{"id":"1295dad0-c7b6-417e-8168-ddcf9069f4cd","arxiv_id":"2506.00984","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"For ARX systems with uniformly quantized output, a CIC-type criterion with least-squares residuals is shown to yield consistent order estimates under small steps and persistent excitation.","lead":"This paper proposes an order-estimation rule for ARX systems whose outputs are only available through a coarse quantizer, and proves that the rule recovers the true orders when the quantization step is small and certain excitation conditions hold. It addresses a niche gap in system identification, but the claimed practical recipe for choosing the tuning constants is not actually derived from observable data.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised concrete penalty choice is unsupported: Theorem 3.1's conditions (26)-(27) depend on the true order p0 and unknown constants, and the numerical example's l_n violates the theorem's own lower bound (26).","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the penalty interval in (26)-(27) depends on the true order and unknown system constants, and the numerical example uses penalties below the theorem's lower bound. I agree with the REJECT verdict. The conditional consistency statement may be mathematically correct, but the abstract's promise of a concrete method for choosing quantization parameters is not fulfilled; Remark 3.3 admits the constants are unknown, and no data-driven procedure is supplied. The numerical example is not independent support because it operates outside the theorem's stated conditions. The concern is not a disagreement with consensus but a failure of the paper's advertised deliverable. A revised version could salvage the idea by deriving a penalty rule based only on p*, q*, epsilon and observed quantities, or by showing that a range of penalties works for small epsilon, and by correcting the simulation to respect (26). Until then, the central claim remains unsupported.","tokens_in":14248,"tokens_out":5392,"duration_ms":55606,"concrete_test":"Using the true Section 4 system so that all constants in (26)-(27) are computable, verify whether the paper's chosen l_n=0.006n satisfies (26) for epsilon=0.001 and p*=3; for c>=1, the lower bound is 0.02n, so it does not. Then re-run the Monte Carlo experiment with l_n=0.03n, which satisfies the lower bound, and check whether the upper bound in (27) also holds; if 0.03n lies outside the interval, pick the midpoint of the computed interval instead. If the order estimates fail to converge to p0, the numerical demonstration is incompatible with the theorem; if they converge, the paper still needs a data-based rule for choosing such a penalty without knowing p0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central advertised contribution is a concrete method for choosing quantization parameters so that the order estimates are consistent (Abstract; Remark 3.2). Theorem 3.1 only proves consistency conditionally on l_n lying in the interval (26)-(27). The lower bound in (26) is 5(1+p*c)epsilon + alpha1, which is computable from p*, c, epsilon, but the upper bound in (27) contains a_{p0}^2, c1, gamma, c3 and the true order p0 itself. Remark 3.3 explicitly states that these quantities are not accessible. No data-based construction of l_n is provided, so the promised concrete method is absent. The numerical example does not fill this gap: for epsilon=0.001 and p*=3, with c >= 1 (since |b1|=1), the required lower bound is at least 0.02n, while the paper sets l_n=0.006n; for epsilon=0.002 and p*=6, the lower bound is at least 0.035n, while l_n=0.012n. Thus the simulation runs outside the theorem's own hypotheses and provides no evidence that the conditions can be met in practice. The central claim of an implementable consistency guarantee therefore rests on an unstated, untested selection mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies order estimation of ARX systems from uniformly quantized outputs. It defines a CIC-type criterion L_n(p,q) based on least-squares estimates from quantized regressors and proves, under Assumptions 3.1-3.5 and provided the penalty rate l_n lies in the interval (26)-(27), that the estimated AR order converges almost surely to p0; Theorem 3.2 gives an analogous result for the exogenous order q0. A numerical example with two quantization steps is presented. The abstract claims a concrete method for choosing quantization parameters to ensure consistency.","tokens_in":14540,"tokens_out":6573,"duration_ms":60627,"significance":"If Theorems 3.1-3.2 were accompanied by a data-based construction of the penalty rate, the paper would be a useful extension of quantized system identification to order estimation. The conditional consistency result is a nontrivial technical step, and the paper is clearly written where it does not depend on unverified constants. However, the advertised practical contribution is not delivered: the conditions (26)-(27) and (53)-(54) depend on the true orders and system constants that the estimator is supposed to discover, and the numerical example runs outside the theorem's hypotheses. The paper therefore does not provide an implementable consistency guarantee.","major_comments":[{"comment":"The abstract promises \"a concrete method is given for choosing quantization parameters to ensure that the system order estimates are consistent.