{
 "bundle_of": "C5",
 "schema_version": "0.40",
 "paper": {
  "id": "entrylib:2026.000001",
  "version": 3,
  "title": "Calibration of the prediction intervals of AutoMSARIMA in smooth on M1, M3, Tourism and M4",
  "licence": "CC-BY-4.0",
  "status": "corrected",
  "date": "2026-11-10",
  "status_notice": {
   "date": "2026-11-10",
   "reason_code": "correction",
   "statement": "The test quantities of the claims bound result entries that version 2 had turned into table cells; they now bind the cells, which keep the full precision the verdicts were computed at, and each table names the CSV of the lab results (A10) as its source. The defining equations of D1, D4, D11 and D13 are set as displayed equations.",
   "issued_by": "Ivan Svetunkov (ORCID 0000-0001-7826-0281)"
  }
 },
 "claims": [
  {
   "id": "C5",
   "statement": "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.",
   "about": {
    "subject": "D8",
    "relation": "lowers, short of the planned threshold, the calibration error of",
    "object": [
     "D13",
     "D4"
    ],
    "comparator": "D10",
    "conditions": [
     "D9"
    ]
   },
   "type": "descriptive",
   "role": "principal",
   "novelty": "no_prior_found",
   "support": [
    {
     "kind": "empirical_quant",
     "evidence": "E12",
     "results": [
      "E12.T1.r1",
      "E12.T1.r2"
     ],
     "artefacts": [
      "A6"
     ]
    }
   ],
   "provenance": "author_stated",
   "hypothesis": {
    "lab_event": "H11",
    "outcome": "inconclusive"
   },
   "planning": "confirmatory",
   "planned_at": "2026-11-09T17:07:13Z",
   "planned_at_anchor": "git_remote",
   "test": {
    "framework": "deterministic",
    "parameters": {
     "sample": "4500 fresh M4 series, one run"
    },
    "quantities": {
     "ms": {
      "variable": "D13",
      "conditions": {
       "data": "D9",
       "initial states": "backcasting",
       "method": "D8"
      }
     },
     "ma": {
      "variable": "D13",
      "conditions": {
       "data": "D9",
       "initial states": "backcasting",
       "method": "D10"
      }
     },
     "ss": {
      "variable": "range of the 80% coverage over the level terciles",
      "conditions": {
       "data": "D9",
       "initial states": "backcasting",
       "method": "D8"
      }
     },
     "sa": {
      "variable": "range of the 80% coverage over the level terciles",
      "conditions": {
       "data": "D9",
       "initial states": "backcasting",
       "method": "D10"
      }
     }
    },
    "supported": "ms <= 0.6*ma and ss <= 0.08",
    "not_supported": "ms >= ma or ss >= sa",
    "verdict": "computed",
    "runs": 1,
    "lab_plan": "PL15"
   },
   "scope": {
    "data_type": "M4 yearly, quarterly and monthly series",
    "conditions": [
     "backcasting",
     "R smooth 4.6"
    ]
   },
   "limitations": [
    "L5",
    "L6"
   ],
   "assessments": [
    {
     "by": "author",
     "strength": "moderate",
     "rationale": "Planned test on 4,500 series that did not suggest the hypothesis; the supported rule failed by 0.0013 of MCE and the refutation rule was not met."
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "id": "D1",
   "term": "MSARIMA analytic interval",
   "kind": "method",
   "source": "B6",
   "definition": "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\n$$\\begin{aligned} y_t &= w^\\top v_{t-l} + \\epsilon_t, \\\\ v_t &= F v_{t-l} + g \\epsilon_t, \\\\ \\epsilon_t &\\sim N(0, \\sigma^2). \\end{aligned}$$\nThe quantile at level $\\tau$ and horizon $h$ is\n$$q_{\\tau,h} = \\hat y_{T+h|T} + \\Phi^{-1}(\\tau) \\sqrt{V_h}, \\qquad V_h = \\hat\\sigma^2 \\frac{T}{T-k} c_h,$$\nwith\n$$\\hat\\sigma^2 = \\frac{1}{T} \\sum_{t=1}^{T} e_t^2, \\qquad c_h = 1 + \\sum_{j=1}^{h-1} \\left(w^\\top F^{j-1} g\\right)^2,$$\nwhere $k$ 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."
