Structured Temporal Causality for Interpretable Multivariate Time Series Anomaly Detection
Authors: Dongchan Cho
Jiho Han
Keumyeong Kang
Minsang Kim
Honggyu Ryu
Namsoon Jung
Venue: NeurIPS 2025 poster ()
Keywords: Multivariate Time Series
Anomaly Detection
Unsupervised learning
Primary area: deep_learning
TL;DR: OracleAD detects multivariate time series anomalies via prediction error and deviation from emergent latent structure, enabling interpretable, fine-grained, and root-cause diagnosis.

Abstract: Real-world multivariate time series anomalies are rare and often unlabeled. Additionally, prevailing methods rely on increasingly complex architectures tuned to benchmarks, detecting only fragments of anomalous segments and overstating performance. In this paper, we introduce OracleAD, a simple and interpretable unsupervised framework for multivariate time series anomaly detection. OracleAD encodes each variable’s past sequence into a single causal embedding to jointly predict the present time point and reconstruct the input window, effectively modeling temporal dynamics. These embeddings then undergo self-attention mechanism to project them into a shared latent space and capture spatial relationships. These relationships are not static, since they are modeled by a property that emerges from each variable's temporal dynamics. The projected embeddings are aligned to a Stable Latent Structure (SLS) representing normal-state relationships. Anomalies are identified using a dual scoring mechanism based on prediction error and deviation from the SLS, enabling fine-grained anomaly diagnosis at each time point and across individual variables. Since any noticeable SLS deviation originates from embeddings that violate the learned temporal causality of normal data, OracleAD directly pinpoints the root-cause variables at the embedding level. OracleAD achieves state-of-the-art results across multiple real-world datasets and evaluation protocols, while remaining interpretable through SLS.

Structured Temporal Causality for Interpretable Multivariate Time Series Anomaly Detection
Dongchan Cho"
Jiho Han
Keumyeong Kang
Minsang Kim
Honggyu Ryu
Namsoon Jung
Industrial Lab, SimPlatform Co. Ltd. Affiliate Research Institute {cdc0213, jihohan, kykang, mskim, hgryu, msj)dssimllatfommccom
Abstract
Real-world multivariate time series anomalies are rare and often unlabeled. Ad- ditionally, prevailing methods rely on increasingly complex architectures tuned to benchmarks, detecting only fragments of anomalous segments and overstating performance. In this paper, we introduce OracleAD, simple and interpretable unsupervised framework for multivariate time series anomaly detection. OracleAD encodes each variable's past sequence into a single causal embedding to jointly predict the present time point and reconstruct the input window, effectively model- ing temporal dynamics. These embeddings then undergo self-attention mechanism to project them into a shared latent space and capture spatial relationships. These relationships are static, since they are modeled by a property that emerges from each variable's 's temporal dynamics. The projected embeddings are aligned to a Stable Latent Structure (SLS) representing normal-state relationships. Anomalies are identified using a dual scoring mechanism based on prediction error and devia- tion from the SLS, enabling fine-grained anomaly diagnosis at each time point across individual variables. Since any noticeable SL. deviation originates from embeddings that violate the learned temporal causality normal data, OracleAD directly pinpoints the root-cause variables at the embedding level. OracleAD achieves state-of-the-ar results across multiple real-world datasets and evaluation protocols, while remaining interpretable through SLS.
Introduction
Unsupervised anomaly detection in multivariate time series (MTSAD) B 46
underpins reliability
and safety domains such as industrial control systems 28 31, healthcare monitoring [3, and cyber-physical security [ In these settings, anomalies are rare, unlabeled, and context-dependeng, posing challenges to conventional methods and requiring actionable insights to maintain operational reliability. Models must therefore not only flag irregularities but also explain their causal origins. Thus, effective MTSAD methods should (i) define anomalies within a multivariate temporal framework, (ii) model the temporal and spatial signals of anomaly formation, and (iii) integrate these signals for both detection and root-cause diagnosis.
Although intrinsic temporal and structural changes occur when anomalies emerge, most MTSAD methods overlook these signals. They often rely on reconstruction errors computed over fixed "normal" and "abnormal" windows, superficial feature-level differences after arbitrary model transformations, or on frequency-domain representations divorced from causal time dynamics. More complex deep-model pipelines (e.g., dual-axis attention) further amplify this issue by creating artificial
"Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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separations between normal and anomalous data through their complex forward- proppaattoonntooonnn outputs, rather than by explicitly modeling how anomalies form and spread. Such complexity rarely translates into practical gains, as these methods frequently fail to outperform simpler baselines or to justify their architectural overhead empirically or theoretically X 47]. Transformer-based approaches exemplify this misalignment by using large temporal contexts and bidirectional attention, which ignore the unidirectional, irreversible nature of time and compromise real-time deployment. Furthermore, recent studies 100 47 demonstrate that compact causal architectures with lightweight designs can match or exceed Transformer performance across diverse forecasting benchmarks while greatly reducing computational costs.
At the core of these issues lies a fundamental misunderstanding regarding what naturally constitutes an anomaly in multivariate time series. We argue that anomalies inherently manifest through two interconnected signals. First, anomalies begin as prediction errors when a variable's current state deviates from the expectations derived from its past states, signifying breakdown in temporal causality within that variable. Second, temporal disruptions propagate into structural deviations by breaking the stable varialle relationships that are typically preserved under normal conditions. Thus, we explicitly define anomalies in multivariate time series as processes originating from a loss of temporal causality in specific variables, which subsequently alters the inter-variable relational structure.
Motivated by these perspectives, we propose OracleAD, an interpretable framework that captures temporal causality and leverages the natural signals inherent in multivariate anomalies. OracleAD processes the time series data with sliding window of length T.. For each variable i, per-variable LSTM encoder processes its past L - 1 observations to produce hidden states [h?....,hh h/--). }. We then apply attention pooling over these hidden states to yield single causal embedding c, which summarizes the temporal information necessary to predict the final timepoint x/ in the window. To instill temporal causality in the encoder, we jointly train it with two objectives: one-step point prediction of o and reconstruction of the window. The learned causal embeddings are projected into a shared latent space via a self-attention mechanism, where attention weights quantify dynamic interaction strengths between variables at each time point.
During training, OracleAD computes pairwise dissimilarities between projected causal embeddings across variables at each time step, forming dissimilarity matrix that explicitly reflects the current spatial organization. At the end of each training epoch, these matrices are aggregated into a Stable Latent Structure (SLS), which serves as a stable, statistically derived reference structure representing consistent vatirlee relationships under normal conditions. At inference time, OracleAD employs a dual scoring system. The prediction score quantifies the error between the actual and predicted states at each time step, signaling temporal irregularities that violate past-informed expectations. The deviation score captures structural inconsistencies by measuring the deviation between the current latent dissimilarity matrix and the SL.S, flagging shifts inter-variable relationships that deviate from learned causal structures. By integrating these complementary signals, OracleAD not only detects anomalies accurately but also transparently pinpoints root-cause variables at the embedding level, addressing critical limitations of existing methods and providing causal explanations of anomaly formation and propagation.
OracleAD demonstrates state-of the-the-art anomaly detection performance across multiple real-world datasets. Experimental results show consistent improvements detection accuracy, anomaly localiza- tion, and robustness under diverse evaluation protocols. Recognizing and explicitly modeling natural anomaly signals enables OracleAD to detect anomalies more efficiently using compact temporal windows. This approach is supported by recent findings 44), which show that anomalies strongly correlate with adjacent time points. OracleAD provides interpretability through a dual scoring system based on prediction error and structural deviation, enabling users to identify anomalies and trace their root causes at both the temporal and variable levels. By grounding anomaly detection in temporal causality and stable latent representations, OracleAD addresses core limitations of existing methods and offers a principled, practical direction for advancing multivariate time series anomaly detection.
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2 Background
2.1 Unsupervised Multivariate Time Series Anomaly Detection
Early approaches to multivariate time-series anomaly detection primarily relied on classical outlier detection methods + 23 36.
However, these early methods treated each time point as an independent multivariate sample and did not leverage the rich temporal dependencies inherent in sequential data. Subsequently, research gradually shifted toward deep sequence modeling based on reconstruction or prediction. EncDec-AD [27]. OmniAnomaly [37], and recurrent ensemble variants [8] train autoencoder or variational models to reproduce normal behavior, flagging large reconstruction or prediction errors as anomalies. Although these approaches are conceptually simple and robust, they treat each channel independently, failing to capture inter-variabl dependencies critical for complex fault patterns. MAD-GAN [ extends this line by applying adversarial training via LSTMMbasee generator-discriinnato networks, enabling joint modeling of temporal dynamics and inter-variable structures. However, unstable GAN optimization often limits robustness and interpretability.
To address this, explicit spatio-temporal modeling techniques emerged. MSCRED (88) uses ConvL- STM (35] reconstruct correlation matrices over sliding windows, and InterFusion [21] employs hierarchical VAEs to disentangle temporal and spatial factors. Transformer-based architectures inject attention mechanisms into MTSAD. Anomaly Transformer 4 characterizes anomalies as discrep- ancies between learned attention weights and prior temporal associations, whereas TranAD [39] combines predictive and adversarial objectives to sharpen sensitivity to subtle deviations. Prior this, Gangopadhyay et al. [12] proposed a apatio-temporal attention mechanism for interpretive forecasting in MTS, showing that attention can offer explanatory cues across time and variable dimensions. PatchTSST [29] and SparseTSSF [22] further improve efficiency by processing fixed-size patches or periodic samples, but at the expense of strict causal consistency and fine- -reiira localization. Graph- wasce and dimension-wist -ise attention methods capture variable interactions more flexibly (50). GDN 8 learns a data-driven adjacency matrix to model dependencies (albeit static at inference), while Crossformer [49] and iTransformer [25] deploy multi-axis attention to fuse temporal and cross-variable signals, though large receptive fields may dilute local anomalies. Recently, Zheng opatial-temporre graph learning model. Contrastive and spectral approaches offer orthogonal gains: DCdetector [45) forgoes reconstruction entirely, using contrastive grouping of normal versus perturbed views; SARAD [6] adds regularization across subsequeeee;; CATCH 43 isolates spectral anomalies via frequency-domat patching.
et al. [51] introduced correlation-awar
2.2 Limitations of Existing Benchmarks
Several widely used MTSAD benchmarks, such SWaT (28), SMAP, and MSL 17 are frequently regarded as fully multivariate datasets. However, practice, most real-world attacks and faults affect only limited number of variables. For example, many scenarios in the SWaT dataset involve perturbations to only a small subset of sensors. Similarly, in the SMAP and MSL datasets, many sensor streams exhibit behavior that closely resembles step functions. A recent study by Liu et al. (24) demonstrates that converting these datasets into univariate formats results in comparable detection performance. This observation raises concerns regarding the actual necessity of using complex models that are specifically designed to capture cross- variable dependencies.
In addition to dataset limitations, the evaluation protocols used in MTSAD research exhibit several inconsistencies 19 42.. The point-adjusted Fl score, for instance, tends to give excessive credit for any detection that falls within an anomaly window, even if the localization is inaccurate. Metrics such as AUC-ROC and AUC-PR are iidheeddnndeeendentt and widely used, but they often produce misleading results in highly imbalanced settings where true negatives dominate. The Affiliation Fl score [16] attempts to compensate for localization errors through post- proccesoc.s,ng but does not fundamentally resolve the problem of poor anomaly localization. More recently, researchers have proposed volume-under made--undr-uuufface metrics such as VUS-ROC and VUS-PR [24] 30), which jointly measure detection consistency and temporal precision across range of thresholds.
Considering these limitations, it is difficult to regard strong performance on a single benchmark or metric as a reliable indicator of general robustness 2 In order to provide an accurate assessment of MTSAD methods, necessary to conduct comprehensive evaluations that include multiple anomaly types, realistic variable correlations, and diverse metrics such VUS-based measures.
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Model Architecture
LSTM Decoder
ESTM Decaker
ESTM Decaeler
Demadler
LSTM
LSTM
+
ESTM
LSTM
Disinlaiy
Ox
LSTM Encoder
Aneetton
Ateatton Pooling
Ateetton Posting
Posting
N
LSSM
LST
LSTH
LSSM
LST3
LSTM
Figure I: Overview of the OracleAD pipeline. Each variable's past window is encoded by an LSTM and pooled via attention. The resulting embeddings refined through multi-hes self-attention to capture inter-variable dependencies. These embeddings are then fed into an LSTM decoder for reconstruction and next-step prediction. Finally, the pairwise distance matrix of the embeddings is compared against the Stable Latent Structure (SLS) to detect anomalies. The decoder uses an all-zero vector its initial input.
3 Method
Let X NxL denote fixed-length input window multivariate observations, defined as
x=
fxt
x
x E
(1)
where each window element is written as
(41,
WT), 1 1sesL.
and x is the most recent observation vector. OracleAD determines whether x- is anomalous based on (i) its prediction error relative to temporal patterns learned within the window, and (ii) its deviation from the stable inter-var variable structure observed under normal conditions.
3.1 Temporal Causality Modeling
OracleAD models each variable independently using per-variable encoder encoder-decoder architecture. This contrasts with prior approaches [41 44 48 that process the multivariate sequence jointly, often with shared model. However, in real-world settings, different variables often follow heterogeneous dynamics, and anomalies may affect only subset of them. Shared architectures can entangle unrelated temporal patterns, hindering both detection sensitivity and interpretability. To address this, adopt modular structure where each variable learns its own causal dynamics.
As shown in Figure for each variable segment x; = (x) - .... of , - e R an LSTM 27 encoder Enc; processes the input window and outputs a sequence of hidden states:
{h} h2 N/ RR.
(3)
where d is the hidden dimension. To distill salient temporal information and suppress noise, we apply learnable attention mechanism the hidden states:
f(h)) w TT,, +b, W Rdd b R.
(4)

