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

Jan Gronewald; Michel Scherer; Nijat Mehdiyev

cs.LGOct 5, 2026

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

Authors: Jan Gronewald, Michel Scherer, Nijat Mehdiyev

Organizations: Saarland University, Campus, 66123, Saarbrücken, Germany · German Research Center for Artificial Intelligence (DFKI), Institute for Information Systems (IWi), Campus D3.2, 66123, Saarbrücken, Germany · Leipzig University, Grimmaische Straße 12, 04109, Leipzig, Germany

Abstract

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.

Figures & tables

Figure 1: Type-aware budget evaluation. Four client ledgers with injected types T1 to T5 and a public ledger with markings M1 to M8 are ranked by nine unsupervised detectors and a supervised LightGBM reference; an auditor reviews the first

k

units. The conventional view counts the anomalies among them, the type-aware view asks which types they are. The example lists are schematic.

Measure	Bound to the budget	Credits breadth over types	Adjusts for type size	Caps over-representation	Order within the budget	Risk weights
$\mathrm{P@}k$	yes	no	no	no	no	no
Intent-aware precision	yes	no (orders as $\mathrm{P@}k$ )	no	no	no	yes
Macro recall at $k$	yes	yes	inversely (favours small types)	only at $n_{t}$	no	possible
$\mathrm{TC@}k$	yes	yes (binary)	not needed	at one hit	no	possible
$\alpha$ -nDCG at $k$	yes	yes	no	soft decay	yes	in intent-aware variant
$\mathrm{FSR@}k$	yes	yes	yes (fair shares)	at the fair share	no (left to $\mathrm{FHR}_{t}$ )	yes

Table 1: Properties of hit-based and type-aware measures for a review budget of

k

units with anomaly types of unequal size. Intent-aware precision is precision under the intent-aware construction of Agrawal et al. (2009) , with anomaly types as intents; macro recall is the mean of

R_{t}\mathrm{@}k

over the types.

	Client ledgers				Public
Property	C	A	B	D	ledger
Scored unit	posting	posting	posting	posting	line
Evaluated unit	posting	posting	posting	posting	entry
Scored rows	3,461	5,080	17,505	279,975	256,232
Evaluated units	3,461	5,080	17,505	279,975	102,429
Fiscal years	1	1	1	1	2

Table 2: Profile of the four client ledgers and the public synthetic ledger. Counts of accounts, contra accounts, tax codes and users are numbers of distinct values. A duplicate row repeats all feature values of another row. Anomalous units are the injected anomalies of the core grid (client ledgers) and the generator-labelled journal entries (public ledger).

(a) Types injected into the client ledgers
Type (Foorthuis subtype)	Modification of a sampled posting	$n_{t}$ : C / A / B / D
T1 extreme amount (Ia)	amount z-score set to $\pm f\,P_{99.9}(\|z\|)$ with a random sign; $f=5,2,1$	22 / 31 / 105 / 1,680
T2 extreme payment period (Ia)	payment period set to $f\,P_{99.9}$ , rows with an observed period only; $f=5,2,1$	10 / 15 / 53 / 840
T3 unseen contra account (IIa)	contra account replaced by a fresh id, one distinct id per row	4 / 6 / 21 / 336
T4 unseen account (IIa)	account replaced by a fresh id, one distinct id per row	4 / 6 / 21 / 336
T5 unusual account pair (Va)	(account, contra account) replaced by a pair that never occurs in the ledger and whose members each occur at least $F$ times; $F=50,20,5$	12 / 18 / 63 / 1,008