\" In the body, the only guidance is Remark 3.2, which asserts that the intervals in (26)-(27) and (53)-(54) are nonempty for suitably chosen ε, α1, α2, β1, β2. No such choice is specified, and the upper bound (27) contains a_{p0}^2, c1, γ, c3 and the true order p0 itself. Remark 3.3 explicitly admits that these quantities are not accessible. Thus the load-bearing hypothesis of the theorem—that l_n lies in the stated interval—cannot be verified from data, and the central advertised contribution is missing. The same problem applies to v_n in Theorem 3.2.","section":"Section 3.2, Theorem 3.1 (Eqs. (26)-(27))"},{"comment":"The simulation chooses l_n=0.006n for ε=0.001, p*=3 and l_n=0.012n for ε=0.002, p*=6. Under Assumption 3.3, c≥1 because |b1|=1, so condition (26) requires l_n ≥ 5(1+p*c)ε n, i.e., at least 0.02n for the first setting and 0.07n for the second. The chosen values are lower by a factor of roughly 3 to 6, so the simulation is not carried out under the hypotheses of Theorem 3.1. The trajectories in Figs. 1-8 therefore do not demonstrate that the consistency conditions can be satisfied, and the paper gives no explanation for these l_n values in terms of (26)-(27).","section":"Section 4, numerical example"},{"comment":"Even the non-emptiness of the interval (26)-(27) is not established. Remark 3.2 merely states that suitable parameters make the intervals nonempty, but it does not quantify how small ε must be relative to a_{p0}^2 c1, γ, c3, and c1 and c3 are themselves not verified in the simulation. A conditional consistency theorem with an unverifiable and possibly empty parameter range does not support the paper's stated claim of a concrete method.","section":"Remark 3.2 and Theorem 3.1"}],"minor_comments":[{"comment":"The term a_{p0}^2 λ_min(p0,q*)(n-1) is used for a matrix whose sum runs from i=0 to n-1, while P_{n+1} in (7) has sum to n; the indexing of λ_min should be made consistent with Assumption 3.5 and Eq. (7).","section":"Equation (40)"},{"comment":"The remainder terms o(n) are used together with \"a.s.\"; the statements should specify whether the o(n) is uniform in the random realization and whether the almost-sure bound is meant to hold along every sample path.","section":"Equations (13), (20), and (33)"},{"comment":"The numerical example uses uniform inputs on [-3,3] for p estimation and [-1,1] for q estimation, but the role of δ from Assumption 3.1 is not explained, and the choice of different δ values in the two simulations is not reconciled with the constants in the theorem.","section":"Section 4"},{"comment":"Several references contain typographical artifacts such as \"S¨ oderstr¨ om\" and \"Birkhauser\"; these should be formatted consistently.","section":"References"}],"recommendation":"reject","confidential_remarks":"The central gap is acknowledged by the authors in Remark 3.3 and in the conclusion, which defers a method for verification to future work. As the abstract advertises a concrete method that is never provided, and as the numerical example does not obey the theorem's own hypotheses, I cannot recommend publication in the present form. If the authors can supply a genuine data-driven construction of l_n and v_n, or substantially reframe the contribution as a conditional consistency theorem, a resubmission could be considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my read on this one. The paper is the first to treat order estimation for ARX systems with quantized output, and that is a real gap in the literature. The criterion itself is a direct adaptation of Chen-Guo's CIC, but that is a sensible starting point. The conditional consistency theorem (Theorem 3.1) is the core contribution, and the proof structure looks plausible: it follows the classic least-squares CIC argument, with the quantization noise bounded and absorbed into the penalty. Lemma 3.4 is the key step, and the bound in (20) is credible. So the theoretical core is not junk.\n\nThe soft spots are in the translation from theory to practice. The theorem's condition on l_n depends on the true order p0 and on constants a_p0, c1, gamma, c3 that are not accessible. Remark 3.3 admits this. The abstract and Remark 3.2 promise a concrete method for choosing the quantization parameters, but no data-based construction is provided. That is a load-bearing gap, because the whole point of an order estimator is to work without knowing p0. The simulation does not fill the gap: the chosen l_n values (0.006n and 0.012n) are lower than the theorem's own lower bound in (26), which for the reported settings is at least 0.02n (with c>=1 and p*=3). So the numerical example runs outside the hypotheses and cannot be taken as evidence that the conditions are satisfiable. That is a serious flaw, not a cosmetic one.