.fd(argvals, xfd)
# Add independent Gaussian noise to the data with std. dev. 0.2
sigerr <- 0.2
y <- x + matrix(rnorm(c(x)),n,N)*sigerr
# Set up spline basis system
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Set up roughness penalty with smoothing parameter 10^(-5)
lambda <- 10^(-5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
# Display the results, note that a gcv value is returned for EACH curve,
# and therefore that a mean gcv result is reported
cat(round(c(df,RMSE,mean(gcv)),3),"\n")
# the fits are plotted interactively by plotfit.fd ... click to advance
# plot
plotfit.fd(y, argvals, xfd)
# Repeat these results for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# Our results were:
# log10 lambda, deg. freedom, RMSE, GCV
# -6.000 30.385 0.140 0.071
# -5.750 26.750 0.131 0.066
# -5.500 23.451 0.123 0.062
# -5.250 20.519 0.116 0.059
# -5.000 17.943 0.109 0.056
# -4.750 15.694 0.104 0.054
# -4.500 13.738 0.101 0.053
# -4.250 12.038 0.102 0.054
# -4.000 10.564 0.108 0.055
# -3.750 9.286 0.120 0.059
# -3.500 8.177 0.139 0.065
# -3.250 7.217 0.164 0.075
# -3.000 6.385 0.196 0.088
# RMSE and gcv both indicate an optimal smoothing level of
# log10(lambda) <- -4.5 corresponding to 13.7 df. When we repeated the
# analysis using regression smoothing with 14 basis functions, we
# obtained RMSE <- 0.110, about 10 percent larger than the value of
# 0.101 in the roughness penalty result. Smooth the data, outputting a
# list containing various quantities
# Homework:
# Sine functions have a curvature that doesn't vary a great deal over
# the range the curve. Devise some test functions with sharp local
# curvature, such as Gaussian densities with standard deviations that
# are small relative to the range of the observations. Compare
# regression and roughness penalty smoothing in these situations.
if(!CRAN()){
##
####### Simulated data example 4: a roughness-penalized smooth ########
## with correlated error
##
# These three examples make GCV look pretty good as a basis for
# selecting the smoothing parameter lambda. BUT GCV is based an
# assumption of independent errors, and in reality, functional data
# often have autocorrelated errors, with an autocorrelation that is
# usually positive among neighboring observations. Positively
# correlated random values tend to exhibit slowly varying values that
# have long runs on one side or the other of their baseline, and
# therefore can look like trend in the data that needs to be reflected
# in the smooth curve. This code sets up the error correlation matrix
# for first-order autoregressive errors, or AR(1).
rho <- 0.9
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
Rerr <- matrix(0,n,n)
for (i in 1:n) {
for (j in 1:n) Rerr[i,j] <- rho^abs(i-j)
}
# Compute the Choleski factor of the correlation matrix
Lerr <- chol(Rerr)
# set up some data
# Generate auto-correlated errors by multipling independent errors by
# the transpose of the Choleski factor
sigerr <- 0.2
err <- as.vector(crossprod(Lerr,matrix(rnorm(x),n,1))*sigerr)
# See the long-run errors that are genrated
plot(argvals, err)
y <- x + err
# Our values of y were:
y <- c(-0.03, 0.36, 0.59, 0.71, 0.97, 1.2, 1.1, 0.96, 0.79, 0.68,
0.56, 0.25, 0.01,-0.43,-0.69, -1, -1.09,-1.29,-1.16,-1.09,
-0.93, -0.9,-0.78,-0.47, -0.3,-0.01, 0.29, 0.47, 0.77, 0.85,
0.87, 0.97, 0.9, 0.72, 0.48, 0.25,-0.17,-0.39,-0.47,-0.71,
-1.07,-1.28,-1.33,-1.39,-1.45, -1.3,-1.25,-1.04,-0.82,-0.55, -0.2)