---

followed by softmax normalization over the temporal axis (1 = =1, I,..., L 1):
L-1 C
a softmax(((h)))
Σ
(5)
The resulting vector 6; R&, termed the causal embedding of variable i, aggregates its past hidden states weighted by their relevance in predicting of. thereby encoding the temporal causality inherent in that variable's s history.
To incorporate inter- variable interactions, we apply Multi-Head Self-Attentic (MHSA) 40 to the stacked embeddings
C C Ca CN! ERNx
(6)
yielding the attention-refined output
C [c.....,cc. E R
(7)
Each of is computed as
C wh softmax W.se:WW o) vdn
W
(8)
where H denotes the number of attention heads, and A W W., W are learnable projec- tion matrices for query Q, key K, value V, and output in head h. The per-variable embeddings fci...., CNl are first projected into a shared latent space via head-specific linear transformations. Subsequent multi-heas self-attention mixes these projections, so each ot absorbs contextual cues from all other variables and captures soft, dynamic dependencies without a predefined static graph.
These context-aware embeddings of then passed into their corresponding LSTM decoders Dec, each trained to both reconstruct the past window and predict the next value:
11:
2 = (ec;(ct))
The outputs each decoder are used to compute the reconstruction loss || prediction loss ||af 112 which together ensure that each causal embedding ej faithfully captures its variable's 's temporal dynamics under causal, windowed regime.
-::2-1113 and the
The final multivariate prediction at time assembled as
x = [ EIERR.
(10)
This per-variable framework guarantees robust modeling of the normal data distribution by (i) condensing all L - 1 past steps into each causal embedding q, thereby covering the full spectrum of normal temporal patterns rather than overfitting to local snippets, (ii) employing independent encoder-decoder branches so that each adapts to its variable's unique dynamics, (iii) fusing these embeddings via MHSA into context-aware vectors oi, which learn the joint normal manifold and soft correlations across variables. and (iv) suppressing the influence of rare spikes through attention pooling and Stable Latent Structure (SLS) structural regularization.
3.2 Stable Latent Structure (SLS)
To detect structural anomalies and promote latent consistency during training, OracleAD maintains a reference structure called the Stable Latent Structure (SLS). The SL.S summarizes stable inter- variable relationships observed in the latent space during normal behavior. Unlike static graphs or manually defined priors, constructed directly from data using the attention-refined latent representations of individual variables.
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Let oj E Rd denote the latent vector of variable (i = 1, ,,,...NN after temporal encoding and relational attention. For each training window form a pairwise dissimilarity matrix
D 2/h =| - (, D E RRNN
(11)
where adopt the L.2 distance because it accounts for both directional mismatch and latent energy differences in the embeddings. Empirically, L2 also provided more stable and discriminative results than alternatives such cosine similarity or LI distance (see Appendix E.4).
Aggregating over the M windows that appear in one training epoch yields stable template
SLS
(())
(12)
M
which provides compact baseline of canonical inter- variable structure. Since the first epoch has prior SLS to build upon, the updating process begins at its end, and the resulting SLS is then used from the second epoch onward. During training, it acts regularizer that encourages the encoders to preserve this relational pattern. At inference time, for each time step / in the entire input sequence, OracleAD computes the deviation matrix
Dr Dimatri
(13)
which measures element-wise deviations from the normal inter-variable structure. High-magnitude rows or columns in Dr mattrx indicate variables that disrupt relational stability, enabling root-cause identification.
3.3 Training Objective
OracleAD is trained with a composite loss function that enforces temporal causality, reconstructs normal patterns, and regularizes inter-variable structure. The total loss is defined
C x² -xx x² +
||x| x1: x1:1-111 1|2 Adev
1 N2
ΣΣ Di - SLS,,) (14)
prediction
reconstruction
deviation
where x" E R. denotes the ε-th element of the window and D is the dissimilarity matrix defined in Eq.
III
The first term encourages each decoder forecast the most recent observation x" using only the preceding window elements, thereby enforcing temporal causality. The second term guides the model to reconstruct the past window x - preserving contextual memory and improving robustness to noise. This reconstruction term prevents the model from relying shorteut predictions based only on local trends, forcing instead to leverage the full temporal context within the window. The third term penalizes deviation between the current dissimilarity matrix D and the reference SLS from Eq. ensuring that latent representations remain aligned with relational patterns observed under normal conditions.
The hyperparameters Arecoa and Adev (set 0.1 and by default, respectively) balance the reconstruc- tion and structural regularization terms against the prediction objective, promoting both predictive accuracy and structural coherence. In the first epoch, the deviation loss omitted because no SL.S available; it is included from the second epoch onward once the initial SLS has been constructed.
3.4 Anomaly Scoring
During inference, OracleAD computes two complementary scores for each time step in the entire input sequence. The prediction score is defined as

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N x
p score
(15)
which measures the average absolute error between the observed variables and their reconstructions. The deviation score is defined as
D D' SLS
||
(16)
where D' the pairwise dissimilarity matrix of the causal-context emheddings at time 1, and SLS is the stable relational template from Eq. 12 Here
|| || denotes the Frobenius norm.
The final anomaly score is then defined as
AScore
P score
Dr
(17)
score
which jointly captures both temporal and spatial anomalies in multivariate time series. By grounding detection in both temporal prediction and relational stability in latent space, OracleAD achieves enhanced sensitivity and interpretability compared to methods relying solely on univariate errors or static dependency graphs. The diagnostic role of each component illustrated with real-world examples in Section
4 Experiments
4.1 Experimental Setup
Datasets. We evaluate OracleAD on three widely adopted benchmark datasets: SMD 37,. PSM [
and SWaT (28). These datasets span diverse industrial scenarios, including cloud server telemetry (SMD, 38 variables), real-world industrial sensors (PSM, 25 variables), and water treatment testbed (SWaT, 51 variables). Each dataset contains labeled anomaly segments with varying temporal and spatial characteristics. Full details are provided in Appendix A
Evaluation Metrics. To comprehensively evaluate model performance, we report results using seven metries: standard F1 (not point-adjusted), range-based F1 (R-F1) (38], affiliation F1 (Aff- FI) 16 AUC-ROC (A-ROC) [Л] AUC-PR (A-PR) 77.. VUS-ROC (V-ROC), and VUS-PR (V- PR) 30J. These metrics capture different aspects of anomaly detection, including temporal accuracy, interval-leve coherence, robustness to class imbalance, and threshold stability. More detailed descriptions of each metrie are provided in Appendix B
Baselines. We benchmark OracleAD against a diverse set of classical and deep learning base- lines. For reconstruction-baasee method, AutoEncoder (A.E) 33 and OnniAnomaly (Omni) 37 are considered. Spatially-structured and iitention-based deep models include Anomaly Trans-
former (A.T) 44 DCdetector (DC) 45), SARAD (6) and CATCH
4 Forecasting-ories oriented
approaches such as PatchTST (Patch) 229 TimesNet (TsNet) [4]. DLinear (DLin) [47] NLinear
(NLin) 47). i'Transfommer (iTrans) 25 and ModernTCN (Modern) ( are also considered,
due to their high performance in general time-series tasks. All baselines are run using publicly available implementations and reimplementations based on the original paper's configurations. The implementation details of baselines provided in Appendix C
OracleAD is implemented in fully unsupervised setting using only normal training data. We use sliding window of length L. 10 for all datasets, and exclude the first L 1 time steps of the test data from evaluation. Following the recommendations from recent benchmark studies [15 24 34), we consider thresholding to orthogonal to the quality of the model itself. Therefore, we either report preshhldd-ineepedden metrics or evaluate detection performance under an optimal threshold better isolate the effectiveness the anomaly scoring function. Further details, sensitivity analysis of hyperparameters, and computational cost provided in Appendix D
Implementation Details
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Table Performance (%) of deep anomaly detection models across seven evaluation metries on three benchmark datasets (PSM, SMD, SWaT). All results are averaged over five random seeds. Boldface indicates the best score and underline indicates the second-best score.
Dataset
Metric
A.E
Omni
A.T
Patch
TsNet
DLin
NLin
DC
iTrans Modern SARAD
CATCH
OracleAD
F1
47.55
45.90
43.45
43.45
43.45
43.45
43.45
43.45
43.45
43.45
45.75
44.33
65.85
R-FI
45.93
40.58
12.17
34,75
37.49
34,74
32.10
21.35
35.91
35.48
13.54
54.19
54,66
Aff-F1
73.70
73.49
69.68
69.43
69.43
69.43
69.43
69.59
69.53
69.43
77.12
79.16
78.07
PSM
A-ROC
66.79
63.95
38.35
58.68
59.09
58.02
58.56
49.86
59.22
59.21
62.86
64.75
84.78
A-PR
47.27
45.00
24.28
37.91
38,95
37.17
37.64
27.78
38.27
38.59
41.67
43.40
68.11
V-ROC
68.20
61.23
50,96
50.62
50.67
50.53
50.69
50.11
51.23
50.68
57.63
57.00
84.24
V-PR
49.66
52.49
49.76
50.31
49.86
49.93
46.71
27.89
44.54
50.55
38.64
45.95
68.17
FI
25.78
32.16
7.98
7.98
7.98
7.98
8.00
7.98
7,98
25.92
7.98
43.03
R-FI
28.09
35.81
5.84
13.81
9.43
13.37
13.51
8.23
9.02
14.43
10.35
2.90
38.88
Aff-FI
83.34
82.57
67.43
67.43
67.43
67.43
67.43
67.49
67.43
67.43
75.87
67.43
84.73
SMD
A-ROC
76.80
71.25
50.40
73.85
59.09
72.73
73.83
49.97
74.57
72.13
72.84
80.96
83.56
A-PR
19.40
27.73
4.57
14.82
13.66
13.88
14.10
4.16
14.57
13.01
25.87
17.09
44.83
V-ROC
74.38
73.26
50.34
51.66
51.10
51.52
51.76
49,92
51.76
51.55
65.57
50.95
69.57
V-PR
22.50
31.18
36.86
41.60
41,21
41.26
39.15
4.27
32.85
41.54
19.33
35.25
47.52
FI
74.46
75.40
21.65
21.65
21.65
7.98
12.14
21.65
21.65
21.65
57.30
21.65
76.50
R-FI
36.63
38.72
15.48
12.23
20.80
5.13
12.06
15.54
11.89
12.88
21.37
14.74
28.15
Aff-F1
75.30
72.09
70.06
69.51
73.47
67.43
69.68
69.25
69.25
72.08
74.88
73.68
71.97
SWaT
A-ROC
81.97
82.15
42.58
24.36
28.97
50.48
23.31
49,89
24.57
24.50
85.40
33.31
82.71
A-PR
67.51
72.73
11.93
844
10.71
4.53
8.00
12.14
8.33
9.00
64.77
13.39
72. 12.39
V-ROC
81.89
81,98
50.72
49.84
20.80
50.36
49.02
50.00
49.90
50.15
86.30
$1.89
82.42
V-PR
65.89
64.42
17.00
11.92
51.20
28.47
10.55
11.92
12.05
13.05
62.72
18.70
74.16
4.2
Main Results
As shown in Table OracleAD achieves the highest scores across most of the reported metrics on SMD (average over all subsets), PSM, and SWaT. This demonstrates strong robustness against diverse anomaly types and dimension of data.
Among the various metrics, we emphasize the standard point-wise Fl score, which directly reflects alignment between detected anomalies and ground truth. OracleAD consistently outperforms all base- lines with large margins in F1 (PSM:+19.955))) ability clearly separate anomalous from normal points. These high scores are often accompanied by strong performance in pareshold-indepencee
AMD:+10.87%p
(WaT:+0.9%pp. confirming its
metrics (AUC, VUS). Recent benchmarks (24) identify VUS-PR key indicator of detection consistency and localization quality. In this metric, OracleAD significantly outperforms all baselines (PSM:+15.68) underscores its superior anomaly localization.
CMD:+5.92%pp SWaT:+8.27%pp which
In contrast, Affiliation Fl tends to yield high scores across most models regardless of their perfor- mance on other metrics. This reveals key limitation of the metric, Affiliation Fl rewards predictions made near the anomaly segment, even if they do not directly overlap with the ground truth. Experi- ments in [24] demonstrate that models can score highly on affiliation even with random predictions or by detecting only small part of an anomaly segment. This explains why some baselines (c.g.. A.T, CATCH) exhibit high affiliation Fl despite poor F1, AUC-PR, and VUS-PR, indicating their failure to fully capture anomaly regions. Overall, these results confirm OracleAD's dual-score design delivers both sharp detection and robust localization across diverse scenarios.
4.3 Visual Interpretation and Anomaly Diagnosis
A key interpretability component is the deviation matrix D Diuunss = - SLS), which highlights each variable's 's disruption of the learned structure. Figure 3 visualizes deviation matrices at representative anomaly timestamps from the SMD dataset. For example, variables 32, 33 in Figure 3 a and 10, 15, 18 in Figure
show dominant deviations. Their corresponding raw signals and scoring trends in Figure 2 validate validatee these candidates, as each shows strong alignment with prediction peaks and anomaly score elevations, confirming OracleAD's ability to localize root causes through unsupervised latent analysis. A practical heuristic is that variables with multiple highlighted rows or columns are more likely to be root causes, as their relational patterns remain consistently perturbed across time. When arranged sequentially, these matrices reveal how the structural disruptions of key variables evolve over the course of an anomaly segment. Quantitative analysis based on this row-wise aggregation and more examples are provided in Appendix G
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30
(a) anomaly range=[14933, 14938]
(b)
anomaly range=[15064, 15066]
Fedeiii
Seatbe Sore
anomaly range=[16535, 16557]
(d) anomaly range=[18640. 18944]
Figure 2: Raw signals of the identified root-cause variables and anomaly score visualization for selected intervals the SMD dataset. Shaded red regions indicate ground-truth anomalies.
(-13265
(=183:3
(-18841
22 22.50-22 1.74
20 222Z22 Werab Tadis
Varaala 11.5i
Vacalle Tadin
Vacalh Ttdaa
(a) D 149236
(b) D 15065
(c) D 16536
(d) D L8641
Figure Visualization for Deviation Matrices D matrix at anomalous time points SMD dataset. Each matrix corresponds respectively to the subfigures in Figure 2 The color bar of each heatmap represents the magnitude of pairwise dissimilarity between variables. A bigger row over column value means a higher chance of an anomaly in that variable. For visualization, each matrix is independently min-man normalized and values are clipped the range [0.3, 0.9] to enhance contrast.
As shown in Figure OracleAD combines prediction and deviation scores into a final anomaly score. The prediction score measures point-level discrepancies between observation and forecast, responding sharply to sudden deviations and achieving high temporal precision. However, its response is typically brief and localized the moment of disruption, potentially missing prolonged structural anomalies. In contrast, the deviation score compares the current dissimilarity matrix to the Stable Latent Structure (SLS) and captures longer-term relational disturbances (see Figure dh that persist even after the raw signal stabilizes. This property aligns with real-world anomaly labeling, where fault intervals span the full duration of system disturbance rather than just its initiation. Consequently, the deviation score remains elevated across an entire fault segment, reflecting sustained relational instability.
Nonetheless, the deviation score may exhibit slight temporal lag because structural disruption can only captured after anomalous patterns have entered the input window. This means the effect of recent anomaly only reflected after it enters the window, and the score can also be

