smooth.basis: Smooth Data with an Indirectly Specified Roughness Penalty

Description

This is the main function for smoothing data using a roughness penalty. Unlike function data2fd, which does not employ a rougness penalty, this function controls the nature and degree of smoothing by penalyzing a measure of rougness. Roughness is definable in a wide variety of ways using either derivatives or a linear differential operator.

Usage

smooth.basis(argvals, y, fdParobj, wtvec=rep(1,n),
             dffactor=1,
             fdnames=list(NULL, dimnames(y)[2], NULL))

Arguments

argvals

a vector of argument values correspond to the observations in array y.

an array containing values of curves at discrete sampling points or argument values. If the array is a matrix, the rows must correspond to argument values and columns to replications, and it will be assumed that there is only one variable per

fdParobj

a functional parameter object, a functional data object or a functional basis object. If the object is a functional parameter object, then the linear differential operator object and the smoothing parameter in this object define the roughness

wtvec

a vector of the same length as argvals containing weights for the values to be smoothed.

dffactor

Chong Gu in his book Smoothing Spline ANOVA Models suggests a modification of the GCV criterion using a factor modifying the effective degrees of freedom of the smooth. He suggests that values like 1.2 are effective at avoiding under

fdnames

a list of length 3 with members containing

a single name for the argument domain, such as 'Time'
a vector of names for the replications or cases
a name for the function, or a vector of names if there are multiple function

Value

a named list of length 7 containing:
fda functional data object that smooths the data.
dfa degrees of freedom measure of the smooth
gcvthe value of the generalized cross-validation or GCV criterion. If there 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.
coefthe coefficient matrix or array for the basis function expansion of the smoothing function
SSEthe error sums of squares. SSE is a vector or a matrix of the same size as GCV.
penmatthe penalty matrix.
y2cMapthe matrix mapping the data to the coefficients.

Details

If the smoothing parameter lambda is zero, there is no penalty on roughness. As lambda increases, usually in logarithmic terms, the penalty on roughness increases and the fitted curves become more and more smooth. Ultimately, the curves are forced to have zero roughness in the sense of being in the null space of the linear differential operator object Lfdobjthat is a member of the fdParobj. For example, a common choice of roughness penalty is the integrated square of the second derivative. This penalizes curvature. Since the second derivative of a straight line is zero, very large values of lambda will force the fit to become linear. It is also possible to control the amount of roughness by using a degrees of freedom measure. The value equivalent to lambda is found in the list returned by the function. On the other hand, it is possible to specify a degrees of freedom value, and then use function df2lambda to determine the equivalent value of lambda. One should not put complete faith in any automatic method for selecting lambda, including the GCV method. There are many reasons for this. For example, if derivatives are required, then the smoothing level that is automatically selected may give unacceptably rough derivatives. These methods are also highly sensitive to the assumption of independent errors, which is usually dubious with functional data. The best advice is to start with the value minimizing the gcv measure, and then explore lambda values a few log units up and down from this value to see what the smoothing function and its derivatives look like. The function plotfit.fd was designed for this purpose. 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.

Examples

Run this code

# A toy example that creates problems with
# data2fd:  (0,0) -> (0.5, -0.25) -> (1,1)
b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1))
# 3 bases, order 2 = degree 1 =
# continuous, bounded, locally linear
fdPar2 <- fdPar(b2.3, Lfdobj=2, lambda=1)
# Penalize excessive slope Lfdobj=1;  
# second derivative Lfdobj=2 is discontinuous.
#fd2.3s0 <- smooth.basis(0:1, 0:1, fdPar2)

b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1))
# 4 bases, order 3 = degree 2 =
# continuous, bounded, locally quadratic 
fdPar3 <- fdPar(b3.4, lambda=1)
# Penalize excessive slope Lfdobj=1;  
# second derivative Lfdobj=2 is discontinuous.
fd3.4s0 <- smooth.basis(0:1, 0:1, fdPar3)
plot(fd3.4s0$fd)


#  Shows the effects of three levels of smoothing
#  where the size of the third derivative is penalized.
#  The null space contains quadratic functions.
x <- seq(-1,1,0.02)
y <- x + 3*exp(-6*x^2) + rnorm(rep(1,101))*0.2
#  set up a saturated B-spline basis
basisobj <- create.bspline.basis(c(-1,1), 101)

fdParobj <- fdPar(basisobj, 2, lambda=1)
result1  <- smooth.basis(x, y, fdParobj)
yfd1     <- result1$fd

with(result1, c(df, gcv, SSE))

fdParobj <- fdPar(basisobj, 2, lambda=1e-4)
result2  <- smooth.basis(x, y, fdParobj)
yfd2     <- result2$fd

with(result2, c(df, gcv, SSE))

fdParobj <- fdPar(basisobj, 2, lambda=0)
result3  <- smooth.basis(x, y, fdParobj)
yfd3     <- result3$fd

with(result3, c(df, gcv, SSE))

plot(x,y)           # plot the data
lines(yfd1, lty=2)  #  add heavily penalized smooth
lines(yfd2, lty=1)  #  add reasonably penalized smooth
lines(yfd3, lty=3)  #  add smooth without any penalty
legend(-1,3,c("1","0.0001","0"),lty=c(2,1,3))
plotfit.fd(y, x, yfd2)  # plot data and smooth

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