fpc {refund}

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.

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.

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.

Author(s)

Jonathan Gellar [e-mail], Phil Reiss [e-mail], Lan Huo [e-mail], and Lei Huang [e-mail]

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

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)