CLAIMSET The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11; highlights C1, C2, C5, C4, C8
declarations funding: none | competing interests: None declared.

DEFINITIONS
D1 [method] single source of error form: State space form of a time series model, also called the innovations form, in which the observation equation and the state transition share one error term $\epsilon_t$ per period: $$\begin{aligned} z_t &= w(\mathbf v_{t-\mathbf l}) + r(\mathbf v_{t-\mathbf l})\epsilon_t \\ \mathbf v_t &= f(\mathbf v_{t-\mathbf l}) + g(\mathbf v_{t-\mathbf l})\epsilon_t. \end{aligned}$$
D2 [concept] unobserved period: A period $t$ at which the potential value $z_t$ of the modelled variable is not observed ($O_t=0$): a period with zero demand in an intermittent demand model, where the demand size is not observed, or a period with a missing value.
D3 [concept] observed-data likelihood: The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values $\mathbf z_{\mathrm{mis}}$ at the unobserved periods: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$
D4 [concept] entropy-augmented likelihood: The objective of the iETS estimator of Svetunkov and Boylan (2023, eq. B.6): the sum of the one-step log-densities of the observed demand sizes, minus the differential entropy $H_z(\hat\mu_t,\sigma)$ of the size at each period with zero demand, plus the Bernoulli log-likelihood of the occurrence. [source B4]
D5 [method] zero-error recursion: The estimator used in ADAM and in iETS (Svetunkov and Boylan 2023, Appendix C): the single source of error recursion run with $\epsilon_t=0$ at the unobserved periods and with the fixed persistence vector $\mathbf g$ at the observed ones, with the likelihood taken as the product of one-step densities at the observed periods. [source B4]
D6 [method] 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 [method] 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 $m$, $$\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\alpha\epsilon_t) \\ b_t &= b_{t-1}^{\phi}(1+\beta\epsilon_t) \\ s_t &= s_{t-m}(1+\gamma\epsilon_t). \end{aligned}$$
D8 [method] SSOE Kalman filter: The recursion for the conditional mean $\mathbf a_t$ and scaled covariance $\mathbf P_t$ of the state of a linear single source of error model: at an observed period $$\begin{aligned} v_t &= z_t-\mathbf w'\mathbf a_t \\ f_t &= \mathbf w'\mathbf P_t\mathbf w+1 \\ \mathbf k_t &= (\mathbf F\mathbf P_t\mathbf w+\mathbf g)/f_t \\ \mathbf a_{t+1} &= \mathbf F\mathbf a_t+\mathbf k_t v_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'-\mathbf k_t\mathbf k_t'f_t; \end{aligned}$$ at an unobserved period $$\begin{aligned} \mathbf a_{t+1} &= \mathbf F\mathbf a_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'; \end{aligned}$$ started from $\mathbf a_1=\mathbf x_0$, $\mathbf P_1=\mathbf 0$.
D9 [concept] 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.
D10 [method] 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 [[D9: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 [method] 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 [[D9:drift of the states]] over the unobserved periods.
D12 [data] 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 [[D6:ADAM multiplicative ETS]] and $$l_t=l_{t-1}(1+\alpha\epsilon_t)$$ in the [[D7:conventional multiplicative ETS]].
D13 [design] 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.
D14 [variable] 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.
D15 [method] fixed-point moment estimator: The estimator of the scale $s$ of Gamma or inverse Gaussian sizes that solves the moment equation $s=h(s)$, built from the moments of the drift factor $D_t$, by the iteration $s\leftarrow h(s)$ from the one-step estimate, stopping at $|s_{new}-s|\le 10^{-10}s$ or after 100 iterations; a fit fails when an iterate is not positive and finite, the iteration does not converge, or the value leaves the admissible range.
D16 [design] Monte Carlo study of the scale: 480 cells of relative errors simulated with the drift ($s\in\{0.05,0.2,0.5,1\}$, $\alpha\in\{0.1,0.3,0.6\}$, intervals fixed at 2, 5 or 10 or geometric with occurrence probability 0.5 or 0.2, 100 or 1000 observed sizes) and 120 cells within fitted models ($s\in\{0.05,0.5\}$, 100 observed sizes), 2000 replications per cell from numpy SeedSequence(20261015) with one child stream per cell; cells grouped by the drift order $\E(j-1)\alpha^2s$ (scale_recovery.py).
D17 [variable] relative bias: The relative bias of a scale estimate, (mean estimate $- s$)$/s$ over the replications of a cell; its median over cells.
D18 [variable] failed fits: The share of replications in which the scale estimator fails.
D19 [method] approximate prediction interval: The prediction interval of pure multiplicative ETS in the smooth package (generate_prediction_interval with var_anal): the point forecast with zero errors times the quantile of the distribution with mean one and variance $$V_h=(1+v)(1+\alpha^2v)^{h-1}-1,$$ $v=\Var(1+\epsilon)$; for the [[D6:ADAM multiplicative ETS]] with the log-normal distribution, exact in logarithms.
D20 [data] ETS(M,N,N) with s = 0.32 and alpha = 0.21: ETS(M,N,N) with $s=0.32$, $\alpha=0.21$, 100 observations and initial level 10, with Gamma, inverse Gaussian or log-normal errors, in the ADAM and conventional forms; in the second case the last six of the 100 periods unobserved.
D21 [design] simulated paths: The 0.95 quantile of 1,000,000 simulated paths divided by the point forecast with zero errors, twelve steps ahead and 18 steps from the last observed period after six unobserved periods at the end of the sample (interval_bounds.py, seed 20261015).
D22 [variable] upper bound over the point forecast: The upper bound of the 90% prediction interval (its 0.95 quantile) divided by the point forecast with zero errors.
D23 [method] low-rank factor form: The [[D8:SSOE Kalman filter]] with the scaled state covariance kept as a low-rank factor $\mathbf P_t=\mathbf U_t\mathbf U_t'$, updated by rank one at each observed period, as in the C++ core of smooth (src/headers/adamCore.h). [a kind of D8]
D24 [method] full covariance form: The [[D8:SSOE Kalman filter]] with the whole scaled state covariance matrix $\mathbf P_t$. [a kind of D8]
D25 [data] hourly series with missing values: 1344 hourly observations, generated by ADAM ETS(A,N,A) with lags 24 and 168 or following a fixed sine pattern, with 1 to 48 nested missing values, fitted by ADAM ETS(A,N,A) with lags 24 and 168 (193 state cells), the normal distribution and backcast initial states.
D26 [design] interleaved timing: 25 interleaved repetitions per case of the fits with the two forms of the filter, both built from smooth commit 53104ee9, on one cloud VM (Intel Xeon 2.80 GHz, 4 vCPUs, Linux 6.18.44, Python 3.11.15) with one thread (lowrank_timing.py).
