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.

a functional parameter object, a functional data object or a functional basis object. In the simplest case, fdParobj may be a functional basis object with class "basisfd" defined previously by one of the "create" functions, and in this case, no roughness penalty is used. No roughness penalty is also the consequence of supplying a functional data object with class "fd", in which case the basis system used for smoothing is that defining this object. However, if the object is a functional parameter object with class "fdPar", then the linear differential operator object and the smoothing parameter in this object define the roughness penalty. For further details on how the roughness penalty is defined, see the help file for "fdPar". In general, better results can be obtained using a good roughness penalty than can be obtained by merely varying the number of basis functions in the expansion.

a list of length 3 containing character vectors of names for the following:

• args name for each observation or point in time at which data are collected for each 'rep', unit or subject.

• reps name for each 'rep', unit or subject.

• fun 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 by 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 curve 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.