# Retaining the above data, now set up a basis system with a knot at
# every data point: the number of basis functions will equal the
# number of interior knots plus the order, which in this case is still
# 4.
# 19 interior knots + order 4 <- 23 basis functions
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
fdParobj <- fdPar(fdobj)
# Smooth these results for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj)
# Our results without weighting were:
# -6.000 30.385 0.261 0.004
# -5.750 26.750 0.260 0.005
# -5.500 23.451 0.259 0.005
# -5.250 20.519 0.258 0.005
# -5.000 17.943 0.256 0.005
# -4.750 15.694 0.255 0.006
# -4.500 13.738 0.252 0.006
# -4.250 12.038 0.249 0.007
# -4.000 10.564 0.246 0.010
# -3.750 9.286 0.244 0.015
# -3.500 8.177 0.248 0.028
# -3.250 7.217 0.267 0.055
# -3.000 6.385 0.310 0.102
# Now GCV still is firm on the fact that log10(lambda) over -4 is
# over-smoothing, but is quite unhelpful about what constitutes
# undersmoothing. In real data applications you will have to make the
# final call. Now set up a weight matrix equal to the inverse of the
# correlation matrix
wtmat <- solve(Rerr)
# Smooth these results for a range of log10(lambda) values using the
# weight matrix
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj,
wtvec=wtmat)
# Our results with weighting were:
# -6.000 46.347 0.263 0.005
# -5.750 43.656 0.262 0.005
# -5.500 40.042 0.261 0.005
# -5.250 35.667 0.259 0.005
# -5.000 30.892 0.256 0.005
# -4.750 26.126 0.251 0.006
# -4.500 21.691 0.245 0.008
# -4.250 17.776 0.237 0.012
# -4.000 14.449 0.229 0.023
# -3.750 11.703 0.231 0.045
# -3.500 9.488 0.257 0.088
# -3.250 7.731 0.316 0.161
# -3.000 6.356 0.397 0.260
# GCV is still not clear about what the right smoothing level is.
# But, comparing the two results, we see an optimal RMSE without
# smoothing of 0.244 at log10(lambda) <- -3.75, and with smoothing 0.229
# at log10(lambda) <- -4. Weighting improved the RMSE. At
# log10(lambda) <- -4 the improvement is 9 percent.
# Smooth the data with and without the weight matrix at log10(lambda) =
# -4
fdParobj <- fdPar(fdobj, 2, 10^(-4))
smoothlistnowt <- smooth.basis(argvals, y, fdParobj)
fdobjnowt <- smoothlistnowt$fd # the curve smoothing the data
df <- smoothlistnowt$df # the degrees of freedom in the smoothing curve
GCV <- smoothlistnowt$gcv # the value of the GCV statistic
RMSE <- sqrt(mean((eval.fd(argvals, fdobjnowt) - x)^2))
cat(round(c(df,RMSE,GCV),3),"\n")
smoothlistwt <- smooth.basis(argvals, y, fdParobj, wtvec=wtmat)
fdobjwt <- smoothlistwt$fd # the curve smoothing the data
df <- smoothlistwt$df # the degrees of freedom in the smoothing curve
GCV <- smoothlistwt$gcv # the value of the GCV statistic
RMSE <- sqrt(mean((eval.fd(argvals, fdobjwt) - x)^2))
cat(round(c(df,RMSE,GCV),3),"\n")
par(mfrow=c(2,1))
plotfit.fd(y, argvals, fdobjnowt)
plotfit.fd(y, argvals, fdobjwt)
par(mfrow=c(1,1))
plot(fdobjnowt)
lines(fdobjwt,lty=2)
points(argvals, y)
# Homework:
# Repeat these analyses with various values of rho, perhaps using
# multiple curves to get more stable indications of relative
# performance. Be sure to include some negative rho's.
##
######## Simulated data example 5: derivative estimation ########
##
# Functional data analyses often involve estimating derivatives. Here
# we repeat the analyses of Example 2, but this time focussing our
# attention on the estimation of the first and second derivative.
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
Dx <- 4*pi*cos(4*pi*argvals)
D2x <- -(4*pi)^2*sin(4*pi*argvals)
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# We now use order 6 splines so that we can control the curvature of
# the second derivative, which therefore involves penalizing the
# derivative of order four.
norder <- 6
nbasis <- n + norder - 2
basisobj <- create.bspline.basis(c(0,1),nbasis,norder)
fdobj <- fd(matrix(0,nbasis,1),basisobj)