D27 [variable] speed-up: The median run time of the [[D24:full covariance form]] divided by that of the [[D23:low-rank factor form]].
D28 [method] family-scaled error: The approximation of the predictive density of a pure additive model with Laplace, S or Generalised Normal errors that keeps the error in its family with scale $sf_t^{1/p}$, $p$ such that the variance is proportional to $s^p$, as in the smooth package.
D29 [method] Gauss-Hermite mixture: The mixture of the error density over the normal uncertainty of the states by a Gauss-Hermite rule with $Q$ nodes (16 in the smooth package).
D30 [data] additive errors with state uncertainty: Laplace, S and Generalised Normal (shape 0.5 to 3) errors with scale $s=1$, convolved with normal state uncertainty of variance $(f_t-1)V(s)$, for 22 values of $f_t$ from 1.01 to 100 (kl_laplace.py).
D31 [design] divergence by numerical integration: The Kullback-Leibler divergence in both directions between the exact convolution and each approximation, by composite 8-node Gauss-Legendre integration (kl_laplace.py).

CLAIMS
C1 [formal; principal; support: formal_proof E1; assumes C30, C31, C32; verification unchecked; verifiability open]
  For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\sim\text{Bernoulli}(p_t)$ independent of a continuous size $z_t$, the [[D3:observed-data likelihood]] is $$\sum_{O_t=1}\log f(z_t\mid Y_{t-1};\theta) + \sum_{O_t=1}\log p_t + \sum_{O_t=0}\log(1-p_t),$$ in which the unobserved sizes do not appear and the size and occurrence parts separate.
  scope: population: models of intermittent demand $y_t=o_tz_t$ (iETS, ADAM with an occurrence part) | data_type: intermittent demand time series | conditions: Bernoulli occurrence independent of a continuous demand size
C2 [formal; principal; refutation; support: formal_proof E2, cited B1; assumes C31, C32; uses C1; verification unchecked; verifiability open]
  The stationary points of the [[D4:entropy-augmented likelihood]] of iETS differ in general from those of the [[D3:observed-data likelihood]], because the entropy term $-H_z(\theta)$ has a non-zero gradient in the parameters while the cross-entropy of the EM E-step has zero gradient at the current estimate.
  scope: population: iETS estimated by the entropy-augmented likelihood | data_type: intermittent demand time series | conditions: sizes unobserved at the periods with zero demand
  contradicts doi doi:10.1016/j.ijpe.2023.109013, at Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm), own earlier work
C3 [formal; principal; support: formal_proof E3; uses C2; verification agent_checked; verifiability open]
  For normally distributed demand sizes with fixed location, the maximiser in the variance of the [[D4:entropy-augmented likelihood]] is $\frac{T_1}{T}\hat\sigma^2$, where $\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}e_t^2$ is the fixed point of the EM algorithm, $T$ the number of periods and $T_1$ the number of periods with observed sizes.
  scope: population: iETS with normally distributed demand sizes | conditions: location of the sizes held fixed; variance as the only free parameter
C4 [formal; principal; extension; support: formal_proof E4; assumes C30, C33, C34, C35; uses C1; verification agent_checked; verifiability open]
  For a pure additive single source of error model with normal errors and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] gives $z_t\mid Y_{t-1}\sim\mathcal N(\mathbf w'\mathbf a_t,\sigma^2 f_t)$ at the observed periods, the exact observed-data log-likelihood $$-\frac12\sum_{O_t=1}\left[\log(2\pi\sigma^2f_t)+\frac{v_t^2}{\sigma^2f_t}\right],$$ and the concentrated variance $$\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}\frac{v_t^2}{f_t}.$$
  scope: population: pure additive single source of error models (ETS with additive components, ARIMA, regression) | data_type: univariate time series with unobserved periods | conditions: normally distributed errors; initial states treated as fixed parameters | excludes: models mixing additive and multiplicative components
  extends doi doi:10.1017/cbo9781107049994, at Kalman filter for state space models with correlated disturbances
  extends doi doi:10.1093/acprof:oso/9780199641178.001.0001, at Kalman filter for state space models with correlated disturbances
C5 [formal; principal; refutation; support: formal_proof E6; assumes C30, C33, C34, C35; uses C4; verification unchecked; verifiability open]
  For pure additive single source of error models with normal errors, [[D2:unobserved period]]s and persistence vector $\mathbf g\neq\mathbf 0$, the maximiser of the [[D5:zero-error recursion]] likelihood is not the maximum likelihood estimator, because the zero-error recursion uses error variance $\sigma^2$ instead of $\sigma^2f_t\geq\sigma^2$ and gain $\mathbf g$ instead of $\mathbf k_t$ at the observed periods after an unobserved one.
  scope: population: pure additive single source of error models | data_type: univariate time series with unobserved periods | conditions: normally distributed errors; persistence vector $\mathbf g\neq\mathbf 0$
  contradicts doi doi:10.1016/j.ijpe.2023.109013, at Appendix C (estimation with zero errors at the unobserved periods), own earlier work
C6 [formal; principal; support: formal_proof E9; assumes C30, C33, C34, C36; uses C4, C12; verification unchecked; verifiability open]
  For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] applied to $\log z_t$, with the known error mean $-s/2$ added to the propagation and the prediction, gives the exact observed-data likelihood, and the maximum likelihood scale $s$ is the positive root of $$\Big(\sum \frac{b_t^2}{f_t}\Big)s^2+T_1s-\sum\frac{(v_t^0)^2}{f_t}=0,$$ where $v_t(s)=v_t^0+sb_t$ is the filter's prediction error.
  scope: population: pure multiplicative ETS in the ADAM form | data_type: positive univariate time series with unobserved periods | conditions: log-normal distribution with $\E(1+\epsilon_t)=1$
C7 [formal; principal; support: formal_proof E10; assumes C30, C36; uses C6; verification agent_checked; verifiability open]
  For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and all periods observed, the maximum likelihood estimate of the scale is $\hat s=2(\sqrt{1+m}-1)$ with $m=\frac{1}{T_1}\sum\log^2(1+\epsilon_t)$.
  scope: population: pure multiplicative ETS in the ADAM form | data_type: positive univariate time series | conditions: log-normal distribution with $\E(1+\epsilon_t)=1$; every period observed
C8 [formal; principal; refutation; support: formal_proof E10; assumes C36; uses C7; verification unchecked; verifiability open]
  The log-normal scale estimator $2(1-\sqrt{1-m})$ used in ADAM, with $m=\frac{1}{T_1}\sum\log^2(1+\epsilon_t)$, is not the maximum likelihood estimator: it overestimates the scale $s$ by about 35% at $s=0.5$ (0.677 against 0.5 when $m=s+s^2/4$) and is undefined for $m>1$.
