Multi-Resolution Decomposable Diffusion Model for Non-Stationary Time Series Anomaly Detection
Authors: Guojin Zhong
pan wang
Jin Yuan
Zhiyong Li
Long Chen
Venue: ICLR 2025 Poster (2025)
Keywords: Diffusion Model
Non-Stationary Time Series
Anomaly Detection
Multi-Resolution
Primary area: generative models
TL;DR: This paper delves into the potential of multi-resolution technique and diffusion model for non-stationary time series anomaly detection, supported by rigorous mathematical proofs.

Abstract: Recently, generative models have shown considerable promise in unsupervised time series anomaly detection. Nonetheless, the task of effectively capturing complex temporal patterns and minimizing false alarms becomes increasingly challenging when dealing with non-stationary time series, characterized by continuously fluctuating statistical attributes and joint distributions. To confront these challenges, we underscore the benefits of multi-resolution modeling, which improves the ability to distinguish between anomalies and non-stationary behaviors by leveraging correlations across various resolution scales. In response, we introduce a **M**ulti-Res**o**lution **De**composable Diffusion **M**odel (MODEM), which integrates a coarse-to-fine diffusion paradigm with a frequency-enhanced decomposable network to adeptly navigate the intricacies of non-stationarity. Technically, the coarse-to-fine diffusion model embeds cross-resolution correlations into the forward process to optimize diffusion transitions mathematically. It then innovatively employs low-resolution recovery to guide the reverse trajectories of high-resolution series in a coarse-to-fine manner, enhancing the model's ability to learn and elucidate underlying temporal patterns. Furthermore, the frequency-enhanced decomposable network operates in the frequency domain to extract globally shared time-invariant information and time-variant temporal dynamics for accurate series reconstruction. Extensive experiments conducted across five real-world datasets demonstrate that our proposed MODEM achieves state-of-the-art performance and can be generalized to other time series tasks.

Published as a conference paper at ICLR 2025
MULTI-RESOLUTION DECOMPOSABLE DIFFUSION MODEL FOR ANOMALY DETECTION
NON-STATIONARY TIME SERIES
Guojin Zhong!* Pan Wang** Jin Yuan Zhiyong Li Long Chen HHman University (gizhong, yuanjin, zhiyong. 1 J@hnu. edu.cn en longchen@cse. ust. .h
"Hong Kong University of Science and Technology 1innnuu
ABSTRACT
Recently, generative models have shown considerable promise in unsupervised time series anomaly detection. Nonetheless, the task of effectively capturing complex temporal patterns and minimizing false alarms becomes increasingly challenging when dealing with non-stationary time series, characterized by continuously fluctu- ating statistical attributes and joint distributions. confront these challenges, we underscore the benefits of multi-resolution modeling, which improves the ability to distinguish between anomalies and stationtioo behaviors by leveraging correla- tions across various resolution scales. In response, we introduce Multi-Resolution Decomposable Diffusion Model (MODEM), which integrates a coarse-to-fine dif- fusion paradigm with navigate the intricacies of non-stationarity. Technically, the coarse-to carse-to--fine to-fine diffusion model embeds cross-resolution correlations into the forward process optimize diffusion transitions mathematically. It then innovatively employs low-resolution recovery to guide the reverse trajectories of high-resolution series in coarse-to to-fine manner, enhancing the model's ability to learn and elucidate underlying temporal patterns. Furthermore, the rrquency-enhanced decomposable network operates in the frequency domain to extract globally shared time-invarian information and timevvrrintt temporal dynamics for accurate series reconstruction. Extensive ex- periments conducted across five real-world datasets demonstrate that our proposed MODEM achieves state-of-the-art performance and can be generalized to other time series tasks.
-enbance-- decomposable network to adeptly
INTRODUCTION
Time series anomaly detection has emerged as critical task in various domains such as industrial manufacturing, finance, and healthcare monitoring, aiming identify anomalous data within specific time intervals (Xu et al. 2022, Chen et al.) 2023 Yang et al. (023). Considering the scarcity of anomaly labels, most existing time series anomaly detection methods operate under an unsupervised paradigm, trained solely on data with normal behavior. These methods are generally divided into three and reconstruction-bat
Density-based methods (Dai & Chen Zhou et al. 2023 2024) detect anomalies based on the assumption that anomalies often lie on densityy regions of data distribution. Forecasting baased methods (Zong et al.) [2018] Audibert al. 2020 Yao et al. 2022) predict future values from past observations, while reconstruction- -aased methods (Su et al. 2019 et al. 2023) Xiao et al.] 2023) aim to reconstruct the entire input series. Both types identify anomalies by comparing the errors between the predicted or reconstructed series and the actual input series, with large discrepancies indicating potential anomalies (Paparrizos et al.)
categories; density-based forecasting-be
2022; Li al. 2023b). Despite their effectiveness, these methods are primarily optimized for stable time series and are prone to false alarms when dealing with normal yet non-stationary data because non-stationary time series continuously change their statistical properties and joint distribution over time (as shown in the gray area of Fig. [a)). Consequently, these methods struggle to accurately
"Contributed equally. 'Coresponding authors.

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Published as a conference paper at ICLR 2025
0:83
Time
(a) Challenges Non-stationarity
(b) Importance Multi-Resolation Modeling
Denoising
Ampetteee Tabte
Roya
00100
00003
Denoising
0.003
Ro:1
Rple-1
120
Franzeer Commenerr
(c) Characteristies of Time Series Diffusion
(d) Potential of Frequency Decomposition
Figure I: Several examples to illustrate the challenges of non-stationarity and the motivation of the use of multi-resolution modeling and frequency decomposition.
model non-stationarity, often resulting in high reconstruction errors that are mistakenly classified as anomalies (as illustrated by the orange and purple curves in Fig. [a)).
To address non-statinn non-stationary time series in anomaly detection, D (Wang et al. 2024) employs a moving average kernel to extract labeled stable components, which then guide the decomposition of stable and trend components for training the diffusion model. However, D is designed for iingle-reoolution data and lacks the capability to effectively utilize multi-scale temporal information to detect anomalies in non-stationary time series. Moreover, its reliance on the moving average strategy, which presupposes linear trend, is often inadequate for handling non-stationary time series characterized by complex seasonal variations or multiple temporal patterns. Consequently, this approach may even introduce erroneous lag signals, potentially compromising the detection accuracy.
To enhance the robustness of anomaly detection, we emphasize the significance of multi-resolution modeling in capturing the underlying non-stationary patterns. Low-resolution time series typically display prominent long-term trends, while nigh-resolution series provide insights into detailed fluctuations and short-term changes, as shown in Fig. [bb. By exploiting correlations across multiple resolution scales, the model gains comprehensive view of data, benefiting from both overarching trend information and granular event details. Furthermore, as depicted in Fig. TT), time series
diffusion models (Li et al. 2022b Shen al. 2024) typically generate the overall trend in the early denoising steps and then refine local temporal patterns in the later steps. This observation intuitively guides our strategy of sequentially reconstructing low-resolution data followed by high-resolution data using cross-resolution correlations, thereby establishing a more accurate reconstruction baseline for anomaly detection.
Motivated by these insights, we propose a DEM) to enhance anomaly detection in non-stationary time series. MODEM leverages intrinsic correlations across different resolution scales to inform the learning process of the diffusion model. explicitly incorporates the cross-scale correlations between high-resolution and low-resolution data into the forward process, serving as the prior guidance to optimize the diffusion transitions. During denoising process, these cross-scale correlations are harnessed to adapt the reverse trajectories, where predictions at lower resolutions aid in the coarse-to-fin to-eine recovery of higher-resolution data. Additionally, we have extended MODEM to the Denoising Diffusion Implicit Models (DDIM) to accelerate the sampling process. To effectively parameterize the denoising process, we design frequency-en enhanced decomposable network, which employs the short-term Fourier transform to analyze the input series. An examination of the spectrum reveals marked disparities in amplitude energy across various frequency components (Fig. [[)). Higher amplitude levels point to globally dominant time-invariant components, whereas lower amplitudes correspond time-variant patterns.
Decomposable Diffusion Model (MO-
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Drawing on these insights, non-stationary series are decomposed into time- invariant and time-varian components. Subsequently, two distinct encoders are deployed to separately extract globally shared information and complex temporal dynamics, thereby enhancing the modeling of hon-stationarity. Extensive experiments conducted on five real-world datasets demonstrate that MODEM significantly surpasses ofte--of-theart (SOTA) unsupervised anomaly detection methods and exhibits strong generalization capabilities across various time series tasks. Our main contributions are listed below:
We introduce MODEM, a diffusion model that investigates correlations across different resolution scales for non-stationary time series anomaly detection. For the first time, we explore enhancing the diffusion model's 's understanding of time-varying properties induced by non-stationarity from diverse temporal resolution perspectives.
We leverage cross-resolution correlations as guidance to optimize both forward and backward trajectories, synthesizing time series in a low-to-high resolution manner. This approach, grounded in mathematical derivations, not only provides more robust reconstruction signals for anomaly detection but also exhibits strong generalization across various tasks.
We propose eehuunecy-enhaceed decomposable network to effectively parameterize the de- noising process. This network operates in the frequency domain to decompose time-invarian and time-variant components, allowing for the extraction of globally shared information and the identification of underlying temporal dynamics.
2 RELATED WORK
Existing unsupervised anomaly detection methods for time series are primarily categorized into density-basee (Dai & Chen Zhou al. 2023), , forecasting- based (ao et al. 2022), and reconstruction-bsse (Zong et al.) 2018 Audibert et al. 2020) approaches. Density- focus on fitting the density of training and test samples to detect anomalies, which are based on the assumption that anomalies often lie on low-density regions of data distribution (Zhou et al. 2024).
methods
Forecasting- based methods utilize past series data to predict future values through various modified neural networks, including Long Short-Term Memory (LSTM) (Hundman et al. 2018), graph neural networks (Deng& & Hooi [021 Zhao al. 2020a),, and Generative Adversarial Networks (GAN) ( (Yo et al. 2022; Zhong et al. 20223). These methods detect anomalies based on the forecasting error between the predicted and actual values. but they are prone interference from historical data. In contrast, reconstruction-bant based methods involve reconstructing the entire input series and identifying anomalies based on the reconstruction error. Early methods in this category use Variational Auto- Encoders (VAE) (Su et al., (019) or GAN (Li et al. 2019) to reconstruct time series. Inspired by the success of transformers (Liu et al. 2023 Chen et al. 2025). ). TranAD (Tuli et al. 2022) introduces the self-attention mechanism to learn temporal patterns and incorporates adversarial training to enhance model robustness. Anomaly Transformer (Xu et al. 2022) utilizes the proximity concentration devia- tion of anomalies to make exceptions more distinguishable. However, the training instability and significant error accumulation in transformers have been challenged by diffusion models (Ho et al. 2020; ZZhng al. 2024 2025), which recently demonstrate superior performance in time series anomaly detection. For instance. ImDiffusion (Chen et al., 2023) employs diffusion model to fill the masked time series and proposes an ensemble strategy for detecting anomalous data. Similarly, DiffAD (Xiao et al. 2023) adopts the mask moask-the-immptation paradigm and designs a conditional weight-incremental diffusion model to improve the interpolation performance of missing series. Due to the drift caused by non-stationary environments, DdR WWang et al. (024)) dynamically decomposes stable and trend components using the mix mix-nttention technique and leverages the diffusion model to reconstruct the input series, achieving significant performance improvements.
