CLAIM C10 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 C10

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]
D3 [method] forecast::tbats: The tbats() function of the R package forecast (Hyndman et al.), De Livera et al.'s TBATS with its own model selection, called with seasonal.periods = c(24, 168), single-threaded, otherwise default settings. [source B5]
D4 [method] AutoTBATS: AutoTBATS of the Python package statsforecast (Nixtla), called with season_length = [24, 168], otherwise default settings. [source B12]
D5 [method] ADAM ETS: ADAM ETS of smooth's Python port, ADAM(model='ZXZ', lags=[24, 168]) with its default settings: ETS with a daily and a weekly seasonal component selected by AICc. [source B8]
D6 [method] seasonal naive 168: Seasonal naive forecast repeating the last 168 observations (the last week) of the training sample.
D7 [data] M4 hourly: The 414 hourly series of the M4 competition (Makridakis et al., 2020), training samples of 700 or 960 observations and test samples of 48, from the Mcompetitions/M4-methods repository (Dataset/Train/Hourly-train.csv, Dataset/Test/Hourly-test.csv). [source B6]
D9 [design] fixed-origin evaluation: Each method is fitted once to each training sample and forecasts h = 1..48 steps ahead from its end; no refits, no rolling origin.
D10 [variable] MASE: Mean absolute scaled error of a series over the horizon $h = 48$,
$$\mathrm{MASE} = \frac{\frac{1}{h} \sum_{j=1}^{h} |y_{n+j} - \hat y_{n+j}|}{\frac{1}{n-24} \sum_{t=25}^{n} |y_t - y_{t-24}|},$$
scaled by the in-sample mean absolute seasonal difference at lag 24, as in M4. [source B10]
D11 [variable] relative MASE: Geometric mean over series of the ratio of the [[D10:MASE]] of method A to that of method B; below 1, A is more accurate. Its 99% confidence interval is the exponential of the t-interval of the mean log ratio over series.
D12 [variable] coverage: Share of the test values of a set of series and horizons that lie inside the 95% prediction intervals of a method.
D13 [variable] elapsed time: Wall-clock seconds to fit a method to one training sample and produce its forecasts and intervals, single-threaded.
D14 [design] formulation checks: Deterministic checks of smooth TBATS against its specification, per seed: (H1) random parameters, initial states and series of 300 observations simulated by De Livera's recursion with normal errors, for the periods {24}, {7.3}, {24, 168}, {7.3, 30.4}, no/additive/damped trend and lambda in {1, 0.5, 0}, fitted by smooth TBATS with all parameters and initial states fixed; (H2) a noise-free series of 300 observations made of a random level and random harmonics of the periods 7.3 and 24, fitted with random fixed persistence and initial='backcasting'; (H3) nine random pairs of alpha in (0.05, 0.95) and |phi| in (0.05, 0.95) for a local level with AR(1), the impulse response w'F^(j-1)g at j = 1..20 read from the fitted model's matrices. Code: claimset-code/smooth-tbats/formulation_checks.py.
D15 [design] Nemenyi test: The Nemenyi test as computed by greybox's rmcb (distribution 'tukey', ties averaged): the methods are ranked within each series by the error measure, and two methods differ at level $1 - \alpha$ when their mean ranks over series differ by more than the critical distance
$$\mathrm{CD} = \frac{q_{1-\alpha}(k, \infty)}{\sqrt{2}} \sqrt{\frac{k (k + 1)}{6 N}}$$
for $k$ methods and $N$ series. [source B11]
D16 [variable] RMSSE: Root mean squared scaled error of a series over the horizon $h = 48$,
$$\mathrm{RMSSE} = \sqrt{\frac{\frac{1}{h} \sum_{j=1}^{h} (y_{n+j} - \hat y_{n+j})^2}{\frac{1}{n-1} \sum_{t=2}^{n} (y_t - y_{t-1})^2}},$$
scaled by the in-sample root mean squared first difference of the fitting sample, as in M5. [source B7]
D17 [variable] relative RMSSE: Geometric mean over series of the ratio of the [[D16:RMSSE]] of method A to that of method B; below 1, A is more accurate. Its 99% confidence interval is the exponential of the t-interval of the mean log ratio over series.
