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Machine learning for journal entry testing: A type-aware evaluation of anomaly detectors under a review budget

Jan Gronewald, Michel Scherer, Nijat Mehdiyev

arXiv:2610.06133v1cs.LGcs.AI

TL;DR

Conventional anomaly-detector evaluations do not account for the review budget or which anomaly types reach the auditor. This paper evaluates rankings by type under a budget and finds that the type-aware view changes the top-ranked detector on three of four client ledgers.

  • Problem

    Conventional measures such as ROC-AUC and precision treat anomalies as exchangeable, while a ledger ranking can contain multiple anomaly types relevant to different audit risks.

    Source: §1 · p. 2

  • Method

    The paper evaluates budgeted rankings using per-type recall, fair-share type recall, type coverage and first-hit rank across client and public ledgers.

    Source: §1 · p. 3

  • Results

    Type-aware evaluation changes the top detector on three of four client ledgers; on the largest, kNN reaches FSR@100 of at least 0.284 against at most 0.226 for hit leaders.

    Source: §1 · pp. 3, 5

  • Takeaways & Limitations

    High hit rates can conceal types left untested, so the types reached within the review budget matter alongside anomaly counts.

    Source: §1 · p. 5

    Client-ledger labels are injected and genuine anomalies are unlabeled, making client-ledger precision a lower bound and limiting how directly injected types represent real misstatements.

    Source: §6.4 · p. 41

Abstract

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Journal entry anomaly detectors are commonly evaluated on the full population with ROC-AUC, precision and recall, ignoring the review budget and which anomaly types are found. We propose a type-aware evaluation combining per-type recall, fair-share type recall (FSR), which caps each type's credit at its budget share, type coverage and first-hit rank. We evaluate nine unsupervised detectors, a supervised reference and feedback-driven Deep Semi-Supervised Anomaly Detection (DeepSAD) on four real client ledgers with injected typed anomalies and a public synthetic ledger. On the largest client ledger, principal component analysis (PCA), an autoencoder (AE) and a variational autoencoder (VAE) each place on average 98 anomalies among the first 100 postings, but at least 95.8 belong to one type. FSR instead favours a nearest-neighbour (kNN) detector and changes the top-ranked detector on three of four client ledgers. Representation also matters: one-hot encoding exposes unseen accounts, whereas frequency encoding leaves unseen contra accounts largely undetected. On the public ledger, the Histogram-Based Outlier Score (HBOS) and Empirical Cumulative Distribution-Based Outlier Detection (ECOD) reach all eight markings within 1,386 entries, whereas kNN, the hit leader at 1,000 entries, first reaches cross-linked clearing at rank 4,641, and the supervised row-level reference misses this marking within 1,000 entries. There, the adaptive DeepSAD review protocol raises mean hits per 100 reviews from 40.0 to 68.3 but type coverage only from 2.7 to 3.0. These findings show that high hit rates can conceal systematic blind spots and suggest that feedback can reinforce existing detection patterns without broadening anomaly coverage.

1. Introduction

Journal-entry anomaly rankings are reviewed under a limited budget, but conventional measures do not show which anomaly types reach the auditor. The paper proposes a type-aware evaluation and finds that it can change which detector ranks highest.

1. Introduction

  • Conventional anomaly-ranking measures count anomalies without distinguishing types, although different types can indicate different audit risks.The first k ranked entries are what the auditor can examine, making the review budget central to evaluation.Source: §1 · pp. 2–3
  • The proposed evaluation combines per-type recall, fair-share type recall, type coverage and first-hit rank to assess rankings under a review budget.It can also incorporate risk weights and uses only detector scores and anomaly types.Source: §1 · p. 3
  • The benchmark combines typed anomalies injected into four real client ledgers with eight generator-defined markings on a public synthetic ledger.It compares nine unsupervised detectors, a supervised LightGBM reference and feedback-driven DeepSAD.Source: §1 · pp. 3–4
  • The study asks whether type-aware rankings alter hit-based comparisons, which types reach reviewers, how evaluation conditions matter, and whether feedback broadens type coverage.Its contributions include the fair-share normalization and evidence on model selection across the benchmark.Source: §1 · pp. 4–5
  • Type-aware FSR changes the top detector on three of four client ledgers; on the largest, kNN reaches FSR@100 of at least 0.284 against at most 0.226 for hit leaders.PCA, AE and VAE each place an average of 98 anomalies in the first 100 postings, at least 95.8 of one type.Source: §1 · pp. 3, 5

2. Related work

Prior work studies anomaly detectors, journal-entry testing, expert feedback and diversity-aware ranking, but does not evaluate which anomaly types reach a fixed review budget in a mixed ranking. The paper adapts ranked-list diversity measures to that setting.