Notably, the importance of multi-resolluiin techniques Li et al. (023a);; Zhang et al. (2024) has been demonstrated in time series forecasting. For instance, MG- TSD Fan et al. (2024) designs coltiigranuularty guidance loss to guide the learning process of diffusion models, while MR- Shen et al. (2024) employs progressive denoising to generate coarser and finer trend signals, improving the accuracy of time series prediction. These works inspire us to be the first to analyze normal and anomalous change patterns in non-stationary data across multiple resolution scales.
In this work, we further explore the potential of diffusion models for detecting anomalies in non- stationary time series. Uniquely, we extend the diffusion model into multi-resolution paradigm,
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MRDDSampler
Coarse-to-Fiie to-eine Diffusion Model
Frequency-Enhann Enhanced
fo
k
1
Diffustos
Step
Decompesable Network
Fornard Froms
Linear
Eo
Hevrro Freum
(0 Output)
Midilxation
Time-Imarian
Time-Varine Enceder
Encoder
k.r
Spectrum
Amplitude Rank
Fourier Transform
Hig Hiihhdd
Resolution
(Input)
Figure 2: Overview of Multi-Resolutinn Decomposable Diffusion Model, consisting of three mod- ules: Aulti-Resolution Data Sampler (MRD-Sane (MRD-Sampler), a Coarse-to-Fine Diffusion Model, and Frequency-Enhan Enhanced Decomposable Network for non-stationary time series anomaly detection.
which explicitly utilizes correlations within different time scales optimize the forward and reverse trajectories, effectively alleviating the interference of non-stationarity on anomaly detection.
METHODOLOGY
3.1 PROBLEM STATEMENT
Let X NXT denotes observed time series, where N and T number of variables and length of timesteps, respectively. The anomaly detection task aims to produce an output vector E IRT based on input series X. where yı temporal patterns of non-stationary time series typically change over time because they originate from dynamic environments, which poses challenges for anomaly detection.
indicates whether the i-th timestep an anomaly. The
3.2 OVERVIEW OF MODEM ARCHITECTURE
The architecture of MODEM is depicted in Fig. 2 which mainly consists of a multi-resolution data sampler, coarse-to-fine diffusion model, and a frequency-enhance, decomposable network.
Restii-Resoltion Data Sampler Given time series Xo and resolution scale [1, R). this step employs average pooling (Wu et al. 2021) to generate multi-resolution data, x,, that is, x (Paddin(Paddi 2"), where 2" denotes the pooling size that increases with r. AvgPool(. computes the average of every 2" non-overlapping points. Padding(-)) is then applied by replicating the pooled data at r resolution times to maintain the same length as the resolutonn series x This strategy provides the necessary data support for our model's subsequent multi-resolution learning, enabling to capture evolutionary characteristics of non-stationary time series across multiple resolutions and enhancing its understanding of normal and anomalous change patterns.
Coarse-to- to-e Diffusion Model Multi-resolution data {xf x R-1 ...., xb} are input into a coarse- to-fine diffusion model. Different from the existing diffusion models, they sequentially disturbed as {x x R-1 ,..., x from low to high resolution in R K forward steps. During the reverse process, noisy samples are progressively restored to their original form in a coarse-to-fine manner, with the clean prediction x aiding in the denoising of the higher-resolution data x A detailed discussion of this model can be found in Section (.3
Brrquency-Enhaneee Decomposable Network It is designed to parameterize the denoising process of the diffusion model, enabling the prediction of clean data at each diffusion step lt.
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comprises three modules specifically tailored to learn about the underlying pattern of son-stationarity. For a detailed description of this design, please refer to Section 3.4
3.3 COARSE-TO-FINE DIFFUSION MODEL
Motivated by (Fan et al. 2024), we observe that the forward process of the diffusion model intuitively mirrors the smoothing of time series from high to low resolution. Additionally, upon revisiting the conventional denoising process, we find that the diffusion model tends to reconstruct the overall sequence first and then refine the local temporal fluctuations, as illustrated in Fig. Tc). These insights lead to extend the time series diffusion model to multi- paradigm that utilizes the intrinsic correlations within different resolution scales as prior guidance for improved performance.
Multi-Resolutiann Forward Process Rather than naively adding random noises on all It resolutions [x].xij x.. each of the K diffusion steps, we execute the diffusion process sequentially from the lowest-reoolution x to the highest modifying the total number diffusion steps R K. Furthermore, we strive incorporate the cross-scale correlation between xij and x5+1 into the forward process. This ensures the diffusion model maintains the underlying normal pattern from x,+1 to guide the synthesis of the higher- resolution time series xT- Specifically, x and x5+1 are input into transformer-batt designed to learn their cross-scale correlation through cross-attention (Yang et al.) 2024). This cross- scale correlation can intuitively serve prior guidance adaptively optimize diffusion trajectories across all steps:
x/. Here, each scale undergoes K' diffusion steps,
module
2o (x X6, x N( Vārxo + (x x5+1 ,1, - (1-1-);))) ā.)
where 7kk Vak (1-vag) is step-varying factor controlling the intensity of prior information. & denotes the lightweight module with parameter é. Inspired by, but distinct from, ShiftDDPM Zhang al. 2023) and ContextDiff Yang et al. (2024), which utilize classifiers and cross-modal contextual information respectively, we are the first to leverage the temporal correlations across multiple resolution scales to modify the diffusion trajectory, ensuring that lower-resolution data effectively facilitate the reconstruction of hhigher-resolution data. Subsequently, we can derive the multi-resolution forward transition given x Xj, and x (see detailed proof in Appendix A.
go (x x6,x N(Vonii + Ex x k) Vaun (X6. x6 kk 1), (2)
At each step k the model explicitly considers cross-scale correlation to optimize the forward transitions, ensuring a more effective alignment with the coarse-to- denoising process. Using Eq. (1) and Eq. (2). we can further derive the posterior distribution of the forward process for k by applying Bayes' rule (see detailed proof in Appendix A.2
v 0hh BK
Vax(1 (k--1)
(x(x Xx Xo, x6) -N(
x +
(x% 7t En Ex x x b))
Ork
Ökk
(1 - (3. 11x
+ 16 x5=1 1),
In this manner, the forward process of MODEM begins with the corruption of low-resolution time series, which display long-term trends and periodic variations. This is followed by the degradation of high-resolution time series that preserve short-term fluctuations. These diffusion transitions are steered by cross-scale correlations, enabling MODEM seamlessly integrate and capture both macroscopic and microscopic details of non-stationary time series across different scales.
Parameterized Reverse Process To align with the aforemeenioned forward process, the re- verse phase of MODEM aims to restore multi-resolution noisy data {x, x Rt1 :..., x back to clean data {xx' Rt- .... x]). Without loss of generality, we revisit the variational bound of log p(x)), and seek parameterized kernels poxx_1lXx xx+1)= = N(uu,, Ep) to approximate (x( 1o(xf_1xx3 |x%, X x in Eq. (3) by minimizing their KL-divergence (Zhong et al. 2023b) DEL (x+ xxf, xT. + >)po - 1)) By combining Eq. [ and Eq. (3). and as-
suming that Eg equals the posterior variance I-
(1- we derive the distribution of the
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parameterized kernels (see detailed proof in Appendix A.3
vI āx fo k) V /aa(1 āu-1)
Be
po x X6 +1
-N(
na (xU: Xo 2+1
-
(4)
x r+1 kk 1), āu au-1)11 I),
7h-1 Ex (x),x)
ai where fo(x; k, denotes the denoising network with parameter 0 that predicts the noise. Thus the Kl-ddivegeecc objective can be equivalent to:
Th Ex (x,, krr) x vāuxi
Co.b.rr Σ
[|²].
(5)
v1 Okk
where x = V V + m x k) + v1 āe and T] 21 (1pp111)1)
9x. is a loss weight.
Accelerated Training and Sampling To enhance the efficiency of MODEM, we accelerate both the training and sampling procedures. Firstly, it is evident from Eq. (5) that x introduces additional cross-scale correlation calculations at each training step. To mitigate this, our denoising network fo(x;, k r) is designed to directly predict the clean data x at resolution r from X rather than estimating the noise. The refined training objective is defined follows:
K
Laiff Σ Exg. [1]5o(X k, x611%].
R=
Secondly, modeling time series at R different resolutions leads to inefficiencies, as the number of sampling steps increases by factor of R. Drawing inspiration from DDIM (Song al., 2020), the reverse process of MODEM is adapted to be deterministic and we can obtain x given x via (see detailed implementations in Appendix A.44)
x - Vav
x v + V āk -4-1-08- of
VI Ork
1 - -a-1 of
ExE xtt11_ kt) v1 V1-áp
+ 1 8,+11 k 1).
Thus, we can employ a sub-sequence of [1, K" of length L, where L < K. As a result, the total number of denoising steps is reduced from R x K to R resulting in faster sampling. In contrast to the conventional reverse process, MODEM utilizes coarser-grained temporal patterns to adaptively constrain the sampling trajectories of finer-grained time series, thereby effectively reducing the interference caused by non-stationarity and produces precise normal baselines for anomaly detection.
3.4 BHEQUENCY-ENHANEE
DECOMPOSABLE NETWORK
To parameterize the denoising process of MODEM, design Frequency-Enhann Decomposable Network that learns the underlying dynamics of non-stationarity. The proposed network incorporates frequency decomposition module disentangle imm-invariant and varivatiaant components. Following this, two distinct encoders are employed to process these components separately.
