Functional Regression with scalar response using basis representation.

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

Usage

fregre.basis(
  fdataobj,
  y,
  basis.x = NULL,
  basis.b = NULL,
  lambda = 0,
  Lfdobj = vec2Lfd(c(0, 0), rtt),
  weights = rep(1, n),
  ...
)

Arguments

fdataobj: fdata class object.
y: Scalar response with length n.
basis.x: Basis for functional explanatory data fdataobj.
basis.b: Basis for functional beta parameter.
lambda: A roughness penalty. By default, no penalty lambda=0.
Lfdobj: See eval.penalty.
weights: weights
...: Further arguments passed to or from other methods.

Value

Return:

call: The matched call.
coefficients: A named vector of coefficients.
residuals: y minus fitted values.
fitted.values: Estimated scalar response.
beta.est: Estimated beta parameter of class fd.
weights: (only for weighted fits) the specified weights.
df.residual: The residual degrees of freedom.
r2: Coefficient of determination.
sr2: Residual variance.
Vp: Estimated covariance matrix for the parameters.
H: Hat matrix.
y: Response.
fdataobj: Functional explanatory data of class fdata.
a.est: Intercept parameter estimated.
x.fd: Centered functional explanatory data of class fd.
basis.b: Basis used for beta parameter estimation.
lambda.opt: A roughness penalty.
Lfdobj: Order of a derivative or a linear differential operator.
P: Penalty matrix.
lm: Return lm object.

Details

$$Y=\big<X,\beta\big>+\epsilon=\int_{T}{X(t)\beta(t)dt+\epsilon}$$ where $ \big< \cdot , \cdot \big>$ denotes the inner product on $L_2$ and $\epsilon$ are random errors with mean zero, finite variance $\sigma^2$ and $E[X(t)\epsilon]=0$.

The function uses the basis representation proposed by Ramsay and Silverman (2005) to model the relationship between the scalar response and the functional covariate by basis representation of the observed functional data $X(t)\approx\sum_{k=1}^{k_{n1}} c_k \xi_k(t)$ and the unknown functional parameter $\beta(t)\approx\sum_{k=1}^{k_{n2}} b_k \phi_k(t)$.

The functional linear models estimated by the expression: $$\hat{y}= \big< X,\hat{\beta} \big> = C^{T}\psi(t)\phi^{T}(t)\hat{b}=\tilde{X}\hat{b}$$ where $\tilde{X}(t)=C^{T}\psi(t)\phi^{T}(t)$, and $\hat{b}=(\tilde{X}^{T}\tilde{X})^{-1}\tilde{X}^{T}y$ and so, $\hat{y}=\tilde{X}\hat{b}=\tilde{X}(\tilde{X}^{T}\tilde{X})^{-1}\tilde{X}^{T}y=Hy$ where $H$ is the hat matrix with degrees of freedom: $df=tr(H)$.

If $\lambda>0$ then fregre.basis incorporates a roughness penalty:
$\hat{y}=\tilde{X}\hat{b}=\tilde{X}(\tilde{X}^{T}\tilde{X}+\lambda R_0)^{-1}\tilde{X}^{T}y= H_{\lambda}y$ where $R_0$ is the penalty matrix.

This function allows covariates of class fdata, matrix, data.frame or directly covariates of class fd. The function also gives default values to arguments basis.x and basis.b for representation on the basis of functional data $X(t)$ and the functional parameter $\beta(t)$, respectively.

If basis=NULL creates the bspline basis by create.bspline.basis.
If the functional covariate fdataobj is a matrix or data.frame, it creates an object of class "fdata" with default attributes, see fdata.
If basis.x$type=``fourier'' and basis.b$type=``fourier'', the basis are orthonormal and the function decreases the number of fourier basis elements on the $min(k_{n1},k_{n2})$, where $k_{n1}$ and $k_{n2}$ are the number of basis element of basis.x and basis.b respectively.

References

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. https://www.jstatsoft.org/v51/i04/

Author

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

Examples

if (FALSE) { # \dontrun{
# fregre.basis
data(tecator)
names(tecator)
absorp=tecator$absorp.fdata
ind=1:129
x=absorp[ind,]
y=tecator$y$Fat[ind]
tt=absorp[["argvals"]]
res1=fregre.basis(x,y)
summary(res1)
basis1=create.bspline.basis(rangeval=range(tt),nbasis=19)
basis2=create.bspline.basis(rangeval=range(tt),nbasis=9)
res5=fregre.basis(x,y,basis1,basis2)
summary(res5)
x.d2=fdata.deriv(x,nbasis=19,nderiv=1,method="bspline",class.out="fdata")
res7=fregre.basis(x.d2,y,basis1,basis2)
summary(res7)
} # }