{ "schema_version": "0.41", "paper": { "title": "The observed-data likelihood of ADAM and iETS with unobserved periods", "licence": "CC-BY-4.0", "declarations": { "funding": [], "competing_interests": "None declared." }, "classification": { "subfield": "openalex:subfields/2613", "topics": [ "openalex:T10974", "openalex:T10567", "openalex:T10698" ], "keywords": [ "exponential smoothing", "single source of error", "intermittent demand", "missing values", "Kalman filter", "EM algorithm", "ADAM", "iETS" ] }, "contributions": [ "C1", "C2", "C3", "C4", "C5", "C6", "C7", "C8", "C9", "C10", "C11" ], "highlights": [ "C1", "C2", "C5", "C4", "C8" ], "autonomy": "human_supervised", "contribution_record": "complete", "origin": "native", "id": "entrylib:2026.000003", "correction_reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed.", "version": 2, "date": "2026-11-11", "status": "corrected", "previous_version": "entrylib:2026.000003@v1", "status_notice": { "date": "2026-11-11", "reason_code": "correction", "statement": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed.", "issued_by": "Ivan Svetunkov (ORCID 0000-0001-7826-0281)" }, "changes": [ { "op": "add", "path": "/artefacts/9", "value": { "id": "A10", "kind": "proof", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks.md", "version": "sha256:2ccd9457ef876f55c27b3b0004c25c5a991507f6667c59795e7515cc552fd7f1", "availability": { "level": "public" }, "licence": "CC-BY-4.0", "description": "Report of the first round of the independent proof check by the contributor P6 (2026-11-11): for C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19, the re-derivation, the numerical spot checks, the outcome and the gaps found." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/10", "value": { "id": "A11", "kind": "proof", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks-2.md", "version": "sha256:1ba26c97d75792bd6d91d375288b585b375e459560b131ad4102b5762e7dbbf7", "availability": { "level": "public" }, "licence": "CC-BY-4.0", "description": "Report of the second round of the independent proof check by P6 (2026-11-11), on C2, C5, C10, C14 and C19 as corrected after the first round, and on the smaller changes to C1, C6, C8 and C16." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/11", "value": { "id": "A12", "kind": "data", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks.json", "version": "sha256:6d51a427bbf6ae28ed6a27fc951122e3f4c75599d524d16a86c2e02440781d8b", "availability": { "level": "public" }, "licence": "CC-BY-4.0", "description": "The outcomes of the first round of the proof check, one record per claim." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/12", "value": { "id": "A13", "kind": "data", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks-2.json", "version": "sha256:65ad6eb15d5da30bf17d10d6cb8d299e4413cd0a193b106673bdae07d179352a", "availability": { "level": "public" }, "licence": "CC-BY-4.0", "description": "The outcomes of the second round of the proof check, one record per claim." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/13", "value": { "id": "A14", "kind": "code", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk1.py", "version": "sha256:88de8878db174afda5c9a385d6d46bd0b00e744ce5fa8402814f47aa9d65db8f", "availability": { "level": "public" }, "licence": "LGPL-2.1-only", "description": "Numerical spot checks of the first round of the proof check (Python with numpy); the report A10 says which claim each serves." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/14", "value": { "id": "A15", "kind": "code", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk2.py", "version": "sha256:e83caee9e69890fd5dbde71fa35400408bff35931dc748cddc9dcf877b1f70b3", "availability": { "level": "public" }, "licence": "LGPL-2.1-only", "description": "Numerical spot checks of the proof check (Python with numpy), used in both rounds; the reports A10 and A11 say which claim each serves." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/15", "value": { "id": "A16", "kind": "code", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk3.py", "version": "sha256:c87a752acc6bbb6816fce17692a752be300da8ab53fa3ddb769164084a8b3833", "availability": { "level": "public" }, "licence": "LGPL-2.1-only", "description": "Numerical spot checks of the proof check (Python with numpy), used in both rounds; the reports A10 and A11 say which claim each serves." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/artefacts/16", "value": { "id": "A17", "kind": "code", "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk4.py", "version": "sha256:aef765b48df441007f23a0a6bce88cc0b3b17256ea1a39f7fcdda49546002237", "availability": { "level": "public" }, "licence": "LGPL-2.1-only", "description": "Numerical spot checks of the second round of the proof check (Python with numpy); the report A11 says which claim each serves." }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/0/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived the prediction-error factorisation: P(y=0|past)=1-p by independence and continuity, and y>0 has density p*f, so the log-likelihood separates into size and occurrence parts. Checked numerically on ETS(A,N,N) with two unobserved periods: the integral of the complete-data density over the unobserved sizes equals the Kalman predictive-density sum to 2e-13. (first round; report A10, scripts A14-A16.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/0/relations", "value": [ { "type": "extends", "reference": "B1", "self": false, "note": "Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood." }, { "type": "extends", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/0/statement", "value": "For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\\sim\\text{Bernoulli}(p_t)$ independent of a positive continuous size $z_t$, the [[D3:observed-data likelihood]] is $$\\sum_{O_t=1}\\log f(z_t\\mid Y_{t-1};\\theta) + \\sum_{O_t=1}\\log p_t + \\sum_{O_t=0}\\log(1-p_t),$$ in which the unobserved sizes do not appear and the size and occurrence parts separate.", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/0/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/1/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived Fisher's identity (gradient of Q at the current estimate equals the observed-data score, valid at every iterate and so at EM fixed points) and the normal-variance example: the observed-data likelihood is stationary at S/T1 and the entropy-augmented one at S/T, different when T0>0 and S>0. Numerically with T1=60, T0=40 the maximisers are 3.318 and 1.991, and the cross-entropy gradient at the current estimate is 0 while that of -H is -0.385. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/1/relations", "value": [ { "type": "contradicts", "reference": "B4", "locator": "Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm)", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/1/statement", "value": "The stationary points of the [[D4:entropy-augmented likelihood]] of iETS differ in general from those of the [[D3:observed-data likelihood]]: at a fixed point of the EM algorithm the gradient of the expected complete-data log-likelihood equals the score of the observed-data likelihood, while the term $-H_z(\\theta)$ of each period with zero demand has a gradient that is not zero in general, and for normal sizes with fixed location the two objectives are stationary at different variances.", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/1/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/2/relations", "value": [ { "type": "extends", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/3/relations", "value": [ { "type": "extends", "reference": "B6", "locator": "Kalman filter for state space models with correlated disturbances", "self": false, "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated." }, { "type": "extends", "reference": "B7", "locator": "Kalman filter for state space models with correlated disturbances", "self": false, "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/4/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived f=1+sum c_i^2>1 for some c_i!