refund (version 0.1-24)

fpc: Construct a FPC regression term

Description

Constructs a functional principal component regression (Reiss and Ogden, 2007, 2010) term for inclusion in an mgcv::gam-formula (or bam or gamm or gamm4:::gamm) as constructed by pfr. Currently only one-dimensional functions are allowed.

Usage

fpc(
  X,
  argvals = NULL,
  method = c("svd", "fpca.sc", "fpca.face", "fpca.ssvd"),
  ncomp = NULL,
  pve = 0.99,
  penalize = (method == "svd"),
  bs = "ps",
  k = 40,
  ...
)

Arguments

X

functional predictors, typically expressed as an N by J matrix, where N is the number of columns and J is the number of evaluation points. May include missing/sparse functions, which are indicated by NA values. Alternatively, can be an object of class "fd"; see fd.

argvals

indices of evaluation of X, i.e. $(t_{i1},.,t_{iJ})$ for subject $i$. May be entered as either a length-J vector, or as an N by J matrix. Indices may be unequally spaced. Entering as a matrix allows for different observations times for each subject. If NULL, defaults to an equally-spaced grid between 0 or 1 (or within X$basis$rangeval if X is a fd object.)

method

the method used for finding principal components. The default is an unconstrained SVD of the $XB$ matrix. Alternatives include constrained (functional) principal components approaches

ncomp

number of principal components. if NULL, chosen by pve

pve

proportion of variance explained; used to choose the number of principal components

penalize

if TRUE, a roughness penalty is applied to the functional estimate. Defaults to TRUE if method=="svd" (corresponding to the FPCR_R method of Reiss and Ogden (2007)), and FALSE if method!="svd" (corresponding to FPCR_C).

bs

two letter character string indicating the mgcv-style basis to use for pre-smoothing X

k

the dimension of the pre-smoothing basis

...

additional options to be passed to lf. These include argvals, integration, and any additional options for the pre-smoothing basis (as constructed by mgcv::s), such as m.

Value

The result of a call to lf.

NOTE

Unlike fpcr, fpc within a pfr formula does not automatically decorrelate the functional predictors from additional scalar covariates.

Details

fpc is a wrapper for lf, which defines linear functional predictors for any type of basis for inclusion in a pfr formula. fpc simply calls lf with the appropriate options for the fpc basis and penalty construction.

This function implements both the FPCR-R and FPCR-C methods of Reiss and Ogden (2007). Both methods consist of the following steps:

1. project $X$ onto a spline basis $B$

2. perform a principal components decomposition of $XB$

3. use those PC's as the basis in fitting a (generalized) functional linear model

This implementation provides options for each of these steps. The basis for in step 1 can be specified using the arguments bs and k, as well as other options via ...; see s for these options. The type of PC-decomposition is specified with method. And the FLM can be fit either penalized or unpenalized via penalize.

The default is FPCR-R, which uses a b-spline basis, an unconstrained principal components decomposition using svd, and the FLM fit with a second-order difference penalty. FPCR-C can be selected by using a different option for method, indicating a constrained ("functional") PC decomposition, and by default an unpenalized fit of the FLM.

FPCR-R is also implemented in fpcr; here we implement the method for inclusion in a pfr formula.

References

Reiss, P. T. (2006). Regression with signals and images as predictors. Ph.D. dissertation, Department of Biostatistics, Columbia University. Available at http://works.bepress.com/phil_reiss/11/.

Reiss, P. T., and Ogden, R. T. (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistical Association, 102, 984-996.

Reiss, P. T., and Ogden, R. T. (2010). Functional generalized linear models with images as predictors. Biometrics, 66, 61-69.

See Also

lf, smooth.construct.fpc.smooth.spec

Examples

# NOT RUN {
data(gasoline)
par(mfrow=c(3,1))

# Fit PFCR_R
gasmod1 <- pfr(octane ~ fpc(NIR, ncomp=30), data=gasoline)
plot(gasmod1, rug=FALSE)
est1 <- coef(gasmod1)

# Fit FPCR_C with fpca.sc
gasmod2 <- pfr(octane ~ fpc(NIR, method="fpca.sc", ncomp=6), data=gasoline)
plot(gasmod2, se=FALSE)
est2 <- coef(gasmod2)

# Fit penalized model with fpca.face
gasmod3 <- pfr(octane ~ fpc(NIR, method="fpca.face", penalize=TRUE), data=gasoline)
plot(gasmod3, rug=FALSE)
est3 <- coef(gasmod3)

par(mfrow=c(1,1))
ylm <- range(est1$value)*1.35
plot(value ~ X.argvals, type="l", data=est1, ylim=ylm)
lines(value ~ X.argvals, col=2, data=est2)
lines(value ~ X.argvals, col=3, data=est3)

# }