fRegress: Functional Regression Analysis
In fda: Functional Data Analysis
fRegress R Documentation
Functional Regression Analysis
Description
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,\ldots,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 = \beta_1 z_i + \int x_i(t) \beta_2(t) \, dt + e_i
and the concurrent functional dependent variable model
y_i(t) = \beta_1(t) z_i + \beta_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 \beta_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.
Usage
fRegress(y, ...)
## S3 method for class 'fd'
fRegress(y, xfdlist, betalist, wt=NULL,
y2cMap=NULL, SigmaE=NULL, returnMatrix=FALSE,
method=c('fRegress', 'model'), sep='.', ...)
## S3 method for class 'double'
fRegress(y, xfdlist, betalist, wt=NULL,
y2cMap=NULL, SigmaE=NULL, returnMatrix=FALSE, ...)
## 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='.', ...)
Arguments
y
the dependent variable object. It may be an object of five
possible classes or attributes:
character or formula
a formula object or a character object that can be
coerced into a formula providing a symbolic description
of the model to be fitted satisfying the following rules:
The left hand side, formula y, must be either a
numeric vector or a univariate object of class fd.
All objects named on the right hand side must be either
numeric or fd (functional data).
The number of replications of fd
object(s) must match each other and the number of observations
of numeric objects named, as well as the number of
replications of the dependent variable object. The right hand
side of this formula is translated into xfdlist,
then passed to another method for fitting (unless method
= 'model'). Multivariate independent variables are allowed in a
formula and are split into univariate independent
variables in the resulting xfdlist. Similarly,
categorical independent variables with k levels are
translated into k-1 contrasts in xfdlist. Any
smoothing information is passed to the corresponding component
of betalist.
numeric
a numeric vector object or a matrix object if the dependent variable
is numeric or a matrix.
fd
a functional data object or an fdPar object if the dependent variable is
functional.
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:
scalar
a numeric vector if the independent variable is scalar.
fd
a (univariate) functional data object.
In either case, the object must have the same number of replications
as the dependent variable object. That is, if it is a scalar, it
must be of the same length as the dependent variable, and if it is
functional, it must have the same number of replications as the
dependent variable. (Only univariate independent variables are
currently allowed in 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.
returnMatrix
logical: If TRUE, a two-dimensional is returned using a
special class from the Matrix package.
...
optional arguments
Details
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).
Value
These functions return either a standard fRegress fit object or
or a model specification:
- The
fRegress fit object case: A list of class
fRegress with the following components:
y: The first argument in the call to fRegress.
This argument is coerced to class fd in fda version 5.1.9.
Prior versions of the package converted it to an fdPar, but the
extra structures in that class were not used in any of the
fRegress codes.
xfdlist: The second argument in the call to fRegress.
betalist: The third argument in the call to fRegress.
betaestlist: A list of length equal to the number of independent
variables and with members having the same functional parameter
structure as the corresponding members of betalist. These are the
estimated regression coefficient functions.
yhatfdobj: A functional parameter object (class fdPar)
if the dependent variable is functional or a vector if the dependent
variable is scalar. This is the set of predicted by the
functional regression model for the dependent variable.
Cmatinv: A matrix containing the inverse of the coefficient
matrix for the linear equations that define the solution to the
regression problem. This matrix is required for function
fRegress.stderr that estimates confidence regions
for the regression coefficient function estimates.
wt: The vector of weights input or inferred.
If class(y) is numeric, the fRegress object also includes:
df: The equivalent degrees of freedom for the fit.
OCV the leave-one-out cross validation score for the model.
gcv: The generalized cross validation score.
If class(y) is fd or fdPar, the
fRegress object returned also includes 5 other components:
y2cMap: An input y2cMap.
SigmaE: An input SigmaE.
betastderrlist: An fd object estimating the standard
errors of betaestlist.
bvar: A covariance matrix for regression coefficient estimates.
c2bMap: A mapping matrix that maps variation in Cmat to variation
in regression coefficients.
