{
 "bundle_of": "C10",
 "schema_version": "0.42",
 "paper": {
  "id": "entrylib:2026.000006",
  "version": 1,
  "title": "Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models",
  "licence": "CC-BY-4.0",
  "status": "active",
  "date": "2026-11-12"
 },
 "claims": [
  {
   "id": "C10",
   "statement": "Across all five data-generating processes in [[D15:Simulation experiment 2 synthetic series]], [[D10:Akram's log-ETS model]] achieves lower mean [[D11:scaled pinball score (SPS)]] over the 48-step horizon than pure multiplicative [[D3:ETS(M,Md,M)]] models with [[D7:Normal error distribution]], [[D6:Inverse Gaussian error distribution]], [[D4:Log-Normal error distribution]], [[D5:Gamma error distribution]], or [[D9:AICc]]-selected distributions, driven primarily by lower pinball loss in the central quantiles ($\\tau \\in [0.2, 0.8]$) while performing similarly in the distribution tails ($\\tau < 0.2$ and $\\tau > 0.8$).",
   "about": {
    "subject": "D10",
    "relation": "achieves lower mean scaled pinball score than pure multiplicative $ETS(M,Md,M)$ models primarily in central quantiles",
    "object": "D11",
    "comparator": "D3",
    "conditions": [
     "D15"
    ]
   },
   "type": "descriptive",
   "role": "principal",
   "novelty": "no_prior_found",
   "planning": "exploratory",
   "scope": {
    "conditions": [
     "$10,000$ simulated series of length $500$ per DGP, $99$ quantile levels from $0.01$ to $0.99$, horizon $48$"
    ]
   },
   "relations": [
    {
     "type": "extends",
     "target": "10.1111/j.1467-842X.2009.00555.x",
     "id_type": "doi",
     "locator": "Section 4",
     "note": "Benchmarks Akram's logarithmic ETS model against pure multiplicative ETS with positive error distributions across full predictive distributions."
    }
   ],
   "support": [
    {
     "kind": "simulation",
     "evidence": "E5",
     "artefacts": [
      "A3"
     ],
     "results": [
      "E5.T1",
      "E5.T2"
     ]
    }
   ],
   "assumptions": [
    "C12",
    "C13"
   ],
   "limitations": [
    "L4"
   ],
   "assessments": [
    {
     "by": "author",
     "strength": "moderate",
     "rationale": "Consistent across all five DGPs in the simulation study, though evaluated at a single noise level where logarithmic transformation remains well-behaved."
    }
   ],
   "provenance": "author_stated",
   "verifiability": "unverifiable",
   "verifiability_flags": []
  },
  {
   "id": "C12",
   "statement": "The multiplicative error term $(1+\\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\\epsilon_t) = 1$.",
   "type": "formal",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "statistical",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C13",
   "statement": "Smoothing parameters $\\alpha, \\beta, \\gamma$ in pure multiplicative ETS models lie within $[0, 1]$ so that state update terms $(1 + g\\epsilon_t)$ remain strictly positive whenever $1 + \\epsilon_t > 0$.",
   "type": "formal",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "methodological",
   "check": "tested",
   "check_evidence": "E1",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "id": "D1",
   "term": "ETS(M,*,*)",
   "kind": "method",
   "definition": "Pure multiplicative exponential smoothing state space model family written in logarithmic state space form as $$\\begin{aligned} \\log y_t &= w' \\log(v_{t-l}) + \\log(1 + \\epsilon_t) \\\\ \\log v_t &= F \\log v_{t-l} + \\log(1_k + g \\epsilon_t) \\end{aligned}$$ with conventional multi-step point forecast $\\hat{y}_{t+h} = \\exp\\left((w_1' F_1^{h-1} + w_m' F_m^{\\lceil h/m \\rceil - 1}) \\log v_t\\right)$.",
   "source": "B7"
  },
  {
   "id": "D3",
   "term": "ETS(M,Md,M)",
   "kind": "method",
   "broader": [
    "D1"
   ],
   "definition": "Pure multiplicative exponential smoothing model with damped multiplicative trend and multiplicative seasonality, defined by $$\\begin{aligned} y_t &= l_{t-1} b_{t-1}^{\\phi} s_{t-m} (1 + \\epsilon_t) \\\\ l_t &= l_{t-1} b_{t-1}^{\\phi} (1 + \\alpha \\epsilon_t) \\\\ b_t &= b_{t-1}^{\\phi} (1 + \\beta \\epsilon_t) \\\\ s_t &= s_{t-m} (1 + \\gamma \\epsilon_t) \\end{aligned}$$ where $\\phi$ is the dampening parameter, $\\alpha, \\beta, \\gamma$ are smoothing parameters, and $m$ is the seasonal lag.",
   "source": "B3"
  },
  {
   "id": "D4",
   "term": "Log-Normal error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with location $\\mu = -\\sigma^2/2$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = e^{\\sigma^2} - 1$."