\n\nA secondary issue is that the proof leans on Theorem 1 of the author's own earlier paper (Jing, 2022) for the crucial bound (23). That is acceptable if the cited result is correct, but the paper does not restate or verify the dependence on the constants, making the proof harder to check.\n\nIn short: the paper identifies a genuinely new question and gives a plausible conditional solution, but it fails to deliver an implementable recipe. A serious referee could ask for a data-driven penalty rule and a corrected simulation, and the conditional theory might survive. So I'd send it out for review, but I'd expect heavy revision. My own verdict is skeptical—I wouldn't cite it as a usable method until the penalty choice is resolved.","headline":"Novel problem, plausible conditional proof, but the advertised concrete penalty choice is self-referential and the simulation violates the theorem's own conditions.","tokens_in":15042,"tokens_out":2764,"would_cite":false,"duration_ms":25879,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F12","62M10","93E12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a penalty-based least-squares criterion can recover the true orders of a stochastic ARX system from quantized output alone, almost surely as sample size grows.","keywords":["quantized output","ARX systems","order estimation","least squares","control systems information criterion","almost sure consistency","quantization step","persistent excitation"],"falsifier":"Take the paper's simulation setup (ARX with $p_0=2$, $q_0=1$, $\\varepsilon=0.001$, $p^*=3$, $q^*=3$) and choose a penalty $l_n$ such that the interval in (26)-(27) is empty, for instance by making the upper bound negative through a larger $\\varepsilon$; if $\\hat p_n$ still converges to 2, the interval condition is not the operative restriction, while failure would confirm that the bound is doing real work.","tokens_in":14036,"feed_emoji":"📊","tokens_out":9950,"duration_ms":84803,"temperature":0.7,"pith_summary":"This paper claims that the unknown orders of a stochastic ARX (autoregressive with exogenous input) system can be recovered consistently from uniformly quantized output alone. The proposed estimator runs least squares at every candidate order pair in a finite grid and minimizes a criterion that adds a linear penalty proportional to the order to the sum of squared prediction errors. Under stability, bounded-input, and persistent-excitation assumptions, the paper proves that the minimizing orders converge almost surely to the true orders, provided the quantization step is small and the penalty rate per sample lies in a specified interval. The contribution is to extend order estimation, which is classically done on precise measurements, to the quantized-data setting where parameter estimates do not converge to the true values; the catch, acknowledged by the paper, is that the allowed penalty interval is expressed through the true order and unknown system constants, so the result is a consistency theorem rather than a directly implementable tuning rule.","feed_headline":"Quantized output still pins down ARX model orders almost surely","feed_subtitle":"A least-squares criterion with a linear penalty selects the true model orders almost surely when the quantization step is small.","key_machinery":"The load-bearing object is the quantized regression vector $\\psi_i(p,q)=[s_i,\\dots,s_{i-p+1},u_i,\\dots,u_{i-q+1}]^\\top$ built from quantized outputs $s_i$, together with the criterion $L_n(p,q)=\\sum_{i=0}^{n-1}(s_{i+1}-\\theta_n^\\top(p,q)\\psi_i(p,q))^2+l_n(p+q)$. The proof decomposes $\\sigma_n(p,q)-\\sigma_n(p_0,q^*)$ into a bias term that is positive and linear in $n$ for wrong orders and a quantization-error term of order $\\varepsilon n$ for the true order; the penalty rate $l_n$ is the knife-edge that lets the bias dominate quantization noise.","core_discovery":"The central claim is Theorem 3.1 and Theorem 3.2: with $L_n(p,q^*)=\\sigma_n(p,q^*)+l_n(p+q^*)$ and $V_n(p^*,q)=\\sigma_n(p^*,q)+v_n(p^*+q)$, where $\\sigma_n$ is the least-squares prediction-error sum, the estimates $\\hat p_n$ and $\\hat q_n$ converge almost surely to the true orders $p_0$ and $q_0$. The proof shows that for over-ordered models the quantized-noise contribution to $\\sigma_n$ is at most $5(1+p^*c)\\varepsilon n$, while for under-ordered models the missing-regressor bias contributes at least $a_{p_0}^2 c_1 n$; a penalty rate growing linearly in $n$ and bounded between these two scales selects the truth asymptotically.","pith_inferences":["Because the penalty interval in (26)-(27) depends on the true order and unknown constants, turning the theorem into a practical rule requires an adaptive choice of the penalty rate; notably, the paper's own numerical values for $l_n$ fall below the theorem's lower bound, suggesting the bound is conservative rather than sharp.","A natural extension would be a data-driven penalty schedule, for instance one scaling like $\\log n$ with constants selected by validation, that