Filter time series with periodic arma filters
pc.filter.xarma.RdFilter time series with periodic arma filters with or options for periodic and non-periodic intercepts.
Usage
pc.filter.xarma(x, eps, phi, theta, period, p, q, n, from,
seasonof1st = 1, intercept = NULL, nintercept = NULL)Arguments
- x
the time series to be filtered, a vector.
- eps
the innovations, a vector.
- phi
the autoregression parameters, a matrix.
- theta
the moving average parameters, a matrix.
- period
the period (number of seasons in a year).
- p
the autoregression orders, recycled to
periodiflength(p)=1.- q
the moving average orders, recycled to
periodiflength(q)=1.- n
a positive integer, the time index of the last observation to be filtered.
- from
a positive integer, the time index of the first observation to be filtered.
- seasonof1st
a positive integer, the season of the time index of
x[1], see Details.- intercept
the intercepts of the seasons, a vector of length
period.- nintercept
intercepts, a vector of the same length as
x.
Details
pc.filter.xarma is somewhat lower level.
The user level function is pc.filter which uses
pc.filter.xarma to do the computations.
pc.filter.xarma filters the time series x by the
following formula (for t=from,...,n):
$$ x_t
= c_t
+ \sum_{i=1}^{p_t} \phi _t(i)x _{t-i}
+ \sum_{i=1}^{q_t} \theta_t(i)\varepsilon_{t-i}
+ \varepsilon_t,
$$
where \(c_t\) is the overall intercept at time \(t\), see
below. Values of x[t] for t outside the range
from,n, if any, are left unchanged. Values for t<from
are used as initial values when needed.
Two intercepts are provided for convenience and some flexibility. The
periodic intercept, intercept, is a vector of length
period. It is replicated to length n, taking care to
ensure that the first element of the resulting vector, say \(a\),
starts with intercept[seasonof1st]. nintercept can be
an arbitrary vector of length n. It can be used to represent
trend or contributions from covariates. nintercept is not
necessarilly periodic and argument seasonof1st does not affect
its use. The overall intercept is obtained as the sum c = a +
nintercept.
Usually x is a numeric vector but it can also be a matrix in
which each column represents the data for one ``year''.
Also, the length of x is typically, but not necessarilly, equal
to n. It is prudent to ensure that length(x) >= n and
this must be done if x is a matrix.
Argument phi is ignored if p==0,
argument theta is ignored if q==0.
pc.filter.xarma is meant to be called by other functions whose
task is to prepare the arguments with proper checks. It does not make
much sense to repeat the checks in pc.filter.xarma. In
particular, no check is made to ensure that from and n
are correctly specified.
This is a low level function meant to be used with basic vectors
and matrices. TODO: Implement in C/C++. In the current
implementation. it accesses the elements of the arguments with
straightforward indexing, so objects from classes may be used as
well, provided that x[t], eps[t], phi[t,i],
theta[t,i], as well as assignment to x[t], are defined
for scalar indices.
Value
Returns x with x[from] to x[n] filled with the
filtered values and values outside the interval from,...,n left unchanged.
The mode of x is left unchanged. In particular, x may be
a matrix with each row representing the data for a season. This is
convenient since periodic time series are often more easily processed
in this form.