  },
  {
   "id": "D2",
   "term": "known-model simulation",
   "kind": "data",
   "definition": "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."
  },
  {
   "id": "D4",
   "term": "coverage",
   "kind": "variable",
   "definition": "For the quantile $q_{\\tau,j}$ at level $\\tau$ and horizon $j$, the share of holdout observations at or below it,\n$$c_\\tau = \\frac{1}{h} \\sum_{j=1}^{h} 1\\left(y_{T+j} \\le q_{\\tau,j}\\right),$$\naveraged 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 $c_{0.5}$."
  },
  {
   "id": "D5",
   "term": "benchmark",
   "kind": "data",
   "definition": "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"
   ]
  },
  {
   "id": "D6",
   "term": "M4 sample",
   "kind": "data",
   "source": "B4",
   "definition": "1,500 series each of the yearly, quarterly and monthly M4 series, drawn with seed 20261105 without replacement, with the M4 holdouts."
  },
  {
   "id": "D7",
   "term": "level terciles",
   "kind": "design",
   "definition": "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."
  },
  {
   "id": "D8",
   "term": "auto.adam error selection",
   "kind": "method",
   "source": "B6",
   "definition": "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\")`."
  },
  {
   "id": "D9",
   "term": "fresh M4 sample",
   "kind": "data",
   "source": "B4",
   "definition": "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."
  },
  {
   "id": "D10",
   "term": "AutoMSARIMA",
   "kind": "method",
   "source": "B6",
   "definition": "`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."
  },
  {
   "id": "D11",
   "term": "log-normal ARIMA",
   "kind": "method",
   "source": "B6",
   "definition": "ADAM with `model=\"NNN\"`, an ARIMA and `distribution=\"dlnorm\"`: the ARIMA applies to the logarithms,\n$$\\begin{aligned} \\log y_t &= w^\\top v_{t-l} + \\log(1+\\epsilon_t), \\\\ v_t &= F v_{t-l} + g \\log(1+\\epsilon_t), \\\\ 1+\\epsilon_t &\\sim \\mathrm{logN}(-\\sigma^2/2, \\sigma^2), \\end{aligned}$$\nso 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."
  },
  {
   "id": "D13",
   "term": "MCE",
   "kind": "variable",
   "definition": "Mean calibration error over the 99 quantile levels $\\tau = 0.01, \\dots, 0.99$:\n$$\\mathrm{MCE} = \\frac{1}{99} \\sum_{\\tau} \\left|\\bar c_\\tau - \\tau\\right|,$$\nwith $\\bar c_\\tau$ the [[D4:coverage]] of $q_\\tau$."
  },
  {
   "id": "D14",
   "term": "scaled pinball",
   "kind": "variable",
   "definition": "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."
  },
  {
   "id": "D15",
   "term": "ETS forced and selected",
   "kind": "method",
   "source": "B6",
   "definition": "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."
  },
  {
   "id": "D17",
   "term": "level ratio",
   "kind": "variable",
   "definition": "Mean of the last third of the in-sample data of a series divided by the mean of all its in-sample data."
  },
  {
   "id": "D18",
   "term": "M1",
   "kind": "data",
   "source": "B1",
   "definition": "The 1,001 series of the M1 competition with their holdouts."
  },
  {
   "id": "D19",
   "term": "M3",
   "kind": "data",
   "source": "B2",
   "definition": "The 3,003 series of the M3 competition with their holdouts."
  },
  {
   "id": "D20",
   "term": "Tourism",
   "kind": "data",
   "source": "B3",
   "definition": "The 1,311 series of the tourism forecasting competition with their holdouts."
  },
  {
   "id": "D21",
   "term": "smooth 1.0.8 and 1.3.0",
   "kind": "setting",
   "definition": "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)."
