refund (version 0.1-23)

# fpcr: Functional principal component regression

## Description

Implements functional principal component regression (Reiss and Ogden, 2007, 2010) for generalized linear models with scalar responses and functional predictors.

## Usage

fpcr(
y,
xfuncs = NULL,
fdobj = NULL,
ncomp = NULL,
pve = 0.99,
nbasis = NULL,
basismat = NULL,
penmat = NULL,
argvals = NULL,
covt = NULL,
mean.signal.term = FALSE,
spline.order = NULL,
family = "gaussian",
method = "REML",
sp = NULL,
pen.order = 2,
cv1 = FALSE,
nfold = 5,
store.cv = FALSE,
store.gam = TRUE,
...
)

## Arguments

y

scalar outcome vector.

xfuncs

for 1D predictors, an $$n \times d$$ matrix of signals/functional predictors, where $$n$$ is the length of y and $$d$$ is the number of sites at which each signal is defined. For 2D predictors, an $$n \times d1 \times d2$$ array representing $$n$$ images of dimension $$d1 \times d2$$.

fdobj

functional data object (class "fd") giving the functional predictors. Allowed only for 1D functional predictors.

ncomp

number of principal components. If NULL, this is chosen by pve.

pve

proportion of variance explained: used to choose the number of principal components.

nbasis

number(s) of B-spline basis functions: either a scalar, or a vector of values to be compared. For 2D predictors, tensor product B-splines are used, with the given basis dimension(s) in each direction; alternatively, nbasis can be given in the form list(v1,v2), in which case cross-validation will be performed for each combination of the first-dimension basis sizes in v1 and the second-dimension basis sizes in v2. Ignored if fdobj is supplied. If fdobj is not supplied, this defaults to 40 (i.e., 40 B-spline basis functions) for 1D predictors, and 15 (i.e., tensor product B-splines with 15 functions per dimension) for 2D predictors.

basismat

a $$d \times K$$ matrix of values of $$K$$ basis functions at the $$d$$ sites.

penmat

a $$K \times K$$ matrix defining a penalty on the basis coefficients.

argvals

points at which the functional predictors and the coefficient function are evaluated. By default, if 1D functional predictors are given by the $$n \times d$$ matrix xfuncs, argvals is set to $$d$$ equally spaced points from 0 to 1; if they are given by fdobj, argvals is set to 401 equally spaced points spanning the domain of the given functions. For 2D (image) predictors supplied as an $$n \times d1 \times d2$$ array, argvals defaults to a list of (1) $$d1$$ equally spaced points from 0 to 1; (2) $$d2$$ equally spaced points from 0 to 1.

covt

covariates: an $$n$$-row matrix, or a vector of length $$n$$.

mean.signal.term

logical: should the mean of each subject's signal be included as a covariate?

spline.order

order of B-splines used, if fdobj is not supplied; defaults to 4, i.e., cubic B-splines.

family

generalized linear model family. Current version supports "gaussian" (the default) and "binomial".

method

smoothing parameter selection method, passed to function gam; see the gam documentation for details.

sp

a fixed smoothing parameter; if NULL, an optimal value is chosen (see method).

pen.order

order of derivative penalty applied when estimating the coefficient function; defaults to 2.

cv1

logical: should cross-validation be performed to select the best model if only one set of tuning parameters provided? By default, FALSE. Note that, if there are multiple sets of tuning parameters provided, cv is always performed.

nfold

the number of validation sets ("folds") into which the data are divided; by default, 5.

store.cv

logical: should a CV result table be in the output? By default, FALSE.

store.gam

logical: should the gam object be included in the output? Defaults to TRUE.

other arguments passed to function gam.

## Value

A list with components

gamObject

if store.gam = TRUE, an object of class gam (see gamObject in the mgcv package documentation).

fhat

coefficient function estimate.

se

pointwise Bayesian standard error.

undecor.coef

undecorrelated coefficient for covariates.

argvals

the supplied value of argvals.

cv.table

a table giving the CV criterion for each combination of nbasis and ncomp, if store.cv = TRUE; otherwise, the CV criterion only for the optimized combination of these parameters. Set to NULL if CV is not performed.

nbasis, ncomp

when CV is performed, the values of nbasis and comp that minimize the CV criterion.

