The observed-data likelihood of ADAM and iETS with unobserved periods: claim C6

The claim

91C6 · formal · novelty: new, by the authors · no close match on EntryLib

For the ADAM multiplicative ETS with the log-normal distribution and unobserved periods, the SSOE Kalman filter applied to $\log z_t$ , with the known error mean $-s/2$ added to the prediction and, multiplied by $\mathbf g$ , to the propagation, 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: positive univariate time series with unobserved periods; conditions: log-normal distribution with $\E(1+\epsilon_t)=1$ · Proof: E9 · Uses: C4, C12 · Assumes: C30, C33, C34, C36 · Extends: B5 · Verification: checked by an agent

95C1 · formal · novelty: new, by the authors · no close match on EntryLib

For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\sim\text{Bernoulli}(p_t)$ independent of a positive continuous size $z_t$ , the 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: intermittent demand time series; conditions: Bernoulli occurrence independent of a continuous demand size · Proof: E1 · Assumes: C30, C31, C32 · Extends: B1 · Extends: B4, Appendix B, eq. B.6 · Verification: checked by an agent

92C4 · formal · extends earlier work · novelty: extends earlier work, by the authors · no close match on EntryLib

For a pure additive single source of error model with normal errors and unobserved periods, the 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: univariate time series with unobserved periods; conditions: normally distributed errors, initial states treated as fixed parameters; not for: models mixing additive and multiplicative components · Proof: E4 · Uses: C1 · Assumes: C30, C33, C34, C35 · Extends: B6, Kalman filter for state space models with correlated disturbances · Extends: B7, Kalman filter for state space models with correlated disturbances · Verification: checked by an agent

Claims it rests on

100C12 · formal · novelty: new, by the authors

In the 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 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 · Proof: E8 · Verification: checked by an agent · Used by: C6

Evidence

E1 · formal · cited by C1

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C31, C32

$P(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t$ (independence and continuity; from C31, C32)
$y_t>0$ has density $p_tf(y_t\mid Y_{t-1})$ (independence; from C31)
The joint density of the observed data is the product of these conditional terms (prediction-error decomposition; from E1.1, E1.2)

E4 · formal · cited by C4

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C30, C33, C34, C35

$(\mathbf x_{t-1},z_t,\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$ (linearity of Gaussian models; from C35, C30)
$\begin{aligned} \E(z_t\mid Y_{t-1}) &= \mathbf w'\mathbf a_t \\ \Var(z_t\mid Y_{t-1}) &= \sigma^2f_t \\ \Cov(\mathbf x_t,z_t\mid Y_{t-1}) &= \sigma^2(\mathbf F\mathbf P_t\mathbf w+\mathbf g) \end{aligned}$ (moments of linear combinations; from E4.1)
Conditioning on $z_t$ gives the update; unobserved periods only propagate (Gaussian conditioning; from E4.2, C33)
The likelihood follows from the observed-data decomposition, and $\sigma^2$ concentrates out because $f_t$ , $\mathbf k_t$ do not depend on it (prediction-error decomposition; from E4.3, C1)

E8 · formal · cited by C12

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms

$\log z_t=\log l_{t-1}+\phi\log b_{t-1}+\log s_{t-m}+u_t$ (logarithm; from D6)
$\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\alpha u_t$ , $\log b_t=\phi\log b_{t-1}+\beta u_t$ in the ADAM form (logarithm; from D6)
$\log l_t=\log l_{t-1}+\phi\log b_{t-1}+\log(1+\alpha\epsilon_t)$ in the conventional form (logarithm; from D7)

E9 · formal · cited by C6

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C30, C33, C34, C36

$\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$ (linearity in logarithms; from C12, C36)
The filter of the linear Gaussian model with the Jacobian $-\sum\log z_t$ gives the exact likelihood (change of variables; from C33, C34, C4, E9.1)
Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$ (linearity; from E9.2)
The score in $s$ times $2s^2$ gives $(\sum b_t^2/f_t)s^2+T_1s-\sum(v_t^0)^2/f_t=0$ (differentiation; from E9.3)

Assumptions

C30 · statistical · unchecked

The errors $\epsilon_t$ of the single source of error model are independent across periods. Rested on by C6, C1, C4, E4, E9.

C31 · theorem_condition · unchecked

In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$ . Rested on by C1, E1.

C32 · theorem_condition · unchecked

The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$ . Rested on by C1, E1.

C33 · statistical · unchecked

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. Rested on by C6, C4, E4, E9.

C34 · theorem_condition · unchecked

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. Rested on by C6, C4, E4, E9.

C35 · theorem_condition · unchecked

The errors of the pure additive model are normally distributed with constant variance $\sigma^2$ . Rested on by C4, E4.

C36 · theorem_condition · unchecked

In the log-normal ADAM multiplicative ETS, $\log(1+\epsilon_t)\sim\mathcal N(-s/2,s)$ , so that $\E(1+\epsilon_t)=1$ . Rested on by C6, E9.

Definitions

Methods 3

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 $m$ , $\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\phi}s_{t-m}(1+\epsilon_t) \\ l_t &= l_{t-1}b_{t-1}^{\phi}(1+\alpha\epsilon_t) \\ b_t &= b_{t-1}^{\phi}(1+\beta\epsilon_t) \\ s_t &= s_{t-m}(1+\gamma\epsilon_t). \end{aligned}$

D8 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 $\mathbf a_1=\mathbf x_0$ , $\mathbf P_1=\mathbf 0$ .

Concepts 2

D2 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 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: $L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$

Context

Extends B5 · own earlier work · from C6: The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed.
Extends B1 · from C1: Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood.
Extends B4, Appendix B, eq. B.6 · own earlier work · from C1: Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand.
Extends B6, Kalman filter for state space models with correlated disturbances · from C4: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated.
Extends B7, Kalman filter for state space models with correlated disturbances · from C4: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations.

Artefacts

Score

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. The rubric.

C6: 91, High

Novelty: new, by the authors · no close match on EntryLib

−3 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.

C1: 95, High

Novelty: new, by the authors · no close match on EntryLib

−3 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.

C4: 92, High

Novelty: extends earlier work, by the authors · no close match on EntryLib

−3 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.
−1 Open problems 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.

C12: 100, High

Novelty: new, by the authors

Nothing lost.

The observed-data likelihood of ADAM and iETS with unobserved periods · C6

The claim

Claims it rests on

Evidence

Assumptions

Definitions

Methods 3

Concepts 2

Context

Artefacts

Score

C6: 91, High

C1: 95, High

C4: 92, High

C12: 100, High