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.

Value

An object of the same class as x (ts, mts, vector or matrix) with the underlying forecasts.

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

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)

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