CLAIM C21 of entrylib:2026.000007 TBATS in smooth: its formulation, and its accuracy against forecast::tbats and AutoTBATS on hourly data
licence CC-BY-4.0; contributions C21

DEFINITIONS
D1 [method] 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 $y_t^{(\lambda)}$ of the series, built on ADAM's state space core,
$$\begin{aligned} y_t^{(\lambda)} &= w^\top v_{t-\boldsymbol{l}} + \epsilon_t, \\ v_t &= F v_{t-\boldsymbol{l}} + g \epsilon_t, \end{aligned}$$
where $v_{t-\boldsymbol{l}}$ takes each state at its own lag. The level $l_t$ and the trend $b_t$ are those of ETS(A,N,N), ETS(A,A,N) or ETS(A,Ad,N),
$$\begin{aligned} l_t &= l_{t-1} + \phi b_{t-1} + \alpha \epsilon_t, \\ b_t &= \phi b_{t-1} + \beta \epsilon_t. \end{aligned}$$
Each harmonic $j = 1, \dots, k_i$ of each period $m_i$, with frequency $\lambda_{ij} = 2 \pi j / m_i$, is two states $v^{(ij)}_1$ and $v^{(ij)}_2$ with lags 1 and 2 in the ARIMA slot of ADAM's core,
$$\begin{aligned} u_{ij,t} &= v^{(ij)}_{1,t-1} + v^{(ij)}_{2,t-2}, \\ v^{(ij)}_{1,t} &= 2 \cos(\lambda_{ij}) \, u_{ij,t} + \gamma_{1i} \epsilon_t, \\ v^{(ij)}_{2,t} &= -u_{ij,t} + \left(\sin(\lambda_{ij}) \gamma_{2i} - \cos(\lambda_{ij}) \gamma_{1i}\right) \epsilon_t, \end{aligned}$$
so that, with the backshift operator $B$,
$$(1 - 2 \cos(\lambda_{ij}) B + B^2) \, u_{ij,t} = \left(\gamma_{1i} B + (\sin(\lambda_{ij}) \gamma_{2i} - \cos(\lambda_{ij}) \gamma_{1i}) B^2\right) \epsilon_t.$$
The measurement equation is
$$y_t^{(\lambda)} = l_{t-1} + \phi b_{t-1} + \sum_i \sum_{j=1}^{k_i} u_{ij,t} + a_t + \epsilon_t,$$
where $a_t$ is an optional ARMA component of ADAM: an additive block of states whose output enters the measurement equation, not ARMA errors. $\lambda \in [0, 1]$ 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 $\epsilon_t$, 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 [method] De Livera TBATS: TBATS of De Livera, Hyndman and Snyder (2011): for the Box-Cox transform $y_t^{(\lambda)}$,
$$\begin{aligned} y_t^{(\lambda)} &= l_{t-1} + \phi b_{t-1} + \sum_i \sum_{j=1}^{k_i} s^{(i)}_{j,t-1} + d_t, \\ l_t &= l_{t-1} + \phi b_{t-1} + \alpha d_t, \\ b_t &= \phi b_{t-1} + \beta d_t, \\ s^{(i)}_{j,t} &= \cos(\lambda_{ij}) s^{(i)}_{j,t-1} + \sin(\lambda_{ij}) s^{*(i)}_{j,t-1} + \gamma_{1i} d_t, \\ s^{*(i)}_{j,t} &= -\sin(\lambda_{ij}) s^{(i)}_{j,t-1} + \cos(\lambda_{ij}) s^{*(i)}_{j,t-1} + \gamma_{2i} d_t, \\ d_t &= \sum_{k=1}^{p} \varphi_k d_{t-k} + \sum_{k=1}^{q} \theta_k \epsilon_{t-k} + \epsilon_t, \end{aligned}$$
with $\lambda_{ij} = 2 \pi j / m_i$: ARMA(p, q) errors $d_t$ drive every state. [source B4]

