Functional Regression with functional response using basis representation.

Computes functional regression between functional explanatory variable $X(s)$ and functional response $Y(t)$ using basis representation.

Usage

fregre.basis.fr(
  x,
  y,
  basis.s = NULL,
  basis.t = NULL,
  lambda.s = 0,
  lambda.t = 0,
  Lfdobj.s = vec2Lfd(c(0, 0), range.s),
  Lfdobj.t = vec2Lfd(c(0, 0), range.t),
  weights = NULL,
  ...
)

Arguments

x: Functional explanatory variable.
y: Functional response variable.
basis.s: Basis related with s and it is used in the estimation of $\beta(s,t)$.
basis.t: Basis related with t and it is used in the estimation of $\beta(s,t)$.
lambda.s: A roughness penalty with respect to s to be applied in the estimation of $\beta(s,t)$. By default, no penalty lambda.s=0.
lambda.t: A roughness penalty with respect to t to be applied in the estimation of $\beta(s,t)$. By default, no penalty lambda.t=0.
Lfdobj.s: A linear differential operator object with respect to s . See eval.penalty.
Lfdobj.t: A linear differential operator object with respect to t. See eval.penalty.
weights: Weights.
...: Further arguments passed to or from other methods.

Value

Return:

call: The matched call.
a.est: Intercept parameter estimated.
coefficients: The matrix of the coefficients.
beta.est: A bivariate functional data object of class bifd with the estimated parameters of $\beta(s,t)$.
fitted.values: Estimated response.
residuals: y minus fitted values.
y: Functional response.
x: Functional explanatory data.
lambda.s: A roughness penalty with respect to s.
lambda.t: A roughness penalty with respect to t.
Lfdobj.s: A linear differential operator with respect to s.
Lfdobj.t: A linear differential operator with respect to t.
weights: Weights.

Details

$$Y(t)=\alpha(t)+\int_{T}{X(s)\beta(s,t)ds+\epsilon(t)}$$

where $\alpha(t)$ is the intercept function, $\beta(s,t)$ is the bivariate resgression function and $\epsilon(t)$ are the error term with mean zero.

The function is a wrapped of linmod function proposed by Ramsay and Silverman (2005) to model the relationship between the functional response $Y(t)$ and the functional covariate $X(t)$ by basis representation of both.

The unknown bivariate functional parameter $\beta(s,t)$ can be expressed as a double expansion in terms of $K$ basis function $\nu_k$ and $L$ basis functions $\theta_l$, $$\beta(s,t)=\sum_{k=1}^{K}\sum_{l=1}^{L} b_{kl} \nu_{k}(s)\theta_{l}(t)=\nu(s)^{\top}\bold{B}\theta(t)$$ Then, the model can be re–written in a matrix version as, $$Y(t)=\alpha(t)+\int_{T}{X(s)\nu(s)^{\top}\bold{B}\theta(t)ds+\epsilon(t)}=\alpha(t)+\bold{XB}\theta(t)+\epsilon(t)$$ where $\bold{X}=\int X(s)\nu^{\top}(t)ds$

This function allows objects of class fdata or directly covariates of class fd. If x is a fdata class, basis.s is also the basis used to represent x as fd class object. If y is a fdata class, basis.t is also the basis used to represent y as fd class object. The function also gives default values to arguments basis.s and basis.t for construct the bifd class object used in the estimation of $\beta(s,t)$. If basis.s=NULL or basis.t=NULL the function creates a bspline basis by create.bspline.basis.

fregre.basis.fr incorporates a roughness penalty using an appropiate linear differential operator; lambda.s, Lfdobj.s for penalization of $\beta$'s variations with respect to $s$ and
lambda.t, Lfdobj.t for penalization of $\beta$'s variations with respect to $t$.

References

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@udc.es

Examples

if (FALSE) { # \dontrun{
rtt<-c(0, 365)
basis.alpha  <- create.constant.basis(rtt)
basisx  <- create.bspline.basis(rtt,11)
basisy  <- create.bspline.basis(rtt,11)
basiss  <- create.bspline.basis(rtt,7)
basist  <- create.bspline.basis(rtt,9)

# fd class
dayfd<-Data2fd(day.5,CanadianWeather$dailyAv,basisx)
tempfd<-dayfd[,1]
log10precfd<-dayfd[,3]
res1 <-  fregre.basis.fr(tempfd, log10precfd,
basis.s=basiss,basis.t=basist)

# fdata class
tt<-1:365
tempfdata<-fdata(t(CanadianWeather$dailyAv[,,1]),tt,rtt)
log10precfdata<-fdata(t(CanadianWeather$dailyAv[,,3]),tt,rtt)
res2<-fregre.basis.fr(tempfdata,log10precfdata,
basis.s=basiss,basis.t=basist)

# penalization
Lfdobjt <- Lfdobjs <- vec2Lfd(c(0,0), rtt)
Lfdobjt <- vec2Lfd(c(0,0), rtt)
lambdat<-lambdas <- 100
res1.pen <- fregre.basis.fr(tempfdata,log10precfdata,basis.s=basiss,
basis.t=basist,lambda.s=lambdas,lambda.t=lambdat,
Lfdobj.s=Lfdobjs,Lfdobj.t=Lfdobjt)

res2.pen <- fregre.basis.fr(tempfd, log10precfd,
basis.s=basiss,basis.t=basist,lambda.s=lambdas,
lambda.t=lambdat,Lfdobj.s=Lfdobjs,Lfdobj.t=Lfdobjt)

plot(log10precfd,col=1)
lines(res1$fitted.values,col=2)
plot(res1$residuals)
plot(res1$beta.est,tt,tt)
plot(res1$beta.est,tt,tt,type="persp",theta=45,phi=30)
} # }