Table 3: Anomaly types and markings. (a) Types injected into the client ledgers, with their subtype in the typology of Foorthuis (2021) , the difficulty parameter for the levels easy, medium and hard, and the number

n_{t}

of injected rows in the core grid. (b) Markings of the public ledger, the rule population into which each was injected, and

n_{t}

Detector	Family	Input	Key settings
IF	isolation, tree ensemble	$E_{\mathrm{base}}$	200 trees, 256 samples per tree
LOF	local density	$E_{\mathrm{base}}$	20 neighbours, the row itself excluded
kNN	distance to neighbours	$E_{\mathrm{base}}$	distance to the fifth nearest other row
HBOS	per-field histogram	$E_{\mathrm{base}}$	10 bins, $\alpha=0.1$ , tolerance 0.5
ECOD	empirical tail probability	$E_{\mathrm{base}}$	no free parameters; computed column-chunked, scores identical to the library
PCA	linear projection	$E_{\mathrm{base}}$	components explaining 90 percent of the variance; score = sum of the distances to the retained eigenvectors divided by their explained-variance ratios

Table 4: Detectors, the supervised reference and DeepSAD with their settings, fixed in advance and never tuned on the labels; higher scores mean more anomalous. IF to VAE are PyOD implementations ( Zhao et al., 2019 ) , AE-emb and DeepSAD are our PyTorch code.

E_{\mathrm{base}}

is the one-hot and standardised encoding,

E_{\mathrm{emb}}

the integer-coded input of the embedding autoencoder.

	Client ledgers (five seeds)				Public
Detector	C	A	B	D	ledger
(a) P@100
IF	0.04 (0.01)	0.03 (0.02)	0.04 (0.04)	0.11 (0.07)	0.08 (0.04)
LOF	0.06 (0.03)	0.01 (0.01)	0.00 (0.00)	0.11 (0.03)	0.11
kNN	0.40 (0.02)	0.29 (0.04)	0.42 (0.03)	0.34 (0.04)	0.17
HBOS	0.03 (0.01)	0.02 (0.02)	0.03 (0.03)	0.04 (0.01)	0.04

Table 5: Conventional, hit-based view: P@100 and average precision (AP), mean (standard deviation) over seeds. Client ledgers: core grid, five seeds; supervised reference as a mean only. Public ledger: three seeds for IF, AE, VAE, AE-emb and the supervised reference, one run for the deterministic detectors. Bold: best unsupervised detector per column. The supervised LightGBM reference uses the labels and is shown for comparison only.

Figure 2: Recall within the first 100 postings per injected type, $R_{t}$ @100, on the four client ledgers (core grid, mean over five seeds): (a) ledger C, (b) ledger A, (c) ledger B, (d) ledger D. Column headers give the type and its number $n_{t}$ of injected rows. The framed bottom row is the budget ceiling $\min(1,100/n_{t})$ .

Figure 3: Hits against fair-share type recall per detector: P@

k

on the horizontal and FSR@

k

on the vertical axis; the grey diagonal marks FSR = P. (a) to (d) Ledgers C, A, B and D at

k=100

(seed means; on ledger D, bars show the rounding bounds). (e), (f) The public ledger at

k=100

and

k=1{,}000

. Axis ranges differ between panels.

	Best by hits				Best by FSR@ $k$
Ledger ( $k$ )	Detector	P@ $k$	TC@ $k$	FSR@ $k$	Detector	FSR@ $k$	$\tau_{b}$
C (100)	kNN	0.40	n/d	0.57	VAE	0.65	0.83
A (100)	kNN	0.29	n/d	0.29	AE	0.38	0.65
B (100)	kNN	0.42	n/d	0.31–0.32	kNN	0.31–0.32	0.97–0.99
D (100)	PCA, AE, VAE	0.98	n/d	0.200–0.226	kNN	0.284–0.344	0.598–0.625
Public (100)	AE-emb	0.40	2.7	0.22	AE-emb	0.22	1.00

Table 6: What changes under the type-aware view. Per ledger: the detector with the highest P@

k

, its type coverage TC@

k

and its FSR@

k

; the detector with the highest FSR@

k

; and Kendall’s

\tau_{b}

between the orderings of the nine detectors by P@

k

and by FSR@

k

. Client ledgers: FSR of the seed-averaged type profile, with ranges given by the bounds that the two-decimal rounding implies; public ledger: mean of the per-seed values.