CLAIMS
C21 [formal; principal; support: formal_proof E6, at S1-S6; assumes C22; uses C23, C24; verification peer_checked; verifiability open]
  For a local level with a stationary ARMA part, the additive ARMA component of [[D1: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 [[D2: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
  explains C3
  extends B8: The book defines ADAM's ARIMA as a component added to the other states in a single-source-of-error model.
C22 [formal; supporting; assumption (theorem_condition, untestable); verifiability not_applicable]
  The autoregressive polynomial $\phi(B)$ of the ARMA part has all its roots outside the unit circle, and its moving average polynomial $\theta(B)$ is invertible.
C23 [formal; supporting; support: formal_proof E6, at S1-S3, S6; assumes C22; verification peer_checked; verifiability open]
  Under the ARMA errors of [[D2:De Livera TBATS]], for a local level with smoothing parameter $\alpha > 0$ and an ARMA part with polynomials $\phi(B)$ and $\theta(B)$, a unit shock moves the level permanently by $\alpha \theta(1) / \phi(1)$, unbounded as a root of $\phi(B)$ 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
  extends B4: De Livera, Hyndman and Snyder's TBATS models the ARMA part as the errors that drive every state, including the level.
C24 [formal; supporting; support: formal_proof E6, at S1, S4-S6; assumes C22; verification peer_checked; verifiability open]
  Under the additive ARMA component of [[D1:smooth TBATS]], for a local level with smoothing parameter $\alpha$ and a stationary ARMA part, a unit shock moves the level permanently by $\alpha$ 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
  extends B8: The book defines ADAM's ARIMA as a component added to the other states in a single-source-of-error model.

EVIDENCE
E6 [formal; foundation linear difference equations and power series in the backshift operator $B$] 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.
  assumes C22
  S1 Under ARMA errors, $\phi(B) d_t = \theta(B) \epsilon_t$ with $\phi(B)$ stationary gives $d_t = \psi(B) \epsilon_t = \sum_{k \ge 0} \psi_k \epsilon_{t-k}$, with $\psi(B) = \theta(B) / \phi(B)$, $\psi_0 = 1$, $\psi_k \to 0$ and $\sum_{k \ge 0} \psi_k = \psi(1) = \theta(1) / \phi(1)$. (by D2, C22; definition)
  S2 The level of the ARMA-errors model moves by $l_t - l_{t-1} = \alpha d_t = \alpha \psi(B) \epsilon_t$, so a unit shock at $t$ moves $l_{t+j-1}$ by $\alpha \sum_{k=0}^{j-1} \psi_k$ and $y_{t+j} = l_{t+j-1} + d_{t+j}$ by $r^{E}_j = \psi_j + \alpha \sum_{k=0}^{j-1} \psi_k$ for $j \ge 1$. (by S1, D2; substitution)
  S3 $\lim_{j \to \infty} r^{E}_j = \alpha \, \theta(1) / \phi(1)$: the level keeps $\alpha \psi(1)$ of every shock, and its increments $\alpha \psi(B) \epsilon_t$ are autocorrelated unless $\psi(B) = 1$, that is, unless there is no ARMA part, given that $\theta(B)$ is invertible. (by S1, S2; limit)
  S4 Under ADAM's additive ARMA component, $y_t = l_{t-1} + a_t + \epsilon_t$ with $a_t + \epsilon_t = \psi(B) \epsilon_t$ and $l_t - l_{t-1} = \alpha \epsilon_t$, so a unit shock at $t$ moves $y_{t+j}$ by $r^{A}_j = \alpha + \psi_j$ for $j \ge 1$. (by D1; definition)
  S5 $\lim_{j \to \infty} r^{A}_j = \alpha$: the level keeps $\alpha$ of every shock, its increments $\alpha \epsilon_t$ are white noise, and the response of the ARMA part, $\psi_j$, decays to zero. (by S1, S4; limit)
  S6 For an AR(1), $\psi_j = \phi^j$: $r^{E}_j = \phi^j + \alpha (1 - \phi^j) / (1 - \phi) \to \alpha / (1 - \phi)$, which grows without bound as $\phi \to 1$ when $\alpha > 0$, while $r^{A}_j = \alpha + \phi^j \to \alpha$. (by S3, S5; substitution)

REFERENCES
B4 10.1198/jasa.2011.tm09771 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C21, in: TBATS in smooth: its formulation, and its accuracy against forecast::tbats and AutoTBATS on hourly data. EntryLib. entrylib:2026.000007@v2#C21. https://entrylib.org/p/2026.000007/C21
The claimset it is from: Svetunkov, I. (guarantor) (2026). TBATS in smooth: its formulation, and its accuracy against forecast::tbats and AutoTBATS on hourly data. EntryLib. entrylib:2026.000007@v2. https://entrylib.org/p/2026.000007@v2