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Claim C3 · methodological · entrylib:2026.000002@v3 · 2026-11-10

Corrected · 10 November 2026 at 22:09 UTC

Limitations L2 and L4 are answered by a planned independent check (new evidence E25): an exact oracle in rational arithmetic with certified root enclosures, written by a new contributor (P5) without the prototype, compared on the models of the planned tests; it agrees with the PBH reference, the cyclotomic cancellation, the step-down test and the bisection on every model it decides, and its four hypotheses are reported as claims C29-C32, one of them inconclusive by its rule. The assumptions C14 (reworded to the band the tests leave out) and C19 are tested by it; claims C1-C5, C10, C11 and C16 carry its checks; claims C3, C4, C16 and C24-C26 relate to the earlier work they extend, supersede or contradict. The lab record (A9) and the results file (A18) are updated to the record's current version. No earlier claim's conclusion or ID changed.

By Ivan Svetunkov (ORCID 0000-0001-7826-0281)

Admissibility of lagged state space models through the ARIMA form of the whole model · C3

Principal claim C3 of this claimset (Svetunkov, I. (guarantor), 2026), with what it rests on. Read the whole claimset.

Copies a message for ChatGPT, Claude or Gemini: paste it there.

The message

Read https://entrylib.org/p/2026.000002/C3, one claim of a research paper with the evidence it rests on. Tell me briefly what it says and how well it is supported. If you cannot open the link, tell me instead of guessing. Then wait for my questions.

The claim

95C3 · methodological · planned test · novelty: new, by the authors · no close match on EntryLib

On 500 random additive ETS and GUM models, the admissibility decision from the roots of the MA polynomial after cyclotomic cancellation differed from the PBH reference on 0 models, and on the 172 models that both found inadmissible the largest mode modulus differed from the reference by at most 4.4e-13.

Scope: population: pure additive lagged SSOE models; conditions: additive ETS: level; no, additive or damped trend; 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; optional constant; some persistence fixed at 0, GUM with F = I and lags from {1, 4, 12}, GUM with random F and lags from {1, 4, 12, 24}, numpy prototype, models within 1e-8 of the unit circle excluded (none occurred); not for: ARIMA components, CES, adaptive regressors, multiplicative models, seasonal harmonics with non-integer periods (TBATS), the largest mode modulus of admissible models, where a common factor that is not cyclotomic can remain in the cancelled MA polynomial, an eigenvalue of modulus above 1 shared by a reachable and observable mode and an unreachable or unobservable one · Simulation: E3 (R1, R2), artefacts A1, A3 · Uses: C6, C8, C17 · Assumes: C14 · Limitations: L2, L3 · Extends: Hyndman et al. (2008), Chapter 11 · Planned: lab plan PL3, planned 21 October 2026 · Outcome: computed from the recorded results

Claims it rests on

100C6 · formal · extends earlier work · novelty: extends earlier work, by the authors

For a lagged SSOE model, ignoring the initial states, det⁡(I−DL(B))ϵt=det⁡(I−FL(B))yt,\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t, so the model has the ARIMA form with AR polynomial ϕ(B)=det⁡(I−FL(B))\phi(B) = \det(I - F L(B)) and MA polynomial θ(B)=det⁡(I−DL(B))\theta(B) = \det(I - D L(B)).

Scope: population: pure additive lagged SSOE models; conditions: identity of lag-operator polynomials, initial states ignored · Proof: E6 · Consensus · Simulation: E10, artefacts A1, A2 · Extends: Hyndman et al. (2008), Chapter 11 · Verification: checked by an agent · Used by: C3, C17

100C8 · formal · novelty: new, by the authors

An eigenvalue λ≠0\lambda \neq 0 of the expanded transition matrix F~\tilde F with an eigenvector unobservable through w~\tilde w or a left eigenvector unreachable from g~\tilde g is an eigenvalue of the expanded discount matrix, and 1/λ1/\lambda is a common root of ϕ(B)=det⁡(I−FL(B))\phi(B) = \det(I - F L(B)) and θ(B)=det⁡(I−DL(B))\theta(B) = \det(I - D L(B)).