=0 and that the zero-error likelihood uses sigma^2 where the exact one uses sigma^2 f_t, so the likelihoods differ; the variance maximisers (1/T1)sum e^2 and (1/T1)sum v^2/f differ because their difference is positive when the first observed period after a gap is the last observation. In an ETS(A,N,N) check the two sums were 7.4975 and 6.7015, and the difference was positive in all 2000 random samples. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/4/relations", "value": [ { "type": "contradicts", "reference": "B4", "locator": "Appendix C (estimation with zero errors at the unobserved periods)", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/4/scope/conditions", "value": [ "normally distributed errors", "$\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval" ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/4/statement", "value": "For pure additive single source of error models with normal errors and [[D2:unobserved period]]s, if $c_i=\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval, the [[D5:zero-error recursion]] likelihood is not the observed-data likelihood and its maximiser in the error variance is in general not the maximum likelihood estimate, because at the observed period after such an interval the recursion uses the error variance $\\sigma^2$ instead of $\\sigma^2f_t>\\sigma^2$.", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/4/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/5/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Derived the filter on log z with mean shift -g s/2 and the Jacobian, and the score in s; the cross terms cancel to give (sum b^2/f)s^2+T1 s-sum v0^2/f=0. Numerically the filter log-likelihood equals the brute-force integral over two unobserved log-sizes (-18.275599457737) and the positive roots (0.29147, 0.25190) equal the grid maxima of the filter likelihood. (first round; report A10, scripts A14-A16.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/5/relations", "value": [ { "type": "extends", "reference": "B5", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/5/statement", "value": "For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] applied to $\\log z_t$, with the known error mean $-s/2$ added to the prediction and, multiplied by $\\mathbf g$, to the propagation, gives the exact observed-data likelihood, and the maximum likelihood scale $s$ is the positive root of $$\\Big(\\sum \\frac{b_t^2}{f_t}\\Big)s^2+T_1s-\\sum\\frac{(v_t^0)^2}{f_t}=0,$$ where $v_t(s)=v_t^0+sb_t$ is the filter's prediction error.", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/5/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/6/relations", "value": [ { "type": "extends", "reference": "B5", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/7/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "With f_t=1 and b_t=1/2 the C6 equation is s^2+4s-4m=0, root 2(sqrt(1+m)-1); the ADAM-type estimator is the smaller root of s^2-4s+4m=0, real only for m<=1 and larger than the MLE for every m in (0,1). At m=0.5625 the values are 0.677 and 0.5 (ratio 1.354); a numerical likelihood maximisation on simulated log-normal errors agreed with the formula. (first round; report A10, scripts A14-A16.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/7/depends_on", "value": [ "C15", "C7" ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/7/relations", "value": [ { "type": "contradicts", "reference": "B5", "locator": "estimator of the scale of the log-normal distribution", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/7/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/8/relations", "value": [ { "type": "extends", "reference": "B4", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/9/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "P(0<=y<=u)=(1-p)+pF_z(u) for positive sizes, so coverage c>1-p needs F_z(u)=(c-(1-p))/p, a quantile of the size distribution whose mean is mu_hat while E(y)=p mu_hat. Monte Carlo with p=0.4, c=0.9 gave coverage 0.8997 and size level 0.7500. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/9/relations", "value": [ { "type": "extends", "reference": "B4", "self": true, "note": "Svetunkov and Boylan forecast intermittent demand as the product of the occurrence probability and the forecast demand size." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/9/scope/conditions", "value": [ "prediction interval of the demand of the form $[0,u]$ that counts zeros as covered", "coverage $c>1-p_t$", "Bernoulli occurrence independent of a positive continuous demand size" ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/9/statement", "value": "For a model with an occurrence part, occurrence probability $p_t$ and positive sizes, a prediction interval $[0,u]$ of the demand with coverage $c>1-p_t$ that counts the zeros as covered has as upper bound $u$ the quantile at level $(c-(1-p_t))/p_t$ of the distribution of the demand size, whose mean, the conditional mean of the size, is the size forecast $\\hat\\mu_t$, not the demand forecast $p_t\\hat\\mu_t$.", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/claims/9/verification", "value": "agent_checked", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/10/relations", "value": [ { "type": "supersedes", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/claims/11/checks", "value": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Taking logs of D6 gives the additive ETS(A,Ad,A) recursion with innovation u_t=log(1+eps); in D7 the innovation is log(1+g eps), and log(1+g eps)/(g log(1+eps)) is not constant for 00$ and $S>0$, so the stationary points differ", "by": [ "E2.3", "E2.4" ], "rule": "comparison" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/evidence/5/steps", "value": [ { "id": "E6.1", "statement": "After $j-1$ unobserved periods from $\\mathbf P=\\mathbf 0$, $f=1+\\sum_{i1$ when $c_i\\neq0$ for some $i1$ and $v_t=e_t$ being present", "by": [ "E6.2", "E6.4", "C4" ], "rule": "differentiation in $\\sigma^2$" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/evidence/6/steps", "value": [ { "id": "E7.1", "statement": "$P=(j-1)\\alpha^2$, $f=(j-1)\\alpha^2+1$ after $j-1$ unobserved periods", "by": [ "C4", "D8" ], "rule": "substitution $w=F=1$, $g=\\alpha$" }, { "id": "E7.2", "statement": "$k=(P+\\alpha)/f$", "by": [ "E7.1", "D8" ], "rule": "substitution" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/evidence/8/steps", "value": [ { "id": "E9.1", "statement": "$\\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$", "by": [ "C12", "C36" ], "rule": "linearity in logarithms" }, { "id": "E9.2", "statement": "The filter of the linear Gaussian model with the Jacobian $-\\sum\\log z_t$ gives the exact likelihood", "by": [ "C33", "C34", "C4", "E9.1" ], "rule": "change of variables" }, { "id": "E9.3", "statement": "Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$", "by": [ "E9.2" ], "rule": "linearity" }, { "id": "E9.4", "statement": "The score in $s$ times $2s^2$ gives $(\\sum b_t^2/f_t)s^2+T_1s-\\sum(v_t^0)^2/f_t=0$", "by": [ "E9.3" ], "rule": "differentiation" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/evidence/11/steps", "value": [ { "id": "E12.1", "statement": "$$P(0\\le y_t\\le u)=(1-p_t)+p_tF_z(u)$$ for positive sizes and occurrence independent of the size", "by": [ "C31", "C32" ], "rule": "law of total probability" }, { "id": "E12.2", "statement": "Setting it to $c$ with $c>1-p_t$ gives $F_z(u)=(c-(1-p_t))/p_t\\in(0,1]$, so $u$ is the quantile of the size at that level", "by": [ "E12.1" ], "rule": "algebra" }, { "id": "E12.3", "statement": "The size has mean $\\hat\\mu_t$, the demand $\\E(o_tz_t)=p_t\\hat\\mu_t$", "by": [ "C31" ], "rule": "independence" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "replace", "path": "/evidence/12/steps", "value": [ { "id": "E13.1", "statement": "$x\\mapsto x^{\\alpha}$ is strictly concave for $0<\\alpha<1$, so for non-degenerate $1+\\epsilon>0$ with $\\E(1+\\epsilon)=1$, $\\E(1+\\epsilon)^{\\alpha}<(\\E(1+\\epsilon))^{\\alpha}=1$", "by": [ "D6" ], "rule": "Jensen's inequality" }, { "id": "E13.2", "statement": "In the ADAM form $l_t=l_{t-1}(1+\\epsilon_t)^{\\alpha}$ and $z_t=l_{t-1}(1+\\epsilon_t)$, so $$\\E(z_{T+h}\\mid Y_T)=l_T\\,[\\E(1+\\epsilon)^{\\alpha}]^{h-1}\\,\\E(1+\\epsilon)=\\hat\\mu_{T+h}[\\E(1+\\epsilon)^{\\alpha}]^{h-1},$$ with $\\hat\\mu_{T+h}=l_T$ the forecast with zero errors", "by": [ "D6", "C30" ], "rule": "independence of the errors" }, { "id": "E13.3", "statement": "For $h\\ge2$ the factor is below one, so the