		Leader by			Kendall’s $\tau_{b}$
Ledger ( $k$ )	Cap $c_{t}$	P@ $k$	Macro	FSR@ $k$	P, macro	macro, FSR
C (100)	$n_{t}$	kNN	VAE	VAE	0.83	1.00
A (100)	$n_{t}$	kNN	AE	AE	0.65	1.00
B (100)	20	kNN	VAE, AE	kNN	0.74	0.69
D (100)	20	PCA, AE, VAE	PCA, AE, VAE	kNN	0.81	0.42
Public (100)	12.5	AE-emb	AE-emb	AE-emb	0.94	0.94

Table 7: Which conclusions need the fair-share normalisation: the leader under P@

k

, macro-averaged recall (the mean of

R_{t}

k

over the types) and FSR@

k

, the fair-share caps (

n_{t}

when every cap equals the type size, in which case FSR equals macro recall, otherwise the common cap

k/T

), and Kendall’s

\tau_{b}

between the orderings of the nine detectors. Client ledgers at

k=100

, from the printed seed-mean per-type recalls (FSR of the seed-averaged type profile, midpoints of the rounding bounds; detectors within 0.005 of the macro-recall leader are listed together). Public ledger exact, mean over seeds.

Figure 4: The review budget on the public ledger (102,429 journal entries, 850 anomalous). (a) P@ $k$ and (b) FSR@ $k$ against the budget $k$ (log scale); the dotted line in (a) is the attainable maximum and the vertical line marks $k=1{,}000$ . (c) First-hit rank per marking (dark: early; bold: within 1,000); All is the rank at which every marking has appeared.

Figure 5: Markings and rule populations on the public ledger. (a) Recall per marking within the first 1,000 journal entries; $n_{t}$ is the number of anomalous entries per marking. (b) P@100 when the entries inside each journal entry testing rule population are ranked by the detector score; the black bar marks the rule precision, the share of anomalies in the population.

Figure 6: Contamination and difficulty on ledgers C, A and B. (a) P@100 of kNN, VAE, AE, PCA and IF at contamination 0.5, 1.5 and 3 percent and three difficulty levels (mean over five seeds); the dotted line is the attainable maximum. (b) Exploratory critical-difference diagram of mean ranks over the 27 seed-averaged sweep cells, which derive from three ledgers and are not independent datasets; bars join detectors within CD = 2.31.

(a) Anomalies found in 100 reviews
	DeepSAD, after $r$ reviews				AE-emb, first 100		Most hits, first 100
Ledger	25	50	75	100	Found	Gain	Detector	Found
C	3.0	21.0	26.2	29.4 (4.6)	12 (2)	17.4	kNN	40
A	8.8	29.0	40.6	47.0 (3.0)	19 (1)	28.0	kNN	29
B	8.0	23.4	39.4	57.6 (11.7)	23 (7)	34.6	kNN	42
D	10.2	29.4	52.8	76.0 (8.9)	27 (11)	49.0	PCA, AE, VAE	98

Table 8: Auditor reviews fed back to DeepSAD. (a) Anomalies found after 25, 50, 75 and 100 reviews (four rounds of 25), mean over seeds with the standard deviation of the final count, against the anomalies among the first 100 units of AE-emb, the list the same reviews follow if the ranking stays static, and of the static detector with the most hits; the gain is that of the adaptive review protocol over the static AE-emb ranking. Client ledgers: core grid, five seeds; the static counts are 100 times the printed P@100 and exact to within 0.5. Public ledger: three seeds, units are journal entries. Public, control: the same protocol with the revealed labels withheld from the retraining, added in revision outside the frozen protocol. (b) Public ledger: markings among the 100 reviewed entries, with and without labels, and among the first 100 entries of AE-emb, mean over three seeds, with type coverage TC@100 and fair-share type recall FSR@100;

n_{t}

is the number of entries per marking.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$N$ , $y_{i}$	number of units; label of unit $i$ (1 anomalous, 0 normal)
$\mathcal{T}$ , $T$	set of anomaly types (T1 to T5, or markings M1 to M8) and its size
$\mathcal{I}_{t}$ , $n_{t}$	anomalies of type $t$ and their number
$s_{i}$ , $\pi(r)$	score of unit $i$ ; unit at rank $r$ (descending score, ties by position)
$k$ , $\rho$	review budget; budget as a share of the ledger, $k=\rho N$
$\operatorname{top}_{k}$ , $f_{t}(k)$	the first $k$ units of the ranking; number of anomalies of type $t$ among them