Scope: population: pure additive lagged SSOE models · Proof: E8 · Uses: C15 · Verification: checked by an agent · Used by: C3

100C15 · formal · novelty: new, by the authors

For the lag-1 expansion M~\tilde M of vt=Mvt−lv_t = M v_{t-l} in a lagged SSOE model, det⁡(λI−M~)=det⁡(Λ(λ)−M),\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M), which equals λ∑jljdet⁡(I−ML(1/λ))\lambda^{\sum_j l_j} \det(I - M L(1/\lambda)) for λ≠0\lambda \neq 0.

Scope: population: pure additive lagged SSOE models; conditions: any square matrix MM of size kk · Proof: E11 · Verification: checked by an agent · Used by: C8, C17

100C17 · formal · novelty: new, by the authors

For a lagged SSOE model with θ(B)=det⁡(I−DL(B))\theta(B) = \det(I - D L(B)) and ϕ(B)=det⁡(I−FL(B))\phi(B) = \det(I - F L(B)), the roots of θ(B)/gcd⁡(θ(B),ϕ(B))\theta(B) / \gcd(\theta(B), \phi(B)) are exactly the reciprocals 1/λ1/\lambda of the nonzero eigenvalues λ\lambda of the reachable and observable part of the expanded discount matrix (its Kalman decomposition), with their multiplicities.

Scope: population: pure additive lagged SSOE models; conditions: the definition D3 agrees with this part unless an eigenvalue is shared by the reachable and observable part and the rest · Proof: E14 · Consensus · Uses: C6, C15 · Verification: checked by an agent · Used by: C3

Evidence

E3 · simulation · cited by C3

Planned test of the cyclotomic cancellation (lab plan PL3, run RN9): confirm.py --test cancel --seed 2026 --n 200 generated 200 additive ETS models (level; no, additive or damped trend with phi in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; beta = 0 with probability 0.1 and each gamma = 0 with probability 0.15; alpha in (0.01, 1.9)), 100 GUM models with F = I, lags from {1, 4, 12}, g uniform on (-0.3, 0.6) and w normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal F (standard deviation 0.35); for each it compared the largest root modulus of the cancelled MA polynomial with the PBH reference.

Setup: studied check of the whole model · cases random additive ETS and GUM models · procedure comparison with a reference, PBH reference · lens admissibility decision, largest mode modulus

  • R1 admissibility decisions that differ from the PBH reference = 0
  • R2 max abs difference of max |lambda| from the PBH reference, inadmissible models = 0.0000 (inadmissible_models: 172)
  • R3 models within 1e-8 of the unit circle, excluded = 0
  • R4 models compared = 500

E10 · simulation · cited by C6

Exploratory run RN2 of explore.py --seed 0: the ARIMA-form polynomials, their cyclotomic cancellation and the PBH reference for ETS(A,N,A,A) with seasonal lags 24 and 168 (alpha = 0.3, gamma = (0.1, 0.05)) and for GUM with orders (2, 2), lags (1, 12), F = I, g = 0.1 and w = 1 (the default starting point of gum()); the determinant identity at random complex B on 100 random GUM models; the expanded discount matrix of a block-diagonal D; and mean times over 50 repeats of three computations for a random GUM with lags (1, 24, 168). The result labelled 'block-diagonal D, lags (1,12)' comes from D = [[0.7, 0], [-0.1, 0.3]], which is lower-triangular rather than block-diagonal.

Setup: studied lagged SSOE model, cyclotomic cancellation, check of the whole model · cases ETS(A,N,A,A) with seasonal lags 24 and 168, starting point of gum() · procedure comparison with a reference, PBH reference · lens largest mode modulus

