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This paper claims that metric landscapes—maps of AUROC, AUPR, F1, and MCC over the sensitivity–specificity square—reveal how these metrics' values and meanings shift with the anomaly ratio, giving practitioners a reference for comparing sco

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2026-08-01 05:13 UTC pith:NCL4GC3K

load-bearing objection The F1/MCC landscapes and null baselines are genuinely useful, but the AUROC/AUPR landscapes rest on an unstated point-to-curve completion rule, so the paper's central guidance for two of its four metrics doesn't follow from the definitions. the 3 major comments →

arxiv 2607.22286 v1 pith:NCL4GC3K submitted 2026-07-24 cs.LG stat.ML

An Insight on Evaluation Metrics Under the Imbalanced Case of Anomaly Detection

Romain Hermary , Nesryne Mejri , Djamila Aouada This is my paper
classification cs.LG stat.ML
keywords anomaly detectionclass imbalanceevaluation metricsmetric landscapesAUROCAUPRF1-scoreMatthews correlation coefficient
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Anomaly detection evaluates models on severely skewed test sets, so a single metric value like 0.9 AUPR can mean different things depending on how rare anomalies are. This paper tries to make that dependence visible by building 'metric landscapes': surfaces that assign each metric's score to every combination of true-positive rate (TPR) and true-negative rate (TNR), at a given anomaly ratio. Its central claim is that these landscapes show each metric's preference—AUROC and MCC staying centered near random baselines while AUPR and F1 climb with the positive share—and that reading scores through such maps lets practitioners interpret and compare anomaly detection results across datasets with very different imbalance ratios. A sympathetic reader would take away a practical reference: the same score can disguise very different classifier behavior, and the landscapes expose which operating regions each metric rewards.

Core claim

The paper's central discovery is that the four standard metrics live on describable, imbalance-dependent surfaces over the (TPR, TNR) square. On these surfaces, AUROC and MCC remain centered around 0.5 independent of the anomaly ratio, while AUPR and F1 have baselines that rise with the proportion of positives—so a high AUPR or F1 value on a positive-dominant dataset may still be close to random performance. The paper further reports that AUROC and MCC retain near-identical structure across imbalance regimes, that AUPR requires no positive detections up to 0.75 at balance, and that in low-anomaly regimes the metrics decorrelate and reward different operating regions. The authors frame metric

What carries the argument

Metric landscapes—the maps of each metric value over the (TPR, TNR) unit square at a fixed imbalance ratio—are the central object. They are computed for AUROC, AUPR, F1, and MCC across ratios from 99:1 to 1:99 and built on a null model of random predictions (100 runs per ratio) to establish baselines. The landscape construction lets the authors compare regions of interest, measure minimal TPR/TNR needed to reach target scores, and run factor analysis on the landscapes to quantify how metrics correlate under low, medium, and high imbalance.

Load-bearing premise

The paper's landscape conclusions for AUROC and AUPR rest on the unstated assumption that these curve-based metrics can be evaluated at a single (TPR,TNR) operating point; if that single-point assignment does not match the standard integral definitions, the AUROC/AUPR landscape values are not the standard metrics.

What would settle it

Take a classifier with known TPR and TNR, construct the full ROC and PR curves (e.g., a single threshold point with linear interpolation), compute the textbook AUROC and AUPR integrals, and compare them to the values the paper's landscapes assign at those coordinates; disagreement would show the landscape's AUROC/AUPR surfaces are not the standard metrics.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the landscapes are right, a given F1 or AUPR value should always be read against the anomaly ratio of the test set: values near 0.7 can be random in positive-dominant settings but meaningful in negative-dominant ones.
  • AUROC and MCC remaining centered near 0.5 across imbalance means those two are the safer metrics for comparing models across datasets with different anomaly ratios, with AUROC and MCC behaving almost identically in high-positive regimes.
  • The minimal-positive analysis implies that in balanced settings a model can reach AUPR 0.75 without detecting any anomalies, so AUPR gains below that threshold should not be credited to detection.
  • In low-anomaly regimes the four metrics decorrelate, so a single metric cannot proxy for the others; evaluation should report several metrics.
  • As anomaly ratio decreases, reaching a fixed F1 score requires an increasing share of correctly classified negatives, so F1 comparisons across datasets implicitly reward models that get the majority class right.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the underlying single-point assignment for AUROC/AUPR is a piecewise-linear interpolation rather than the standard curve integrals, the paper's quantitative AUROC/AUPR landscape values (e.g., 0.75 baseline at balance) are about a different quantity; the qualitative preference structure may still hold.
  • The landscape framework could be extended to threshold-dependent operating-point selection: practitioners could pick the (TPR,TNR) region their application values and read off the minimal imbalance-ratio-corrected score needed to trust a model.
  • The same landscape construction could be applied to other metrics (e.g., balanced accuracy, G-mean, PR-AUC variants, cost-weighted scores) to check whether their imbalance behavior is similarly structured.
  • A testable extension: on a real dataset, split predictions by anomaly ratio and build empirical landscapes; they should match the paper's synthetic surfaces if the framework captures actual metric behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper analyzes four anomaly-detection evaluation metrics (AUROC, AUPR, F1, MCC) under varying class-imbalance ratios. Its main instrument is the 'metric landscape': a visualization of each metric as a function of (TPR, TNR) over the unit square, plotted at different anomaly ratios. The paper reports random-baseline statistics, landscape-difference plots, region-of-interest averages, and factor analyses. The central claim is that these landscapes provide an intuitive view of metric preferences and stability, and offer practical guidance for interpreting anomaly-detection results across datasets with differing imbalance ratios.