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influenced by residual effects from recent anomalies (see Figure 2 potentially leading to false positives in recovery periods. The multiplicative combination of these two scores enables OracleAD to balance their respective strengths and mitigate their individual weaknesses. While the prediction score anchors detection to the precise moment of temporal disruption, the deviation score extends sensitivity across the whole anomaly segment. This joint scoring mechanism also suppresses false positives and recovers false negatives. High deviation scores that arise in non-anomalous contexts suppressed by low prediction error, while false negatives from prediction-only detection are compensated by the sustained elevation of structural deviations. Together, they offer both sharpness and stability, enabling OracleAD detect anomalies with improved precision, temporal coverage, and interpretability across diverse anomaly types and durations. More case studies are provided in appendix F
4.4 Ablation Study
Table 2: Ablation study results (%) on the PSM, SMD, and SWaT datasets. All results are averaged over five random seeds.
Component
Variant
PSM
SMD
SWaT
F1
V-ROC
V-PR
F1
V-ROC
V.PR
Fl
V-ROC
V-PR
Loss Function(Sec 3.3
w/o Reconstruction Loss
58.03
74.56
54.40
56.47
75.59
54.29
76.61
82.92
71.95
Anomaly Score(Sec Sec
3.4
puue Ascon
P
D
tue
59.06
55.33
60.99
60.99
58.98
47.32
87.39
77.50
53.71
37.02
76,92
70.49
82.68
68.50
70.77
OracleAD (Full Model)
65.85
84.24
60.19
56.63
76.50
74.16
We conduct ablation study evaluate the contributions of reconstruction loss and anomaly scoring strategies. Results on PSM, SWaT, and SMD (average of 4 subsets) are summarized in Table 2 Per-subset results of SMD are provided in Appendix E
Loss Function. The full model achieves the highest performance across most of the datasets and metrics. Removing the reconstruction loss results in moderate degradation, especially in PSM (F1:-7.82ppt, V-ROC:-9.68% V-PR:-13.77%pt and SMD (F1:-3.72%pp V-ROC:-5.22pp,, V- PR:-2.34%pt) indicating that reconstruction helps stabilize temporal representations and improve robustness.
While deviation oonly scoring (Atumme = D performs well on SWaT, it un- derperforms on PSM and SMD, likely because it lacks temporal context. Prediction-only scoring (Abeare Piccree can detect sharp temporal deviations but fails to capture relational disruptions, leading to weak performance on SWaT (F1:-6.01% %pt, 6.P..-).6)ppt).. As discussed Section 2.2 SWaT anomalies often affect only a small subset of variables, which can limit the effectiveness of prediction-based scoring. However, structural disruptions caused by these localized anomalies still alter variable relationships, allowing deviation- scoring to retain strong performance. The combined scoring (A'core Psccoee De D consistently performs best overall, confirming the complementarity of temporal and structural dimensions in OracleAD's design.
Anomaly Score.
5 Conclusion
We introduced OracleAD, unsupervised framework that jointly models temporal causality and inter-variable structure through unified latent representation. OracleAD derives a compact causal embedding for each variable, aligns them via attention into shared latent space, and detects anomalies through deviations from a learned Stable Latent Structure (SL.).
OracleAD achieves state-of-thed th-the- performance across multiple real-world datasets and evaluation protocols, demonstrating strong robustness, localization accuracy, and interpretability. Its dual scoring mechanism captures both prediction errors and structural inconsistencies, enabling precise detection and fine-grain une-grained root-cause analysis even without labels. We also found limitations in our framework when applied to complex multimodal systems, where globally consistent inter-variable relationships and continuous input assumptions may not hold.
Our paper highlights the importance of grounding anomaly detection in temporal causality and latent relational consistency, establishing principled foundation for generalizable and interpretable multivariate time series analysis.
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Appendix for "Structured Temporal Causality for Interpretable Multivariate Time Series Anomaly Detection"
A Datasets
A.1 Dataset Statisties
Table S1: Statistics of the benchmark datasets used for multivariate time-series anomaly detection.
Datasel
Subsets
Features
Avg Set Size
Anomalies
Segmentt Lengths
Sampling
Train
Test
Count
Ratio
Med
Max
Period
PSM
25
132,481
87,841
71
27.73%
5
8,861
min
SMD
28
38
25,300
25,301
327
4.16%
11
3,161
min
SWaT
51
496,800
449,919
34
12.02%
101
447
35,900
sec
Table ST summarizes the key characteristics of the benchmark datasets used in our experiments. These datasets differ substantially in terms of dimensionality, time series length, anomaly density, and segment structure.
PSM features relatively few but lengthy anomaly segments, resulting in high anomaly ratio despite a limited number of anomalous events. In contrast, SMD comprises numerous short anomaly segments with a low overall anomaly density, making it sensitive to brief temporal disruptions. SWaT falls between these extremes, containing moderate number of anomalies with segment lengths ranging from under 200 to tens of thousands of timestamps.
Together, these datasets span spectrum of dense vs. sparse anomalies, short vs. long-term disruptions, and uni-source vs. multi-entity dynamics. This diversity allows comprehensive evaluation of model robustness under heterogeneous anomaly scenarios.
A.2 Dataset Descriptions
PSM (Pooled Server Metries) 1 The PSM dataset comprises server-side metries collected from production servers at eBay. It includes 25 variables recorded over approximately 21 weeks, totaling 132,481 training points and 87,841 testing points. The test contains both short-duration spikes and prolonged system-level anomalies, with an anomaly ratio of 27.73%. The combination of high density and mixed-length segments makes PSM challenging benchmark.
SMD (Server Machine Dataset) [37] The SMD comprises univariate telemetry from 28 individual machines, each monitored with 38 variables. Every machine's 's data spans 10 days, with the first days containing only normal behavior (training set) and the last 5 days including injected anomalies (testing set), resulting in average anomaly rate of 4. 4.1..
Following the original authors' recommendation, we treat each machine as an independent subset and train a separate model per subset. This approach respects the natural isolation of system dynamics and failure modes across machines. For ablation studies and hyperparameter tuning, we exclusively utilize a few selected subsets, without overlap with those used for final evaluation.
The SWaT dataset captures sensor and actuator data from realistic water treatment plant testbed. It contains 51 variables over 11 days, with the first 7 days considered normal and the remaining days including 41 attack scenarios. Around 12% of the test points are labeled anomalous. SWaT's closed-loop dynamics and complex multivariate dependencies
SWaT (Secure Water Treatment) 28
21

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make it especially suitable for evaluating the causal reasoning and interpretability capabilities of detection models.
A.3
Dataset Sources
We use three widely adopted, publicly available MTSAD datasets. Each dataset is accessible for academic research, subject to the terms listed below:
PSM (Pooled Server Metrics):
Released by eBay and available at htttp:///githbb. com/ eBay/RAMSynCoderal Although no explicit license is provided, the dataset is publicly hosted and widely referenced in research.
SMD (Server Machine Dataset): First introduced in the OmniAnomaly paper 37 the dataset is hosted License.
under the MIT
SWaT (Secure Water Treatment): Developed by iTrrust, SUTD. The dataset requires formal access request via http:::/ttrutt...ettt sutd. edu.sg/ Use permitted under non-commercial academic research agreement.
22

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B
Metrics
Table S2: Comparison of evaluation metrics for seme-series anomaly detection. 'Partial' denotes
limited or conditional support depending on implementation details (c.g., decay functions, overlap thresholds).
Metric
Granularity
Threshold-Free
Temporal-Aware
Partial Credit
Imbulance-Robust
Standard FI
Point-wise
No
No
Partial
Range-based Fl
Segment-wise
No
Partial
Yes
Partial
Atilistion Fl
Segment-wise
Yes
Partial
LUC-ROC
Point wise
Yes
No
No
No
AUC-PR
Point-wise
No
Yos
VUS ROC
Point+Time
Yes
Yes
Yes
Yes
VUS
Point+ Time
Yes
Yes
Yes
Yes
Table S2 summarizes the core properties of evaluation metrics commonly used in time-series anomaly detection. Each metric captures different facets of detection quality, including temporal alignment, threshold sensitivity, and tolerance to misclassification. Notably, no single metric is universally optimal: metries vary in granularity (point- vs. segment-level), and in their capacity to reflect delay, calibration, or imbalance.
In our view, using multiple complementary metries is essential for fair and comprehensive assess- ment. Different datasets, anomaly types, and operational goals may favor different evaluation criteria. For instance, segment- based metrics may benefit coarse anomaly annotations, while ranking- -ased metrics offer insight into score separation. However, overreliance on permissive or post hoc-adjusted metries may mask modeling deficiencies, especially in time-sensitive applications.
We assume that multivariate time-series anomaly detection (MTSAD) demands stricter evaluation. Point-wise and delay-sensitive metrics, such as Standard F1 and VUS, impose higher requirements for detection precision, temporal consistency, and calibration. While these metrics may be unforgiving, they serve as catalysts for improving the model itself, encouraging detection systems that not merely explainable or reactive, but fundamentally precise.
B.1
Standard F1 Score (Point-wise)
The standard Fl score is defined as the harmonic mean of point-level precision and recall:
TP
TP
Precision
Recall
(S1)
TP + FP
TP+FN
F1 2 Precision Recall
(S2)
Precision Recall
Despite its well-known limitations, Fl remains one of the most rigorous indicators of model capability in pointwise anomaly detection. Evaluating each time step independently avoids assumptions about anomaly structure and instead forces the model to separate normal and anomalous scores clearly. This property can be harsh for weak models, but acts as fair and objective measure for well-calibrated ones.
Temporal fragmentation and misalignment often cited as weaknesses of F1. However, argue that a strong model should inherently produce temporally coherent anomaly scores, without relying on the metric compensate for structural noise. In this view, fragmentation bias is not failure of the metric, but meaningful signal about a model's inability to maintain temporal consistency.
Fl does require thresholding, which introduces variability across datasets. However, in controlled evaluation settings, identifying the optimal threshold gives insight into a model's best achievable distinction and reflects the strength of score contrast between normal and abnormal patterns.
Finally, Fl partially robust to class imbalance due to its exclusion of true negatives, though its reliability degrades in extremely sparse regimes. Overall, we consider F1 a high-precision, low-tolerance metric that is particularly suitable for isolating model quality in tightly controlled, point-level evaluations.
23

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B.2 Range-based F1 Score [88
We adopt a range-aware variant the Fl score implemented in the TSB-AD benchmark suite [24] which provides finer-gree finer-grained assessment of segment-level anomaly detection. Unlike standard range-based Fl that uses binary overlap rules match predicted and ground-trune segments, this version assigns partial credit based on three interpretable components: existence, overlap, and cardinality.
The range-based recall defined as:
Recallrange
(w,,,, CF(R,, + ER((R)))
(S3)
R
and Precisionrang is computed symmetrically by swapping the roles of predictions and ground truth (with 0). Here, P PPm denotes the set of predicted anomaly time steps, and R {RI.... R the of ground-truth anomaly ranges.
For each ground-truth range R, a weighted overlap score w (R; P) measures the fraction of the range covered by predictions using bias-adjusted coverage function. cardinality factor CF penalizes cases where multiple predicted segments correspond to single ground-truth anomaly, while an existence reward ER grants partial credit if any prediction intersects the range. The parameter [0, balances the overlap and existence terms.
The final score is computed as the harmonic mean:
2 Precisionaagee Recallrange
Fl Fange
(S4)
Precision Recall
While this formulation enables smoother credit assignment and interpretable decomposition of detection quality, its reliance hyperparameters (e.g., bias function, a) and the aggregation procedure makes it sensitive to implementation details. Therefore, standardized parameter settings are essential for fair cross- comparisons.
B.3 Affiliation F1 Score 16
The Affiliation Fl score evaluates prediction quality by assigning partial credit based temporal proximity between predictions and ground sounh- anomaly segments. Using soft decay functions f(-), it rewards predictions that "close enough," even if not perfectly aligned.
The score is computed as:
R
Precision
1 m
ΣΣ
Jp
min d(p,, R.)) Recall
1
min d d(Ri:.))) PEEP
(S5)
R.ER
R
2. Precision Recallan
Flair
(S6)
Precisionaa Recallaft
where function d(p,, R₂) computes the temporal distance between prediction p, and range R,
Though often described as temporally sensitive, Affiliation F1 tends to produce inflated scores when predictions are loosely aligned with anomaly segments. The decay function ʃ(.), which grants high credit to any prediction within short temporal window, effectively smooths over temporal emror.. As a result, predicting only a small portion of segment or merely grazing its edge may yield disproportionately high Fl.
This behavior is analogous to post hoc adjustments like point- or range-based scoring. does not penalize imprecise or fragmented detection, and may mask deficiencies in model's ability to produce temporally coherent scores. While this soft matching is useful when ground-trui labels are coarse or ambiguous, we argue that it should be treated primary measure of detection capability.
modellinddeendent alignment compensator, not
In our view, high-performing model should not rely on the metric to explain away timing enrors. Affiliation Fl may serve as a secondary diagnostic tool to assess tolerance to misalignment, but we do not consider it sufficient for rigorous performance evaluation.

---

B.4 AUC-ROC and AUC-PR 7
AUC-ROC and AUC-PR measure the gireshold-indeeedden ranking quality of anomaly scores. For a continuous anomaly score function, they integrate model performance across all thresholds.
The AUC-ROC computes the under the TPR-FPR curve:
TP
FP
TPR
FPR FP+TN
(S7)
+P+
AUC-ROC = [(' ((PP(()))))
(S8)
while AUC-PR evaluates the precision-recall trade-off:
AUC-PR Precision(r) d(Recall(r)).
(S9)
Although widely adopted, these metrics ignore the critical temporal structure in time-series anomaly detection (TSAD). In particular, they are insensitive to detection continuity, localization lag, and whether anomalies are detected early, partially, at all. As emphasized by Liu et al. [24), model that correctly ranks only single point in a long anomaly segment can still achieve high AUC, even if it fails detect the anomaly holistically.
Additionally, AUC-ROC is known to be unreliable in highly imbalanced settings, since the false positive rate denominator includes the large pool of true negatives. This allows naive models to achieve deceptively high AUC-ROC scores by merely avoiding false positives, without demonstrating meaningful detection. AUC-PR partially addresses this by focusing only the positive class, making it more appropriate for sparse anomaly regimes.
Nonetheless, both AUC metrics are temporally agnostic and may obscure model behavior over time. We report them for completeness, but caution against over-reliance in TSAD tasks where the timing and continuity of detection are central.
B.5 VUS-ROC and VUS-PR (30
Volume Under the Surface (VUS) metrics evaluate the consistency and timeliness of anomaly score rankings by integrating performance over both threshold levels and temporal tolerance windows. Given a set of allowable lags Ω fwi, w2, ... }. where each w defines symmetrie temporal buffer, the predicted points are matched to ground-truth anomalies if they fall within dow time steps. For each w, we compute a corresponding AUC score:
VUS-ROC
Σ
AUC-ROC,,,
(S10)
WER
VUS-PR Σ
AUC-PR...
(S11)
WER
VUS metrics address two limitations of classical AUC: their insensitivity to detection delay and their lack of temporal context. By explicitly modeling detection within variable tolerance windows, VUS captures how consistently model produces timely, high-ranking anomaly scores across a range of thresholds. This makes it strong tool for evaluating delay-aware detection performance, especially in streaming or real-time settings.
Unlike segment-level metrics such Affiliation F1, which apply smooth decay functions to soften the penalty for temporal misalignment, VUS does not interpolate between correct and incorrect predictions. It avoids heuristic credit assignment and instead tests whether the model can repeatedly detect anomalies within strict or relaxed time boundaries, without requiring threshold or decay tuning. This makes it better measure of intrinsic anomaly scoring quality.
We find that Affiliation F1 and VUS serve complementary purposes: Affiliation Fl suitable for analyzing misaligned segment predictions in the presence of noisy labels or lag-prone detection. At the same time, VUS offers a more principled, high-resolution assessment of detection precision and consistency across time and score thresholds.
25

