{"id":"cb77e6b0-8e0c-4ca9-b4ea-e7e5e745d4bd","arxiv_id":"2502.05719","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"EHBOS averages univariate HBOS scores with pairwise bivariate histogram scores, improving ROC AUC over HBOS on some interaction-dependent datasets.","lead":"This paper proposes EHBOS, an outlier detection method that extends HBOS by adding two-dimensional histograms to capture relationships between pairs of features. Tests on 17 benchmark datasets show it beats plain HBOS on several datasets where feature interactions matter, though it loses on others.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Empty-bin handling in the 2D histogram score is unspecified; since -log(0) is infinite, the reported AUC improvements over HBOS depend on an undocumented implementation choice.","rationale":"The reader's weakest assumption exactly identifies the most load-bearing issue: the 2D histogram density estimate is not well-defined when bins are empty, and the paper gives no handling rule. The proposed score uses -log h, which is infinite at h=0, yet Table 2 reports finite ROC AUC values; therefore some undocumented finite approximation must have been used. Because the empirical comparison to HBOS is the main support for the central claim, and because high-dimensional datasets in the benchmark make empty bins common, the reported improvements cannot be reproduced or trusted without this implementation detail. The paper's synthetic demonstrations (Figures 3 and 4) provide some support for the qualitative behavior on two-dimensional data, and the published version in the Journal of Statistical Theory and Practice adds credibility, but neither addresses the empty-bin ambiguity in the benchmark evaluation. The concern is concrete and addressable: a code release or an explicit smoothing rule would settle it. This does not by itself refute the method — the idea of adding pairwise histograms to HBOS is coherent — but it makes the current empirical evidence conditional. Since the reader already reached CONDITIONAL for this reason, the stress-test pass does not change the verdict; it sharpens the required condition: specify or release the exact empty-bin and normalization procedure used to produce Table 2.","tokens_in":7032,"tokens_out":4095,"duration_ms":46839,"concrete_test":"Obtain or reimplement the Section 3 EHBOS procedure and recompute Table 2 under two explicit empty-bin conventions: (a) add-one smoothing, h = (count+1)/(n*area + area); and (b) skip empty bins, ignoring feature pairs where either point falls in an empty 2D cell. Compare ROC AUC on glass, cardio, satellite, lympho, mnist, and optdigits. If any of the claimed improvements (glass +0.15, cardio +0.05, satellite +0.04) changes by more than 0.05 or reverses under either convention, the central empirical claim is implementation-dependent rather than a property of the proposed score.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central empirical claim — EHBOS improves ROC AUC over HBOS on glass, cardio, and satellite — rests on the 2D score defined in Section 3 as s2(i) = sum over feature pairs (j,k) of -log h_jk(x_ij, x_ik). No rule is given for empty bins. If h_jk = 0, then -log(h_jk) is +infinity for any sample landing in an empty cell, so the averaged final score is infinite and ROC AUC is undefined. The finite values in Table 2 therefore require an implicit convention — pseudocount, floor, skip-empty, or similar — that is never stated. With d(d-1)/2 pairwise histograms, and with datasets like mnist (d=100), optdigits (d=64), and musk (d=166), many 2D cells are empty under any fixed binning, so the undocumented convention can change which samples dominate the score and can even make min-max normalization of s2 ill-defined. Thus the reported improvements may be artifacts of the particular empty-bin handling rather than of the EHBOS construction itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces EHBOS, an extension of the histogram-based outlier score HBOS that adds pairwise two-dimensional histograms to the original independent-feature histograms. The final EHBOS score is defined as the average of a normalized one-dimensional HBOS score and a normalized two-dimensional score over all feature pairs. The authors motivate the method with two synthetic examples where HBOS fails to detect contextual and dependency-driven outliers, present an algorithm and a claimed O(n*d^3) time complexity, and report ROC AUC results on 17 benchmark datasets, mostly in comparison with HBOS alone. The central claim is that EHBOS outperforms HBOS on datasets where feature interactions matter, such as glass, cardio, and satellite, while remaining competitive elsewhere.","tokens_in":7225,"tokens_out":4409,"duration_ms":46443,"significance":"If the empirical claims hold, EHBOS would be a simple, interpretable, and computationally modest extension of a widely used baseline, and the two synthetic demonstrations are pedagogically useful. The method is defined before evaluation and does