This function constructs a functional kernel model and performs the
estimation of it's bandwidth.
One nonparametric way to deal with the conditional expectation
\(\rho(x)=\mbox{I$\!$E}\left[ X_{i}\left| X_{i-1}=x\right. \right]\), where \(\left(X_{i}\right)\) is a
$H$-valued process, is to consider a predictor inspired by the
classical kernel regression, as in Nadaraja and Watson. This estimator
is defined by :
$$\hat{\rho}_{h_{n}}(x)=\frac{\sum\limits_{i=1}^{n-1}X_{i+1}\cdot%
K\left(\frac{\left\| X_{i}-x\right\| _{H}}{h_{n}}\right)%
}{\sum\limits_{i=1}^{n-1}K\left( \frac{\left\| X_{i}-x\right\|%
_{H}}{h_{n}}\right)},x\in H$$
Where K is a kernel, \(\left\|.\right\| _{H}\) is the
norm in H, and \(h_n\) is the bandwidth (\(\in
\mbox{I$\!$R}^+_*\)).
The function kerfon use the cross validation to determinate a
value for \(h_n\). This method have been chosen because of the
lack of theoretical results about this model. The parameters
hmin and hmax are used, when provided, to control the
permissible values of \(h_n\). By default, those parameters are
respectively equals to \(\sigma/8\) and \(4*\sigma\), where
\(\sigma\) is the estimated squared root of the variance operator of
X. To choose the value of \(h_n\), you need to provide the same
value for both hmin and hmax.
During the cross-validation, considering that the fdata object
x contains \(n\) observations, the function use the first
\((n-r)\) observations as the past values, and compute the mean
square norm of the errors on the last \(r\) observations.
Of course, if the model created is then used to compute prediction
through predict.kerfon, the whole set of observations (the
\(n\) observations) are used as the past values.
As fdata object may contains several variables, a way is
provided to select the studied variable (the function only works
with one variable for the moment).