fRegress {fda} | R Documentation |
This function carries out a functional regression analysis, where
either the dependent variable or one or more independent variables are
functional. Non-functional variables may be used on either side
of the equation. In a simple problem where there is a single scalar
independent covariate with values z_i, i=1,...,N and a single
functional covariate with values x_i(t), the two versions of the
model fit by fRegress
are the scalar dependent variable
model
y_i = β_1 z_i + int x_i(t) β_2(t) , dt + e_i
and the concurrent functional dependent variable model
y_i(t) = β_1(t) z_i + β_2(t) x_i(t) + e_i(t).
In these models, the final term e_i or e_i(t) is a residual, lack of fit or error term.
In the concurrent functional linear model for a functional dependent variable, all functional variables are all evaluated at a common time or argument value $t$. That is, the fit is defined in terms of the behavior of all variables at a fixed time, or in terms of "now" behavior.
All regression coefficient functions β_j(t) are considered to be functional. In the case of a scalar dependent variable, the regression coefficient for a scalar covariate is converted to a functional variable with a constant basis. All regression coefficient functions can be forced to be smooth through the use of roughness penalties, and consequently are specified in the argument list as functional parameter objects.
fRegress(y, ...) ## S3 method for class 'formula': fRegress(y, data=NULL, betalist=NULL, wt=NULL, y2cMap=NULL, SigmaE=NULL, method=c('fRegress', 'model'), sep='.', ...) ## S3 method for class 'character': fRegress(y, data=NULL, betalist=NULL, wt=NULL, y2cMap=NULL, SigmaE=NULL, method=c('fRegress', 'model'), sep='.', ...) ## S3 method for class 'fd': fRegress(y, xfdlist, betalist, wt=NULL, y2cMap=NULL, SigmaE=NULL, ...) ## S3 method for class 'fdPar': fRegress(y, xfdlist, betalist, wt=NULL, y2cMap=NULL, SigmaE=NULL, ...) ## S3 method for class 'numeric': fRegress(y, xfdlist, betalist, wt=NULL, y2cMap=NULL, SigmaE=NULL, ...)
y |
the dependent variable object. It may be an object of five
possible classes:
|
data |
an optional list or data.frame containing names of
objects identified in the formula or character
y .
|
xfdlist |
a list of length equal to the number of independent
variables (including any intercept). Members of this list are the
independent variables. They can be objects of either of these two
classes:
xfdlist .)
|
betalist |
For the fd , fdPar , and numeric methods,
betalist must be a list of length equal to
length(xfdlist) . Members of this list are functional
parameter objects (class fdPar ) defining the regression
functions to be estimated. Even if a corresponding independent
variable is scalar, its regression coefficient must be functional if
the dependent variable is functional. (If the dependent variable is
a scalar, the coefficients of scalar independent variables,
including the intercept, must be constants, but the coefficients of
functional independent variables must be functional.) Each of these
functional parameter objects defines a single functional data
object, that is, with only one replication.
For the formula and character methods, betalist
can be either a list , as for the other methods, or
NULL , in which case a list is created. If betalist is
created, it will use the bases from the corresponding component of
xfdlist if it is function or from the response variable.
Smoothing information (arguments Lfdobj , lambda ,
estimate , and penmat of function fdPar ) will
come from the corresponding component of xfdlist if it is of
class fdPar (or for scalar independent variables from the
response variable if it is of class fdPar ) or from optional
... arguments if the reference variable is not of class
fdPar .
|
wt |
weights for weighted least squares |
y2cMap |
the matrix mapping from the vector of observed values to the
coefficients for the dependent variable. This is output by function
smooth.basis . If this is supplied, confidence limits are
computed, otherwise not.
|
SigmaE |
Estimate of the covariances among the residuals. This can only be
estimated after a preliminary analysis with fRegress .
|
method |
a character string matching either fRegress for functional
regression estimation or mode to create the argument lists
for functional regression estimation without running it.
|
sep |
separator for creating names for multiple variables for
fRegress.fdPar or fRegress.numeric created from single
variables on the right hand side of the formula y .