not fit parameters to the anomaly labels, so there is no circularity in the main comparison. The paper is also honest about cases where EHBOS underperforms, notably on lympho, mnist, and optdigits. However, the central claim is currently not well supported: the 2D score has an undefined behavior for empty histogram bins, the normalization of scores is unspecified, the reported AUC values lack uncertainty quantification, and no comparison is made with the HBOS extensions cited in the introduction. These issues are load-bearing for the paper's main conclusion and need to be resolved before the reported improvements can be accepted.","major_comments":[{"comment":"The definition s2(i) = sum_{j<k} -log h_jk(x_ij, x_ik) is undefined when a 2D histogram bin is empty, because -log(0) is +infinity and such a score would dominate the final average. The manuscript states no pseudocount, floor value, or skip-empty rule, yet Table 2 reports finite ROC AUC values. For high-dimensional datasets in Table 1 (mnist with d=100, optdigits with d=64, musk with d=166), many feature pairs have empty cells under any reasonable binning, so the reported improvements may be artifacts of an implicit convention rather than of the EHBOS construction. Please specify the empty-bin handling and report the sensitivity of Table 2 to that choice.","section":"Section 3, 2D score equation"},{"comment":"The algorithm says to normalize the 1D scores and to accumulate normalized 2D scores, but no normalization procedure is defined. If min-max normalization is used, an infinite 2D score makes the normalized score undefined; if rank or z-score normalization is used, the relative contribution of the 2D component changes. Since the final score is the average of these normalized components, the exact form of normalization can determine whether EHBOS improves over HBOS on a given dataset. Please state the normalization formulas and justify them.","section":"Section 3, Algorithm steps 02 and 07"},{"comment":"Table 2 reports a single ROC AUC value per method and dataset, with no repeated runs, confidence intervals, or significance tests. Several reported differences are very small (pima 0.6956 vs 0.6954, pendigits 0.9283 vs 0.9238, wbc 0.9583 vs 0.9533) and cannot support claims of improvement or robustness. Moreover, the paper cites three existing HBOS extensions (multi-step histogram scores, subspace histograms, and PCA-based histograms) but does not compare against any of them, so the incremental contribution over the closest baselines is not established. Please add uncertainty estimates and comparisons with at least the most directly related HBOS variants.","section":"Section 4.2, Table 2"},{"comment":"The paper claims that EHBOS has time complexity O(n*d^3), but the algorithm computes one 2D histogram for each of the d(d-1)/2 feature pairs, which gives O(n*d^2) after the per-pair binning. The text in Section 4.3 also says 'scaling quadratically in O(n*d^3)', which is internally inconsistent. The complexity claim should be corrected to O(n*d^2), and the empirical scaling plot in Figure 6 should be redrawn with the correct theoretical cost metric.","section":"Section 4.3 and Section 3, complexity claim"}],"minor_comments":[{"comment":"Several equations use corrupted mathematical notation (e.g., X∈R!×#, h$+x%,$-, s%(3-), O(n⋅d3)), making it difficult for a reader to parse the exact definitions. Please use standard typography for all formulas.","section":"Section 3, formulas"},{"comment":"Figures 3 and 4 are referenced as showing comparisons for both example datasets, but the text only describes one figure at a time and refers to 'the second example dataset' without explicitly distinguishing the two figures. Please make the figure references and captions clearer.","section":"Section 4.1, Figures 3 and 4"},{"comment":"For the lympho dataset, EHBOS drops from a perfect HBOS AUC of 1.0000 to 0.8556; this is a substantial degradation, not a negligible one, and it deserves explicit discussion rather than the blanket statement that underperformance is 'slight'.","section":"Section 4.2, Table 2"},{"comment":"The text says the runtime evaluation uses a fixed bin-width setting, whereas the earlier experiments use the Birgé-Rozenholc adaptive binning. Please clarify whether the runtime experiments use a different binning protocol and whether this affects the complexity comparison.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a Version of Record in the Journal of Statistical Theory and Practice, so the issues raised here may have already been through a review cycle. However, the empty-bin and normalization problems in Section 3 are fundamental enough that I would have expected them to be addressed before publication. The editor may want to consider whether a correction or an appendix specifying the implementation details is feasible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is simple: take HBOS's univariate scores, add pairwise 2D histogram scores, normalize both, and average. That is a natural extension, and the paper is honest that it is an extension rather than a new paradigm. The two synthetic examples clearly illustrate why HBOS misses contextual outliers, and the 2D score does fix those cases. The empirical section compares against plain HBOS on 17 datasets with adaptive binning, and the paper cites the closest prior work (Sathe & Aggarwal, Aryal et al.) rather than ignoring it. So the author knows the literature and the positioning is fair.