Frequency Decomposition Previous works (Yang& & Hong 2022 He al. 2023 have demon- strated that spectral responses can more robustly capture underlying temporal patterns compared to time-domain representations. Motivated by these insights, we can utilize spectral statistics to disentangle the time series components: Time-invariant components, which represent dominant sta- tionary factors, typically manifest as high-magnitude frequency components when transformed into the frequency domain, while time-variant components reflecting non-stationarity are characterized by low-magni frequency components. Specifically, we employ the short-term Fourier transform (STFT), which is well-suited for time-varying signals, to compute the mean amplitude for each spectrum, denoted as A = {0, 1, ..., 2 subset A... which captures the dominant iine-invariant information (Liu et al. 2024). The disentanglement is performed as follows:
I }. We then select the top m percent of the highest amplitudes
Xine "SeleeSeett(A... F(x)), Xvar = Xinv:
8
where F denotes STFT and F-1 its inverse transformation, Select(-) allows only the selected frequency components to pass through.

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Time- Invariat Encoder Given the time-invarinn component Xine. utilize several stacked residual blocks to encode the globally shared information. Recognizing that understanding the relationships between different variables is advantageous for modeling non-stationarity (Tashiro et al. 2021; Liu et al. 2023), each residual block includes hierarchical transformers (Chen et al. (003) capture both intra-series temporal correlations and inter-series interactions among variables (see Appendix B.5 for detailed architecture).
Time- Variant Encoder To model underlying temporal dynamics, we develop the üime-variant encoder process Xear. As illustrated in Fig. 3 it comprises multiple Di- lated ModemnTN Blocks (DMTB). DMTB is based on the ModemnTCN module (Luo & Wang 2024), which con- nects ID depthwise convolutional layer (DWConv) and convolutional feed forward network (ConvFFN) in series, demonstrating excellent performance time series tasks, To uncover the complex periodic characteristics of Xyars we expand the kernel size of DWConv using a dilated factor d, forming c parallel ModernTCN modules. These convolution kernels of varying sizes, with different recep- tive fields, slide across multiple time scales, effectively capturing periodic variations of different magnitudes. No- tably, they share the same parameters without introducing additional parameters. Subsequently, we utilize additive concatenation (Hu et al., 2023) to fuse their output results,
Output
Linear
Spatial Trundfammr
DMTB
Trundoruise
DMTB
DMTB
Input
Figure 3: Architecture of -ame--ariant Encoder, consisting of multiple stacked Dilated ModernTCN Blocks (DMTB). Moreover, diffusion step and resolution-scale embeddings are added as complementary information and hierarchical transformers are incorporated to enhance temporal representation similar to the time-invariant encoder.
EXPERIMENTS
4.1 DATASETS AND EVALUATION METRICS
Datasets. We evaluate the performance of MODEM five real-world datasets: SMD (Server Machine Dataset) (Su et al. 2019). PSM (Pooled Server Metrics) (Abdulaal et al. 2021 MSL (Mars Science Laboratory) (Hundman et al., 2018). SWaT (Secure Water Treatment) (Mathur & Tippenhauer 2016), and SMAP (Soil Moisture Active Passive satellite) (Entekhabi et al. 2010).
Evaluation Metrics. For performance evaluation, we employ Precision (P), Recall (R), and F1-score metrics (Deng & Hooi] [202]] /hhao et al.] (020b). Additionally, we introduce the Average Sequence Detection Delay (ADD) metric (Tuli et al. 20022 to evaluate the speed and timeliness of each time series anomaly detection algorithm.
4.2 IMPLEMENTATION DETAILS
The proposed MODEM is implemented using the PyTorch framework and optimized using the Adam optimizer (Kingma& & Ba 2014) with a learning rate of 1e - 3 and a weight decay rate of 1e - 6. We train MODEM for 100 epochs datasets that contain only normal time series four NVIDIA A6000 GPUs For anomaly detection, we utilize the ensemble inference strategy (Chen et al. 2023 to improve the robustness and accuracy of MODEM. Specifically, this involves integrating the denoising results at each diffusion step and resolution scale to vote on whether a point is anomalous (see Appendix for detailed explanation). and we the voting threshold to 10 for all datasets. Our method employs square noise schedule, uses 50 diffusion steps, and operates across 4 resolution scales. Further details on the hyperparameters of MODEM can be found in Appendix B.6
4.3 DETECTION RESULTS
The detection performance of our MODEM is presented alongside the baseline methods in Tab. where F1" denotes the average F1-score. Notably, these metrics calculated using the point
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adjustment strategy (Xu et al. 2022). in line with the approach used in most previous studies (Deng & Hooi 2021 Zhao et al. 2020b). Additionally, the timeliness of these methods is detailed in Tab. 2 and the comparison of model efficiency is illustrated in Tab. 14 of Appendix B.10 Based on these
results, we can draw the following conclusions:
Firstly, thanks to the optimized diffusion and denoising trajectories in and the decomposition of time-variant components in the frequency domain, our MODEM effectively identifies anomalous data. This is evidenced by its consistently superior performance in four out of the five datasets, with an average Fl-score improvement of 2.11% (from 91 91.35% to 93.46%). The only exception is the MSL dataset, where MODEM is slightly less effective compared to TranAD which focuses more on exploring variable correlations, better matching the characteristics of MSL. Additionally, on the SMD and SWaT datasets, known for their high non-stationarity, our approach achieves significant performance improvements. This suggests that MODEM can effectively learn underlying temporal dynamics and mitigate the impact of non-stationarity on anomaly detection.
time series
Secondly, anomaly detection methods that utilize generative models, such as VAEs and diffusion, to reconstruct the time series consistently outperform prediction methods like LSTM- or GNN- based methods. This indicates that generative models with robust reconstruction capabilities may provide a more effective paradigm for detecting anomalies in non-stationary time series.
Thirdly, benefiting from both the general trend information provided by low- www-resoution data and the detailed event information from high-resolution data, our model is able to detect anomalies more promptly, resulting in lower ADD. Moreover, despite processing time series at multiple resolution scales, our MODEM still demonstrates competitive computational efficiency. This is attributed to the mathematical extension of the multi-resolution diffusion process the DDIM paradigm, which enables MODEM to sample a sub-sequence at each resolution, significantly accelerating the sampling process. Further discussions on computational efficiency have been included in Appendix B.10
Table Comparison of detection performance across five real-world datasets using point adjustment strategy. The best and second-best performances are highlighted in bold and underlined, respectively.
Method
SMD
F1
P
PSM
F1
P
MSI
F1
P
SWaT R
F1
SMAP R
F1
F1'
IForest
20.30
21
17.98
66.30
19.19
56.11
60.59
53.28
53.31
97.65
66.50
79.07
28.85
76.71
41.63
29.69
LSTM-AD
32.29
26.39
83.13
63.78
99.25
67.37
78.41
56.
MSCRED
90.38
36
79.65
50.08
84.23
84.60
78.22
78.65
85.64
80.27
83.42
13.11
5.85
MTAD-GAT
88.36
83.30
$4.63
87.63
87.25
87.44
$4.68
82.24
83.84
84.68
82.24
63.44
97.18
52. 22.59
68.24
$1.44
OmniAnoutaly
87.51
90.52
87.35
92.55
95.51
91.11
83.21
82.15
82.68
97.49
75.00
84.70
84.07
96.74
59.95
87:24
InterFusioon
88.15
90.71
$7.32
91.28
93.26
92.26
76.88
94.64
84.42
96.83
85.30
90.50
87.88
77.04
$2.47
57.49
ReaiAAN
90.13
58.94
87.97
92.04
87.67
89.75
96.06
70.20
81.07
96.06
70.20
81.07
89.15
67.81
76.63
83.30
MAD-GAN
$5.96
79.41
54.25
63.85
96.40
54.74
AnomalyTrame TramolyTannSomerr
68.01
TEAD
92.42
91.39
93.83
$7.15
73.26
84.59
78.52
86.72
85.16
87.56
TranAD
89.06
89.82
$7.85
95.06
91.17
92.20
$9.51
92.97
91.15
70.25
72.66
68.86
82.24
85.02
83.60
84.73
KPSK
87.09
90.63
89.15
87.61
94.73
91.02
53.62
85.53
84.56
79.35
83.29
81.25
82.64
86.42
84.49
86.09
DiMTA
90.52
91.71
91 40
23
97.70
96. 95
85.59
88.32
86.93
86.64
88.35
87.40
91.08
87 30
80 19
90.79
InDiiftusion
94.53
94 48
94.
97.72
96
27.27
87.74
84.65
86.16
89.88
84.65
87.00
87.71
96 18
95
91.35
D'R
94.33
40
86.80
91.
97.21
To ensure a more comprehensive comparison and provide more convincing validation of MODEM's effectiveness, we also evaluate the detection performance using affiliation- based metrics (Wang et al. 2024 to avoid the illusion of progress caused by point adjustment (Kim et al. 2022). As shown in Tab. of Appendix [.7]] our MODEM achieves the highest F1 F1-scores across three datasets with high non-stationarity, with an average improvement of 3.86% (from 80.34% to 84. 440). This suggests that the cross-resolution correlation enables robust reconstruction signal in a coarse-to-fine fine manner, allowing most anomalous points to be effectively distinguished from normal non-stationary behavior.
4.4 ABLATION STUDIES
Coarse-to-Fine Diffusion Model We extended the diffusion model into a multi-resolltion frame- work, incorporating cross-scale correlations into both the forward and reverse processes. The effects of various modifications are detailed in Tab. 3 where "w/o resolution". "w/o M-forward", and "w/o A-reverse" denote variants of MODEM that lack multi-resolution diffusion settings, modifications to the forward process, and adaptations to the reverse process, respectively. The data shows that removing the multi-resolution settings leads to a significant performance decrease across all datasets,
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Table 2: Comparison of ADD (mean + std.) performance across all five datasets. The best and second-best performances are highlighted in bold and underlined, respectively.