This happens with multidimensional fd objects as well as with
categorical variables.
|
... |
optional arguments |
Alternative forms of functional regression can be categorized with traditional least squares using the following 2 x 2 table:
explanatory | variable | |||
response | | | scalar | | | function |
| | | | |||
scalar | | | lm | | | fRegress.numeric |
| | | | |||
function | | | fRegress.fd or | | | fRegress.fd or |
| | fRegress.fdPar | | | fRegress.fdPar or linmod |
For fRegress.numeric
, the numeric response is assumed to be the
sum of integrals of xfd * beta for all functional xfd terms.
fRegress.fd or .fdPar
produces a concurrent regression with
each beta
being also a (univariate) function.
linmod
predicts a functional response from a convolution
integral, estimating a bivariate regression function.
In the computation of regression function estimates in
fRegress
, all independent variables are treated as if they are
functional. If argument xfdlist
contains one or more vectors,
these are converted to functional data objects having the constant
basis with coefficients equal to the elements of the vector.
Needless to say, if all the variables in the model are scalar, do NOT
use this function. Instead, use either lm
or lsfit
.
These functions provide a partial implementation of Ramsay and Silverman (2005, chapters 12-20).
These functions return either a standard fRegress
fit object or
or a model specification:
fRegress fit |
a list of class fRegress with the following components:
If class(y) is numeric, the fRegress object also includes:
If class(y) is either fd or fdPar , the
fRegress object returned also includes 5 other components:
|
model specification |
The fRegress.formula and fRegress.character
functions translate the formula into the argument list
required by fRegress.fdPar or fRegress.numeric .
With the default value 'fRegress' for the argument method ,
this list is then used to call the appropriate other
fRegress function.
Alternatively, to see how the formula is translated, use
the alternative 'model' value for the argument method . In
that case, the function returns a list with the arguments
otherwise passed to these other functions plus the following
additional components:
|
J. O. Ramsay, Giles Hooker, and Spencer Graves
Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.
fRegress.formula
,
fRegress.stderr
,
fRegress.CV
,
linmod
### ### ### scalar response and explanatory variable ### ... to compare fRegress and lm ### ### # example from help('lm') ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14) trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69) group <- gl(2,10,20, labels=c("Ctl","Trt")) weight <- c(ctl, trt) lm.D9 <- lm(weight ~ group) fRegress.D9 <- fRegress(weight ~ group) (lm.D9.coef <- coef(lm.D9)) (fRegress.D9.coef <- sapply(fRegress.D9$betaestlist, coef)) all.equal(as.numeric(lm.D9.coef), as.numeric(fRegress.D9.coef)) ### ### ### vector response with functional explanatory variable ### ### ## ## set up ## annualprec <- log10(apply(CanadianWeather$dailyAv[,,"Precipitation.mm"], 2,sum)) # The simplest 'fRegress' call is singular with more bases # than observations, so we use a small basis for this example smallbasis <- create.fourier.basis(c(0, 365), 25) # There are other ways to handle this, # but we will not discuss them here tempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], smallbasis)$fd ## ## formula interface ## precip.Temp.f <- fRegress(annualprec ~ tempfd) ## ## Get the default setup and modify it ## precip.Temp.mdl <- fRegress(annualprec ~ tempfd, method='m') # set up a smaller basis than for temperature nbetabasis <- 21 