Smooth (argvals, y) data with roughness penalty defined by the
remaining arguments. This function acts as a wrapper for those who want
to bypass the step of setting up a functional parameter object before invoking
function smooth.basis. This function simply does this setup for the
user. See the help file for functions smooth.basis and fdPar
for further details, and more complete descriptions of the arguments.
smooth.basisPar(argvals, y, fdobj=NULL, Lfdobj=NULL,
lambda=0, estimate=TRUE, penmat=NULL,
wtvec=NULL, fdnames=NULL, covariates=NULL,
method="chol", dfscale=1)The output of a call to smooth.basis, which is an object of
class fdSmooth, being a list of length 8 with the following
components:
a functional data object that smooths the data.
a degrees of freedom measure of the smooth
the 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.
the error sums of squares. SSE is a vector or a matrix of the same size as 'gcv'.
the penalty matrix.
the matrix mapping the data to the coefficients.
input arguments
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.
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.
One of the following:
a functional data object (class fd)
a functional basis object (class basisfd), which is
converted to a functional data object with the identity matrix
as the coefficient matrix.
a functional parameter object (class fdPar)
a positive integer giving the order of a B-spline basis, which is further converted to a functional data object with the identity matrix as the coefficient matrix.
replaced by fd(fdobj)
Defaults to fdobj = create.bspline.basis(argvals).
either a nonnegative integer or a linear differential operator object.
If NULL, Lfdobj depends on fdobj[['basis']][['type']]:
Lfdobj <- int2Lfd(max(0, norder-2)), where norder = norder(fdobj).
Lfdobj = a harmonic acceleration operator:
Lfdobj <- vec2Lfd(c(0,(2*pi/diff(rng))^2,0), rng)
where rng = fdobj[['basis']][['rangeval']].
Lfdobj <- int2Lfd(0)
a nonnegative real number specifying the amount of smoothing to be applied to the estimated functional parameter.
a logical value: if TRUE, the functional parameter is
estimated, otherwise, it is held fixed.
a roughness penalty matrix. Including this can eliminate the need to compute this matrix over and over again in some types of calculations.
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.
a list of length 3 containing character vectors of names for the following:
name for each observation or point in time at which data are collected for each 'rep', unit or subject.
name for each 'rep', unit or subject.
name for each 'fun' or (response) variable measured repeatedly (per 'args') for each 'rep'.
the observed values in 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 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 cueve and each
variable.
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.
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.
1. if(is.null(fdobj))fdobj <- create.bspline.basis(argvals). Else if(is.integer(fdobj)) fdobj <- create.bspline.basis(argvals, norder = fdobj)
2. fdPar
3. smooth.basis
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.
df2lambda,
fdPar,
lambda2df,
lambda2gcv,
plot.fd,
project.basis,
smooth.basis,
smooth.fd,
smooth.monotone,
smooth.pos