# fdPar: Define a Functional Parameter Object In fda: Functional Data Analysis

## Description

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.

## Usage

 ```1 2``` ```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 elseLfdobj <- 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.

## Value

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

`cca.fd`, `density.fd`, `fRegress`, `intensity.fd`, `pca.fd`, `smooth.fdPar`, `smooth.basis`, `smooth.basisPar`, `smooth.monotone`, `int2Lfd`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58``` ```## ## 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 ## ```