# Note that there are more basis functions than observed values. A
# basis like this is called "super-saturated." We have to use a
# positive smoothing parameter for it to work. Set up an object of
# class "fdPar" that penalizes the total squared fourth derivative. The
# smoothing parameter that is set here to 10^(-10), because the squared
# fourth derivative is a much larger number than the squared second
# derivative.
lambda <- 10^(-10)
fdParobj <- fdPar(fdobj, Lfdobj=4, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
Dxhat <- eval.fd(argvals, xfd, Lfd=1)
D2xhat <- eval.fd(argvals, xfd, Lfd=2)
RMSED <- sqrt(mean((Dxhat - Dx )^2))
RMSED2 <- sqrt(mean((D2xhat - D2x)^2))
cat(round(c(df,RMSED,RMSED2,gcv),3),"\n")
# Four plots of the results row-wise: data fit, first derivative fit,
# second derivative fit, second vs. first derivative fit
# (phase-plane plot)
par(mfrow=c(2,2))
plotfit.fd(y, argvals, xfd)
plot(argvals, Dxhat, type="p", pch="o")
points(argvals, Dx, pch="*")
title("first derivative approximation")
plot(argvals, D2xhat, type="p", pch="o")
points(argvals, D2x, pch="*")
title("second derivative approximation")
plot(Dxhat, D2xhat, type="p", pch="o")
points(Dx, D2x, pch="*")
title("second against first derivative")
# This illustrates an inevitable problem with spline basis functions;
# because they are not periodic, they fail to capture derivative
# information well at the ends of the interval. The true phase-plane
# plot is an ellipse, but the phase-plane plot of the estimated
# derivatives here is only a rough approximtion, and breaks down at the
# left boundary.
# Homework:
# Repeat these results with smaller standard errors.
# Repeat these results, but this time use a fourier basis with no
# roughness penalty, and find the number of basis functions that gives
# the best result. The right answer to this question is, of course, 3,
# if we retain the constant term, even though it is here not needed.
# Compare the smoothing parameter preferred by RMSE for a derivative to
# that preferred by the RMSE for the function itself, and to that
# preferred by gcv.
## Simulated data example 6:
## a better roughness penalty for derivative estimation
##
# We want to see if we can improve the spline fit.
# We know from elementary calculus as well as the code above that
# (4*pi)^2 sine(2*p*x) <- -D2 sine(2*p*x), so that
# Lx <- D2x + (4*pi)^2 x is zero for a sine or a cosine curve.
# We now penalize roughness using this "smart" roughness penalty
# Here we set up a linear differential operator object that defines
# this penalty
constbasis <- create.constant.basis(c(0,1))
xcoef.fd <- fd((4*pi)^2, constbasis)
Dxcoef.fd <- fd(0, constbasis)
bwtlist <- vector("list", 2)
bwtlist[[1]] <- xcoef.fd
bwtlist[[2]] <- Dxcoef.fd
Lfdobj <- Lfd(nderiv=2, bwtlist=bwtlist)
# Now we use a much larger value of lambda to reflect our confidence
# in power of calculus to solve problems!
lambda <- 10^(0)
fdParobj <- fdPar(fdobj, Lfdobj=Lfdobj, lambda=lambda)
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
Dxhat <- eval.fd(argvals, xfd, Lfd=1)
D2xhat <- eval.fd(argvals, xfd, Lfd=2)
RMSED <- sqrt(mean((Dxhat - Dx )^2))
RMSED2 <- sqrt(mean((D2xhat - D2x)^2))
cat(round(c(df,RMSED,RMSED2,gcv),3),"\n")
# Four plots of the results row-wise: data fit, first derivative fit,
# second derivative fit, second versus first derivative fit
# (phase-plane plot)
par(mfrow=c(2,2))
plotfit.fd(y, argvals, xfd)
plot(argvals, Dxhat, type="p", pch="o")
points(argvals, Dx, pch="*")
title("first derivative approximation")
plot(argvals, D2xhat, type="p", pch="o")
points(argvals, D2x, pch="*")
title("second derivative approximation")
plot(Dxhat, D2xhat, type="p", pch="o")
points(Dx, D2x, pch="*")
title("second against first derivative")
