fda (version 2.4.0)

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 in setting up a call to function smooth.basis. Be warned that the result may not be a satisfactory smooth of the data, and consequently that it may be necessary to use function smooth.basis instead, the help file for which provides a great deal more information than is provided here. Also, function Data2fd can swap the first two arguments, argvals and y if it appears that they have been included in reverse order. A warning message is returned if this swap takes place. Any such automatic decision, though, has the possibility of being wrong, and the results should be carefully checked. Preferably, the order of the arguments should be respected: argvals comes first and y comes second.

Usage

Data2fd(argvals=NULL, y=NULL, basisobj=NULL, nderiv=NULL,
        lambda=3e-8/diff(as.numeric(range(argvals))),
        fdnames=NULL, covariates=NULL, method="chol",
        dfscale=1)

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 argume

y

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

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 extract

Value

• an object of the fd class containing:
• coefs
  the coefficient array
• basis
  a basis object
• fdnames
  a list containing names for the arguments, function values and variables

item

• nderiv
• lambda
• fdnames
• repname
• value
• covariates
• method
• dfscale

code

n

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.

See Also

smooth.basisPar, smooth.basis, project.basis, smooth.fd, smooth.monotone, smooth.pos day.5

Examples

##
## Simplest possible example:  constant function
##
# 1 basis, order 1 = degree 0 = constant function
b1.1 <- create.bspline.basis(nbasis=1, norder=1)
# data values: 1 and 2, with a mean of 1.5
y12 <- 1:2
# smooth data, giving a constant function with value 1.5
fd1.1 <- Data2fd(y12, basisobj=b1.1)
plot(fd1.1)
# now repeat the analysis with some smoothing, which moves the
# toward 0.
fd1.1.5 <- Data2fd(y12, basisobj=b1.1, lambda=0.5)
#  values of the smooth:
# fd1.1.5 = sum(y12)/(n+lambda*integral(over arg=0 to 1 of 1))
#         = 3 / (2+0.5) = 1.2
eval.fd(seq(0, 1, .2), fd1.1.5)

##
## step function smoothing
##
sessionInfo()
find("crossprod")

# 2 step basis functions: order 1 = degree 0 = step functions
b1.2 <- create.bspline.basis(nbasis=2, norder=1)
#  fit the data without smoothing
fd1.2 <- Data2fd(1:2, basisobj=b1.2)
# plot the result:  A step function:  1 to 0.5, then 2
op <- par(mfrow=c(2,1))
plot(b1.2, main='bases')
plot(fd1.2, main='fit')
par(op)

##
## Simple oversmoothing
##
# 3 step basis functions: order 1 = degree 0 = step functions
b1.3 <- create.bspline.basis(nbasis=3, norder=1)
#  smooth the data with smoothing
fd1.3.5 <- Data2fd(y12, basisobj=b1.3, lambda=0.5)
#  plot the fit along with the points
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
##
# 3 bases, order 2 = degree 1
b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1))
# interpolate the values 0, 2, 1
fd2.3 <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=0)
#  display the coefficients
round(fd2.3$coefs, 4)
# plot the results
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3, main='fit')
par(op)
# apply some smoothing
fd2.3. <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=1)
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3., main='fit', ylim=c(0,2))
par(op)

##
## quadratic spline fit
##
# 4 bases, order 3 = degree 2 = continuous, bounded, locally quadratic
b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1))
# fit values c(0,4,2,3) without interpolation
fd3.4 <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=0)
round(fd3.4$coefs, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4)
points(c(0,1/3,2/3,1), c(0,4,2,3))
par(op)
#  try smoothing
fd3.4. <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=1)
round(fd3.4.$coef, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4., ylim=c(0,4))
points(seq(0,1,len=4), c(0,4,2,3))
par(op)

##
##  Two simple Fourier examples
##
gaitbasis3 <- create.fourier.basis(nbasis=5)
gaitfd3    <- Data2fd(gait, basisobj=gaitbasis3)
plotfit.fd(gait, seq(0,1,len=20), gaitfd3)
#    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.
tempfd <- Data2fd(CanadianWeather$dailyAv[,,"Temperature.C"],
                  day.5, daybasis)
#  plot the temperature curves
par(mfrow=c(1,1))
plot(tempfd)

##
## argvals of class Date and POSIXct
##
#  Date
invasion1 <- as.Date('1775-09-04')
invasion2 <- as.Date('1812-07-12')
earlyUS.Canada <- c(invasion1, invasion2)
BspInvasion <- create.bspline.basis(earlyUS.Canada)

earlyYears <- seq(invasion1, invasion2, length.out=7)
(earlyQuad <- (as.numeric(earlyYears-invasion1)/365.24)^2)
fitQuad <- Data2fd(earlyYears, earlyQuad, BspInvasion)

# POSIXct
AmRev.ct <- as.POSIXct1970(c('1776-07-04', '1789-04-30'))
BspRev.ct <- create.bspline.basis(AmRev.ct)
AmRevYrs.ct <- seq(AmRev.ct[1], AmRev.ct[2], length.out=14)
(AmRevLin.ct <- as.numeric(AmRevYrs.ct-AmRev.ct[1]))
fitLin.ct <- Data2fd(AmRevYrs.ct, AmRevLin.ct, BspRev.ct)