Appendix

Table 1: Notation of the type-aware budget evaluation, the benchmark and the review protocol.

Seed	List	M1	M2	M3	M4	M5	M6	M7	M8	F@25	F@100	TC@100	FSR@100
0	AE-emb, first 100	0	0	0	0	1	0	18	0	6	19	2	0.135
0	DeepSAD, reviewed	5	0	0	0	6	0	60	0	6	71	3	0.235
0	Control, reviewed	25	0	0	0	1	0	35	0	6	61	3	0.260
1	AE-emb, first 100	26	0	0	0	6	0	5	0	15	37	3	0.235
1	DeepSAD, reviewed	58	0	0	0	1	0	5	0	15	64	3	0.185
1	Control, reviewed	28	0	0	0	2	0	5	0	15	35	3	0.195

Appendix

Table 1: Public ledger, DeepSAD per seed: markings among the 100 reviewed entries with labels and in the no-feedback control (labels withheld, added in revision), and among the first 100 entries of the AE-emb model both runs start from, with the anomalies found after 25 and 100 reviews (F@25, F@100), type coverage TC@100 and fair-share type recall FSR@100. Round 1 reviews the first 25 entries of AE-emb, so the three lists agree after 25 reviews.

(a) Point estimate [95 percent interval]
Detector	P@100	FSR@100	P@1,000	FSR@1,000	TC@1,000
IF	0.08 [0.03, 0.14]	0.08 [0.03, 0.14]	0.083 [0.067, 0.100]	0.111 [0.087, 0.136]	7 [6, 7]
LOF	0.11 [0.05, 0.18]	0.11 [0.05, 0.13]	0.074 [0.059, 0.092]	0.128 [0.112, 0.145]	4 [3, 4]
kNN	0.17 [0.10, 0.26]	0.17 [0.10, 0.26]	0.215 [0.189, 0.240]	0.301 [0.275, 0.323]	6 [5, 7]
HBOS	0.04 [0.01, 0.08]	0.04 [0.01, 0.08]	0.067 [0.052, 0.083]	0.104 [0.077, 0.131]	7 [5, 7]
ECOD	0.04 [0.01, 0.08]	0.04 [0.01, 0.08]	0.083 [0.065, 0.100]	0.136 [0.110, 0.158]	6 [5, 6]

Appendix

Table 1: Public ledger, bootstrap over journal entries: 1,000 resamples of the entries with replacement, the same resamples for every detector, percentile 95 percent intervals. The stochastic detectors enter with their seed-0 scores, so the point estimates can differ from the three-seed means in Table 5 . (a) Intervals per detector. (b) Paired differences kNN minus the named detector; italic marks an interval that contains zero, and the second line gives the share of resamples in which kNN is strictly ahead.

Figure 1: Bootstrap intervals on the public ledger: 1,000 resamples of the 102,429 journal entries with replacement, the same resamples for every detector, percentile 95 percent intervals; the stochastic detectors enter with their seed-0 scores. (a) Point estimate and interval of P@100, FSR@100, P@1,000 and FSR@1,000. (b) Paired difference kNN minus each other detector; open markers are intervals that contain zero.