betabasis2. <- create.fourier.basis(c(0, 365), nbetabasis) betafd2. <- fd(rep(0, nbetabasis), betabasis2.) # add smoothing betafdPar2. <- fdPar(betafd2., lambda=10) # Now do it. precip.Temp.m <- do.call('fRegress', precip.Temp.mdl) # With the change in betalist, the answers may be different # Without that change, the answes will be the same. all.equal(precip.Temp.m, precip.Temp.f) ## ## Manual construction of xfdlist and betalist ## xfdlist <- list(const=rep(1, 35), tempfd=tempfd) # The intercept must be constant for a scalar response betabasis1 <- create.constant.basis(c(0, 365)) betafd1 <- fd(0, betabasis1) betafdPar1 <- fdPar(betafd1) betafd2 <- with(tempfd, fd(basisobj=basis, fdnames=fdnames)) # convert to an fdPar object betafdPar2 <- fdPar(betafd2) betalist <- list(const=betafdPar1, tempfd=betafdPar2) precip.Temp <- fRegress(annualprec, xfdlist, betalist) all.equal(precip.Temp, precip.Temp.f) ### ### ### functional response with vector explanatory variables ### ### ## ## simplest: formula interface ## daybasis65 <- create.fourier.basis(rangeval=c(0, 365), nbasis=65, axes=list('axesIntervals')) Temp.fd <- with(CanadianWeather, smooth.basisPar(day.5, dailyAv[,,'Temperature.C'], daybasis65)$fd) TempRgn.f <- fRegress(Temp.fd ~ region, CanadianWeather) ## ## Get the default setup and possibly modify it ## TempRgn.mdl <- fRegress(Temp.fd ~ region, CanadianWeather, method='m') # make desired modifications here # then run TempRgn.m <- do.call('fRegress', TempRgn.mdl) # no change, so match the first run all.equal(TempRgn.m, TempRgn.f) ## ## More detailed set up ## region.contrasts <- model.matrix(~factor(CanadianWeather$region)) rgnContr3 <- region.contrasts dim(rgnContr3) <- c(1, 35, 4) dimnames(rgnContr3) <- list('', CanadianWeather$place, c('const', paste('region', c('Atlantic', 'Continental', 'Pacific'), sep='.')) ) const365 <- create.constant.basis(c(0, 365)) region.fd.Atlantic <- fd(matrix(rgnContr3[,,2], 1), const365) region.fd.Continental <- fd(matrix(rgnContr3[,,3], 1), const365) region.fd.Pacific <- fd(matrix(rgnContr3[,,4], 1), const365) region.fdlist <- list(const=rep(1, 35), region.Atlantic=region.fd.Atlantic, region.Continental=region.fd.Continental, region.Pacific=region.fd.Pacific) beta1 <- with(Temp.fd, fd(basisobj=basis, fdnames=fdnames)) beta0 <- fdPar(beta1) betalist <- list(const=beta0, region.Atlantic=beta0, region.Continental=beta0, region.Pacific=beta0) TempRgn <- fRegress(Temp.fd, region.fdlist, betalist) all.equal(TempRgn, TempRgn.f) ### ### ### functional response with ### (concurrent) functional explanatory variable ### ### ## ## predict knee angle from hip angle; from demo('gait', package='fda') ## ## formula interface ## (gaittime <- as.numeric(dimnames(gait)[[1]])*20) gaitrange <- c(0,20) gaitbasis <- create.fourier.basis(gaitrange, nbasis=21) harmaccelLfd <- vec2Lfd(c(0, (2*pi/20)^2, 0), rangeval=gaitrange) gaitfd <- smooth.basisPar(gaittime, gait, gaitbasis, Lfdobj=harmaccelLfd, lambda=1e-2)$fd hipfd <- gaitfd[,1] kneefd <- gaitfd[,2] knee.hip.f <- fRegress(kneefd ~ hipfd) ## ## manual set-up ## # set up the list of covariate objects const <- rep(1, dim(kneefd$coef)[2]) xfdlist <- list(const=const, hipfd=hipfd) beta0 <- with(kneefd, fd(basisobj=basis, fdnames=fdnames)) beta1 <- with(hipfd, fd(basisobj=basis, fdnames=fdnames)) betalist <- list(const=fdPar(beta0), hipfd=fdPar(beta1)) fRegressout <- fRegress(kneefd, xfdlist, betalist) all.equal(fRegressout, knee.hip.f) #See also the following demos: #demo('canadian-weather', package='fda') #demo('gait', package='fda') #demo('refinery', package='fda') #demo('weatherANOVA', package='fda') #demo('weatherlm', package='fda')