Method
SMD
PSM
MSL
SMAP
SWaT
Average ADD
IForest
90
191 + 17
123 28
394 1: 93
539 + 20
257 27
LSTM-AD
87 37=1
224 = 54
115 = 29
541 + 51
627 =
284 + 23
MSCRED
32=
218 + 35
109 = 30
622 + 48
1065 + 339
365 + 76
GDN
+8+1
148
106 +2
402
1478 +
383 +
MTAD-GAT
182
96 17
1:
d: 80
256
OmmiAnomaly
26
121
116 1
550 4:8
173 18
InterFusion
22
40 = 10
32 +
423 +
454 141
185 +: 29
BeatGAN
3
166 +
68 + 24
345 + 23
607
226 13
MAD-GAN
59
122 =2
=
404 + 20
926 337
293
TranAD
127
56 46+12
291 +
657 246
210
DifIA
48 1:
134
7
368 +: 22
437 1:50
212 1: 18
ImDiffusion
24 1
28
1 1
46
98 1
350 4:3
104 1 14
D²R
65 65- 16
172 6
90 12
514 32
643 + 46
297 22
Ours
27 2
26 +:
0
52.55
102 8
343 + 26
109 9
with the average F1-score declining from 95.26% 92.22%. This underscores the critical importance of integrating multi-resolution information for accurate anomaly detection. The guidance provided by lower-resolution data enables MODEM to better identify underlying temporal patterns and more effectively recover higher-resolution data. The alterations to the forward process and the adjustments to the reverse process contribute to an average performance improvement of 1.50% (from 73376% to 95.26%) and 2.18% (from 93.08% to 95.26%) respectively. These outcomes confirm the efficacy of optimizing diffusion and sampling trajectories by leveraging correlations across different time scales.
Frequency-Enhance Decomposable Network We conduct ablation experiments on three key components: frequency decomposition (w/o decomposition), the time-invariant encoder (w/o invari- ant), and the time-variant encoder (w/o variant), with results detailed in Tab. 3 The integration of frequency decomposition effectively disentangles time-varying dynamics in the frequency domain, enhancing our understanding of non-stationarity and resulting in an average F1 F1-score improvement of 1.1% (from 94.16% to 95. 95..6%) Furthermore, removing the time-invariant and time-vane variant encoders leads to a decline in performance across all datasets, with average Fl-scores dropping to 94.40% and 93.10% respectively. The time-invariant encoder captures globally shared information, while the time-variant encoder designed to capture dynamic temporal patterns, working synergistically to achieve optimal performance. The more significant impact observed upon removing the time-vane variant encoder highlights its crucial role in Tab. 3 accounts for an average improvement of 1.05% by adeptly learning complex periodic patterns across different receptive fields (see w/o DBTM). Additionally, the introduction of hierarchical transformers (w/o temporal and w/o spatial) within MODEM allows for the exploration of dependencies both temporally and across variables, leading to substantial improvements in Fl-scores. Additional ablation studies on the number of DBTM and residual block are presented in Appendix B.6
Table 3: Results of ablation studies on SMD, PSM, and SWaT datasets measured by Fl-scores.
Dataset
MODEM
w/o M-forward
w/o A-reverse
w/o invariant
w/o DBTM
w/o spatial
resolution
decomposition
variant
temporal
SMD
96.01
93.02
94.88
93.84
94.73
94.96
93.15
95.01
93.33
94.69
PSM
97.65
95.20
96.67
97.09
95.77
96.96
SWaT
91.21
87.73
89.13
88.39
90.14
90.45
89.57
90.05
89.04
89.69
Averago
94.96
91.98
93.56
92.85
93.92
94.16
92.83
94.01
92.61
93.35
4.5 EFFECTIVENESS ANALYSIS
Coarse-to-Fin Diffusion Model We model non-stationary time series across multiple resolutions (R) and have extended MODEM to the DDIM framework, which enables accelerated sampling through sub-sequence of length L at each resolution. As illustrated in Fig, 4 explore the impact of varying and L on the Fl-scores in the SMD dataset. The results indicate that detection performance improves R increases from to 4, highlighting that utilizing correlations across

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Published as a conference paper at ICLR 2025
different time scales enables the model to differentiate between anomalies and normal patterns more accurately. However, increasing R beyond this point leads to a decline in performance, as some short- term anomalies erroneously filtered out. Furthermore, also improves, suggesting that a more comprehensive sampling process bolsters detection capabilities. The optimal configuration is found with L. 20 and R 4, which not only maintains nearly intact performance but also achieves processing speeds comparable to previous methods (Chen et al.) (023). This balance between speed and accuracy renders MODEM highly effective for practical applications. Additional case studies and experimental results validating the effectiveness of multi-resolution modeling on the other four datasets can be found Appendix B.8 and Appendix BB6 respectively.
L increases, MODEM's performance
96
Thee Invelts Componce Compusst
9:20
1.500e
C11
8.15-
91-
3.10-
0.00
846-
Resolation
Figure 4: Results of various resolution and sampling steps on SMD dataset.
Figure 5: Results of standard deviation.
Frequency Decomposition To evaluate the effectiveness of frequency decomposition, sample 50 subsets from various periods in accordance with the methodology described in (Liu et al. (024). We apply frequency decomposition these subsets extract the time-invariant and time-varian components. Separate linear regressions are conducted on each component, and the standard deviation of the regression weights is utilized to assess the interdependencies between the components. The results, illustrated in Fig. [ show that the standard deviations for the time-variant components are consistently larger than those for the time-invariant components across all datasets. This demonstrates that frequency decomposition effectively separates the two types of components, confirming its utility in enhancing time series analysis. More experimental results and clarifications on our frequency- enhanced decomposable network are included in Appendix B.8
4.6 GENERALIZATION PERFORMANCE
To assess the generalization ability of MODEM, we conduct forecasting and imputation experiments across multiple non-stationary datasets. We apply MODEM in unconditional manner non- stationary time series forecasting and imputation following (Kollovich et al. 2024), and report the average continuous ranked probability scores (CRPS) (Gneiting & Raftery 2007) across three independent runs in Tab. 15] and Tab. of Appendix B.11 as well as the detailed discussion on experimental setup can be found in Appendix B). The results demonstrate that MODEM performs competitively against SOTA methods, despite not being specifically designed for forecasting tasks. We also extend the proposed multi-resolution diffusion paradigm to CSDI (Tashiro et al., (02T) and TSDiff (Kollovich et al.) (224) (denoted as CSDI-MR and TSDiff-MR). The improved forecasting and imputation performance validate that correlations across different time scales facilitate the model to better capture diverse temporal patterns.
5 CONCLUSION
This paper introduces novel "Multi-Resolutionn Decomposable Diffusion Model" that delves deeply into non-stationary time series to enhance anomaly detection performance. Our method innovatively incorporates multi-resolution correlation information into Coarse-to-Fine Diffusion Model to optimize the diffusion trajectories, effectively capturing the non-stationarity to better differentiate between anomalies and normal patterns. Additionally, we design a frequency frequency-enhanced decomposable network, which separates time-invann invariant and time-variant components in the frequency domain. Extensive experiments conducted on five real-world datasets demonstrate that MODEM significantly outperforms of-the--r- unsupervised anomaly detection methods.
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ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (No. 62272157, No. U21A20518, No. U23A20341).
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A
MATHEMATICAL DERIVATIONS
A.1 DERIVATION OF AULTI-RESOLUTION DIFFUSION TRANSITIONS
Given the diffusion trajectory modified by cross-scale correlation, which is defined Eq. (..
now derive the multi-resolution diffusion transitions:
qo (xxx 1,, x xb=
([ x5= .k) & x k 1), 3
We can achieve this by proving that Lemma. D which has been proposed in (Zhag et al. 2023
Yang
al. 2024).
lemma
Civen the forward process defined as q(x), x(x;,x2,.... x2...., xx1xx)) = II-= ((x) x x x ")
where the diffusion transitions q(x;, [x x5, x6 are defined as:
2o xx x (x V / + (x xb" k)). )
(10)
Proof 1 Inspired by previous work (Yang al. 2024). we prove this lemma by induction. Assume that at time k; both g(x%) xx___ k- X6: x + and q 1(x%-1/x5 1-. x6t1) adhere to their respective
distributions in Eq. 3) and Eq.
7 We need prove that q(x] xx,xxxxxx777 x5+1) -N(xF vanxo
ThE (Xh, x k)), (1 axk II)
We can rewrite ( - Xby x and x(x [x, x5+1) follows:
1,
x
- Eg (x5,, x +1 Rb) VaLTE- 1 x k 1) + BaCL,
(11)
x Va Tk Tk- E4 x kk - 1) + 1- a
(12)
where 61 and 62 denote independent standard Gaussian variables. Substituting x from the latter equation into the former, we obtain:
X = Vā + (x) X k) + BB + Vax * (1-āx-1) * c2
(13)
= Vaxxi + (x x5 k) + aa. - āk 1) Bbe,
where is Gaussian noise resulting from linear combination of 61 and 62. To this end, (xx x x with mean VouXi + "xE,(x), x,+,,,+,,,, and variance e admits the expected
distribution.
A.2 DERIVATION OF POSTERIOR DISTRIBUTIONS OF MULTT-RESOLLUIOO DIFFUSION PROCESS
Given the modified diffusion trajectories in Eq. T and the diffusion transitions in Eq. we now derive the posterior distributions of multi-resolution diffusion process:
(x X X 1) =N( 02 33
/as(1 (1 - xx-1)
x
(x% E x6x1 kk)
O
Ckk
(14)
A 1), (1 σ% -+ 1).3k I).
+ 1 X5"
1 ās
Proof 2 By Bayes's 'ss rule (Ho el al. 2020), we have:
q(x,
(xp )x(x% xX- Xb: x5
q(x 1x X/ x1)
(15)
<(x x x6" +11
Given that the numerator and denominator are both Gaussian, the posterior distribution is also Gaussian (Song et al. 2020), and we can proceed to calculate its mean and variance:
+ Th Ex x k), (1 1)
q(x}_ XT, x5)
N(x; v V + 7k- (x/ x k 1), (1 - ā, 1,-1)
N(a,, Varti-1 + 74 x
k) Voutk- X r+1 k B,T)
N(x2, /o x6 74
Es x x5+1 k 1), (1 1
(16)
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Dropping the constants that are unrelated to xb. X - and Xo r+1 have:
(x/-1 V 1xx T (x6: x6" .))
a(xk-1|xi X5, x xxtt)) oc exp(
2(1 Ōx-
+ (x% Vāxx N/ (xb. xx)
2(1 (1k)
X Vauxi-1 1 Ta (X), X
k) + & ,,,,
28c
1
exx((C(k, 2
+
x
x
xx-1
3k
(Vak-ix6 + (xE xo 1+1 k))
(xcs -r x
x+1 ,,k)
+
(x xb xx1 1)1
+ vau
a
8x
=exp(C(k,r) 5 1
+
ak
x + X
ci--
3x
(vak-1 Cx
x
VOlk (xk (xEo(No
x,=1 k)) +
1
+
Ok
" 1 Ex x5 ++1 k 1)]}
Be
Gx-1
3x
(17)
where Ckk, r) a constant term with respect to x 2024), this can be simplified to:
With some algebraic derivation
Yang et al.
