The models
A Functional AutoRegressive of order 1 (FAR(1)) process is, in a
general way, defined by the following equation:
$$T_{n}=\rho\left(T_{n-1}\right)+\epsilon_{n}, n \in
Z$$
where \(T_{n}\) and \(\epsilon_{n}\) take their values in a
functional space (for instance an Hilbertian one), and \(\rho\)
is a linear operator. \(\epsilon_{n}\) is a strong white noise.
Now, let us consider a vector of observations, for instance:
$$\left(T_{1,n},...,T_{i,n},...,T_{m,n}\right)$$
where each \(T_{i,n}\) lives in a one dimension functional
space (not necessary the same). In the following, we will cut this
list into two parts: the endogeneous variables \(Y_{n}\) (the
ones we are interested in), and the exogeneous variables
\(X_{n}\) (which influence the endogeneous ones).
Then an order 1 Functional AutoRegressive process with eXogeneous
variables (FARX(1)) is defined by the equation:
$$Y_{n}=\rho\left(Y_{n-1}\right)+a\left(X_{n}\right)+\epsilon_{n}
, n \in Z$$
where \(\rho\) and \(a\) are linear operators in the
adequate spaces.
Estimation
This function estimates the parameters of FAR and FARX models.
First, if the mean of the data is not zero (which is required
by the model), you can substance this mean using the center
option. Moreover, if the data contains NA values, you
can work with it using the na.rm option.
FAR Estimation
The estimation is mainly about estimating the \(\rho\)
operator. This estimation is done in a appropriate subspace (computed
from the variance of the observations). What is important to know is
that the best dimension kn for this subspace is not determined
by this function. So the user have to supply this dimension using the
kn option. A way to chose this dimension is to first use the
far.cv function on the history.
FARX Estimation
The FARX estimation can be realized by two methods: joined or not.
The joined estimation is done by ``joining'' the variables
into one and estimating a FAR model on the resulting variable. For
instance, with the previous notations, the transformation is:
$$T_{n}=\left(Y_{n},X_{n+1}\right)$$
and \(T_{n}\) is then a peculiar FAR(1) process. In such a case,
you have to use the joined=TRUE oto the interpretation of
this operatorption and specify one
value for kn (corresponding to the \(T_{n}\) variable).
Alternatively, you can choose not to estimate the FARX model by the
joined procedure, then kn need to be a vector with a length
equal to the number of variables involved in the FARX model
(endogeneous and exogeneous).
In both procedures, the endogeneous and exogeneous variables are
provided through the y and x options respectively.
Results
The function returns a far object. Use the print,
coef and predict functions to get more informations
about the model.