CLAIM C5 of entrylib:2026.000001 Calibration of the prediction intervals of AutoMSARIMA in smooth on M1, M3, Tourism and M4
licence CC-BY-4.0; contributions C5

DEFINITIONS
D1 [method] MSARIMA analytic interval: The quantiles of `predict(interval="prediction")` of a pure ARIMA in smooth 1.3.0, which dispatches to the approximate interval. The ARIMA is the state space model
$$\begin{aligned} y_t &= w^\top v_{t-l} + \epsilon_t, \\ v_t &= F v_{t-l} + g \epsilon_t, \\ \epsilon_t &\sim N(0, \sigma^2). \end{aligned}$$
The quantile at level $\tau$ and horizon $h$ is
$$q_{\tau,h} = \hat y_{T+h|T} + \Phi^{-1}(\tau) \sqrt{V_h}, \qquad V_h = \hat\sigma^2 \frac{T}{T-k} c_h,$$
with
$$\hat\sigma^2 = \frac{1}{T} \sum_{t=1}^{T} e_t^2, \qquad c_h = 1 + \sum_{j=1}^{h-1} \left(w^\top F^{j-1} g\right)^2,$$
where $k$ is the number of estimated parameters and $c_h$ the $h$-th diagonal element of the analytic covariance of the state space model, written here for a model with lag 1. [source B6]
D2 [data] known-model simulation: Gaussian ARIMA series with $\sigma = 10$ generated from 11 data generating processes (white noise with a constant, random walk with and without drift, AR(1), MA(1), ARIMA(0,1,1), ARIMA(1,1,0), ARIMA(1,1,1), ARIMA(0,2,2), seasonal SARIMA(0,1,1)(0,1,1)$_{12}$ and SARIMA(1,0,0)(1,0,0)$_{12}$) plus an IMA(1,1) with Student-t errors on 4 degrees of freedom; in-sample lengths $T$ of 15, 25, 50, 100 and 300, holdout of 18; 2,000 replicates per process and length with seed 20261105 unless stated.
D4 [variable] coverage: For the quantile $q_{\tau,j}$ at level $\tau$ and horizon $j$, the share of holdout observations at or below it,
$$c_\tau = \frac{1}{h} \sum_{j=1}^{h} 1\left(y_{T+j} \le q_{\tau,j}\right),$$
averaged first within a series and then over series. cov80 is $c_{0.9} - c_{0.1}$, the coverage of the central 80% interval; q50 is $c_{0.5}$.
D5 [data] benchmark: The 5,315 series of M1 (1,001), M3 (3,003) and Tourism (1,311) from fcompdata 0.1.4, with their own holdouts and seasonal periods; lags $(1, m)$ for seasonal series. [members D18, D19, D20]
D6 [data] M4 sample: 1,500 series each of the yearly, quarterly and monthly M4 series, drawn with seed 20261105 without replacement, with the M4 holdouts. [source B4]
D7 [design] level terciles: Series split into terciles of the [[D17:level ratio]] across the series of a data set; cov80 reported for each tercile, and the spread as the largest minus the smallest of the three.
