fregre.plm | R Documentation |
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),
...
)
formula |
an object of class |
data |
List that containing the variables in the model. |
h |
Bandwidth, |
Ker |
Type of asymmetric kernel used, by default asymmetric normal kernel. |
metric |
Metric function, by default |
type.CV |
Type of cross-validation. By default generalized
cross-validation |
type.S |
Type of smothing matrix |
par.CV |
List of parameters for |
par.S |
List of parameters for |
... |
Further arguments passed to or from other methods. |
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 =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
.
call
: The matched call.
fitted.values
: Estimated scalar response.
residuals
: y
minus fitted values
.
df.residual
: 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.
Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@udc.es
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. https://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)
## End(Not run)
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