fda.usc (version 1.1.0)

fregre.basis: Functional Regression with scalar response using basis representation.

Description

Computes functional regression between functional explanatory variable $X(t)$ and scalar response $Y$ using basis representation. $$Y=\big+\epsilon=\int_{T}{X(t)\beta(t)dt+\epsilon}$$ 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 $E[X(t)\epsilon]=0$.

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.
• beta.est
  beta coefficient estimated of class fd
• a.est
  Intercept parameter estimated
• fitted.values
  Estimated scalar response.
• H
  Hat matrix.
• residuals
  y minus fitted values.
• df
  The residual degrees of freedom.
• r2
  Coefficient of determination.
• sr2
  Residual variance.
• y
  Response.
• fdataobj
  Functional explanatory data of class fdata.
• x.fd
  Centered functional explanatory data of class fd.
• basis.b
  Basis used for beta parameter estimation.
• lm
  Return lm object

Details

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

See Also

See Also as: fregre.basis.cv, summary.fregre.fd and predict.fregre.fd. Alternative method: fregre.pc and fregre.np.

Examples

# 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)