D18 [data] electricity hourly: The 370 hourly series of electricity consumption (kW) of clients of a Portuguese utility in GluonTS's electricity_nips archive (SR3), drawn from UCI ElectricityLoadDiagrams20112014: each series is the archive's training series extended by its last test window, ending 2014-09-08 01:00, 1249 to 6001 hours long. [source B9]
D19 [design] rolling-origin evaluation: Seven daily forecast origins over the last week of each series: the last one 48 hours before its end, the others 24, 48, ..., 144 hours earlier. At each origin each method is fitted to the 1008 preceding hours (six weeks) and forecasts 48 hours ahead; the RMSSE and the interval coverage of a series are averaged over its seven origins.
D20 [method] smooth TBATS dnorm: [[D1:smooth TBATS]] with the normal distribution of $\epsilon_t$: `TBATS(lags=[1, 24, 168], distribution="dnorm")` of the Python smooth package, every other setting at its default. The structure is selected and fitted with the normal distribution only, instead of the generalised normal distribution followed by the closest named distribution of the default `distribution="auto"`. [a kind of D1]

CLAIMS
C10 [descriptive; principal; support: empirical_quant E3 (E3.R1, E3.T3.r1) artefacts A8; verifiability open; planning confirmatory; outcome supported]
  On [[D7:M4 hourly]] with [[D9:fixed-origin evaluation]], [[D1:smooth TBATS]] is not less accurate than [[D3:forecast::tbats]] in [[D16:RMSSE]] beyond the non-inferiority margin of 1.05: the [[D17:relative RMSSE]] of smooth TBATS to forecast::tbats is 0.848 (99% confidence interval 0.7969 to 0.9024).
  scope: data_type: the 414 hourly series of the M4 competition | conditions: one fixed origin per series: the M4 test split; horizon 48; periods 24 and 168; smooth (Python port) at the commit of the run; comparator: forecast::tbats; measure: RMSSE
  extends B4: De Livera, Hyndman and Snyder introduced TBATS; this claim is about the implementation of smooth, which places the harmonics in ADAM's state space core, compared with two other implementations.
  test: frequentist; alpha 0.01; test one-sided one-sample t-test on the per-series log ratios (smooth TBATS over the comparator) of RMSSE (H10-H12), H0: mean log ratio >= log(1.05) against forecast::tbats and AutoTBATS (H10, H11), >= 0 against seasonal naive 168 (H12); their intervals are 99% t-intervals. H13: the Nemenyi test (greybox rmcb, distribution='tukey', level 0.99, ranks averaged over ties) on the per-series RMSSE of the five methods, with its own family-wise control over the pairwise comparisons (not in the Holm family); sides less; correction holm over the four one-sided t-tests the script computes (H10-H12 and the elapsed time against forecast::tbats, reported but not deciding here); the Nemenyi test's own control for H13; margin 1.05; sample All 414 series of the official M4 hourly test split, fixed by the data set; one run, no interim analysis. For orientation: with a standard deviation of the per-series log ratio of 0.3, the standard error of the mean log ratio is 0.015, so a true relative RMSSE of 1 lies 3.3 standard errors below the non-inferiority margin log(1.05).; supported if p_ftbats < alpha; not supported if lower_ftbats > margin; verdict computed; flags code_changed_after_plan
  plan: lab plan PL5; planned 2026-11-12T00:02:39Z; anchor git_remote

EVIDENCE
E1 [simulation] The [[D14:formulation checks]] of [[D1:smooth TBATS]], run once with the seeds 101 to 110 (lab run RN6, plan PL1): the largest relative difference of the one-step fitted values from an independent implementation of the recursion of [[D2:De Livera TBATS]] over 360 cases (H1), the largest absolute in-sample error of backcasting on 10 noise-free series (H2), and, over nine pairs of smoothing parameter and AR(1) coefficient, the largest distance of the impulse response of smooth TBATS from that of an additive ARMA component and the smallest from that of ARMA errors (H3). The per-case values are in the CSV of the lab results (A8).
