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Claim C21 · formal · entrylib:2026.000007@v2 · 2026-11-13

Corrected · 13 November 2026 at 12:34 UTC

The paper had no summary and did not state why the ARMA part of smooth's TBATS differs from De Livera's in effect: a summary in plain language is added; a formal claim (C21, with its condition C22 and proof E6) shows that ARMA errors pass the ARMA's persistence into the level, while ADAM's additive ARMA component leaves the level a fixed share of each shock; the highlights are amended; and the equation of the harmonic states in D1 is set on a line of its own.

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

TBATS in smooth: its formulation, and its accuracy against forecast::tbats and AutoTBATS on hourly data · C21

Principal claim C21 of this paper (Svetunkov, I. (guarantor), 2026), with what it rests on. Read the whole paper.

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

The message

Read https://entrylib.org/p/2026.000007/C21, 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

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

For a local level with a stationary ARMA part, the additive ARMA component of smooth TBATS keeps the permanent effect of a shock in the level and its transitory effect in the ARMA part, unlike the ARMA errors of De Livera TBATS that pass the persistence of the ARMA part into the level.

Scope: conditions: a single source of error, a local level without trend or seasonal states, an ARMA part with stationary autoregressive polynomial, both formulations compared · Proof: E6 · Uses: C23, C24 · Assumes: C22 · Explains: C3 · Extends: B8 · Verification: checked by a person

Claims it rests on

SupportS 100C23 · formal · novelty: new, by the authors

Under the ARMA errors of De Livera TBATS, for a local level with smoothing parameter α>0\alpha > 0 and an ARMA part with polynomials ϕ(B)\phi(B) and θ(B)\theta(B), a unit shock moves the level permanently by αθ(1)/ϕ(1)\alpha \theta(1) / \phi(1), unbounded as a root of ϕ(B)\phi(B) approaches 1.

Scope: conditions: a single source of error, a local level without trend or seasonal states, an ARMA part with stationary autoregressive polynomial, ARMA errors · Proof: E6 · Assumes: C22 · Extends: De Livera et al. (2011) · Verification: checked by a person · Used by: C21

SupportS 100C24 · formal · novelty: new, by the authors

Under the additive ARMA component of smooth TBATS, for a local level with smoothing parameter α\alpha and a stationary ARMA part, a unit shock moves the level permanently by α\alpha and the response of the ARMA part decays to zero.

Scope: conditions: a single source of error, a local level without trend or seasonal states, an ARMA part with stationary autoregressive polynomial, an additive ARMA component · Proof: E6 · Assumes: C22 · Extends: B8 · Verification: checked by a person · Used by: C21

Evidence

E6 · formal · cited by C21, C23, C24

Derivation of the impulse responses of a local level with an ARMA part under ARMA errors and under ADAM's additive ARMA component, and their limits.

Foundation: linear difference equations and power series in the backshift operator $B$ · Assumes: C22

