Fit linear regression with functional responses and scalar predictors, with efficient selection of optimal smoothing parameters.

```
fosr(
formula = NULL,
Y = NULL,
fdobj = NULL,
data = NULL,
X,
con = NULL,
argvals = NULL,
method = c("OLS", "GLS", "mix"),
gam.method = c("REML", "ML", "GCV.Cp", "GACV.Cp", "P-REML", "P-ML"),
cov.method = c("naive", "mod.chol"),
lambda = NULL,
nbasis = 15,
norder = 4,
pen.order = 2,
multi.sp = ifelse(method == "OLS", FALSE, TRUE),
pve = 0.99,
max.iter = 1,
maxlam = NULL,
cv1 = FALSE,
scale = FALSE
)
```

formula

Formula for fitting fosr. If used, data argument must not be null.

Y, fdobj

the functional responses, given as either an \(n\times d\)
matrix `Y`

or a functional data object (class `"fd"`

)
as in the fda package.

data

data frame containing the predictors and responses.

X

the model matrix, whose columns represent scalar predictors. Should ordinarily include a column of 1s.

con

a row vector or matrix of linear contrasts of the coefficient functions, to be constrained to equal zero.

argvals

the \(d\) argument values at which the coefficient functions will be evaluated.

method

estimation method: `"OLS"`

for penalized ordinary least
squares, `"GLS"`

for penalized generalized least squares, `"mix"`

for mixed effect models.

gam.method

smoothing parameter selection method, to be passed to
`gam`

: `"REML"`

for restricted maximum likelihood,
`"GCV.Cp"`

for generalized cross-validation.

cov.method

covariance estimation method: the current options are naive or modified Cholesky. See Details.

lambda

smoothing parameter value. If `NULL`

, the smoothing
parameter(s) will be estimated. See Details.

nbasis, norder

number of basis functions, and order of splines (the
default, 4, gives cubic splines), for the B-spline basis used to represent
the coefficient functions. When the functional responses are supplied using
`fdobj`

, these arguments are ignored in favor of the values pertaining
to the supplied object.

pen.order

order of derivative penalty.

multi.sp

a logical value indicating whether separate smoothing
parameters should be estimated for each coefficient function. Currently
must be `FALSE`

if `method = "OLS"`

.

pve

if `method = 'mix'`

, the percentage of variance explained
by the principal components; defaults to 0.99.

max.iter

maximum number of iterations if `method = "GLS"`

.

maxlam

maximum smoothing parameter value to consider (when
`lamvec=NULL`

; see `lofocv`

).

cv1

logical value indicating whether a cross-validation score should
be computed even if a single fixed `lambda`

is specified (when
`method = "OLS"`

).

scale

logical value or vector determining scaling of the matrix
`X`

(see `scale`

, to which the value of this argument is
passed).

An object of class `fosr`

, which is a list with the following
elements:

object of class `"fd"`

representing the
estimated coefficient functions. Its main components are a basis and a
matrix of coefficients with respect to that basis.

if
`method = "mix"`

, an object representing a functional PCA of the
residuals, performed by `fpca.sc`

if the responses are in raw
form or by `pca.fd`

if in functional-data-object form.

if `method = "mix"`

, an \(n\times m\) matrix of random
effects, where \(m\) is the number of functional PC's needed to explain
proportion `pve`

of the residual variance. These random effects can be
interpreted as shrunken FPC scores.

objects of the same
form as the functional responses (see arguments `Y`

and `fdobj`

),
giving the fitted values and residuals.

matrix of values
of the coefficient function estimates at the points given by
`argvals`

.

matrix of values of the standard error
estimates for the coefficient functions, at the points given by
`argvals`

.

points at which the coefficient functions are evaluated.

fit object outputted by `amc`

.

effective degrees of freedom of the fit.

smoothing parameter, or vector of smoothing parameters.

cross-validated integrated squared error if `method="OLS"`

,
otherwise `NULL`

.

value of the roughness penalty.

`"raw"`

or `"fd"`

, indicating whether the
responses were supplied in raw or functional-data-object form.

The GLS method requires estimating the residual covariance matrix, which
has dimension \(d\times d\) when the responses are given by `Y`

, or
\(nbasis\times nbasis\) when they are given by `fdobj`

. When
`cov.method = "naive"`

, the ordinary sample covariance is used. But
this will be singular, or nonsingular but unstable, in high-dimensional
settings, which are typical. `cov.method = "mod.chol"`

implements the
modified Cholesky method of Pourahmadi (1999) for estimation of covariance
matrices whose inverse is banded. The number of bands is chosen to maximize
the p-value for a sphericity test (Ledoit and Wolf, 2002) applied to the
"prewhitened" residuals. Note, however, that the banded inverse covariance
assumption is sometimes inappropriate, e.g., for periodic functional
responses.

There are three types of values for argument `lambda`

:

if a scalar, this value is used as the smoothing parameter (but only for the initial model, if

`method = "GLS"`

);if a vector, this is used as a grid of values for optimizing the cross-validation score (provided

`method = "OLS"`

; otherwise an error message is issued).

Please note that currently, if `multi.sp = TRUE`

, then `lambda`

must be `NULL`

and `method`

must be `"GLS"`

.

Ledoit, O., and Wolf, M. (2002). Some hypothesis tests for the
covariance matrix when the dimension is large compared to the sample size.
*Annals of Statistics*, 30(4), 1081--1102.

Pourahmadi, M. (1999). Joint mean-covariance models with applications to
longitudinal data: unconstrained parameterisation. *Biometrika*,
86(3), 677--690.

Ramsay, J. O., and Silverman, B. W. (2005). *Functional Data
Analysis*, 2nd ed., Chapter 13. New York: Springer.

Reiss, P. T., Huang, L., and Mennes, M. (2010). Fast function-on-scalar
regression with penalized basis expansions. *International Journal of
Biostatistics*, 6(1), article 28. Available at
https://works.bepress.com/phil_reiss/16/

# NOT RUN { require(fda) # The first two lines, adapted from help(fRegress) in package fda, # set up a functional data object representing daily average # temperatures at 35 sites in Canada daybasis25 <- create.fourier.basis(rangeval=c(0, 365), nbasis=25, axes=list('axesIntervals')) Temp.fd <- with(CanadianWeather, smooth.basisPar(day.5, dailyAv[,,'Temperature.C'], daybasis25)$fd) modmat = cbind(1, model.matrix(~ factor(CanadianWeather$region) - 1)) constraints = matrix(c(0,1,1,1,1), 1) # Penalized OLS with smoothing parameter chosen by grid search olsmod = fosr(fdobj = Temp.fd, X = modmat, con = constraints, method="OLS", lambda=100*10:30) plot(olsmod, 1) # Test use formula to fit fosr set.seed(2121) data1 <- pffrSim(scenario="ff", n=40) formod = fosr(Y~xlin+xsmoo, data=data1) plot(formod, 1) # Penalized GLS glsmod = fosr(fdobj = Temp.fd, X = modmat, con = constraints, method="GLS") plot(glsmod, 1) # }