Significance. If the construction were well-defined, the proposed landscapes could serve as a useful visual reference for practitioners. The paper does provide a straightforward empirical characterization of F1 and MCC as pointwise functions of (TPR, TNR), and the random-baseline analysis is reproducible in principle. However, the central tool is fatally under-specified: AUROC and AUPR are defined as integrals over full ROC/PR curves, but the paper plots and interprets them at single (TPR, TNR) operating points without stating any rule for obtaining a scalar value from one point. The reported behaviors are inconsistent with the stated definitions, and no code or computation details are supplied. Consequently, the main results about AUROC and AUPR — including the minimal-positives analysis and the factor-analysis conclusions — do not pertain to the metrics as defined.

major comments (3)
  1. [§III-B, §V-A] The definitions of AUROC and AUPR in §III-B are curve integrals: AUROC = ∫ TPR(FPR^{-1}(x)) dx and AUPR = ∫ PPV(TPR^{-1}(x)) dx. A single operating point (TPR, TNR) does not determine either curve, so the value of AUROC or AUPR at that point is not identifiable. Yet Figures 2, 3, 4, 5, and 8 plot these metrics over the (TPR, TNR) square. The observed features, such as the AUROC baseline along TPR+TNR = 1 (§V-A), are not derivable from the integrals and appear to correspond to an ad hoc surrogate (e.g., (TPR+TNR)/2). The paper never states the interpolation or completion rule that maps a point to a curve. All AUROC/AUPR landscape conclusions are therefore about an unspecified quantity, not about the metrics defined in §III-B.
  2. [§V-C] The 'Minimal Positives Detection' analysis for AUROC and AUPR is built directly on the same undefined pointwise mapping. For instance, the claim that AUPR can reach 0.75 with zero TPR in the balanced case, or that AUROC requires no positive detection up to 0.5, is a property of the hidden interpolation rule, not of the standard AUROC/AUPR definitions. Without an explicit definition of the pointwise surrogate, these results are unreproducible and cannot be used as practical guidance. This is a load-bearing flaw, as the section's title and conclusions explicitly concern AUROC and AUPR.
  3. [§VI and Appendix B] The factor analysis (§VI) and score-coverage analysis (Appendix B) use landscape values as inputs. Since the AUROC/AUPR landscapes are not properly defined, the Pearson-correlation and Tucker-congruence results involving AUROC or AUPR are not meaningful as statements about the standard metrics. This undermines the cross-metric relationship claims, including the reported AUROC–MCC agreement and AUPR–AUROC decorrelation.
minor comments (5)
  1. [§V-D, R5] The text says AUROC and MCC 'remain stably centred around 0' in the 45–55% TPR/TNR region, but the plotted landscapes and region averages (Figure 13) indicate values around 0.5. This appears to be a typographical error and should be corrected.
  2. [Figure 7] The lower-triangle label 'Person's correlation' should be 'Pearson's correlation'.
  3. [§V-A] The phrase 'diagonal (TPR+TNR)/2 = 0.5' is confusing; the baseline line is TPR+TNR = 1, which is equivalent but should be written consistently.
  4. [Figure 1] The x-axis ordering runs from 99:1 to 1:99, while the text says 'as the proportion of positive samples increases (left side of the plot)'. Verify the direction and clarify the axis label.
  5. [Abstract/Introduction] There are minor grammatical issues, e.g., 'this work therefore study how metric values evolve' should be 'this work therefore studies...'.

Circularity Check

1 steps flagged

AUROC/AUPR landscapes are not the defined curve-integral metrics; the minimal-positive results reduce to an unstated pointwise construction.

specific steps
  1. self definitional [§III-B (AUROC/AUPR integral definitions); §V-A (landscapes); §V-C (minimal positives detection)]
    "AUROC:=∫_0^1 TPR(FPR−1(x)) dx ... AUPR:=∫_0^1 PPV(TPR−1(x)) dx ... For AUROC, no positive sample needs to be detected up to a score of 0.5 ... For AUPR, no positive detection is required up to a score of 0.75 in a balanced dataset."