1o X X6: x ) -N(
0k 0
x
v
vas(1 - (h--1)
(x]
T x5=1 kk)
1 0
0kk
(18)
k 1), (1
ax 1) 1).
+ 16-1E X6
A.3 DERIVATION OF TRAINING OBJECTIVE
According to Eq. T x can be rewritten as:
x
[ Th x ++1 ,k)) VI age),
(19)
Vas
where denotes gaussian noise. Then we can obatin:
x
(x,, x5
(20)
VI
a
VI
Since the second term is available, we employ denoising network fo fo k. r) to predict the first term for training. Then obtain the predicted posterior distributions:
x1 -N(
[x]
3g VT
(o(x,,krr))
v Van(1 Ōx-1)
ThE EnE xp xxt11k)
Val
akk
an
(21)
1 xb" A 1), (1 1) I),
2
Revisiting the objective of diffusion model (Song et al. 2020). we insteadly minimize the KL-
divergence (Van Erven & Harremos 2014) D ( [x]_
x X x (lpo)) x x)+))).
4 we can obtain the training objective: With Eq. 3 and Eq.
Lo Co.b =D DDxz(gs(x)
Xx xy, x
xx x(+1))
nr E Exjer (1loo( k,) xx
Voxx 112]
(22)
Σ
v1 Ci
where x
Vagxh 76E6 xx r+1 (,,,k) + vI age and 7 2a.(1-an-)
By
is a loss weight.
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A.4 DERIVATION OF ACCELERATED SAMPLING
Given X we can obtain x via:
x x = ===v-x-1 + V1- ax-1
X -vax
V1 Ork
(23)
0--1 of
+ 12 x-+1 k 1).
v1 - 0k
Proof 3 We can prove this by induction following (Zhang et al. 2023). Assume that at time k. the posterior and marginal distributions admit the expected distributions. then need to prove that at time k- 1, 4o (xxxx Xx x6+1 + also has the expected distribution. We can rewrite the posterior and marginal distribution:
x = 120 + V1 0k-1 - of
X VORx
v1
(24)
- Xo x0=11111111111111 .h) v 1 Tk
Tk- (x) X0 x0=1 K 1) + oREI,
VI Ck.
x Vayxi T x 1 k) + vI VI-24e2.
(25)
where 61, C2 are standard gaussian noises. Plugging Xx, we have:
x-_1 1
+ (x)) x x11,111 .k) VI ax-1 oz
-x, (xxxxxxxx1 ,k) V1
āz o
v1 Ok
O
(26)
+ 16 x0 k 1) + OREI + 11-ax- Cx-1 - -
=Von- + 7h- E (x x b 1) + 0R61 + v1 1-
- 1-0k-1 o2
Since the variance of (a,c, + v شي- --11 02222 = (1 - (.-1)) we have the expected sampling.
B EXPERIMENTAL DETAILS
B.1 DATASETS
We use the following five datasets for anomaly detection experiments:
SMD (Server Machine Dataset) (Su ct al. 2019): The SMD dataset is collected from a large internet company and includes 5 weeks of data from 28 server machines with 38 sensors each. The initial 5 days consist solely of normal data, while anomalies are intermittently introduced over the last 5 days.
PSM (Pooled Server Metries) (Abdulaal et al. 2021): The PSM dataset is collected internally from multiple application server nodes at eBay. It consists of 13 weeks of training data and weeks of testing data.
MSL (Mars Science Laboratory) (Hundman et al. 2018) and SMAP (Soil Moisture Active Passive satellite) (Entekhabi al. 2010):: The MSL and SMAP datasets are publicly available datasets collected by NASA. They contain telemetry anomaly data derived from the Incident Surprise Anomaly (ISA) reports of spacecraft monitoring systems. The MSL dataset has 55 dimensions, while the SMAP dataset has 25 dimensions. The training sets for both datasets include unlabeled anomalies.
The SWaT dataset is collected over 11 days from a scaled-down water treatment testbed with 51 sensors. For the first 7 days, only normal data were generated. During the last days, 41 anomalies were injected using various attack methods.
SWaT (Secure Water Treatment) (Mathur & Tippenhauer 2016);
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Table 4: Datasets used for anomaly detection experiments.
Dataset
Entities
Dimensions
Train #
Test
Anomaly Rate (%)
SMD
28
38
708405
708420
4.16
PSM
25
132481
87841
27.76
MSL
27
55
58317
73729
10.48
SMAP
55
25
140825
444035
12.83
SWaT
495000
449919
12.14
We present the statistics of these datasets in Table. I Train # and Test denote the number of training and testing data, respectively. Anomaly Rate is the ratio between the sum of all anomaly points and sum of all test points.
We use the following three non-stationary datasets for forecasting and imputation experiments to assess the generalization ability of MODEM.
Traffic (Tashiro et al. 2021 The Traffic dataset records the hourly road occupancy rates generated by sensors in the San Francisco Bay area freeways.
Exchange (Shen & Kwok 2023): The Exchange dataset describes the daily exchange rates eight countries (Australia, British, Canada, Switzerland, China, Japan, New Zealand, and Singapore).
KDDCup (Kollovieh et al. 2024): The KDDCup is dataset of the air quality indices (AQIs) of Beijing and London used the KDD Cup 2018
For imputation performance evaluation, examine three scenarios following (Kollovieh et.
2024):
(1) random missing, where values are missing sporadically, (2) blackout missing at the beginning of the context window, involving sequence of consecutive missing values, and (3) blackout missing the end of the context window. We report the average performance of three conditions.
B.2 BASELINES
We introduce the following state-of-the-arr time series anomaly detection methods for extensive comparisons:
Isolation Forest (Li et al. 2022ay: constructs 3D features (text, reviewer behavior, deceptive ratings) and integrates feature selection detect fake reviews.
LSTM-AD (Malhotra et al. 2015): possesses long-term memory capabilities, and for the first time, hierarchical recurrent processing layers have been combined to detect anomalies in univariate time series without using labels for training.
MSCRED (Zhang et al., 2019): designs attention- ConvLSTM network to capture temporal trends, and a convolutional autoencoder is used encode and reconstruct the signature matrix instead of relying on the time series explicitly.
OmniAnomall (Su et al.) 2019): is Variational Autoencoder that performs anomaly detection by computing the reconstruction probability and quantifying interpretability based on the reconstruction probability of each feature.
InterFusion (Li et al.) 202T): uses a hierarchical variational autoencoder with two ran- dom latent variables learn metrics and temporal representations and by relying "reconstruction input" to compress the MTS.
GDN (Deng Hooi 202T): utilizes the nodes and edges of the GNN to capture sensor features and spatial information, respectively. It then leverages this data to predict sensor behavior based on the attention function of adjacent sensors.
MST-GAT (Ding et al. 2023): uses multimodal graph attention network and temporal convolutional network to capture spatiotemporal correlations in multimodal time series.
BeatGAN (Zhou et al. 2019): uses a group of autoencoders and GANs in cases where tags are not available, which accurately detect anomalies in both ECG and sensor data.
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MAD-GAN eLi 2019) uses LSTM-RNN generator and discriminator to capture temporal relationships in Gans, while using reconstruction and discriminant losses to detect anomalies.
Anomaly Transformer (Xu al. p:22 captures association differences by modeling prior and sequential associations for each timestamp, making rare exceptions easier to distinguish.
TFAD (Zhng et al., 2022): introduces time series decomposition and data augmentation mechanisms into the designed time-frequency architecture, enhancing both performance and interpretability.
TranAD TTlli et al. 2022): ) adopts a two-step reconstruction method, introduces the attention mechanism into Transformer model, and integrates adversarial training.
NPSR (Lai
2024): proposes a framework for unsupervised time series anomaly detection using based and sequence-based reconstruction models.
DiffAD (Xiao et al. improve the imputation performance of missing values with conditional weight-incremental diffusion.
2023): designs novel denoising diffusion-based imputation method to
ImDiffusion (Chen et al. 2023): combines time series imputation and diffusion models to achieve accurate and robust anomaly detection.
D*R (Wang et al. 2024): tackles the drift via decomposition and reconstruction, overcoming the limitation of the local sliding window.
We introduce the following state-of-the-art time series forecasting and imputation methods for comparison assess the generalization ability of MODEM:
CSDI (Tashiro et 2021
sele-supervised method that uses the observed value as conditional information to imputation the masked time series.
TSDiff (Kollovieh et al., 2024): is an unconditionally trained diffusion model for time series and a mechanism to condition TSDiff during inference for arbitrary forecasting tasks (observation self-guidance).
To ensure fair comparisons, first reference the best-reported values from the original papers other publications. these values are unavailable, we report the results reproduced on our machine using the publicly available codes. Moreover, assume that the thresholds of these SOTA methods have been optimized for optimal performance.
Notably, TSDiff is developed for univariate time series generation, and the selected Matasets-Traffe Exchange, and KDDCup-and are widely used, preprocessed univariate datasets derived from traffic, financial exchange rates, and climate domains, respectively. These datasets, after processing, can be obtained from Appendix B. of the TSDiff paper. When extending the proposed multi-resolution approach to TSDiff, we preserved its original network structure and hyperparameter settings as presented in the original paper and code. The only additions were the resolution scale R and an extra cross-attention layer to capture correlations across different resolutions. In practical implementation. we modified the original diffusion and sampling processes so that TSDiff first generates low-resolution data and then refines to high-resolution data.
Moreover, similar to TSDiff, our proposed MODEM is also unconditional diffusion model. When applying MODEM to univariate datasets, the Spatial Transformer components in both the time-invariin invarinnt and time-varian encoders were omitted, and the model's input dimensions were adjusted based on the specific context length and prediction length each dataset. The resolution scale, context length, and prediction length for the Traffic, Exchange, and KDDCup datasets set to (3, 336, 24), (3, 360, 30), and (4, 312, 48), respectively. These modifications ensure seamless deployment of MODEM univariate data.
B.3 EVALUATION METRICS
We use the following metrics for evaluating the performance of time series anomaly detection methods:
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Precision: It measures the proportion of correctly detected anomalies among all time points flagged as anomalies:
TP
Precision
(27)
TP + FP
where TP denotes the number of true anomalies correctly detected, and FP denotes the number of normal time points incorrectly identified anomalies.
