fregre.pc

From fda.usc v2.0.1 by Manuel Oviedo de la Fuente

Functional Regression with scalar response using Principal Components Analysis

Computes functional (ridge or penalized) regression between functional explanatory variable $X(t)$ and scalar response $Y$ using Principal Components Analysis. $$Y=\big<X,\beta\big>+\epsilon=\int_{T}{X(t)\beta(t)dt+\epsilon}$$ 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 $E[X(t)\epsilon]=0$.

Keywords: regression

Usage

fregre.pc(
  fdataobj,
  y,
  l = NULL,
  lambda = 0,
  P = c(0, 0, 1),
  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.

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) by P=P.penalty(fdataobj["argvals"],P), $$\hat{y}=\nu_k^{\top}(\nu_k\top \nu_k+\lambda \nu_k^{\top} \textbf{P}\nu_k)^{-1}\nu_k^{\top}y$$

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.

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

Examples

# NOT RUN {
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)
# }
# NOT RUN {
# }

Documentation reproduced from package fda.usc, version 2.0.1, License: GPL-2

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