2. Related work

  • The related work spans four areas: detector evaluation, journal-entry machine learning, expert feedback and diversity-aware ranked-list evaluation.The paper positions its contribution as evaluating which anomaly types reach a fixed review budget.Source: §2 · p. 6, §2.4 · p. 8

2.1. Evaluating anomaly detectors

  • Benchmarks show detector rankings depend on anomaly type, but typically score whole rankings or datasets containing only one type.Standard multi-detector comparisons also use mean ranks with Friedman and Nemenyi tests.Source: §2.1 · p. 6

2.2. Machine learning for journal entry testing

  • Journal-entry studies have applied outlier detection, autoencoders and other machine-learning approaches, but typically score whole rankings rather than the types in a fixed review list.The public synthetic ledger provides shared data with eight anomaly markings, though it was not created by an independent group.Source: §2.2 · pp. 6–7

2.3. Learning from expert feedback

  • Expert-feedback methods use labels from selected top-ranked points, which are unrepresentative; whether they broaden the types found was unmeasured.Prior work notes that feedback labels concentrate on points the ranking already places first.Source: §2.3 · pp. 7–8

2.4. Diversity- and coverage-aware evaluation

  • Ranked-list diversity measures cover different query intents, but audit evaluation must account for full-list review, unequal type sizes and risk-based weights.The paper’s contribution is a budget-constrained type-aware evaluation, including fair-share normalization for mixed rankings.Source: §2.4 · p. 8

3. Type-aware budget evaluation

The evaluation reframes anomaly detection around what an auditor can inspect, combining hit yield with type-specific breadth, balance, and exposure order. It is model-agnostic and requires only detector scores and typed anomaly labels.

3. Type-aware budget evaluation

  • The framework formalizes what a review list contains and applies to unsupervised, supervised and feedback-driven models using scores and typed anomaly labels.It is an evaluation criterion, not a training objective.Source: §3 · p. 9, §3.1 · p. 9

3.1. Hits, types and the budget ceiling

  • Precision at k counts anomalies among reviewed units, per-type recall shows each type’s share of anomalies found, and type coverage records categories seen at least once.Per-type recall is read against the maximum that the review budget can hold for that type.Source: §3.1 · pp. 9–10

3.2. Fair-share type recall

  • FSR@k credits each type’s hits only up to its water-filled fair-share cap, redistributing unused shares from smaller types; a full score requires every type to receive its fair share.When the budget covers all anomalies, FSR@k is the weighted mean per-type recall; when all types have at least k/T anomalies, equal-weight FSR@k discounts hits beyond k/T per type.Source: §3.2 · pp. 10–11

3.3. First-hit rank and a size-scaled budget

  • First-hit rank records units reviewed until each type’s first anomaly; k=ρN scales the review budget to ledger size.Full rankings are available for the public ledger, while retained client-ledger results provide per-type recalls only at k=100.Source: §3.3 · pp. 11–12

3.4. Relation to existing measures

  • Intent-aware precision preserves P@k’s ordering, while macro recall favors smaller types and has attainable values that vary with budget and type sizes.With disjoint types and equal weights, intent-aware precision equals P@k/T and cannot reveal concentration.Source: §3.4 · p. 12
  • Unlike α-nDCG’s position discounts or binary type coverage, FSR caps over-representation and adjusts for type size; first-hit ranks add review order.FSR is not a replacement for the other measures, and it reduces to macro recall when the budget covers every anomaly.Source: §3.4 · pp. 12–13

3.5. Aggregation over seeds and uncertainty

  • FSR of seed-averaged counts is at least mean per-seed FSR, while seed-mean coverage overstates typical coverage; client-ledger FSR therefore has ledger-dependent interpretation.Public-ledger FSR is averaged per seed; client-ledger estimates equal expected single-run FSR on ledgers A and C and bound it from above on B and D.Source: §3.5 · pp. 13–14

3.6. Reading a type profile

  • P@k, FSR@k, TC@k and first-hit ranks are complementary measures of yield, within-budget balance, breadth and exposure speed, not one score.Equal weights are a risk-neutral benchmarking convention, not a claim that types matter equally in audits; risk weights can reflect an engagement’s assessed risks.Source: §3.6 · pp. 14–15

4. Benchmark

The benchmark combines typed anomaly injections in real client ledgers with a public synthetic ledger, then evaluates unsupervised, supervised-reference, and feedback-driven review regimes under a shared protocol. Its data and design cover different ledger settings while fixing detector choices and evaluation procedures in advance.

