{"id": "entrylib:2026.000001", "version": 3, "status": "corrected", "date": "2026-11-10", "title": "Calibration of the prediction intervals of AutoMSARIMA in smooth on M1, M3, Tourism and M4", "url": "https://entrylib.org/p/2026.000001.json", "reading_view": "https://entrylib.org/p/2026.000001.txt", "principal_claims": [{"id": "C1", "statement": "With the true orders and true parameters, the [[D1:MSARIMA analytic interval]] is calibrated over the 99 quantiles in the [[D2:known-model simulation]] at $T$ of 100 and 300: [[D13:MCE]] lies between 0.0007 and 0.0024 and the ratio of the interval width to the theoretical width between 0.9965 and 1.0019, with backcasting and with optimal initial states."}, {"id": "C2", "statement": "On the [[D6:M4 sample]], the 80% [[D4:coverage]] of [[D10:AutoMSARIMA]] falls with the [[D17:level ratio]]: 0.813, 0.700 and 0.554 from the lowest to the highest of the [[D7:level terciles]], while ETS(MNN) of [[D15:ETS forced and selected]], whose error scale follows the level, covers 0.764, 0.778 and 0.814."}, {"id": "C3", "statement": "On the [[D6:M4 sample]], the [[D11:log-normal ARIMA]] with the orders [[D10:AutoMSARIMA]] selects removes the fall of 80% [[D4:coverage]] over the [[D7:level terciles]] (0.774, 0.783 and 0.752, a spread of 0.031) and lowers [[D13:MCE]] from 0.0386 to 0.0177."}, {"id": "C4", "statement": "On known models, the [[D10:AutoMSARIMA]] interval of smooth 1.3.0 is calibrated no worse than that of 1.0.8 ([[D21:smooth 1.0.8 and 1.3.0]]): at $T = 100$ in the [[D2:known-model simulation]] its [[D13:MCE]] is 0.0100 against 0.0120 with estimated parameters, 0.0139 against 0.0172 with selected orders and 0.0085 against 0.0246 with optimal initial states."}, {"id": "C5", "statement": "On the [[D9:fresh M4 sample]], [[D8:auto.adam error selection]] lowers [[D13:MCE]] from 0.0352 to 0.0224 and the spread of 80% [[D4:coverage]] over the [[D7:level terciles]] from 0.272 to 0.077 against R auto.msarima ([[D10:AutoMSARIMA]]), short of the planned reduction of MCE to 0.6 times that of auto.msarima, so the planned test is inconclusive."}, {"id": "C6", "statement": "The wider intervals of ETS than of [[D10:AutoMSARIMA]] on the [[D5:benchmark]] come from the multiplicative-error models in the ETS pool of [[D15:ETS forced and selected]], not from how either package computes the interval."}], "keywords": ["prediction intervals", "probabilistic forecast calibration", "ARIMA", "state space models", "forecasting software", "M competitions"], "subfield": "openalex:subfields/1803", "topics": ["openalex:T10282", "openalex:T10243", "openalex:T11918"]}
{"id": "entrylib:2026.000002", "version": 3, "status": "corrected", "date": "2026-11-10", "title": "Admissibility of lagged state space models through the ARIMA form of the whole model", "url": "https://entrylib.org/p/2026.000002.json", "reading_view": "https://entrylib.org/p/2026.000002.txt", "principal_claims": [{"id": "C3", "statement": "On [[D19:500 random additive ETS and GUM models]], the [[D29:admissibility decision]] from [[D10:the roots of the MA polynomial after cyclotomic cancellation]] differed from the [[D5:PBH reference]] on 0 models, and on the 172 models that both found inadmissible the [[D30:largest mode modulus]] differed from the reference by at most 4.4e-13."}, {"id": "C25", "statement": "On the same [[D22:1,000 random ETS+ARIMA models]], [[D10:the check of the whole model]] (the MA polynomial of the ARIMA form of the whole lagged SSOE after [[D6:cyclotomic cancellation]], with the AR polynomial of the ARIMA part stationary), gave the same [[D29:admissibility decision]] as the [[D5:PBH reference]] on every model, and the same [[D30:largest modulus]] to within rounding error on the models both found inadmissible."}, {"id": "C4", "statement": "On [[D20:575 random additive ETS and GUM models]], including ETS models with up to two seasonal lags from {24, 168, 365} whose generator allows MA polynomial degrees up to 536, the [[D7:step-down test]] on the MA polynomial after [[D6:cyclotomic cancellation]] gave an [[D29:admissibility decision]] different from [[D15:the roots of that polynomial]] on 0 models."}, {"id": "C16", "statement": "On [[D21:800 random additive ETS and GUM models in four families]], [[D12:the implementation of the check in smooth (smoothEigensCpp() through smoothEigensR())]] gave the same [[D29:decision]] as [[D10:the numpy prototype]] on every model, a [[D30:largest root modulus]] within the bisection tolerance of it on the models both found inadmissible, and the same modulus and MA polynomial on three consecutive calls with each model."