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C
Baselines
C.1 Implementation Details
All bascline models were implemented using official or widely adopted public repositories, with minor formatting adjustments to ensure evaluation consistency. Hyperparameters were either taken from the original papers or selected via lightweight tuning. All input features were normalized using StandardScaler fitted on the training set (zero mean and unit variance), and no separate validation set was used. Each model was trained and evaluated using only the provided training/test splits. To ensure robustness, we repeated all experiments with five independent random seeds and report the average performance. As described in Section 4.1, we apply optimal threshold selection for each method to decouple detection performance from threshold sensitivity. The threshold search procedure is described in Appendix D.3 and standard deviations across seeds are reported in Appendix E..
C.2 Codebase of Implementation
Most baseline models, excluding OmniAnomaly and SARAD, are implemented using the unified codebase of CATCH:
CATCH (ICLR2025); github. fet.tb/ccmmeeeeiiiiiiieellllllee
Below, we list the official implementation links for all baseline models used in our study;
OmniAnomaly (KDD 2019): github. hatp:///.c.//eeaaaala//////aaaaa
GooomllyTranffommn (ICLR 2021): httpa//....//ooo/aaaaooooooaaaaa
PatchTST (ICLR 2022):
TimesNet (ICLR 2022): g.thub.ccamm/humm///nneeeell
DLinear (AAAI 2023): gLhhubb.cca//oonyye11111//11111
NLinear (AAAT 2023): LiLubb..cco/hooeyyee111/////Lbrr
DCdetector (KDD 2023): gbapboa.co//aaae0.a.1//000030100
iTransformer (ICLR 2023): g.thub.caa/hhmml///raatfoooo
TCN (ICLR 2023): gtttdb.ccm//uoochh///odeer11111
SARAD (NeurIPS 2024): githbb..cm/daaldhhha//AARDD
26

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D More Details of Experiment
D.I Implementation
We trained all models using the A (26] optimizer with default hyperparameters, provided stable convergence across datasets. To account for dataset-specific scale differences, we set the learning rate to 5e-5 for PSM, while higher value of 4 was used for both SMD and SWaT. Prior to training, all input features were standardized using a SraadardScales fitted on the training set, ensuring zero mean and unit variance. Since our evaluation protocol does not rely on validation-based model selection, we did not use a separate validation set. Instead, each model was trained and evaluated solely on the provided training/test splits.
D.2
Hyperparameter Sensitivity
We evaluated the sensitivity of OracleAD to four key hyperparameters: the window length, the batch size, the deviation weighting factor Adev and the number of encoder and decoder layers. The analysis was performed on three benchmark datasets, and the results are summarized in Figures ST through S4. using F1, VUS-ROC, and VUS-PR as evaluation metries.
For the SMD dataset, we followed the protocol from OmniAnomaly!! treating each machine as an independent subset and selecting representative subsets for this analysis to avoid overlap with the evaluation set. The reported performance is averaged over these selected subsets.
The default configuration used across all datasets fixes the window length to 10, the batch size to 1024, the deviation weight to 3, and the number of layers 2. While this setting does not always achieve the highest performance, it offers a stable balance between accuracy and training efficiency.
Window length. As shown in Figure ST OracleAD achieves its best performance with window lengths of 10 or 20 across most datasets, including PSM, SMD, and SWaT. Larger windows such as 40 occasionally improve VUS-ROC, but often lead to reduced precision in VUS-PR, likely due the dilution of localized anomaly patterns. On the other hand, smaller windows such as 5 may fail to capture sufficient temporal context. Overall, OracleAD performs robustly within the range of 10 20, suggesting that this moderate window length is well suited for capturing both short-term dynamics and broader structural dependencies.
Batch Size. Figure S2 shows that as batch size increases, performance improves in general. For instance, in the case PSM, Fl score rises from 45.77 65.85 as batch size increases from 64 to 1024. This trend suggests that OracleAD benefits from larger batch statistics during training, potentially due more stable gradient estimates.
Deviation Weight Adev- As shown in Figure S3 increasing the structural regularization weight Adev consistently improves performance across metrics. This effect is most prominent in PSM and SMD, where larger À values enhance VUS-ROC and VUS-PR score. The results confirm that deviation scoring based on the Stable Latent Structure plays a key role in anomaly localization and benefits from stronger guidance. In our main experiments, we set Adev = as offers the best balance between prediction and structural alignment.
Model Depth. Figure illustrates the impact of varying the number of encoder and decoder layers. Using two layers generally leads the best overall performance across datasets. This setting provides sufficient model capacity to capture temporal and structural dependencies without overfitting, Increasing the number of layers three does not consistently improve performance and, in some cases such as PSM and SMD, leads to degradation in Fl score. These results suggest that additional depth may introduce unnecessary complexity or optimization instability, particularly in datasets with limited variability or high noise. Therefore, a moderate depth of two layers strikes good balance between expressiveness and generalization.
27

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2
Window Length
Window Length
Window Length
PSM
SMD
SWaT
Figure S1: Effect of input window length on OracleAD performance. Each subplot shows a different evaluation metric. All results averaged five random seeds.
V-BLOC (%
1:PR
21
U0D4
25
Batch Size
Batch Size SMD
Batch Size
PSM
SWaT
Figure S2: Effect of training batch size on OracleAD performance. Each subplot shows a different evaluation metric. All results averaged five random seeds.
V-RDC (%C)
V-PRR
Lambda
Lambdu SMD
Lambda
PSM
SWaT
Figure S3: Effect of the deviation regularization weight Adev on OracleAD performance. Each subplot shows different evaluation metric. All results averaged over five random seeds.
150
V-ROC.OM (%
V-PR (96)
F35
650
535
Depth
Depth
Depth
PSM
SMD
SWaT
Figure S4: Effect of the encoder and decoder depth on OracleAD performance. Each subplot shows different evaluation metric. All results are averaged over five random seeds.
28

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D.3 Thresholding Strategy
We adopt score calibration procedure that selects the optimal threshold based on a target evaluation metric to convert continuous anomaly scores into binary predictions.
Given sequence of anomaly scores { and corresponding binary ground-truth labels (t Y the optimal threshold r* is obtained by maximizing an evaluation metric M over discrete candidate set T:
arg max TET
M ,y),
(S12)
where V/ Hfss 7| denotes the binary prediction time based on threshold 7.
The candidate set 7 is constructed by uniformly partitioning the score range into N bins:
i min(s) + N (max(s) min(s)). i 0, 1..,,.....NN N-1
T
(S13)
We set N = 200 all experiments, following the protocol used by Liu et al. [24) and adopted in the official TSB-UAD benchmark suite" This value offers good balance between threshold granularity and computational cost, and has been shown to yield stable evaluation results in large-scale studies. The threshold search performed independently for each model and dataset to ensure fair and metric-sensitive calibration.
D.4 System Configuration
All experiments were conducted on single workstation equipped with NVIDIA GeForce RTX 5090 GPU (32GB) and an Intel Core Ultra 7 265K CPU (20 cores). The system had 96GB of DDR5 RAM and ran Ubuntu 24.04 LTS (64-bit). The software environment included Python 3.12, PyTorch 2.7.0, and CUDA Toolkit 12.8.
http::///gtaabb.oo//ootttttt///
29

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D.5
Computational Cost
separate LSTM encoder decoder for each of the N variables, giving a per- mncoder-decoder window temporal cost of O(N T). Multi-head elff-attention requires O pairwise comparisons, leading an overall per- per-window complexity of O ( + Although this scales superlinearly with N, the quadratic cost is comparable to basic correlation computations and is common requirement models that capture explicit inter-variable structure.
OracleAD employs
We empirically profiled training and inference efficiency across real-world (PSM, SMD, SWaT) and synthetic datasets (up N 400 variables). Training results (Table (S3) show that while memory usage increases with dimensionality, throughput remains practical. Inference throughput (Table(S4) demonstrates that OracleAD remains feasible for real-time monitoring in systems with hundreds of variables.
Table S3:
Training costs across datasets and batch sizes.
Dataset (N)
Batch Size
Memory Usage
Training Speed (it/s)
PSM (25)
64
1.5 GB
12.6
1024
8.5 GB
9.6
SMD (38)
64
1.9 GB
8.0
1024
13.0 GB
6.8
SWaT (51)
64
2.4 GB
6.0
1024
18.5 GB
5.0
Syn-PSM (100)
64
4.7 GB
3.1
512
23.3 GB
2.9
Syn-PSM (400)
32
20.4 GB
1.35
Table S4: Inference throughput by dataset dimensionality.
Dataset (N)
Inference Speed (time steps/s)
PSM (25)
54.0
SMD (38)
37.0
SWaT (51)
28.0
Syn-PSM (100)
13.5
Syn-PSM (400)
3.15
30

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E Additional Experiments
E.1
Full Evaluation Results with Classical and Deep Baselines
Table S5: Performance (%) of anomaly detection models the PSM. SMD, and SWaT datasets. Boldface indicates the best score and underline indicates the second-ber score.
Duizsel
Metric
PCA
HBOS
1F
A.E
Omzii
A.T
Paich
T:Net
DLin
NLin
DC
iTans
Modem
SARAD
CATCH
OrackeAD
R-FI
46.18
47.74
45.78
43.93
40.58
12.17
34.75
37,49
34.74
32,10
21.35
35.91
35.48
13.54
54.19
$4.66
ATT-F1
73.90
72.77
$1.98
73 70
73.49
69.6 68
69
69.43
69.43
69.43
69.59
69.
69.43
77.12
78.16
78.07
PSM
A AOOC
64.82
61.98
54.23
66
63:95
38.35
58.68
59.09
58.02
58.56
49.96
59.
59.21
02.86
64.75
$4.78
A-RR
46.75
39.34
33.41
47.27
45.00
24.28
37.91
36,95
37.17
37.64
27.78
38.27
38.59
41.67
43.40
68.11
V.RUC
56.47
08
84.24
40
46.71
44.54
45.95
F1
21,97
17.66
14.91
25. 25.78
32.16
7,95
7,98
7,95
7,95
7.98
8.00
7.96
7,98
25.92
7.98
43,03
AEE-1
$3.50
67.43
75.87
67.88
71:25
71.85
59.09
72.73
73.83
49.97
74.57
72.13
50.96
A-PR
12.75
14.53
12:17
19 40
27.73
4.57
14.82
13.66
13.85
14.10
4.16
14:57
13.01
25.87
17.09
44.83
V-ROC
66.50
65.3 24.24
68.58
74.38
75.26
50.34
51.66
$1.10
51.52
$1.76
49.92
51.76
51.55
655.57
50.95
69 57
V-PR
15,02
16.09
16.02
22.50
31.18
36. 86
41.5 11.60
41,21
41.36
79.15
4.27
32.85
41.55
19.33
35.25
47.52
Fl
75.36
76.04
21.65
74.46
75:40
21.65
21.65
21.65
7.98
12.14
21.65
21.65
21.65
$7.30
21.65
76.50
R-F1
29.84
20.18
14:77
56.63
38.73
15.48
12.25
20,50
5.13
12.06
15.54
11.89
12.88
21.37
14,74
28.
AIT-FI
72.18
70.63
72.04
75.30
72.09
70.06
69.51
73.47
67.43
69.68
69.25
69.
72.05
74.88
73.68
71.97
V-R0C
81.75
83,49
41.37
81
81.98
50. 73
49.5
20.80
50 36
49.00
50.00
49. 90
86.30
51.89
R2.42
VPR
32.31
73.90
10.17
63
89
64.42
17.00
11.92
31,20
28.47
10.55
11.92
12.05
13.05
62.72
18.70
74.16
This section presents the complete evaluation results of OracleAD alongside both deep learning and classical anomaly detection baselines on the PSM, SMD, and SWaT datasets. While deep learning models were trained and evaluated over five random seeds to capture stochastic variation, classical baselines such as Principal Component Analysis (PCA) (36), Histogram -Based Outlier Score (HBOS) [14,, and Isolation Forest (IF) [33 were evaluated under fixed random seed. This because they are either fully deterministic or only marginally affected by random initialization, and thus produce effectively consistent outputs.
Interestingly, as shown in Table S5 several classical baselines outperform many deep learning models across range of metries. This reflects broader phenomenon we also highlighted in Section [ complex or novel architectures do not necessarily lead to better performance. We observe that many deep models achieve their highest Fl scores when recall maximized, meaning recall is close to one. This implies that thresholding tends to favor high precision, often at the cost of missing many true anomalies, especially those that fall within labeled anomaly segments. Consequently, these models often struggle on strict evaluation metrics such as VUS-PR and VUS-ROC, which emphasize consistent detection quality across thresholds and accurate alignment with ground-truth structure.
In contrast, classical models tend to produce relatively strong AUC-ROC and VUS-ROC scores. This indicates that while they have high overall detection rates, they often lack precision stability, which leads to weaker performance on precision-sensitive metrics like AUC-PR. The combination of high recall and low precision reveals that these models are often overly sensitive, generating many false positives.
Taken together, these findings clarify OracleAD's distinctive advantage. It is explicitly designed to address both detection coverage and accuracy through a principled dual-scoring mechanism that combines prediction error with structural deviation. Unlike many deep models that overfit to single scoring heuristic, OracleAD integrates temporal and inter-variable signals, resulting in stable performance across broad range of evaluation criteria. This is particularly evident in metrics such as VUS-PR and VUS-ROC, where OracleAD consistently outperforms both classical and deep learning baselines.
31

---

E.2
Ablation study on the SMD dataset
Table S6: Ablation results (%) on selected SMD subsets. All results are averaged over five random seeds.
machine
machine 2-3
machine 2-7
machine 3-6
Component
Variant
Loss Function
w/o Recon Loas
59.61
81.82
49.24
51.94
73.70
55.60
80.07
81.33
76.94
34.25
65.51
35.37
w/o Pt
23,31
72.35
11.13
50.29
80.26
51.84
81.50
79.14
62.17
34.18
78.23
22.95
Anomaly Score
w/o D
54.32
86,25
49.60
52.89
86.60
42.86
82,66
86.94
71.66
46.03
89.76
50.70
OracleAD (Full Model)
62.03
82.44
52.15
53.99
81.71
54.83
80.35
81.75
76,42
44,39
77.32
43.11
The SMD dataset is characterized by its diversity in anomaly types, ranging from abrupt point anoma- lies to gradually developing structural failures. This heterogeneity results in variable performance across scoring mechanisms depending on the subset. For instance, the prediction- banded score Prr score often excels when anomalies manifest as sharp temporal deviations, whereas the deviation-basse based score Lecore is more effective when anomalies involve shifts in inter-variable structure. Table S6 shows that each score alone captures certain types of anomalies well, but fails to generalize across the full spectrum. In contrast, combining both scores consistently yields more stable performance across all subsets, striking balance between temporal precision and structural sensitivity.
The role of the reconstruction loss also varies across subsets. In machine 2-7, excluding the recon- struction objective has little impact performance, suggesting that prediction alone suffices model normal dynamics. However, in subsets like machine 2-3 and 3-6, removing reconstruction objective significantly degrades structural metries such as VUS-PR and causes instability in overall detection performance. This indicates that the reconstruction loss is a form of regularization that stabilizes the latent space and improves robustness, particularly when the data contains noise or complex normal variation.
These findings confirm that OracleAD's complete model design offers the most consistent and generalizable performance across the varied anomaly profiles present in SMD.
E.3
Statistical Significance
Table S7: Standard deviation (%) of anomaly detection performance across seven evaluation metrics on three benchmark datasets (PSM, SMD, and SWaT). All values are computed over five random seeds for each model. Lower values indicate higher stability and consistency across runs. Metrics include F1, R-FI, Aff-F1, AUC-ROC, AUC-PR, VUS- ROC. and VUS-PR.
Dataset
Metric
A.E
Omni
A.T
Patch
TaNet
DLin
NLin
iTrnn
Modern
SARAD
CATCH OracicAD
1.29
0.00
0.00
2.9%
R-F1
0.48
2.49
7.50
0.44
0.61
0.61
0.11
24.03
1.67
0.38
7.54
3.46
2.93
Aff-FI
0.24
0.65
0.55
0.00
0.00
0.00
0.00
0.36
0.23
0.00
4.30
4.76
0.94
PSM
A-ROC
1.07
1.89
2.10
0.06
0.56
0.07
0.05
0.29
0.11
0.14
1.52
0.78
2.49
A-PR
0.59
2.34
0.88
0.10
0.46
0.05
0.05
0.24
0.11
0.18
1.26
0.75
4.30
V-ROC
0.96
0.56
0.95
0.05
0.03
0.08
0.01
0.56
0.24
0.03
4.98
5.14
2.47
V-PR
0.56
0.52
13.09
0.76
1.40
0,80
0.51
0.82
1.02
0.46
1.75
0.87
4.31
1.60
0.11
0.00
0.00
0.00
0.00
0.00
0.04
0.00
0.00
0.35
0.00
0.92
R-F1
2.48
0.26
3.69
0.44
0.84
0.48
0.15
5.40
1.21
0.18
2,28
0.14
0.93
AIl-F1
1.28
0.29
0.00
0.00
0.00
0,00
0.00
0.11
0.00
0.00
0.35
0.00
0.78
SMD
A-ROC
0.98
0.07
0.27
0.24
0.42
0.11
0.11
0.15
0.47
A.PR
1.78
0.11
0.10
0.13
0.32
0.06
0.01
0.25
0.18
0.06
2.53
0.22
0.90
V-PR
2.20
0.10
4.35
0.60
0.52
0.47
0.26
0.07
0.45
0.34
0.37
0.89
1.34
FI
2.34
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3.29
0.00
0.31
0.12
2.72
FI
0.77
0.05
0.96
0.40
0.64
0.00
0.08
0.00
0.00
0.72
1.51
0.94
1,80
SWaT
A-PR
6.48
0.02
0.41
0.02
0.09
0.01
1.10
0.03
0.01
0.18
12.61
0.35
1.99
V-ROC
0.67
0.12
0.02
0.10
0.06
0.00
0.08
0.06
0.16
0.90
V.PR
7.38
3.58
0.42
0.03
0.76
0.04
0.00
0.31
0.20
0.19
9.02
0.73
4.38
32