- The model specification object case:
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:
xfdlist0: A list of the objects named on the right hand side of
formula. This will differ from xfdlist for
any categorical or multivariate right hand side object.
type: the type component of any fd object on the
right hand side of formula.
nbasis: A vector containing the nbasis components of
variables named in formula having such components.
xVars: An integer vector with all the variable names on the right
hand side of formula containing the corresponding
number of variables in xfdlist. This can exceed 1 for
any multivariate object on the right hand side of class either
numeric or fd as well as any categorical variable.
Author(s)
J. O. Ramsay, Giles Hooker, and Spencer Graves
References
Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009)
Functional Data Analysis in R and Matlab, Springer,
New York.
Ramsay, James O., and Silverman, Bernard W. (2005),
Functional Data Analysis, 2nd ed., Springer,
New York.
See Also
fRegress.stderr,
fRegress.CV,
linmod
Examples
oldpar <- par(no.readonly=TRUE)
###
###
### vector response with functional explanatory variable
###
###
# data are in Canadian Weather object
# print the names of the data
print(names(CanadianWeather))
# set up log10 of annual precipitation for 35 weather stations
annualprec <-
log10(apply(CanadianWeather$dailyAv[,,"Precipitation.mm"], 2,sum))
# The simplest 'fRegress' call is singular with more bases
# than observations, so we use only 25 basis functions, for this example
smallbasis <- create.fourier.basis(c(0, 365), 25)
# The covariate is the temperature curve for each station.
tempfd <- smooth.basis(day.5,
CanadianWeather$dailyAv[,,"Temperature.C"], smallbasis)$fd
##
## formula interface: specify the model by a formula, the method
## fRegress.formula automatically sets up the regression coefficient functions,
## a constant function for the intercept,
## and a higher dimensional function
## for the inner product with temperature
##
precip.Temp1 <- fRegress(annualprec ~ tempfd)
# the output is a list with class name fRegress, display names
names(precip.Temp1)
# the vector of fits to the data is object precip.Temp1$yfdPar,
# but since the dependent variable is a vector, so is the fit
annualprec.fit1 <- precip.Temp1$yhatfdobj
# plot the data and the fit
plot(annualprec.fit1, annualprec, type="p", pch="o")
lines(annualprec.fit1, annualprec.fit1, lty=2)
# print root mean squared error
RMSE <- sqrt(mean((annualprec-annualprec.fit1)^2))
print(paste("RMSE =",RMSE))
# plot the estimated regression function
plot(precip.Temp1$betaestlist[[2]])
# This isn't helpful either, the coefficient function is too
# complicated to interpret.
# display the number of basis functions used:
print(precip.Temp1$betaestlist[[2]]$fd$basis$nbasis)
# 25 basis functions to fit 35 values, no wonder we over-fit the data
##
## Get the default setup and modify it
## the "model" value of the method argument causes the analysis
## to produce a list vector of arguments for calling the
## fRegress function
##
precip.Temp.mdl1 <- fRegress(annualprec ~ tempfd, method="model")
# First confirm we get the same answer as above by calling
# function fRegress() with these arguments:
precip.Temp.m <- do.call('fRegress', precip.Temp.mdl1)
all.equal(precip.Temp.m, precip.Temp1)
# set up a smaller basis for beta2 than for temperature so that we
# get a more parsimonious fit to the data
nbetabasis2 <- 21 # not much less, but we add some roughness penalization
betabasis2 <- create.fourier.basis(c(0, 365), nbetabasis2)
betafd2 <- fd(rep(0, nbetabasis2), betabasis2)
# add smoothing
betafdPar2 <- fdPar(betafd2, lambda=10)
# replace the regress coefficient function with this fdPar object
precip.Temp.mdl2 <- precip.Temp.mdl1
precip.Temp.mdl2[['betalist']][['tempfd']] <- betafdPar2
# Now do re-fit the data
precip.Temp2 <- do.call('fRegress', precip.Temp.mdl2)