  • R11 max relative error of the determinant identity = 0.0000 (models: 100)
  • R12 max |lambda|, per-lag check = 0.7 (model: block-diagonal D, lags (1,12))
  • R13 max |lambda|, all modes of the expanded discount matrix = 0.9045 (model: block-diagonal D, lags (1,12))
  • R14 mean time, ms = 4.6963 (task: coefficients by DFT, degree: 193)
  • R15 mean time, ms = 0.0069 (task: step-down test, degree: 193)
  • R16 mean time, ms = 22.2257 (task: all roots, numpy companion eigenvalues, degree: 193)
E10.T1 · Degree of the MA polynomial theta and of its shared cyclotomic factor, and the largest mode modulus from the roots of theta before and after cancellation and from the PBH reference, for ETS(A,N,A,A) with seasonal lags 24 and 168 and for the default starting point of gum() with orders (2, 2) and lags (1, 12)
Rowmodeldegree of thetadegree of the shared cyclotomic factormax |lambda|, roots of theta before cancellationmax |lambda|, roots of theta after cancellationmax |lambda|, controllable and observable modes (PBH)
r1ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05)19325111
r2GUM start, orders (2,2), lags (1,12)261410.98290.9829

E6 · formal · cited by C6

Derivation of the ARIMA form of a lagged SSOE model by the matrix determinant lemma.

Foundation: linear algebra over the field of rational functions in $B$

  1. ϵt=yt−w′vt−l⇒vt=Dvt−l+gyt,D=F−gw′\epsilon_t = y_t - w' v_{t-l} \Rightarrow v_t = D v_{t-l} + g y_t,\ D = F - g w' (substitution; from D1)
  2. vt−l=L(B)vt⇒(I−DL(B))vt=gytv_{t-l} = L(B) v_t \Rightarrow (I - D L(B)) v_t = g y_t (definition of $L(B)$; from S1, D1)
  3. det⁡(I−DL(B))|B=0=1⇒I−DL(B) is invertible over rational functions\det(I - D L(B))|_{B=0} = 1 \Rightarrow I - D L(B) \text{ is invertible over rational functions} (evaluation at $B = 0$; from S2)
  4. ϵt=(1−w′L(B)(I−DL(B))−1g)yt\epsilon_t = \left(1 - w' L(B) (I - D L(B))^{-1} g\right) y_t (substitution; from S1, S2, S3)
  5. I−DL(B)−gw′L(B)=I−FL(B)I - D L(B) - g\, w' L(B) = I - F L(B) (definition of $D$; from D1)
  6. det⁡(I−FL(B))=det⁡(I−DL(B))(1−w′L(B)(I−DL(B))−1g)\det(I - F L(B)) = \det(I - D L(B))\left(1 - w' L(B)(I - D L(B))^{-1} g\right) (matrix determinant lemma $\det(A - g u') = \det(A)(1 - u' A^{-1} g)$; from S3, S5)
  7. det⁡(I−DL(B))ϵt=det⁡(I−FL(B))yt\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t (substitution; from S4, S6)

E8 · formal · cited by C8

Unobservable or unreachable modes of the expanded transition matrix are modes of the expanded discount matrix and give common roots of the two ARIMA-form polynomials.

Foundation: linear algebra over the complex numbers

  1. F~u=λu,w~′u=0⇒D~u=F~u−g~(w~′u)=λu\tilde F u = \lambda u,\ \tilde w' u = 0 \Rightarrow \tilde D u = \tilde F u - \tilde g (\tilde w' u) = \lambda u (substitution of $\tilde D = \tilde F - \tilde g \tilde w'$; from D2)
  2. u′F~=λu′,u′g~=0⇒u′D~=u′F~−(u′g~)w~′=λu′u' \tilde F = \lambda u',\ u' \tilde g = 0 \Rightarrow u' \tilde D = u' \tilde F - (u' \tilde g) \tilde w' = \lambda u' (substitution of $\tilde D = \tilde F - \tilde g \tilde w'$; from D2)
  3. det⁡(λI−M~)=λ∑jljdet⁡(I−ML(1/λ)),λ≠0\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0 (identity of the lag-1 expansion; from C15)
  4. ϕ(1/λ)=det⁡(I−FL(1/λ))=0,θ(1/λ)=det⁡(I−DL(1/λ))=0\phi(1/\lambda) = \det(I - F L(1/\lambda)) = 0,\ \theta(1/\lambda) = \det(I - D L(1/\lambda)) = 0 (substitution; from S1, S2, S3)

E11 · formal · cited by C15

Schur-complement derivation of the characteristic polynomial of the lag-1 expansion.

