{"id":"57d66513-890b-4351-9ce5-e1f505483058","arxiv_id":"2411.12265","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Moving averages for Wi-Fi frame delivery ratio estimation have explicit error variances that match simulation well and match real data only when the filter is fast relative to channel changes.","lead":"This paper derives closed-form formulas for the estimation error of simple and exponential moving averages used to monitor Wi-Fi link quality, and checks them against simulated and real one-week traces. The formulas are proposed as a cheap baseline against which machine-learning estimators of link quality can be compared.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eqs. (24)/(30) rely on uncorrelated transmission outcomes; the paper's 0.5 s spacing is asserted but never validated on the experimental traces, so correlated burst errors would change the predicted MSE.","rationale":"The reader identified the same weakest assumption, and I agree. It is load-bearing because the closed forms are the paper's main quantitative contribution; if correlation is present, they are not the MSE even in stationary conditions. I did not find a derivation error: Eqs. (24) and (30) follow from the stated independence assumptions, and the synthetic stationary validation is convincing. Table III provides some real-data support for fast filters, which is why a conditional verdict, not rejection, is appropriate. The non-stationary 'one decade' rule is also heuristic, but it is explicitly presented as a rule of thumb from Section IV and is less central than the stationary formulas. The proposed test would either confirm the assumption (leaving the paper's claims intact) or show the need for a correlation correction or a stated limitation. Hence no change to the reader's verdict is needed.","tokens_in":14174,"tokens_out":10677,"duration_ms":127495,"concrete_test":"Generate synthetic binary trials with a two-state Markov chain having the per-channel mean ϵ from Table III and lag-1 autocorrelation ρ∈{0,0.1,0.2,0.4}; compute the SMA/EMA error MSE empirically for m=10,100,1000 (α=2/m) and compare with Eqs. (24)/(30). If the relative discrepancy exceeds 10% for ρ=0.1 at m=100, the uncorrelated assumption is load-bearing; then redo the comparison on the four experimental logs using the estimated autocorrelation function (or block bootstrap) to obtain corrected variances and check whether Table III entries shift by more than 20%.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section II-C states: 'Probabilistic variance can be easily computed under the assumption that random variables xi are uncorrelated. This is not unrealistic, provided that transmission attempts used for probing the link are spaced adequately (as in the experiments we did, where we set Ts = 0.5 s).' The adequacy of that spacing is never quantified. Eqs. (10)–(12) add only marginal variances, so Eqs. (24) and (30) inherit the independence assumption. For a stationary correlated binary process, the variance of an m-sample mean is (p(1-p)/m)[1+2Σ_{k=1}^{m-1}(1-k/m)ρ_k], not p(1-p)/m, and the EMA/block-difference terms in (24)/(30) also omit cross-block covariances. Real Wi-Fi failures are often bursty (interference episodes, slow fading), so nonzero ρ_k at lags up to the filter length is plausible. Table III's fast-filter agreement is encouraging for those four logs at 2 Hz, but the paper generalizes to 'Wi-Fi link quality' without autocorrelation diagnostics or released traces. The central stationary-MSE claim is therefore contingent on an unverified independence assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes the accuracy and precision of simple moving average (SMA) and exponential moving average (EMA) estimators of Wi-Fi link quality, quantified as the frame delivery ratio (FDR) over an interval centered on the current time. Under stationary conditions with uncorrelated transmission outcomes, it derives closed-form expressions for the mean squared error (MSE) of the SMA and EMA estimation errors: Eq. (24) for the SMA and Eq. (30) for the EMA. The derivations are validated against synthetic i.i.d. Bernoulli processes with more than ten million samples, where the sample MSE closely matches the formulas. The paper then considers non-stationary conditions, both synthetic (sinusoidal failure probability) and experimental (one week of logs from four Wi-Fi channels), and claims that the stationary formulas provide an acceptable approximation to the MSE provided the filter cutoff is about a decade below the disturbance frequency, with larger deviations for slow filters. The paper positions the results as a parameter-free baseline against which machine-learning-based link quality estimators can be compared.","tokens_in":14458,"tokens_out":4687,"duration_ms":50581,"significance":"If the central claims hold, the paper provides simple, closed-form expressions for the MSE of SMA and EMA link-quality estimators that require no fitted parameters and can serve as a baseline for runtime monitoring and for evaluating more complex ML-based approaches. The algebraic derivation is self-contained and the synthetic validation is convincing: the formulas match simulation results across a wide range of filter settings and failure probabilities. The main significance is in the utility of the formulas as a design tool, rather than in methodological novelty. However, the result is conditional on the assumption that transmission outcomes are uncorrelated, which is asserted but not verified on the experimental traces, and