4. Benchmark

  • The benchmark pairs typed injections in four real client ledgers with eight generator-labelled markings in a public synthetic ledger, comparing three learning regimes.The client-ledger core grid uses 1.5% contamination and medium difficulty; the public ledger is independently generated.Source: §4 · p. 15, §4.1 · p. 17, §4.2 · p. 17

4.1. Client ledgers and typed injection

  • Client ledgers range from 3,461 to 279,975 postings; injected anomalies vary across five types, contamination rates, and difficulty levels.Each injection modifies one sampled posting in place, and no posting carries more than one type.Source: §4.1 · pp. 15, 17

4.2. Public synthetic ledger

  • The public ledger contains 102,429 journal entries, including 850 generator-labelled entries across eight markings; entries are ranked by their most suspicious line.The fixed review budget is 100, with k = 1,000 representing about 1% of entries.Source: §4.2 · pp. 17, 19

4.3. Unsupervised detectors and the supervised reference

  • Nine unsupervised detectors span isolation, local density, marginal rarity, subspace, reconstruction, and entity-embedding approaches; LightGBM provides a label-trained reference.Ebase uses one-hot and standardised features, while AE-emb uses learned categorical embeddings; the supervised reference uses Ebase.Source: §4.3 · pp. 19, 21, Table 4 · p. 20

4.4. Review protocol with auditor feedback

  • DeepSAD adaptively reranks after four rounds of 25 reviewed units, using revealed labels to retrain and compare against the static AE-emb ranking.A public-ledger no-feedback control separates the combined effects of label feedback and the change in scoring approach.Source: §4.4 · pp. 21–22

4.5. Protocol and reproducibility

  • Detector settings and evaluation procedures were fixed in advance, with client-ledger results rerun independently and neural-detector means differing by up to 0.09.The design is transductive, and client-ledger type coverage is not reported.Source: §4.5 · p. 22

5. Results

Across the results, hit-based leaders vary by ledger, while type-aware measures change the preferred detector on three of four client ledgers. On the public ledger, broader marking coverage and first-hit ranks diverge from hit counts, and adaptive reviews increase hits more than coverage.

5.1. The conventional view

  • kNN leads P@100 on ledgers C, A and B at 0.40, 0.29 and 0.42, while PCA, AE and VAE each place 98 anomalies in ledger D’s first 100 postings.The supervised reference reaches P@100 of 0.97 or more on ledgers B and D and 0.83 on the public ledger.Source: §5.1 · p. 23, Table 5 · p. 24

5.2. The type-aware view on the client ledgers

  • kNN recovers extreme amounts and payment periods on ledger C; PCA, AE and VAE recover unseen accounts, but every detector family largely misses unseen contra accounts.On ledger B, PCA, AE and VAE find almost nothing beyond unseen accounts, with recall at most 0.03 on the other four types.Source: §5.2 · p. 25
  • Type-aware ranking changes the hit leader on three of four client ledgers, favoring reconstruction models on two smaller ledgers and kNN on the largest.On the largest ledger, PCA, AE and VAE concentrate hits on unseen accounts, while kNN spreads hits across types.Source: §5.2 · pp. 25, 27, 29, Table 6 · p. 27
  • On ledger D, macro recall keeps PCA, AE and VAE ahead, but fair-share caps place kNN first by limiting credit for concentrated hits.On ledgers C and A, where the budget can hold every anomaly, FSR coincides with macro recall.Source: §5.2 · p. 29

