Admissibility of lagged state space models through the ARIMA form of the whole model: claim C22

The claim

99C22 · descriptive · refutes earlier work · novelty: refutes earlier work, by the authors · no close match on EntryLib

For ETS(A,N,A) with $m = 12$ , $\alpha = 1.2$ and $\gamma = 0.9$ , the stability condition of Section 5.4 of the ADAM monograph holds, while in the full discount recursion the effect of the initial states on the fitted values and the weight of a past observation both grow from period to period, and the largest root modulus of the cancelled MA polynomial exceeds one.

Scope: population: ETS(A,N,A) with m = 12; conditions: alpha = 1.2, gamma = 0.9, initial states set to one or one unit observation, 2400 periods · Simulation: E18 (T1.r2), artefacts A1, A6 · Uses: C21 · Limitations: L6 · Contradicts: Svetunkov (2023), Section 5.4, equations 5.20 to 5.22

Claims it rests on

100C21 · formal · novelty: new, by the authors

The per-lag discount matrices $D_{m_i} = F_{m_i} - g_{m_i} w_{m_i}'$ of Section 5.4 of the ADAM monograph have as nonzero eigenvalues those of the diagonal block of $D = F - g w'$ for the lag $m_i$ (the per-lag check), and their sum is $D + \sum_{i \neq j} g_{m_i} w_{m_j}',$ which differs from $D$ whenever $g w'$ links components of distinct lags.

Scope: population: pure additive lagged SSOE models with more than one distinct lag; conditions: F split by the columns, and g and w by the components, of each lag, as in Section 5.2 of the monograph, which writes g_{m_i} and w_{m_i} without defining their split; with g kept whole the first part holds and the sum equals D · Proof: E17 · Verification: checked by an agent · Used by: C22

Evidence

E1 · simulation

Planned test of the per-lag check (lab plan PL1, run RN7): confirm.py --test perlag --seed 2026 --n 200 computed, for 3042 ETS(A,N,A) models (m = 4 and 12, alpha and gamma on the grid 0.05, 0.10, ..., 1.95) and 200 GUM models with lags (1, 12) (entries of F normal with mean 0 and standard deviation 0.35, of g with mean 0 and standard deviation 0.175, of w with mean 1 and standard deviation 0.175), the per-lag check's largest eigenvalue modulus and the PBH reference, and counted the models on which the two decisions disagree beyond a band of 1e-8 around 1.

Setup: studied per-lag check · cases ETS(A,N,A) grid, random GUM models with lags (1, 12) · procedure comparison with a reference, PBH reference · lens admissibility decision

R1 models accepted by the per-lag check that do not forget their initial states = 1482 (family: ETS(A,N,A), models: 3042)
R2 models rejected by the per-lag check that forget their initial states = 0 (family: ETS(A,N,A), models: 3042)
R3 models accepted by the per-lag check = 3042 (family: ETS(A,N,A), models: 3042)
R4 models accepted by the per-lag check that do not forget their initial states = 10 (family: GUM, models: 200)
R5 models rejected by the per-lag check that forget their initial states = 1 (family: GUM, models: 200)
R6 models accepted by the per-lag check = 193 (family: GUM, models: 200)

E17 · formal · cited by C21

The per-lag discount matrices of Section 5.4 of the ADAM monograph against D.

Foundation: linear algebra over the complex numbers

$F_{m_i} = F_{\cdot S_i},\ g_{m_i} = g_{S_i},\ w_{m_i} = w_{S_i} \text{ (zero elsewhere), } S_i \text{ the components of lag } m_i$ (the split of Section 5.2 of the monograph; from D1)
$D_{m_i} = F_{m_i} - g_{m_i} w_{m_i}' \text{ has non-zero columns only in } S_i$ (definition; from S1)
$D_{m_i} \sim \begin{pmatrix} D_{S_i S_i} & 0 \\ \ast & 0 \end{pmatrix} \Rightarrow \mathrm{spec}(D_{m_i}) \setminus \{0\} = \mathrm{spec}(D_{S_i S_i}) \setminus \{0\}$ (a symmetric permutation and the eigenvalues of a block-triangular matrix; from S2)
$\sum_i D_{m_i} = F - \sum_i g_{m_i} w_{m_i}' = D + \sum_{i \neq j} g_{m_i} w_{m_j}'$ ($g w' = \sum_i \sum_j g_{m_i} w_{m_j}'$; from S1)
$(a, b) \text{ with lags } m_i \neq m_j: \left(\sum_{i \neq j} g_{m_i} w_{m_j}'\right)_{ab} = g_a w_b$ (a single term, so the sum differs from D exactly where g w' links components of different lags; from S4)