(a) Client ledgers, $k=100$
Ledger	P@100 leader	FSR leader, equal weights	Reversal fixed (of 11 schemes)	Exceptions	Random vectors with reversal fixed
C	kNN	VAE	10	T4 halved	91%
A	kNN	AE	11	none	89%
B	kNN	kNN (no reversal)	2	reversal to VAE when T1 is halved or T4 doubled	26%
D	PCA, AE, VAE	kNN	9	T1 halved, T4 doubled (undetermined)	63%

Appendix

Table 1: Sensitivity of FSR@

k

to the type weights ( D ). (a) Client ledgers at

k=100

: the leader by P@100, the FSR leader under equal weights, the number of the eleven structured schemes (equal weights and each type doubled or halved) in which a reversal between the P@100 leader and FSR is fixed by the rounding bounds, the schemes that are exceptions, and the share of 300 random weight vectors (each weight between half and twice its equal value) with a reversal fixed by the bounds. (b) Public ledger: the FSR leader under equal weights, the number of the seventeen structured schemes in which it stays the FSR leader, the smallest Kendall

\tau_{b}

between the orderings by P@

k

and FSR@

k

over these schemes, and, over 2,000 random vectors, the share in which the equal-weight FSR leader stays the leader and the median

\tau_{b}

Detector	P@25	P@50	P@100	AP	ROC-AUC
(a) Ledger C: 3,461 postings, 52 injected anomalies, 5 seeds
IF	0.04 (0.03)	0.04 (0.02)	0.04 (0.01)	0.04 (0.01)	0.67 (0.03)
LOF	0.18 (0.07)	0.11 (0.05)	0.06 (0.03)	0.11 (0.02)	0.83 (0.04)
kNN	0.65 (0.02)	0.64 (0.02)	0.40 (0.02)	0.52 (0.02)	0.98 (0.00)
HBOS	0.02 (0.04)	0.02 (0.03)	0.03 (0.01)	0.03 (0.01)	0.66 (0.03)
ECOD	0.02 (0.04)	0.08 (0.02)	0.08 (0.01)	0.05 (0.01)	0.79 (0.02)

Appendix

Table S1: Core grid on the client ledgers (contamination 1.5 percent, medium difficulty): precision at budgets of 25, 50 and 100 postings, average precision and ROC-AUC, mean (standard deviation) over five seeds. Bold marks the best unsupervised detector per column within a ledger, on the printed values. The supervised reference is available as a mean for P@25, P@100 and AP only (n/a: not recorded).

Area	Deviation or implementation note
Injection	Ledger A, easy level, $c=3$ %: the pool holds 19 valid account pairs for the 36 T5 rows the cell needs, so 17 rows reuse a pair. The protocol permits drawing with replacement once the pool is exhausted; the reuse is recorded per cell.
Typology	The protocol called T5 a multidimensional mixed-data anomaly (Foorthuis Type VI). T5 changes two categorical fields only, so it is classified as Type V, subtype Va.
PCA	The score is PyOD’s own: the sum of the distances between the standardised row and the retained eigenvectors, each divided by its explained-variance ratio, not a literal reconstruction error. The covariance eigensolver replaces the full singular value decomposition (same components, scores agree to $4\times 10^{-4}$ relative or better, identical P@ $k$ and AP).
ECOD	PyOD’s algorithm recomputed column chunk by column chunk, because the library implementation exhausts memory on ledger D; maximum difference to the library $10^{-14}$ on ledgers C, A and B. Used on every ledger.
Networks on ledger D	AE, VAE and AE-emb train with batch 256 for 30 epochs instead of batch 64 for 100; the protocol grants this for AE-emb, and a runtime probe extended it to the two PyOD networks.
Execution on ledger D	One thread per process for the linear algebra libraries and PyTorch; every detector runs in a checkpointing worker process; LOF and kNN are fitted on all 279,975 rows, the subsampling rule being implemented but never triggered. Scores are unaffected.

Appendix

Table S2: Protocol deviations and implementation notes. None changes an injection definition, a metric or a detector hyperparameter on ledgers C, A and B, and each is recorded per cell in the harness output.

