Semi-functional partially linear model with scalar response.

Computes functional regression between functional (and non functional) explanatory variables and scalar response using asymmetric kernel estimation.

Usage

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),
  ...
)

Arguments

formula: 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.
data: List that containing the variables in the model.
h: 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.
Ker: Type of asymmetric kernel used, by default asymmetric normal kernel.
metric: Metric function, by default metric.lp.
type.CV: Type of cross-validation. By default generalized cross-validation GCV.S method.
type.S: Type of smothing matrix S. By default S is calculated by Nadaraya-Watson kernel estimator (S.NW).
par.CV: List of parameters for type.CV: trim, the alpha of the trimming
and draw=TRUE.
par.S: List of parameters for type.S: w, the weights.
...: Further arguments passed to or from other methods.

Value

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.

Details

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.

References

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/

Author

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

Examples

if (FALSE) { # \dontrun{
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)
} # }