E18 · simulation · cited by C22

Exploratory run RN44 of book_check.py --periods 2400: for ETS(A,N,A) with m = 12 and (alpha, gamma) = (0.1, 0.1) and (1.2, 0.9), the largest nonzero eigenvalue modulus of the per-lag discount matrices of the monograph, the effect $|w' v_{t-l}|$ of initial states set to one in the full recursion $v_t = D v_{t-l}$ ( $y = 0$ ) over 2400 periods and its growth per period over the second half, the growth of the weight of an observation (the same recursion from zero states with one unit observation), and the largest root modulus of the cancelled MA polynomial.

Setup: studied stability condition of the ADAM monograph · cases ETS(A,N,A) with m = 12 · procedure discount recursion over 2400 periods · lens largest mode modulus, growth per period

E18.T1 · For ETS(A,N,A) with m = 12 at two parameter pairs: the largest nonzero eigenvalue modulus of the monograph's per-lag discount matrices, the effect of the initial states on the fitted value at period 2400 and its growth per period over the second half, the growth of the weight of an observation, and the largest root modulus of the cancelled MA polynomial (model: ETS(A,N,A), m: 12)
Row	alpha	gamma	largest nonzero eigenvalue modulus of the book's per-lag discount matrices	effect of the initial states on the fitted value at the last period (period: 2400)	growth of the effect of the initial states per period over the second half	growth of the weight of an observation per period over the second half	largest root modulus of the cancelled MA polynomial
r1	0.1	0.1	0.9	0.0000	0.9896	0.9909	0.9911
r2	1.2	0.9	0.2	5.96878e+16	1.0164	1.0164	1.0165

Limitations

L6 · other · reduces transferability · affects C22 · mitigation: partial

The recursion of the counterexample to the monograph's condition was run for one parameter pair, with the initial states set to one and no data.

Definitions

Methods 6

D1 lagged SSOE model: A pure additive single-source-of-error state space model with $k$ components, $\begin{aligned} y_t &= w' v_{t-l} + \epsilon_t \\ v_t &= F v_{t-l} + g \epsilon_t, \end{aligned}$ in which component $j$ of $v_{t-l}$ enters with its own lag $l_j$ ; its discount matrix is $D = F - g w'$ and its lag matrix is $L(B) = \mathrm{diag}(B^{l_1}, \dots, B^{l_k})$ for the backshift operator $B$ ; $\Lambda(\lambda) = \mathrm{diag}(\lambda^{l_1}, \dots, \lambda^{l_k})$ for a complex number $\lambda$ .

D4 per-lag check: The test applied by smoothEigens() of smooth 4.6.0 and of its repository until commit 4a44c1c (src/headers/eigenCalc.h) to models without adaptive regressors: every eigenvalue of every diagonal block of $D$ , one block per unique lag, has modulus below 1; reimplemented in numpy as per_lag_max().

D5 PBH reference: The largest modulus among the eigenvalues $\lambda$ of the expanded discount matrix for which the smallest singular values of $[\lambda I - \tilde D;\ \tilde w']$ and of $[\lambda I - \tilde D,\ \tilde g]$ both exceed $10^{-7}(\|\tilde D\|_2 + \|\tilde g\| + \|\tilde w\|)$ (Popov-Belevitch-Hautus rank tests), computed with numpy as pbh_max().

D8 GUM: Generalised Univariate Model of the smooth package: a lagged SSOE model whose $F$ , $g$ and $w$ are all estimated. Source: Svetunkov (2016).

D9 ETS(A,N,A): Exponential smoothing with additive error, no trend and additive seasonality of period $m$ ; here in the lagged form with a level of lag 1 and one seasonal state of lag $m$ , $F = I$ , $w = (1, 1)'$ , $g = (\alpha, \gamma)'$ . Source: Hyndman et al. (2008).

D16 stability condition of the ADAM monograph: The condition of Section 5.4 of the ADAM monograph (eqs 5.20-5.22): every nonzero eigenvalue of each per-lag discount matrix $D_{m_i} = F_{m_i} - g_{m_i} w_{m_i}'$ has modulus below 1. Source: Svetunkov (2023).