conditional mean is below the forecast with zero errors", "by": [ "E13.1", "E13.2" ], "rule": "substitution" }, { "id": "E13.4", "statement": "For $1+\\epsilon$ Gamma with shape $1/s$ and scale $s$ (mean one, variance $s$), $\\E(1+\\epsilon)^{\\alpha}=\\Gamma(1/s+\\alpha)s^{\\alpha}/\\Gamma(1/s)$", "by": [ "E13.2" ], "rule": "Gamma moments" } ], "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/evidence/14", "value": { "id": "E15", "profile": "formal", "description": "Derivation of the sign of the term iETS adds for each period with zero demand (source draft, artefact A1, the remark after the corollary on the shrunk scale).", "foundation": "Lebesgue integration", "steps": [ { "id": "E15.1", "statement": "$$H_z = -\\E\\log f_z = -\\int f_z(z)\\log f_z(z)\\,dz$$", "by": [ "D32" ], "rule": "definition of the differential entropy" }, { "id": "E15.2", "statement": "$$\\int f_z(z)\\log f_z(z)\\,dz = -H_z$$", "by": [ "E15.1" ], "rule": "multiplying by $-1$" }, { "id": "E15.3", "statement": "The term added for each period with zero demand is the expected log-density of the unobserved size under its own distribution, $\\int f_z\\log f_z\\,dz$", "by": [ "D4" ], "rule": "definition of the objective" }, { "id": "E15.4", "statement": "$$\\int f_z\\log f_z\\,dz = -H_z,$$ so the entropy enters the objective with a negative sign", "by": [ "E15.2", "E15.3" ], "rule": "substitution" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/models/1", "value": { "id": "M2", "provider": "Anthropic", "name": "Claude Sonnet", "version": "5.5", "identifier": "claude-sonnet-5-5", "weights": "api_only", "accessed": "2026-11-11" }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/references/4", "value": { "id": "B5", "target": "10.1201/9781003452652", "id_type": "doi" }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/references/5", "value": { "id": "B6", "target": "10.1017/cbo9781107049994", "id_type": "doi" }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "add", "path": "/references/6", "value": { "id": "B7", "target": "10.1093/acprof:oso/9780199641178.001.0001", "id_type": "doi" }, "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." }, { "op": "remove", "path": "/warrants/1", "reason": "The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\\mathbf w'\\mathbf F^{i-1}\\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed." } ] }, "contributors": [ { "id": "P1", "type": "agent", "name": "Claude Code", "positions": [ "author" ], "models": [ { "model": "M1", "role": "sole" } ], "credit_roles": [ "writing_original_draft" ], "ai_tasks": [ { "task": "json_preparation", "extent": "full" } ], "tools": [ "code_execution", "file_access", "retrieval" ], "record_source": "harness", "system": { "name": "Claude Code" }, "run_period": { "start": "2026-10-15", "end": "2026-10-15" } }, { "id": "P2", "type": "human", "name": "Ivan Svetunkov", "orcid": "0000-0001-7826-0281", "positions": [ "supervisor", "guarantor" ], "credit_roles": [ "conceptualization", "methodology", "supervision" ] }, { "type": "agent", "positions": [ "author" ], "models": [ { "model": "M1", "role": "sole" } ], "tools": [ "code_execution", "file_access" ], "record_source": "user_reported", "id": "P3", "name": "Claude Code (drafting agent)", "credit_roles": [ "formal_analysis", "methodology", "software", "writing_original_draft" ], "ai_tasks": [ { "task": "analysis", "extent": "substantial" }, { "task": "code", "extent": "substantial" }, { "task": "writing", "extent": "substantial" } ], "system": { "name": "Claude Code" } }, { "type": "agent", "positions": [ "contributor" ], "models": [ { "model": "M1", "role": "sole" } ], "tools": [ "code_execution", "file_access" ], "record_source": "user_reported", "id": "P4", "name": "Claude Code (numerical evidence agent)", "credit_roles": [ "investigation", "software", "validation" ], "ai_tasks": [ { "task": "analysis", "extent": "full" }, { "task": "code", "extent": "full" } ], "system": { "name": "Claude Code" } }, { "id": "P5", "type": "agent", "name": "Claude Code (independent proof checker)", "positions": [ "contributor" ], "models": [ { "model": "M1", "role": "subagent" } ], "credit_roles": [ "validation" ], "ai_tasks": [ { "task": "review", "extent": "full" } ], "tools": [ "code_execution", "file_access" ], "record_source": "self_reported", "run_period": { "start": "2026-10-15", "end": "2026-10-15" }, "system": { "name": "Claude Code" } }, { "id": "P6", "type": "agent", "name": "Claude Code (independent proof checker, second model)", "positions": [ "contributor" ], "models": [ { "model": "M2", "role": "sole" } ], "credit_roles": [ "validation" ], "ai_tasks": [ { "task": "review", "extent": "full" } ], "tools": [ "code_execution", "file_access" ], "record_source": "self_reported", "run_period": { "start": "2026-11-11", "end": "2026-11-11" }, "system": { "name": "Claude Code" } }, { "id": "P7", "type": "agent", "name": "Claude Code (correction agent)", "positions": [ "author" ], "models": [ { "model": "M1", "role": "sole" } ], "credit_roles": [ "formal_analysis", "writing_review_editing" ], "ai_tasks": [ { "task": "analysis", "extent": "substantial" }, { "task": "writing", "extent": "full" }, { "task": "json_preparation", "extent": "full" } ], "tools": [ "code_execution", "file_access" ], "record_source": "self_reported", "run_period": { "start": "2026-11-11", "end": "2026-11-11" }, "system": { "name": "Claude Code" } } ], "models": [ { "id": "M1", "provider": "Anthropic", "name": "Claude Opus", "version": "5.5", "identifier": "claude-opus-5-5", "weights": "api_only", "accessed": "2026-10-15" }, { "id": "M2", "provider": "Anthropic", "name": "Claude Sonnet", "version": "5.5", "identifier": "claude-sonnet-5-5", "weights": "api_only", "accessed": "2026-11-11" } ], "definitions": [ { "id": "D1", "term": "single source of error form", "definition": "State space form of a time series model, also called the innovations form, in which the observation equation and the state transition share one error term $\\epsilon_t$ per period: $$\\begin{aligned} z_t &= w(\\mathbf v_{t-\\mathbf l}) + r(\\mathbf v_{t-\\mathbf l})\\epsilon_t \\\\ \\mathbf v_t &= f(\\mathbf v_{t-\\mathbf l}) + g(\\mathbf v_{t-\\mathbf l})\\epsilon_t. \\end{aligned}$$", "kind": "method" }, { "id": "D2", "term": "unobserved period", "definition": "A period $t$ at which the potential value $z_t$ of the modelled variable is not observed ($O_t=0$): a period with zero demand in an intermittent demand model, where the demand size is not observed, or a period with a missing value.", "kind": "concept" }, { "id": "D3", "term": "observed-data likelihood", "definition": "The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values $\\mathbf z_{\\mathrm{mis}}$ at the unobserved periods: $$L(\\theta)=\\int f(\\mathbf y_{\\mathrm{obs}},\\mathbf z_{\\mathrm{mis}};\\theta)\\,d\\mathbf z_{\\mathrm{mis}}.$$", "kind": "concept" }, { "id": "D4", "term": "entropy-augmented likelihood", "definition": "The objective of the iETS estimator of Svetunkov and Boylan (2023, eq. B.6): the sum of the one-step log-densities of the observed demand sizes, plus, for each period with zero demand, the expected log-density of the unobserved size under its own distribution, $\\int f_z\\log f_z\\,dz$, plus the Bernoulli log-likelihood of the occurrence.", "kind": "concept", "source": "B4" }, { "id": "D5", "term": "zero-error recursion", "definition": "The estimator used in ADAM and in iETS (Svetunkov and Boylan 2023, Appendix C): the single source of error recursion run with $\\epsilon_t=0$ at the unobserved periods and with the fixed persistence vector $\\mathbf g$ at the observed ones, with the likelihood taken as the product of one-step densities at the observed periods.", "kind": "method", "source": "B4" }, { "id": "D6", "term": "ADAM multiplicative ETS", "definition": "Pure multiplicative exponential smoothing in the form of the Augmented Dynamic Adaptive Model, in which each component $k$ is updated by the factor $(1+\\epsilon_t)^{g_k}$; for ETS(M,Md,M) with seasonal period $m$, $$\\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\\phi}s_{t-m}(1+\\epsilon_t) \\\\ l_t &= l_{t-1}b_{t-1}^{\\phi}(1+\\epsilon_t)^{\\alpha} \\\\ b_t &= b_{t-1}^{\\phi}(1+\\epsilon_t)^{\\beta} \\\\ s_t &= s_{t-m}(1+\\epsilon_t)^{\\gamma}. \\end{aligned}$$", "kind": "method" }, { "id": "D7", "term": "conventional