Description Usage Arguments Details Value Author(s) See Also Examples

Modelization of `fdata`

using functional kernel.

1 |

`data` |
A |

`x` |
The name of the studied variable. |

`r` |
Number of observations used to cross validate the model. |

`hmin` |
Minimal value of the bandwidth. |

`hmax` |
Maximal value of the bandwidth. |

`na.rm` |
Logical. Does the |

This function constructs a functional kernel model and performs the estimation of it's bandwidth.

One nonparametric way to deal with the conditional expectation
*
r(x)=E[X(i)|X(i-1)=x]*, where *X(i)* 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 :

*r*(x,hn)=sum
of(X(i+1)*K(||X(i)-x||/hn)))/((n-1) * sum
of(K(||X(i)-x||/hn))))*

Where K is a kernel, *||.||* is the
norm in H, and *hn* is the bandwidth (*in R+**).

The function `kerfon`

use the cross validation to determinate a
value for *hn*. 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 *hn*. By default, those parameters are
respectively equals to *σ/8* and *4*σ*, where
*σ* is the estimated squared root of the variance operator of
X. To choose the value of *hn*, 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).

A kerfon object. A method for the `print`

function is
provided.

For information, the object is a list with the following elements :

`call` |
the call of the function. |

`h` |
the bandwidth (three values : optimal, minimum, maximum) |

`x` |
the name of the chosen variable |

`xdata` |
the past values for |

`ydata` |
the associated values for |

J. Damon

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
# Simulation of a FARX process
data1 <- simul.farx(m=10,n=400,base=base.simul.far(20,5),
base.exo=base.simul.far(20,5),
d.a=matrix(c(0.5,0),nrow=1,ncol=2),
alpha.conj=matrix(c(0.2,0),nrow=1,ncol=2),
d.rho=diag(c(0.45,0.90,0.34,0.45)),
alpha=diag(c(0.5,0.23,0.018)),
d.rho.exo=diag(c(0.45,0.90,0.34,0.45)),
cst1=0.0)
# Cross validation
model1 <- kerfon(data=data1, x="X", r=10, na.rm=TRUE)
print(model1)
``` |

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