---

E,4
Distance Metrics for Dissimilarity Matrix
Table S8: Performance comparison of cosine similarity, LI, and L2 distances for latent vector dissimilarity the PSM dataset. All results are averaged over five random seeds. Boldface indicates the best score.
Method
F1
Aff-F1
A-ROC
V-ROC
Cosine Sim.
46.81
43.68
78.18
66.46
48.59
65.83
48.82
57.72
51,54
75.87
76.33
54.29
73.33
53.52
65.85
54.66
78.07
84.78
68.11
84.24
68.17
In Eq. 11 we define the pairwise dissimilarity matrix using the 1.2 distance. We choose I.2 over other metrics because captures both directional mismatch and magnitude differences (latent energy variation) in the embeddings. To validate this choice, we empirically compared L2 against cosine similarity and LI distance across multiple evaluation metrics. As shown in Table S8 L2 consistently outperforms the alternatives, yielding higher and more stable performance across live random seeds.
E.5 Variance Dynamics of Dissimilarity Matrices
Variance Across Training Steps
0.0004
Variance
0.0003
0.0002
0.0001
0.0000
2
3
5
8
9
11 Epoch
10
12
13
14
15
16
17
18
19
20
Figure S5: Variance of dissimilarity matrices across epochs on the PSM dataset. All results are averaged over five random seeds.
To further analyze the stability of the Stable Latent Structure (SLS), track the variance of the dissimilarity matrices (A) across training epochs. This variance reflects how much the learned inter-variable relationships fluctuate from one sliding window to another within the same epoch.
Figure S5 shows the variance trend on the PSM dataset. The trajectory can be divided into two phases: in the early epochs, variance increases rapidly as the model explores latent relational geometry from sliding windows, reflecting active adaptation consistent with the reviewer's ' observation that rising variance indicates meaningful structural learning; later epochs, variance gradually decreases and converges, showing that the latent representations stabilize toward a consistent inter- variable template. This stabilization confirms that the learned SLS serves as robust baseline for normal relational structure.
33

---

E.6
Comparison of Anomaly Score Fusion Strategies
Table S9: Performance comparison between multiplicative and additive anomaly score fusion on PSM and SWaT datasets. Results are averaged over five random seeds.
Dataset
Anomaly Score
R-FI
AfF-FI
A-ROC
A-PR
V-ROC
V-PR
PSM
Plome
54.66
78.07
84.24
68.36
55.48
78.52
85.03
67.19
84.82
67.43
P koms
DI
76.50
28.15
71.97
82.71
72.39
82.42
74.16
P
Soore D
76.91
37.75
73.43
81.73
71.73
81.42
71.07
To find an effective anomaly score construction, we compare two fusion strategies that combine the temporal prediction score(P secre) and the structural deviation score(Di (Dooo)). Their fusion determines how OracleAD balances temporal precision and structural sensitivity.
Table $9 presents results for multiplicative and additive fusion on PSM and SWaT. While additive fusion also achieves strong performance, it requires dataset-specific tuning to balance the relative score and D Desonee their scales and sensitivities can differ across domains. In contrast, multiplicative fusion inherently balances the two by emphasizing co-occurring signals, leading to sharper and more consistent anomaly responses. Furthermore, higher A-PR and V-PR scores indicate that the multiplicative form is more effective reducing false positives, making it better suited for real-world applications where reliability and precision are critical.
influence of P
Overall, both methods achieve competitive performance, but the multiplicative design better aligns with OracleAD's 's objective, providing improved stability and interpretability in multivariate detection.
34

---

Case Studies for Interpretability
F1
Complex Multivariate Anomaly
1
0.3
0.8
Var 15
0.775
Prediction Score
0.50
0.25
Deviation Soore
(Ame Momamn
1.0-
1.00
Anomaly
maat Mon will Mnmmend Wnndoman
50000
Figure S6: Raw signals of the identified root-cause variables and anomaly score visualization for selected intervals the PSM dataset. Shaded red regions indicate ground-truth anomalies.
[-48300
I-48700
Vertable
65
-5
22
24
Variable Indox
Variable Tndex
Vartable Indox
Variable Tadex
matrix at anomalous time points in PSM dataset. Each matrix corresponds respectively to the subfigures in Figure (S6] Each matrix is independently min-max normalized and values are clipped to the range [0.3, 0.9] to enhance contrast.
Figure S7: Visualization for Deviation Matrices D
Figure presents case from the PSM dataset which a single anomaly segment contains multiple overlapping anomaly signals across different variables. While the majority of variables in the PSM follow relatively stable patterns, several variables (e.g.. 1, 3, 15, 24) exhibit subtle yet complex dynamics that makes it difficult to detect through prediction alone. In the highlighted anomaly segment, these variables show anomalous behavior either concurrently or sequentially, creating challenging detection scenario.
The prediction score captures local deviations in the latter part of the segment but remains relatively low fragmented in the earlier phase. In contrast, the deviation score maintains more consistent profile, reflecting sustained disruptions the latent structure. This contrast demonstrates how the deviation score more effective in representing the internal disorganization of the system, especially when multiple variables interact in non-trivial way.
Overall, this case illustrates that OracleAD's 's dual-scoring mechanism enables anomaly detection that both sensitive and robust across a variety of complex conditions. The prediction score is responsive to short-term deviations in individual variable behavior, while the deviation score captures structural inconsistencies across variables over time. Together, these two signals allow OracleAD handle heterogencous and overlapping anomaly patterns effectively.
35

---

F.2 Long-Duration Anomaly
300
250
200
Prediction Soore
200
Devitton Soore
1000
Anomaly Soore
220000
225000
230000
235000
240000
245000
250000
255000
260000
265000
270000
275000
Time
Figure S8: Raw signals of the identified root-cause variables and anomaly score visualization for selected intervals the SWaT dataset. Shaded red regions indicate ground-truth anomalies.
I=245000
Figure $9: Visualization for Deviation Matrices Dr manis anomalous time points in SWaT dataset. Each matrix corresponds respectively to the subfigures in Figure (8 Each matrix is independently min-max normalized and values are clipped the range [0.3, 0.9] to enhance contrast.
Figure S8 illustrates a long anomaly segment in the SWaT dataset, notable for persistent and structurally coherent anomaly behavior. Among all available sequences, this segment contains longest continuous anomaly window, allowing us to evaluate OracleAD's interpretability sustained abnormal context.
As shown in Figure S8 variable deviates sharply and persistently from its normal operating range throughout the entire anomaly segment. This deviation is visually evident and positions variable 26 as the dominant root cause of the anomaly. Both the prediction and the deviation scores rise significantly within the anomaly interval, the deviation score plays pivotal role in this case.
The deviation matrices, visualized in Figure 59 further articulate the identification of root-cause variable. Across multiple time stamps within the anomaly segment, variable 26 consistently dominates the matrix in terms of deviation strength. This pattern indicates its sustained role the structural root cause.
This case demonstrates how OracleAD leverages deviation scoring to detect sustained and structure- driven anomalies that may not cxhibit dynamic variation within individual variables. By modeling the relational structure among variables, OracleAD maintains sensitivity even when traditional prediction-based detectors may fail due to temporal flatness.
36

---

G Quantitative Analysis of Interpretability
Table S10: Top-3 predicted variables ranked by aggregate deviation scores compared with ground- truth causal variables across representative anomaly timestamps in the SMD dataset. Each case is drawn from a distinct anomaly segment.
2106
4-49
t=6032
t=
14936
15005
Var. Score
Var.
Score
Var.
Score
Var.
Score
Var.
Score
Var.
Score
Var.
Score
Var.
Score
Top-3 Predicted
13 9.00
13
9.49
32
15.00
29
15.99
33
12,84
13
9.39
13
13.55
15 55.97
12 3.46
7.36
33
10.16
33
4.48
32
6.11
8
7.50
12
6.62
49.98
8 3.91
5.79
31
3.49
34
2.95
35
3.42
12
5.57
8
5.56
45.22
Ground Truth
12, 13
8, 12, 13
32,
29
32, 33
12,
13.
14
8.
12,
13,
14
10. 15, 18
tw 21 106
Iw4506
[we60 32
[w76687
Verabll onder
Variafle Trdees
Varable Infwx
Vessbbl Index
Varoeble
Indee
Vustilo Index
Varable Indox
Vusibbl Index
Mariable Indos
Figure S10: Deviation matrices correspond to deviation matrix visualization in Table S10 Row and column activations highlight the variables most responsible for abnormal behavior, and red boxes mark the ground-truth root-cause variables. For visualization, each matrix is independently min-max normalized and values are clipped to the range [0.3, 0.9] to enhance contrast.
A distinctive strength of OracleAD lies in its ability to localize root-cause variables through the deviation matrix D Piaanix D SLS| Each entry measures the extent which variable pair's relationship deviates from the stable latent structure, and row-wise aggregation highlights which variables most strongly disrupt normal interactions.
Figure shows deviation matrices at representative anomaly timestamps from the SMD dataset. Each anomaly produces characteristic disruption pattern. As discussed in Section 4.3 variables associated with multiple high-maagittue rows or columns are more likely to correspond to root causes. This enables not only root-cause analysis but also a clear visualization of how structural disruptions in key variables evolve over the course of an anomaly segment.
For precise localization, however, rely on the aggregate deviation score obtained by summing each row (or column). This yields a variable-level anomaly score that can be ranked to produce candidate root causes. As shown in Table [S1) most cases quantitatively validate this approach. The top-3 ranked variables coincide with the ground-truth causal set, and even for multivariable anomalies, OracleAD reliably recovers the majority of causal variables.
37

---

H
Broader Impacts
Anomaly detection in multivariate time series data has wide-reaching applications across domains such as industrial monitoring, cybersecurity. healthcare, and environmental systems. The ability detect abnormal behavior timely and interpretable manner can improve operational safety, prevent financial or physical damage, and assist in early diagnostics. OracleAD aims to address key limitations of existing models by offering interpretable, structure-aware, and segment-sensitive anomaly detection.
However, with all data-driven systems, there potential risks. If deployed in high-stakes domains such as healthcare or autonomous control systems, undetected anomalies or false alarms may lead to significant consequences. Furthermore, the model's performance relies heavily on the quality and representativeness of the training data. If the training data lacks diversity or contains unannotated anomalies, the model may reinforce biases or exhibit degraded performance underrepresented conditions.
To mitigate such risks, practitioners should carefully evaluate the model under realistic operational conditions and complement automated decisions with human oversight, particularly in critical appli- cations. We believe that OracleAD can contribute positively by promoting reliable and interpretable anomaly detection, but responsible deployment and continuous evaluation are necessary to ensure its benefits are realized without unintended harms.
I Limitations and Future Works
While OracleAD demonstrates strong generalization and interpretability across diverse benchmarks, several limitations remain. First, Stable Latent Structure (SL.S) assumes globally consistent inter-variable relationships learned from continuous-valued inputs. This assumption may not hold in complex or multimodal systems such as SWaT, where heterogeneous subsystems or mixed variable types (e.g., continuous and discrete sensors) coexist. Under such conditions, the deviation matrix may lose discriminative sharpness, and the anomaly scoring can become less reliable when one component signal (prediction deviation) weakens.
To address these challenges, future work will focus on developing (I) an anomaly detection framework that is agnostic to variable types, seamlessly handling continuous, categorical, and discrete signals; and (2) dynamically updated SLS mechanism that adapts over time rather than remaining static during inference. These extensions would enhance OracleAD's applicability to large-scale, dynamic, and heterogeneous industrial systems.
38