multiplicative ETS", "definition": "Pure multiplicative exponential smoothing in the form of Hyndman et al. (2008), in which each component $k$ is updated by the factor $1+g_k\\epsilon_t$; for ETS(M,Md,M) with seasonal period $m$, $$\\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\\phi}s_{t-m}(1+\\epsilon_t) \\\\ l_t &= l_{t-1}b_{t-1}^{\\phi}(1+\\alpha\\epsilon_t) \\\\ b_t &= b_{t-1}^{\\phi}(1+\\beta\\epsilon_t) \\\\ s_t &= s_{t-m}(1+\\gamma\\epsilon_t). \\end{aligned}$$", "kind": "method" }, { "id": "D8", "term": "SSOE Kalman filter", "definition": "The recursion for the conditional mean $\\mathbf a_t$ and scaled covariance $\\mathbf P_t$ of the state of a linear single source of error model: at an observed period $$\\begin{aligned} v_t &= z_t-\\mathbf w'\\mathbf a_t \\\\ f_t &= \\mathbf w'\\mathbf P_t\\mathbf w+1 \\\\ \\mathbf k_t &= (\\mathbf F\\mathbf P_t\\mathbf w+\\mathbf g)/f_t \\\\ \\mathbf a_{t+1} &= \\mathbf F\\mathbf a_t+\\mathbf k_t v_t \\\\ \\mathbf P_{t+1} &= \\mathbf F\\mathbf P_t\\mathbf F'+\\mathbf g\\mathbf g'-\\mathbf k_t\\mathbf k_t'f_t; \\end{aligned}$$ at an unobserved period $$\\begin{aligned} \\mathbf a_{t+1} &= \\mathbf F\\mathbf a_t \\\\ \\mathbf P_{t+1} &= \\mathbf F\\mathbf P_t\\mathbf F'+\\mathbf g\\mathbf g'; \\end{aligned}$$ started from $\\mathbf a_1=\\mathbf x_0$, $\\mathbf P_1=\\mathbf 0$.", "kind": "method" }, { "id": "D9", "term": "drift of the states", "definition": "The change $d_t=\\log(\\mu_t/\\hat\\mu_t)$ of the logarithm of the location of a pure multiplicative model accumulated over the unobserved periods since the last observed period, relative to the point forecast $\\hat\\mu_t$ with zero errors.", "kind": "concept" }, { "id": "D10", "term": "normal-drift mixture density", "definition": "Approximate predictive density of a demand size after $j_t-1$ unobserved periods: the one-step density $f_z(z_t\\mid\\hat\\mu_t e^{d})$ mixed over a normal [[D9:drift of the states]] $d\\sim\\mathcal N(m_t,w_t)$, with $m_t$, $w_t$ computed by 32-node Gauss-Legendre quadrature on the quantile function of $1+\\epsilon$ and the mixture by Gauss-Hermite quadrature.", "kind": "method" }, { "id": "D11", "term": "one-step density", "kind": "method", "definition": "The density of a demand size after an interval taken as the one-step density $f_z(z_t\\mid\\hat\\mu_t)$ of the model, without the [[D9:drift of the states]] over the unobserved periods." }, { "id": "D12", "term": "iETS(M,N,N)", "kind": "data", "definition": "Intermittent demand sizes $$z_t=l_{t-1}(1+\\epsilon_t)$$ with $\\E(1+\\epsilon_t)=1$ and $\\Var(1+\\epsilon_t)=s$, Gamma, log-normal or inverse Gaussian $1+\\epsilon_t$ (the greybox parameterisation of the package) and the level moving at every period, observed or not: $$l_t=l_{t-1}(1+\\epsilon_t)^\\alpha$$ in the [[D6:ADAM multiplicative ETS]] and $$l_t=l_{t-1}(1+\\alpha\\epsilon_t)$$ in the [[D7:conventional multiplicative ETS]]." }, { "id": "D13", "term": "exact density by convolution", "kind": "design", "definition": "The exact density of the size after an interval, computed on a grid of step 0.001 in $\\log z$ by FFT convolution with tails cut at 1e-12, compared with an approximation by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20 (kl_mixture.py); a refinement of 164 cells halved the step and cut the tails at 1e-14." }, { "id": "D14", "term": "Kullback-Leibler divergence", "kind": "variable", "definition": "The divergence $\\int g\\log(g/h)$ of an approximate density $h$ from the exact density $g$, per observation; over a grid, its largest value." }, { "id": "D15", "term": "fixed-point moment estimator", "kind": "method", "definition": "The estimator of the scale $s$ of Gamma or inverse Gaussian sizes that solves the moment equation $s=h(s)$, built from the moments of the drift factor $D_t$, by the iteration $s\\leftarrow h(s)$ from the one-step estimate, stopping at $|s_{new}-s|\\le 10^{-10}s$ or after 100 iterations; a fit fails when an iterate is not positive and finite, the iteration does not converge, or the value leaves the admissible range." }, { "id": "D16", "term": "Monte Carlo study of the scale", "kind": "design", "definition": "480 cells of relative errors simulated with the drift ($s\\in\\{0.05,0.2,0.5,1\\}$, $\\alpha\\in\\{0.1,0.3,0.6\\}$, intervals fixed at 2, 5 or 10 or geometric with occurrence probability 0.5 or 0.2, 100 or 1000 observed sizes) and 120 cells within fitted models ($s\\in\\{0.05,0.5\\}$, 100 observed sizes), 2000 replications per cell from numpy SeedSequence(20261015) with one child stream per cell; cells grouped by the drift order $\\E(j-1)\\alpha^2s$ (scale_recovery.py)." }, { "id": "D17", "term": "relative bias", "kind": "variable", "definition": "The relative bias of a scale estimate, (mean estimate $- s$)$/s$ over the replications of a cell; its median over cells." }, { "id": "D18", "term": "failed fits", "kind": "variable", "definition": "The share of replications in which the scale estimator fails." }, { "id": "D19", "term": "approximate prediction interval", "kind": "method", "definition": "The prediction interval of pure multiplicative ETS in the smooth package (generate_prediction_interval with var_anal): the point forecast with zero errors times the quantile of the distribution with mean one and variance $$V_h=(1+v)(1+\\alpha^2v)^{h-1}-1,$$ $v=\\Var(1+\\epsilon)$; for the [[D6:ADAM multiplicative ETS]] with the log-normal distribution, exact in logarithms." }, { "id": "D20", "term": "ETS(M,N,N) with s = 0.32 and alpha = 0.21", "kind": "data", "definition": "ETS(M,N,N) with $s=0.32$, $\\alpha=0.21$, 100 observations and initial level 10, with Gamma, inverse Gaussian or log-normal errors, in the ADAM and conventional forms; in the second case the last six of the 100 periods unobserved." }, { "id": "D21", "term": "simulated paths", "kind": "design", "definition": "The 0.95 quantile of 1,000,000 simulated paths divided by the point forecast with zero errors, twelve steps ahead and 18 steps from the last observed period after six unobserved periods at the end of the sample (interval_bounds.py, seed 20261015)." }, { "id": "D22", "term": "upper bound over the point forecast", "kind": "variable", "definition": "The upper bound of the 90% prediction interval (its 0.95 quantile) divided by the point forecast with zero errors." }, { "id": "D23", "term": "low-rank factor form", "kind": "method", "definition": "The [[D8:SSOE Kalman filter]] with the scaled state covariance kept as a low-rank factor $\\mathbf P_t=\\mathbf U_t\\mathbf U_t'$, updated by rank one at each observed period, as in the C++ core of smooth (src/headers/adamCore.h).", "broader": [ "D8" ] }, { "id": "D24", "term": "full covariance form", "kind": "method", "definition": "The [[D8:SSOE Kalman filter]] with the whole scaled state covariance matrix $\\mathbf P_t$.", "broader": [ "D8" ] }, { "id": "D25", "term": "hourly series with missing values", "kind": "data", "definition": "1344 hourly observations, generated by ADAM ETS(A,N,A) with lags 24 and 168 or following a fixed sine pattern, with 1 to 48 nested missing values, fitted by ADAM ETS(A,N,A) with lags 24 and 168 (193 state cells), the normal distribution and backcast initial states." }, { "id": "D26", "term": "interleaved timing", "kind": "design", "definition": "25 interleaved repetitions per case of the fits with the two forms of the filter, both built from smooth commit 53104ee9, on one cloud VM (Intel Xeon 2.80 GHz, 4 vCPUs, Linux 6.18.44, Python 3.11.15) with one thread (lowrank_timing.py)." }, { "id": "D27", "term": "speed-up", "kind": "variable", "definition": "The median run time of the [[D24:full covariance form]] divided by that of the [[D23:low-rank factor form]]." }, { "id": "D28", "term": "family-scaled error", "kind": "method", "definition": "The approximation of the predictive density of a pure additive model with Laplace, S or Generalised Normal errors that keeps the error in its family with scale $sf_t^{1/p}$, $p$ such that the variance is proportional to $s^p$, as in the smooth package." }, { "id": "D29", "term": "Gauss-Hermite mixture", "kind": "method", "definition": "The mixture of the error density over the normal uncertainty of the states by a Gauss-Hermite rule with $Q$ nodes (16 in the smooth package)." }, { "id": "D30", "term": "additive errors with state uncertainty", "kind": "data", "definition": "Laplace, S and Generalised Normal (shape 0.5 to 3) errors with scale $s=1$, convolved with normal state uncertainty of variance $(f_t-1)V(s)$, for 22 values of $f_t$ from 1.01 to 100 (kl_laplace.py)." }, { "id": "D31", "term": "divergence by numerical integration", "kind": "design", "definition": "The Kullback-Leibler divergence in both directions between the exact convolution and each approximation, by composite 8-node Gauss-Legendre integration (kl_laplace.py)." }, { "id": "D32", "term": "differential entropy", "kind": "concept", "definition": "For a size with density $f_z$, $$H_z = -\\E\\log f_z = -\\int f_z(z)\\log f_z(z)\\,dz.