the non-stationary/experimental validation relies on subjective color-coded criteria without confidence intervals. These limitations currently bound the paper's generality.","major_comments":[{"comment":"The closed-form variance formulas are derived under the explicit assumption that the random variables xi are uncorrelated, but this assumption is never validated for the experimental data. The text asserts that Ts = 0.5 s is 'spaced adequately', yet no autocorrelation diagnostics are reported. Real Wi-Fi transmission failures are frequently bursty (e.g., due to interference episodes or slow fading), and for a correlated binary process the variance of an m-sample mean contains additional terms of the form (p(1-p)/m)[1+2Σ(1-k/m)ρ_k]; similar corrections apply to the block-difference terms in Eqs. (24) and (30). Without quantifying the autocorrelation of the experimental logs, the claim that the formulas describe the MSE of real Wi-Fi FDR estimation is not established. I recommend adding (a) sample autocorrelation plots or lag-1/lag-k correlation estimates for the four experimental logs, and (b) a sensitivity analysis using a correlated binary process with realistic ρ_k to show how much the true MSE can deviate from Eqs. (24)/(30).","section":"Section IV-B, Table III"},{"comment":"The experimental evaluation uses a single week-long log per channel and reports no confidence intervals or quantitative criterion for when the approximation is 'acceptable'. Table III shows large discrepancies for slow filters; for example, on channel 13 with m = 10000, Var(e) = 0.000012 while the MSE is 0.000320, and on channel 1 with m = 1000, Var(d) = 0.000033 while the MSE is 0.000127. Even for faster filters where the match is better, the sampling variability of the MSE estimate from a single trace is not quantified. The paper should define an explicit acceptable-error threshold (e.g., relative error between MSE and Var), and report bootstrap confidence intervals for the experimental MSE values so that the reader can assess whether the observed agreement is statistically meaningful.","section":"Section IV-A, Table II"},{"comment":"The conclusion that the stationary formulas are 'acceptable' only when the filter cutoff is about a decade below the disturbance frequency is based on a small, hand-picked set of parameter combinations (f ∈ {0.0001, 0.001} Hz, m ∈ {10, 100, 1000, 10000}) and on a color code that is never defined in the table or text. The text states that in some rows 'the MSE becomes poorer but still acceptable' without giving a quantitative rule. To support the 'one decade' claim, the paper should sweep the ratio of filter cutoff to disturbance frequency more finely and plot the relative discrepancy between the simulated MSE and the stationary variance as a function of that ratio, or alternatively derive an analytic bound for the non-stationary case. As written, the claim is an observation about a few rows rather than a demonstrated property.","section":"Section II-C, Eq. (13)"}],"minor_comments":[{"comment":"The color code (green, yellow, red) mentioned in the text of Section IV-A is not reproduced or defined in the table itself; please add a legend or explicit thresholds for what counts as acceptable.","section":"Section II-C, Eq. (13)"},{"comment":"The term 'uncorrected variance s²_ε[i]' is used but never defined; please define it as the sample variance of the ε_j values in the interval without Bessel's correction.","section":"Section II"},{"comment":"The notation xi is used both for the observed binary sequence and for the random variable model. Using uppercase X_i for the random variable would avoid ambiguity, especially in the probabilistic derivations of Section II-C.","section":"Section IV-B"},{"comment":"The experimental setup does not state the actual probing rate Ts used for the four channel logs, even though the adequacy of Ts = 0.5 s is a key assumption in Section II-C; please report the probing period for the experimental traces.","section":"References"},{"comment":"Several references used for motivation and prior art are self-citations (refs. 10, 11, 12, 15); please consider citing independent evaluations of moving-average or regression-based link-quality estimation to strengthen the literature context.","section":"Eq. (13)"},{"comment":"In Eq. (13), the expansion from ϵ_j(1-ϵ_j) to the terms involving ¯ϵ[i], ¯ϵ²[i], and s²_ϵ[i] has a skipped step; adding one intermediate line would improve readability.","section":"Section IV-A"}],"recommendation":"major_revision","confidential_remarks":"The paper is a competent derivation with a useful practical baseline, and the synthetic validation is thorough. The main risk is that the central variance formulas are only valid under an uncorrelated-outcomes assumption that is asserted but not verified for the experimental data; the experimental section also lacks statistical rigor (no confidence intervals, undefined color-coding). These are fixable with additional analyses within the manuscript's scope. The paper is somewhat incremental relative to the authors' prior work and leans on self-citations, but it does offer a self-contained closed-form result that could be useful to the community. I see no concerns about fabrication or circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi,\n\nQuick take: this is a clean, limited-scope paper. The new thing is Eq. (30), a closed-form expression for the estimation error variance of an EMA when the target is the FDR over a centered window. The SMA result in Eq. (24) is textbook, and the authors' own prior work covered prediction rather than estimation. The derivation of the cross-term between the EMA and the past window is legitimate; I checked the algebra and it holds. The synthetic validation with 10^7 samples matches the formulas to four digits, which gives real confidence that the math is right.