Implements functional principal component regression (Reiss and Ogden, 2007, 2010) for generalized linear models with scalar responses and functional predictors.
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,
...
)scalar outcome vector.
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 \(d1 \times d2\).
functional data object (class "fd") giving
the functional predictors. Allowed only for 1D functional predictors.
number of principal components. If NULL, this is chosen
by pve.
proportion of variance explained: used to choose the number of principal components.
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.
a \(d \times K\) matrix of values of \(K\) basis functions at the \(d\) sites.
a \(K \times K\) matrix defining a penalty on the basis coefficients.
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.
covariates: an \(n\)-row matrix, or a vector of length \(n\).
logical: should the mean of each subject's signal be included as a covariate?
order of B-splines used, if fdobj is not
supplied; defaults to 4, i.e., cubic B-splines.
generalized linear model family. Current version supports
"gaussian" (the default) and "binomial".
a fixed smoothing parameter; if NULL, an optimal value is
chosen (see method).
order of derivative penalty applied when estimating the
coefficient function; defaults to 2.
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.
the number of validation sets ("folds") into which the data are divided; by default, 5.
logical: should a CV result table be in the output? By
default, FALSE.
logical: should the gam object be
included in the output? Defaults to TRUE.
other arguments passed to function gam.
A list with components
if store.gam = TRUE,
an object of class gam (see gamObject in the
mgcv package documentation).
coefficient function estimate.
pointwise Bayesian standard error.
undecorrelated coefficient for covariates.
the supplied value of argvals.
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.
when CV is performed, the values
of nbasis and comp that minimize the CV criterion.
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.
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.
# 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))
# }