$$" } ], "claims": [ { "id": "C1", "statement": "For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\\sim\\text{Bernoulli}(p_t)$ independent of a positive continuous size $z_t$, the [[D3:observed-data likelihood]] is $$\\sum_{O_t=1}\\log f(z_t\\mid Y_{t-1};\\theta) + \\sum_{O_t=1}\\log p_t + \\sum_{O_t=0}\\log(1-p_t),$$ in which the unobserved sizes do not appear and the size and occurrence parts separate.", "about": { "subject": "D3", "relation": "separates into a size part and an occurrence part for", "object": "models of intermittent demand", "comparator": null, "conditions": [ "Bernoulli occurrence independent of a continuous size" ] }, "type": "formal", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "models of intermittent demand $y_t=o_tz_t$ (iETS, ADAM with an occurrence part)", "data_type": "intermittent demand time series", "conditions": [ "Bernoulli occurrence independent of a continuous demand size" ] }, "support": [ { "kind": "formal_proof", "evidence": "E1" } ], "assumptions": [ "C30", "C31", "C32" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B1", "self": false, "note": "Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood." }, { "type": "extends", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived the prediction-error factorisation: P(y=0|past)=1-p by independence and continuity, and y>0 has density p*f, so the log-likelihood separates into size and occurrence parts. Checked numerically on ETS(A,N,N) with two unobserved periods: the integral of the complete-data density over the unobserved sizes equals the Kalman predictive-density sum to 2e-13. (first round; report A10, scripts A14-A16.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C2", "statement": "The stationary points of the [[D4:entropy-augmented likelihood]] of iETS differ in general from those of the [[D3:observed-data likelihood]]: at a fixed point of the EM algorithm the gradient of the expected complete-data log-likelihood equals the score of the observed-data likelihood, while the term $-H_z(\\theta)$ of each period with zero demand has a gradient that is not zero in general, and for normal sizes with fixed location the two objectives are stationary at different variances.", "about": { "subject": "D4", "relation": "has stationary points that differ in general from those of", "object": "D3", "comparator": null, "conditions": [] }, "type": "formal", "role": "principal", "novelty": "refutation", "scope": { "population": "iETS estimated by the entropy-augmented likelihood", "data_type": "intermittent demand time series", "conditions": [ "sizes unobserved at the periods with zero demand" ] }, "support": [ { "kind": "formal_proof", "evidence": "E2" }, { "kind": "cited", "reference": "B1", "self": false } ], "assumptions": [ "C31", "C32" ], "depends_on": [ "C1" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "contradicts", "reference": "B4", "locator": "Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm)", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived Fisher's identity (gradient of Q at the current estimate equals the observed-data score, valid at every iterate and so at EM fixed points) and the normal-variance example: the observed-data likelihood is stationary at S/T1 and the entropy-augmented one at S/T, different when T0>0 and S>0. Numerically with T1=60, T0=40 the maximisers are 3.318 and 1.991, and the cross-entropy gradient at the current estimate is 0 while that of -H is -0.385. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C3", "statement": "For normally distributed demand sizes with fixed location, the maximiser in the variance of the [[D4:entropy-augmented likelihood]] is $\\frac{T_1}{T}\\hat\\sigma^2$, where $\\hat\\sigma^2=\\frac{1}{T_1}\\sum_{O_t=1}e_t^2$ is the fixed point of the EM algorithm, $T$ the number of periods and $T_1$ the number of periods with observed sizes.", "about": { "subject": "D4", "relation": "is maximised at T1/T times the EM fixed point in", "object": "the variance of the sizes", "comparator": null, "conditions": [ "normal sizes", "location held fixed" ] }, "type": "formal", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "iETS with normally distributed demand sizes", "conditions": [ "location of the sizes held fixed", "variance as the only free parameter" ] }, "support": [ { "kind": "formal_proof", "evidence": "E3" } ], "depends_on": [ "C2" ], "verification": "agent_checked", "checks": [ { "by": "P5", "outcome": "confirmed", "extent": "full", "notes": "The independent proof checker re-derived the proof of E3 step by step and confirmed it." } ], "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C4", "statement": "For a pure additive single source of error model with normal errors and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] gives $z_t\\mid Y_{t-1}\\sim\\mathcal N(\\mathbf w'\\mathbf a_t,\\sigma^2 f_t)$ at the observed periods, the exact observed-data log-likelihood $$-\\frac12\\sum_{O_t=1}\\left[\\log(2\\pi\\sigma^2f_t)+\\frac{v_t^2}{\\sigma^2f_t}\\right],$$ and the concentrated variance $$\\hat\\sigma^2=\\frac{1}{T_1}\\sum_{O_t=1}\\frac{v_t^2}{f_t}.$$", "about": { "subject": "D8", "relation": "gives the exact observed-data likelihood of", "object": "pure additive SSOE models with normal errors", "comparator": null, "conditions": [ "D2" ] }, "type": "formal", "role": "principal", "novelty": "extension", "scope": { "population": "pure additive single source of error models (ETS with additive components, ARIMA, regression)", "data_type": "univariate time series with unobserved periods", "conditions": [ "normally distributed errors", "initial states treated as fixed parameters" ], "excludes": [ "models mixing additive and multiplicative components" ] }, "support": [ { "kind": "formal_proof", "evidence": "E4" } ], "assumptions": [ "C30", "C33", "C34", "C35" ], "depends_on": [ "C1" ], "verification": "agent_checked", "checks": [ { "by": "P5", "outcome": "confirmed", "extent": "full", "notes": "The independent proof checker re-derived the proof of E4 step by step and confirmed it." } ], "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B6", "locator": "Kalman filter for state space models with correlated disturbances", "self": false, "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated." }, { "type": "extends", "reference": "B7", "locator": "Kalman filter for state space models with correlated disturbances", "self": false, "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations." } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C5", "statement": "For pure additive single source of error models with normal errors and [[D2:unobserved period]]s, if $c_i=\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval, the [[D5:zero-error recursion]] likelihood is not the observed-data likelihood and its maximiser in the error variance is in general not the maximum likelihood estimate, because at the observed period after such an interval the recursion uses the error variance $\\sigma^2$ instead of $\\sigma^2f_t>\\sigma^2$.", "about": { "subject": "D5", "relation": "is not the maximum likelihood estimator of", "object": "pure additive SSOE models with normal errors", "comparator": null, "conditions": [ "D2", "nonzero persistence vector" ] }, "type": "formal", "role": "principal", "novelty": "refutation", "scope": { "population": "pure additive single source of error models", "data_type": "univariate time series with unobserved periods", "conditions": [ "normally distributed errors", "$\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval" ] }, "support": [ { "kind": "formal_proof", "evidence": "E6" } ], "assumptions": [ "C30", "C33", "C34", "C35" ], "depends_on": [ "C4" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "contradicts", "reference": "B4", "locator": "Appendix C (estimation with zero errors at the unobserved periods)", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Re-derived f=1+sum c_i^2>1 for some c_i!