  },
  {
   "id": "D5",
   "term": "Gamma error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with shape $\\xi = \\sigma^{-2}$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = \\sigma^2$."
  },
  {
   "id": "D6",
   "term": "Inverse Gaussian error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with location $\\mu = 1$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = \\sigma^2$."
  },
  {
   "id": "D7",
   "term": "Normal error distribution",
   "kind": "concept",
   "definition": "Classical Gaussian distribution on $(-\\infty, \\infty)$ applied to $1+\\epsilon_t$ with mean $\\mu = 1$ and variance $\\sigma^2$.",
   "source": "B3"
  },
  {
   "id": "D9",
   "term": "AICc",
   "kind": "variable",
   "unit": "1",
   "definition": "Corrected Akaike Information Criterion used to select among ETS component structures and error distributions.",
   "source": "B6"
  },
  {
   "id": "D10",
   "term": "Akram's log-ETS model",
   "kind": "method",
   "definition": "Additive error ETS model $ETS(A,Ad,A)$ with Normal errors applied to the natural logarithms of strictly positive time series and exponentiated back to the original scale.",
   "source": "B4"
  },
  {
   "id": "D11",
   "term": "scaled pinball score (SPS)",
   "kind": "variable",
   "unit": "1",
   "definition": "Quantile forecast accuracy measure scaled by the in-sample mean $\\bar{y}$, defined for probability level $\\tau$ over horizon $h$ as $$SPS = \\frac{1}{\\bar{y}} \\left( (1-\\tau) \\sum_{y_{t+j} < \\hat{q}_{\\tau,t+j}} (\\hat{q}_{\\tau,t+j} - y_{t+j}) + \\tau \\sum_{y_{t+j} \\ge \\hat{q}_{\\tau,t+j}} (y_{t+j} - \\hat{q}_{\\tau,t+j}) \\right).$$"
  },
  {
   "id": "D15",
   "term": "Simulation experiment 2 synthetic series",
   "kind": "data",
   "definition": "10,000 simulated time series of length 500 generated for each of five data-generating processes ($ETS(M,Md,M)$ with Normal, Inverse Gaussian, Gamma, Log-Normal errors, and Akram's exponentiated $ETS(A,Ad,A)$) with $\\sigma=0.1$, $\\alpha=0.1$, $\\beta=0.05$, $\\gamma=0.2$, $\\phi=0.95$, $l_0=100$, $b_0=1$, and holdout horizon $48$."
  },
  {
   "id": "D17",
   "term": "quantile bias",
   "kind": "variable",
   "unit": "1",
   "definition": "Mean across quantile levels $\\tau \\in \\{0.01, 0.02, \\dots, 0.99\\}$ of empirical quantile coverage minus nominal target level $\\tau$."
  },
  {
   "id": "D18",
   "term": "quantile precision",
   "kind": "variable",
   "unit": "1",
   "definition": "Mean across quantile levels $\\tau \\in \\{0.01, 0.02, \\dots, 0.99\\}$ of the absolute difference between empirical quantile coverage and nominal target level $\\tau$."
  },
  {
   "id": "D20",
   "term": "48-step quantile forecast simulation procedure",
   "kind": "design",
   "definition": "Estimation of $ETS(M,Md,M)$ with each candidate distribution, AICc-based automatic selection (`smooth::auto.adam`), and Akram's log-ETS model on 500 observations, evaluating 99 quantile levels ($\\tau = 0.01, \\dots, 0.99$) over a 48-step holdout."