Data 3

D17 ETS(A,N,A) grid: The 3042 ETS(A,N,A) models with $m = 4$ and $m = 12$ and $\alpha$ and $\gamma$ on the grid 0.05, 0.10, ..., 1.95 (confirm.py, plan PL1).

D18 random GUM models with lags (1, 12): 200 GUM models with lags (1, 12): entries of $F$ normal with mean 0 and standard deviation 0.35, of $g$ with mean 0 and standard deviation 0.175, of $w$ with mean 1 and standard deviation 0.175 (confirm.py, seed 2026).

D35 ETS(A,N,A) with m = 12: ETS(A,N,A) with $m = 12$ at $(\alpha, \gamma) = (0.1, 0.1)$ and $(1.2, 0.9)$ (book_check.py).

Variables and measures 3

D29 admissibility decision: Whether a method accepts a model as admissible (decaying weights); compared between a method and a reference as the number of models on which the two differ.

D30 largest mode modulus: The largest modulus of the modes that decide admissibility: for the PBH reference the largest $|\lambda|$ of the reachable and observable eigenvalues of the expanded discount matrix, for the check of the whole model $1/\min|B|$ over the roots $B$ of the cancelled MA polynomial, for the per-lag check the largest $|\lambda|$ of its blocks; a model is admissible when it is below 1.

D34 growth per period: The growth per period, over the second half of the recursion, of the effect $|w' v_{t-l}|$ of initial states set to one on the fitted value, and of the weight of one unit observation.

Study design 2

D26 comparison with a reference: Each model's admissibility decision, and its largest mode modulus where both find it inadmissible, computed by the method studied and by a reference on the same model (the PBH reference, the roots by numpy, the numpy prototype of the check of the whole model or an exact oracle); the result is the number of models whose decisions differ and the largest absolute difference of the modulus. In the planned tests, models whose reference modulus lies within 1e-8 of 1 are left out and counted.

D28 discount recursion over 2400 periods: The full discount recursion $v_t = D v_{t-l}$ of a lagged SSOE model run for 2400 periods, from initial states set to one with no data and from zero states with one unit observation, its growth per period measured over the second half (book_check.py).

Concepts 2

D2 expanded discount matrix: The transition matrix $\tilde D$ of size $\sum_j l_j$ of the lag-1 system equivalent to $v_t = D v_{t-l}$ , in which component $j$ carries itself and its $l_j - 1$ lagged copies; $\tilde F$ , $\tilde g$ and $\tilde w$ are expanded in the same way, so that $\tilde D = \tilde F - \tilde g \tilde w'$ .

D3 decaying weights: A lagged SSOE model has decaying weights when the weights $\tilde w' \tilde D^j \tilde g$ of the past observations in its fitted values tend to zero, that is, when every eigenvalue $\lambda$ of its expanded discount matrix that is reachable from $\tilde g$ and observable through $\tilde w$ has $|\lambda| < 1$ ; the largest such $|\lambda|$ is its largest mode modulus. A mode that $\tilde g$ does not reach but $\tilde w$ sees carries an initial state whose effect is not covered: with $|\lambda| > 1$ it grows.

Context

Contradicts Svetunkov (2023), Section 5.4, equations 5.20 to 5.22 · own earlier work · from C22: The monograph states that the model is stable when all the nonzero eigenvalues of the per-lag discount matrices D_{m_i} are below one in absolute value, and forecastable when the effect of the initial states through each D_{m_i} tends to a constant (eq. 5.22).

References

Hyndman et al. (2008) Hyndman, R., Koehler, A., Ord, K., & Snyder, R. (2008). Forecasting with Exponential Smoothing. Springer Series in Statistics. doi:10.1007/978-3-540-71918-2 Used by: D9
Svetunkov (2016) Svetunkov, I. (2016). smooth: Forecasting Using State Space Models. CRAN: Contributed Packages. doi:10.32614/CRAN.package.smooth Used by: D8
Svetunkov (2023) Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM). Chapman and Hall/CRC. doi:10.1201/9781003452652 Used by: Context, D16

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.

C22: 99, High

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

−1 Open problems Limitation L6 (reduces transferability, mitigation partial): The recursion of the counterexample to the monograph's condition was run for one parameter pair, with the initial states set to one and no data. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

C21: 100, High

Novelty: new, by the authors

Nothing lost.