5.3. The budget on the public ledger

  • HBOS and ECOD show all eight public-ledger markings within 1,386 and 1,103 entries; kNN first reaches M8 at rank 4,641, while unsupervised recall for M3, M6 and M8 is at most 0.05 within 1,000 entries.For HBOS, ECOD, PCA, AE and VAE, 96 to 99 of the first 100 entries touch rare accounts but are legitimate.Source: §5.3 · p. 30, Figure 4 · p. 31

5.4. Contamination and difficulty

  • At 3 percent contamination, kNN’s P@100 falls from easy to hard difficulty by 0.11 on C, 0.24 on A and 0.36 on B; it leads 24 of 27 sweep cells.The per-type effects of contamination and difficulty were not tested because per-type recalls were retained only for the core grid.Source: §5.4 · p. 32, Figure 6 · p. 34

5.5. Auditor reviews and the type gaps

  • On the public ledger, DeepSAD reaches 68.3 anomalies per 100 reviews versus 40.0 for static AE-emb, while type coverage rises only from 2.7 to 3.0 markings.The control ends at 40.3 anomalies; the authors read it as evidence that labels matter, not as an exact decomposition of the gain. No reviewed entry in any seed is M2, M3, M6 or M8.Source: §5.5 · pp. 35, 37, Table 8 · p. 36

5.6. Robustness

  • kNN leads every public-ledger resample at P@1,000 and FSR@1,000, but its comparison with AE-emb at k = 100 is unresolved; moderate alternative type weights preserve many, but not all, reversals.Across 300 random weight vectors, reversals hold in 91, 89 and 63 percent on ledgers C, A and D, and appear in 26 percent on B.Source: §5.6 · pp. 37–38

6. Discussion

The discussion shows that detector rankings depend on whether evaluation counts hits or assesses anomaly types, and relates those differences to model bias, data representation and expert-label sampling. It draws implications for audit testing while qualifying the evidence by its data and design limits.

6.1. What the type-aware view changes

  • On three of four client ledgers, the highest-FSR detector differs from the hit leader; on the public ledger, hit and coverage rankings at 1,000 entries are unrelated.On ledger D, only fair-share normalization reveals the reversal; on the public ledger, detectors with fewer hits can reach all markings sooner.Source: §6.1 · pp. 38–39

6.2. Learning mechanisms behind the results

  • At least 96 of the first 100 public-ledger entries ranked by HBOS, ECOD, PCA, AE and VAE are normal entries on rarely used accounts.Whether rarity explains the near-chance precision of several detectors on client ledgers remains untested because the contents of their top ranks were not examined.Source: §6.2 · p. 39
  • One-hot encoding lets PCA, AE and VAE find every unseen account within 100 postings on ledgers C, A and B, while AE-emb recalls at most 0.20 for that type.No detector finds unseen contra accounts, which have no dedicated column in the representation.Source: §6.2 · p. 39
  • On the public ledger, every unsupervised detector has recall at most 0.05 within 1,000 entries for delayed, weekend and cross-linked entries, whose evidence lies in document-chain relations.The row-level supervised reference also misses every cross-linked entry, while a post-hoc relational diagnostic places them all within its first 1,000.Source: §6.2 · pp. 39–40
  • On the public ledger, auditor labels add an average of 28 anomalies over the no-label protocol, but reviewed lists reach no more markings with labels than without: 3.0 versus 3.3 on average.With labels, 55 to 60 of 64 to 71 reviewed hits fall on one marking already present in the starting list; the authors limit this evidence to three seeds on one ledger.Source: §6.2 · p. 40

6.3. Implications for AI-based journal entry testing

  • The authors recommend judging review lists by their labelled anomaly-category profile alongside hits, scaling budgets to ledger size, and selecting models by the risks they cover.They did not test model combinations and recommend relational features for relational risks, while retaining document-chain rule-based tests until detectors have those features.Source: §6.3 · pp. 40–41

6.4. Limitations

  • Evidence is limited to four client ledgers with one fiscal year each and structured fields, plus a synthetic public ledger that is not an independent replication.Client data cannot be shared; client-ledger feedback gains cannot be separated from retraining because the no-feedback control exists only for the public ledger.Source: §6.4 · pp. 41–42

7. Conclusion

The conclusion argues that journal-entry anomaly evaluation should account for which categories reach the review list, not just the number of hits. Its findings describe changed rankings, relational blind spots and feedback that mostly increases hits in already-detected types.

