Computes functional regression between functional (and non functional) explanatory variables and scalar response using asymmetric kernel estimation.
fregre.plm(
formula,
data,
h = NULL,
Ker = AKer.norm,
metric = metric.lp,
type.CV = GCV.S,
type.S = S.NW,
par.CV = list(trim = 0, draw = FALSE),
par.S = list(w = 1),
...
)an object of class formula (or one that can be coerced
to that class): a symbolic description of the model to be fitted. The
details of model specification are given under Details.
List that containing the variables in the model.
Bandwidth, h>0. Default argument values are provided as the
sequence of length 51 from 2.5%--quantile to 25%--quantile of the distance
between the functional data, see h.default.
Type of asymmetric kernel used, by default asymmetric normal kernel.
Metric function, by default metric.lp.
Type of cross-validation. By default generalized
cross-validation GCV.S method.
Type of smothing matrix S. By default S is
calculated by Nadaraya-Watson kernel estimator (S.NW).
List of parameters for type.CV: trim, the alpha
of the trimming and draw=TRUE.
List of parameters for type.S: w, the weights.
Further arguments passed to or from other methods.
call The matched call.
fitted.values Estimated scalar response.
residuals y minus fitted values.
df The residual degrees of freedom.
H Hat matrix.
r2 Coefficient of determination.
sr2 Residual variance.
y Scalar response.
fdataobj Functional explanatory data.
XX Non functional explanatory data.
mdist Distance matrix between curves.
betah beta coefficient estimated
data List that containing the variables in the model.
Ker Asymmetric kernel used.
h.opt Value that minimizes CV or GCV method.
h Smoothing parameter or bandwidth.
data List that containing the
variables in the model.
gcv GCV values.
formula formula.
An extension of the non-parametric functional regression models is the
semi-functional partial linear model proposed in Aneiros-Perez and Vieu
(2005). This model uses a non-parametric kernel procedure as that described
in fregre.np. The output \(y\) is scalar. A functional
covariate \(X\) and a multivariate non functional covariate \(Z\) are
considered.
$$y =\emph{r(X)}+\sum_{j=1}^{p}{Z_j\beta_j}+\epsilon$$
The unknown smooth real function \(r\) is estimated by means of $$\hat{r}_{h}(X)=\sum_{i=1}^{n}{w_{n,h}(X,X_{i})(Y_{i}-Z_{i}^{T}\hat{\beta}_{h})}$$ where \(W_h\) is the weight function:
\(w_{n,h}(X,X_{i})=\frac{K(d(X,X_i)/h)}{\sum_{j=1}^{n}K(d(X,X_j)/h)}\) with smoothing
parameter \(h\), an asymmetric kernel \(K\) and a metric or semi-metric
\(d\). In fregre.plm() by default \(W_h\) is a functional
version of the Nadaraya-Watson-type weights (type.S=S.NW) with
asymmetric normal kernel (Ker=AKer.norm) in \(L_2\)
(metric=metric.lp with p=2). The unknown parameters
\(\beta_j\) for the multivariate non functional covariates are estimated
by means of
\(\hat{\beta}_j=(\tilde{Z}_{h}^{T}\tilde{Z}_{h})^{-1}\tilde{Z}_{h}^{T}\tilde{Z}_{h}\) where \(\tilde{Z}_{h}=(I-W_{h})Z\) with the
smoothing parameter \(h\). The errors \(\epsilon\) are independent, with
zero mean, finite variance \(\sigma^2\) and
\(E[\epsilon|Z_1,\ldots,Z_p,X(t)]=0\).
The first item in the data list is called "df" and is a data
frame with the response and non functional explanatory variables, as
link{lm}. If non functional data into the formula then
lm regression is performed. Functional variable
(fdata or fd class) is introduced in the second item in the
data list. If only functional variable into the formula then
fregre.np.cv is performed.
The function estimates the value of smoothing parameter or the bandwidth
h through Generalized Cross-validation GCV criteria. It
computes the distance between curves using the metric.lp,
although you can also use other metric function. Different asymmetric
kernels can be used, see Kernel.asymmetric.
Aneiros-Perez G. and Vieu P. (2005). Semi-functional partial linear regression. Statistics & Probability Letters, 76:1102-1110.
Ferraty, F. and Vieu, P. (2006). Nonparametric functional data analysis. Springer Series in Statistics, New York.
Hardle, W. Applied Nonparametric Regression. Cambridge University Press, 1994.
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 as: predict.fregre.plm and
summary.fregre.fd Alternative methods:
fregre.lm, fregre.np and
fregre.np.cv
# NOT RUN {
data(tecator)
x=tecator$absorp.fdata[1:129]
dataf=tecator$y[1:129,]
f=Fat~Water+x
ldata=list("df"=dataf,"x"=x)
res.plm=fregre.plm(f,ldata)
summary(res.plm)
# with 2nd derivative of functional data
x.fd=fdata.deriv(x,nderiv=2)
f2=Fat~Water+x.fd
ldata2=list("df"=dataf,"x.fd"=x.fd)
res.plm2=fregre.plm(f2,ldata2)
summary(res.plm2)
# }
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