    The paper defines AUROC and AUPR as integrals over entire ROC/PR curves, but then plots and interprets scalar values for each single (TPR,TNR) operating point without specifying how a point determines a full curve. A single operating point is compatible with infinitely many ROC/PR curves and hence with infinitely many AUROC/AUPR values. The 'minimal positives detection' claims in §V-C are therefore not consequences of the §III-B integral definitions; they are consequences of the unstated pointwise completion rule used to build the landscapes. The paper's own baseline observation, 'AUROC and MCC exhibit an approximately symmetric structure, with the baseline behaviour aligned along the diagonal (TPR+TNR)/2 = 0.5', reveals that the plotted AUROC object is a pointwise construction, not the in

full rationale

There is no fitting-to-data circularity and no load-bearing self-citation chain: the null analysis in §IV is descriptive and reproduces known random-baseline properties, F1 and MCC landscapes are genuine mathematical consequences of their confusion-matrix formulas, and the factor analysis is a post-hoc descriptive summary. The central circularity concern is limited to AUROC and AUPR. Because the paper never defines how a single (TPR,TNR) pair determines an entire ROC/PR curve, the landscape values for these two metrics are not the quantities defined in §III-B. Consequently, the practical guidance in §V-C and the coverage analysis in Appendix B inherit an unstated auxiliary construction. This is a partial, definitional circularity affecting the two threshold-independent metrics, while the F1/MCC portions retain independent content. A score of 4 reflects that the central visualization framework is compromised for AUROC/AUPR but is not a global reduction of the paper to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The main analysis rests on two load-bearing choices: (1) the unstated pointwise construction of AUROC/AUPR landscapes, and (2) the representativeness of the imbalance ratios extracted from ten benchmarks. There are no fitted parameters in the mathematical plots; the dataset ratios are inputs. The factor-analysis section adds further methodological assumptions that are not fully documented.

free parameters (1)
  • factor-analysis settings
    Three factors and 90% explained-variance retention are chosen in §VI without a stated criterion; affects the reported Tucker/correlation matrices but not the main landscape claim.
axioms (3)
  • ad hoc to paper AUROC and AUPR are pointwise functions of a single (TPR,TNR) operating point
    Section V-A constructs landscapes without stating how scalar AUROC/AUPR follows from one operating point; under §III-B definitions this is false unless an extra curve-shape assumption is made.
  • domain assumption Imbalance ratios from Table I are representative of real anomaly-detection test distributions
    The paper grounds its guidance in ten dataset ratios, but provides no formal argument that these cover the space of practical AD settings.
  • domain assumption Factor analysis with three factors and orthogonal rotation captures metric relationships
    Section VI reports factor loadings and Tucker coefficients but omits sampling, input matrix, extraction details, and stability checks; conclusions depend on these choices.

pith-pipeline@v1.3.0-alltime-deepseek · 11611 in / 14997 out tokens · 125312 ms · 2026-08-01T05:13:55.844616+00:00 · methodology

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read the original abstract

Anomaly detection is inherently characterised by severe class imbalance, making the interpretation of evaluation metrics challenging. Although metrics such as AUROC, AUPR, F1-score, and MCC are widely used, their values convey different meanings depending on the anomaly ratio. In this work, we analyse the behaviour of those four common anomaly detection metrics under varying levels of imbalance. We focus on the study of metric landscapes, visualisations that relate metric values to true positive and true negative rates, providing an intuitive view of metric preferences and stability. Our analysis offers practical guidance for interpreting and comparing anomaly detection results across datasets with different imbalance ratios.

Figures

Figures reproduced from arXiv: 2607.22286 by Djamila Aouada, Nesryne Mejri, Romain Hermary.

Figure 1
Figure 1. Figure 1: Random baselines are evaluated across different target classes balance. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Metrics landscapes in the balanced case (50:50). Colour follows scores [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Landscape evolution between the balanced (50:50) and negative-majority settings (5:95) shown as the landscapes difference. Surface colour follows [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Minimum TPR (y-axis) required to achieve the target score (x-axis). Right: Corresponding TNR (y-axis) at minimum TPR required to achieve the score (x-axis). Values at the different datasets imbalance ratios are shown in colour, while the balanced case is represented with dashed lines. 4) AUPR: AUPR measures the trade-off between precision and recall across all thresholds, emphasising the quality of p… view at source ↗
Figure 7
Figure 7. Figure 7: Upper triangle: Tucker’s congruence coefficients between metrics factor loadings ( [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Proportion of surface area (%) from landscape corresponding to each [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Landscape evolution (difference) between balanced and positive majority cases. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Metrics landscapes in the negative majority case (5:95). [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: APPENDIX D REGIONS OF INTEREST: COMPLETE STUDY The full plot (all datasets) of the mean values in the five regions of interest is presented in [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: Metrics landscapes in the negative majority case (95:5). [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Landscapes from different ratios superposed on z-axis. Downsampling was applied and sub-regions replaced with dots to improve visibility. [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Average score within 5 different 10% × 10% (TPR×TNR) metric landscapes regions. The evaluated landscapes are taken at the datasets ratios. The radial scale is gamma-corrected (γ = .6). AUROC AUPR F1 MCC 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 3 2 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 UCR (0.19%) ViSA (L) (0.61%) MVTec (L) (3.33%) WaDi (5.77%) PUMP (10.05%) SWaT (12.14%) PSM (27.76%) ViSA… view at source ↗

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