Fitting Functional Linear Models
Computes functional regression between functional (and non functional) explanatory variables and scalar response using basis representation.
fregre.lm( formula, data, basis.x = NULL, basis.b = NULL, rn, lambda, weights = rep(1, n), ... )
formula |
an object of class |
data |
List that containing the variables in the model. |
basis.x |
List of basis for functional explanatory data estimation. |
basis.b |
List of basis for functional beta parameter estimation. |
rn |
List of Ridge parameter. |
lambda |
List of Roughness penalty parameter. |
weights |
weights |
... |
Further arguments passed to or from other methods. |
This section is presented as an extension of the linear regression models:
fregre.pc, fregre.pls and
fregre.basis. Now, the scalar response Y is estimated by
more than one functional covariate X^j(t) and also more than one non
functional covariate Z^j. The regression model is given by:
E[Y|X,Z]=α+∑_j β_j Z^j + ∑_k <X^k,β_k>
where Z=[Z^1,...,Z^p] are the non functional covariates, X(t)=[X^1(t),...,X^q(t)] are the functional ones and ε are random errors with mean zero , finite variance σ^2 and E[X(t)ε]=0.
The first item in the data list is called "df" and is a data
frame with the response and non functional explanatory variables, as
lm. Functional covariates of class fdata or fd
are introduced in the following items in the data list.
basis.x is a list of basis for represent each functional covariate.
The basis object can be created by the function:
create.pc.basis, pca.fd
create.pc.basis, create.fdata.basis or
create.basis.basis.b is a list of basis for
represent each functional β_k parameter. If basis.x is a
list of functional principal components basis (see
create.pc.basis or pca.fd) the argument
basis.b (is unnecessary and) is ignored.
The user can penalty the basis elements by: (i) lambda is a list of
rough penalty values for the second derivative of each functional covariate,
see fregre.basis for more details.
(ii) rn is a list
of Ridge penalty value for each functional covariate, see
fregre.pc, fregre.pls and
P.penalty for more details.
Note: For the case of the
Functional Principal Components basis two penalties are allowed (but not the
two together).
Return lm object plus:
sr2 Residual variance.
Vp Estimated covariance matrix for the parameters.
lambda A roughness penalty.
basis.x Basis used for fdata or fd covariates.
basis.b Basis used for beta parameter estimation.
beta.l List of estimated beta parameter of functional covariates.
data List that containing the variables in the model.
formula formula.
Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@usc.es
Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.
Febrero-Bande, M., Oviedo de la Fuente, M. (2012). Statistical Computing in Functional Data Analysis: The R Package fda.usc. Journal of Statistical Software, 51(4), 1-28. http://www.jstatsoft.org/v51/i04/
See Also as: predict.fregre.lm and
summary.lm.
Alternative method: fregre.glm.
data(tecator)
x=tecator$absorp.fdata
y=tecator$y$Fat
tt=x[["argvals"]]
dataf=as.data.frame(tecator$y)
nbasis.x=11
nbasis.b=7
basis1=create.bspline.basis(rangeval=range(tt),nbasis=nbasis.x)
basis2=create.bspline.basis(rangeval=range(tt),nbasis=nbasis.b)
f=Fat~Protein+x
basis.x=list("x"=basis1)
basis.b=list("x"=basis2)
ldata=list("df"=dataf,"x"=x)
res=fregre.lm(f,ldata,basis.x=basis.x,basis.b=basis.b)
summary(res)
f2=Fat~Protein+xd
xd=fdata.deriv(x,nderiv=2,class.out='fdata',nbasis=nbasis.x)
ldata2=list("df"=dataf,"xd"=xd)
basis.x2=list("xd"=basis1)
basis.b2=list("xd"=basis2)
res2=fregre.lm(f2,ldata2,basis.x=basis.x2,basis.b=basis.b2)
summary(res2)
par(mfrow=c(2,1))
plot(res$beta.l$x,main="functional beta estimation")
plot(res2$beta.l$xd,col=2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.