\n\nThe soft spots are real but concentrated. The load-bearing issue is that the 2D score in Section 3 is defined as -log h_jk, with no rule for empty bins. If a sample lands in an empty cell, its score is +infinity, which makes the averaged final score infinite and ROC AUC undefined. On high-dimensional datasets like mnist (d=100), optdigits (d=64), and musk (d=166), most pairwise bins will be empty under any fixed binning, so the finite numbers in Table 2 require an undocumented convention. The fact that EHBOS does much worse than HBOS on exactly those high-dimensional datasets suggests the empty-bin handling is not a minor detail but is affecting the results. This needs to be specified and the experiments re-run with a stated pseudocount, floor, or skip rule.\n\nThe secondary issues are thinner evidence: no error bars, no significance tests, no code, and no comparison to the subspace-histogram or PCA-based HBOS variants. The complexity notation is also sloppy: it says O(n·d^3) but all feature pairs are O(n·d^2); the text calls it quadratic, which is right, so d^3 is a typo. The arbitrary 0.5 weighting is a design choice, not a flaw.\n\nWho is this for? Practitioners who want a quick, interpretable way to add pairwise interactions to HBOS. The method is well motivated and the idea is sound, but the current paper does not provide enough detail to reproduce the claimed improvements. I would not cite it as a reliable empirical result until the empty-bin issue is resolved, but I would reference it as a related extension.\n\nMy recommendation: this deserves a serious referee rather than a desk reject, but only with major revisions. The author should specify the empty-bin handling, provide code or at least a precise algorithmic description, and add confidence intervals or significance tests on the key comparisons. Without those, the central empirical claim is not supported.","headline":"EHBOS is a plausible but thinly supported incremental extension of HBOS; the missing empty-bin rule for 2D histograms makes the reported AUC gains unverifiable as stated.","tokens_in":7762,"tokens_out":1670,"would_cite":false,"duration_ms":20219,"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":"EHBOS detects dependency-driven outliers by averaging normalized 1D and 2D histogram scores, beating HBOS on glass, cardio, and satellite.","keywords":["outlier detection","anomaly detection","histogram-based outlier score","HBOS","EHBOS","two-dimensional histogram","feature dependencies","ROC AUC"],"falsifier":"Take a two-feature dataset where a known outlier lies in an empty 2D bin while all normal points have nonzero bin counts: if EHBOS gives that outlier an infinite or default-capped score, the result is decided by the binning grid rather than by dependency structure. Alternatively, rerun the reported mnist and optdigits experiments with an explicit empty-bin rule such as Laplace smoothing, a floor density, or skipping empty bins; if the drops from 0.6140 to 0.4647 and from 0.8279 to 0.6267 change materially, the averaging claim is not robust to binning artifacts.","tokens_in":6802,"feed_emoji":"📊","tokens_out":11424,"duration_ms":103047,"temperature":0.7,"pith_summary":"The paper proposes EHBOS, a variant of the histogram-based outlier score HBOS. HBOS scores each feature independently, so it misses points that are anomalous only in how features combine. EHBOS adds a two-dimensional histogram for every feature pair, normalizes the resulting score, and averages it with the normalized HBOS score. On 17 benchmark datasets, EHBOS raises ROC AUC on several sets where feature interactions matter, including glass, cardio, and satellite, while staying close to HBOS on most others. The payoff is a cheap, interpretable detector that catches contextual and dependency-driven anomalies without fitting a model.","feed_headline":"Pairing features catches outliers single-feature HBOS misses","feed_subtitle":"EHBOS averages 1D and 2D histogram scores, lifting ROC AUC on glass, cardio, and satellite.","key_machinery":"The load-bearing object is the EHBOS score, the average of a normalized one-dimensional histogram score and a normalized two-dimensional histogram score. The 2D histogram of a feature pair estimates joint density by counting points in rectangular bins and dividing by bin area; taking the negative logarithm turns sparse joint regions into large outlier contributions, and summing over all pairs lets the score respond to deviations in the relationship between two features rather than in either feature