\name{Data2fd}
\alias{Data2fd}
\title{
  Create a functional data object from data
}
\description{
  This function converts an array \code{y} of function values plus an
  array \code{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 \code{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 \code{smooth.basis} instead, the help
  file for which provides a great deal more information than is provided
  here.
  Also, function \code{Data2fd} can swap the first two arguments,
  \code{argvals} and \code{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: \code{argvals} comes first and
  \code{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{
  \item{argvals}{
    a set of argument values.  If this is a vector, the same set of
    argument values is used for all columns of \code{y}.  If
    \code{argvals} is a matrix, the columns correspond to the columns of
    \code{y}, and contain the argument values for that replicate or
    case.

    Dimensions for \code{argvals} must match the first dimensions of
    \code{y}, though \code{y} can have more dimensions.  For example, if
    dim(y) = c(9, 5, 2), \code{argvals} can be a vector of length 9 or a
    matrix of dimensions c(9, 5) or an array of dimensions c(9, 5, 2).
  }
  \item{y}{
    an array containing sampled values of curves.

    If \code{y} is a vector, only one replicate and variable are
    assumed.  If \code{y} is a matrix, rows must correspond to argument
    values and columns to replications or cases, and it will be assumed
    that there is only one variable per observation.  If \code{y} is a
    three-dimensional array, the first dimension (rows) corresponds to
    argument values, the second (columns) to replications, and the third
    (layers) to variables within replications.  Missing values are
    permitted, and the number of values may vary from one replication to
    another.  If this is the case, the number of rows must equal the
    maximum number of argument values, and columns of \code{y} having
    fewer values must be padded out with NA's.
  }
  \item{basisobj}{
    One of the following:

    \itemize{
      \item{basisfd}{
	a functional basis object (class \code{basisfd}).
      }
      \item{fd}{
	a functional data object (class \code{fd}), from which its
	\code{basis} component is extracted.
      }
      \item{fdPar}{
	a functional parameter object (class \code{fdPar}), from which
	its \code{basis} component is extracted.
      }
      \item{integer}{
	an integer giving the order of a B-spline basis,
	create.bspline.basis(argvals, norder=basisobj)
      }
      \item{numeric vector}{
	specifying the knots for a B-spline basis,
	create.bspline.basis(basisobj)
      }
      \item{NULL}{
	Defaults to create.bspline.basis(argvals).
      }
    }
  }
  \item{nderiv}{
    Smoothing typically specified as an integer order for the derivative
    whose square is integrated and weighted by \code{lambda} to smooth.
    By default, if basisobj[['type']] == 'bspline', the smoothing
    operator is int2Lfd(max(0, norder-2)).

    A general linear differential operator can also be supplied.
  }
  \item{lambda}{
    weight on the smoothing operator specified by \code{nderiv}.
  }
  \item{fdnames}{
    Either a charater vector of length 3 or a named list of length 3.
    In either case, the three elements correspond to the following:

    \itemize{
      \item{argname}{
	name of the argument, e.g. "time" or "age".
      }
      \item{repname}{
	a description of the cases, e.g. "reps" or "weather stations"
      }
      \item{value}{
	the name of the observed function value, e.g. "temperature"
      }
    }

    If fdnames is a list, the components provide labels for the levels
    of the corresponding dimension of \code{y}.
  }
  \item{covariates}{
    the observed values in \code{y} are assumed to be primarily determined
    the 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 \code{n} by
    \code{p} matrix, where \code{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 cueve and each
    variable.
  }
  \item{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 \code{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 \code{n} is large, meaning more than 500 or so.  The default
    is "chol", referring the Choleski decomposition of a symmetric positive
    definite matrix.
  }
  \item{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 \code{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.
  }
}
\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 \code{\link{smooth.basisPar}}.
}
\value{
  an object of the \code{fd} class containing:

  \item{coefs}{
    the coefficient array
  }
  \item{basis}{
    a basis object
  }
  \item{fdnames}{
    a list containing names for the arguments, function values
    and variables
  }
}
\references{
  Ramsay, James O., and Silverman, Bernard W. (2006), \emph{Functional
    Data Analysis, 2nd ed.}, Springer, New York.

  Ramsay, James O., and Silverman, Bernard W. (2002), \emph{Applied
    Functional Data Analysis}, Springer, New York.
}
\seealso{
  \code{\link{smooth.basisPar}},
  \code{\link{smooth.basis}},
  \code{\link{project.basis}},
  \code{\link{smooth.fd}},
  \code{\link{smooth.monotone}},
  \code{\link{smooth.pos}}
  \code{\link{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)

}

% docclass is function
\keyword{smooth}