Foundation: linear algebra over the complex numbers

  1. λI−M~=(λIk−A−B−CλI−S)\lambda I - \tilde M = \begin{pmatrix} \lambda I_k - A & -B \\ -C & \lambda I - S \end{pmatrix} (partition of the states into the $k$ current values (heads) and the $\sum_j (l_j - 1)$ lagged copies (tails); $S$ shifts each tail chain, $C$ feeds each head into its first copy, $A$ and $B$ hold the entries of $M$ that act on heads with lag 1 and on last copies; from D2)
  2. det⁡(λI−S)=λ∑j(lj−1)\det(\lambda I - S) = \lambda^{\sum_j (l_j - 1)} (determinant of a lower-triangular matrix with diagonal $\lambda$; from S1)
  3. A+B(λI−S)−1C=Mdiag(λ1−l1,…,λ1−lk),λ≠0A + B (\lambda I - S)^{-1} C = M\, \mathrm{diag}(\lambda^{1 - l_1}, \dots, \lambda^{1 - l_k}),\ \lambda \neq 0 ($(\lambda I - S)^{-1} C$ maps head $j$ to its copies $\lambda^{-s}$, $s = 1, \dots, l_j - 1$, and $B$ reads copy $l_i - 1$ of component $i$; from S1)
  4. det⁡(λI−M~)=λ∑j(lj−1)det⁡(λIk−Mdiag(λ1−lj))=det⁡(Λ(λ)−M),λ≠0\det(\lambda I - \tilde M) = \lambda^{\sum_j (l_j - 1)} \det(\lambda I_k - M\, \mathrm{diag}(\lambda^{1 - l_j})) = \det(\Lambda(\lambda) - M),\ \lambda \neq 0 (Schur complement, then factoring $\lambda\, \Lambda(\lambda)^{-1}$ out on the right; from S2, S3)
  5. det⁡(λI−M~)=det⁡(Λ(λ)−M) for all λ\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M) \text{ for all } \lambda (two polynomials equal on $\lambda \neq 0$ are equal; from S4)
  6. det⁡(λI−M~)=λ∑jljdet⁡(I−ML(1/λ)),λ≠0\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0 (factoring $\Lambda(\lambda)$ out on the right: $\Lambda(\lambda) - M = (I - M \Lambda(\lambda)^{-1}) \Lambda(\lambda)$; from S5, D1)

E14 · formal · cited by C17

The reduced form of the ARIMA polynomials through the Kalman decomposition of the expanded system.

Foundation: linear systems theory over the complex numbers

  1. ∃T:T−1F~T block-triangular with a reachable and observable block (Fco,gco,wco),w~′(zI−F~)−1g~=wco′(zI−Fco)−1gco\exists T: T^{-1}\tilde F T \text{ block-triangular with a reachable and observable block } (F_{co}, g_{co}, w_{co}),\ \tilde w'(zI - \tilde F)^{-1}\tilde g = w_{co}'(zI - F_{co})^{-1} g_{co} (Kalman canonical decomposition; from D2)
  2. D~=F~−g~w~′ keeps the reachable and the unobservable subspaces, so Dco=Fco−gcowco′\tilde D = \tilde F - \tilde g \tilde w' \text{ keeps the reachable and the unobservable subspaces, so } D_{co} = F_{co} - g_{co} w_{co}' (reachability is invariant under state feedback and observability under output injection; from S1, D2)
  3. 1+wco′(zI−Fco)−1gco=det⁡(zI−Dco)det⁡(zI−Fco),gcd⁡(det⁡(zI−Dco),det⁡(zI−Fco))=11 + w_{co}'(zI - F_{co})^{-1} g_{co} = \frac{\det(zI - D_{co})}{\det(zI - F_{co})},\ \gcd(\det(zI - D_{co}), \det(zI - F_{co})) = 1 (matrix determinant lemma; a single-input single-output realisation is minimal if and only if the numerator and denominator of its transfer function are coprime; from S2)
  4. 1+w~′(zI−F~)−1g~=det⁡(zI−D~)det⁡(zI−F~)=θ(1/z)ϕ(1/z)1 + \tilde w'(zI - \tilde F)^{-1}\tilde g = \frac{\det(zI - \tilde D)}{\det(zI - \tilde F)} = \frac{\theta(1/z)}{\phi(1/z)} (matrix determinant lemma and the identity of the lag-1 expansion; from C15, D1)
  5. θ(B)gcd⁡(θ,ϕ)(B)=0⟺B=1/λ,λ∈spec(Dco)∖{0} (with multiplicity)\frac{\theta(B)}{\gcd(\theta, \phi)}(B) = 0 \iff B = 1/\lambda,\ \lambda \in \mathrm{spec}(D_{co}) \setminus \{0\} \text{ (with multiplicity)} (uniqueness of the reduced fraction, with z = 1/B: the eigenvalues at zero give no root, theta(0) = phi(0) = 1 makes theta / gcd the reduced numerator, and the reversal keeps coprimality away from B = 0; from S3, S4)