  scope: population: pure multiplicative ETS in the ADAM form | data_type: positive univariate time series | conditions: log-normal distribution with $\E(1+\epsilon_t)=1$; every period observed
  contradicts doi doi:10.1201/9781003452652, at estimator of the scale of the log-normal distribution, own earlier work
C9 [descriptive; principal; support: simulation E20 (E20.T1.r1, E20.T1.r2, E20.T1.r3) artefacts A3, A4, A5; assumes C37; strength moderate (author); verifiability open]
  For [[D12:iETS(M,N,N)]] in the [[D7:conventional multiplicative ETS]] form, the [[D10:normal-drift mixture density]] of the demand size after an interval had a largest [[D14:Kullback-Leibler divergence]] per observation from the exact density at least three orders of magnitude smaller than that of [[D11:the one-step density]], for the Gamma, the log-normal and the inverse Gaussian distribution.
  scope: population: iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes | conditions: scale $s\in[0.05,1]$; smoothing parameter $\alpha\in[0.1,0.6]$; interval length $j\in[2,20]$; states known at the previous observed period | excludes: models with trend or seasonality; ADAM multiplicative ETS
C10 [formal; principal; support: formal_proof E12; assumes C31, C32; verification unchecked; verifiability open]
  For a model with an occurrence part and occurrence probability $p_t$, a prediction interval of the demand with coverage $c$ that counts the zeros as covered is an interval of the demand sizes with coverage $(c-(1-p_t))/p_t$, centred on the size forecast $\hat\mu_t$ rather than on the demand forecast $p_t\hat\mu_t$.
  scope: population: models with an occurrence part (iETS, ADAM) | conditions: prediction interval of the demand that counts zeros as covered; Bernoulli occurrence independent of a continuous demand size
C11 [methodological; principal; support: demonstration artefacts A2; uses C1, C4, C6, C7; strength moderate (author); verifiability open]
  The smooth package for R and Python implements the [[D3:observed-data likelihood]] without the entropy term, the [[D8:SSOE Kalman filter]] for the pure additive models with the normal distribution and in logarithms for the [[D6:ADAM multiplicative ETS]] with the log-normal distribution, the [[D10:normal-drift mixture density]] with moment-equation scales for the other pure multiplicative models, and the corrected log-normal scale, in one C++ routine shared by both languages.
  scope: setting: smooth package for R and Python, master branch at commit 8a5fbbb3
C12 [formal; supporting; support: formal_proof E8; verification unchecked; verifiability open]
  In the [[D6:ADAM multiplicative ETS]], the logarithms of the observation and of the components follow a linear single source of error model with innovation $u_t=\log(1+\epsilon_t)$ and the measurement, transition and persistence of the corresponding additive model, whereas in the [[D7:conventional multiplicative ETS]] the innovation of component $k$ is $\log(1+g_k\epsilon_t)$, which is not proportional to $u_t$.
  scope: population: pure multiplicative ETS in the ADAM and conventional forms
C30 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  The errors $\epsilon_t$ of the single source of error model are independent across periods.
C31 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$.
C32 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$.
C33 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  Whether a period is observed carries no information about the unobserved value $z_t$ beyond the model, so that the missing values are missing at random.
C34 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The initial states of the single source of error model are fixed parameters, so that the state covariance is zero at the start of the sample.
C35 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The errors of the pure additive model are normally distributed with constant variance $\sigma^2$.
C36 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$.
C37 [descriptive; supporting; assumption (methodological, tested); verifiability not_applicable]
  The [[D9:drift of the states]] of a pure multiplicative model over an interval is approximately normally distributed, with the states known at the previous observed period.
C14 [formal; secondary; refutation; support: argument W2, cited B4, at eq. 29 and eq. B.6, own earlier work; warrant W2; verification unchecked; verifiability open]
  With the differential entropy defined as $H=-\E\log f$, the sign of the entropy term in equations 29 and B.6 of Svetunkov and Boylan (2023) should be negative.
  scope: population: iETS objective of Svetunkov and Boylan (2023)
  contradicts doi doi:10.1016/j.ijpe.2023.109013, at eq. 29 and eq. B.6 (sign of the entropy term), own earlier work
C15 [formal; secondary; support: formal_proof E5, cited B2, cited B3; uses C4; verification agent_checked; verifiability open]
  When every period is observed, the [[D8:SSOE Kalman filter]] keeps $\mathbf P_t=\mathbf 0$, has gain $\mathbf k_t=\mathbf g$ and $f_t=1$, and coincides with the single source of error recursion and the likelihood of the one-step errors.
  scope: population: pure additive single source of error models | data_type: univariate time series | conditions: every period observed
C16 [formal; secondary; support: formal_proof E7; uses C4; verification unchecked; verifiability open]
  For ETS(A,N,N) with smoothing parameter $\alpha$ and zero state covariance before an interval of $j$ periods, the gain of the [[D8:SSOE Kalman filter]] at the end of the interval is $$\alpha_j=\frac{(j-1)\alpha^2+\alpha}{(j-1)\alpha^2+1},$$ which equals $\alpha$ for $j=1$ and tends to one as $j$ grows.
  scope: population: ETS(A,N,N) | conditions: normally distributed errors; zero state covariance before an interval of $j$ periods
C17 [formal; secondary; support: formal_proof E11; uses C12; verification agent_checked; verifiability open]
  For pure multiplicative models with the Gamma, log-normal or inverse Gaussian distribution, if $1+e_t=(1+\epsilon_t)D_t$ with $D_t=e^{d_t}$ independent of $\epsilon_t$, $\E(1+\epsilon_t)=1$ and $\Var(1+\epsilon_t)=s$, then $$\E(e_t^2)=(1+s)\E(D_t^2)-2\E(D_t)+1,$$ and for the inverse Gaussian distribution $$\E\big(e_t^2/(1+e_t)\big)=\E(D_t)-2+(1+s)\E(D_t^{-1}).$$
  scope: population: pure multiplicative models with Gamma, log-normal or inverse Gaussian errors | conditions: drift factor $D_t$ independent of $\epsilon_t$
C18 [descriptive; secondary; support: simulation E21 (E21.T1.r1, E21.T1.r2, E21.T1.r3, E21.T1.r4) artefacts A3, A6; assumes C37; uses C17; limitations L1; strength moderate (author); verifiability open]
  On errors simulated with the [[D9:drift of the states]] of [[D12:iETS(M,N,N)]] and 1000 observed sizes, the [[D15:fixed-point moment estimator]] of the Gamma and inverse Gaussian scale had a median [[D17:relative bias]] of +0.02% with no [[D18:failed fits]] when the drift order $\E(j-1)\alpha^2s$ was at most 0.02, and failed in 18%, 52% and 71% of fits for drift orders of 0.02 to 0.1, 0.1 to 0.3 and above 0.3.