  setup: studied D1, D2; procedure D14
  E1.R1 max relative difference of fitted values over all cases, smooth TBATS vs De Livera recursion (max) = 5.1122857914327425e-14 {seeds: [101, 102, 103, 104, 105, 106, 107, 108, 109, 110], cases: 360}
  E1.R2 max absolute in-sample error over all noise-free series, backcasting (max) = 1.7696635268293903e-09 {seeds: [101, 102, 103, 104, 105, 106, 107, 108, 109, 110], n: 300}
  E1.R3 max absolute difference of smooth's impulse response from alpha + phi^j over all cases (max) = 2.220446049250313e-16
  E1.R4 min over cases of the max absolute difference of smooth's impulse response from the ARMA-errors response (min) = 0.01651176558114527
E2 [quant; data_origin secondary] Benchmark of [[D1:smooth TBATS]] against [[D3:forecast::tbats]], [[D4:AutoTBATS]], [[D5:ADAM ETS]] and [[D6:seasonal naive 168]] on the 414 series of [[D7:M4 hourly]] with [[D9:fixed-origin evaluation]] (the M4 test split, 48 hours ahead), run once under plan PL3 (lab run RN8). Each fit is single-threaded; a failed fit gets the seasonal naive 168 forecasts and no interval. Per series, [[D10:MASE]], the coverage of the 95% interval and the elapsed time (averaged over the origins where there are several); the relative MASE of smooth TBATS to each comparator is the geometric mean over series of the ratio of their MASE, with a 99% t-interval on the log scale; the one-sided t-tests on the mean log ratios are adjusted by Holm over four; the [[D15:Nemenyi test]] ranks the five methods within each series. The sMAPE that the run also computed is in the CSV of the lab results only (decision DC4 of the lab record).
  setup: studied D1, D3, D4, D5, D6; cases D7; procedure D9, D15; lens D10, D11, D12, D13
  E2.R1 D11 (geometric mean) = 0.8317576290033031 99% confidence interval 0.7793039397579915 to 0.8877418913345124 {comparator: D3}
  E2.R2 D11 (geometric mean) = 0.780315891451513 99% confidence interval 0.7242363686872306 to 0.840737798842475 {comparator: D4}
  E2.R3 D11 (geometric mean) = 0.9260655272734568 99% confidence interval 0.824281045050103 to 1.0404186362821646 {comparator: D5}
  E2.R4 D11 (geometric mean) = 0.5067448313602549 99% confidence interval 0.44786668083806935 to 0.5733633134525995 {comparator: D6}
  E2.R5 relative elapsed time (geometric mean) = 0.5790894532352115 99% confidence interval 0.5389759363891272 to 0.6221884358973419 {comparator: D3}
  E2.R6 critical distance, Nemenyi test (MASE) = 0.35767939611022115 {level: 0.99}
  E2.R7 series in Nemenyi test (count) = 414
  E2.R8 p-value, rmcb test of equal mean ranks (MASE) (p_value) = 1.1039717573166142e-32
  E2.T1 Per method: mean and median MASE over series, coverage of the 95% prediction intervals, median elapsed time per series in seconds, series with a failed fit, and the mean rank of MASE in the Nemenyi test with the bounds of its 99% interval.
  | row | method | D10 (mean) | D10 (median) | D12 {nominal level: 0.95} (mean) | D13 (median) | series failed (count) | mean rank of MASE, Nemenyi test (mean) | mean rank of MASE, Nemenyi test {bound: 99% interval, lower} | mean rank of MASE, Nemenyi test {bound: 99% interval, upper} |
  | r1 | D1 | 1.029 | 0.87 | 0.9273349436392914 | 14.18 | 0 | 2.5724637681159437 | 2.394 | 2.751 |
  | r2 | D3 | 1.304 | 0.9941 | 0.9101 | 24.4 | 0 | 2.9541062801932383 | 2.775 | 3.133 |
  | r3 | D4 | 1.516 | 0.9836 | 0.9183 | 21.42 | 0 | 3.07729468599034 | 2.898 | 3.256 |
  | r4 | D5 | 2.11 | 0.7984 | 0.9054 | 1.31 | 0 | 2.618 | 2.44 | 2.797 |
  | r5 | D6 | 2.867 | 1.397 |  | 4.323e-05 | 0 | 3.778 | 3.599 | 3.957 |
  E2.T2 Per comparator: the series the relative MASE of smooth TBATS to it is computed on, and those left out because the MASE of smooth TBATS or of the comparator is zero. {method: D1}
  | row | comparator | series in relative MASE (count) | series excluded by zero MASE (count) |
  | r1 | D3 | 414 | 0 |
  | r2 | D4 | 414 | 0 |
  | r3 | D5 | 414 | 0 |
  | r4 | D6 | 414 | 0 |
  E2.T3 The one-sided t-tests of the plan on the per-series log ratios of smooth TBATS to the comparator: p-value and p-value adjusted by Holm over the four tests.