7. Conclusion

  • The type-aware view changes the top-ranked model on three of four client ledgers for their seed-averaged type profiles.On the largest ledger, only fair-share normalization reverses the hit-based ranking; macro-averaged recall would leave the hit leaders on top.Source: §7 · p. 42
  • On the public ledger, detectors with few hits reach all anomaly types sooner than the hit leader, while one relational anomaly type escapes row-level learning even with labels.Rarity-based models also rank legitimate rare entries highly, and adaptive auditor feedback mostly multiplies hits in types already found.Source: §7 · p. 42
  • The authors recommend reporting a profile of labelled anomaly categories alongside hit counts and treating representation and exploration of unseen types as design concerns.The risk interpretation applies where anomaly categories map to engagement risks.Source: §7 · p. 42

CRediT authorship contribution statement

The authors report their individual contributions to the study.

CRediT authorship contribution statement

  • Jan Gronewald contributed to conceptualization, methodology, software, data curation, resources, visualization, writing, project administration and funding acquisition.Michel Scherer contributed to software, validation and investigation; Nijat Mehdiyev contributed to conceptualization, methodology, formal analysis and writing.Source: CRediT authorship contribution statement · p. 43

Declaration of competing interest

The authors disclose no known competing interests beyond the stated co-authorship connection to the dataset.

Declaration of competing interest

  • Jan Gronewald co-authored the synthetic ledger dataset used in the study; the authors declare no other known competing financial interests or personal relationships that could have influenced the paper.Source: Declaration of competing interest · p. 43

Appendix A. Notation

The notation treats postings and journal entries as the ranked evaluation units, and its type measures apply to labelled anomaly categories while representing audit risks only when an engagement maps risks to them.

Appendix A. Notation

  • A posting is the unit on client ledgers, while a journal entry is the unit on the public ledger.Measures use detector-induced rankings, with ties broken by ascending file position.Source: Appendix A · p. 44
  • Type symbols apply to injected client types T1–T5 and generator-defined public markings M1–M8, both treated as labelled anomaly categories.Source: Appendix A · p. 44
  • The categories measure audit risks only where an engagement maps its risks to them.Source: Appendix A · p. 44

Appendix B. DeepSAD in the review protocol

The review protocol adapts DeepSAD from the AE-emb representation, using reviewed labels to move units toward or away from a centre and retraining between review rounds. The appendix documents its initialization, label handling, scoring and public-ledger controls.

Appendix B. DeepSAD in the review protocol

  • DeepSAD starts from the same-setting, same-seed AE-emb embedding and encoder weights, making the first 100 AE-emb units the static-ranking comparison for reviews.The centre is the mean initial code, with small coordinates pushed away from zero.Source: Appendix B · p. 44
  • Unlabelled and reviewed normal units are pulled toward the centre, while reviewed anomalies are pushed away.The method scores units by their distance from the centre.Source: Appendix B · p. 46
  • Later rounds retrain from the initial weights using only revealed labels; public-ledger entry scores take the maximum score over an entry’s lines.The no-feedback control withholds labels but still removes reviewed units from the ranking.Source: Appendix B · p. 46
  • Across public-ledger seeds, one marking accounts for most reviewed hits, more so with labels, and no run reviews M2, M3, M6 or M8.The control’s reviewed-hit count moves in either direction depending on the seed.Source: Appendix B · p. 46, Table B.1 · p. 46

Appendix C. Bootstrap intervals on the public ledger

Paired bootstrap resampling on the public ledger supports kNN’s lead at the 1,000-unit budget, but not a separated lead over AE-emb or LOF at 100. Seed variation can exceed the resampling uncertainty, and the analysis is limited to the public ledger.

Appendix C. Bootstrap intervals on the public ledger

  • At k = 1,000, kNN leads every other unsupervised detector on P@k and FSR@k in all 1,000 paired resamples.The resamples draw journal entries with replacement and use percentile 95 percent intervals.Source: Appendix C · p. 47
  • At k = 100, kNN is not separated from AE-emb or LOF because their paired intervals contain zero, though it is separated from the other six detectors.Source: Appendix C · p. 47
  • AE-emb’s P@100 ranges from 0.19 to 0.64 across three seeds, a wider variation than any bootstrap interval.Its lead in the three-seed mean therefore reflects seed variation as well as sampled entries.Source: Appendix C · p. 47
  • The bootstrap is confined to the public ledger because per-seed and per-unit client-ledger results needed for paired resampling were not retained.Source: Appendix C · p. 47

Appendix D. Sensitivity to the type weights

The sensitivity analysis finds that moderate changes to type weights leave major reversals intact, although discounting the type that separates two detectors can change their preference. The supplement also records the detailed results, implementation protocol and public-ledger field mapping.