  scope: population: iETS(M,N,N) with Gamma and inverse Gaussian sizes | conditions: errors simulated with the drift, outside a fitted model; 1000 observed sizes per series; $s\in\{0.05,0.2,0.5,1\}$, $\alpha\in\{0.1,0.3,0.6\}$
C19 [formal; secondary; support: formal_proof E13; assumes C30; verification unchecked; verifiability open]
  The point forecast with zero errors of the [[D6:ADAM multiplicative ETS]] is not the conditional mean, because $\E(1+\epsilon_t)^{g_k}<1$ for $0<g_k<1$; for ETS(M,N,N) with Gamma errors of mean one and variance $s$ the conditional mean $h$ steps ahead is $$\hat\mu_{T+h}\left[\frac{\Gamma(1/s+\alpha)s^{\alpha}}{\Gamma(1/s)}\right]^{h-1}.$$
  scope: population: pure multiplicative ETS in the ADAM form
C20 [descriptive; secondary; support: simulation E22 (E22.R1, E22.R2, E22.R3, E22.R4) artefacts A3, A7; uses C19; strength moderate (author); verifiability open]
  For [[D20:ETS(M,N,N)]] in the [[D6:ADAM multiplicative ETS]] form with Gamma errors, $s=0.32$ and $\alpha=0.21$, the [[D19:approximate upper bound of the 90% prediction interval]] divided by the point forecast was 2.431 against 1.807 from [[D21:simulated paths]] twelve steps ahead, and 2.616 against 1.638 after six unobserved periods at the end of the sample.
  scope: population: ETS(M,N,N) in the ADAM form | conditions: Gamma errors; $s=0.32$; $\alpha=0.21$; 100 observations
C21 [formal; secondary; support: formal_proof E14; uses C4; verification agent_checked; verifiability open]
  With a single source of error, the scaled state covariance $\mathbf P_t$ of the [[D8:SSOE Kalman filter]] has rank at most the number of [[D2:unobserved period]]s since it last vanished, which allows a low-rank factor $\mathbf P_t=\mathbf U_t\mathbf U_t'$ with a rank-one update at each observed period.
  scope: population: linear single source of error models | data_type: univariate time series with unobserved periods
C22 [descriptive; secondary; support: empirical_quant E23 (E23.R1, E23.R2, E23.R3) artefacts A3, A8; uses C21; limitations L2; strength moderate (author); verifiability open]
  For ADAM ETS(A,N,A) with lags 24 and 168 (193 state cells) on [[D25:1344 hourly observations]] generated by ETS(A,N,A), the [[D23:low-rank factor form]] of the [[D8:SSOE Kalman filter]] was 10.55 times faster than the [[D24:full covariance form]] for one missing value and 3.50 times faster for 48, with log-likelihoods equal to within 4.5e-13.
  scope: population: ADAM ETS(A,N,A) with double seasonality | data_type: hourly time series with missing values | conditions: lags 24 and 168; 1344 observations; 1 to 48 missing values; data generated by ETS(A,N,A); one filter pass, one thread
C23 [methodological; secondary; support: demonstration artefacts A2, argument W3; uses C15; warrant W3; strength moderate (author); verifiability open]
  In the smooth package, the [[D8:SSOE Kalman filter]] replaces the fitter only in the last forward pass after backcasting of the initial states, so that fits of series without [[D2:unobserved period]]s are unchanged.
  scope: setting: smooth package for R and Python
C24 [descriptive; secondary; support: simulation E24 (E24.R1, E24.R2) artefacts A3, A9; uses C4; strength moderate (author); verifiability open]
  For [[D30:the pure additive models with the Laplace, S and Generalised Normal distributions]], the smooth package keeps [[D28:the error in its family]] with scale $sf_t^{1/p}$, and the [[D14:Kullback-Leibler divergence]] of this approximation from the exact density is 0.0491 for the Laplace distribution and 0.260 for the S distribution at $f_t=4$.
  scope: population: pure additive ADAM models with Laplace, S or Generalised Normal errors
C25 [descriptive; secondary; support: simulation E20 (E20.T1.r4, E20.T1.r5, E20.T1.r6) artefacts A3, A4, A5; strength moderate (author); verifiability open]
  For [[D12:iETS(M,N,N)]] in the [[D6:ADAM multiplicative ETS]] form, the [[D10:normal-drift mixture density]] of the demand size after an interval had a largest [[D14:Kullback-Leibler divergence]] per observation from the exact density at least two orders of magnitude smaller than that of [[D11:the one-step density]] for the Gamma and the inverse Gaussian distribution, and is exact for the log-normal distribution.
  scope: population: iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes | conditions: scale $s\in[0.05,1]$; smoothing parameter $\alpha\in[0.1,0.6]$; interval length $j\in[2,20]$; states known at the previous observed period | excludes: models with trend or seasonality; conventional multiplicative ETS
C26 [descriptive; secondary; support: simulation E21 (E21.R9, E21.R10, E21.R11) artefacts A3, A6; uses C18; strength moderate (author); verifiability open]
  Within [[D12:iETS(M,N,N)]] models fitted by the [[D5:zero-error recursion]], updating the states with the persistence vector $\mathbf g$ after an interval raised the estimate of the Gamma and inverse Gaussian scale from the moment equation solved by bisection by more than three Monte Carlo standard errors in 97 of 120 cells, with a median increase in [[D17:relative bias]] of 12.8 percentage points.
  scope: population: iETS(M,N,N) with Gamma and inverse Gaussian sizes | conditions: $s\in\{0.05,0.5\}$; 100 observed sizes per series
C27 [descriptive; secondary; support: simulation E22 (E22.R5, E22.R6, E22.R7, E22.R8) artefacts A3, A7; strength moderate (author); verifiability open]
  For [[D20:ETS(M,N,N)]] in the [[D7:conventional multiplicative ETS]] form with Gamma errors, $s=0.32$ and $\alpha=0.21$, the [[D19:approximate upper bound of the 90% prediction interval]] divided by the point forecast was 2.431 against 2.393 from [[D21:simulated paths]] twelve steps ahead, and 2.616 against 2.536 after six unobserved periods at the end of the sample.
  scope: population: ETS(M,N,N) in the conventional form | conditions: Gamma errors; $s=0.32$; $\alpha=0.21$; 100 observations
C28 [descriptive; secondary; support: empirical_quant E23 (E23.R4, E23.R5) artefacts A3, A8; uses C21; limitations L2; strength moderate (author); verifiability open]
  For ADAM ETS(A,N,A) with lags 24 and 168 on [[D25:1344 hourly observations]] following a fixed sine pattern, the [[D23:low-rank factor form]] of the [[D8:SSOE Kalman filter]] was 2.96 times faster than the [[D24:full covariance form]] for one missing value and 0.72 times as fast for 48, and slower than the full form above about 30 missing values.