Reviews:
[{"review_id": "UzSrOCPmoo", "reviewer": "Reviewer_chjA", "summary": "This paper proposes a multivariate time series anomaly detection framework named OracleAD. Its core contribution lies in proposing a novel perspective on anomaly evolution: an anomaly first manifests as a disruption in a single variable's \"temporal causality,\" which then propagates to cause a \"structural\" anomaly in the relationships between variables. Overall, the paper presents a novel concept, demonstrates excellent model performance, and possesses a degree of interpretability. However, OracleAD relies on a specific model architecture (LSTM) and a key assumption (a stable latent structure), and the generalizability of these components warrants further investigation.", "questions": "1. The window length used in the experiments is notably short. Can you elaborate on the rationale for this choice? Does this imply that OracleAD's performance is insensitive to long-range dependencies, or that the studied anomalies are primarily local in nature? Can you provide a sensitivity analysis on the window length T to show its impact on model performance?\n2. The static assumption for the SLS may limit the model's applicability in scenarios with concept drift. Do you consider a mechanism for dynamically updating the SLS, for instance, by using a moving average window or by introducing an online update step during inference?\n3. The model employs a separate LSTM encoder-decoder for each variable, and its computational complexity appears to scale linearly or super-linearly with the number of variables, N. Can you provide a computational complexity analysis with respect to N and the window length T? Is the model feasible for large-scale datasets with hundreds or thousands of variables?\n4. Does the SLS, obtained via \"aggregation at the end of an epoch,\" converge to a consistent structure across different initialization conditions? Furthermore, can you justify why this specific aggregation method is preferable to alternatives, such as per-batch aggregation?\n5. When the deviation score of a variable A is high, how does OracleAD distinguish between the scenario where \"A itself is the source of the anomaly\" and the scenario where \"neighboring variables are anomalous and affect A through the self-attention mechanism\"?", "limitations": "Yes. While the authors indicate that limitations are discussed in the appendix, it is recommended that the following points be addressed more explicitly:\n1. Explicitly state that the static nature of the SLS is a core model assumption and discuss scenarios where this assumption may not hold (e.g., in systems with concept drift).\n2. Explicitly discuss the computational and memory overhead of the model when the number of variables, N, is large.\n3. The current model appears to be designed for continuous-valued time series. Its applicability to mixed-type time series that include discrete or categorical variables warrants discussion.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n1. The research proposes a novel conceptual framework where anomalies originate from a loss of \"temporal causality\" and subsequently propagate into \"structural\" deviations. This viewpoint offers a new perspective for anomaly detection.\n2. The paper evaluates the proposed method on widely used benchmark datasets, and the experimental results demonstrate its effectiveness.\n3. The separation of prediction and deviation scores enables the model to distinguish between transient temporal spikes and persistent structural anomalies. Furthermore, analysis of the deviation matrix allows for the direct identification of root-cause variables that disrupt the stable inter-variable relationships. This fine-grained diagnostic capability holds practical value for real-world applications.\n\nWeaknesses\n1. Although the central concept is novel, the individual technical components of OracleAD are established technologies. This factor diminishes the work's originality from a foundational model perspective.\n2. SLS is aggregated over the entire training set and remains static during inference. This stability assumption may not hold in real-world scenarios, particularly for systems exhibiting concept drift. The paper does not discuss how the model adapts to such dynamically changing normal patterns.\n3. The model employs a separate LSTM encoder-decoder for each of the N variables. This design incurs significant computational and memory overhead as the number of variables increases. The paper does not discuss the model's computational complexity or its scalability for high-dimensional time series.", "was_revised": true, "final_justification": "The authors responded in detail to the opinions I gave twice, which basically solved all my questions, but in terms of explanation of the novelty of the method, the explanation of the plain text still seems to lack persuasion. Since the core innovations of the article are entirely focused on a new conceptual framework, it is recommended that the authors can add more ablation experiments to verify the effectiveness of this new framework, such as trying to integrate other different basic models into this framework, indicating that the fundamental source of performance improvement is the proposed framework rather than the existing basic models.\n\nThe overall quality of this article is good, but given that the original innovation is slightly insufficient and it seems that it cannot be solved in a short period of time, I finally decided to keep the initial review score (4) unchanged.", "rebuttal": "## Q1. On the Choice of Window Length and Sensitivity to Long-Range Dependencies\nWe appreciate the reviewer’s question regarding window length. OracleAD was intentionally designed with relatively short temporal windows to capture both abrupt and subtle anomalies. In real‑world datasets (e.g., SWaT, SMD, PSM), anomalies often appear suddenly or evolve gradually in ways that normal temporal patterns cannot capture. While some faults span very long periods, our short-window dual-scoring mechanism enables OracleAD to detect both short-term and long-term anomalies effectively without requiring lengthy input sequences.\n\nWe also wish to clarify that a detailed window length sensitivity analysis is already included in **Appendix D.2**. The analysis shows that OracleAD maintains robust performance across various window sizes, further validating that our model is not overly sensitive to this hyperparameter. Moreover, on some datasets, using long windows does not yield additional benefits.\n\n---\n---\n\n## Q2 & W2 & L1. On the Static Assumption of SLS\nThank you for highlighting this insightful point. We fully acknowledge that the assumption of a static Stable Latent Structure (SLS) may indeed limit the model's adaptability in scenarios involving concept drift, where the underlying normal behavior evolves over time.\n\nIn our current framework, the choice of a static SLS was motivated by the observation that anomalies typically represent pronounced deviations from normal behavior, whereas data drift is often more gradual and subtle. Our empirical results validated this choice within the benchmark scenarios considered, showing strong anomaly detection performance despite the static assumption.\n\nHowever, there are real-world deployment scenarios, particularly in industrial applications, where normal operational patterns slowly but significantly evolve. We have indeed encountered such challenges while deploying OracleAD in practice, which further underscores the importance of your suggestion regarding dynamic SLS updates.\n\nTherefore, enhancing OracleAD’s ability to adaptively update the SLS—such as periodically recalculating it from inferred normal samples—is a promising future research direction that we will explicitly highlight in the camera-ready version.\n\n---\n---\n\n## Q3 & W3 & L2. On Model Complexity and Scalability\nOur design employs a separate LSTM encoder-decoder for each of the $N$ variables, resulting in a per-window computational cost of $N \\cdot O(T) = O(N \\cdot T)$ for temporal modeling. Additionally, multi-head self-attention involves an $O(N^2)$ pairwise comparison, which is computed once per window. Thus, the overall per-window complexity becomes $O(N \\cdot T + N^2)$.\n\nWe agree that this results in superlinear growth with respect to $N$, and the $O(N^2)$ component could be a concern as dimensionality increases. However, this quadratic cost is a common minimum requirement in models that explicitly model inter-variable relationships -- it is worth noting that even basic binary correlation computations scale as $\\binom{N}{2} \\approx O(N^2)$.\n\nDespite the theoretical cost, we observed no practical limitations when applying OracleAD to datasets with up to several dozen variables. To further assess scalability, we conducted extensive resource profiling on both real-world and synthetic datasets with variable counts ranging up to 400. These results confirm that OracleAD remains computationally feasible in industrial applications involving hundreds of variables.\n\nThis discussion will be included in the Appendix of the camera-ready version for completeness, as we believe it will be useful for readers concerned with large-scale deployment.  \n\n|**Dataset (N)**|**Batch Size**|**Memory Usage**|**Training Speed (it/s)**|\n|:-------------:|:------------:|:--------------:|:-------------------:|\n|**PSM (25)**|64|1.5 GB|12.6|\n| |1024|8.5 GB|9.6|\n|**SMD (38)**|64|1.9 GB|8.0|\n| |1024|13.0 GB|6.8|\n|**SWaT (51)**|64|2.4 GB|6.0|\n| |1024|18.5 GB|5.0|\n|**Synthetic PSM (100)**|64|4.7 GB|3.1|\n| |512|23.3 GB|2.9|\n|**Synthetic PSM (400)**|32|20.4 GB|1.35|\n\n*Table 1: Training Efficiency Across Datasets and Batch Sizes*\n\n|**Dataset (N)**|**Inference Speed (time point/s)**|\n|:-------------:|:------------------:|\n|**PSM (25)**|54.0|\n|**SMD (38)**|37.0|\n|**SWaT (51)**|28.0|\n|**Synthetic PSM (100)**|13.5|\n|**Synthetic PSM (400)**|3.15|\n\n*Table 2: Inference Speed by Dataset Dimensionality*\n\n---\n---\n\n## Q4. On SLS Convergence and Aggregation Strategy\nThank you for this thoughtful and technical question. We address both aspects below.\n\n### Convergence of SLS across Seeds\nTo assess the robustness of the learned Stable Latent Structure (SLS), we trained OracleAD with five different random seeds and visualized the resulting SLS matrices as heatmaps. Across all runs, the learned SLS exhibited **extremely high consistent inter-variable structural patterns**, confirming that OracleAD reliably captures stable relationships regardless of initialization.\n\n### Justification for Aggregation Strategy\nWhile our original implementation adopted an epoch-level aggregation for conceptual clarity, we do not believe this approach is inherently superior. In fact, as the underlying computation is a simple mean over latent variable pairs, both epoch-level and batch-level aggregation are mathematically equivalent under typical conditions (e.g., uniform batch sizes).\n\n*Our initial choice of epoch-level aggregation was purely for its intuitive interpretation, not due to performance advantages.* After reviewing implementation trade-offs, we updated the code to perform aggregation incrementally at the mini-batch level and compute the final SLS at the end of each epoch. This version is more memory-efficient and offers improved training throughput, **without affecting the results**. We confirmed that both aggregation strategies yield identical SLS matrices and detection performance.\n\nAccordingly, we are considering revising the description of SLS computation in the Method section to reflect its practical implementation better.\n\n---\n---\n\n## Q5. On Identifying Root-Cause Variables\n\nTo determine whether a variable $A$ is the root cause of an anomaly or merely affected by others, we examine the structure of the deviation matrix. A true root-cause variable typically exhibits elevated values across both its row and column, reflecting broad disruptions in inter-variable relationships. In contrast, if a variable is only influenced by another anomalous source, deviation will be localized to a few entries, while the rest of its row and column remain stable.\n\nBased on this principle,we compute a deviation score for each variable by summing its corresponding row in $\\mathcal{D^t_{matrix}}$. A higher aggregate score indicates that the variable’s interactions are broadly disrupted, suggesting it is more likely to be the anomaly’s origin.\n\nTo empirically validate this, we analyzed deviation matrices using ground-truth root-cause annotations from the SMD dataset. Below are the Top-3 predicted variables per case (scores in parentheses represent total deviation per variable), alongside the true sources:\n\n- **Case (a)**:  \n  - **Top-3 Predicted**: 33 (12.8), 32 (6.1), 35 (3.4)  \n  - **Ground Truth**: 32, 33  \n\n- **Case (b)**:  \n  - **Top-3 Predicted**: 13 (9.4), 8 (7.5), 12 (5.6)  \n  - **Ground Truth**: 8, 12, 13, 14  \n\n- **Case (c)**:  \n  - **Top-3 Predicted**: 13 (13.5), 12 (6.6), 8 (5.6)  \n  - **Ground Truth**: 8, 12, 13, 14  \n\n- **Case (d)**:  \n  - **Top-3 Predicted**: 15 (56.0), 10 (50.0), 18 (45.2)  \n  - **Ground Truth**: {0–3, 5–6, 10, 15, 18, 19–22, 24, 27–28, 30–31, 33–35}\n\nIn all four examples from Section 4.3, top-ranked variables from our deviation matrix consistently overlap with known root causes, substantiating the interpretability of our method.\n\n---\n---\n\n## W1. On the Originality of OracleAD\nAs also noted in the Strengths section, we believe OracleAD introduces clear conceptual novelty by framing anomaly detection. This perspective, supported by the Stable Latent Structure (SLS) and the dual-scoring mechanism, is the core contribution of our work.\n\nWe acknowledge that the technical components of OracleAD, such as LSTM and attention, are widely used. However, our primary goal was to focus on validating a new conceptual framework for anomaly detection. By applying this framework to existing modules, we demonstrated strong gains in both performance and interpretability. This highlights that the improvement arises from the design principle itself, not just from technical modifications.\n\nThat said, we fully agree that future work should include more technically original elements, particularly to broaden applicability from a foundation model perspective. Having verified the validity of our approach, we believe that extending OracleAD into a scalable multivariate time-series foundation model is a promising next step.\n\n---\n---\n\n## L3. On Applicability to Mixed-type Time Series\nWe thank the reviewer for highlighting this important point. While OracleAD primarily targets continuous-valued multivariate time series, we note that benchmark datasets such as SWaT, SMD, and PSM all include some categorical variables as well. In our experiments, these categorical features were treated numerically or encoded using standard embedding methods, which demonstrated effective performance without explicit categorical modeling.\n\nMoreover, OracleAD’s parallel architecture naturally enables the use of encoders explicitly tailored to the data distribution of individual variables, including categorical or mixed-type data. Therefore, we consider explicit modeling of categorical variables an insightful and promising direction, and plan to explore specialized encoders optimized for mixed-type data distributions in future work. We will clearly discuss this perspective as a valuable limitation and research opportunity in our camera-ready version."}, {"review_id": "csfLby3XPe", "reviewer": "Reviewer_tFFt", "summary": "The work proposes a framework for anomaly detection by combining temporal causality modeling and inter-variable structure learning. The framework provides interpretable deviation matrices that highlight which variables are likely responsible for detected anomalies.", "questions": "1. Could the authors kindly explain the novelty in their method\n2. How does this work compare to foundation models for time-series anomaly detection such as MOMENT [1]\n\n[1] Goswami, M., Szafer, K., Choudhry, A., Cai, Y., Li, S., & Dubrawski, A. (2024, July). MOMENT: A Family of Open Time-series Foundation Models. In International Conference on Machine Learning (pp. 16115-16152). PMLR.", "limitations": "yes", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Interpretable method that allows pin-pointing likely root-cause variables through a deviation matrix.\n2. Comprehensive experiments and evaluations are performed, demonstrating improvements over baselines\n3. Transparent discussion on limitations of existing benchmarks and corresponding results\n\nWeaknesses:\n1. Certain writing issues, such as (1) line 204 \"regularication\", and (2) line 182 \"summarices\".\n2. Unclear novelty: Modeling each channel independently (also known as channel independence in several works) is a well explored area [1, 2]. Applying attention across variates is similar to iTransformer [2]. \n\n[1] Nie, Y., Nguyen, N. H., Sinthong, P., & Kalagnanam, J. (2022). A time series is worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2211.14730.\n[2] Nie, Y., Nguyen, N. H., Sinthong, P., & Kalagnanam, J. (2022). A time series is worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2211.14730.\n[3] Liu, Y., Hu, T., Zhang, H., Wu, H., Wang, S., Ma, L., & Long, M. (2023). itransformer: Inverted transformers are effective for time series forecasting. arXiv preprint arXiv:2310.06625.", "was_revised": false, "final_justification": "", "rebuttal": "## W1. On Correcting Typos\n\nThank you for pointing these out. We have fixed the following:\n\n* Line 182: *summarices* → **summaries**\n* Line 204: *regularication* → **regularization**\n\nAll such corrections will be reflected in the camera-ready version.