E25 · simulation

Planned test of the contributor P5 (lab plan AC5.PL1, run AC5.RN8, with the breakdown run AC5.RN9): the exact oracle, written without the prototype, on the models of the planned tests exported by runs RN57-RN61, against the prototype's values stored with each model. The oracle was tried beforehand only on models of other seeds (AC5.RN2, AC5.RN6, AC5.RN7) and on hand-checkable models. AC5.RN8 took 30 s.

Setup: studied PBH reference, check of the whole model, step-down test, bisection for the largest root modulus, per-lag check · cases models of the planned tests, ETS(A,N,A) grid, random GUM models with lags (1, 12), random additive ETS and GUM models with long seasonal lags · procedure comparison with the exact oracle, exact oracle · lens admissibility decision, largest mode modulus, non-cyclotomic shared factor in the closed unit disc

E25.T1 · Each set of models of the planned tests (D42) given to the exact oracle (D41), and over all sets: models given, left out by the edge band and undecided; decisions of the PBH reference and of the prototype's cancellation that differ from the oracle, and the largest absolute difference of the largest mode modulus on the models both find inadmissible; models with a non-cyclotomic shared factor in the closed unit disc (D44); models whose cancelled MA degree differs from the prototype's. Empty cells: the set has no such value of the prototype (perlag holds the PBH reference only; stepdown and cpp the cancellation only).
Rowsetmodels given to the oraclemodels on which the oracle failedmodels comparedmodels whose shared non-cyclotomic factor could not be locatedmodels with a shared non-cyclotomic factor, all roots in |B| > 1models excluded by edgemodels excluded by undecidedadmissibility decisions of the PBH reference that differ from the exact oraclemax abs difference of the largest mode modulus, PBH reference against the exact oracle, models both find inadmissibleadmissibility decisions of the prototype's cancellation that differ from the exact oraclemax abs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracle, models both find inadmissiblemodels with a shared factor of theta and phi that is not a product of Phi_n, n dividing a lag, with a root in |B| <= 1models whose cancelled MA degree differs from the prototype's
r1perlag3242031640078000.00000
r2cancel5000500080000.000000.000000
r3stepdown5750575080000.000000
r4cpp8000797083000.000000
r5examples2020000000000
r6all51190503802481000.000000.000000
E25.T2 · The 575 models of PL4 and PL5 (MA degree up to 535 before cancellation): step-down decisions of the prototype that differ from the exact oracle; the largest absolute difference of the bisection value from the certified largest mode modulus, over the models that have one; and the degrees. (set: stepdown)
Rowsetstep-down decisions of the prototype that differ from the exact oraclemax abs difference of the largest mode modulus, prototype's bisection against the exact oraclemodels with a bisection valuemax degree of theta, exactmax degree of the cancelled MA polynomial, exact
r1stepdown00.0000198535533
E25.T3 · The models of PL1 and PL2 (3042 ETS(A,N,A) grid models, 200 GUM models with lags (1, 12)): the counts of C1 and C2 with the exact oracle in place of the PBH reference and the band of 1e-8. (set: perlag)
Rowfamilymodels accepted by the per-lag check that do not forget their initial states, exact oraclemodels rejected by the per-lag check that forget their initial states, exact oraclemodels accepted by the per-lag checkmodels of the family
r1ETS(A,N,A)1482030423042
r2GUM101193200
E25.T4 · The models in the edge band, by set and family (run AC5.RN9): their number, the largest distance of the certified largest mode modulus from 1, those with roots proved on the unit circle, and those whose decision by the method of the original test (the PBH reference for perlag, the prototype's cancellation for cpp) differs from the oracle's, with the largest modulus difference.
Rowsetfamilymodels in the edge bandmax distance of the certified largest mode modulus from 1, edge modelsedge models with roots proved on the unit circleedge models whose decision differs from the exact oraclemax abs difference of the largest mode modulus against the exact oracle, edge models
r1perlagETS(A,N,A)780.0000260.0000
r2cppETS high degree30.0000000.0000
E25.T5 · The examples of C10 and C11 (run AC5.RN9): the certified largest mode modulus, its difference from the PBH reference and from the prototype's roots after cancellation, and the exact degrees of theta, of the shared cyclotomic factor and of the gcd of theta and phi.
Rowmodellargest mode modulus, exact oracle (theta_G)abs difference of the largest mode modulus, PBH reference against the exact oracleabs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracledegree of theta, exactdegree of the shared cyclotomic factor, exactdegree of the exact gcd of theta and phi
r1ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05)10.00000.00001932525
r2GUM start, orders (2,2), lags (1,12)0.98290.00000.0000261414

