# Define a Functional Parameter Object Functional parameter objects are used as arguments to functions that estimate functional parameters, such as smoothing functions like `smooth.basis`. A functional parameter object is a functional data object with additional slots specifying a roughness penalty, a smoothing parameter and whether or not the functional parameter is to be estimated or held fixed. Functional parameter objects are used as arguments to functions that estimate functional parameters. ```r fdPar(fdobj=NULL, Lfdobj=NULL, lambda=0, estimate=TRUE, penmat=NULL) ``` ## Arguments - `fdobj`: a functional data object, functional basis object, a functional parameter object or a matrix. If it a matrix, it is replaced by fd(fdobj). If class(fdobj) == 'basisfd', it is converted to an object of class `fd` with a coefficient matrix consisting of a single column of zeros. - `Lfdobj`: either a nonnegative integer or a linear differential operator object. If `NULL`, Lfdobj depends on fdobj[['basis']][['type']]: - **bspline**: Lfdobj <- int2Lfd(max(0, norder-2)), where norder = norder(fdobj). - **fourier**: Lfdobj = a harmonic acceleration operator: `Lfdobj <- vec2Lfd(c(0,(2*pi/diff(rng))^2,0), rng)` where rng = fdobj[['basis']][['rangeval']]. - **anything else**: Lfdobj <- int2Lfd(0) - `lambda`: a nonnegative real number specifying the amount of smoothing to be applied to the estimated functional parameter. - `estimate`: not currently used. - `penmat`: a roughness penalty matrix. Including this can eliminate the need to compute this matrix over and over again in some types of calculations. ## Details Functional parameters are often needed to specify initial values for iteratively refined estimates, as is the case in functions `register.fd` and `smooth.monotone`. Often a list of functional parameters must be supplied to a function as an argument, and it may be that some of these parameters are considered known and must remain fixed during the analysis. This is the case for functions `fRegress` and `pda.fd`, for example. ## Returns a functional parameter object (i.e., an object of class `fdPar`), which is a list with the following components: - **fd**: a functional data object (i.e., with class `fd`) - **Lfd**: a linear differential operator object (i.e., with class `Lfd`) - **lambda**: a nonnegative real number - **estimate**: not currently used - **penmat**: either NULL or a square, symmetric matrix with penmat[i, j] = integral over fd[['basis']][['rangeval']] of basis[i]*basis[j] ## Source 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 `cca.fd`, `density.fd`, `fRegress`, `intensity.fd`, `pca.fd`, `smooth.fdPar`, `smooth.basis`, `smooth.monotone`, `int2Lfd` ## Examples ```r oldpar <- par(no.readonly=TRUE) ## ## Simple example ## # set up range for density rangeval <- c(-3,3) # set up some standard normal data x <- rnorm(50) # make sure values within the range x[x < -3] <- -2.99 x[x > 3] <- 2.99 # set up basis for W(x) basisobj <- create.bspline.basis(rangeval, 11) # set up initial value for Wfdobj Wfd0 <- fd(matrix(0,11,1), basisobj) WfdParobj <- fdPar(Wfd0) WfdP3 <- fdPar(seq(-3, 3, length=11)) ## ## smooth the Canadian daily temperature data ## # set up the fourier basis nbasis <- 365 dayrange <- c(0,365) daybasis <- create.fourier.basis(dayrange, nbasis) dayperiod <- 365 harmaccelLfd <- vec2Lfd(c(0,(2*pi/365)^2,0), dayrange) # Make temperature fd object # Temperature data are in 12 by 365 matrix tempav # See analyses of weather data. # Set up sampling points at mid days daytime <- (1:365)-0.5 # Convert the data to a functional data object daybasis65 <- create.fourier.basis(dayrange, nbasis, dayperiod) templambda <- 1e1 tempfdPar <- fdPar(fdobj=daybasis65, Lfdobj=harmaccelLfd, lambda=templambda) #FIXME #tempfd <- smooth.basis(CanadianWeather$tempav, daytime, tempfdPar)$fd # Set up the harmonic acceleration operator Lbasis <- create.constant.basis(dayrange); Lcoef <- matrix(c(0,(2*pi/365)^2,0),1,3) bfdobj <- fd(Lcoef,Lbasis) bwtlist <- fd2list(bfdobj) harmaccelLfd <- Lfd(3, bwtlist) # Define the functional parameter object for # smoothing the temperature data lambda <- 0.01 # minimum GCV estimate #tempPar <- fdPar(daybasis365, harmaccelLfd, lambda) # smooth the data #tempfd <- smooth.basis(daytime, CanadialWeather$tempav, tempPar)$fd # plot the temperature curves #plot(tempfd) ## ## with rangeval of class Date and POSIXct ## par(oldpar) ```

fdPar function