alone.","core_discovery":"The central claim is that $s_{\\mathrm{EHBOS}} = (s_{\\mathrm{1D}} + s_{\\mathrm{2D}})/2$ identifies outliers that HBOS misses, specifically when anomalies are defined by pairwise feature interactions. EHBOS computes the usual per-feature density estimates as HBOS does; for every pair $(j,k)$ with $j<k$ it builds a two-dimensional histogram, converts low joint density into high scores via negative log density, and accumulates normalized pairwise scores. In the reported experiments EHBOS beats HBOS in ROC AUC on glass (0.8515 vs. 0.7003), cardio (0.9037 vs. 0.8511), satellite (0.7886 vs. 0.7516), satimage-2, shuttle, and vowels, while losing on lympho, mnist, and optdigits. The paper reads the gains as evidence that pairwise dependence carries anomaly structure that univariate histograms cannot see, and it reports empirical runtimes consistent with $O(n d^2)$ complexity.","pith_inferences":["A direct extension would replace the fixed equal-weight average with learned, rank-based, or weighted fusion of the 1D and 2D scores; the paper does not test whether the equal-weight average is optimal.","The same 1D-plus-2D averaging idea transfers to other density estimators, such as kernel density estimates, which would soften the empty-bin problem at higher computational cost.","On high-dimensional datasets like mnist, optdigits, and musk, most pairwise bins are likely empty; an explicit empty-bin policy could change rankings substantially, meaning the reported underperformance may be a binning artifact rather than a property of the averaging.","Selecting feature pairs before building 2D histograms could recover some of the $O(n d^2)$ cost and focus scoring on genuinely dependent pairs; the paper lists adaptive feature selection as a future direction."],"forward_implications":["On datasets whose anomalies are defined by feature interactions, EHBOS assigns higher outlier scores than HBOS, as shown on the two synthetic examples and on glass, cardio, and satellite.","EHBOS's ROC AUC gains come at a computational cost of $O(n d^2)$ rather than HBOS's $O(n d)$, so the method is most practical when the number of features is moderate.","The method stays close to HBOS on datasets where anomalies are mostly univariate, with small differences on pima, pendigits, and wbc, and it degrades on lympho, mnist, and optdigits.","The 1D/2D mixture can be treated as a tunable hyperparameter, so a user can weight the pairwise component more heavily when dependencies are known to matter.","Because EHBOS is built from counting and binning, it remains interpretable and requires no training, fitting the same niche as HBOS."],"supporting_citations":[{"why":"Defines the HBOS algorithm that EHBOS extends and provides the 1D score that forms half of the final average.","marker":"Goldstein & Dengel, 2012"},{"why":"Supplies the adaptive bin-count rule used to construct both the 1D and 2D histograms in every experiment.","marker":"Birgé & Rozenholc, 2006"},{"why":"The closest prior multi-step HBOS extension; the paper cites it to frame the open problem of dependency-aware histogram scoring that EHBOS addresses.","marker":"Aguilera-Martos et al., 2023"}],"fun_headline_variants":["2D histograms boost outlier detection beyond HBOS","EHBOS pairs features to spot hidden anomalies","Pairwise histograms catch outliers HBOS overlooks","Extended HBOS: dependency-aware outlier scoring","Feature pairs unlock anomalies univariate misses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes every two-dimensional histogram bin it scores has at least one data point; the paper never says what happens to the score when a bin is empty, and an empty bin would make the log-density term infinite and dominate the average.","fun_headline_variants_meta":{"raw":{"variants":["2D histograms boost outlier detection beyond HBOS","EHBOS pairs features to spot hidden anomalies","Pairwise histograms catch outliers HBOS overlooks","Extended HBOS: dependency-aware outlier scoring","Feature pairs unlock anomalies univariate misses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1420,"prompt_tokens":941,"completion_tokens":479,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":423}},"tokens_in":557,"tokens_out":479,"duration_ms":4750,"temperature":1.0,"reasoning_tokens":423,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:12:27.202367+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-feature dataset where a known outlier lies in an empty 2D bin while all normal points have nonzero bin counts: if EHBOS gives that outlier an infinite or default-capped score, the result is decided by the binning grid rather than by dependency structure. Alternatively, rerun the reported mnist and optdigits experiments with an explicit empty-bin rule such as Laplace smoothing, a floor density, or skipping empty bins; if the drops from 0.6140 to 0.4647 and from 0.8279 to 0.6267 change materially, the averaging claim is not robust to binning artifacts.","supporting_citations":[],"review_version":1}