refund: fpc – R documentation

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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`.

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:

project X onto a spline basis B
perform a principal components decomposition of XB
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 JGellar@mathematica-mpr.com, Phil Reiss phil.reiss@nyumc.org, Lan Huo lan.huo@nyumc.org, and Lei Huang huangracer@gmail.com

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.

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)

refund

Regression with Functional Data

v0.1-23

GPL (>= 2)

Authors

Jeff Goldsmith [aut], Fabian Scheipl [aut], Lei Huang [aut], Julia Wrobel [aut, cre], Chongzhi Di [aut], Jonathan Gellar [aut], Jaroslaw Harezlak [aut], Mathew W. McLean [aut], Bruce Swihart [aut], Luo Xiao [aut], Ciprian Crainiceanu [aut], Philip T. Reiss [aut], Yakuan Chen [ctb], Sonja Greven [ctb], Lan Huo [ctb], Madan Gopal Kundu [ctb], So Young Park [ctb], David L. Miller [ctb], Ana-Maria Staicu [ctb]

Initial release

2020-12-03

fpc