Recall: This metric is also named as sensitivity or true positive rate. It measures the proportion of actual anomalies that are correctly detected by the algorithm:
TP
Recall
(28)
+ FN'
where FN denotes the number of true anomalies that are not detected.
F1-score: Fl-score is the harmonic mean of Precision and Recall, which can balance both false positives and false negatives:
Fi-score = 2 Precision x Recall
(29)
Precision + Recall
Average Sequence Detection Delay (ADD) Tuuli al. 2022): It used for evaluating the speed and timeliness of time series anomaly detection algorithm:
1 ADD
S (T. - pi);
(30)
where p; denotes the initial time of anomalous span i, Ti 2. P: denotes the corresponding detection delay time by the anomaly detection algorithm. S is the total number of anomalous spans. A small ADD signifies a more timely detection of anomalies.
B.4 ENSEMBLE INFERENCE
Considering the multi-resolution setting, we extend the ensemble inference (Chen al., 2023) to
combine the reconstruction eтrors [E1.1.... E......, Et...... ET,3} at different diffusion steps t and resolution scales r for enhanced anomaly detection. First, the anomaly label Yr,- at step I, r computed based on the reconstruction error:
EETR EEr
Ytr 1(E,r Ter)), where Te.
.T.RR
(31)
where æT, denotes the upper percentage of reconstruction errors final step T, R. This approach allows us to assess the reconstruction quality at each step relative to the final reconstruction error. If the ratio EEr.a is small, it indicates poor reconstruction quality, and the upper percentage of anomaly ΣE.. labels decreases accordingly, retaining only the time points with high reconstruction errors and confidence. After obtaining the prediction labels Y,r- = (vi.r- .i,, V(,r) (here vf,- = denotes that the data at time point / predicted as an anomaly, and sf,- 0 otherwise),
voting scheme aggregate them. If the number of anomaly votes vi = ΣT_ D vl.- at time point / exceeds the voting threshold (, that is:
M 1(v > (),
(32)
it will be classified as anomaly data. Since the reconstruction errors in the carlier diffusion steps are typically larger, we only aggregate the denoising results from the last 10 diffusion steps for each resolution.
is well known that there is an inherent trade-ofi off between precision and recall. By setting a higher voting threshold the model reduces false positives (i.e., higher precision) but risks missing some actual anomalies (i.e., lower recall). Conversely, lower threshold allows the model to detect more anomalies (i.e., higher recall) but may result in more false positives (i.e., lower precision).
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B.5 NETWORK STRUCTURE
The detailed architecture of Time-Invariant Encoder is presented in Figure. it comprises multiple residual blocks. The residual block first uses 1 convolution kernel to encode the time-invariant components obtained from frequency decomposition. The diffusion step &: and resolution scale are fed into simple MLPs to obtain corresponding embeddings, which are added to the convolution result as supplementary information. Subsequently, it employs hierarchical transformers to further explore the intra-scries temporal features and inter-series dependencies among various variables.
Output
Linear
Conv-1x1
Spatial Transformer
Residual
Block
Temporal Transformer
MLP
Residual
Conv-1x1
MLP
Block
k
Residual
Block
Input
Figure 6: Architecture of Time-Invariant Encoder, consisting of multiple stacked Residual Block.
B.6 HYPERPARAMETER SETTINGS
The detailed hyperparameter settings of our MODEM are presented in Tab. Due to the large number of hyperparameters and resource constraints, selection to determine the parameter combination. The diffusion step K is set to 50. while the resolution scale R indicating the number of steps for the forward process. The sampling step is set to 20 for the denoising process. 20% of the frequency components are used for decomposition, represented by m. Our model incorporates 2 residual blocks and 4 DMTBs. Additionally, a dilation factor of 2 is applied, and our model is equipped with TCN modules, which are temporal convolutional networks that help handle non-stationary time series. For ensemble inference, the upper percentage at final step RT.R is set to 0.02, and the ensemble threshold is set to 10 for all datasets. These hyperparameter values collectively define our model's structure and its ability to process son-stationarity.
empirical tuning combined with Bayesian
B.7 COMPARISON RESULTS USING AFFILIATON -ASSDD METRICS
To ensure a more comprehensive comparison and provide more convincing validation of MODEM's effectiveness, we also evaluate the detection performance using affiliation-based metrics (Wang et al. 2024) to avoid the illusion of progress caused by point adjustment (Kim al. 2022 As shown in Tab. 6 our MODEM achieves the highest Fi-scores across all three datasets with high non-stationarity. with average improvement of 3.86% (from 80.34% 84.20%). This suggests that the cross-resolution correlation enables a robust reconstruction signal in a coarse-to sonfine manner, allowing most anomalous points be effectively distinguished from normal non-stationary behavior.
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Table 5: Detailed hyperparameter settings of MODEM.
Hyperparameter
Value
Diffusion step K
50
Sampling step L Resolution scale
20
Percentage of frequency decomposition Number of residual blocks
20
Number of DMTBs
4
Dilated Factor
2
Number of Modern modules
5
Upper Percentage ит,
0.02
Ensemble Threshold Ç
10
Table 6: Comparison of detection performance across three datasets with high non-stationarity using affiliation-basee metrics. The best results are bold, and the second-best are underlined.
PSM
SMD
SWaT
Method
F1"
F1
F1
Sampling
84.39
$1.65
64.08
74.53
3144
44.22
60.62
84,66
70.65
59.65
ECOD
74.60
33.84
46.56
73.98
16.15
26.51
97.61
11.51
20.59
31.22
PCA
92.20
37.71
53.53
83.88
40.19
54.34
63.58
78.18
67.61
58.49
CBLOF
59.90
98.45
74.49
86.67
33.52
48.34
63.08
70.91
66.77
63.20
[Forest
100.0
3.35
6.48
100.0
9.37
17.13
61.17
70.14
65.35
29.65
LODA
92.66
40.17
56.05
59.02
66.18
62.60
61.77
70.41
65.35
61.33
VAE
62.21
87.72
72.80
82.09
43.49
58.66
73.19
66.02
65.79
DeeDSVDD
74.05
50.64
60.15
64, 98
46.77
54.13
59.11
72.35
65.69
$9.99
ATM-AE
75.11
75.86
75.48
84.96
43.49
57.53
60.18
72.19
65.64
66.22
MTAD-GAT
79.90
60.14
68.63
85.90
67.69
75.71
65.09
77.51
71.23
71,86
TEAD
79.14
71.63
75.20
56.32
97.88
71.49
60.38
71.96
69.53
72.07
AnomalyTransfome
52.01
85.04
64.55
100.0
3.19
6.19
55.41
59.94
57.59
42.78
D'R
62.94
76.09
99.26
86.82
85.29
78.12
MODEM
73.48
87.55
79.90
89.18
95.82
92.38
74.36
87.32
80.32
84,20
B.8 ADDITIONAL EFFECTIVENESS ANALYSIS
Multi-Resolution Modeling We present several case studies in Fig. 7 which quantitatively validate the effectiveness of multi-resolution modeling for anomaly detection. In the figure, the pink and yellow areas represent true anomalies and false anomalies (normal data). respectively. The green and purple lines correspond to the input and reconstructed time series, respectively, with the red bold frame highlighting anomalies detected by the model. These examples demonstrate that models operating single resolution struggle to detect true anomalies and frequently trigger false alarms. Benefiting from an expansion to multi-resolution settings, our model learns normal temporal patterns more effectively, providing more reliable reconstructed baseline that significantly reduces false alarms while accurately detecting true anomalies.
Frequency-Enhane Decomposable Network Previous works (Yang & Hong 2022 He et al.
2023) have demonstrated that spectral responses can more robustly capture underlying temporal patterns compared to time-domain representations. Despite this foundation, our approach introduces several distinct contributions and innovations for non-stationary time series anomaly detection.
Firstly, we explore different frequency computation strategies and utilize the Short-Term Fourier Transform (STFT) to extract the spectral information. STFT not only provides a detailed view of how the frequency content of the signal changes over time but also captures the transient behavior and dynamics of time series through overlapping windows. Compared to the conventional Fast Fourier Transform (FFT) and the Discrete Fourier Transform (DFT), STFT is more suitable for decomposing non-stationary time series, which quantitatively verified through experiments on PSM and SMD datasets, shown in Tab. 7
Secondly, the proposed Frequency-Enhanced Enhanced Decomposable Network differs significantly in its network structure from the MI.P-hased block like Koopa (Liu 2024). Specifically, it incorporates
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Input Tume Sories
True Aommlyy
Predicted A nomaly
Reconstructed Series
False A Aoomaly
Imput Time Series
Single Resolution
Resolution
1.0
02-
Resolution
Resolution
Figure Comparison of detection performance across different resolution settings originates from the SMD dataset, where the pink and yellow areas represent true anomalies and false anomalies (normal data), respectively. The green and purple lines indicate the input time series and the reconstructed time series, respectively, while the red bold frame highlights anomalies detected by the model.
hierarchical transformers the time-variant and time-invariant encoders to capture both intra-series temporal dependencies and inter-series correlations among variables. Additionally, it designs novel ModernTCN block enhanced by dilated convolution (named DMTBs) to uncover complex periodic patterns across multiple time scales. These unique designs, compared to Koopa's all-MLP architecture, are more suitable for long-series reconstruction, thus providing reliable reconstruction signals for anomaly detection, as demonstrated by the quantitative validations in Tab.
Table 7:
Comparison of different fre- quency statisties approaches PSM and SMD datasets.
Table 8: Comparison of different network types on PSM and SMD datasets.
Dataset
Approach
P
R
F1
Dataset
Network
P
R
F1
FFT
96.64
97.98
97.30
Koopa (MLP)
95.68
97.53
96.59
PSM
DFT
95.86
97.12
96.48
PSM
MODEM
96.97
98.35
97.65
STFT
96.97
98.35
97.65
Koopa (MLP)
94.42
95.43
94.92
FFT
95.46
96.07
95.76
SMD
MODEM
95.70
96.32
96.01
SMD
DFT
94.73
95.44
95.08
STFT
95.70
96.32
96.01
In our method, we select frequency components with the top m percent of amplitudes stationary factors based on spectral statistics, with the remaining frequency components treated as non-stationary factors. As mentioned in Appendix B.6, we set the percentage m 20, because the frequencies corresponding to the top 20% amplitudes account for over 90% of all frequency components, which aligns with the reality where stationary factors dominant. To investigate the impact of
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percentage the detection performance further, conduct sensitivity analysis on the SMD dataset regarding m, as illustrated in Tab. Variations within a reasonable range m do not cause drastic changes in performance, demonstrating that this frequency- basyd selection approach is robust. m is set too low (see m = 2) or too high (see m = 50), the model's performance significantly decreases because the time-variant and time-invariant variables not effectively separated.