D8 [method] auto.adam error selection: R `auto.adam(y, model="NNN", lags, distribution=c("dnorm","dlnorm"), initial="backcasting")` of smooth 4.6: the ARIMA orders and constant are selected by AICc for each distribution with the maxima of auto.msarima (AR (3,3), I (2,1), MA (3,3)), then the distribution with the lower AICc is kept; with dnorm the model is the [[D10:AutoMSARIMA]] model, with dlnorm it is the [[D11:log-normal ARIMA]]. Quantiles from `forecast(interval="prediction")`. [source B6]
D9 [data] fresh M4 sample: 1,500 series each of the yearly, quarterly and monthly M4 series, drawn with seed 20261107 from the series not in the [[D6:M4 sample]], so disjoint from it. [source B4]
D10 [method] AutoMSARIMA: `AutoMSARIMA(lags, initial)` in Python and `auto.msarima()` in R of smooth 1.3.0 / 4.6: ARIMA orders up to AR (3,3), I (2,1), MA (3,3) and a constant selected by AICc, normal errors, the [[D1:MSARIMA analytic interval]] for the quantiles. Backcasting initial states unless stated. [source B6]
D11 [method] log-normal ARIMA: ADAM with `model="NNN"`, an ARIMA and `distribution="dlnorm"`: the ARIMA applies to the logarithms,
$$\begin{aligned} \log y_t &= w^\top v_{t-l} + \log(1+\epsilon_t), \\ v_t &= F v_{t-l} + g \log(1+\epsilon_t), \\ 1+\epsilon_t &\sim \mathrm{logN}(-\sigma^2/2, \sigma^2), \end{aligned}$$
so the scale of the error is proportional to the level. In the comparisons with fixed orders, the orders and constant are those [[D10:AutoMSARIMA]] selects on the same series. [source B6]
D13 [variable] MCE: Mean calibration error over the 99 quantile levels $\tau = 0.01, \dots, 0.99$:
$$\mathrm{MCE} = \frac{1}{99} \sum_{\tau} \left|\bar c_\tau - \tau\right|,$$
with $\bar c_\tau$ the [[D4:coverage]] of $q_\tau$.
D14 [variable] scaled pinball: Pinball loss averaged over the 99 quantile levels and the horizons, divided by the mean absolute first difference of the in-sample data for real series; in the simulation, by $\sigma$, and relpin is its ratio to the pinball of the true predictive distribution.
D15 [method] ETS forced and selected: ETS of smooth with backcasting: AutoADAM-style selection among ETS models (`ES(model="ZXZ")`, or ZXN for non-seasonal series), or ETS(ANN) or ETS(MNN) fitted to every series. [source B6]
D17 [variable] level ratio: Mean of the last third of the in-sample data of a series divided by the mean of all its in-sample data.
D18 [data] M1: The 1,001 series of the M1 competition with their holdouts. [source B1]
D19 [data] M3: The 3,003 series of the M3 competition with their holdouts. [source B2]
D20 [data] Tourism: The 1,311 series of the tourism forecasting competition with their holdouts. [source B3]
D21 [setting] smooth 1.0.8 and 1.3.0: Python smooth at tag v1.0.8-py (a separate worktree with its own environment) and at 1.3.0 (commit 634657d8 of openforecast-org/smooth).

CLAIMS
C5 [descriptive; principal; support: empirical_quant E12 (E12.T1.r1, E12.T1.r2) artefacts A6; limitations L5, L6; strength moderate (author); verifiability open; planning confirmatory; outcome inconclusive]
  On the [[D9:fresh M4 sample]], [[D8:auto.adam error selection]] lowers [[D13:MCE]] from 0.0352 to 0.0224 and the spread of 80% [[D4:coverage]] over the [[D7:level terciles]] from 0.272 to 0.077 against R auto.msarima ([[D10:AutoMSARIMA]]), short of the planned reduction of MCE to 0.6 times that of auto.msarima, so the planned test is inconclusive.
  scope: data_type: M4 yearly, quarterly and monthly series | conditions: backcasting; R smooth 4.6
  test: deterministic; sample 4500 fresh M4 series, one run; supported if ms <= 0.6*ma and ss <= 0.08; not supported if ms >= ma or ss >= sa; verdict computed
  plan: lab plan PL15; planned 2026-11-09T17:07:13Z; anchor git_remote

EVIDENCE
E12 [quant; design benchmark] R auto.msarima ([[D10:AutoMSARIMA]]) and [[D8:auto.adam error selection]] fitted to the 4,500 series of the [[D9:fresh M4 sample]] with backcasting; 99 quantiles per series scored on the M4 holdouts; MCE, cov80 by [[D7:level terciles]], scaled pinball, and the share of series choosing dlnorm (run RN48, plan PL15, which repeats PL14 after its failed run).
