fda.usc (version 1.2.3)

fregre.pc: Functional Regression with scalar response using Principal Components Analysis.

Description

Computes functional (ridge or penalized) regression between functional explanatory variable $X(t)$ and scalar response $Y$ using Principal Components Analysis. $$Y=\big+\epsilon=\int_{T}{X(t)\beta(t)dt+\epsilon}$$ 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.pc(fdataobj, y, l =NULL,lambda=0,P=c(1,0,0),
 weights = rep(1, len = n),...)

Arguments

fdataobj

fdata class object or fdata.comp class object created by create.pc.basis function.

y

Scalar response with length n.

l

Index of components to include in the model.If is null l (by default), l=1:3.

lambda

Amount of penalization. Default value is 0, i.e. no penalization is used.

P

If P is a vector: P are coefficients to define the penalty matrix object, see P.penalty. If P is a matrix: P is the penalty matrix object.

weights

weights

...

Further arguments passed to or from other methods.

Value

• Return:
• call
  The matched call of fregre.pc function.
• coefficients
  A named vector of coefficients.
• residuals
  y-fitted values.
• fitted.values
  Estimated scalar response.
• beta.est
  beta coefficient estimated of class fdata
• df
  The residual degrees of freedom. In ridge regression, df(rn) is the effective degrees of freedom.
• r2
  Coefficient of determination.
• sr2
  Residual variance.
• Vp
  Estimated covariance matrix for the parameters.
• H
  Hat matrix.
• l
  Index of principal components selected.
• lambda
  Amount of shrinkage.
• P
  Penalty matrix.
• fdata.comp
  Fitted object in fdata2pc function.
• lm
  lm object.
• fdataobj
  Functional explanatory data.
• y
  Scalar response.

Details

The function computes the $\left{\nu_k\right}_{k=1}^{\infty}$ orthonormal basis of functional principal components to represent the functional data as $X_i(t)=\sum_{k=1}^{\infty}\gamma_{ik}\nu_k$ and the functional parameter as $\beta(t)=\sum_{k=1}^{\infty}\beta_k\nu_k$, where $\gamma_{ik}=\Big< X_i(t),\nu_k\Big>$ and $\beta_{k}=\Big<\beta,\nu_k\big>$. The response can be fitted by:

• $\lambda=0$, no penalization,$$\hat{y}=\nu_k^{\top}(\nu_k^{\top}\nu_k)^{-1}\nu_k^{\top}y$$
• Ridge regression,$\lambda>0$and$P=1$,$$\hat{y}=\nu_k^{\top}(\nu_k\top \nu_k+\lambda I)^{-1}\nu_k^{\top}y$$
• Penalized regression,$\lambda>0$and$P\neq0$. For example,$P=c(0,0,1)$penalizes the second derivative (curvature) byP=P.penalty(fdataobj["argvals"],P),$$\hat{y}=\nu_k^{\top}(\nu_k\top \nu_k+\lambda \nu_k^{\top} \textbf{P}\nu_k)^{-1}\nu_k^{\top}y$$

References

Cai TT, Hall P. 2006. Prediction in functional linear regression. Annals of Statistics 34: 2159{-}2179. Cardot H, Ferraty F, Sarda P. 1999. Functional linear model. Statistics and Probability Letters 45: 11{-}22. Hall P, Hosseini{-}Nasab M. 2006. On properties of functional principal components analysis. Journal of the Royal Statistical Society B 68: 109{-}126. 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/ N. Kraemer, A.-L. Boulsteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94, 60 - 69. http://dx.doi.org/10.1016/j.chemolab.2008.06.009

See Also

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

Examples

data(tecator)
absorp=tecator$absorp.fdata
ind=1:129
x=absorp[ind,]
y=tecator$y$Fat[ind]
res=fregre.pc(x,y)
summary(res)
res2=fregre.pc(x,y,l=c(1,3,4))
summary(res2)
# Functional Ridge Regression
res3=fregre.pc(x,y,l=c(1,3,4),lambda=1,P=1)
summary(res3)
# Functional Regression with 2nd derivative penalization
res4=fregre.pc(x,y,l=c(1,3,4),lambda=1,P=c(0,0,1))
summary(res4)
betas<-c(res$beta.est,res2$beta.est,res3$beta.est,res4$beta.est)
plot(betas)