Table Impact of different percentages m on detection performance.
Percentage m
Frequency Ratio
P
R
F1
66.89%
95.34
95.67
95.50
10
85.33%
95.68
96.10
96.88
15
88.62%
95.74
96.21
95.97
20
91.45%
95.70
96.32
96.01
25
93.56%
95.66
96.25
95.95
50
98.73%
95.40
95.76
95.57
B.9 HYPERPARAMETER EXPERIMENTS
Effectiveness of Resolution Scale Regarding the hyperparameter resolution scale R, we have validated its impact on detection performance in the SMD dataset in Section 4.3. From this, we selected the recommended value 4, which we then generalized the other four datasets to report performance. To more comprehensively evaluate the effectiveness of the multi-resolution setting and the sensitivity of hyperparameter R. we fixed the sampling steps of accuracy in favor of inference efficiency, while adjusting R on the other four datasets to observe performance changes. As shown in Tab. 10 the optimal resolution scale R varies slightly across different datasets. We attribute this variation differences in anomaly rates and the types of anomaly patterns contained in each dataset, which require tuning R to allow the model to effectively benefit from multi-resolution data. In general, for datasets with low anomaly rate and many short-duration anomalies (lasting only a few time points), we recommend setting a lower resolution scale π (e.g.. overlooked in the low-resolution
to sacrifice a small amount
for the SMD and SMAP datasets) to ensure that anomalies data. For datasets with higher anomaly rate and longer-duration anomalies, a higher Rt should be used allow the model to fully leverage the temporal correlations across different resolution scales and capture both normal and anomalous event patterns.
Table 10: Results of various resolution scales on PSM, MSL, SWaT, and SMAP datasets.
PSM R
MSL R
SWa'T R
SMAP
R
P
F1
P
F1
P
F1
P
R
F!
I
94.27
96.14
95.20
87.84
86.25
87.03
86.30
89.21
87.73
87.21
95.40
91.12
2
95.02
96.81
95.91
89.76
87.02
88.37
87.44
90.65
89.02
88.92
96.32
91.92
3
96.18
97.70
96.93
90.55
87.74
89.12
88.92
92.43
90.63
90.68
96.93
93.04
4
96.97
98.35
97.65
91.28
88.32
89.77
89.42
93.08
91.21
89.50
97.40
92.66
5
97.46
98.27
97.86
91.26
88.84
90.03
89.51
93.43
91.43
88.50
97.22
92.66
6
97.83
98.23
98.03
90.92
88.41
89.65
89.30
93.49
91.34
87.92
97.38
92.68
7
97.12
97.96
97.54
88.8
87.75
88.29
88.75
92.51
90.55
87.22
96.17
91.47
8
96.27
97.41
96.83
88.05
86.87
87.46
86.23
90.40
88.57
86.53
95.49
90.78
Trade-off between Precision and Recall It well-known that there an inherent trade-off between precision and recall. As described in Appendix B.4 we can balance this trade-off by adjusting the voting threshold Ç curing the ensemble inference process. Specifically, we ensure that the optimal resolution scale R is maintained across all datasets, and then dynamically vary the value to observe the impact on anomaly detection performance. As shown Tab, [T our approach does not solely focus on optimizing recall to improve detection performance. Regardless of whether is increased or decreased (except when Ç is set to a very low value, such as Ç = 2), our model consistently maintains stable performance, with significant drop in Fl-score. In practical applications, by setting a higher value for Ç, our model reduces false positives and thus achieves higher precision. Conversely, lower ( allows the model to detect more anomalies, leading
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higher recall. This enables better balance between precision and recall, tailoring the model to meet different real-world requirements.
Table 11: Results of various voting thresholds for ensemble inference on trade-off between precision and recall.
SMD
(R
4)
SMAP (R
4)
SWaT (R
5)
P
F1
R
F1
R
F1
2 92.51
98.03
95.19
87.62
98.12
92.57
87.26
95.03
90.98
5
94.68
97.28
95.96
88.20
97.95
92.81
88.27
94.50
91.27
95.19
96.77
95.98
88.78
97.71
93.02
88.90
93.87
91.31
10
95.70
96.32
96.01
89.05
97.40
93.04
89.51
93.43
91.43
13
96.15
95.93
96.03
89.63
96.83
93.09
90.16
92.93
91.52
96.82
95.08
95.94
90.07
96.29
93.07
90.94
92.01
91.47
20
97.27
94.76
95.99
90.81
95.27
92.98
92.07
91.05
91.55
25
97.62
94.18
95.86
92.10
93.70
92.89
92.95
90.08
91.49
Number of Network Layer
As shown in Fig. 3 and Fig. 6 our time-variant and time-invariant encoders are primarily composed of the proposed Dilated ModernTCN Blocks (DMTB) and residual blocks, respectively. We next explore how varying the number of DMTBs and residual blocks affects anomaly detection performance, providing practical insights for fine-tuning network architecture.
We begin by investigating the relationship between the number of DMTBs and detection performance, as illustrated in Tab. [2 Keeping other hyperparameters constant, we start by setting the number of DMTBs to 0, which means that only linear layers and hierarchical Transformers are used to model the non-stationarity. In this configuration, the model performs the worst on both datasets. As the number of DMTBs increases from 4, the model's detection performance steadily improves, reaching its highest point. Specifically, the F1-score on the SMD and SWaT datasets increased by 1.05% (from 95.01 96.01) and 1.29% (from 90.05 91.21), respectively. However, when the number of DMTBs is further increased beyond 4, the performance no longer improves and instead slightly declines both datasets. This suggests that the benefit of decoupling the time-invariant and time-variant components from complex non-stationary temporal patterns, based on spectral information, diminishes with excessive use of DMTBs. Overusing DMTBs may lead overfitting. where normal non non-stationarity is misidentified as anomalous behavior.
Table 12: Results of various number of Dilated Modern MCN Blocks (DBTMss on SMD and SWaT datasets.
SMD R
SWaT
Number of DMTBs
P
F1
F1
0
94.59
95.44
95.01
88.30
91.86
90.05
1
94.82
95.75
95.28
88.82
92.45
90.60
2
95.15
95.93
95.54
89.16
92.73
90.91
3
95.48
96.21
95.84
89.30
92.94
91.08
95.70
96.32
96.01
89.42
93.08
91.21
5
95.64
96.28
95.95
89.33
93.16
91.20
6 95.52
96.04
95.78
89.17
92.85
90.97
The residual blocks in the time-invariant encoder are designed to capture global dynamic information. As illustrated in Tab. 13 since time-invariant patterns tend to be simpler than time-varying compo- nents, we found that increasing the number of residual blocks to 2 yielded satisfactory detection performance. Increasing the number beyond 2 did not result in significant performance gains but did incur additional computational cost. Therefore, we decided to set the number of residual modules in the time-invariant encoder to 2.
B.10 MODEL EFFICIENCY
We provide a comparison of various time series anomaly detection methods, including both diffusion- based and non-diffusion methods, in terms of computational cost and efficiency the SMD dataset.
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Published as a conference paper at ICLR 2025
Table 13: Results of various number of Residual Blocks on SMD and SWaT datasets.
SMD
SWaT
Number of Residual Blocks
P
R
F1
P
R
F1
0
94.38
95.54
94.96
88.54
92.45
90.45
95.37
96.16
95.76
88.97
93.01
90.94
2
95.70
96.32
96.01
89.42
93.08
91.21
3
95.62
96.26
95.94
89.56
92.98
91.23
4
95.54
96.12
95.83
89.22
92.90
91.02
The results are summarized in Tab. 14 Specifically, Training Time represents the time taken to train the model for epochs with the same batch size, while Inference Time indicates the duration required to process the entire test dataset. Additionally, we report the total number of trainable parameters of the model and the FLOPs (floating operations per second) per input unit. As models naturally require longer training and inference times due to the iterative nature solving each diffusion step. However, considering that the training and test data on the SMD dataset spans approximately 16 days, the extra overhead introduced by the diffusion process is acceptable given the significant performance gains provides for real-world applications. Furthermore, training within the noise space enables diffusion-based methods detect anomalous data more swiftly, as reflected in their lower ADD.
observed, diffusion-basse
Even with multi-resolution time series modeling, MODEM method outperforms the state-of- -ftthe- art method D3R, saving 9.62% in training time, 22.8% in inference time, and 82.77% in memory usage. These gains are primarily due to the lighter denoising network and the mathematical extension of the multi-resolltion diffusion process to DDIM sampling, which significantly reduces the number of sampling iterations. Regarding the computational overhead of our MODEM, it incorporates hierarchical transformers, and the calculation of attention matrices contributes to higher FLOPs. However, this is acceptable, given the support of highly parallelized attention mechanisms and the continuous advancement of hardware resources.
Moreover, the ongoing development fast sampling techniques in the diffusion domain, such as DPM Solver++ and single-step sampling, ensures that our multi-resolution paradigm can be further optimized and scaled for wider adoption.
Table 14:
Comparison various methods with respect training time, inference time, parameters. FLOPs, and ADD.
Method MTAD-GAT
Training Time (s)
Inference Time (s)
Total Params (MB)
FLOPs (M)
ADD
213.92
76.54
0.98
4.46
90 10
TEAD
353.26
46.82
1.56
5.20
52
Anomaly Transformer
487.85
97.24
23.28
106.38
31 1
TranAD
256.77
44.60
1.97
8.56
24 14
DifIAD
620.19
153.76
15,38
93.30
48 1
ImDiffusion
442.57
102.35
8.23
55.59
24 +
D3R
527.48
164.28
68.27
303.27
65 + 16
MODEM (ours)
476.73
126.79
11.76
76.65
27 2
B.11
GENERALIZATION PERFORMANCE
To assess the generalization ability of MODEM, we conduct forecasting and imputation experiments across multiple non-stationary datasets. We apply MODEM in an unconditional manner non- stationary time series forecasting and imputation following (Kollovich et al. 2024), and report the average continuous ranked probability scores (CRPS) (Gneiting & Raftery 2007) across three independent runs in Tab. 15 1 and Tab. T6 as well the detailed discussion on experimental setup can be found in Appendix B. The results demonstrate that MODEM performs competitively against SOTA methods, despite not being specifically designed for forecasting tasks. We also extend the proposed multi-resolution diffusion paradigm to CSDI (Tashiro et al.) 202T) and TSDiff (Kollovieh et al., 2024) (denoted CSDI-MR and TSDiff-MR) The improved forecasting and imputation performance validate that correlations across different time scales facilitate the model to better capture diverse temporal patterns.
