fda.usc: fregre.pls – R documentation

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fregre.pls

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.
ν_1,...,ν_∞ orthonormal basis of PLS to represent the functional data as X(t)=∑_(k=1:∞) γ_k ν_k.

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),
  
  y.est=ν'(ν'ν+λ v'Pv)^{-1}ν'y

Value

Return:

call The matched call of fregre.pls function.
beta.est Beta coefficient estimated of class fdata.
coefficients A named vector of coefficients.
fitted.values Estimated scalar response.
residualsy-fitted values.
H Hat matrix.
df The residual degrees of freedom.
r2 Coefficient of determination.
GCV GCV criterion.
sr2 Residual variance.
l Index of components to include in the model.
lambda Amount of shrinkage.
fdata.comp Fitted object in fdata2pls function.
lm Fitted object in lm function
fdataobj Functional explanatory data.
y Scalar response.

Author(s)

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

References

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/

Examples

## Not run: 
data(tecator)
x<-tecator$absorp.fdata
y<-tecator$y$Fat
res=fregre.pls(x,y,c(1:8),lambda=10)
summary(res)

## End(Not run)

fda.usc

Functional Data Analysis and Utilities for Statistical Computing

v2.0.2

GPL-2

Authors

Manuel Febrero Bande [aut], Manuel Oviedo de la Fuente [aut, cre], Pedro Galeano [ctb], Alicia Nieto [ctb], Eduardo Garcia-Portugues [ctb]

Initial release

2020-02-17