  }
 ],
 "evidence": [
  {
   "id": "E12",
   "profile": "quant",
   "design": "benchmark",
   "description": "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).",
   "availability": {
    "level": "public"
   },
   "lab_runs": [
    "RN48"
   ],
   "setup": {
    "studied": [
     "D8",
     "D10"
    ],
    "cases": [
     "D9"
    ],
    "procedure": [
     "D7"
    ],
    "lens": [
     "D4",
     "D13",
     "D14",
     "D17"
    ]
   },
   "execution": {
    "code_version": "5ca96a0fb806fcabad975937952e4a1330c3759d",
    "software": {
     "python": "3.14.4",
     "platform": "Linux-7.0.0-42-generic-x86_64-with-glibc2.43"
    }
   },
   "tables": [
    {
     "id": "T1",
     "description": "R auto.msarima and auto.adam error selection on the fresh M4 sample, backcasting",
     "conditions": {
      "data": "D9",
      "initial states": "backcasting"
     },
     "row_labels": [
      "method"
     ],
     "columns": [
      {
       "variable": "D4",
       "conditions": {
        "nominal level": 0.8
       }
      },
      "D13",
      {
       "variable": "D14",
       "statistic": "median"
      },
      {
       "variable": "D4",
       "conditions": {
        "nominal level": 0.8,
        "D7": "low"
       }
      },
      {
       "variable": "D4",
       "conditions": {
        "nominal level": 0.8,
        "D7": "mid"
       }
      },
      {
       "variable": "D4",
       "conditions": {
        "nominal level": 0.8,
        "D7": "high"
       }
      },
      "range of the 80% coverage over the level terciles",
      "share of series for which dlnorm is selected"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "D8"
       ],
       "values": [
        0.7461,
        0.02235705823668769,
        0.7514,
        0.7865,
        0.7425,
        0.7092,
        0.07729629629629642,
        0.6702
       ]
      },
      {
       "id": "r2",
       "label": [
        "D10"
       ],
       "values": [
        0.6923,
        0.035202331961592524,
        0.6872,
        0.8193,
        0.7098,
        0.5477,
        0.2715277777777778,
        null
       ]
      }
     ],
     "artefact": "A10"
    }
   ],
   "artefacts": [
    "A10"
   ]
  },
  {
   "id": "E20",
   "profile": "quant",
   "design": "benchmark",
   "description": "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).",
   "availability": {
    "level": "public"
   },
   "lab_runs": [
    "RN49"
   ],
   "setup": {
    "studied": [
     "D10",
     "D8",
     "D15"
    ],
    "cases": [
     "D2",
     "D5",
     "D9"
    ],
    "procedure": [
     "D7"
    ],
    "lens": [
     "D13"
    ]
   },
   "execution": {
    "code_version": "5ca96a0fb806fcabad975937952e4a1330c3759d",
    "software": {
     "python": "3.14.4",
     "platform": "Linux-7.0.0-42-generic-x86_64-with-glibc2.43"
    }
   },
   "tables": [
    {
     "id": "T1",
     "description": "MCE of each curve of figure A3",
     "row_labels": [
      "panel",
      "line"
     ],
     "columns": [
      "D13"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "T=15",
        "true parameters"
       ],
       "values": [
        0.01439
       ]
      },
      {
       "id": "r2",
       "label": [
        "T=15",
        "estimated parameters"
       ],
       "values": [
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       ]
      },
      {
       "id": "r3",
       "label": [
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        "selected orders"
       ],
       "values": [
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       ]
      },
      {
       "id": "r4",
       "label": [
        "T=25",
        "true parameters"
       ],
       "values": [
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       ]
      },
      {
       "id": "r5",
       "label": [
        "T=25",
        "estimated parameters"
       ],
       "values": [
        0.03414
       ]
      },
      {
       "id": "r6",
       "label": [
        "T=25",
        "selected orders"
       ],
       "values": [
        0.03659
       ]
      },
      {
       "id": "r7",
       "label": [
        "T=50",
        "true parameters"
       ],
       "values": [
        0.001008
       ]
      },
      {
       "id": "r8",
       "label": [
        "T=50",
        "estimated parameters"
       ],
       "values": [
        0.01962
       ]
      },
      {
       "id": "r9",
       "label": [
        "T=50",
        "selected orders"
       ],
       "values": [
        0.02353
       ]
      },
      {
       "id": "r10",
       "label": [
        "T=100",
        "true parameters"
       ],
       "values": [
        0.002348
       ]
      },
      {
       "id": "r11",
       "label": [
        "T=100",
        "estimated parameters"
       ],
       "values": [
        0.009999
       ]
      },
      {
       "id": "r12",
       "label": [