# The results are nearly perfect in spite of the fact that we are not using
# periodic basis functions. Notice, too, that we have used 2.03
# degrees of freedom, which is close to what we would use for the ideal
# fourier series basis with the constant term dropped.
# Homework:
# These results depended on us knowing the right period, of course.
# The data would certainly allow us to estimate the period 1/2 closely,
# but try various other periods by replacing 1/2 by other values.
# Alternatively, change x by adding a small amount of, say, linear trend.
# How much trend do you have to add to seriously handicap the results?
##
######## Simulated data example 7: Using covariates ########
##
# Now we simulate data that are defined by a sine curve, but where the
# the first 20 observed values are shifted upwards by 0.5, and the
# second shifted downwards by -0.2. The two covariates are indicator
# or dummy variables, and the estimated regression coefficients will
# indicate the shifts as estimated from the data.
n <- 51
argvals <- seq(0,1,len=n)
x <- sin(4*pi*argvals)
sigerr <- 0.2
y <- x + rnorm(x)*sigerr
# the n by p matrix of covariate values, p being here 2
p <- 2
zmat <- matrix(0,n,p)
zmat[ 1:11,1] <- 1
zmat[11:20,2] <- 1
# The true values of the regression coefficients
beta0 <- matrix(c(0.5,-0.2),p,1)
y <- y + zmat
# The same basis system and smoothing process as used in Example 2
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
fdParobj <- fdPar(fdobj, 2, lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj, covariates=zmat)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
beta <- smoothlist$beta # the regression coefficients
RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
cat(round(c(beta,df,RMSE,gcv),3),"\n")
par(mfrow=c(1,1))
plotfit.fd(y, argvals, xfd)
points(argvals,x, pch="*")
print(beta)
# The recovery of the smooth curve is fine, as in Example 2. The
# shift of the first 10 observations was estimated to be 0.62 in our run,
# and the shift of the second 20 was estimated to be -0.42. These
# estimates are based on only 10 observations, and these estimates are
# therefore quite reasonable.
# Repeat these analyses for a range of log10(lambda) values
loglamout <- plotGCVRMSE.fd(-6, -3, 0.25, x, argvals, y, fdParobj,
covariates=zmat)
# Homework:
# Try an example where the covariate values are themselves are
# generated by a smooth known curve.
##
## Simulated data example 8:
## a roughness-penalized smooth of a sample of curves and
## variable observation points
##
n <- 51 # number of observations per curve
N <- 100 # number of curves
argvals <- matrix(0,n,N)
for (i in 1:N) argvals[,i] <- sort(runif(1:n))
# The true curve values are linear combinations of fourier function
# values.
# Set up the fourier basis
nfourierbasis <- 13
fourierbasis <- create.fourier.basis(c(0,1),nfourierbasis)
# Set up some random coefficients, with declining contributions from
# higher order basis functions
basiswt <- matrix(exp(-(1:nfourierbasis)/3),nfourierbasis,N)
xcoef <- matrix(rnorm(nfourierbasis*N),nfourierbasis,N)*basiswt
xfd <- fd(xcoef, fourierbasis)
x <- matrix(0,n,N)
for (i in 1:N) x[,i] <- eval.fd(argvals[,i], xfd[i])
# Add independent Gaussian noise to the data with std. dev. 0.2
sigerr <- 0.2
y <- x + matrix(rnorm(c(x)),n,N)*sigerr
# Set up spline basis system
nbasis <- n + 2
basisobj <- create.bspline.basis(c(0,1),nbasis)
fdobj <- fd(matrix(0,nbasis,1), basisobj)
# Set up roughness penalty with smoothing parameter 10^(-5)
lambda <- 10^(-5)
fdParobj <- fdPar(fdobj, Lfdobj=2, lambda=lambda)
# Smooth the data, outputting a list containing various quantities
smoothlist <- smooth.basis(argvals, y, fdParobj)
xfd <- smoothlist$fd # the curve smoothing the data
df <- smoothlist$df # the degrees of freedom in the smoothing curve
gcv <- smoothlist$gcv # the value of the gcv statistic
#RMSE <- sqrt(mean((eval.fd(argvals, xfd) - x)^2))
eval.x <- eval.fd(argvals, xfd)
e.xfd <- (eval.x-x)
mean.e2 <- mean(e.xfd^2)
RMSE <- sqrt(mean.e2)
# Display the results, note that a gcv value is returned for EACH
# curve, and therefore that a mean gcv result is reported
cat(round(c(df,RMSE,mean(gcv)),3),"\n")
# Function plotfit.fd is not equipped to handle a matrix of argvals,
# but can always be called within a loop to plot each curve in turn.
# Although a call to function plotGCVRMSE.fd works, the computational
# overhead is substantial, and we omit this here.
##
## Real data example 9. gait
##
# These data involve two variables in addition to multiple curves
gaitnbasis <- 21
gaittime <- (1:20)/21
gaitrange <- c(0,1)
gaitbasis <- create.fourier.basis(gaitrange, gaitnbasis)
gaitfd <- fd(matrix(0,gaitnbasis,1), gaitbasis)
lambda <- 10^(-11.5)
harmaccelLfd <- vec2Lfd(c(0, 0, (2*pi)^2, 0))
gaitfdPar <- fdPar(gaitfd, harmaccelLfd, lambda)
gaitSmooth <- smooth.basis(gaittime, gait, gaitfdPar)
gaitfd <- gaitSmooth$fd
if (!CRAN()){
# by default creates multiple plots, asking for a click between plots
plotfit.fd(gait, gaittime, gaitfd)
}
}
# end of if (!CRAN)
par(oldpar)
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