  1. Under ARMA errors, ϕ(B)dt=θ(B)ϵt\phi(B) d_t = \theta(B) \epsilon_t with ϕ(B)\phi(B) stationary gives dt=ψ(B)ϵt=∑k≥0ψkϵt−kd_t = \psi(B) \epsilon_t = \sum_{k \ge 0} \psi_k \epsilon_{t-k}, with ψ(B)=θ(B)/ϕ(B)\psi(B) = \theta(B) / \phi(B), ψ0=1\psi_0 = 1, ψk→0\psi_k \to 0 and ∑k≥0ψk=ψ(1)=θ(1)/ϕ(1)\sum_{k \ge 0} \psi_k = \psi(1) = \theta(1) / \phi(1). (definition; from D2, C22)
  2. The level of the ARMA-errors model moves by lt−lt−1=αdt=αψ(B)ϵtl_t - l_{t-1} = \alpha d_t = \alpha \psi(B) \epsilon_t, so a unit shock at tt moves lt+j−1l_{t+j-1} by α∑k=0j−1ψk\alpha \sum_{k=0}^{j-1} \psi_k and yt+j=lt+j−1+dt+jy_{t+j} = l_{t+j-1} + d_{t+j} by rjE=ψj+α∑k=0j−1ψkr^{E}_j = \psi_j + \alpha \sum_{k=0}^{j-1} \psi_k for j≥1j \ge 1. (substitution; from S1, D2)
  3. limj→∞⁡rjE=αθ(1)/ϕ(1)\lim_{j \to \infty} r^{E}_j = \alpha \, \theta(1) / \phi(1): the level keeps αψ(1)\alpha \psi(1) of every shock, and its increments αψ(B)ϵt\alpha \psi(B) \epsilon_t are autocorrelated unless ψ(B)=1\psi(B) = 1, that is, unless there is no ARMA part, given that θ(B)\theta(B) is invertible. (limit; from S1, S2)
  4. Under ADAM's additive ARMA component, yt=lt−1+at+ϵty_t = l_{t-1} + a_t + \epsilon_t with at+ϵt=ψ(B)ϵta_t + \epsilon_t = \psi(B) \epsilon_t and lt−lt−1=αϵtl_t - l_{t-1} = \alpha \epsilon_t, so a unit shock at tt moves yt+jy_{t+j} by rjA=α+ψjr^{A}_j = \alpha + \psi_j for j≥1j \ge 1. (definition; from D1)
  5. limj→∞⁡rjA=α\lim_{j \to \infty} r^{A}_j = \alpha: the level keeps α\alpha of every shock, its increments αϵt\alpha \epsilon_t are white noise, and the response of the ARMA part, ψj\psi_j, decays to zero. (limit; from S1, S4)
  6. For an AR(1), ψj=ϕj\psi_j = \phi^j: rjE=ϕj+α(1−ϕj)/(1−ϕ)→α/(1−ϕ)r^{E}_j = \phi^j + \alpha (1 - \phi^j) / (1 - \phi) \to \alpha / (1 - \phi), which grows without bound as ϕ→1\phi \to 1 when α>0\alpha > 0, while rjA=α+ϕj→αr^{A}_j = \alpha + \phi^j \to \alpha. (substitution; from S3, S5)

Assumptions

C22 · theorem_condition · untestable

The autoregressive polynomial ϕ(B)\phi(B) of the ARMA part has all its roots outside the unit circle, and its moving average polynomial θ(B)\theta(B) is invertible. Rested on by C21, C23, C24, E6.

Definitions

Methods 2

D1 smooth TBATS: TBATS of the smooth package (R tbats(), Python TBATS): a linear single-source-of-error state space model of the Box-Cox transform yt(λ)y_t^{(\lambda)} of the series, built on ADAM's state space core, yt(λ)=w⊤vt−𝒍+ϵt,vt=Fvt−𝒍+gϵt,\begin{aligned} y_t^{(\lambda)} &= w^\top v_{t-\boldsymbol{l}} + \epsilon_t, \\ v_t &= F v_{t-\boldsymbol{l}} + g \epsilon_t, \end{aligned} where vt−𝒍v_{t-\boldsymbol{l}} takes each state at its own lag. The level ltl_t and the trend btb_t are those of ETS(A,N,N), ETS(A,A,N) or ETS(A,Ad,N), lt=lt−1+ϕbt−1+αϵt,bt=ϕbt−1+βϵt.\begin{aligned} l_t &= l_{t-1} + \phi b_{t-1} + \alpha \epsilon_t, \\ b_t &= \phi b_{t-1} + \beta \epsilon_t. \end{aligned} Each harmonic j=1,…,kij = 1, \dots, k_i of each period mim_i, with frequency λij=2πj/mi\lambda_{ij} = 2 \pi j / m_i, is two states v1(ij)v^{(ij)}_1 and v2(ij)v^{(ij)}_2 with lags 1 and 2 in the ARIMA slot of ADAM's core, uij,t=v1,t−1(ij)+v2,t−2(ij),v1,t(ij)=2cos⁡(λij)uij,t+γ1iϵt,v2,t(ij)=−uij,t+(sin⁡(λij)γ2i−cos⁡(λij)γ1i)ϵt,\begin{aligned} u_{ij,t} &= v^{(ij)}_{1,t-1} + v^{(ij)}_{2,t-2}, \\ v^{(ij)}_{1,t} &= 2 \cos(\lambda_{ij}) \, u_{ij,t} + \gamma_{1i} \epsilon_t, \\ v^{(ij)}_{2,t} &= -u_{ij,t} + \left(\sin(\lambda_{ij}) \gamma_{2i} - \cos(\lambda_{ij}) \gamma_{1i}\right) \epsilon_t, \end{aligned} so that, with the backshift operator BB, (1−2cos⁡(λij)B+B2)uij,t=(γ1iB+(sin⁡(λij)γ2i−cos⁡(λij)γ1i)B2)ϵt.(1 - 2 \cos(\lambda_{ij}) B + B^2) \, u_{ij,t} = \left(\gamma_{1i} B + (\sin(\lambda_{ij}) \gamma_{2i} - \cos(\lambda_{ij}) \gamma_{1i}) B^2\right) \epsilon_t. The measurement equation is yt(λ)=lt−1+ϕbt−1+∑i∑j=1kiuij,t+at+ϵt,y_t^{(\lambda)} = l_{t-1} + \phi b_{t-1} + \sum_i \sum_{j=1}^{k_i} u_{ij,t} + a_t + \epsilon_t, where ata_t is an optional ARMA component of ADAM: an additive block of states whose output enters the measurement equation, not ARMA errors. λ∈[0,1]\lambda \in [0, 1] is estimated by likelihood with the Jacobian of the transform. The harmonics are preselected by information criterion on a global regression of the transformed series on Fourier terms, and the trend and the ARMA orders are selected by AICc. With the default distribution="auto", the structure is selected with the generalised normal distribution of ϵt\epsilon_t, and the closest of the normal, Laplace and S distributions to its estimated shape is then fitted on that structure. Code: R/adam-tbats.R and python/src/smooth/adam_general/core/tbats/ of the smooth repository, at the code versions the evidence names.