}, {"id": "C26", "statement": "On [[D22:1,000 further random ETS+ARIMA models]], [[D14:the bounds checker of adam() in R and of ADAM in Python]] with bounds=\"admissible\", which run smoothEigensCpp() on the whole model, gave the same [[D29:admissibility decision]] as [[D10:the prototype check of the whole model]] on every model, and a penalty slope ([[D30:the largest root modulus]]) within the bisection tolerance of it in both families."}, {"id": "C1", "statement": "The [[D4:per-lag check]] accepted all [[D17:3042 ETS(A,N,A) grid models]], 1482 of which do not have [[D3:decaying weights]], and 193 of [[D18:200 random GUM models with lags (1, 12)]], 10 of which do not have decaying weights."}, {"id": "C24", "statement": "On [[D22:1,000 new random ADAM ETS+ARIMA models]] (500 with ARIMA at lag 1, 500 with seasonal ARIMA), [[D13:checking the ETS part and the ARIMA polynomials apart]], as adam() did with bounds=\"admissible\", gave an [[D29:admissibility decision]] different from the [[D5:PBH reference]] of the whole model on about a third of the models in each family, accepting models that are not forecastable and rejecting models that are."}, {"id": "C22", "statement": "For [[D35:ETS(A,N,A)]] with $m = 12$, $\\alpha = 1.2$ and $\\gamma = 0.9$, the [[D16:stability condition of Section 5.4 of the ADAM monograph]] holds, while in the full discount recursion the effect of the initial states on the fitted values and the weight of a past observation both grow from period to period, and the largest root modulus of the cancelled MA polynomial exceeds one."}], "keywords": ["exponential smoothing", "state space models", "admissible parameter space", "invertibility", "ARIMA reduced form", "Schur-Cohn stability test", "cyclotomic polynomials", "time series forecasting", "ARIMA", "forecastability"], "subfield": "openalex:subfields/1803", "topics": ["openalex:T10142", "openalex:T11236", "openalex:T11413"]}
{"id": "entrylib:2026.000003", "version": 2, "status": "corrected", "date": "2026-11-11", "title": "The observed-data likelihood of ADAM and iETS with unobserved periods", "url": "https://entrylib.org/p/2026.000003.json", "reading_view": "https://entrylib.org/p/2026.000003.txt", "principal_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."}, {"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."}, {"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."}, {"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 $$\\hat\\sigma^2=\\frac{1}{T_1}\\sum_{O_t=1}\\frac{v_t^2}{f_t}.$$"}, {"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 $\\sigma^2f_t>\\sigma^2$."}, {"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."}, {"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 $m=\\frac{1}{T_1}\\sum\\log^2(1+\\epsilon_t)$."}, {"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 $m>1$."}, {"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."}, {"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 $p_t\\hat\\mu_t$."}, {"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."}], "keywords": ["exponential smoothing", "single source of error", "intermittent demand", "missing values", "Kalman filter", "EM algorithm", "ADAM", "iETS"], "subfield": "openalex:subfields/2613", "topics": ["openalex:T10974", "openalex:T10567", "openalex:T10698"]}
{"id": "entrylib:2026.000004", "version": 1, "status": "active", "date": "2026-11-11", "title": "What MuseScore users ask for: unmet needs around arranging and parts", "url": "https://entrylib.org/p/2026.000004.json", "reading_view": "https://entrylib.org/p/2026.000004.txt", "principal_claims": [{"id": "C1", "statement": "Requests in the [[D5:arranging and instrumentation theme]] make up less than 10% of [[D1:MuseScore feature requests]]: the [[D7:share of issues in theme]] was 0.0459 on the [[D3:held-out set]]."}, {"id": "C2", "statement": "Requests in the [[D6:notation and engraving theme]] did not receive more [[D8:total reactions on issues in theme]] than every other theme on the [[D3:held-out set]]: they received 215 reactions, against 250 for the input, workflow and interface theme."}], "keywords": ["MuseScore", "music notation software", "feature requests", "user needs", "music arranging", "GitHub issues"], "subfield": "openalex:subfields/1210", "topics": ["openalex:T11349", "openalex:T11309", "openalex:T11948"]}