Retrieve underlying forecasts corresponding to the asymmetric filters

Function to retrieve the underlying forecasts corresponding to the asymmetric filters: the asymmetric filtering produce the same result than the symmetric filters applied to the extended series.

Usage

underlying_forecasts(x, coefs)

Arguments

x

a univariate or multivariate time series.

coefs

a matrix or a list that contains all the coefficients of the asymmetric and symmetric filters. (from the symmetric filter to the shortest). See details.

Details

Let \(h\) be the bandwidth of the symmetric filter, \(v_{-h}, \ldots, v_h\) the coefficients of the symmetric filter and \(w_{-h}^q, \ldots, w_h^q\) the coefficients of the asymmetric filter used to estimate the trend when \(q\) future values are known (with the convention \(w_{q+1}^q=\ldots=w_h^q=0\)). Let denote \(y_{-2h+1},\ldots, y_0\) the last \(2h\) available values of the input times series. The underlying forecast \(y_{1}^*,\dots y_h^*\) induced by \(w^0,\dots w^{h-1}\) are defined by: $$ \forall q\in\{0,...,h-1\}, \quad \sum_{i=-h}^q v_iy_{i-q} + \sum_{i=q+1}^h v_iy_{i-q}^* =\sum_{i=-h}^q w_i^qy_{i-q}. $$ Note that this is solved numerically: the solution isn't exact.

Examples

if (FALSE) { # \dontrun{

x <- retailsa$AllOtherGenMerchandiseStores
ql <- lp_filter(horizon = 6, kernel = "Henderson", endpoints = "QL")
lc <- lp_filter(horizon = 6, kernel = "Henderson", endpoints = "LC")
f_ql <- underlying_forecasts(x, ql)
f_lc <- underlying_forecasts(x, lc)

plot(window(x, start = 2007),
     xlim = c(2007,2012))
lines(ts(c(tail(x,1), f_ql), frequency = frequency(x), start = end(x)),
      col = "red", lty = 2)
lines(ts(c(tail(x,1), f_lc), frequency = frequency(x), start = end(x)),
      col = "blue", lty = 2)
} # }