Root Cause Analysis of Anomalies in Multivariate Time Series through Granger Causal Discovery
Authors: Xiao Han
Saima Absar
Lu Zhang
Shuhan Yuan
Venue: ICLR 2025 Oral (2025)
Keywords: root cause analysis
Granger causality
multivariate time series
Primary area: interpretability and explainable AI

Abstract: Identifying the root causes of anomalies in multivariate time series is challenging due to the complex dependencies among the series. In this paper, we propose a comprehensive approach called AERCA that inherently integrates Granger causal discovery with root cause analysis. By defining anomalies as interventions on the exogenous variables of time series, AERCA not only learns the Granger causality among time series but also explicitly models the distributions of exogenous variables under normal conditions. AERCA then identifies the root causes of anomalies by highlighting exogenous variables that significantly deviate from their normal states. Experiments on multiple synthetic and real-world datasets demonstrate that AERCA can accurately capture the causal relationships among time series and effectively identify the root causes of anomalies.

Published as a conference paper at ICLR 2025
RoOT CAUSE ANALYSIS OF ANOMALIES IN MULTI- VARIATE TIME SERIES THROUGH GRANGER CAUSAL DISCOVERY
Xiao Han' Saima Absar? Lu Zhang
Shuhan Yuan!
Utah State University, "University of Arkansas {xiao. han, shuhan. yuaanOusu. edu, {sa059, 9, 1z006 @uuark.ecu
ABSTRACT
Identifying the root causes of anomalies in multivariate time series is challeng- ing due the complex dependencies among the series. In this paper, we pro- pose comprehensive approach called AERCA that inherently integrates Granger causal discovery with root cause analysis. By defining anomalies as interventions on the exogenous variables of time series, AERCA not only learns the Granger causality among time series but also explicitly models the distributions of exoge- nous variables under normal conditions. AERCA then identifies the root causes anomalies by highlighting exogenous variables that significantly deviate from their normal states. Experiments multiple synthetic and real- world datasets demonstrate that AERCA can accurately capture the causal relationships among time series and effectively identify the root causes of anomalies.
INTRODUCTION
Root cause analysis on multivariate time series data, which is to identify the underlying causes of an anomaly, has wide spectrum of applications in various domains, such as diagnosing the fault of online cloud-based systems cyher-physical systems (Jayathilaka et al. 2017: Jeyakuma et al.) 2019 Soldani & Brogi, 2022, Yu al., (022). Traditional approaches, which manually trace the root cause based on the topology of the systems, have become impractical due to increasing system complexity, leading to greater focus on data-driven methods. One promising direction is based on causal framework, which models system components and their dependencies via causal graph and then traces how the failure of one component might propagate through the system (Assaad ee al. 2023a Li et al.) 2022 Zhhang et al.) [2021] Ikram al. 2022 Wang et al. [2023b] Okati al. 2024 For instance, for a cyber-physical system like a water treatment plant equipped with multiple sensors such water level, PH level, and electrical conductivity- generate multivariate time series data, if an attacker overdoses sodium hydroxide, it could lead to abnormal readings in metrics like pH level and electrical conductivity. Root cause analysis aims identify the root cause such abnormal behavior, even when the time series data create ripple effects across other metrics-for example, an increase in sodium hydroxide leading to abnormalities in additional measurements.
Despite the advantages of providing scalable and systematic way of understanding the relation- ships and causal chains in complex systems. existing causal inference- based root cause analysis approaches usually suffer from various limitations. For example, BBuhhthhkii et al. (2022) and As- saad et al. (2023b) assume the causal relationships as prior knowledge, which may not be feasible in real cases, On the other hand, although some approaches (Yang et al. [022 Meng al.) 2020 Wang ctt al. (2018) try to learn the causal structures from the observational data, they usually leverage the existing causal discovery algorithms, which do not consider the need for identifying root causes.
In this paper, we propose a comprehensive approach that inherently integrates Granger causal dis- covery with root cause analysis. We treat the root cause of the anomaly, such overdose of sodium hydroxide as an intervention on the exogenous variables a structural causal model (SCM). We refer this the exogenous intervention, where the exogenous variables follow stable distri-