  }
 ],
 "evidence": [
  {
   "id": "E1",
   "profile": "formal",
   "description": "Analytical derivation of recursive conditional expectations and point forecasts for pure multiplicative ETS models in logarithmic state space form (Section 2, Eqs. 2.1-2.11).",
   "foundation": "classical_probability",
   "assumptions": [
    "C12",
    "C13"
   ],
   "steps": [
    {
     "id": "E1.1",
     "statement": "Reformulate pure multiplicative ETS in logarithmic state space form: $\\log y_t = w' \\log(v_{t-l}) + \\log(1 + \\epsilon_t)$ and $\\log v_t = F \\log v_{t-l} + \\log(1_k + g \\epsilon_t)$.",
     "by": "Section 2, Eq. (2.1)",
     "rule": "logarithmic_transformation"
    },
    {
     "id": "E1.2",
     "statement": "Expand $h$-steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation $E(\\log y_{t+h}|t)$.",
     "by": "Section 2, Eqs. (2.5)-(2.6)",
     "rule": "recursive_substitution"
    },
    {
     "id": "E1.3",
     "statement": "Apply Jensen's inequality and non-negativity of $g$ to show point forecast $\\hat{y}_{t+h} \\le \\tilde{y}_{t+h} \\le \\mu_{y,t+h}$, with equality only at $h=1$, $h \\le m$ without trend, or for $ETS(M,N,N)$.",
     "by": "Section 2, Eqs. (2.8)-(2.11)",
     "rule": "jensens_inequality"
    }
   ],
   "availability": {
    "level": "public"
   }
  },
  {
   "id": "E5",
   "profile": "simulation",
   "description": "Monte Carlo simulation of $10,000$ series of length $500$ for each of five DGPs ($ETS(M,Md,M)$ with Normal, Inverse Gaussian, Gamma, Log-Normal, and Akram's exponentiated $ETS(A,Ad,A)$), evaluating 48-step quantile forecasts across 99 probability levels via scaled pinball score, quantile bias, and quantile precision (Section 5, Tables 4-5).",
   "setup": {
    "studied": [
     "D7",
     "D6",
     "D5",
     "D4",
     "D10"
    ],
    "cases": [
     "D15"
    ],
    "procedure": [
     "D20"
    ],
    "lens": [
     "D11",
     "D17",
     "D18"
    ]
   },
   "model": "$ETS(M,Md,M)$ and Akram $ETS(A,Ad,A)$ via `smooth::adam` and `smooth::auto.adam` v3.2.0",
   "runs": 10000,
   "tables": [
    {
     "id": "T1",
     "description": "Mean scaled pinball scores ($SPS$) across $10,000$ simulated series for applied models (rows) against data-generating processes (columns) over forecast horizon $48$.",
     "artefact": "A3",
     "row_labels": [
      "applied_model"
     ],
     "columns": [
      {
       "variable": "D11",
       "conditions": {
        "DGP": "Normal"
       }
      },
      {
       "variable": "D11",
       "conditions": {
        "DGP": "Inverse Gaussian"
       }
      },
      {
       "variable": "D11",
       "conditions": {
        "DGP": "Gamma"
       }
      },
      {
       "variable": "D11",
       "conditions": {
        "DGP": "Log-Normal"
       }
      },
      {
       "variable": "D11",
       "conditions": {
        "DGP": "Akram"
       }
      }
     ],
     "units": [
      "1",
      "1",
      "1",
      "1",
      "1"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "Normal"
       ],
       "values": [
        5.255,
        5.204,
        5.193,
        9.417,
        5.192
       ]
      },
      {
       "id": "r2",
       "label": [
        "Inverse Gaussian"
       ],
       "values": [
        5.3,
        5.221,
        5.224,
        9.464,
        5.229
       ]
      },
      {
       "id": "r3",
       "label": [
        "Gamma"
       ],
       "values": [
        5.287,
        5.192,
        5.231,
        9.388,
        5.214
       ]
      },
      {
       "id": "r4",
       "label": [
        "Log-Normal"
       ],
       "values": [
        5.215,
        5.295,
        5.224,
        9.423,
        5.22
       ]
      },
      {
       "id": "r5",
       "label": [
        "Akram"
       ],
       "values": [
        5.11,
        5.052,
        5.074,
        9.18,
        5.089
       ]
      },
      {
       "id": "r6",
       "label": [
        "Auto (AICc)"
       ],
       "values": [
        5.258,
        5.175,
        5.208,
        9.332,
        5.178
       ]
      }
     ]
    },
    {
     "id": "T2",