\n\nWhat the paper does well: it positions the result as a baseline for ML link-quality estimators, which is a sensible use. It is honest about the stationarity assumption and about the fact that in non-stationary conditions the formulas only work when the filter cutoff is about a decade below the disturbance frequency. The experimental section uses a week-long real-world log, which is more than many papers in this area do.\n\nThe soft spots are in proportion. The biggest one is the uncorrelated-outcomes assumption in Section II-C. The paper asserts that Ts=0.5s makes this \"not unrealistic\" but never checks autocorrelation on the experimental traces. Real Wi-Fi failures are often bursty, so the variance formulas could understate the true error at lags on the order of the filter length. Table III's agreement on four channels at fast filter settings is encouraging but does not establish the assumption holds generally. Second, the experimental claims have no confidence intervals and the traces are not released, so the reader cannot verify the discrepancy between measured and predicted MSE for slow filters. Third, the \"acceptable\" color code in Table II is subjective. None of these are fatal; they are addressable.\n\nIf I were referee I'd ask for an autocorrelation diagnostic on the experimental logs and for the variance of the MSE estimates. The core result is sound and the paper deserves peer review. It is not a world-beater, but it is a correct, reusable baseline.\n\nBest.","headline":"A correct, modest analytical result: closed-form MSE for SMA/EMA estimation of Wi-Fi FDR, well validated on synthetic data but with an unaddressed independence assumption and thin experimental support.","tokens_in":14961,"tokens_out":1675,"would_cite":true,"duration_ms":17192,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives closed-form mean squared error formulas for simple and exponential moving averages that estimate Wi-Fi frame delivery ratio, exact in steady channels and good when the filter cutoff is a decade below the disturbance.","keywords":["Wi-Fi link quality","frame delivery ratio","simple moving average","exponential moving average","mean squared error","variance","non-stationary channel","filter cutoff frequency"],"falsifier":"Generate a stationary binary process with fixed mean failure probability but controlled positive autocorrelation between consecutive outcomes, run the SMA and EMA filters, and compare the measured MSE against Eq. (24) and Eq. (30); systematic overshoot of the formulas would show the uncorrelated-outcome assumption is the weak point.","tokens_in":14022,"feed_emoji":"📶","tokens_out":7999,"duration_ms":71823,"temperature":0.7,"pith_summary":"Wi-Fi links are monitored at runtime by averaging recent transmission successes and failures, but every averaging filter trades noise suppression against responsiveness. This paper derives closed-form formulas for the mean squared error (MSE) of the two standard filters, the simple moving average (SMA) and the exponential moving average (EMA), when they estimate the centered frame delivery ratio of a link. In stationary channels the formulas give the exact error variance, and in slowly varying channels they remain a good approximation provided the filter cutoff is about a decade below the disturbance frequency. These formulas turn a heuristic tuning problem into a calculable one, and they provide a baseline against which machine-learning estimators of link quality can be judged.","feed_headline":"Simple formulas pin down Wi-Fi link-quality estimation error","feed_subtitle":"The error of SMA and EMA link-quality estimators is exactly predictable in steady channels, and closely in slow-moving ones.","key_machinery":"The load-bearing identities are the variance decompositions of the estimation errors. For the SMA, the error $d=z-u$ splits into a future half-window and a past half-window, yielding $\\mathrm{Var}(d)=\\mathrm{Var}(x)/(2m)$. For the EMA, the error $e=z-y$ splits into $v/2$ and $y-u/2$, and because the two sums run over disjoint outcome indices they are uncorrelated, which gives $\\mathrm{Var}(e)=\\mathrm{Var}(x)\\left[\\frac{\\alpha}{2-\\alpha}+\\frac{1}{m}\\left((1-\\alpha)^m-\\frac12\\right)\\right]$ in terms of the smoothing parameter $\\alpha$ and window half-width $m$. The paper uses these identities to predict the sample MSE from the per-attempt variance $\\epsilon(1-\\epsilon)$, and to state the cutoff-frequency rule for non-stationary validity.","core_discovery":"The paper's central claim is that the mean squared error of SMA and EMA estimates of the centered frame delivery ratio equals the probabilistic variance of the estimation error under stationary conditions, with explicit closed forms given by Eq. (24) for the SMA and Eq. (30) for the EMA. The target quantity is the frame delivery ratio over a window of $2m$ transmission attempts centered on the current time, which is non-causal and so only usable offline; the SMA estimates it from the most recent $m$ attempts, while the EMA weights recent attempts exponentially. Because both filters are unbiased in stationary conditions, the MSE reduces to the variance, which is expressed in terms of the per-attempt variance $\\mathrm{Var}(x)=\\epsilon(1-\\epsilon)$ and the filter parameters $m$ and $\\alpha$. Numerical runs on synthetic stationary processes confirm the formulas, and tests on synthetic and experimental non-stationary logs show that the approximation stays acceptable as long as the filter cutoff frequency is roughly one decade below the disturbance frequency.","pith_inferences":["If real transmission outcomes are positively correlated, for instance under bursty interference, the independence assumption will make the closed-form variance understate the true error; an extension with an autocorrelation correction would cover such channels.","The same variance-splitting technique applies to other linear filters, such as multi-pole IIR designs, so closed-form MSE baselines for the more sophisticated estimators mentioned in the conclusions seem within reach.","Because the centered frame delivery ratio is the target used to train neural-network predictors, these formulas also provide a lower bound on what such predictors can achieve in stationary conditions."],"forward_implications":["In stationary channels, SMA error variance is inversely proportional to window size $m$, and EMA error variance grows roughly linearly with $\\alpha$, giving a direct rule for sizing the filters.","For slowly varying channels, keeping the filter cutoff about one decade below the disturbance frequency keeps the closed-form formulas within acceptable accuracy, so the formulas double as a tuning rule.","Because MSE equals variance for the unbiased estimators, the sample variance of observed estimation errors is a legitimate proxy for MSE in stationary conditions, and the formulas give the value before any data is collected.","Any machine-learning link-quality estimator can be held to this baseline: to justify its added complexity, its MSE should beat the SMA and EMA values at the same responsiveness."],"supporting_citations":[{"why":"supplies the earlier moving-average and regression predictors that this paper formalizes with error-analysis formulas.","marker":"[10]"},{"why":"extends the EMA family to linear combinations, providing the motivation for analyzing a single EMA as a baseline.","marker":"[11]"},{"why":"provides the neural-network prediction counterpart that motivates using SMA and EMA as interpretable baselines.","marker":"[12]"},{"why":"supplies the per-frame outcome logging mechanism used to collect the experimental Wi-Fi datasets.","marker":"[14]"},{"why":"gives the neural-network link-quality predictor and the training-phase logic that the paper invokes for parameter selection.","marker":"[15]"}],"fun_headline_variants":["Closed-form error for Wi-Fi moving-average estimators","Exact MSE for SMA and EMA link-quality estimators","Wi-Fi link quality: moving-average error exactly known","Predicting Wi-Fi link-quality estimator error precisely","Closed-form MSE for SMA and EMA in steady Wi-Fi channels"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The variance formulas assume that successive transmission outcomes are uncorrelated, so if real failures cluster in bursts, the true estimation error will be larger than the formula predicts even in stationary conditions.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form error for Wi-Fi moving-average estimators","Exact MSE for SMA and EMA link-quality estimators","Wi-Fi link quality: moving-average error exactly known","Predicting Wi-Fi link-quality estimator error precisely","Closed-form MSE for SMA and EMA in steady Wi-Fi channels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000356,"raw_usage":{"total_tokens":1887,"prompt_tokens":857,"completion_tokens":1030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":954}},"tokens_in":473,"tokens_out":1030,"duration_ms":8366,"temperature":1.0,"reasoning_tokens":954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:43:43.553398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate a stationary binary process with fixed mean failure probability but controlled positive autocorrelation between consecutive outcomes, run the SMA and EMA filters, and compare the measured MSE against Eq. (24) and Eq. (30); systematic overshoot of the formulas would show the uncorrelated-outcome assumption is the weak point.","supporting_citations":[{"cited_title":"Predicting Wire- less Channel Quality by Means of Moving Averages and Regression Models,","cited_arxiv_id":null,"evidence_quote":"supplies the earlier moving-average and regression predictors that this paper formalizes with error-analysis formulas."},{"cited_title":"Linear Combination of Exponential Moving Averages for Wire- less Channel Prediction,","cited_arxiv_id":null,"evidence_quote":"extends the EMA family to linear combinations, providing the motivation for analyzing a single EMA as a baseline."},{"cited_title":"Predicting Wi-Fi link quality through artificial neural networks,","cited_arxiv_id":null,"evidence_quote":"provides the neural-network prediction counterpart that motivates using SMA and EMA as interpretable baselines."},{"cited_title":"SDMAC: A Software-Defined MAC for Wi-Fi to Ease Implementation of Soft Real-Time Applica- tions,","cited_arxiv_id":null,"evidence_quote":"supplies the per-frame outcome logging mechanism used to collect the experimental Wi-Fi datasets."},{"cited_title":"On the Use of Artificial Neural Networks to Predict the Quality of Wi-Fi Links,","cited_arxiv_id":null,"evidence_quote":"gives the neural-network link-quality predictor and the training-phase logic that the paper invokes for parameter selection."}],"review_version":1}