The observed-data likelihood of ADAM and iETS with unobserved periods · C9

D6 ADAM multiplicative ETS: Pure multiplicative exponential smoothing in the form of the Augmented Dynamic Adaptive Model, in which each component $k$ is updated by the factor $(1+\epsilon_t)^{g_k}$; for ETS(M,Md,M) with seasonal period $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\epsilon_t)^{\alpha} \\ b_t &= b_{t-1}^{\phi}(1+\epsilon_t)^{\beta} \\ s_t &= s_{t-m}(1+\epsilon_t)^{\gamma}. \end{aligned}$$

D7 conventional multiplicative ETS: Pure multiplicative exponential smoothing in the form of Hyndman et al. (2008), in which each component $k$ is updated by the factor $1+g_k\epsilon_t$; for ETS(M,Md,M) with seasonal period $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\alpha\epsilon_t) \\ b_t &= b_{t-1}^{\phi}(1+\beta\epsilon_t) \\ s_t &= s_{t-m}(1+\gamma\epsilon_t). \end{aligned}$$

D10 normal-drift mixture density: Approximate predictive density of a demand size after $j_t-1$ unobserved periods: the one-step density $f_z(z_t\mid\hat\mu_t e^{d})$ mixed over a normal drift of the states $d\sim\mathcal N(m_t,w_t)$, with $m_t$, $w_t$ computed by 32-node Gauss-Legendre quadrature on the quantile function of $1+\epsilon$ and the mixture by Gauss-Hermite quadrature.

D11 one-step density: The density of a demand size after an interval taken as the one-step density $f_z(z_t\mid\hat\mu_t)$ of the model, without the drift of the states over the unobserved periods.

D12 iETS(M,N,N): Intermittent demand sizes $$z_t=l_{t-1}(1+\epsilon_t)$$ with $\E(1+\epsilon_t)=1$ and $\Var(1+\epsilon_t)=s$, Gamma, log-normal or inverse Gaussian $1+\epsilon_t$ (the greybox parameterisation of the package) and the level moving at every period, observed or not: $$l_t=l_{t-1}(1+\epsilon_t)^\alpha$$ in the ADAM multiplicative ETS and $$l_t=l_{t-1}(1+\alpha\epsilon_t)$$ in the conventional multiplicative ETS.

D14 Kullback-Leibler divergence: The divergence $\int g\log(g/h)$ of an approximate density $h$ from the exact density $g$, per observation; over a grid, its largest value.

D13 exact density by convolution: The exact density of the size after an interval, computed on a grid of step 0.001 in $\log z$ by FFT convolution with tails cut at 1e-12, compared with an approximation by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20 (kl_mixture.py); a refinement of 164 cells halved the step and cut the tails at 1e-14.

D9 drift of the states: The change $d_t=\log(\mu_t/\hat\mu_t)$ of the logarithm of the location of a pure multiplicative model accumulated over the unobserved periods since the last observed period, relative to the point forecast $\hat\mu_t$ with zero errors.