        "T=100",
        "selected orders"
       ],
       "values": [
        0.01388
       ]
      },
      {
       "id": "r13",
       "label": [
        "T=300",
        "true parameters"
       ],
       "values": [
        0.0006806
       ]
      },
      {
       "id": "r14",
       "label": [
        "T=300",
        "estimated parameters"
       ],
       "values": [
        0.002644
       ]
      },
      {
       "id": "r15",
       "label": [
        "T=300",
        "selected orders"
       ],
       "values": [
        0.004572
       ]
      }
     ],
     "artefact": "A10"
    },
    {
     "id": "T2",
     "description": "MCE of each curve of figure A4",
     "row_labels": [
      "panel",
      "line"
     ],
     "columns": [
      "D13"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "benchmark, backcasting",
        "1.0.8"
       ],
       "values": [
        0.03258
       ]
      },
      {
       "id": "r2",
       "label": [
        "benchmark, backcasting",
        "1.3.0"
       ],
       "values": [
        0.04024
       ]
      },
      {
       "id": "r3",
       "label": [
        "benchmark, optimal",
        "1.0.8"
       ],
       "values": [
        0.03312
       ]
      },
      {
       "id": "r4",
       "label": [
        "benchmark, optimal",
        "1.3.0"
       ],
       "values": [
        0.03899
       ]
      },
      {
       "id": "r5",
       "label": [
        "simulation T=100, true parameters",
        "1.0.8"
       ],
       "values": [
        0.01229
       ]
      },
      {
       "id": "r6",
       "label": [
        "simulation T=100, true parameters",
        "1.3.0"
       ],
       "values": [
        0.002348
       ]
      },
      {
       "id": "r7",
       "label": [
        "simulation T=100, estimated",
        "1.0.8"
       ],
       "values": [
        0.01202
       ]
      },
      {
       "id": "r8",
       "label": [
        "simulation T=100, estimated",
        "1.3.0"
       ],
       "values": [
        0.009999
       ]
      },
      {
       "id": "r9",
       "label": [
        "simulation T=100, selected",
        "1.0.8"
       ],
       "values": [
        0.01724
       ]
      },
      {
       "id": "r10",
       "label": [
        "simulation T=100, selected",
        "1.3.0"
       ],
       "values": [
        0.01388
       ]
      }
     ],
     "artefact": "A10"
    },
    {
     "id": "T3",
     "description": "MCE of each curve of figure A5",
     "row_labels": [
      "panel",
      "line"
     ],
     "columns": [
      "D13"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "all series",
        "D10"
       ],
       "values": [
        0.04024
       ]
      },
      {
       "id": "r2",
       "label": [
        "all series",
        "D8"
       ],
       "values": [
        0.01866
       ]
      },
      {
       "id": "r3",
       "label": [
        "all series",
        "ETS (AutoADAM ZXZ, saved)"
       ],
       "values": [
        0.04424
       ]
      },
      {
       "id": "r4",
       "label": [
        "low level tercile",
        "D10"
       ],
       "values": [
        0.0281
       ]
      },
      {
       "id": "r5",
       "label": [
        "low level tercile",
        "D8"
       ],
       "values": [
        0.0334
       ]
      },
      {
       "id": "r6",
       "label": [
        "mid level tercile",
        "D10"
       ],
       "values": [
        0.05052
       ]
      },
      {
       "id": "r7",
       "label": [
        "mid level tercile",
        "D8"
       ],
       "values": [
        0.03523
       ]
      },
      {
       "id": "r8",
       "label": [
        "high level tercile",
        "D10"
       ],
       "values": [
        0.0623
       ]
      },
      {
       "id": "r9",
       "label": [
        "high level tercile",
        "D8"
       ],
       "values": [
        0.06547
       ]
      },
      {
       "id": "r10",
       "label": [
        "auto.adam by selected distribution",
        "chose dnorm"
       ],
       "values": [
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       ]
      },
      {
       "id": "r11",
       "label": [
        "auto.adam by selected distribution",
        "chose dlnorm"
       ],
       "values": [
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       ]
      }
     ],
     "artefact": "A10"
    },
    {
     "id": "T4",
     "description": "MCE of each curve of figure A6",
     "row_labels": [
      "panel",
      "line"
     ],
     "columns": [
      "D13"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "all series",
        "D10"
       ],
       "values": [
        0.0352
       ]
      },
      {
       "id": "r2",
       "label": [
        "all series",
        "D8"
       ],
       "values": [
        0.02236
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