Limitations

L2 · measurement · could reverse · affects C1, C2, C3, C10, C11, C16 · mitigation: addressed

The PBH reference and the cyclotomic cancellation were written by the same agent and share the lag-1 expansion code and numpy's LAPACK eigenvalue routines.

L3 · sampling · reduces transferability · affects C2, C3, C4, C5, C16 · mitigation: none

The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data.

Assumptions

C14 · methodological · tested

The PBH reference with relative rank tolerance 1e-7 classifies every eigenvalue of modulus at least 1 of the expanded discount matrix of the tested models correctly as both reachable and observable or not, outside the band of 1e-8 around the unit circle that the tests leave out. Rested on by C3.

Definitions

Methods 10

D1 lagged SSOE model: A pure additive single-source-of-error state space model with kk components, yt=w′vt−l+ϵtvt=Fvt−l+gϵt,\begin{aligned} y_t &= w' v_{t-l} + \epsilon_t \\ v_t &= F v_{t-l} + g \epsilon_t, \end{aligned} in which component jj of vt−lv_{t-l} enters with its own lag ljl_j; its discount matrix is D=F−gw′D = F - g w' and its lag matrix is L(B)=diag(Bl1,…,Blk)L(B) = \mathrm{diag}(B^{l_1}, \dots, B^{l_k}) for the backshift operator BB; Λ(λ)=diag(λl1,…,λlk)\Lambda(\lambda) = \mathrm{diag}(\lambda^{l_1}, \dots, \lambda^{l_k}) for a complex number λ\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 DD, 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 λ\lambda of the expanded discount matrix for which the smallest singular values of [λI−D~;w~′][\lambda I - \tilde D;\ \tilde w'] and of [λI−D~,g~][\lambda I - \tilde D,\ \tilde g] both exceed 10−7(‖D~‖2+‖g~‖+‖w~‖)10^{-7}(\|\tilde D\|_2 + \|\tilde g\| + \|\tilde w\|) (Popov-Belevitch-Hautus rank tests), computed with numpy as pbh_max().

D6 cyclotomic cancellation: The procedure that computes the coefficients of θ(B)=det⁡(I−DL(B))ϕ(B)=det⁡(I−FL(B))\begin{aligned} \theta(B) &= \det(I - D L(B)) \\ \phi(B) &= \det(I - F L(B)) \end{aligned} by an inverse discrete Fourier transform of their values at the NN-th roots of unity (NN the smallest power of 2 above ∑jlj\sum_j l_j), and divides both by every cyclotomic polynomial Φn(B)\Phi_n(B), nn dividing one of the lags, for as long as it divides both with a remainder at most 10−910^{-9} times the sum of absolute coefficients; the quotient of θ\theta is the cancelled MA polynomial θ*(B)\theta^*(B).