## Details

One-dimensional functional predictors can be given either in functional data object form, using argument fdobj (see the fda package of Ramsay, Hooker and Graves, 2009, and Method 1 in the example below), or explicitly, using xfuncs (see Method 2 in the example). In the latter case, arguments basismat and penmat can also be used to specify the basis and/or penalty matrices (see Method 3).

For two-dimensional predictors, functional data object form is not supported. Instead of radial B-splines as in Reiss and Ogden (2010), this implementation employs tensor product cubic B-splines, sometimes known as bivariate O-splines (Ormerod, 2008).

For purposes of interpreting the fitted coefficients, please note that the functional predictors are decorrelated from the scalar predictors before fitting the model (when there are no scalar predictors other than the intercept, this just means the columns of the functional predictor matrix are de-meaned); see section 3.2 of Reiss (2006) for details.

## References

Ormerod, J. T. (2008). On semiparametric regression and data mining. Ph.D. dissertation, School of Mathematics and Statistics, University of New South Wales.

Ramsay, J. O., Hooker, G., and Graves, S. (2009). Functional Data Analysis with R and MATLAB. New York: Springer.

Reiss, P. T. (2006). Regression with signals and images as predictors. Ph.D. dissertation, Department of Biostatistics, Columbia University. Available at https://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.

Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Boca Raton, FL: Chapman & Hall.

## Examples

# NOT RUN {
require(fda)
### 1D functional predictor example ###

######### Octane data example #########
data(gasoline)

# Create the requisite functional data objects
bbasis = create.bspline.basis(c(900, 1700), 40)
wavelengths = 2*450:850
nir <- t(gasoline$NIR) gas.fd = smooth.basisPar(wavelengths, nir, bbasis)$fd

# Method 1: Call fpcr with fdobj argument
gasmod1 = fpcr(gasoline$octane, fdobj = gas.fd, ncomp = 30) plot(gasmod1, xlab="Wavelength") # } # NOT RUN { # Method 2: Call fpcr with explicit signal matrix gasmod2 = fpcr(gasoline$octane, xfuncs = gasoline$NIR, ncomp = 30) # Method 3: Call fpcr with explicit signal, basis, and penalty matrices gasmod3 = fpcr(gasoline$octane, xfuncs = gasoline$NIR, basismat = eval.basis(wavelengths, bbasis), penmat = getbasispenalty(bbasis), ncomp = 30) # Check that all 3 calls yield essentially identical estimates all.equal(gasmod1$fhat, gasmod2$fhat, gasmod3$fhat)
# But note that, in general, you'd have to specify argvals in Method 1
# to get the same coefficient function values as with Methods 2 & 3.
# }
# NOT RUN {
### 2D functional predictor example ###

n = 200; d = 70

# Create true coefficient function
ftrue = matrix(0,d,d)
ftrue[40:46,34:38] = 1

# Generate random functional predictors, and scalar responses
ii = array(rnorm(n*d^2), dim=c(n,d,d))
iimat = ii; dim(iimat) = c(n,d^2)
yy = iimat %*% as.vector(ftrue) + rnorm(n, sd=.3)

mm = fpcr(yy, ii, ncomp=40)

image(ftrue)
contour(mm$fhat, add=TRUE) # } # NOT RUN { ### Cross-validation ### cv.gas = fpcr(gasoline$octane, xfuncs = gasoline$NIR, nbasis=seq(20,40,5), ncomp = seq(10,20,5), store.cv = TRUE) image(seq(20,40,5), seq(10,20,5), cv.gas$cv.table, xlab="Basis size",
ylab="Number of PCs", xaxp=c(20,40,4), yaxp=c(10,20,2))

# }