  | row | comparator | measure | p-value, one-sided t-test (p_value) | Holm-adjusted p-value, one-sided t-test (p_value) |
  | r1 | D3 | MASE | 5.799e-19 | 5.798511501263025e-19 |
  | r2 | D4 | MASE | 1.349e-22 | 2.6978102344588355e-22 |
  | r3 | D6 | MASE | 5.031e-38 | 1.5092330045971394e-37 |
  | r4 | D3 | elapsed time | 1.154e-61 | 4.6151072509330304e-61 |
E3 [quant; data_origin secondary] Benchmark of [[D1:smooth TBATS]] against [[D3:forecast::tbats]], [[D4:AutoTBATS]], [[D5:ADAM ETS]] and [[D6:seasonal naive 168]] on the 414 series of [[D7:M4 hourly]] with [[D9:fixed-origin evaluation]] (the M4 test split, 48 hours ahead), run once under plan PL5 (lab run RN11). Each fit is single-threaded; a failed fit gets the seasonal naive 168 forecasts and no interval. Per series, [[D16:RMSSE]], the coverage of the 95% interval and the elapsed time (averaged over the origins where there are several); the relative RMSSE of smooth TBATS to each comparator is the geometric mean over series of the ratio of their RMSSE, with a 99% t-interval on the log scale; the one-sided t-tests on the mean log ratios are adjusted by Holm over four; the [[D15:Nemenyi test]] ranks the five methods within each series.
  setup: studied D1, D3, D4, D5, D6; cases D7; procedure D9, D15; lens D16, D17, D12, D13
  E3.R1 D17 (geometric mean) = 0.8480372827132892 99% confidence interval 0.7969221555458947 to 0.9024309687802653 {comparator: D3}
  E3.R2 D17 (geometric mean) = 0.7959442609119876 99% confidence interval 0.741071448751255 to 0.854880143535765 {comparator: D4}
  E3.R3 D17 (geometric mean) = 0.9482580389950931 99% confidence interval 0.8470034046237921 to 1.0616171122927285 {comparator: D5}
  E3.R4 D17 (geometric mean) = 0.535834976024436 99% confidence interval 0.47846285660893406 to 0.600086542905422 {comparator: D6}
  E3.R5 relative elapsed time (geometric mean) = 0.568175851157546 99% confidence interval 0.5281131840262834 to 0.6112776722925655 {comparator: D3}
  E3.R6 critical distance, Nemenyi test (RMSSE) = 0.35767939611022115 {level: 0.99}
  E3.R7 series in Nemenyi test (count) = 414
  E3.R8 p-value, rmcb test of equal mean ranks (RMSSE) (p_value) = 4.096753399289539e-34
  E3.T1 Per method: mean and median RMSSE over series, coverage of the 95% prediction intervals, median elapsed time per series in seconds, series with a failed fit, and the mean rank of RMSSE in the Nemenyi test with the bounds of its 99% interval.
  | row | method | D16 (mean) | D16 (median) | D12 {nominal level: 0.95} (mean) | D13 (median) | series failed (count) | mean rank of RMSSE, Nemenyi test (mean) | mean rank of RMSSE, Nemenyi test {bound: 99% interval, lower} | mean rank of RMSSE, Nemenyi test {bound: 99% interval, upper} |
  | r1 | D1 | 0.8294 | 0.6242 | 0.9273 | 12.25 | 0 | 2.562801932367151 | 2.384 | 2.742 |
  | r2 | D3 | 0.9094 | 0.698 | 0.9101 | 21.84 | 0 | 2.9371980676328517 | 2.758 | 3.116 |
  | r3 | D4 | 1.063 | 0.6854 | 0.9183 | 19.7 | 0 | 3.0434782608695667 | 2.865 | 3.222 |
  | r4 | D5 | 2.1 | 0.6314 | 0.9055 | 1.112 | 0 | 2.645 | 2.466 | 2.824 |
  | r5 | D6 | 1.453 | 1.095 |  | 4.024e-05 | 0 | 3.812 | 3.633 | 3.99 |
  E3.T2 Per comparator: the series the relative RMSSE of smooth TBATS to it is computed on, and those left out because the RMSSE of smooth TBATS or of the comparator is zero. {method: D1}
  | row | comparator | series in relative RMSSE (count) | series excluded by zero RMSSE (count) |
  | r1 | D3 | 414 | 0 |
  | r2 | D4 | 414 | 0 |
  | r3 | D5 | 414 | 0 |
  | r4 | D6 | 414 | 0 |
  E3.T3 The one-sided t-tests of the plan on the per-series log ratios of smooth TBATS to the comparator: p-value and p-value adjusted by Holm over the four tests.