# Compare the two fits:
# degrees of freedom
precip.Temp1[['df']] # 26
precip.Temp2[['df']] # 8
# root-mean-squared errors:
RMSE1 <- sqrt(mean(with(precip.Temp1, (yhatfdobj-yfdobj)^2)))
RMSE2 <- sqrt(mean(with(precip.Temp2, (yhatfdobj-yfdobj)^2)))
# display further results for the more parsimonious model
annualprec.fit2 <- precip.Temp2$yhatfdobj
plot(annualprec.fit2, annualprec, type="p", pch="o")
lines(annualprec.fit2, annualprec.fit2, lty=2)
# plot the estimated regression function
plot(precip.Temp2$betaestlist[[2]])
# now we see that it is primarily the temperatures in the
# early winter that provide the fit to log precipitation by temperature
##
## 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 <- create.bspline.basis(c(0, 365),7)
# convert to an fdPar object
betafdPar2 <- fdPar(betafd2)
betalist <- list(const=betafdPar1, tempfd=betafdPar2)
precip.Temp3 <- fRegress(annualprec, xfdlist, betalist)
annualprec.fit3 <- precip.Temp3$yhatfdobj
# plot the data and the fit
plot(annualprec.fit3, annualprec, type="p", pch="o")
lines(annualprec.fit3, annualprec.fit3)
plot(precip.Temp3$betaestlist[[2]])
###
###
### 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$place, c('const',
paste('region', c('Atlantic', 'Continental', 'Pacific'), sep='.')) )
const365 <- create.constant.basis(c(0, 365))
region.fd.Atlantic <- fd(matrix(rgnContr3[,,2], 1), const365)
# str(region.fd.Atlantic)
region.fd.Continental <- fd(matrix(rgnContr3[,,3], 1), const365)
region.fd.Pacific <- fd(matrix(rgnContr3[,,4], 1), const365)
region.fdlist <- list(const=rep(1, 35),
region.Atlantic=region.fd.Atlantic,
region.Continental=region.fd.Continental,
region.Pacific=region.fd.Pacific)
# str(TempRgn.mdl$betalist)
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 functional explanatory variable
###
###
##
## predict knee angle from hip angle;
## from demo('gait', package='fda')
##
## formula interface
##
gaittime <- as.matrix((1:20)/21)
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)
par(oldpar)
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| fRegress | R Documentation |
Functional Regression Analysis
Description
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,\ldots,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 = \beta_1 z_i + \int x_i(t) \beta_2(t) \, dt + e_i
and the concurrent functional dependent variable model
y_i(t) = \beta_1(t) z_i + \beta_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 \beta_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.
Usage
fRegress(y, ...)
## S3 method for class 'fd'
fRegress(y, xfdlist, betalist, wt=NULL,
y2cMap=NULL, SigmaE=NULL, returnMatrix=FALSE,
method=c('fRegress', 'model'), sep='.', ...)
## S3 method for class 'double'
fRegress(y, xfdlist, betalist, wt=NULL,
y2cMap=NULL, SigmaE=NULL, returnMatrix=FALSE, ...)
## 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='.', ...)
Arguments
y |
the dependent variable object. It may be an object of five possible classes or attributes:
|
data |
an optional |
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:
In either case, the object must have the same number of replications
as the dependent variable object. That is, if it is a scalar, it
must be of the same length as the dependent variable, and if it is
functional, it must have the same number of replications as the
dependent variable. (Only univariate independent variables are
currently allowed in |
betalist |
For the For the |
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
|
SigmaE |
Estimate of the covariances among the residuals. This can only be
estimated after a preliminary analysis with |
method |
a character string matching either |
sep |
separator for creating names for multiple variables for
|
returnMatrix |
logical: If TRUE, a two-dimensional is returned using a special class from the Matrix package. |
... |
optional arguments |
Details
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).