Role	Client ledgers	Public ledger
Scored row	one posting	one posting line
Evaluated unit	one posting	one journal entry; its score is the maximum over its lines, and labels are constant within an entry
Account	Kontonummer	gl_account
Contra account	Gegenkonto	derived: the account of the largest-amount line on the opposite side of the same entry, 21 levels
Amount	Umsatz_mod_z , the modified z-score supplied with the data	amount ; $\log(1+x)$ and standardised in the encodings
Payment period	Zahlungszeitraum	entry lag in days, entered_date minus document_date ; used in the encodings

Appendix

Table S3: Mapping of the benchmark onto the public ledger, field by field. The protocol expects the eleven columns of the client ledgers; the table states which field of the public ledger takes each role, what had to be derived and what was added because the public ledger records timestamps. The rule attributes supplied with the dataset are excluded from every feature set, as its authors recommend. All detector settings are those of ledger D.

Detector	M1	M2	M3	M4	M5	M6	M7	M8	All
IF	1	51	626	309	46	217	3,415	774	3,415
min	1	37	327	74	12	140	2,157	661	2,157
max	1	62	1,108	654	92	279	4,486	944	4,486
LOF	239	3,116	472	11,107	192	3,098	1	6,895	11,107
kNN	43	185	824	15	70	1,001	53	4,641	4,641
HBOS	1	429	1,386	661	456	479	30	703	1,386

Appendix

Table S4: Public ledger, first-hit rank per marking: the number of journal entries an auditor reviews until the first anomaly of the marking appears. All is the rank at which every marking has appeared at least once (the largest first-hit rank). Stochastic detectors and the supervised reference: mean over three seeds, with the smallest and largest value over the seeds in italics; the mean of All is taken over the per-seed maxima.

Detector	M1	M2	M3	M4	M5	M6	M7	M8
$n_{t}$	120	60	120	120	100	250	60	20
(a) Recall within the first 100 entries
IF	0.02	0.04	0.00	0.00	0.02	0.00	0.00	0.00
LOF	0.00	0.00	0.00	0.00	0.00	0.00	0.18	0.00
kNN	0.02	0.00	0.00	0.07	0.03	0.00	0.07	0.00
HBOS	0.02	0.00	0.00	0.00	0.00	0.00	0.03	0.00

Appendix

Table S5: Public ledger, recall per marking within the first 100 and the first 1,000 journal entries, mean over seeds for the stochastic detectors. Bold marks the best unsupervised detector per column (none where all are zero). M1 high amount, M2 high cash disbursement, M3 delayed entry, M4 very delayed entry, M5 non-working hours, M6 weekend, M7 shadow bank account, M8 cross-linked clearing;

n_{t}

is the number of anomalous entries per marking.

Detector	T1	T2	T3	T4	T5	FSR@100
(a) Ledger C
$n_{t}$	22	10	4	4	12
IF	0.12	0.16	0.05	0.00	0.00	0.07
LOF	0.22	0.00	0.00	0.30	0.03	0.11
kNN	1.00	1.00	0.15	0.10	0.58	0.57
HBOS	0.06	0.04	0.05	0.10	0.03	0.06

Appendix

Table S6: Recall within the first 100 postings per injected type (

R_{t}

@100) on the core grid of ledgers C and A, mean over five seeds as printed in the result tables, with the fair-share type recall FSR@100 derived from them.

n_{t}

is the number of injected rows of type

t

and the budget ceiling

\min(1,100/n_{t})

the largest attainable

R_{t}

@100. Because the per-type recalls are seed means rounded to two decimals, FSR@100 is given as the tightest bounds consistent with the printed values and the printed P@100 (a single value where both bounds agree at the printed precision). Type coverage is not reported, because the seed-mean recalls do not determine it. T1 extreme amount, T2 extreme payment period, T3 unseen contra account, T4 unseen account, T5 unusual account pair.