     "description": "Quantile bias and quantile precision across 99 quantile levels ($0.01$ to $0.99$) for applied models against data-generating processes.",
     "artefact": "A3",
     "row_labels": [
      "applied_model"
     ],
     "columns": [
      {
       "variable": "D17",
       "conditions": {
        "DGP": "Normal"
       }
      },
      {
       "variable": "D17",
       "conditions": {
        "DGP": "Inverse Gaussian"
       }
      },
      {
       "variable": "D17",
       "conditions": {
        "DGP": "Gamma"
       }
      },
      {
       "variable": "D17",
       "conditions": {
        "DGP": "Log-Normal"
       }
      },
      {
       "variable": "D17",
       "conditions": {
        "DGP": "Akram"
       }
      },
      {
       "variable": "D18",
       "conditions": {
        "DGP": "Normal"
       }
      },
      {
       "variable": "D18",
       "conditions": {
        "DGP": "Inverse Gaussian"
       }
      },
      {
       "variable": "D18",
       "conditions": {
        "DGP": "Gamma"
       }
      },
      {
       "variable": "D18",
       "conditions": {
        "DGP": "Log-Normal"
       }
      },
      {
       "variable": "D18",
       "conditions": {
        "DGP": "Akram"
       }
      }
     ],
     "units": [
      "1",
      "1",
      "1",
      "1",
      "1",
      "1",
      "1",
      "1",
      "1",
      "1"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "Normal"
       ],
       "values": [
        -0.002,
        0.003,
        0.011,
        0.007,
        -0.01,
        0.007,
        0.011,
        0.005,
        0.009,
        0.01
       ]
      },
      {
       "id": "r2",
       "label": [
        "Inverse Gaussian"
       ],
       "values": [
        -0.004,
        0.009,
        0.001,
        0.001,
        -0.013,
        0.009,
        0.009,
        0.006,
        0.009,
        0.015
       ]
      },
      {
       "id": "r3",
       "label": [
        "Gamma"
       ],
       "values": [
        -0.004,
        0.008,
        0.001,
        0.003,
        -0.012,
        0.011,
        0.009,
        0.005,
        0.008,
        0.014
       ]
      },
      {
       "id": "r4",
       "label": [
        "Log-Normal"
       ],
       "values": [
        -0.005,
        0.008,
        0.0,
        0.001,
        -0.013,
        0.008,
        0.004,
        0.008,
        0.004,
        0.014
       ]
      },
      {
       "id": "r5",
       "label": [
        "Akram"
       ],
       "values": [
        0.0,
        0.004,
        0.01,
        0.006,
        -0.007,
        0.004,
        0.01,
        0.005,
        0.008,
        0.007
       ]
      },
      {
       "id": "r6",
       "label": [
        "Auto (AICc)"
       ],
       "values": [
        -0.003,
        0.008,
        0.0,
        0.002,
        -0.012,
        0.008,
        0.004,
        0.009,
        0.007,
        0.013
       ]
      }
     ]
    }
   ],
   "artefacts": [
    "A3"
   ],
   "availability": {
    "level": "unavailable",
    "reason": "other"
   }
  }
 ],
 "limitations": [
  {
   "id": "L4",
   "category": "model_specification",
   "statement": "Simulation experiment 2 uses a single fixed parameter vector with error standard deviation $\\sigma = 0.1$, where multiplicative errors remain small enough that Normal and logarithmic approximations behave regularly.",
   "affects": [
    "C9",
    "C10"
   ],
   "impact": "reduces_transferability",
   "mitigation": "none"
  }
 ],
 "artefacts": [
  {
   "id": "A3",
   "kind": "table",
   "uri": "https://entrylib.org/p/2026.000006/v1/files/files/simulation-results.csv",
   "version": "sha256:c12d5bad7a350e03b8f6ef8606cb1ad7f12b8baca1b4daa50a482b99570f2296",
   "licence": "CC-BY-4.0",
   "availability": {
    "level": "public"
   },
   "description": "Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV."
  }
 ],
 "references": [
  {
   "id": "B3",
   "target": "978-3-540-71918-2",
   "id_type": "isbn"
  },
  {
   "id": "B4",
   "target": "10.1111/j.1467-842X.2009.00555.x",
   "id_type": "doi"
  },
  {
   "id": "B6",
   "target": "10.1080/03610927808837599",
   "id_type": "doi"
  },
  {
   "id": "B7",
   "target": "10.1201/9781003452652",
   "id_type": "doi"
  }
 ]
}