Value
These functions return either a standard fRegress fit object or
or a model specification:
- The
fRegressfit object case: A list of class
fRegresswith the following components:y: The first argument in the call to
fRegress. This argument is coerced toclassfdin fda version 5.1.9. Prior versions of the package converted it to anfdPar, but the extra structures in that class were not used in any of thefRegresscodes.xfdlist: The second argument in the call to
fRegress.betalist: The third argument in the call to
fRegress.betaestlist: A list of length equal to the number of independent variables and with members having the same functional parameter structure as the corresponding members of
betalist. These are the estimated regression coefficient functions.yhatfdobj: A functional parameter object (class
fdPar) if the dependent variable is functional or a vector if the dependent variable is scalar. This is the set of predicted by the functional regression model for the dependent variable.Cmatinv: A matrix containing the inverse of the coefficient matrix for the linear equations that define the solution to the regression problem. This matrix is required for function
fRegress.stderrthat estimates confidence regions for the regression coefficient function estimates.wt: The vector of weights input or inferred.
If
class(y)is numeric, thefRegressobject also includes:df: The equivalent degrees of freedom for the fit.
OCV the leave-one-out cross validation score for the model.
gcv: The generalized cross validation score.
If
class(y)isfdorfdPar, thefRegressobject returned also includes 5 other components:y2cMap: An input
y2cMap.SigmaE: An input
SigmaE.betastderrlist: An
fdobject estimating the standard errors ofbetaestlist.bvar: A covariance matrix for regression coefficient estimates.
c2bMap: A mapping matrix that maps variation in Cmat to variation in regression coefficients.
- The model specification object case:
The
fRegress.formulaandfRegress.characterfunctions translate theformulainto the argument list required byfRegress.fdParorfRegress.numeric. With the default value 'fRegress' for the argumentmethod, this list is then used to call the appropriate otherfRegressfunction.
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:
xfdlist0: A list of the objects named on the right hand side of
formula. This will differ fromxfdlistfor any categorical or multivariate right hand side object.type: the
typecomponent of anyfdobject on the right hand side offormula.nbasis: A vector containing the
nbasiscomponents of variables named informulahaving such components.xVars: An integer vector with all the variable names on the right hand side of
formulacontaining the corresponding number of variables inxfdlist. This can exceed 1 for any multivariate object on the right hand side of class eithernumericorfdas well as any categorical variable.
Author(s)
J. O. Ramsay, Giles Hooker, and Spencer Graves
References
Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009) Functional Data Analysis in R and Matlab, Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.
See Also
fRegress.stderr,
fRegress.CV,
linmod
Examples
oldpar <- par(no.readonly=TRUE)
###
###
### vector response with functional explanatory variable
###
###
# data are in Canadian Weather object
# print the names of the data
print(names(CanadianWeather))
# set up log10 of annual precipitation for 35 weather stations
annualprec <-
log10(apply(CanadianWeather$dailyAv[,,"Precipitation.mm"], 2,sum))
# The simplest 'fRegress' call is singular with more bases
# than observations, so we use only 25 basis functions, for this example
smallbasis <- create.fourier.basis(c(0, 365), 25)
# The covariate is the temperature curve for each station.
tempfd <- smooth.basis(day.5,
CanadianWeather$dailyAv[,,"Temperature.C"], smallbasis)$fd
##
## formula interface: specify the model by a formula, the method
## fRegress.formula automatically sets up the regression coefficient functions,
## a constant function for the intercept,
## and a higher dimensional function
## for the inner product with temperature
##
precip.Temp1 <- fRegress(annualprec ~ tempfd)
# the output is a list with class name fRegress, display names
names(precip.Temp1)
# the vector of fits to the data is object precip.Temp1$yfdPar,
# but since the dependent variable is a vector, so is the fit
annualprec.fit1 <- precip.Temp1$yhatfdobj
# plot the data and the fit
plot(annualprec.fit1, annualprec, type="p", pch="o")
lines(annualprec.fit1, annualprec.fit1, lty=2)
# print root mean squared error
RMSE <- sqrt(mean((annualprec-annualprec.fit1)^2))
print(paste("RMSE =",RMSE))
# plot the estimated regression function
plot(precip.Temp1$betaestlist[[2]])