\n\n---\n---\n\n## Q1 & W2. On the Novelty of OracleAD\n\nThank you for the thoughtful question. The outline below highlights OracleAD's core contributions that we believe distinguish it from prior work in multivariate time series anomaly detection:\n\n1. **A Unified Hypothesis Linking Temporal and Structural Anomalies**\n\n   OracleAD is based on the hypothesis that anomalies originate from disruptions in temporal causality and subsequently propagate through inter-variable structural deviations. This hypothesis unifies two major anomaly modalities—temporal and structural—into a coherent framework, influencing both model design and scoring formulation.\n\n2. **Dual-Scoring System**\n\n   OracleAD introduces two separate scores:\n\n   * Temporal prediction error $\\mathcal{P_score}$\n   * Structural deviation $\\mathcal{D_score}$\n\n   These are combined via multiplication $\\mathcal{A_score} = \\mathcal{P_score} \\cdot \\mathcal{D_score}$, highlighting anomalies that are both temporally and structurally deviant.\n\n3. **Per-Variable Encoding with Explicit Structural Modeling**\n\n   While existing models such as PatchTST [1] also adopt channel-independent encoding to prevent unintended information mixing across variables, OracleAD takes a step further by explicitly utilizing these independently encoded latent vectors to construct a structured latent representation of the variable interaction space (i.e., the Stable Latent Structure). Rather than merely enforcing independence, our design leverages this **explicit structural modeling** with a clear objective: systematically capturing inter-variable relationships and deviations for precise anomaly detection and root-cause diagnosis.\n\n\n4. **Stable Latent Structure (SLS)**\n\n   Specifically, after independently encoding each variable through per-variable LSTM encoders to capture fine-grained temporal dynamics, we project these encoded latent vectors into a shared latent space via a multi-head attention mechanism. \n\n   Importantly, attention in OracleAD is solely intended as a projection mechanism to build the Stable Latent Structure (SLS), explicitly differentiating it from existing models such as iTransformer [2], where attention weights themselves implicitly encode relational information.\n\n   The SLS explicitly aggregates and stabilizes pairwise latent distances among variables during training, forming a reference structural pattern of normal inter-variable interactions. This structured latent representation serves two key purposes:\n     - **Training phase:** It acts as a reference structure, guiding the model toward consistent inter-variable representations under normal conditions.\n     - **Inference phase:** Deviations from the SLS are quantified at each timestep to form a Deviation Matrix, clearly indicating anomalies through disrupted inter-variable relationships.\n   \n   This structured deviation information is not only used for anomaly scoring but also proves critical for **root-cause analysis**. In summary, OracleAD’s  layered design(independent modeling plus structured latent comparison) is what distinguishes OracleAD from prior channel-independent or attention-based models \n\n\n5. **Interpretable Deviation Matrices for Root-Cause Analysis**\n\n   Our model produces a matrix-valued signal at each timestep, identifying disrupted variable interactions. By aggregating row-wise deviations, we infer the most anomalous variables. We validate this in Section 4.3 using ground-truth labels from SMD:\n\n   * **Case (a)**:\n\n     * **Top-3 Predicted**: 33 (12.8), 32 (6.1), 35 (3.4)\n     * **Ground Truth**: 32, 33\n   * **Case (b)**:\n\n     * **Top-3 Predicted**: 13 (9.4), 8 (7.5), 12 (5.6)\n     * **Ground Truth**: 8, 12, 13, 14\n   * **Case (c)**:\n\n     * **Top-3 Predicted**: 13 (13.5), 12 (6.6), 8 (5.6)\n     * **Ground Truth**: 8, 12, 13, 14\n   * **Case (d)**:\n\n     * **Top-3 Predicted**: 15 (56.0), 10 (50.0), 18 (45.2)\n     * **Ground Truth**: {0–3, 5–6, 10, 15, 18, 19–22, 24, 27–28, 30–31, 33–35}\n\n   Values in parentheses indicate total deviation scores per variable. Across these examples, OracleAD's Top-3 predictions consistently capture relevant variables. We will expand this evaluation across datasets in future work.\n\n\n6. **Rigorous Multi-Metric Evaluation for Objective Benchmarking**  \n\n   We rigorously evaluate OracleAD across seven distinct anomaly detection metrics (Standard F1, Range-F1, Affiliation-F1, AUC-ROC, AUC-PR, VUS-ROC, VUS-PR) on multiple real-world benchmarks. This comprehensive and robust benchmarking approach not only overcomes evaluation limitations observed in prior works but also empirically demonstrates OracleAD's clear performance advantage over existing models such as PatchTST [1] and iTransformer [2]. Specifically, our results indicate that OracleAD consistently outperforms these previous approaches across multiple metrics, highlighting how our *methodological innovations directly translate into substantial performance gains*.\n\nWe believe that this combination of conceptual clarity, principled design, and structured interpretability makes OracleAD a valuable contribution to the field of multivariate anomaly detection.\n\n---\n---\n\n## Q2. On Comparison with Foundation Models such as MOMENT\n\nOracleAD and MOMENT [3] differ fundamentally in design principles, representational scope, and intended use cases. While MOMENT positions itself as a foundation model—emphasizing pretraining on diverse sources and transfer across tasks—OracleAD is purpose-built for interpretable multivariate anomaly detection in high-stakes, real-world systems such as industrial control or cyber-physical environments. The table below summarizes key differences:\n\n| Aspect                    | OracleAD                                            | MOMENT                                       |\n|---------------------------|-----------------------------------------------------|----------------------------------------------|\n| **Objective**             | Root-cause-aware MTS anomaly detection              | Universal time-series representation         |\n| **Input Format**          | Multivariate                | Mostly univariate              |\n| **Architecture**          | Variable-wise LSTM + inter-variable attention       | Patchified Transformer encoder               |\n| **Cross-variable modeling** | Explicit | Not modeled |\n| **Output**                | Anomaly score + Deviation matrix | Reconstruction or classification score       |\n| **Use Case**              | Post-deployment, traceable fault detection          | Pretraining for downstream tasks             |\n\n*Table 1: Conceptual Comparison of OracleAD and MOMENT*\n\n\nWhile MOMENT aims for broad transferability across tasks, OracleAD is tailored for **diagnostic fidelity and real-world deployment** in complex multivariate settings.\n\nIn particular, MOMENT primarily targets short-segment, univariate anomaly detection, with no built-in mechanism for root-cause localization or inter-variable relational modeling. In contrast, OracleAD explicitly captures cross-variable dependencies and provides a deviation matrix for pinpointing which variables are most responsible for anomalies. This is critical in industrial sensor networks, where understanding anomaly origin is as important as detection accuracy.\n\nWe view the two approaches as complementary—MOMENT serves as a versatile pretraining backbone, while OracleAD offers a specialized, interpretable solution for high-stakes multivariate anomaly scenarios.\n\n---\n---\n\n\n[1] Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A Time Series is Worth 64 Words: Long-term Forecasting with Transformers. In Proc. of ICLR, September 2022.\n\n[2] Yong Liu et al. iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. In Proc. of ICLR, October 2023.\n\n[3] Mononito Goswami, Konrad Szafer, Arjun Choudhry, Yifu Cai, Shuo Li, and Artur Dubrawski. MOMENT: A Family of Open Time-series Foundation Models. In Proc. of the 41st ICML, pages 16115-16152. PMLR, July 2024."}, {"review_id": "iN0Iv4XNGb", "reviewer": "Reviewer_GXza", "summary": "An unsupervised multi-variate anomaly detection framework that captures both temporal and structural aspects for improved performance.", "questions": "1. Please describe how localization is achieved.\n2. Please detail the assumptions about the inference operation (size of window, etc.).\n3. It would be useful to include the training costs associated with training this model on the three benchmarks?  Just to get a sense of what is needed to deploy this method in the industrial settings.\n4. How can we support the claim that the heatmaps shown in Fig. 3 aids in interpretability.\n5. A discussion around the construction of anomaly score, i.e., why use multiplication there, why not simply add P and D values?", "limitations": "The paper doesn't include a limitations section and no limitations are discussed in Sec. 5.  It would be useful to put some thought into where/why the system might underperform.  See, for example, the performance on SWaT data.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is well-written.\n\nThe discussion around setting up an LSTM+attention scheme for capturing variable-specific temporal dynamics makes sense.\n\nThe discussion around setting up a multi-headed attention scheme to capture inter-variable dynamics makes sense.\n\nThe introduction of Stable Latent Structure that captures the \"normal, stationary\" behaviour of the multi-variate data is new.\n\nThe training objective that consists of prediction loss, reconstruction loss, and deviation loss seems to make sense.\n\nThe model is evaluated on three commonly used multi-variate anomaly detection benchmarks using standard metrics. \n\n# Weaknesses\n\nThe intuition behind how anomaly score is constructed by multiplying P^t_{score} and D^t_{score} isn't clear.  \n\nIt is not explicitly stated anywhere that inference operates over windows of length T and not the data item at time t.  Or am I mistaken?  Is it possible to performance inference on a single data item x^t?  If inference requires an input window, say of size T, how is localization achieved?  \n\nThe claim about interpretability is weak.  How can we confirm it that highlighted regions in Fig. 3 heatmaps actually correspond to offending variables?  Is it possible to setup an experiment to confirm this claim?  What data would we need?\n\nThe results achieved on PSM and SMD datasets improve upon the state-of-the-art; however, the results on SWaT benchmark do not match the state-of-the-art results according to a number of metrics.  Is there a reason why?", "was_revised": true, "final_justification": "The authors rebuttal addressed my primary concerns regarding the lack of a limitations section and the poor performance on SWaT dataset.  It seems that SLS is well-suited for problems where it is possible to \"learn\" inter-variable patterns.  The discussion around results on SWaT dataset highlights SLS's inability to deal with complex systems or multi-modal data.  The results around interpretability still seems somewhat ad hoc.  See the post-rebuttal comment #5 of Reviewer xLg3.", "rebuttal": "## Q1 & Q2 & W2. On Inference Window Assumptions and Localization\n\nThank you for highlighting this point. Your observation is valid, and we acknowledge that our formulation diverges slightly from conventional practices.\n\nTypically, when referring to a sliding window of length $T$, the entire window is used to predict a value that falls outside the window. In contrast, our approach uses the first $T-1$ steps to predict the value at step $T$:\n\nGiven inputs from $t = 1$ to $t = T-1$, the model predicts ${\\hat{y}}$ at ${t=T}$, and we compare it with the actual value $y$ at ${t=T}$.\n\n**Motivations:**\n\n1. **Temporal Focus**: We wanted to semantically emphasize that the model focuses on predicting what is happening **at the present moment** (i.e., $T$) based on the historical context (i.e., from $1$ to $T-1$).\n2. **Training Design**: This setup simplifies training and evaluation, as it pairs input ($1$ to $T-1$) with the corresponding supervision label (step $T$).\n\nInference on a single data point $x^t$ is possible, especially if trained with minimal window length (e.g., $T = 2$).\n\n**On localization:** A window from $t = n$ to $n + T - 1$ predicts the state at $t = n + T$, offering direct temporal localization.\n\n---\n---\n## Q3. Training Costs of OracleAD\n\nThank you for the suggestion. To provide a clearer picture of OracleAD’s computational cost and feasibility in real-world applications, we have conducted and summarized empirical resource usage measurements across a range of datasets, including real-world and synthetic multivariate time series benchmarks.\n\nThese results demonstrate that OracleAD is practical for deployment in industrial scenarios involving dozens to hundreds of variables. While high-dimensional settings naturally require more memory, inference remains efficient and can be parallelized in real-time systems. We have added these resource profiles to the Appendix to help practitioners estimate deployment requirements.\n\n|     **Dataset (N)**      | **Batch Size** | **Memory Usage** | **Training Speed (it/s)** |\n|:------------------------:|:--------------:|:-----------------:|:--------------------------:|\n|        **PSM (25)**      |      64        |      1.5 GB       |           12.6            |\n|                          |     1024       |      8.5 GB       |           9.6             |\n|        **SMD (38)**      |      64        |      1.9 GB       |           8.0             |\n|                          |     1024       |     13.0 GB       |           6.8             |\n|       **SWaT (51)**      |      64        |      2.4 GB       |           6.0             |\n|                          |     1024       |     18.5 GB       |           5.0             |\n| **Synthetic PSM (100)**  |      64        |      4.7 GB       |           3.1             |\n|                          |     512        |     23.3 GB       |           2.9             |\n| **Synthetic PSM (400)**  |      32        |     20.4 GB       |           1.35            |\n\n*Table 1: Training Efficiency Across Datasets and Batch Sizes*\n\n\n|     **Dataset (N)**      | **Inference Speed (time point/s)** |\n|:------------------------:|:----------------------------------:|\n|        **PSM (25)**      |               54.0                |\n|        **SMD (38)**      |               37.0                |\n|       **SWaT (51)**      |               28.0                |\n| **Synthetic PSM (100)**  |               13.5                |\n| **Synthetic PSM (400)**  |               3.15                |\n\n*Table 2: Inference Speed by Dataset Dimensionality*\n\n---\n---\n## Q4 & W3. On Evaluating the Interpretability of Heatmap\n\nWe appreciate the reviewer’s request for clearer support of our interpretability claims. To address this, we conducted a quantitative evaluation using ground-truth variable-level anomaly labels provided by the official SMD GitHub repository.\n\nWe examined four representative cases:\n\n* **Case (a):**\n\n  * **Top-3 Predicted:** 33 (12.8), 32 (6.1), 35 (3.4)\n  * **Ground Truth:** 32, 33\n* **Case (b):**\n\n  * **Top-3 Predicted:** 13 (9.4), 8 (7.5), 12 (5.6)\n  * **Ground Truth:** 8, 12, 13, 14\n* **Case (c):**\n\n  * **Top-3 Predicted:** 13 (13.5), 12 (6.6), 8 (5.6)\n  * **Ground Truth:** 8, 12, 13, 14\n* **Case (d):**\n\n  * **Top-3 Predicted:** 15 (56.0), 10 (50.0), 18 (45.2)\n  * **Ground Truth:** multiple variables including 10, 15, 18 {0–3, 5–6, 10, 15, 18, 19–22, 24, 27–28, 30–31, 33–35}\n    \nHere, the numbers in parentheses represent the total deviation score of each variable, calculated by summing the values along its row in the deviation matrix. In all four examples, at least one of the top-3 predicted rows overlapped with labeled anomalies, achieving a 100% hit rate. These results support our claim that the deviation matrix highlights meaningful variables.\n\nWe will incorporate this finding in Section 4.3 and plan to include Top-$K$ hit rate comparisons in future work. Furthermore, future work will involve systematic comparisons using Top-$K$ hit rates across interpretable baselines, thereby strengthening the evaluation of root-cause variable localization.\n\n---\n---\n## Q5 & W1. On Constructing the Anomaly Score via Fusion of $\\mathcal{P_score}$ and $\\mathcal{D_score}$\n\nThank you for bringing this important question to our attention. We defined the anomaly score as the **product** of the temporal prediction score $\\mathcal{P_score}$ and structural deviation score $\\mathcal{D_score}$, guided by the following conceptual and empirical considerations:\n\n1. **Motivation: Dynamic Reinforcement**\n   The two scores capture orthogonal anomaly signals:\n\n   * $\\mathcal{P_score}$: deviation from individual temporal dynamics.\n   * $\\mathcal{D_score}$: deviation from inter-variable latent structure.\n\n   Multiplying these ensures high anomaly scores only when both signals co-occur, thus suppressing noise from isolated fluctuations. Here, $\\mathcal{D_score}$ serves as a *dynamic weighting factor* for $\\mathcal{P_score}$.\n\n2. **Why not addition?**\n      We initially speculated that additive scoring ($\\mathcal{P_score} + \\mathcal{D_score}$) may suffer from **balancing issues**, where the dominant score can disproportionately influence the overall result. This motivated our use of multiplicative scoring, which naturally emphasizes aligned signals and produces sharper and more interpretable anomaly curves. We also evaluated additive scoring (Table 3) as a baseline and found its performance to be competitively strong, with each approach exhibiting clear trade‑offs; accordingly, we plan to include various strategies in future investigations of scoring functions.