fregre.pls | R Documentation |
Computes functional linear regression between functional explanatory variable X(t)
and scalar response Y
using penalized Partial
Least Squares (PLS)
Y=\big<\tilde{X},\beta\big>+\epsilon=\int_{T}{\tilde{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[\tilde{X}(t)\epsilon]=0
.
\left\{\nu_k\right\}_{k=1}^{\infty}
orthonormal basis of PLS to represent the functional data as X_i(t)=\sum_{k=1}^{\infty}\gamma_{ik}\nu_k
.
fregre.pls(fdataobj, y = NULL, l = NULL, lambda = 0, P = c(0, 0, 1), ...)
fdataobj |
|
y |
Scalar response with length |
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 |
... |
Further arguments passed to or from other methods. |
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 \left\{\tilde{\nu}_k\right\}_{k=1}^{\infty}
the functional PLS components and \tilde{X}_i(t)=\sum_{k=1}^{\infty}\tilde{\gamma}_{ik}\tilde{\nu}_k
and \beta(t)=\sum_{k=1}^{\infty}\tilde{\beta}_k\tilde{\nu}_k
. The functional linear model is estimated by:
\hat{y}=\big< X,\hat{\beta} \big> \approx \sum_{k=1}^{k_n}\tilde{\gamma}_{k}\tilde{\beta}_k
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
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
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.
residuals
: y
minus fitted values
.
H
: Hat matrix.
df.residual
: 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.
Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@udc.es
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("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. https://www.jstatsoft.org/v51/i04/
See Also as: P.penalty
and
fregre.pls.cv
.
Alternative method: fregre.pc
.
## Not run:
data(tecator)
x <- tecator$absorp.fdata
y <- tecator$y$Fat
res <- fregre.pls(x,y,c(1:4))
summary(res)
res1 <- fregre.pls(x,y,l=1:4,lambda=100,P=c(1))
res4 <- fregre.pls(x,y,l=1:4,lambda=1,P=c(0,0,1))
summary(res4)#' plot(res$beta.est)
lines(res1$beta.est,col=4)
lines(res4$beta.est,col=2)
## End(Not run)
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