  setup: studied D8, D10; cases D9; procedure D7; lens D4, D13, D14, D17
  E12.T1 R auto.msarima and auto.adam error selection on the fresh M4 sample, backcasting {data: D9, initial states: backcasting}
  | row | method | D4 {nominal level: 0.8} | D13 | D14 (median) | D4 {nominal level: 0.8, D7: low} | D4 {nominal level: 0.8, D7: mid} | D4 {nominal level: 0.8, D7: high} | range of the 80% coverage over the level terciles | share of series for which dlnorm is selected |
  | r1 | D8 | 0.7461 | 0.02235705823668769 | 0.7514 | 0.7865 | 0.7425 | 0.7092 | 0.07729629629629642 | 0.6702 |
  | r2 | D10 | 0.6923 | 0.035202331961592524 | 0.6872 | 0.8193 | 0.7098 | 0.5477 | 0.2715277777777778 |  |
E20 [quant; design benchmark] MCE of each calibration curve of the 99 quantiles drawn from recorded outputs: the [[D2:known-model simulation]] (runs RN8, RN11, RN39), [[D21:smooth 1.0.8 and 1.3.0]] (RN16, RN19, RN43, RN44), the [[D5:benchmark]] (RN45 and the saved ETS benchmark) and the [[D9:fresh M4 sample]] (RN48), overall and by [[D7:level terciles]] (exploratory run RN49).
  setup: studied D10, D8, D15; cases D2, D5, D9; procedure D7; lens D13
  E20.T1 MCE of each curve of figure A3
  | row | panel | line | D13 |
  | r1 | T=15 | true parameters | 0.01439 |
  | r2 | T=15 | estimated parameters | 0.06533 |
  | r3 | T=15 | selected orders | 0.05793 |
  | r4 | T=25 | true parameters | 0.006361 |
  | r5 | T=25 | estimated parameters | 0.03414 |
  | r6 | T=25 | selected orders | 0.03659 |
  | r7 | T=50 | true parameters | 0.001008 |
  | r8 | T=50 | estimated parameters | 0.01962 |
  | r9 | T=50 | selected orders | 0.02353 |
  | r10 | T=100 | true parameters | 0.002348 |
  | r11 | T=100 | estimated parameters | 0.009999 |
  | r12 | T=100 | selected orders | 0.01388 |
  | r13 | T=300 | true parameters | 0.0006806 |
  | r14 | T=300 | estimated parameters | 0.002644 |
  | r15 | T=300 | selected orders | 0.004572 |
  E20.T2 MCE of each curve of figure A4
  | row | panel | line | D13 |
  | r1 | benchmark, backcasting | 1.0.8 | 0.03258 |
  | r2 | benchmark, backcasting | 1.3.0 | 0.04024 |
  | r3 | benchmark, optimal | 1.0.8 | 0.03312 |
  | r4 | benchmark, optimal | 1.3.0 | 0.03899 |
  | r5 | simulation T=100, true parameters | 1.0.8 | 0.01229 |
  | r6 | simulation T=100, true parameters | 1.3.0 | 0.002348 |
  | r7 | simulation T=100, estimated | 1.0.8 | 0.01202 |
  | r8 | simulation T=100, estimated | 1.3.0 | 0.009999 |
  | r9 | simulation T=100, selected | 1.0.8 | 0.01724 |
  | r10 | simulation T=100, selected | 1.3.0 | 0.01388 |
  E20.T3 MCE of each curve of figure A5
  | row | panel | line | D13 |
  | r1 | all series | D10 | 0.04024 |
  | r2 | all series | D8 | 0.01866 |
  | r3 | all series | ETS (AutoADAM ZXZ, saved) | 0.04424 |
  | r4 | low level tercile | D10 | 0.0281 |
  | r5 | low level tercile | D8 | 0.0334 |
  | r6 | mid level tercile | D10 | 0.05052 |
  | r7 | mid level tercile | D8 | 0.03523 |
  | r8 | high level tercile | D10 | 0.0623 |
  | r9 | high level tercile | D8 | 0.06547 |
  | r10 | auto.adam by selected distribution | chose dnorm | 0.04425 |
  | r11 | auto.adam by selected distribution | chose dlnorm | 0.02725 |
  E20.T4 MCE of each curve of figure A6
  | row | panel | line | D13 |
  | r1 | all series | D10 | 0.0352 |
  | r2 | all series | D8 | 0.02236 |
  | r3 | low level tercile | D10 | 0.03481 |
  | r4 | low level tercile | D8 | 0.04103 |
  | r5 | mid level tercile | D10 | 0.04118 |
  | r6 | mid level tercile | D8 | 0.02004 |
  | r7 | high level tercile | D10 | 0.07568 |
  | r8 | high level tercile | D8 | 0.1052 |
  | r9 | auto.adam by selected distribution | chose dnorm | 0.02093 |
  | r10 | auto.adam by selected distribution | chose dlnorm | 0.03506 |

LIMITATIONS
L5 [computational; affects C5, C20; reduces_transferability; mitigation none] The auto.adam comparison ran in R, because Python's AutoADAM removes dlnorm from the candidate distributions of a pure ARIMA whatever the user passes, while R's auto.adam removes it only from its default list.