  | row | comparator | measure | p-value, one-sided t-test (p_value) | Holm-adjusted p-value, one-sided t-test (p_value) |
  | r1 | D3 | RMSSE | 9.427e-18 | 9.427033177562526e-18 |
  | r2 | D4 | RMSSE | 1.181e-21 | 2.3628603624600894e-21 |
  | r3 | D6 | RMSSE | 4.342e-38 | 1.302649972219819e-37 |
  | r4 | D3 | elapsed time | 4.715e-63 | 1.886e-62 |
E4 [quant; data_origin secondary] Benchmark of [[D1:smooth TBATS]] against [[D3:forecast::tbats]], [[D4:AutoTBATS]], [[D5:ADAM ETS]] and [[D6:seasonal naive 168]] on the 370 series of [[D18:electricity hourly]] with [[D19:rolling-origin evaluation]] (7 daily origins of the last week, 1008-hour fitting windows, 48 hours ahead; series 322, constant in its windows, has an undefined RMSSE and is left out), run once under plan PL7 (lab run RN13). Each fit is single-threaded; a failed fit gets the seasonal naive 168 forecasts and no interval. Per series, [[D16:RMSSE]], the coverage of the 95% interval and the elapsed time (averaged over the origins where there are several); the relative RMSSE of smooth TBATS to each comparator is the geometric mean over series of the ratio of their RMSSE, with a 99% t-interval on the log scale; the one-sided t-tests on the mean log ratios are adjusted by Holm over four; the [[D15:Nemenyi test]] ranks the five methods within each series.
  setup: studied D1, D3, D4, D5, D6; cases D18; procedure D19, D15; lens D16, D17, D12, D13
  E4.R1 D17 (geometric mean) = 0.9752899990850776 99% confidence interval 0.9470396739420402 to 1.0043830353548469 {comparator: D3}
  E4.R2 D17 (geometric mean) = 0.990872986502657 99% confidence interval 0.9753390861558684 to 1.006654290099668 {comparator: D4}
  E4.R3 D17 (geometric mean) = 1.0042706046509193 99% confidence interval 0.7972644036763257 to 1.2650250565750576 {comparator: D5}
  E4.R4 D17 (geometric mean) = 0.8243857583084898 99% confidence interval 0.7980923108751354 to 0.8515454531276542 {comparator: D6}
  E4.R5 relative elapsed time (geometric mean) = 0.6094493108259951 99% confidence interval 0.5799927267313179 to 0.6404019315199876 {comparator: D3}
  E4.R6 critical distance, Nemenyi test (RMSSE) = 0.37886188095533324 {level: 0.99}
  E4.R7 series in Nemenyi test (count) = 369
  E4.R8 p-value, rmcb test of equal mean ranks (RMSSE) (p_value) = 1.5420804210195766e-81
  E4.R9 series excluded by undefined RMSSE (count) = 1
  E4.T1 Per method: mean and median RMSSE over series, coverage of the 95% prediction intervals, median elapsed time per series in seconds, series with a failed fit, and the mean rank of RMSSE in the Nemenyi test with the bounds of its 99% interval.