Appendix D. Sensitivity to the type weights

  • Moderate departures from equal type weights leave the main FSR reversals in place, but discounting the type separating two models can change which is preferred.On ledger A the reversal survives every structured scheme; on ledger D it is fixed by bounds in nine of eleven schemes, while ledger B has reversals toward VAE in two schemes.Source: Appendix D · p. 50

Supplementary material for: Machine learning for journal

S1. Protocol details and deviations

  • The protocol fixes injection, encodings, detector settings and measures before runs; each seed uses a shared random posting permutation, with T2–T5 assigned fixed shares and T1 receiving the remainder.No posting carries two types; T1 and T2 set amount and payment-period modified z-scores to multiples of the ledger’s 99.9th percentile, while T3 and T4 draw fresh identifiers and T5 draws account pairs that occur often but never together.Source: S1. Protocol details and deviations · pp. 57–58
  • The supplement records deviations and omissions, including no optional contextual extreme-amount type or random-label sensitivity analysis, and a client-ledger sweep limited to C, A and B.For client-ledger type-aware measures, seed-mean per-type recalls printed to two decimals were used because per-seed values were not retained.Source: S1. Protocol details and deviations · p. 58

S2. Field mapping of the public ledger

  • Public-ledger features map client fields, deriving contra account from the largest-amount opposite-side line and converting payment period to entry lag in days.Rule attributes, labels, markings, identifiers, free text and account names are excluded from feature sets.Source: S2. Field mapping of the public ledger · p. 58

S3. Cross-machine reproduction

A second analyst’s complete benchmark run broadly reproduces the first, with exact agreement for most compared cells and stable type-aware rankings despite neural-model variation.

S3. Cross-machine reproduction

  • The comparison covers 648 cells, excludes ledger D and one ledger B sweep cell, and does not compare dataset hashes byte by byte.The runs used different numbers of seeds for the excluded cells; data audits agree on row counts, distinct values, percentiles and account-pair pools.Source: S3. Cross-machine reproduction · pp. 58–59
  • 435 of 435 compared cells agree for IF, LOF, kNN, HBOS, ECOD, PCA and LightGBM, while AE, VAE and AE-emb agree in fewer than half of theirs.AE and VAE move by at most 0.06; AE-emb moves by up to 0.09 in a core-grid mean and 0.10 in a per-type recall.Source: S3. Cross-machine reproduction · p. 59
  • Network training is probably not bitwise reproducible across processors and linear algebra libraries, although it is reproducible on one machine.The authors identify this as the probable cause of the neural-model differences.Source: S3. Cross-machine reproduction · p. 59
  • FSR@100 selects the same highest-ranked detector on ledgers C, A and B in both runs, and PCA ranks above kNN on C and A.The ranking comparison uses per-type recalls from the earlier run; PCA reproduces exactly.Source: S3. Cross-machine reproduction · p. 59

S4. Runtimes

Fit times vary by ledger and detector, with some methods taking substantially longer on ledger D. Neural-network runtimes on ledger D and the public ledger are not directly comparable with those on ledgers C, A and B because training settings differ.

S4. Runtimes

  • LOF and kNN take about 17 minutes each on ledger D, and AE-emb about 86 minutes; on the public ledger, LOF and kNN take about eight minutes and every other detector less.Every fit on ledgers C, A and B takes under three minutes.Source: S4. Runtimes · p. 59
  • Fit times are measured on two CPU cores with 7 GB RAM and no GPU; client-ledger times use five seeds on C, A and B and three on D.Public-ledger times average runs, using three seeds for stochastic detectors and the supervised reference.Source: Table S10 · p. 69
  • Neural-network times on ledger D and the public ledger are not comparable with C, A and B because they use 30 epochs with batch size 256 instead of 100 epochs with batch size 64.The networks also train for fewer epochs on ledger D and the public ledger.Source: S4. Runtimes · p. 59, Table S10 · p. 69
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