\n\n| Dataset | Anomaly Score | F1    | R-F1  | Aff-F1 | A-ROC | A-PR  | V-ROC | V-PR  |\n| ------- | --------------| ----- | ----- | ------ | ----- | ----- | ----- | ----- |\n| PSM     |     $\\mathcal{P_score} \\cdot \\mathcal{D_score}$     | 65.85 | 54.66 | 78.07  | 84.78 | 68.11 | 84.24 | 68.17 |\n|         |     $\\mathcal{P_score} + \\mathcal{D_score}$    | 68.36 | 55.48 | 78.52  | 85.03 | 67.19 | 84.82 | 67.43 |\n| SWaT    |     $\\mathcal{P_score} \\cdot \\mathcal{D_score}$     | 76.50 | 28.15 | 71.97  | 82.71 | 72.39 | 82.42 | 74.16 |\n|         |     $\\mathcal{P_score} + \\mathcal{D_score}$     | 76.91 | 37.75 | 73.43  | 81.73 | 71.73 | 81.42 | 71.07 |\n\n*Table 3: Performance comparison between multiplicative and additive anomaly score constructions on PSM and SWaT datasets. All results are averaged over five random seeds.*\n\nThese results will be included in Appendix E as part of an ablation study. We acknowledge additive fusion's viability and plan future exploration of scoring functions.\n\n---\n---\n\n## W4 & L1. On Performance in the SWaT Dataset\n\nThank you for this thoughtful question. We believe the discrepancy observed on SWaT arises from two primary factors:\n\n---\n\n### (1) Limitations in SLS generalization for high-dimensional or heterogeneous systems\n\nThe Stable Latent Structure (SLS) is built under the assumption that a consistent inter-variable relational pattern can be learned during training. This assumption holds well in datasets like PSM and SMD, where variable dynamics are relatively homogeneous and interactions remain stable. However, **SWaT contains significantly more variables** and represents a more complex physical system, involving different subsystems.\n\nThis diversity makes it more difficult for the SLS to converge to a single stable representation that generalizes across the whole dataset. As a result, the deviation score $\\mathcal{D_score}$ may be less discriminative, leading to a drop in overall detection performance.\n\n\n### (2) Sensitivity of multiplicative scoring to mixed (discrete/continuous) variable types\n\nMore critically, SWaT presents a **multimodal data challenge**, as it combines both continuous and discrete sensor types. Our scoring mechanism uses a multiplicative formulation:\n\n$\\mathcal{A_score} = \\mathcal{P_score} \\cdot \\mathcal{D_score}$\n\nFor this multiplicative design to succeed, both components must provide meaningful signals. However, for **discrete variables**, forecasting is inherently noisy under regression-based models (e.g., LSTMs), as these variables tend to exhibit abrupt, non-continuous behavior. This can lead to diminished prediction scores, which in turn suppress the final anomaly score -- even when the deviation signal is strong. \n\n\nWe plan to include these limitations explicitly in Section 5 as part of our broader discussion on modeling challenges."}, {"review_id": "evzTkIhot7", "reviewer": "Reviewer_xLg3", "summary": "The author proposed OracleAD, a framework for multivariate time series anomaly detection. OracleAD uses an encoder-decoder architecture using a proposed loss. The inner latent vectors will be used to detect the anomalies during inference time.", "questions": "1) Label equations. Use `\\begin{equation}\\end{equation}` instead of `\\[ \\]`\n2) Equation after Line 154. My understanding here is $h_i^t \\in R^d$, $w\\in R^d$ and softmax is applied across the t-th dimension. Is this correct? In any case, we should describe it more clearly, especially for the softmax part.\n3) Section 3.2, what is \"normal training window\" k? What is \"M windows\" in SLS computation?\n4) I think Section 3.2 is lacking some justifications. I think there are several \"choices\" made in this section that was not explained or justified. Not to say that this choices are not correct, but some simple justifications or discussions would be helpful. Example of choices are \n- The choice of D and SLS: As we are comparing the latent vectors from the \"baseline\" vs the inference time, why do we opt it to compute the distances. i.e. the choice was to check for \"inter-variable relational stability\" instead of \"variable stability\"? (I noticed that are some almost relevant discussion in section 4.3\n\n  > This score captures anomalies that manifest as disruptions in inter-variable relationships, which often persist beyond the initial signal deviation. Notably, we find that this behavior aligns well with real-world anomaly labeling practices, as seen in the shaded regions of Figure 2, \n\n  but the results showing a spike in the anomaly score does not match the statement.\n\n- Choosing the L2 distances between latent vectors: Is there justification of using both the directional mismatch and the latent enrgy differences?\n- The choice of the baseline being the average of \"k\" dissimilarity matrices. In case that the dissimilarity matrix are fluctuating normally, the deviation matrix may not take account to the \"variances\" of the $D^{(k)}$'s.\n5) Similar to 4), Figure 2 and 3 does not really shows the clear interpretability from the deviation matrix. From Fig 3a), we can see that feature 33 deviates from the baseline by quite a lot, which was shown in Figure 2a). But for Fig 3b), we see feataure 13, say has a high deviation with feature 8 or 12, but not the other features. How do you interpret the difference between case a) and b)?", "limitations": "The authors should include a section regarding limitations.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper is relatively easy to follow, except for the parts highlighted in the \"questions\" below.\n- The idea introduced are relatively novel and non-trivial.\n- Adequate experiments and baselines are included.\n- Examples and simple ablation studies are included.\n\nWeaknesses:\n- I think there are some justifications are lacking in certain parts of the paper. See \"questions\" below.\n- There are also minor unclear details in Section 3. See \"questions\" below.\n- Limitations and Discussions are slightly lacking.", "was_revised": true, "final_justification": "The work is solid. As I mentioned in the strengths and weaknesses above. The authors also actively engaged in discussions and resolves all of my questions, mostly on clarity and justification of design choices. Some of the responses are found in discussions with other reviewers as well (the multiplicative scoring function).\n\nI have raised the rating from 4 to 5. However, despite working on time series for a short time, I am not familiar with the field of anomaly detection and recent developments in time series. Thus I do not have enough knowledge on competing methods or literature comparison. Thus my confidence score remains at 3.", "rebuttal": "## Q1. On the Labeling of Equations\nThank you for pointing this out. The minor fixes in the submission have already been made (e.g., citation consistency, equation format) and will be applied accordingly for the camera-ready version.\n\n---\n---\n\n## Q2. On the Clarification of Dimensionality and Softmax Application in Attention Computation\n\nIn our implementation:\n - Each LSTM output $h_i^t \\in \\mathbb{R}^d$ is passed through a learnable **linear projection** with bias: $\\ f(h_i^t) = w^\\top h_i^t + b \\$, where $w \\in \\mathbb{R}^d$, $b \\in \\mathbb{R}$.\n - The attention weights are computed by applying softmax over the temporal axis($t=1,\\dots, T{-}1$): \n    $\n    \\alpha_i^t = \\frac{\\exp(w^\\top h_i^t + b)}{\\sum_{s=1}^{T-1} \\exp(w^\\top h_i^s + b)}.\n    $\n - The causal embedding is then computed as a weighted sum: $ c_i = \\sum_{t=1}^{T-1} \\alpha_i^t h_i^t \\$.\n \nIn the final version, we will revise the equation to match this implementation more faithfully and explicitly mention:\n - the dimensionality of $h_i^t$ and $w$,\n - that $f(h_i^t)$ is a learnable scalar scoring function,\n - and that softmax is applied over the temporal dimension.\n\nThank you very much for bringing this to our attention. It's a very fair and helpful observation.\n \n---\n---\n\n## Q3. On the Definition and Clarification of the Training Window Terminology\n\nWe acknowledge that the term “normal training window” may cause ambiguity. What we intended to convey is that, during training, the model is only exposed to time series segments that consist of normal (i.e., non-anomalous) behavior. To clarify this, we will revise the phrase to read “training window” in the camera-ready version.\n\nRegarding your question about “$M$ windows” in the SLS computation: this refers to the total number of sliding windows extracted from the training data in each epoch, where each of these windows comprises a segment of the time series.\n\n---\n---\n\n## Q4. On the Choices made in Section 3.2\n### On variable stability vs inter-variable relational stability:\nWe fully agree that both forms of stability are important. In our formulation, **variable-level stability** is captured by the prediction score $\\mathcal{P}_{score}$, which measures whether each variable's current value deviates from the trajectory predicted from its own past. That is, we interpret the “stability” of a variable as temporal consistency — whether it behaves in accordance with its historical dynamics. \n \nOn the other hand, $\\mathcal{D_score}$ reflects the inter-variable relational stability, i.e., whether the pattern of interactions between variables remains consistent. Thus, OracleAD jointly tracks both: $\\mathcal{P_score}$ handles per-variable dynamics, while $\\mathcal{D_score}$ monitors structural coherence.\n\n\n### On the Rationale for Modeling Inter-variable Structure via Pairwise Latent Distances\n  \nThe latent pairwise distance matrix $\\mathcal{D_score}$ at inference time and the Stable Latent Structure (SLS) during training both model *inter-variable relations* — not individual embeddings, but their geometry. Comparing $\\mathcal{D_score}$ with SLS is thus not a direct vector comparison, but a comparison between **relational structures**. We found that this abstraction is more robust to latent noise and more effective at capturing cross-sensor disruptions. The SLS represents the standard inter-variable relationship patterns observed in normal data, and deviations from this structure indicate potential structural anomalies.\n\n### On the Clarification of Score Alignment with Anomaly Labels in Figure 2\n \nThank you for this important point. Among the examples in Figure 2, subfigure (d) corresponds explicitly to the claim made in Section 4.3 — namely, that structural deviation scores align well with real-world labeling practices. In this case, although the prediction score remains low during the anomaly segment, the deviation score ($\\mathcal{D_score}$) rises and stays elevated, helping to detect sustained faults that prediction-only methods often miss.  \n\nWe acknowledge that presenting four different cases in Figure 2 without explicitly linking them to specific text may have caused confusion. To address this, we will revise the text in Section 4.3 to explicitly reference **Figure 2(d)**.\n\nWe appreciate the reviewer for identifying this ambiguity and prompting a clearer justification of our modeling choices.\n\n\n### On the Justification of Using L2 Distance over Other Metrics for Latent Vector Comparison \n\nWe opted for the L2 distance as it incorporates both directional discrepancy and magnitude variation (latent energy difference), enabling a more comprehensive measure of dissimilarity. Empirically, we compared several distance metrics, including cosine similarity and L1 distance, and found that L2 consistently produced more stable and discriminative results. This choice reflects a balance between interpretability and performance, and we will add a brief mention of this empirical comparison in the camera-ready version.\n\n|Dataset| Metrics  |   F1   |  R-F1  | Aff-F1 |  A-ROC  |  A-PR  |  V-ROC  |  V-PR  |\n|----------|----------|--------|--------|--------|---------|--------|---------|--------|\n|PSM| cos Sim. | 46.81  | 43.68  | **78.18**  | 66.46   | 48.59  | 65.83   | 48.82  |\n|| L1       | 57.72  | 51.54  | 75.87  | 76.33   | 54.29  | 73.33 | 53.52  |\n|| L2       | **65.85**  | **54.66**  | 78.07  | **84.78**   | **68.11**  | **84.24**   | **68.17**  |\n\n*Table 1: Performance comparison of cosine similarity, L1, and L2 distances for latent vector dissimilarity. L2 consistently provides higher and more stable performance across evaluation metrics. All results are averaged over five random seeds.*\n\n### On the baseline construction using the average of dissimilarity matrices\nWe appreciate the reviewer’s insightful remark regarding variance. Indeed, if the averaged dissimilarity matrix were not directly involved in training the subsequent epoch, accounting for the potential variance across $D^{(k)}$ would have been critical. However, we considered using the average in our design to be acceptable because the averaged matrix is itself used as supervision for training in the following epoch. As a result, we anticipated that the model would progressively learn to minimize variance across training iterations, effectively converging toward a stable representation of normality.\n\n| Dataset | Epoch | Variance |\n|-----|-------|----------|\n| PSM | 1     | 1.10E-10 |\n|     | 2     | 2.85E-05 |\n|     | 3     | 6.55E-05 |\n|     | 6     | 5.99E-04 |\n|     | 10    | 3.06E-04 |\n|     | 20    | 1.95E-04 |\n\n*Table 2: Variance of dissimilarity matrices at selected training epochs. The variance gradually decreases, indicating stabilization of the latent representation. All results are averaged over five random seeds.*\n\n---\n---\n\n## Q5. On the Interpretation and Visual Representation of Deviation Matrices\n\nThank you for this valuable question. We agree that the difference in heatmap sharpness between cases (a) and (b) in Figures 2 and 3 deserves clarification, and we now provide both qualitative and quantitative explanations.\n\n### (1) Visual normalization effects  \nFirst, the color scales in Figures 3(a) and 3(b) are independently normalized per matrix. This means that even if both cases contain substantial deviations, the highest values are mapped to full-intensity in each plot, which may obscure global comparison. In (b), the deviation between two jointly anomalous variables (e.g., features 8 and 13) dominates the matrix. In contrast, deviations between anomalous and normal variables appear relatively small in the same plot — even though they may still be significant in absolute terms. We will revise the figures using a unified color scale to improve cross-comparability.\n\n### (2) Structural contrast difference\nCase (a) involves a smaller number of root-cause variables (32, 33), creating a sparse and localized deviation pattern — easier to interpret visually. Case (b), on the other hand, includes multiple correlated variables (8, 12, 13, 14), which results in a more diffuse deviation structure with several inter-anomalous pairings and overlapping effects. This leads to a matrix where contrast is distributed across multiple entries rather than concentrated in a single block.\n\n### (3) Quantitative validation \nTo support this interpretation, we computed Top-3 variable rankings from the deviation matrix in each case and compared them with the ground-truth root causes (obtained from SMD's official repository):\n\n* **Case (a):**\n\n  * **Top-3 Predicted:** 33 (12.8), 32 (6.1), 35 (3.4)\n  * **Ground truth:** 32, 33\n* **Case (b):**\n\n  * **Top-3 Predicted:** 13 (9.4), 8 (7.5), 12 (5.6)\n  * **Ground truth:** 8, 12, 13, 14\n* **Case (c):**\n\n  * **Top-3 Predicted:** 13 (13.5), 12 (6.6), 8 (5.6)\n  * **Ground truth:** 8, 12, 13, 14\n* **Case (d):**\n\n  * **Top-3 Predicted:** 15 (56.0), 10 (50.0), 18 (45.2)\n  * **Ground truth:** {0–3, 5–6, 10, 15, 18, 19–22, 24, 27–28, 30–31, 33–35}\n\nDespite the visual difference in heatmap texture, the model successfully highlights true root-cause variables among the top deviating rows in both cases. This demonstrates that the heatmap reflects valid diagnostic information even in visually diffuse scenarios. We will update Section 4.3 to include this discussion and quantitative evaluation.\n\n---\n---\n\n## W3. On the Explicit Discussion of Limitations and Future Directions\n\nThank you for raising this point. While a limitations discussion is currently included in Appendix H, we agree that it deserves more emphasis in the main paper. We will move the key points to Section 5 and expand them in the camera-ready version. Additionally, we will incorporate insights and observations gained during the rebuttal period to further refine this discussion."}]

Author rebuttal:

Meta-review: The paper proposes a structured temporal causality approach for interpretable time series anomaly detection. Reviewers agreed the problem is timely and the idea of combining causal structure with anomaly detection is appealing. They appreciated the interpretability angle and the empirical evaluation. Concerns were mainly about novelty (some parts overlap with existing causal discovery methods), clarity of technical details, and the strength of baselines. The rebuttal and final clarifications did help address several points, especially on experiments and interpretability claims, though not every doubt was fully resolved. 

While not the strongest submission, the work is methodologically sound, has solid experiments, and makes a useful contribution. Given the competitive bar at NeurIPS, I lean to a weak accept.
Decision: Accept (poster)