  | row | method | D16 (mean) | D16 (median) | D12 {nominal level: 0.95} (mean) | D13 (median) | series failed (count) | fits failed (count) | mean rank of RMSSE, Nemenyi test (mean) | mean rank of RMSSE, Nemenyi test {bound: 99% interval, lower} | mean rank of RMSSE, Nemenyi test {bound: 99% interval, upper} |
  | r1 | D1 | 0.7948 | 0.6644 | 0.954 | 12.22 | 0 | 0 | 2.4823848238482347 | 2.293 | 2.672 |
  | r2 | D3 | 0.7985 | 0.6878 | 0.9433 | 20.26 | 0 | 0 | 2.558265582655823 | 2.369 | 2.748 |
  | r3 | D4 | 0.7949 | 0.671 | 0.9372 | 15.07 | 0 | 0 | 2.38753387533875 | 2.198 | 2.577 |
  | r4 | D5 | 0.8795 | 0.7065 | 0.9776 | 1.245 | 0 | 0 | 3.276 | 3.087 | 3.466 |
  | r5 | D6 | 1.003 | 0.8002 |  | 3.08e-05 | 0 | 0 | 4.295 | 4.106 | 4.485 |
  E4.T2 Per comparator: the series the relative RMSSE of smooth TBATS to it is computed on, and those left out because the RMSSE of smooth TBATS or of the comparator is zero. {method: D1}
  | row | comparator | series in relative RMSSE (count) | series excluded by zero RMSSE (count) |
  | r1 | D3 | 369 | 0 |
  | r2 | D4 | 369 | 0 |
  | r3 | D5 | 369 | 0 |
  | r4 | D6 | 368 | 1 |
  E4.T3 The one-sided t-tests of the plan on the per-series log ratios of smooth TBATS to the comparator: p-value and p-value adjusted by Holm over the four tests.
  | row | comparator | measure | p-value, one-sided t-test (p_value) | Holm-adjusted p-value, one-sided t-test (p_value) |
  | r1 | D3 | RMSSE | 0.01407 | 0.02814260346047013 |
  | r2 | D4 | RMSSE | 0.06692 | 0.06691757930915539 |
  | r3 | D6 | RMSSE | 4.861e-42 | 1.4582738129581806e-41 |
  | r4 | D3 | elapsed time | 3.609e-85 | 1.444e-84 |
E5 [quant; data_origin secondary] [[D20:smooth TBATS dnorm]] against [[D1:smooth TBATS]] (distribution "auto") on the 370 series of [[D18:electricity hourly]] with [[D19:rolling-origin evaluation]] (the windows of run RN13), run once under plan PL8 (lab run RN14). Both are fitted to each window one after the other in the same single-threaded worker, so that their elapsed times are measured under the same load; a failed fit gets the seasonal naive 168 forecasts and no interval. Per series, the RMSSE, the coverage of the 95% interval and the elapsed time averaged over the 7 origins; one-sided t-tests over series on the mean log ratio of RMSSE (auto over dnorm), the mean coverage of dnorm minus 0.95 and the mean log ratio of elapsed time (dnorm over auto), adjusted by Holm over the three; the [[D15:Nemenyi test]] on the six methods (A1). Series 322 has an undefined RMSSE and is left out.
  setup: studied D20, D1; cases D18; procedure D19; lens D16, D17, D12, D13
  derived from E4 (reanalysis)
  E5.R1 D17 (geometric mean) = 0.9274901866520837 99% confidence interval 0.782479221333724 to 1.099374939144907 {method: D20, comparator: D1}
  E5.R2 relative elapsed time (geometric mean) = 0.636836353873358 99% confidence interval 0.6189769713949341 to 0.6552110342663261 {method: D20, comparator: D1}
  E5.R3 coverage of smooth TBATS dnorm minus 0.95 (mean) = -0.01679894179894176 99% confidence interval -0.02364168981181017 to -0.00995619378607335
  E5.R4 log ratio of RMSSE, smooth TBATS to smooth TBATS dnorm (mean) = 0.07527306498554022 99% confidence interval -0.09474178115542094 to 0.2452879111265014
  E5.R5 log ratio of elapsed time, smooth TBATS dnorm to smooth TBATS (mean) = -0.4512425577124759 99% confidence interval -0.47968720990568653 to -0.42279790551926527
  E5.R6 critical distance, Nemenyi test (RMSSE) = 0.46329573710090965 {level: 0.99}
  E5.R7 series in Nemenyi test (count) = 369
  E5.R8 p-value, rmcb test of equal mean ranks (RMSSE) (p_value) = 1.0372190587495083e-94
  E5.R9 series excluded by undefined RMSSE (count) = 1
  E5.T1 Per method: mean and median RMSSE over series, coverage of the 95% prediction intervals, median elapsed time per series in seconds, and the series and fits that failed.