  scope: population: ADAM ETS(A,N,A) with double seasonality | data_type: hourly time series with missing values | conditions: lags 24 and 168; 1344 observations; 1 to 48 missing values; data following a fixed sine pattern; one filter pass, one thread
C29 [descriptive; secondary; support: simulation E24 (E24.R1, E24.R2, E24.R3, E24.R4) artefacts A3, A9; uses C24; strength moderate (author); verifiability open]
  At variance ratio $f_t=4$, the [[D29:16-node Gauss-Hermite mixture]] of the error density over normal state uncertainty had a [[D14:Kullback-Leibler divergence]] from the exact density of 0.0081 for the Laplace and 0.113 for the S distribution, against 0.0491 and 0.260 for [[D28:the error kept in its family]] with scale $sf_t^{1/p}$.
  scope: population: pure additive models with Laplace or S errors | conditions: $f_t=4$; $s=1$

CLAIMS WITHOUT EVIDENCE
C13 [formal; secondary; support: argument W1; uses C2; warrant W1; verification unchecked; verifiability open]
  With a dynamic model for the scale, the [[D4:entropy-augmented likelihood]] of iETS is unbounded, because the entropy terms of the periods with zero demand reward a scale going to zero for the normal distribution and to infinity for the Gamma distribution.
  scope: population: iETS with a dynamic model for the scale of the demand sizes | conditions: normal or Gamma distribution of the sizes

EVIDENCE
E1 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E1.1 $P(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t$ (by C31, C32; independence and continuity)
  E1.2 $y_t>0$ has density $p_tf(y_t\mid Y_{t-1})$ (by C31; independence)
  E1.3 The joint density of the observed data is the product of these conditional terms (by E1.1, E1.2; prediction-error decomposition)
E2 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E2.1 $Q(\theta\mid\theta^{(k)})$ contains at each zero the cross-entropy $\int f_z^{(k)}\log f_z\,dz=-H_z(\theta^{(k)})-\KL(f_z^{(k)}\|f_z)$ (by D4; definition of the E-step)
  E2.2 $$\partial_\theta\int f_z\log f_z^{\theta}dz\Big|_{\theta=\theta^{(k)}}=\partial_\theta\int f_z\,dz=0$$ (by E2.1; differentiation under the integral)
  E2.3 $\nabla Q(\theta\mid\theta^{(k)})|_{\theta^{(k)}}=\nabla\ell(\theta^{(k)})$, so EM fixed points are stationary points of $\ell$ (by E2.2, B1, C1; EM identity)
  E2.4 $-\partial_\theta H_z(\theta)\neq0$ in general (e.g. in the scale), adding to the score of $\ell$ (by D4; differentiation)
E3 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E3.1 $H_z=\frac12\log(2\pi e\sigma^2)$ for the normal distribution (by D4; definition)
  E3.2 $$\frac{\partial\ell_H}{\partial\sigma^2}=-\frac{T_1}{2\sigma^2}+\frac{\sum e_t^2}{2\sigma^4}-\frac{T_0}{2\sigma^2},$$ zero at $\sigma^2=\sum e_t^2/T$ (by E3.1; differentiation)
  E3.3 EM iterates $\sigma^2_{(k+1)}=(\sum e_t^2+T_0\sigma^2_{(k)})/T$ with fixed point $\frac{1}{T_1}\sum e_t^2$ (by B1; fixed point)
E4 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30, C33, C34, C35
  E4.1 $(\mathbf x_{t-1},z_t,\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$ (by C35, C30; linearity of Gaussian models)
  E4.2 $$\begin{aligned} \E(z_t\mid Y_{t-1}) &= \mathbf w'\mathbf a_t \\ \Var(z_t\mid Y_{t-1}) &= \sigma^2f_t \\ \Cov(\mathbf x_t,z_t\mid Y_{t-1}) &= \sigma^2(\mathbf F\mathbf P_t\mathbf w+\mathbf g) \end{aligned}$$ (by E4.1; moments of linear combinations)
  E4.3 Conditioning on $z_t$ gives the update; unobserved periods only propagate (by E4.2, C33; Gaussian conditioning)
  E4.4 The likelihood follows from the observed-data decomposition, and $\sigma^2$ concentrates out because $f_t$, $\mathbf k_t$ do not depend on it (by E4.3, C1; prediction-error decomposition)
E5 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E5.1 With $\mathbf P_t=\mathbf 0$, $f_t=1$ and $\mathbf k_t=\mathbf g$ (by D8; substitution)
  E5.2 $\mathbf P_{t+1}=\mathbf g\mathbf g'-\mathbf g\mathbf g'=\mathbf 0$ (by E5.1; substitution)
  E5.3 By induction from $\mathbf P_1=\mathbf 0$ the filter is the SSOE recursion (by E5.2; induction)
E6 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E6.1 After $j-1$ unobserved periods from $\mathbf P=\mathbf 0$, $f=1+\sum_{i<j}c_i^2$ with $c_i=\mathbf w'\mathbf F^{i-1}\mathbf g$ (by D8; iteration of the propagation)
  E6.2 $f_t>1$ when $c_1\neq0$, and $\mathbf k_t\neq\mathbf g$ while $\mathbf P_t\neq\mathbf 0$ (by E6.1; substitution)
  E6.3 The zero-error recursion uses variance $\sigma^2$ and gain $\mathbf g$, so its objective differs from the exact likelihood (by E6.2, D5, C4; comparison)
E7 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E7.1 $P=(j-1)\alpha^2$, $f=(j-1)\alpha^2+1$ after $j-1$ unobserved periods (by D8; substitution $w=F=1$, $g=\alpha$)
  E7.2 $k=(P+\alpha)/f$ (by E7.1, D8; substitution)
E8 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E8.1 $\log z_t=\log l_{t-1}+\phi\log b_{t-1}+\log s_{t-m}+u_t$ (by D6; logarithm)
  E8.2 $\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\alpha u_t$, $\log b_t=\phi\log b_{t-1}+\beta u_t$ in the ADAM form (by D6; logarithm)
  E8.3 $\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\log(1+\alpha\epsilon_t)$ in the conventional form (by D7; logarithm)
E9 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30, C33, C34, C36
  E9.1 $\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$ (by C12, C36; linearity in logarithms)
  E9.2 The filter of the linear Gaussian model with the Jacobian $-\sum\log z_t$ gives the exact likelihood (by E9.1, C4; change of variables)
  E9.3 Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$ (by E9.2; linearity)
  E9.4 The score in $s$ times $2s^2$ gives $(\sum b_t^2/f_t)s^2+T_1s-\sum(v_t^0)^2/f_t=0$ (by E9.3; differentiation)
E10 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C36
  E10.1 $f_t=1$, $b_t=1/2$ give $s^2+4s-4m=0$ (by C6, C15; substitution)
  E10.2 The positive root is $2(\sqrt{1+m}-1)$; $2(1-\sqrt{1-m})$ solves $s^2-4s+4m=0$ instead (by E10.1; quadratic formula)
  E10.3 At $s=0.5$, $\E m=s+s^2/4=0.5625$ and $2(1-\sqrt{1-0.5625})=0.677$ (by E10.2, C36; arithmetic)
E11 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E11.1 $e_t^2=(1+e_t)^2-2(1+e_t)+1$ and $e_t^2/(1+e_t)=(1+e_t)-2+(1+e_t)^{-1}$ (by none; algebra)