L6 [analytic_flexibility; affects C5; could_bias_magnitude; mitigation none] The thresholds of plan PL15 were fixed after the exploratory benchmark run of auto.adam (RN45) was seen, on M4 series that run did not use.

ARTEFACTS
A3 [figure, data_plot] Calibration curves of the 99 quantiles in the known-model simulation, cases T, E and S, at T = 15, 25, 50, 100, 300: coverage of each quantile (top) and coverage minus level (bottom). shows E20.T1.r1, E20.T1.r2, E20.T1.r3, E20.T1.r4, E20.T1.r5, E20.T1.r6, E20.T1.r7, E20.T1.r8, E20.T1.r9, E20.T1.r10, E20.T1.r11, E20.T1.r12, E20.T1.r13, E20.T1.r14, E20.T1.r15
A4 [figure, data_plot] Calibration curves of the 99 quantiles of AutoMSARIMA 1.0.8 and 1.3.0 on the benchmark (backcasting, optimal) and in the simulation at T = 100. shows E20.T2.r1, E20.T2.r2, E20.T2.r3, E20.T2.r4, E20.T2.r5, E20.T2.r6, E20.T2.r7, E20.T2.r8, E20.T2.r9, E20.T2.r10
A5 [figure, data_plot] Calibration curves of the 99 quantiles on the 5,315 benchmark series: auto.msarima, auto.adam with error selection and the selected ETS, overall, by level tercile and by the distribution auto.adam chose. shows E20.T3.r1, E20.T3.r2, E20.T3.r3, E20.T3.r4, E20.T3.r5, E20.T3.r6, E20.T3.r7, E20.T3.r8, E20.T3.r9, E20.T3.r10, E20.T3.r11
A6 [figure, data_plot] Calibration curves of the 99 quantiles on the fresh M4 sample: auto.msarima and auto.adam with error selection, overall, by level tercile and by the distribution auto.adam chose. shows E20.T4.r1, E20.T4.r2, E20.T4.r3, E20.T4.r4, E20.T4.r5, E20.T4.r6, E20.T4.r7, E20.T4.r8, E20.T4.r9, E20.T4.r10
A7 [lab_record] Lab record of the study (branch claimset/msarima-interval-calibration): questions, hypotheses, plans, approvals, runs, results and notes as an append-only, hash-chained event log (lab record 0.20, claimset 0.39.0).
A10 [data] Every result recorded in the lab record (A7), one row per result: the lab result ID, the run that produced it, the quantity as the analysis script names it, its conditions as JSON and its full-precision value. The tables of the evidence objects hold these values rounded to four significant digits and at least four decimals.

REFERENCES
B1 10.1002/for.3980010202 (doi)
B2 10.1016/S0169-2070(00)00057-1 (doi)
B3 10.1016/j.ijforecast.2010.04.009 (doi)
B4 10.1016/j.ijforecast.2019.04.014 (doi)
B6 10.1201/9781003452652 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C5, in: Calibration of the prediction intervals of AutoMSARIMA in smooth on M1, M3, Tourism and M4. EntryLib. entrylib:2026.000001@v3#C5. https://entrylib.org/p/2026.000001/C5
The claimset it is from: Svetunkov, I. (guarantor) (2026). Calibration of the prediction intervals of AutoMSARIMA in smooth on M1, M3, Tourism and M4. EntryLib. entrylib:2026.000001@v3. https://entrylib.org/p/2026.000001@v3