Data2fd: Create a functional data object from data

Description

This function converts an array y of function values plus an array argvals of argument values into a functional data object. This function tries to do as much for the user as possible. NOTE: Interpolation with data2fd(...) can be shockingly bad, as illustrated in one of the examples.

Usage

Data2fd(argvals=NULL, y=NULL, basisobj=NULL, nderiv=NULL,
        lambda=3e-8/diff(range(argvals)), fdnames=NULL)

Arguments

argvals

a set of argument values. If this is a vector, the same set of argument values is used for all columns of y. If argvals is a matrix, the columns correspond to the columns of y, and contain the argument

an array containing sampled values of curves.

If y is a vector, only one replicate and variable are assumed. If y is a matrix, rows must correspond to argument values and columns to replications or cases, and it wil

basisobj

One of the following:

basisfd

{ a functional basis object (class basisfd). } fd{ a functional data object (class fd), from which its basis component is extracted.

Value

an object of the fd class containing:
coefsthe coefficient array
basisa basis object
fdnamesa list containing names for the arguments, function values and variables

item

nderiv
lambda
fdnames
repname
value

code

y

itemize

argname

Details

This function tends to be used in rather simple applications where there is no need to control the roughness of the resulting curve with any great finesse. The roughness is essentially controlled by how many basis functions are used. In more sophisticated applications, it would be better to use the function smooth.basisPar.

References

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

Examples

Run this code

##
## Simplest possible example:  step function
##
b1.1 <- create.bspline.basis(nbasis=1, norder=1)
# 1 basis, order 1 = degree 0 = step function

y12 <- 1:2
fd1.1 <- Data2fd(y12, basisobj=b1.1)
plot(fd1.1)
# fd1.1 = mean(y12) = 1.5

fd1.1.5 <- Data2fd(y12, basisobj=b1.1, lambda=0.5)
eval.fd(seq(0, 1, .2), fd1.1.5)
# fd1.1.5 = sum(y12)/(n+lambda*integral(over arg=0 to 1 of 1))
#         = 3 / (2+0.5) = 1.2

##
## 3 step functions
##
b1.2 <- create.bspline.basis(nbasis=2, norder=1)
# 2 bases, order 1 = degree 0 = step functions
fd1.2 <- Data2fd(1:2, basisobj=b1.2)

op <- par(mfrow=c(2,1))
plot(b1.2, main='bases')
plot(fd1.2, main='fit')
par(op)
# A step function:  1 to 0.5, then 2

##
## Simple oversmoothing
##
b1.3 <- create.bspline.basis(nbasis=3, norder=1)
fd1.3.5 <- Data2fd(y12, basisobj=b1.3, lambda=0.5)
plot(0:1, c(0, 2), type='n')
points(0:1, y12)
lines(fd1.3.5)
# Fit = penalized least squares with penalty =
#          = lambda * integral(0:1 of basis^2),
#            which shrinks the points towards 0.
# X1.3 = matrix(c(1,0, 0,0, 0,1), 2)
# XtX = crossprod(X1.3) = diag(c(1, 0, 1))
# penmat = diag(3)/3
#        = 3x3 matrix of integral(over arg=0:1 of basis[i]*basis[j])
# Xt.y = crossprod(X1.3, y12) = c(1, 0, 2)
# XtX + lambda*penmat = diag(c(7, 1, 7)/6
# so coef(fd1.3.5) = solve(XtX + lambda*penmat, Xt.y)
#                  = c(6/7, 0, 12/7)

##
## linear spline fit
##
b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1))
# 3 bases, order 2 = degree 1 =
# continuous, bounded, locally linear

fd2.3 <- Data2fd(0:1, basisobj=b2.3, lambda=0)
round(fd2.3$coefs, 4)
# (0, 0, 1),
# though (0, a, 1) is also a solution for any 'a'
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3, main='fit')
par(op)

# smoothing?
fd2.3. <- Data2fd(0:1, basisobj=b2.3, lambda=1)
stopifnot(
all.equal(as.vector(round(fd2.3.$coefs, 4)),
          c(0.0159, -0.2222, 0.8730) )
)
# The default smoothing with spline of order 2, degree 1
# has nderiv = max(0, norder-2) = 0.
# Direct computations confirm that the optimal B-spline
# weights in this case are the numbers given above.

op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3., main='fit')
par(op)

##
## quadratic spline fit
##
b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1))
# 4 bases, order 3 = degree 2 =
# continuous, bounded, locally quadratic

fd3.4 <- Data2fd(0:1, basisobj=b3.4, lambda=0)
round(fd3.4$coefs, 4)
# (0, 0, 0, 1),
# but (0, a, b, 1) is also a solution for any 'a' and 'b'
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4)
par(op)

#  try smoothing?
fd3.4. <- Data2fd(0:1, basisobj=b3.4, lambda=1)
round(fd3.4.$coef, 4)

op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4.)
par(op)

##
##  A simple Fourier example
##
gaitbasis3 <- create.fourier.basis(nbasis=3)
# note:  'names' for 3 bases
gaitfd3 <- Data2fd(gait, basisobj=gaitbasis3)
# Note: dimanes for 'coefs' + basis[['names']]
# + 'fdnames'

#    set up the fourier basis
daybasis <- create.fourier.basis(c(0, 365), nbasis=65)
#  Make temperature fd object
#  Temperature data are in 12 by 365 matrix tempav
#    See analyses of weather data.

#  Convert the data to a functional data object
tempfd <- Data2fd(CanadianWeather$dailyAv[,,"Temperature.C"],
                  day.5, daybasis)
#  plot the temperature curves
plot(tempfd)

##
## Terrifying interpolation
##
hgtbasis <- with(growth, create.bspline.basis(range(age),
                                              breaks=age, norder=6))
girl.data2fd <- with(growth, Data2fd(hgtf, age, hgtbasis, lambda=0))
age2 <- with(growth, sort(c(age, (age[-1]+age[-length(age)])/2)))
girlPred <- eval.fd(age2, girl.data2fd)
range(growth$hgtf)
range(growth$hgtf-girlPred[seq(1, by=2, length=31),])
# 5.5e-6 0.028 <
# The predictions are consistently too small
# but by less than 0.05 percent

matplot(age2, girlPred, type="l")
with(growth, matpoints(age, hgtf))
# girl.data2fd fits the data fine but goes berzerk
# between points

# Smooth
girl.data2fd1 <- with(growth, Data2fd(age, hgtf, hgtbasis))
girlPred1 <- eval.fd(age2, girl.data2fd1)

matplot(age2, girlPred1, type="l")
with(growth, matpoints(age, hgtf))

# problems splikes disappear with the default lambda

##
## multivariate argvals
##
ageMat <- with(growth, array(age, dim(hgtf)))

girl.data2fd2 <- Data2fd(ageMat, growth$hgtf, hgtbasis)

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