  E11.2 Independence of $\epsilon_t$ and $D_t$ with $\E(1+\epsilon)^2=1+s$ and, for the inverse Gaussian, $\E(1+\epsilon)^{-1}=1+s$ (by E11.1; independence and moments)
E12 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E12.1 $P(y_t\in[0,u])=(1-p_t)+p_tF_z(u)$ when zeros count as covered (by C31, C32; law of total probability)
  E12.2 Setting it to $c$ gives $F_z$ coverage $(c-(1-p_t))/p_t$, a quantile of the sizes around $\hat\mu_t$ (by E12.1; algebra)
E13 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30
  E13.1 $x\mapsto x^{g}$ is concave for $0<g<1$, so $\E(1+\epsilon)^{g}<(\E(1+\epsilon))^{g}=1$ (by D6; Jensen's inequality)
  E13.2 For ETS(M,N,N), $\E(z_{T+h}\mid Y_T)=\hat\mu_{T+h}[\E(1+\epsilon)^{\alpha}]^{h-1}$ by independence of the errors (by E13.1, C30; independence)
  E13.3 For Gamma with mean one and variance $s$, $\E(1+\epsilon)^{\alpha}=\Gamma(1/s+\alpha)s^{\alpha}/\Gamma(1/s)$ (by E13.2; Gamma moments)
E14 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  E14.1 An unobserved period adds the rank-one term $\mathbf g\mathbf g'$ (by D8; propagation)
  E14.2 An observed period removes the direction it reveals while adding the new error, so the rank does not increase (by D8; update)
  E14.3 With $\mathbf M=[\mathbf U_t^F,\mathbf g]$ and $\mathbf M(\mathbf a,1)'=f_t\mathbf k_t$, a Householder reflection gives $$\mathbf U_{t+1}=\mathbf U_t^F-\frac{(f_t\mathbf k_t+\sqrt{f_t}\mathbf g)\mathbf a'}{f_t+\sqrt{f_t}}$$ (by E14.2; Householder reflection)
E20 [simulation; data_origin primary; runs 1] Comparison, for iETS(M,N,N) under the ADAM and conventional multiplicative ETS forms, of the normal-drift mixture density (drift moments by 32-node Gauss-Legendre, mixture by 16-node Gauss-Hermite) and of the one-step density against 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, 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; a 164-cell refinement run halved the step and cut tails at 1e-14. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.
  setup: studied D10, D11; cases D12, D6, D7; procedure D13; lens D14
  E20.T1 Largest Kullback-Leibler divergence per observation from the exact density of the size after an interval, of the normal-drift mixture density and of the one-step density, for iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes in the conventional and the ADAM form {model: iETS(M,N,N)}
  | row | distribution | ets_form | largest KL divergence per observation, normal-drift mixture vs exact density | largest KL divergence per observation, one-step vs exact density |
  | r1 | dgamma | conventional | 0.00114767 | 2.35934 |
  | r2 | dlnorm | conventional | 0.00447805 | 5.94732 |
  | r3 | dinvgauss | conventional | 0.00399623 | 47.5452 |
  | r4 | dgamma | adam | 0.0145197 | 4.58358 |
  | r5 | dlnorm | adam | 9.63871e-07 | 18.6354 |
  | r6 | dinvgauss | adam | 0.000446034 | 630.772 |
  outputs: largest KL per observation, conventional form: mixture 0.00115 (Gamma), 0.00448 (log-normal), 0.00400 (inverse Gaussian), one-step 2.359, 5.947, 47.545; ADAM form: mixture 0.01452, 0 (closed form), 0.00045, one-step 4.584, 18.635, 630.772
  seeds: [20261015]
E21 [simulation; data_origin primary; runs 2000] Monte Carlo study of the scale of Gamma and inverse Gaussian iETS(M,N,N) sizes under the ADAM and conventional forms. Part A: 480 cells of simulated relative errors with drift ($s\in\{0.05,0.2,0.5,1\}$, $\alpha\in\{0.1,0.3,0.6\}$, intervals fixed at 2, 5 or 10 or geometric with occurrence probability 0.5 or 0.2, 100 or 1000 observed sizes), 2000 replications per cell, fixed-point iteration with tolerance 1e-10 and at most 100 iterations; a fit fails when an iterate is not positive and finite, the iteration does not converge, or the value leaves the admissible range. Part B: 120 cells within fitted models with $s\in\{0.05,0.5\}$ and 100 observed sizes. Cells are grouped by the drift order $\E(j-1)\alpha^2s$. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.
  setup: studied D15, D5; cases D12, D6, D7; procedure D16; lens D17, D18
  E21.R9 cells where the gain update raised the scale estimate by more than three Monte Carlo standard errors = 97 {estimator: moment equation solved by bisection, observed_sizes: 100, comparison: fitted model with gain update vs simulated drift errors}
  E21.R10 number of cells = 120 {estimator: moment equation solved by bisection, observed_sizes: 100, comparison: fitted model with gain update vs simulated drift errors}
  E21.R11 median increase in relative bias from the gain update = 0.127928 {estimator: moment equation solved by bisection, observed_sizes: 100, comparison: fitted model with gain update vs simulated drift errors}
  E21.T1 Median relative bias and share of failed fits of the fixed-point estimator of the Gamma and inverse Gaussian scale on 1000 observed sizes simulated with the drift, by drift order {estimator: fixed point, observed_sizes: 1000}
  | row | drift_order | median relative bias of the scale estimate | share of failed fits |
  | r1 | <=0.02 | 0.000159064 | 0 |
  | r2 | 0.02-0.1 | -0.000599351 | 0.176735 |
  | r3 | 0.1-0.3 | -0.00418249 | 0.523375 |
  | r4 | >0.3 | -0.00430986 | 0.710058 |
  outputs: fixed-point estimator, 1000 observed sizes, by drift order: <=0.02 median relative bias +0.02%, 0% failed; 0.02-0.1 -0.06%, 18%; 0.1-0.3 -0.42%, 52%; >0.3 -0.43%, 71%. Within fitted models the gain update raised the relative bias in 97 of 120 cells by a median of 12.8 percentage points
  seeds: ["numpy SeedSequence(20261015), one child stream per cell"]
E22 [simulation; data_origin primary; runs 1000000] Comparison of the approximate upper bound of the 90% prediction interval (0.95 quantile) of ETS(M,N,N) with $s=0.32$, $\alpha=0.21$, 100 observations and initial level 10, divided by the point forecast, with the quantile of 1,000,000 simulated paths, twelve steps ahead and 18 steps from the last observation after six unobserved periods at the end of the sample, for Gamma, inverse Gaussian and log-normal errors under the ADAM and conventional forms. The run recorded smooth commit 3eb93bb1 with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.
