R: Functional Penalized PLS regression with scalar response
fregre.pls
R Documentation
Functional Penalized PLS regression with scalar response
Description
Computes functional linear regression between functional explanatory variable X(t) and scalar response Y using penalized Partial Least Squares (PLS)
Y=<\tilde{X},β>+ε
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.pls(fdataobj, y=NULL, l = NULL,lambda=0,P=c(0,0,1),...)
Arguments
fdataobj
fdata class object.
y
Scalar response with length n.
l
Index of components to include in the model.
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. By default P=c(0,0,1) penalize the second derivative (curvature) or acceleration.
If P is a matrix: P is the penalty matrix object.
...
Further arguments passed to or from other methods.
Details
Functional (FPLS) algorithm maximizes the covariance between X(t) and the scalar response Y via the partial least squares (PLS) components. The functional penalized PLS are calculated in fdata2pls by alternative formulation of the NIPALS algorithm proposed by Kraemer and Sugiyama (2011).
Let {ν_k}_k=1:∞ the functional PLS components and X_i(t)=∑{k=1:∞} γ_{ik} ν_k and β(t)=∑{k=1:∞} β_k ν_k. The functional linear model is estimated by:
y.est=< X,β.est > approx ∑{k=1:k_n} γ_k β_k
The response can be fitted by:
λ=0, no penalization,
y.est= ν'(ν'ν)^{-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),
Preda C. and Saporta G. PLS regression on a stochastic process. Comput. Statist. Data Anal. 48 (2005): 149-158.
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
Martens, H., Naes, T. (1989) Multivariate calibration. Chichester: Wiley.
Kraemer, N., Sugiyama M. (2011). The Degrees of Freedom of Partial Least Squares Regression. Journal of the American Statistical Association. Volume 106, 697-705.
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: P.penalty and fregre.pls.cv.
Alternative method: fregre.pc.