# This isn't helpful either, the coefficient function is too
# complicated to interpret.
# display the number of basis functions used:
print(precip.Temp1$betaestlist[[2]]$fd$basis$nbasis)
# 25 basis functions to fit 35 values, no wonder we over-fit the data
##
## Get the default setup and modify it
## the "model" value of the method argument causes the analysis
## to produce a list vector of arguments for calling the
## fRegress function
##
precip.Temp.mdl1 <- fRegress(annualprec ~ tempfd, method="model")
# First confirm we get the same answer as above by calling
# function fRegress() with these arguments:
precip.Temp.m <- do.call('fRegress', precip.Temp.mdl1)
all.equal(precip.Temp.m, precip.Temp1)
# set up a smaller basis for beta2 than for temperature so that we
# get a more parsimonious fit to the data
nbetabasis2 <- 21 # not much less, but we add some roughness penalization
betabasis2 <- create.fourier.basis(c(0, 365), nbetabasis2)
betafd2 <- fd(rep(0, nbetabasis2), betabasis2)
# add smoothing
betafdPar2 <- fdPar(betafd2, lambda=10)
# replace the regress coefficient function with this fdPar object
precip.Temp.mdl2 <- precip.Temp.mdl1
precip.Temp.mdl2[['betalist']][['tempfd']] <- betafdPar2
# Now do re-fit the data
precip.Temp2 <- do.call('fRegress', precip.Temp.mdl2)
# Compare the two fits:
# degrees of freedom
precip.Temp1[['df']] # 26
precip.Temp2[['df']] # 8
# root-mean-squared errors:
RMSE1 <- sqrt(mean(with(precip.Temp1, (yhatfdobj-yfdobj)^2)))
RMSE2 <- sqrt(mean(with(precip.Temp2, (yhatfdobj-yfdobj)^2)))
# display further results for the more parsimonious model
annualprec.fit2 <- precip.Temp2$yhatfdobj
plot(annualprec.fit2, annualprec, type="p", pch="o")
lines(annualprec.fit2, annualprec.fit2, lty=2)
# plot the estimated regression function
plot(precip.Temp2$betaestlist[[2]])
# now we see that it is primarily the temperatures in the
# early winter that provide the fit to log precipitation by temperature
##
## 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 <- create.bspline.basis(c(0, 365),7)
# convert to an fdPar object
betafdPar2 <- fdPar(betafd2)
betalist <- list(const=betafdPar1, tempfd=betafdPar2)
precip.Temp3 <- fRegress(annualprec, xfdlist, betalist)
annualprec.fit3 <- precip.Temp3$yhatfdobj
# plot the data and the fit
plot(annualprec.fit3, annualprec, type="p", pch="o")
lines(annualprec.fit3, annualprec.fit3)
plot(precip.Temp3$betaestlist[[2]])
###
###
### 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$place, c('const',
paste('region', c('Atlantic', 'Continental', 'Pacific'), sep='.')) )
const365 <- create.constant.basis(c(0, 365))
region.fd.Atlantic <- fd(matrix(rgnContr3[,,2], 1), const365)
# str(region.fd.Atlantic)
region.fd.Continental <- fd(matrix(rgnContr3[,,3], 1), const365)
region.fd.Pacific <- fd(matrix(rgnContr3[,,4], 1), const365)
region.fdlist <- list(const=rep(1, 35),
region.Atlantic=region.fd.Atlantic,
region.Continental=region.fd.Continental,
region.Pacific=region.fd.Pacific)
# str(TempRgn.mdl$betalist)
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 functional explanatory variable
###
###
##
## predict knee angle from hip angle;
## from demo('gait', package='fda')
##
## formula interface
##
gaittime <- as.matrix((1:20)/21)
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)
par(oldpar)
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