D2 De Livera TBATS: TBATS of De Livera, Hyndman and Snyder (2011): for the Box-Cox transform yt(λ)y_t^{(\lambda)}, yt(λ)=lt−1+ϕbt−1+∑i∑j=1kisj,t−1(i)+dt,lt=lt−1+ϕbt−1+αdt,bt=ϕbt−1+βdt,sj,t(i)=cos⁡(λij)sj,t−1(i)+sin⁡(λij)sj,t−1*(i)+γ1idt,sj,t*(i)=−sin⁡(λij)sj,t−1(i)+cos⁡(λij)sj,t−1*(i)+γ2idt,dt=∑k=1pφkdt−k+∑k=1qθkϵt−k+ϵt,\begin{aligned} y_t^{(\lambda)} &= l_{t-1} + \phi b_{t-1} + \sum_i \sum_{j=1}^{k_i} s^{(i)}_{j,t-1} + d_t, \\ l_t &= l_{t-1} + \phi b_{t-1} + \alpha d_t, \\ b_t &= \phi b_{t-1} + \beta d_t, \\ s^{(i)}_{j,t} &= \cos(\lambda_{ij}) s^{(i)}_{j,t-1} + \sin(\lambda_{ij}) s^{*(i)}_{j,t-1} + \gamma_{1i} d_t, \\ s^{*(i)}_{j,t} &= -\sin(\lambda_{ij}) s^{(i)}_{j,t-1} + \cos(\lambda_{ij}) s^{*(i)}_{j,t-1} + \gamma_{2i} d_t, \\ d_t &= \sum_{k=1}^{p} \varphi_k d_{t-k} + \sum_{k=1}^{q} \theta_k \epsilon_{t-k} + \epsilon_t, \end{aligned} with λij=2πj/mi\lambda_{ij} = 2 \pi j / m_i: ARMA(p, q) errors dtd_t drive every state. Source: De Livera et al. (2011).

Context

Extends B8 · from C21
The book defines ADAM's ARIMA as a component added to the other states in a single-source-of-error model.
Extends De Livera et al. (2011) · from C23
De Livera, Hyndman and Snyder's TBATS models the ARMA part as the errors that drive every state, including the level.
Extends B8 · from C24
The book defines ADAM's ARIMA as a component added to the other states in a single-source-of-error model.

References

  1. De Livera et al. (2011) De Livera, A. M., Hyndman, R. J., & Snyder, R. D. (2011). Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing. Journal of the American Statistical Association, 106(496), 1513–1527. doi:10.1198/jasa.2011.tm09771 Used by: Context, D2

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.4): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C21: 100, Well supported

Novelty: new, by the authors

Nothing lost.

C23: 100, Well supported

Novelty: new, by the authors

Nothing lost.

C24: 100, Well supported

Novelty: new, by the authors

Nothing lost.