D7 step-down test: The Levinson-Durbin step-down (Schur-Cohn) recursion on 1+c1B+…+cnBn1 + c_1 B + \dots + c_n B^n: for j=n,…,1j = n, \dots, 1, κj=cj,ci←ci−κjcj−i1−κj2;\kappa_j = c_j, \qquad c_i \leftarrow \frac{c_i - \kappa_j c_{j-i}}{1 - \kappa_j^2}; the test passes when every |κj|<1|\kappa_j| < 1, which holds if and only if all roots lie outside the unit circle (the κj\kappa_j are the partial autocorrelations of the autoregression with this polynomial). Source: Barndorff-Nielsen and Schou (1973).

D8 GUM: Generalised Univariate Model of the smooth package: a lagged SSOE model whose FF, gg and ww are all estimated. Source: Svetunkov (2016).

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

D10 check of the whole model: Admissibility of a lagged SSOE model from the MA polynomial θ(B)=det⁡(I−DL(B))\theta(B) = \det(I - D L(B)) of its ARIMA form: θ\theta is divided by its common factors with ϕ(B)=det⁡(I−FL(B))\phi(B) = \det(I - F L(B)) by cyclotomic cancellation, and the model is accepted when the cancelled polynomial θ*\theta^* has no root in |B|≤1|B| \le 1 and, for a model with ARIMA components, the AR polynomial of its ARIMA part is stationary; its largest mode modulus is 1/min⁡|B|1/\min|B| over the roots of θ*\theta^*. In the numpy prototype (admissibility.py, confirm.py, ets_arima.py) the roots come from numpy; in smooth the step-down test decides and bisection gives the modulus.

D11 · part of D10 bisection for the largest root modulus: Bisection on rr over the step-down test applied to θ*(B/r)\theta^*(B/r), which passes exactly when 1/|B|<r1/|B| < r for every root BB of θ*\theta^*, so that the smallest such rr is the largest mode modulus; run to a width of 1e-10 in the prototype and to a tolerance of 1e-6 in smooth.

D41 exact oracle: Admissibility decided in exact rational arithmetic with certified root enclosures (independent/oracle.py, python-flint 0.9.0), written by the contributor P5 from the definitions of this claimset without the prototype: each float of FF, gg, ww is converted exactly to a rational, D=F−gw′D = F - g w', and θ(B)=det⁡(I−DL(B)),ϕ(B)=det⁡(I−FL(B))\theta(B) = \det(I - D L(B)), \qquad \phi(B) = \det(I - F L(B)) by fraction-free elimination over ℚ[B]\mathbb{Q}[B]; G=gcd⁡(θ,ϕ)G = \gcd(\theta, \phi) exactly and θG=θ/G\theta_G = \theta / G; CC is the product of the cyclotomic polynomials Φn\Phi_n, nn dividing a lag, at the multiplicity dividing both, and θC=θ/C\theta_C = \theta / C; the roots of θG\theta_G, θC\theta_C and G/CG / C are enclosed in Arb balls of 128 bits, raised to 512 and 2048 bits while a modulus ball contains 1, and a root counts as on the unit circle only when it is proved a root of unity whose Φn\Phi_n divides the polynomial. The largest mode modulus is 1/min⁡|B|1/\min|B| over the roots; a model is admissible when it is below 1.

Data 7

D17 ETS(A,N,A) grid: The 3042 ETS(A,N,A) models with m=4m = 4 and m=12m = 12 and α\alpha and γ\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 FF normal with mean 0 and standard deviation 0.35, of gg with mean 0 and standard deviation 0.175, of ww with mean 1 and standard deviation 0.175 (confirm.py, seed 2026).

D20 random additive ETS and GUM models with long seasonal lags: The 575 models of plan PL4 (confirm.py --test stepdown --seed 2026 --n 200): the 500 models of plan PL3 generated afresh, 50 additive ETS models with seasonal lags from {24, 168, 365} and 25 GUM models with lags (1, 24, 168) and normal FF (standard deviation 0.3). The generator allows MA polynomial degrees up to 536 (level, trend, seasonal lags 168 and 365, constant).

