Selected exponential smoothing methods within the state-space forecasting framework.

All exponential smoothing (ES) methods within the state-space forecasting (ETS) framework (Equations 2 & 3) were optimized with a likelihood function analog as new Schistosoma haematobium time-series (TS) observations for the district of Niono, Mali, became available; the best-performing method was continuously re-selected with the Akaike's Information Criterion (AIC) to generate optimum forecasts (Methods). Throughout the investigational period, only 3 from a total of 15 ES methods considered within the ETS framework were re-selected; they are: the multiplicative error/ trendless/ aseasonal (MNN); multiplicative error/ damped additive trend/ aseasonal (MAdN); and, multiplicative error/ damped multiplicative trend/ aseasonal (MMdN) ES methods. Notice that none of them are seasonal. Although a full portrayal of the ETS state-space framework (Equations 2 & 3) encapsulating all 30 ES methods [11], [21]–[31] is beyond the scope of this investigation, those ES methods which have been selected at least once during the TS analysis are described herein in terms of E[F(yt|It-1)] and xt recursions—α, β , and φ control smoothing of level (lt), trend (rt), and rt-dampening, respectively. Large α, β, and φ values confer greater weights to recent information and effectively shorten the smoothing “memory”, i.e. the recent-past has a more pronounced influence on estimated components than does the distant-past [11], [21]–[31]. For example, MAdN state-space Eqs. 2 & 3 may be written in explicit matrix form as: F(yt|It-1) = A•xt-1•(1+εt) & xt = B•xt-1+A•xt•C•εt where A = (1, φ)′, xt-1 = (lt-1, rt-1), C = (α, β), and B is a 2×2 matrix whose entries b1,1, b1,2, b2,1, b2,2 are 1, φ, 0, φ, respectively.