Detector	T1	T2	T3	T4	T5	FSR@100
(c) Ledger B
$n_{t}$	105	53	21	21	63
IF	0.02	0.04	0.00	0.00	0.00	0.04
LOF	0.00	0.00	0.00	0.00	0.00	0.00
kNN	0.29	0.20	0.01	0.00	0.01	0.31 to 0.32
HBOS	0.02	0.01	0.00	0.02	0.01	0.03

Appendix

Table S7: Recall within the first 100 postings per injected type, continued from Table S6 for ledgers B and D. On ledger D every type holds at least 336 injected rows, so no per-type recall can exceed 0.30.

(a) Precision at review budgets
Detector	P@25	P@50	P@100	P@1,000
IF	0.09 (0.02)	0.07 (0.03)	0.08 (0.04)	0.067 (0.024)
LOF	0.12	0.14	0.11	0.074
kNN	0.08	0.12	0.17	0.215
HBOS	0.08	0.08	0.04	0.067
ECOD	0.08	0.08	0.04	0.083

Appendix

Table S8: Public ledger, global ranking of all 102,429 journal entries, of which 850 are anomalous. Mean (standard deviation) over seeds 0 to 2 for the stochastic detectors and the supervised reference; the data are fixed, so deterministic detectors have a single run. Bold marks the best unsupervised detector per column on the printed values. The last three columns split the first 100 entries into anomalous entries, normal entries on a rare account (fewer than 1,000 lines) and other normal entries (means over seeds).

	J1	J2	J3	J4
Rule	top-n amounts	high cash	delayed entry	weekend or non-working hours
Entries	2,000	402	8,207	16,493
Anomalies	196	190	310	415
Rule precision	0.098	0.473	0.038	0.025
(a) P@100 inside the population
IF	0.40	0.53	0.06	0.08

Appendix

Table S9: Public ledger, ranking inside the four journal entry testing rule populations. Each population is the entry set that one rule returns; inside it the rule attribute is constant, so the ranking cannot use it. Rule precision is the share of anomalous entries in the population, which is what plain rule matching delivers. Means over seeds for the stochastic detectors; bold marks the best unsupervised detector per column.

	Client ledgers				Public
Detector	C	A	B	D	ledger
IF	14.1	14.7	0.6	9.8	5.1
LOF	0.1	0.1	1.5	1,022.6	478.2
kNN	0.0	0.1	1.5	1,029.0	467.1
HBOS	1.3	1.3	0.2	10.5	4.6
ECOD	0.1	0.1	0.4	25.1	4.8

Appendix

Table S10: Wall-clock seconds for one detector fit (2 CPU cores, 7 GB RAM, no GPU). Client ledgers: core-grid cells of the first execution of the benchmark, the one whose hardware is documented, mean over 5 seeds on ledgers C, A and B and over 3 seeds on ledger D. Public ledger: mean over the runs of a detector (three seeds for the stochastic detectors and the supervised reference), on two CPU cores. On ledger D and on the public ledger the three networks train for 30 epochs with batch 256 instead of 100 epochs with batch 64, so their times there are not comparable with ledgers C, A and B. n/a: not timed.

	$c=0.5$ %			$c=1.5$ %			$c=3$ %
Detector	easy	medium	hard	easy	medium	hard	easy	medium	hard
(a) Ledger C
IF	0.02	0.01	0.01	0.04	0.04	0.04	0.09	0.09	0.08
LOF	0.04	0.03	0.02	0.11	0.06	0.03	0.07	0.04	0.02
kNN	0.15	0.14	0.14	0.43	0.40	0.37	0.75	0.69	0.64
HBOS	0.01	0.01	0.01	0.03	0.03	0.03	0.05	0.06	0.06

Appendix

Table S11: Contamination and difficulty sweep on ledgers C, A and B: P@100 for every combination of the contamination

c

and the difficulty level, mean over five seeds. The row Maximum is the attainable P@100,

\min(1,n/100)

for

n

injected anomalies; at

c=0.5

percent every ledger holds fewer than 100 anomalies. The 27 cells are the input of the rank analysis in Section 5.1, which uses the seed-averaged cells because per-seed values were not retained. Bold marks the highest P@100 per column within a ledger.

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