  setup: studied D19; cases D20, D6, D7; procedure D21; lens D22
  E22.R1 approximate upper bound of the 90% interval over the point forecast = 2.43087 {distribution: Gamma, ets_form: adam, h: 12, unobserved_at_end: 0, s: 0.32, alpha: 0.21}
  E22.R2 simulated upper bound of the 90% interval over the point forecast = 1.80713 {distribution: Gamma, ets_form: adam, h: 12, unobserved_at_end: 0, s: 0.32, alpha: 0.21}
  E22.R3 approximate upper bound of the 90% interval over the point forecast = 2.61587 {distribution: Gamma, ets_form: adam, h: 12, unobserved_at_end: 6, s: 0.32, alpha: 0.21}
  E22.R4 simulated upper bound of the 90% interval over the point forecast = 1.63792 {distribution: Gamma, ets_form: adam, h: 12, unobserved_at_end: 6, s: 0.32, alpha: 0.21}
  E22.R5 approximate upper bound of the 90% interval over the point forecast = 2.43087 {distribution: Gamma, ets_form: conventional, h: 12, unobserved_at_end: 0, s: 0.32, alpha: 0.21}
  E22.R6 simulated upper bound of the 90% interval over the point forecast = 2.39273 {distribution: Gamma, ets_form: conventional, h: 12, unobserved_at_end: 0, s: 0.32, alpha: 0.21}
  E22.R7 approximate upper bound of the 90% interval over the point forecast = 2.61587 {distribution: Gamma, ets_form: conventional, h: 12, unobserved_at_end: 6, s: 0.32, alpha: 0.21}
  E22.R8 simulated upper bound of the 90% interval over the point forecast = 2.53649 {distribution: Gamma, ets_form: conventional, h: 12, unobserved_at_end: 6, s: 0.32, alpha: 0.21}
  outputs: Gamma, bound over point forecast, approximate vs simulated: ADAM form 2.431 vs 1.807 (h=12), 2.616 vs 1.638 (six trailing unobserved); conventional form 2.431 vs 2.393, 2.616 vs 2.537
  seeds: [20261015]
E23 [quant; design benchmark; data_origin primary; n 25] Timing of the SSOE Kalman filter in the smooth package with the low-rank factor against the full covariance, both built from smooth commit 53104ee9, for ADAM ETS(A,N,A) with lags 24 and 168 (193 profile cells), normal distribution and backcast initial states, on 1344 hourly observations with 1 to 48 nested missing values, for data generated by ETS(A,N,A) and by a fixed sine pattern; 25 interleaved repetitions per case on one cloud VM (Intel Xeon 2.80 GHz, 4 vCPUs, Linux 6.18.44, Python 3.11.15) with one thread.
  setup: studied D23, D24; cases D25; procedure D26; lens D27
  E23.R1 speed-up of low-rank over full covariance filter = 10.5511 ratio {data: ets, missing_values: 1, timed_call: filter pass}
  E23.R2 speed-up of low-rank over full covariance filter = 3.50461 ratio {data: ets, missing_values: 48, timed_call: filter pass}
  E23.R3 largest absolute difference in log-likelihood, low-rank vs full = 4.54747e-13 {data: ets}
  E23.R4 speed-up of low-rank over full covariance filter = 2.96089 ratio {data: sine, missing_values: 1, timed_call: filter pass}
  E23.R5 speed-up of low-rank over full covariance filter = 0.723544 ratio {data: sine, missing_values: 48, timed_call: filter pass}
  E23.R6 largest absolute difference in log-likelihood, low-rank vs full = 0 {data: sine}
E24 [simulation; data_origin primary; runs 1] Kullback-Leibler divergence, in both directions, between the exact convolution of Laplace, S and Generalised Normal (shape 0.5 to 3) errors with normal state uncertainty and three approximations (the error kept in its family with scale $sf_t^{1/p}$, the one-step density, and Gauss-Hermite mixtures with 8 to 64 nodes), for 22 values of $f_t$ from 1.01 to 100, by composite 8-node Gauss-Legendre integration. The run recorded smooth commit 3eb93bb1 with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.
  setup: studied D28, D29; cases D30; procedure D31; lens D14
  E24.R1 KL divergence, exact density vs approximation = 0.049114 {distribution: dlaplace, approximation: family_scaled, f_t: 4, s: 1}
  E24.R2 KL divergence, exact density vs approximation = 0.2598 {distribution: ds, approximation: family_scaled, f_t: 4, s: 1}
  E24.R3 KL divergence, exact density vs approximation = 0.00809702 {distribution: dlaplace, approximation: gauss_hermite 16 nodes, f_t: 4, s: 1}
  E24.R4 KL divergence, exact density vs approximation = 0.112709 {distribution: ds, approximation: gauss_hermite 16 nodes, f_t: 4, s: 1}
  outputs: Laplace at f_t=4, family-scaled: 0.0491 (exact||approximation), 0.0563 (reverse); above 0.05 for f_t above about 4.5, levelling off at 0.056; S at f_t=4: 0.260; 16-node Gauss-Hermite at f_t=4: 0.0081 (Laplace), 0.113 (S)
  seeds: [20261015]

WARRANTS
W1 [deductive; premises C2] With a dynamic scale model, the scale at each period with zero demand is a free function of the parameters and enters the entropy-augmented likelihood only through the term $-H_z$; for the normal distribution $-H_z=-\frac12\log(2\pi e\sigma^2)$ increases without bound as $\sigma\to0$, and for the Gamma distribution the entropy decreases in the scale for large values, so the objective has no finite maximum.
W2 [deductive] The expected log-density of an unobserved size is $\int f_z\log f_z\,dz$, which equals $-H_z$ when the differential entropy is $H_z=-\E\log f_z$; the term added for each zero therefore enters the objective with a negative sign.
W3 [deductive; premises C15] The SSOE Kalman filter coincides with the single source of error recursion whenever its covariance is zero; the covariance starts at zero after backcasting and becomes non-zero only at an unobserved period, so on series without unobserved periods the last forward pass performs the same operations as the recursion.

