R: Double-Sided Symmetric Linear Filter

filtering {tsdecomp}

R Documentation

Double-Sided Symmetric Linear Filter

Description

Double-sided symmetric linear filter.

Usage

filtering(x, mod, 
  trend = list(ar=1, ma=1, sigma2=NULL), 
  transitory = list(ar=1, ma=1, sigma2=NULL), 
  seasonal = list(ar=1, ma=1, sigma2=NULL),
  irregular.sigma2 = NULL,
  extend = 16, drift = FALSE)
dsfilter(x, w, mod, extend = 16)

Arguments

`x`	a univariate time series.
`mod`	an object of class `Arima`. See `arima`.
`trend`	a list containing the coefficients and variance of the ARIMA model related to the trend component.
`transitory`	a list containing the coefficients and variance of the ARIMA model related to the transitory component.
`seasonal`	a list containing the coefficients and variance of the ARIMA model related to the seasonal component.
`irregular.sigma2`	numeric, variance of the irregular component. If `NULL`, the estimate of the irregular component is not computed.
`extend`	integer; if greater than zero, the series is extended by means of forecasts and backcasts based on the fitted model `mod`.
`drift`	logical, if `TRUE` the `intercept` in the fitted model `mod` or an external regressor named `"drift"` is treated as a deterministic linear trend.
`w`	a vector of filter coefficients (one side).

Details

These functions perform the convolution of the time series and the double-sided symmetric filter. They perform:

stats::filter(c(rep(0, n-1), x, rep(0, n-1)),

+ filter=c(rev(w[-1]), w), method="convolution", sides=1)

where n is length(x).

The design of the filter in the ARIMA-model-based decomposition procedure relies on the following result. The minimum mean squared error estimator of the component is given by the ACGF of the model:

% \theta(L)x_t = \phi_n(L)\theta_s(L)a_t \,,

where \theta(L) is the MA of the model fitted to the observed data, \theta_s(L) is the MA of the component (signal) to be estimated and \phi_n(L) is the product of the AR polynomials of the remaining components. The estimate of the signal, \hat{s}_t, is obtained by means of a double-sided symmetrical filter where the weights, w, are the theoretical autocovariances of the model above:

% \hat{s}_t = \sum_{i=-\infty}^\infty w_i x_{t-i} \,.

Value

filtering returns a list of class tsdecFilter containing the series extended with forecasts (if extend > 0) (based on the ARMA model given as input), the weights of one side of the filter for each component and the corresponding estimate of the components.