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fregre.pc {fda.usc}

Functional Regression with scalar response using Principal Components Analysis.
Package: 
fda.usc
Version: 
1.1.0

Description

Computes functional (ridge or penalized) regression between functional explanatory variable X(t) and scalar response Y using Principal Components Analysis.     

Y=

where <.,.> denotes the inner product on L_2 and ε are random errors with mean zero , finite variance σ^2 and E[X(t)ε]=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.

Details

The function computes the ν_1,...,ν_∞ orthonormal basis of functional principal components to represent the functional data as X(t)=∑_(k=1:∞) γ_k ν_k and the functional parameter as β(t)=∑_(k=1:∞) β_k ν_k, where γ_k= < X(t), ν_k > and β_k=<β,ν_k>.

The response can be fitted by:

  • λ=0, no penalization, y.est= ν'(ν'ν)^{-1}ν'y
  • Ridge regression, λ>0 and P=1, y.est=ν'(ν'ν+λ I)^{-1}ν'y
  • Penalized regression, λ>0 and P!=0. For example, P=c(0,0,1) penalizes the second derivative (curvature) by P=P.penalty(fdataobj["argvals"],P), y.est=ν'(ν'ν+λ v'Pv)^{-1}ν'y

Values

Return:

call
The matched call of fregre.pc function.
beta.est
beta coefficient estimated of class fdata
coefficients
A named vector of coefficients.
fitted.values
Estimated scalar response.
residuals
y-fitted values.
H
Hat matrix.
df
The residual degrees of freedom. In ridge regression, df(rn) is the effective degrees of freedom.
r2
Coefficient of determination.
GCV
GCV criterion.
sr2
Residual variance.
l
Index of principal components selected.
lambda
Amount of shrinkage.
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

See Also

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

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)
## End(Not run)

Author(s)

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

Documentation reproduced from package fda.usc, version 1.1.0. License: GPL-2