LIMITATIONS
L1 [model_specification; affects C18; could_bias_magnitude; mitigation none] Within a model fitted by the zero-error recursion, the fixed-point moment estimate of the Gamma and inverse Gaussian scale is biased upwards, because the states after an interval are updated with gain $\mathbf g$ and part of the drift leaks into the following errors.
L2 [computational; affects C22, C28; reduces_transferability; mitigation none] The low-rank timing comparison uses one model configuration and one series length on one cloud VM (Intel Xeon 2.80 GHz, 4 vCPUs).

ARTEFACTS
A1 [manuscript] LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).
A2 [software] smooth package for R and Python, including the SSOE Kalman filter in the shared C++ core (src/headers/adamCore.h).
A3 [code] Python scripts (kl_mixture.py, scale_recovery.py, interval_bounds.py, lowrank_timing.py with build_lowrank_variants.sh, kl_laplace.py) that produce the numerical results on the observed-data likelihood of ADAM and iETS.
A4 [table] Kullback-Leibler divergence per observation of the normal-drift mixture density and of the one-step density from the exact density of the iETS(M,N,N) demand size after an interval, for 25,080 cells of distribution, ETS form, scale, smoothing parameter and interval length.
A5 [table] Refinement check of 164 cells of the iETS(M,N,N) Kullback-Leibler grid with half the grid step and a tail cut of 1e-14, with differences from the base run.
A6 [table] Mean, bias, Monte Carlo standard error, relative bias and number of failed fits of scale estimators for Gamma and inverse Gaussian iETS(M,N,N) sizes, on simulated drift errors (480 cells) and within fitted models (120 cells).
A7 [table] Upper bound of the 90% prediction interval of ETS(M,N,N) divided by the point forecast, approximate against simulated, for Gamma, inverse Gaussian and log-normal errors under the ADAM and conventional forms, twelve steps ahead and after six unobserved periods at the end of the sample.
A8 [table] Median and interquartile range of run times of the low-rank and full-covariance SSOE Kalman filter for ADAM ETS(A,N,A) with lags 24 and 168 on 1344 hourly observations with 1 to 48 missing values, with speed-ups and the largest differences in log-likelihood and fitted values.
A9 [table] Kullback-Leibler divergence, in both directions, between the exact convolution of Laplace, S and Generalised Normal errors with normal state uncertainty and its family-scaled, one-step and Gauss-Hermite approximations, for variance ratios $f_t$ from 1.01 to 100.

REFERENCES
B1 10.1111/j.2517-6161.1977.tb01600.x (doi)
B2 10.1080/01621459.1997.10473684 (doi)
B3 10.1111/j.2517-6161.1985.tb01355.x (doi)
B4 10.1016/j.ijpe.2023.109013 (doi)

SCORES
Scores are computed by rules from the claimset file (EntryLib rubric 0.2): they check the record, not the science, and may change as the rubric and reviews develop.
Reliability of the paper: 61 (medium), the score of its lowest principal claim, C6.
C1: reliability 65 (medium); novelty: new, by the authors · no close match on EntryLib
C2: reliability 65 (medium); novelty: refutes earlier work, by the authors · no close match on EntryLib
C3: reliability 65 (medium); novelty: new, by the authors · no close match on EntryLib
C4: reliability 62 (medium); novelty: extends earlier work, by the authors · no close match on EntryLib
C5: reliability 62 (medium); novelty: refutes earlier work, by the authors · no close match on EntryLib
C6: reliability 61 (medium); novelty: new, by the authors · no close match on EntryLib
C7: reliability 61 (medium); novelty: new, by the authors · no close match on EntryLib
C8: reliability 61 (medium); novelty: refutes earlier work, by the authors · no close match on EntryLib
C9: reliability 95 (high); novelty: new, by the authors · no close match on EntryLib
C10: reliability 66 (medium); novelty: new, by the authors · no close match on EntryLib
C11: reliability 61 (medium); novelty: new, by the authors · no close match on EntryLib
C12: reliability 75 (medium); novelty: new, by the authors
C13: reliability 60 (medium); novelty: new, by the authors
C14: reliability 80 (high); novelty: refutes earlier work, by the authors
C15: reliability 62 (medium); novelty: new, by the authors
C16: reliability 62 (medium); novelty: new, by the authors
C17: reliability 75 (medium); novelty: new, by the authors
C18: reliability 63 (medium); novelty: new, by the authors
C19: reliability 72 (medium); novelty: new, by the authors
C20: reliability 72 (medium); novelty: new, by the authors
C21: reliability 62 (medium); novelty: new, by the authors
C22: reliability 60 (medium); novelty: new, by the authors
C23: reliability 62 (medium); novelty: new, by the authors
C24: reliability 62 (medium); novelty: new, by the authors
C25: reliability 100 (high); novelty: new, by the authors
C26: reliability 63 (medium); novelty: new, by the authors
C27: reliability 100 (high); novelty: new, by the authors
C28: reliability 60 (medium); novelty: new, by the authors
C29: reliability 62 (medium); novelty: new, by the authors
Open problems in the principal claims and what they rest on:
P1 [support; up to -30; C1, C2, C5, C6, C8, C10, C12] Rests on an unchecked proof. To remove it: Support the claim with evidence the claimset records, or have its proof checked.
P2 [context; up to -5; C1, C3, C6, C7, C9, C10, C11] A principal claim related to no earlier work. To remove it: Relate the claim to the earlier work it extends, addresses or departs from.
P3 [open; up to -3; C1, C4, C5, C6, C7] Assumption C30 is not checked: The errors $\epsilon_t$ of the single source of error model are independent across periods. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P4 [open; up to -3; C1, C2, C10] Assumption C31 is not checked: In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P5 [open; up to -1; C1, C2, C10] Assumption C32 is not checked: The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P6 [open; up to -1; C4, C5, C6] Assumption C33 is not checked: Whether a period is observed carries no information about the unobserved value $z_t$ beyond the model, so that the missing values are missing at random. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P7 [open; up to -1; C4, C5, C6] Assumption C34 is not checked: The initial states of the single source of error model are fixed parameters, so that the state covariance is zero at the start of the sample. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P8 [open; up to -1; C4, C5] Assumption C35 is not checked: The errors of the pure additive model are normally distributed with constant variance $\sigma^2$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
P9 [open; up to -3; C6, C7, C8] Assumption C36 is not checked: In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$, so that $\E(1+\epsilon_t)=1$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

HOW TO CITE
This claimset: Svetunkov, I. (guarantor) (2026). The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v1. https://entrylib.org/p/2026.000003@v1
One claim of it: by its identifier, as entrylib:2026.000003@v1#C1 for claim C1. Each principal claim has its own page, as https://entrylib.org/p/2026.000003/C1.