D19 · part of D20 random additive ETS and GUM models: The 500 models of plan PL3 (confirm.py --test cancel --seed 2026 --n 200): 200 additive ETS models (level; no, additive or damped trend with ϕ\phi in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; β=0\beta = 0 with probability 0.1 and each γ=0\gamma = 0 with probability 0.15; α\alpha in (0.01, 1.9)), 100 GUM models with F=IF = I, lags from {1, 4, 12}, gg uniform on (-0.3, 0.6) and ww normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal FF (standard deviation 0.35).

D24 ETS(A,N,A,A) with seasonal lags 24 and 168: The additive ETS model with a level and two seasonal states of lags 24 and 168, α=0.3\alpha = 0.3 and γ=(0.1,0.05)\gamma = (0.1, 0.05), in the lagged form.

D25 starting point of gum(): The default starting point of gum() with orders (2, 2) and lags (1, 12): F=IF = I, g=0.1g = 0.1, w=1w = 1.

D42 models of the planned tests: The models of the planned tests PL1-PL6 and the two examples of C10 and C11, drawn again by export_models.py exactly as confirm.py and cpp_check.py draw them (the counts of the original tests are reproduced), with the prototype's values for each: 3242 models of PL1 and PL2, 500 of PL3, 575 of PL4 and PL5 (MA degree up to 535), 800 of PL6 and 2 examples; 5119 in all.

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 |λ||\lambda| of the reachable and observable eigenvalues of the expanded discount matrix, for the check of the whole model 1/min⁡|B|1/\min|B| over the roots BB of the cancelled MA polynomial, for the per-lag check the largest |λ||\lambda| of its blocks; a model is admissible when it is below 1.

D44 non-cyclotomic shared factor in the closed unit disc: Whether G/CG / C, the factor shared by θ\theta and ϕ\phi after removing the cyclotomic polynomials Φn\Phi_n (nn dividing a lag), has a root in |B|≤1|B| \le 1, certified by the exact oracle; counted over models.

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.

D43 comparison with the exact oracle: For each model, the admissibility decision and largest mode modulus of a method of the study against those of the exact oracle: the PBH reference against θG\theta_G; the prototype's cancellation, step-down test and bisection against θC\theta_C. Counted are the models whose decisions differ and the largest absolute difference of the modulus on the models both find inadmissible (for the bisection, on every model with a value). Models the oracle cannot decide, and models whose certified largest modulus lies within 10−810^{-8} of 1, are left out and counted.

Concepts 2

D2 expanded discount matrix: The transition matrix D~\tilde D of size ∑jlj\sum_j l_j of the lag-1 system equivalent to vt=Dvt−lv_t = D v_{t-l}, in which component jj carries itself and its lj−1l_j - 1 lagged copies; F~\tilde F, g~\tilde g and w~\tilde w are expanded in the same way, so that D~=F~−g~w~′\tilde D = \tilde F - \tilde g \tilde w'.

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

Context

Extends Hyndman et al. (2008), Chapter 11 · from C3
Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.
Extends Hyndman et al. (2008), Chapter 11 · from C6
Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.

References

  1. 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: Context, D9
  2. Barndorff-Nielsen and Schou (1973) Barndorff-Nielsen, O., & Schou, G. (1973). On the parametrization of autoregressive models by partial autocorrelations. Journal of Multivariate Analysis, 3(4), 408–419. doi:10.1016/0047-259X(73)90030-4 Used by: D7
  3. Svetunkov (2016) Svetunkov, I. (2016). smooth: Forecasting Using State Space Models. CRAN: Contributed Packages. doi:10.32614/CRAN.package.smooth Used by: D8

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.

C3: 95, High

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

  1. −3 Plan The plan's time is anchored by a git push only. To remove it: Anchor plans with the platform, a timestamp authority (RFC 3161) or a registry.
  2. −2 Open problems Limitation L3 (reduces transferability, not mitigated): The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

C6: 100, High

Novelty: extends earlier work, by the authors

Nothing lost.

C8: 100, High

Novelty: new, by the authors

Nothing lost.

C15: 100, High

Novelty: new, by the authors

Nothing lost.

C17: 100, High

Novelty: new, by the authors

Nothing lost.