achieves the same consistency without prior knowledge of $p_0$.","The proof structure should transfer to other quantization schemes whose error is uniformly bounded by $O(\\varepsilon)$ and to noise sequences satisfying the same moment assumptions, broadening the class of observation models for which order-consistent estimation is possible.","A falsifiable consequence of the consistency claim is that the order estimates should still converge for a range of penalty constants around the numerical choices used in the example; failure in some part of that range would locate the true restriction."],"forward_implications":["If the assumptions hold, the estimated AR order $\\hat p_n$ converges almost surely to the true $p_0$, so the selected order can be trusted once enough quantized samples are collected.","A non-empty penalty interval forces the quantization step $\\varepsilon$ to be small relative to the smallest-regressor eigenvalue, quantifying the resolution the quantizer must provide for consistency.","Larger search bounds $p^*, q^*$ preserve consistency but slow the convergence rate, as the numerical trajectories show.","The criterion can be evaluated on every candidate pair $(p,q)$ in a finite grid, making the method straightforward to implement once the penalty rate is chosen.","Order consistency is obtained without knowing the parameters, so the estimated order can be fed into an existing quantized parameter estimator as the correct model structure."],"supporting_citations":[{"why":"Supplies Theorem 1, the bound on quantized least-squares parameter error used in Lemma 3.2 and Lemma 3.4.","marker":"(Jing, 2022)"},{"why":"Provides Lemma B.3.3 bounding the accumulated output energy, used in Lemma 3.1 to bound the largest eigenvalue of the quantized regression matrix.","marker":"(Goodwin,&Sin, 1989)"},{"why":"Introduces the control systems information criterion whose prediction-error-plus-penalty form Ln and Vn adapt to quantized data.","marker":"(Guo,Chen&Zhang, 1989)"},{"why":"Cited in Remark 3.3 as the precedent that the allowable penalty range depends on unknown model quantities.","marker":"(Chen,&Guo, 1991)"},{"why":"Earlier consistent order estimation for stochastic feedback control systems by least squares, the template the quantized version extends.","marker":"(Chen,&Guo, 1987)"}],"fun_headline_variants":["Quantized data still suffice for consistent ARX order estimation","ARX order estimation works even with coarse quantization","Quantized least squares reliably recovers true ARX orders","Small quantization steps enable consistent ARX model order selection","Almost sure order estimation from quantized ARX observations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The consistency proof requires the penalty rate per sample, $l_n/n$, to lie between a lower bound set by the quantization step and an upper bound whose value depends on the true order $p_0$ and on unknown constants $a_{p_0}, c_1, \\gamma, c_3$; no data-based rule is given to place $l_n$ there, so the theorem guarantees consistency only for a penalty interval one cannot compute before identifying the system.","fun_headline_variants_meta":{"raw":{"variants":["Quantized data still suffice for consistent ARX order estimation","ARX order estimation works even with coarse quantization","Quantized least squares reliably recovers true ARX orders","Small quantization steps enable consistent ARX model order selection","Almost sure order estimation from quantized ARX observations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000463,"raw_usage":{"total_tokens":2255,"prompt_tokens":829,"completion_tokens":1426,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":1320}},"tokens_in":445,"tokens_out":1426,"duration_ms":10273,"temperature":1.0,"reasoning_tokens":1320,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:55:32.957030+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the paper's simulation setup (ARX with $p_0=2$, $q_0=1$, $\\varepsilon=0.001$, $p^*=3$, $q^*=3$) and choose a penalty $l_n$ such that the interval in (26)-(27) is empty, for instance by making the upper bound negative through a larger $\\varepsilon$; if $\\hat p_n$ still converges to 2, the interval condition is not the operative restriction, while failure would confirm that the bound is doing real work.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 1, the bound on quantized least-squares parameter error used in Lemma 3.2 and Lemma 3.4."},{"cited_title":"F., & Zhang , J","cited_arxiv_id":null,"evidence_quote":"Introduces the control systems information criterion whose prediction-error-plus-penalty form Ln and Vn adapt to quantized data."},{"cited_title":"F., & Guo , L","cited_arxiv_id":null,"evidence_quote":"Cited in Remark 3.3 as the precedent that the allowable penalty range depends on unknown model quantities."},{"cited_title":"F., & Guo , L","cited_arxiv_id":null,"evidence_quote":"Earlier consistent order estimation for stochastic feedback control systems by least squares, the template the quantized version extends."}],"review_version":1}