  | row | method | D16 (mean) | D16 (median) | D12 {nominal level: 0.95} (mean) | D13 (median) | series failed (count) | fits failed (count) |
  | r1 | D1 | 0.7948 | 0.6644 | 0.954 | 12.64 | 0 | 0 |
  | r2 | D20 | 0.7836 | 0.6554 | 0.9332 | 8.031 | 0 | 0 |
  E5.T2 Nemenyi test on the per-series RMSSE of the six methods: smooth TBATS and smooth TBATS dnorm from this run, the other four from run RN13 on the same windows; mean rank with the bounds of its 99% interval.
  | row | method | mean rank of RMSSE, Nemenyi test (mean) | mean rank of RMSSE, Nemenyi test {bound: 99% interval, lower} | mean rank of RMSSE, Nemenyi test {bound: 99% interval, upper} |
  | r1 | D1 | 3.03 | 2.798 | 3.261 |
  | r2 | D3 | 3.122 | 2.89 | 3.354 |
  | r3 | D4 | 2.921 | 2.69 | 3.153 |
  | r4 | D5 | 3.995 | 3.763 | 4.226 |
  | r5 | D6 | 5.165 | 4.934 | 5.397 |
  | r6 | D20 | 2.767 | 2.535 | 2.999 |
  E5.T3 The three one-sided t-tests of plan PL8 over series: p-value, p-value adjusted by Holm over the three, series in the test and series left out for a non-finite value.
  | row | measure | p-value, one-sided t-test (p_value) | Holm-adjusted p-value, one-sided t-test (p_value) | series in the t-test (count) | series excluded by a non-finite value (count) |
  | r1 | RMSSE | 0.8738 | 0.8738080346463214 | 369 | 0 |
  | r2 | coverage | 3.054e-10 | 6.107568971470071e-10 | 369 | 0 |
  | r3 | elapsed time | 8.144e-140 | 2.4431847468486607e-139 | 369 | 0 |

ARTEFACTS
A1 [figure, data_plot] Nemenyi test on the per-series RMSSE of the six methods on the electricity data (369 series, level 0.99): mean rank of each method with its 99% interval, from run RN14 (smooth TBATS, smooth TBATS dnorm) and run RN13 (the other four). shows E5.T2.r1, E5.T2.r2, E5.T2.r3, E5.T2.r4, E5.T2.r5, E5.T2.r6
A2 [table] CSV written by run RN14 with the per-method values of smooth TBATS and smooth TBATS dnorm (E5.T1), at full precision.
A3 [table] CSV written by run RN14 with the three t-tests of plan PL8 and the relative RMSSE and elapsed time with their intervals (E5.T3, E5.R1 to E5.R5), at full precision.
A4 [table] CSV written by run RN14 with the Nemenyi test of the six methods: mean ranks, intervals, critical distance (E5.T2, E5.R6 to E5.R8), at full precision.
A8 [data] Every result of the lab record (1308 results of 14 runs) at full precision: the lab result's ID, its run, the quantity as the run's script named it, its conditions and its value (both as JSON). The evidence's tables give the values to four significant digits.
A10 [code] The study's scripts: formulation_checks.py (E1), m4_hourly.py and m4_forecast_tbats.R (E2, E3), electricity_hourly.py (E4) and electricity_dnorm.py (E5), at the last commit of the study's code; each evidence names the commit it ran at.

REFERENCES
B4 10.1198/jasa.2011.tm09771 (doi)
B5 10.18637/jss.v027.i03 (doi)
B6 10.1016/j.ijforecast.2019.04.014 (doi)
B7 10.1016/j.ijforecast.2021.11.013 (doi)
B8 10.1201/9781003452652 (doi)
B9 10.24432/C58C86 (doi)
B10 10.1016/j.ijforecast.2006.03.001 (doi)
B11 https://jmlr.org/papers/v7/demsar06a.html (url)
B12 https://github.com/Nixtla/statsforecast (url)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C10, in: TBATS in smooth: its formulation, and its accuracy against forecast::tbats and AutoTBATS on hourly data. EntryLib. entrylib:2026.000007@v2#C10. https://entrylib.org/p/2026.000007/C10
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