=0 and that the zero-error likelihood uses sigma^2 where the exact one uses sigma^2 f_t, so the likelihoods differ; the variance maximisers (1/T1)sum e^2 and (1/T1)sum v^2/f differ because their difference is positive when the first observed period after a gap is the last observation. In an ETS(A,N,N) check the two sums were 7.4975 and 6.7015, and the difference was positive in all 2000 random samples. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C6", "statement": "For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] applied to $\\log z_t$, with the known error mean $-s/2$ added to the prediction and, multiplied by $\\mathbf g$, to the propagation, gives the exact observed-data likelihood, and the maximum likelihood scale $s$ is the positive root of $$\\Big(\\sum \\frac{b_t^2}{f_t}\\Big)s^2+T_1s-\\sum\\frac{(v_t^0)^2}{f_t}=0,$$ where $v_t(s)=v_t^0+sb_t$ is the filter's prediction error.", "about": { "subject": "D8", "relation": "gives in logarithms the exact observed-data likelihood of", "object": "D6", "comparator": null, "conditions": [ "log-normal distribution", "D2" ] }, "type": "formal", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "pure multiplicative ETS in the ADAM form", "data_type": "positive univariate time series with unobserved periods", "conditions": [ "log-normal distribution with $\\E(1+\\epsilon_t)=1$" ] }, "support": [ { "kind": "formal_proof", "evidence": "E9" } ], "assumptions": [ "C30", "C33", "C34", "C36" ], "depends_on": [ "C4", "C12" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B5", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Derived the filter on log z with mean shift -g s/2 and the Jacobian, and the score in s; the cross terms cancel to give (sum b^2/f)s^2+T1 s-sum v0^2/f=0. Numerically the filter log-likelihood equals the brute-force integral over two unobserved log-sizes (-18.275599457737) and the positive roots (0.29147, 0.25190) equal the grid maxima of the filter likelihood. (first round; report A10, scripts A14-A16.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C7", "statement": "For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and all periods observed, the maximum likelihood estimate of the scale is $\\hat s=2(\\sqrt{1+m}-1)$ with $m=\\frac{1}{T_1}\\sum\\log^2(1+\\epsilon_t)$.", "about": { "subject": "D6", "relation": "has the maximum likelihood scale 2(sqrt(1+m)-1)", "object": null, "comparator": null, "conditions": [ "log-normal distribution", "every period observed" ] }, "type": "formal", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "pure multiplicative ETS in the ADAM form", "data_type": "positive univariate time series", "conditions": [ "log-normal distribution with $\\E(1+\\epsilon_t)=1$", "every period observed" ] }, "support": [ { "kind": "formal_proof", "evidence": "E10" } ], "assumptions": [ "C30", "C36" ], "depends_on": [ "C6" ], "verification": "agent_checked", "checks": [ { "by": "P5", "outcome": "confirmed", "extent": "full", "notes": "The independent proof checker re-derived the proof of E10 step by step and confirmed it." } ], "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B5", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C8", "statement": "The log-normal scale estimator $2(1-\\sqrt{1-m})$ used in ADAM, with $m=\\frac{1}{T_1}\\sum\\log^2(1+\\epsilon_t)$, is not the maximum likelihood estimator: it overestimates the scale $s$ by about 35% at $s=0.5$ (0.677 against 0.5 when $m=s+s^2/4$) and is undefined for $m>1$.", "about": { "subject": "the log-normal scale estimator 2(1-sqrt(1-m)) used in ADAM", "relation": "is not the maximum likelihood estimator of", "object": "the scale s", "comparator": null, "conditions": [] }, "type": "formal", "role": "principal", "novelty": "refutation", "scope": { "population": "pure multiplicative ETS in the ADAM form", "data_type": "positive univariate time series", "conditions": [ "log-normal distribution with $\\E(1+\\epsilon_t)=1$", "every period observed" ] }, "support": [ { "kind": "formal_proof", "evidence": "E10" } ], "assumptions": [ "C36" ], "depends_on": [ "C15", "C7" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "contradicts", "reference": "B5", "locator": "estimator of the scale of the log-normal distribution", "self": true, "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "With f_t=1 and b_t=1/2 the C6 equation is s^2+4s-4m=0, root 2(sqrt(1+m)-1); the ADAM-type estimator is the smaller root of s^2-4s+4m=0, real only for m<=1 and larger than the MLE for every m in (0,1). At m=0.5625 the values are 0.677 and 0.5 (ratio 1.354); a numerical likelihood maximisation on simulated log-normal errors agreed with the formula. (first round; report A10, scripts A14-A16.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C9", "statement": "For [[D12:iETS(M,N,N)]] in the [[D7:conventional multiplicative ETS]] form, the [[D10:normal-drift mixture density]] of the demand size after an interval had a largest [[D14:Kullback-Leibler divergence]] per observation from the exact density at least three orders of magnitude smaller than that of [[D11:the one-step density]], for the Gamma, the log-normal and the inverse Gaussian distribution.", "about": { "subject": "D10", "relation": "has a smaller Kullback-Leibler divergence from the exact density than", "object": [ "D14" ], "comparator": "D11", "conditions": [ "D12", "D7" ] }, "type": "descriptive", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes", "conditions": [ "scale $s\\in[0.05,1]$", "smoothing parameter $\\alpha\\in[0.1,0.6]$", "interval length $j\\in[2,20]$", "states known at the previous observed period" ], "excludes": [ "models with trend or seasonality", "ADAM multiplicative ETS" ] }, "support": [ { "kind": "simulation", "evidence": "E20", "artefacts": [ "A3", "A4", "A5" ], "results": [ "E20.T1.r1", "E20.T1.r2", "E20.T1.r3" ] } ], "assumptions": [ "C37" ], "assessments": [ { "by": "author", "strength": "moderate", "rationale": "Deterministic archived computation with a refinement check; one model, one grid, and values that differ from those first reported in the source draft." } ], "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B4", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C10", "statement": "For a model with an occurrence part, occurrence probability $p_t$ and positive sizes, a prediction interval $[0,u]$ of the demand with coverage $c>1-p_t$ that counts the zeros as covered has as upper bound $u$ the quantile at level $(c-(1-p_t))/p_t$ of the distribution of the demand size, whose mean, the conditional mean of the size, is the size forecast $\\hat\\mu_t$, not the demand forecast $p_t\\hat\\mu_t$.", "about": { "subject": "a prediction interval of the demand that counts zeros as covered", "relation": "is an interval of the demand sizes centred on", "object": "the size forecast", "comparator": null, "conditions": [ "models with an occurrence part" ] }, "type": "formal", "role": "principal", "novelty": "no_prior_found", "scope": { "population": "models with an occurrence part (iETS, ADAM)", "conditions": [ "prediction interval of the demand of the form $[0,u]$ that counts zeros as covered", "coverage $c>1-p_t$", "Bernoulli occurrence independent of a positive continuous demand size" ] }, "support": [ { "kind": "formal_proof", "evidence": "E12" } ], "assumptions": [ "C31", "C32" ], "verification": "agent_checked", "provenance": "author_stated", "relations": [ { "type": "extends", "reference": "B4", "self": true, "note": "Svetunkov and Boylan forecast intermittent demand as the product of the occurrence probability and the forecast demand size." } ], "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "P(0<=y<=u)=(1-p)+pF_z(u) for positive sizes, so coverage c>1-p needs F_z(u)=(c-(1-p))/p, a quantile of the size distribution whose mean is mu_hat while E(y)=p mu_hat. Monte Carlo with p=0.4, c=0.9 gave coverage 0.8997 and size level 0.7500. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)" } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C11", "statement": "The smooth package for R and Python implements the [[D3:observed-data likelihood]] without the entropy term, the [[D8:SSOE Kalman filter]] for the pure additive models with the normal distribution and in logarithms for the [[D6:ADAM multiplicative ETS]] with the log-normal distribution, the [[D10:normal-drift mixture density]] with moment-equation scales for the other pure multiplicative models, and the corrected log-normal scale, in one C++ routine shared by both languages.", "about": { "subject": "the smooth package for R and Python", "relation": "implements", "object": "D3", "comparator": null, "conditions": [] }, "type": "methodological", "role": "principal", "novelty": "no_prior_found", "scope": { "setting": "smooth package for R and Python, master branch at commit 8a5fbbb3" }, "support": [ { "kind": "demonstration", "artefacts": [ "A2" ] } ], "depends_on": [ "C1", "C4", "C6", "C7" ], "assessments": [ { "by": "author", "strength": "moderate", "rationale": "Implementation exists in the public repository; correctness relies on the package tests, not on independent verification." } ], "provenance": "author_stated", "relations": [ { "type": "supersedes", "reference": "B4", "locator": "Appendix B, eq. B.6", "self": true, "note": "Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand." } ], "verifiability": "open", "verifiability_flags": [] }, { "id": "C12", "statement": "In the [[D6:ADAM multiplicative ETS]], the logarithms of the observation and of the components follow a linear single source of error model with innovation $u_t=\\log(1+\\epsilon_t)$ and the measurement, transition and persistence of the corresponding additive model, whereas in the [[D7:conventional multiplicative ETS]] the innovation of component $k$ is $\\log(1+g_k\\epsilon_t)$, which is not proportional to $u_t$.", "about": { "subject": "D6", "relation": "is a linear SSOE model in logarithms, unlike", "object": null, "comparator": "D7", "conditions": [] }, "type": "formal", "role": "supporting", "novelty": "no_prior_found", "support": [ { "kind": "formal_proof", "evidence": "E8" } ], "verification": "agent_checked", "provenance": "author_stated", "scope": { "population": "pure multiplicative ETS in the ADAM and conventional forms" }, "checks": [ { "by": "P6", "outcome": "confirmed", "extent": "full", "notes": "Taking logs of D6 gives the additive ETS(A,Ad,A) recursion with innovation u_t=log(1+eps); in D7 the innovation is log(1+g eps), and log(1+g eps)/(g log(1+eps)) is not constant for 00$ has density $p_tf(y_t\\mid Y_{t-1})$", "by": [ "C31" ], "rule": "independence" }, { "id": "E1.3", "statement": "The joint density of the observed data is the product of these conditional terms", "by": [ "E1.1", "E1.2" ], "rule": "prediction-error decomposition" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C31", "C32" ] }, { "id": "E2", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov 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$$\\ell_H(\\sigma^2)=\\ell(\\sigma^2)-\\frac{T_0}{2}\\log(2\\pi e\\sigma^2),$$ stationary at $\\sigma^2=S/T$ with $T=T_1+T_0$", "by": [ "D4", "D32", "E2.3" ], "rule": "normal entropy and differentiation" }, { "id": "E2.5", "statement": "$S/T\\neq S/T_1$ whenever $T_0>0$ and $S>0$, so the stationary points differ", "by": [ "E2.3", "E2.4" ], "rule": "comparison" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C31", "C32" ] }, { "id": "E3", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E3.1", "statement": "$H_z=\\frac12\\log(2\\pi e\\sigma^2)$ for the normal distribution", "by": [ "D4" ], "rule": "definition" }, { "id": "E3.2", "statement": "$$\\frac{\\partial\\ell_H}{\\partial\\sigma^2}=-\\frac{T_1}{2\\sigma^2}+\\frac{\\sum e_t^2}{2\\sigma^4}-\\frac{T_0}{2\\sigma^2},$$ zero at $\\sigma^2=\\sum e_t^2/T$", "by": [ "E3.1" ], "rule": "differentiation" }, { "id": "E3.3", "statement": "EM iterates $\\sigma^2_{(k+1)}=(\\sum e_t^2+T_0\\sigma^2_{(k)})/T$ with fixed point $\\frac{1}{T_1}\\sum e_t^2$", "by": [ "B1" ], "rule": "fixed point" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E4", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E4.1", "statement": "$(\\mathbf x_{t-1},z_t,\\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$", "by": [ "C35", "C30" ], "rule": "linearity of Gaussian models" }, { "id": "E4.2", "statement": "$$\\begin{aligned} \\E(z_t\\mid Y_{t-1}) &= \\mathbf w'\\mathbf a_t \\\\ \\Var(z_t\\mid Y_{t-1}) &= \\sigma^2f_t \\\\ \\Cov(\\mathbf x_t,z_t\\mid Y_{t-1}) &= \\sigma^2(\\mathbf F\\mathbf P_t\\mathbf w+\\mathbf g) \\end{aligned}$$", "by": [ "E4.1" ], "rule": "moments of linear combinations" }, { "id": "E4.3", "statement": "Conditioning on $z_t$ gives the update; unobserved periods only propagate", "by": [ "E4.2", "C33" ], "rule": "Gaussian conditioning" }, { "id": "E4.4", "statement": "The likelihood follows from the observed-data decomposition, and $\\sigma^2$ concentrates out because $f_t$, $\\mathbf k_t$ do not depend on it", "by": [ "E4.3", "C1" ], "rule": "prediction-error decomposition" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C30", "C33", "C34", "C35" ] }, { "id": "E5", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E5.1", "statement": "With $\\mathbf P_t=\\mathbf 0$, $f_t=1$ and $\\mathbf k_t=\\mathbf g$", "by": [ "D8" ], "rule": "substitution" }, { "id": "E5.2", "statement": "$\\mathbf P_{t+1}=\\mathbf g\\mathbf g'-\\mathbf g\\mathbf g'=\\mathbf 0$", "by": [ "E5.1" ], "rule": "substitution" }, { "id": "E5.3", "statement": "By induction from $\\mathbf P_1=\\mathbf 0$ the filter is the SSOE recursion", "by": [ "E5.2" ], "rule": "induction" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E6", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E6.1", "statement": "After $j-1$ unobserved periods from $\\mathbf P=\\mathbf 0$, $f=1+\\sum_{i1$ when $c_i\\neq0$ for some $i1$ and $v_t=e_t$ being present", "by": [ "E6.2", "E6.4", "C4" ], "rule": "differentiation in $\\sigma^2$" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E7", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E7.1", "statement": "$P=(j-1)\\alpha^2$, $f=(j-1)\\alpha^2+1$ after $j-1$ unobserved periods", "by": [ "C4", "D8" ], "rule": "substitution $w=F=1$, $g=\\alpha$" }, { "id": "E7.2", "statement": "$k=(P+\\alpha)/f$", "by": [ "E7.1", "D8" ], "rule": "substitution" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E8", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E8.1", "statement": "$\\log z_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\log s_{t-m}+u_t$", "by": [ "D6" ], "rule": "logarithm" }, { "id": "E8.2", "statement": "$\\log l_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\alpha u_t$, $\\log b_t=\\phi\\log b_{t-1}+\\beta u_t$ in the ADAM form", "by": [ "D6" ], "rule": "logarithm" }, { "id": "E8.3", "statement": "$\\log l_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\log(1+\\alpha\\epsilon_t)$ in the conventional form", "by": [ "D7" ], "rule": "logarithm" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E9", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E9.1", "statement": "$\\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$", "by": [ "C12", "C36" ], "rule": "linearity in logarithms" }, { "id": "E9.2", "statement": "The filter of the linear Gaussian model with the Jacobian $-\\sum\\log z_t$ gives the exact likelihood", "by": [ "C33", "C34", "C4", "E9.1" ], "rule": "change of variables" }, { "id": "E9.3", "statement": "Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$", "by": [ "E9.2" ], "rule": "linearity" }, { "id": "E9.4", "statement": "The score in $s$ times $2s^2$ gives $(\\sum b_t^2/f_t)s^2+T_1s-\\sum(v_t^0)^2/f_t=0$", "by": [ "E9.3" ], "rule": "differentiation" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C30", "C33", "C34", "C36" ] }, { "id": "E10", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E10.1", "statement": "$f_t=1$, $b_t=1/2$ give $s^2+4s-4m=0$", "by": [ "C6", "C15" ], "rule": "substitution" }, { "id": "E10.2", "statement": "The positive root is $2(\\sqrt{1+m}-1)$; $2(1-\\sqrt{1-m})$ solves $s^2-4s+4m=0$ instead", "by": [ "E10.1" ], "rule": "quadratic formula" }, { "id": "E10.3", "statement": "At $s=0.5$, $\\E m=s+s^2/4=0.5625$ and $2(1-\\sqrt{1-0.5625})=0.677$", "by": [ "E10.2", "C36" ], "rule": "arithmetic" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C36" ] }, { "id": "E11", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E11.1", "statement": "$e_t^2=(1+e_t)^2-2(1+e_t)+1$ and $e_t^2/(1+e_t)=(1+e_t)-2+(1+e_t)^{-1}$", "by": [], "rule": "algebra" }, { "id": "E11.2", "statement": "Independence of $\\epsilon_t$ and $D_t$ with $\\E(1+\\epsilon)^2=1+s$ and, for the inverse Gaussian, $\\E(1+\\epsilon)^{-1}=1+s$", "by": [ "E11.1" ], "rule": "independence and moments" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E12", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E12.1", "statement": "$$P(0\\le y_t\\le u)=(1-p_t)+p_tF_z(u)$$ for positive sizes and occurrence independent of the size", "by": [ "C31", "C32" ], "rule": "law of total probability" }, { "id": "E12.2", "statement": "Setting it to $c$ with $c>1-p_t$ gives $F_z(u)=(c-(1-p_t))/p_t\\in(0,1]$, so $u$ is the quantile of the size at that level", "by": [ "E12.1" ], "rule": "algebra" }, { "id": "E12.3", "statement": "The size has mean $\\hat\\mu_t$, the demand $\\E(o_tz_t)=p_t\\hat\\mu_t$", "by": [ "C31" ], "rule": "independence" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C31", "C32" ] }, { "id": "E13", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E13.1", "statement": "$x\\mapsto x^{\\alpha}$ is strictly concave for $0<\\alpha<1$, so for non-degenerate $1+\\epsilon>0$ with $\\E(1+\\epsilon)=1$, $\\E(1+\\epsilon)^{\\alpha}<(\\E(1+\\epsilon))^{\\alpha}=1$", "by": [ "D6" ], "rule": "Jensen's inequality" }, { "id": "E13.2", "statement": "In the ADAM form $l_t=l_{t-1}(1+\\epsilon_t)^{\\alpha}$ and $z_t=l_{t-1}(1+\\epsilon_t)$, so $$\\E(z_{T+h}\\mid Y_T)=l_T\\,[\\E(1+\\epsilon)^{\\alpha}]^{h-1}\\,\\E(1+\\epsilon)=\\hat\\mu_{T+h}[\\E(1+\\epsilon)^{\\alpha}]^{h-1},$$ with $\\hat\\mu_{T+h}=l_T$ the forecast with zero errors", "by": [ "D6", "C30" ], "rule": "independence of the errors" }, { "id": "E13.3", "statement": "For $h\\ge2$ the factor is below one, so the conditional mean is below the forecast with zero errors", "by": [ "E13.1", "E13.2" ], "rule": "substitution" }, { "id": "E13.4", "statement": "For $1+\\epsilon$ Gamma with shape $1/s$ and scale $s$ (mean one, variance $s$), $\\E(1+\\epsilon)^{\\alpha}=\\Gamma(1/s+\\alpha)s^{\\alpha}/\\Gamma(1/s)$", "by": [ "E13.2" ], "rule": "Gamma moments" } ], "artefacts": [ "A1" ], "availability": { "level": "public" }, "assumptions": [ "C30" ] }, { "id": "E14", "profile": "formal", "description": "Analytical derivation in the source draft (artefact A1).", "foundation": "Kolmogorov probability axioms", "steps": [ { "id": "E14.1", "statement": "An unobserved period adds the rank-one term $\\mathbf g\\mathbf g'$", "by": [ "D8" ], "rule": "propagation" }, { "id": "E14.2", "statement": "An observed period removes the direction it reveals while adding the new error, so the rank does not increase", "by": [ "D8" ], "rule": "update" }, { "id": "E14.3", "statement": "With $\\mathbf M=[\\mathbf U_t^F,\\mathbf g]$ and $\\mathbf M(\\mathbf a,1)'=f_t\\mathbf k_t$, a Householder reflection gives $$\\mathbf U_{t+1}=\\mathbf U_t^F-\\frac{(f_t\\mathbf k_t+\\sqrt{f_t}\\mathbf g)\\mathbf a'}{f_t+\\sqrt{f_t}}$$", "by": [ "E14.2" ], "rule": "Householder reflection" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E15", "profile": "formal", "description": "Derivation of the sign of the term iETS adds for each period with zero demand (source draft, artefact A1, the remark after the corollary on the shrunk scale).", "foundation": "Lebesgue integration", "steps": [ { "id": "E15.1", "statement": "$$H_z = -\\E\\log f_z = -\\int f_z(z)\\log f_z(z)\\,dz$$", "by": [ "D32" ], "rule": "definition of the differential entropy" }, { "id": "E15.2", "statement": "$$\\int f_z(z)\\log f_z(z)\\,dz = -H_z$$", "by": [ "E15.1" ], "rule": "multiplying by $-1$" }, { "id": "E15.3", "statement": "The term added for each period with zero demand is the expected log-density of the unobserved size under its own distribution, $\\int f_z\\log f_z\\,dz$", "by": [ "D4" ], "rule": "definition of the objective" }, { "id": "E15.4", "statement": "$$\\int f_z\\log f_z\\,dz = -H_z,$$ so the entropy enters the objective with a negative sign", "by": [ "E15.2", "E15.3" ], "rule": "substitution" } ], "artefacts": [ "A1" ], "availability": { "level": "public" } }, { "id": "E20", "profile": "simulation", "description": "Comparison, for iETS(M,N,N) under the ADAM and conventional multiplicative ETS forms, of the normal-drift mixture density (drift moments by 32-node Gauss-Legendre, mixture by 16-node Gauss-Hermite) and of the one-step density against the exact density of the size after an interval, computed on a grid of step 0.001 in log z by FFT convolution with tails cut at 1e-12, by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20; a 164-cell refinement run halved the step and cut tails at 1e-14. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.", "setup": { "studied": [ "D10", "D11" ], "cases": [ "D12", "D6", "D7" ], "procedure": [ "D13" ], "lens": [ "D14" ] }, "model": "iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes", "outputs": "largest KL per observation, conventional form: mixture 0.00115 (Gamma), 0.00448 (log-normal), 0.00400 (inverse Gaussian), one-step 2.359, 5.947, 47.545; ADAM form: mixture 0.01452, 0 (closed form), 0.00045, one-step 4.584, 18.635, 630.772", "artefacts": [ "A3", "A4", "A5" ], "data_origin": "primary", "preregistered": false, "availability": { "level": "public" }, "runs": 1, "seeds": [ 20261015 ], "execution": { "code_version": "f9d062ee850675acc2160d851a34093d035a474e", "notes": "Run with the claimset scripts not yet committed; the scripts are archived at f1de3ea1 and the package code is identical at both commits (user-reported)." }, "tables": [ { "id": "T1", "description": "Largest Kullback-Leibler divergence per observation from the exact density of the size after an interval, of the normal-drift mixture density and of the one-step density, for iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes in the conventional and the ADAM form", "conditions": { "model": "iETS(M,N,N)" }, "row_labels": [ "distribution", "ets_form" ], "columns": [ "largest KL divergence per observation, normal-drift mixture vs exact density", "largest KL divergence per observation, one-step vs exact density" ], "rows": [ { "id": "r1", "label": [ "dgamma", "conventional" ], "values": [ 0.00114767, 2.35934 ] }, { "id": "r2", "label": [ "dlnorm", "conventional" ], "values": [ 0.00447805, 5.94732 ] }, { "id": "r3", "label": [ "dinvgauss", "conventional" ], "values": [ 0.00399623, 47.5452 ] }, { "id": "r4", "label": [ "dgamma", "adam" ], "values": [ 0.0145197, 4.58358 ] }, { "id": "r5", "label": [ "dlnorm", "adam" ], "values": [ 9.63871e-07, 18.6354 ] }, { "id": "r6", "label": [ "dinvgauss", "adam" ], "values": [ 0.000446034, 630.772 ] } ], "artefact": "A4" } ] }, { "id": "E21", "profile": "simulation", "description": "Monte Carlo study of the scale of Gamma and inverse Gaussian iETS(M,N,N) sizes under the ADAM and conventional forms. Part A: 480 cells of simulated relative errors with drift ($s\\in\\{0.05,0.2,0.5,1\\}$, $\\alpha\\in\\{0.1,0.3,0.6\\}$, intervals fixed at 2, 5 or 10 or geometric with occurrence probability 0.5 or 0.2, 100 or 1000 observed sizes), 2000 replications per cell, fixed-point iteration with tolerance 1e-10 and at most 100 iterations; a fit fails when an iterate is not positive and finite, the iteration does not converge, or the value leaves the admissible range. Part B: 120 cells within fitted models with $s\\in\\{0.05,0.5\\}$ and 100 observed sizes. Cells are grouped by the drift order $\\E(j-1)\\alpha^2s$. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.", "setup": { "studied": [ "D15", "D5" ], "cases": [ "D12", "D6", "D7" ], "procedure": [ "D16" ], "lens": [ "D17", "D18" ] }, "model": "iETS(M,N,N) relative errors with drift", "artefacts": [ "A3", "A6" ], "data_origin": "primary", "preregistered": false, "availability": { "level": "public" }, "runs": 2000, "seeds": [ "numpy SeedSequence(20261015), one child stream per cell" ], "outputs": "fixed-point estimator, 1000 observed sizes, by drift order: <=0.02 median relative bias +0.02%, 0% failed; 0.02-0.1 -0.06%, 18%; 0.1-0.3 -0.42%, 52%; >0.3 -0.43%, 71%. 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