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Published as a conference paper at ICLR 2025
Table 15: Comparison of forecasting performance on three datasets.
Table 16: Comparison of imputation perfor-
mance on two datasets.
Method
Traffic
Exchange
KDDCup
Method
Traffic
Exchange
CSDI
0.159 + 0.002
0.033 + 0.014
0.318 0.002
CSDI-MR
0,148 = 0.004
0.029 0.008
TSDiff TSDifI-MR
0.120 + 0.008
0.015 = 0.001
TSDiT
0.098 1 0.002
0.011
0.001
0.311 0.026
0.115 0.015
0.011 = 0.001
TSD
0.090 + 0.003
0.006
0.000
0.295 0.018
MODEM
0,117 + 0.007
0.017 = 0.001
MODEM
0.096
0.002
C LIMITATIONS AND FUTURE WORK
We extend the diffusion model into multi-resolution paradigm, leading directly to R-fold increase in the number of sampling steps, which results in longer training convergence times and prolonged inference durations. While introduce accelerated sampling strategies for each resolution R, these are predicated on the acceptance of loss in precision. In the future, it may prove beneficial investigate sampling along deterministic trajectory within mixed resolutions improve both accuracy and efficiency. Furthermore, the hyperparameters Rt and m in our proposed frequency- enhanced decomposition network require customization for different datasets, thereby adding the complexity of parameter tuning. Additionally, in the future, it may be advantageous to consider employing dynamic smoothing strategies instead of fixed pooling sizes to better reveal the periodic and varying characteristics of non-stationary time series.
D POTENTIAL NEGATIVE SOCIETAL IMPACTS
The diffusion models, like other generative technologies, have inherent risks. Our model, case in point, could potentially have negative socictal effects. For instance, might memorize private data and be exploited to fabricate misleading or false information.
27


Reviews:
[{"review_id": "out2CB3Lr4", "reviewer": "Reviewer_cQvr", "summary": "The authors introduce a multiscale diffusion model for anomaly detection in non-stationary time series that relies on combining representations of the time series at multiple temporal resolutions.", "questions": "- see Weaknesses", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is well written, mathematically sound and clearly formulates subcomponents of the method. \n\n- The experimental evaluation and comparisons are rather extensive.", "weaknesses": "- My main concern centres around hyperparameter settings for which the authors claim to follow empirical tuning with Bayesian selection without providing further details. I understand the difficulties stated by the authors in properly tuning the large number of hyperparameters; however without an objective method for setting parameters based on training data alone it is not clear whether the selected parameter settings will generalise well to other datasets. Here, the authors should at least provide some analysis of how sensitive the results are to variations of individual parameter values by keeping the rest of the parameters at their suggested values. \n- The resolution parameter R is set to a rather low value of 4 in the experiments and hence covers only a narrow range of resolutions which seems contrary to the overall claim of the authors that the method leverages correlations across various resolution scales. \n- The method seems to be mainly improving on the recall value while trailing behind interns of precision, sometimes significantly, compared to other methods. I believe further clarification of this tradeoff in practical scenarios is needed.  \n- The authors should provide more details on the pooling operation in the MRD sampler.\n- No code is provided which makes proper assesment of the experimental results and method difficult.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cEZTjPNodn", "reviewer": "Reviewer_meLm", "summary": "In this paper, authors design a modal named MODEM（Multi-Resolution Decomposable Diffusion Model）for anomaly detection in non-stationary time series data. The proposed model integrates a coarse-to-fine diffusion paradigm with a frequency-enhanced decomposable network to enhance the modal’s ability of handing the non-stationary time series data. Through this approach, the proposed model allows for utilizing the multiple resolutions information of non-stationary time series data, which varies for each datasets.", "questions": "see Weaknesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1)The proposed method MODEM is novel , it can utilize the multiple resolutions information of non-stationary time series data to improve the modal’s ability of anomaly detection.\n\n2)The authors conducted experiments on diverse datasets, and their proposed method demonstrated excellent performance.\n\n3)The authors presented a clear exposition of their research motivation and proposed solution. The analyze of difference between low-resolution time series and high-resolution time series helps understand the effect of the proposed modules.", "weaknesses": "1)While the model achieved the best F-score performance due to higher recall values, its precision values were comparatively lower than other models (Table 3). The authors should explain this observation.\n\n2)There is no comparison of computational cost(# of params, Gflops...) compared to existing models, while the diffusion model is extended into a multi-resolution paradigm, leading directly to an R-fold increase in the number of sampling steps.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ePCS7Pg6AZ", "reviewer": "Reviewer_hGSS", "summary": "The paper introduces a novel Multi-Resolution Decomposable Diffusion Model (MODEM), designed for unsupervised non-stationary time-series anomaly detection. It addresses the challenge of capturing complex temporal patterns in non-stationary data by leveraging multi-resolution modeling. MODEM employs a coarse-to-fine diffusion process to handle cross-resolution correlations, improving anomaly detection and time-series reconstruction. A frequency-enhanced decomposable network in MODEM extracts both global and time-variant dynamics. Extensive experiments across five real-world datasets demonstrate its superior performance.", "questions": "1.\tRegarding the ablation study, for SMD, increasing R from 1 to 4 improves the performance in Figure 4. Does this generalize to other datasets? For datasets with many short-lasting anomalies (spanning only a few time points), would these anomalies only be captured in high-resolution data? If so, the utility of low-resolution data in such cases remains questionable.\n\n2.\tIs the time series in Figure 1 are real-world data? If so, which domain does the data belong to, and do similar time series challenges appear across other domains?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well-organized with clear writing and visualization. The experiments are comprehensive.", "weaknesses": "1.\tConsidering the FFT process and the multi-resolution, it may cast a concern on the training time and the computational complexity, which are not sufficiently discussed in the paper, since ADD seems to be a metric for responsiveness of the method, not its computational efficiency.\n\n2.\tIn addition to reconstructed-based and forecasting-based methods, there are also some density-based methods can be discussed [1,2].\n\nReference:\n\n[1] Graph-Augmented Normalizing Flows for Anomaly Detection of Multiple Time Series\n\n[2] Detecting Multivariate Time Series Anomalies with Zero Known Label", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oNcBvw9psH", "reviewer": "Reviewer_VctK", "summary": "This paper introduces MODEM, a conditional diffusion model for non-stationary time series anomaly detection that incorporates (1) frequency-enhanced decomposable network, (2) multi-resolution guidance and (3) contextualized (cross-scale contextual) diffusion. The underlying network consists of two parallel encoders using residual Transformer and dilated ModernTCN blocks respectively, while a frequency decomposition module is built to disentangle time-invariant and time-variant components. Empirical studies on a variety of datasets show that this coarse-to-fine diffusion framework achieves good reconstruction performance and captures the key information in the non-stationary data, while ablations show that the specific design of the MODEM outperforms reasonable alternatives.", "questions": "1. See the \"Weaknesses\" section.\n2.  From `Section 3.2 Multi-Resolution Data Sampler`: \"This strategy enables our model to learn the evolutionary characteristics of non-stationary time series across multiple resolutions, ... \"  $\\rightarrow$ This statement describes multi-resolution as the contributor of non-stationary time series learning, but how does this facilitation occur ?\n3. The computational cost is a concern of diffusion models, especially under real-time anomaly detection. Therefore, comparison of the efficiency of different methods (diffusion and non-diffusion) is of great interest. Can the authors provide more accurate estimates w.r.t. benchmarks within a reasonable time and parameters ?\n4. As far as I know, TSDiff was developed for univariate time series generation. Can the authors give more details of your changes in its original paper ?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-structured and the main ideas are clearly presented. The figures and supplementary details are relevant and informative.\n- To my best knowledge, this is the first work to introduce a multi-granularity mechanism into time series anomaly detection. The same applies to the contextualized forward and reverse conditional diffusion processes.\n- The approach is studied through a broad set of experiments that show it consistently outperforms recent, relevant competitor methods in terms of some standard quality metrics for both real-world and non-stationary time series anomaly detection. The additional experiments are conducted, e.g. multivariate time series forecasting and imputation, despite being relatively weak.\n- This paper also provides a series of ablation experiments to test the effect of each important choice.", "weaknesses": "1. The novelty of the proposal is somehow modest overall. The main body of the proposed method is like a stack of existing works (framework $\\rightarrow$ CONTEXTDIFF [1], Top-K iFFT $\\rightarrow$ ETSFormer [2], network $\\rightarrow$ ModernTCN [3] and D$^3$R [4], multi-resolution $\\rightarrow$ MG-TSD [5]), although applying algorithms from different tasks (text-to-image, prediction) to a new task (anomaly detection) can also be seen as a valid contribution.\n2. The content of Section 3.3 with theoretical  derivation is very similar to CONTEXTDIFF [1], where the difference is just *coarse-resolution sequence $\\leftarrow$ text (condition)* and *high-resolution sequence $\\leftarrow$ image (target)*. However, the authors did not clearly tell this point in their paper where the proper citation is not given. In particular, the entire paper cites [1] in `Section 3.3 Multi-Resolution Forward Process` for only once (in an inconspicuous place), and [1] is completely omitted in the theoretical results in Appendix A. I understand that these sections make the paper more solid, but I suggest the authors emphasize contributions of the prior work when necessary. \n3. Although important works in the field of anomaly detection are analyzed in Section 2, there are some other diffusion-based approaches that should be included, such as related conditional diffusion model [1], diffusion model for time series [5,6] and so on.\n\n**References:**\n> [1] Yang, Ling, et al. \"Cross-modal contextualized diffusion models for text-guided visual generation and editing.\"  \n[2] Woo, Gerald, et al. \"Etsformer: Exponential smoothing transformers for time-series forecasting.\"  \n[3] Luo, Donghao, and Xue Wang. \"Moderntcn: A modern pure convolution structure for general time series analysis.\"  \n[4] Wang, Chengsen, et al. \"Drift doesn't matter: dynamic decomposition with diffusion reconstruction for unstable multivariate time series anomaly detection.\"  \n[5] Fan, Xinyao, et al. \"MG-TSD: Multi-Granularity Time Series Diffusion Models with Guided Learning Process.\"  \n[6] Shen, Lifeng, et al. \"Multi-Resolution Diffusion Models for Time Series Forecasting.\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)