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bution under normal conditions but undergo interventions when anomalies occur Under this core idea, we identify the key to root cause identification, which is to model the normality of exoge- nous variables for multivariate time series and then highlight abnormal exogenous variables. The current causal discovery approaches mainly focus on identifying the causal structures among time series/endogenous variables without explicitly modeling the impact of exogenous variables, making them unsuitable for locating the root cause of an anomaly due to exogenous interventions. There- fore, achieve our goal, we propose novel autoencoder-based framework for root cause analysis, referred to as AERCA. This framework identifies Granger causal relationships in time series by ex- plicitly modeling the distributions of exogenous variables, which serves as the foundation for our root cause localization approach.
Specifically, to model the data generation process, i.e., the causal relationships as well as the dis- tributions of exogenous variables, the encoder models the abductive reasoning process to derive the exogenous variable for each time series. Based on our core assumption that the exogenous variables are mutually independent, we establish effective constraints to ensure this independence. Meanwhile, the decoder learns a deductive reasoning process to infer the observed data from the exogenous variables. We theoretically show that to predict the input at time t, rather than using exogenous variables of all time steps before t, the decoder only needs to take in the exogenous vari- ables and observed time series from a window prior to I. We train AERCA on the normal data. Then, upon deployment, if the encoder-derived values of exogenous variables significantly deviate from the norm, the corresponding time series are highly likely be the root cause of the anomaly.
The contributions of this paper are as follows: 1) we propose novel encoder-decoder structure for Granger causal discovery, which can not only learn the causal relationships between time series but also capture the distribution of exogenous variables; 2) based on the learned structural causal model, AERCA can not only identify the root cause time series but also highlight the root cause time steps: 3) experimental results on multiple datasets show that AERCA can achieve state-of-the art performance on both Granger causal discovery and root cause identification.
2 RELATED WORK
Understanding the root cause of an anomaly has received increasing attention because of wide real- world applications. Accurate root cause localization can help domain users understand and mitigate abnormal behaviors.
The mainstream approaches in root cause analysis follow two-step framework: identifying the dependency between variables from observational data and then localizing the root cause by explor- ing the dependency graph. Therefore, the key step is to build the dependency graph. Traditionally, domain knowledge a systems tool can be leveraged to build the dependency graph. For example, in a microservice system, a directed edge between two nodes usually indicates a system call (Kim)
al. 2013 Weng et al. 2018. Wang et al. 2018a Yu et al. 2021).
However, as the system becomes sophisticated, it becomes impractical to build the dependency graph based on domain knowledge, and the call graph learned by system tools may not represent the true dependency between sensors (Kim et al.] 2013). Therefore, data-driven approaches are now commonly used for learning the dependency between variables. For example, various deep neural networks are developed to capture the temporal and spatial correlations in the multivariate time series for root cause analysis (Zhang al. 2019 Tuli al. 2022 Zhao et al. 2020).
Recently, causal inference- based root cause analysis has received increasing attention, which models the anomaly as data under intervention (Assaad al. 2023a Li et al. 2022). Under this assump- tion. root cause localization is to identify the intervention on observational data (Li et al. 2022). Several approaches leverage the PC algorithm (Spirtes et al. [|00]] or its variance build the causal graph by using the conditional independent test (Zhang et al.) [021] [kram et al.] 2022). Some ap- proaches also leverage the graph neural networks learn the causal relationships between nodes by simulating the data generation process (Wang et al. 2023b a)
Note that not all attacks can be treated as interventions exogenous variables. Understanding the nature of anomalies is crucial before applying our method real-world applications.
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In this work, we propose comprehensive approach that inherently integrates Granger causal dis- covery with root cause analysis. By assuming that anomalies are caused by exogenous interventions, introduce novel method for Granger causal discovery that explicitly models the distribution of exogenous variables. Consequently, unlike existing studies that can only locate the root-cause time series without specifying the abnormal time steps, our approach identifies the root cause the time series receiving exogenous interventions at specific time steps, providing much more informative and precise localization.
PRELIMINARY: GRANGER CAUSALITY
Granger causality (Granger 1969 Dahlhaus & Eichler 2003) is commonly used for modeling causal relationships multivariate time series. The key assumption that if the prediction of the future value Y can be improved by knowing past elements of X, then X "Granger causes"
Y. Granger causality was originally defined for linear relationships, while recently, the non-linear Granger causality has been proposed (Tank et al. 2021
Assaad et al. 2022):
Let stationary time-series as X (x1,,...... (X1: Nc:... XT), where x, E Rd is d-dimensional vector (e.g., d-dimensional time series data from d sensors) specific time t. Suppose that the true data generation mechanism is defined in the form of
Ft (i) p()
(d) (x 51-11" 51-1) uy
(1)
for 10r10 s d,
(1) ( denotes the past of series j: ut uli) indicates exoge- where (1) I... 2(22 x-r-1 nous variable for time series j at time step t; (-))() is function for time series j that cap- tures how the past values impact the future values of x(1). The time series i GGanger causes
(c) if fO) depends on x SS-- i.e. Ex((6)
St-1 xc(- (x) ,xx( 5t-1 x
x
(6)
(d)
* (()
(d) (1-1). Tank et al. 2021
Marcinkevičs & Vogt 2021 Shojaie
St-1""
Fox 2022)
Limitations of Granger Causality. While Granger causality is a valuable method for detecting temporal causal dependencies, it is important understand its limitations. Specifically, Granger causality assumes no hidden confounding, i.e., all relevant variables influencing the causal relation- ship are observed and included in the model, and no instantaneous effects between variables, i.e., the influence of one variable on another is not immediate but occurs with some time lag. Violating these assumptions can lead erroneous conclusions in Granger causality analysis, highlighting the importance of careful assessment of assumptions and consideration of alternative models.
4 METHODOLOGY
4.1 PROBLEM FORMULATION AND FRAMEWORK
Based on the structural equation of multivariate time series defined in Eq. [ in this work, we focus on the anomaly 2,(1) caused by exogenous interventions on a single or multiple time series, leading to significantly deviating value in its exogenous variable a(?)) which can be defined as
(1) (0) = ) 11-1 x + at ( (x) S1-1""" (t u + +
()) for isidd, (2)
where 2. = ur + c with an anomaly term e Note that the abnormal time series caused by exogenous interventions can be either point anomaly or sequential anomaly. The point anomaly can be due to an exogenous intervention on a specific time series at a time step. In contrast, a sequence anomaly be caused by the propagation of an exogenous intervention through time by following the causal structural model or consistent exogenous intervention over time steps.
Therefore, an informative root cause analysis shows not just the time series but also the time steps receiving the exogenous intervention. Based on this motivation, we define the task of root cause identification below.
Definition 1. The root cause identification is to locate the time series/variables (j) at specific time step(s) with the abnormal exogenous variable uif))
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For the anomaly caused by the exogenous interventions, achieve the root cause analysis, we learn the Granger causality in multivariate time series by explicitly modeling the distribution of exoge- nous variables. To this end, develop encoder-decoder structure for root cause analysis, called AERCA, which can calculate the exogenous variable for each time series at a specific time step- AERCA explicitly computes the exogenous variables via encoder, and decoder predicts the current value by simulating the data generation mechanism defined by the Granger causality. By training the encoder-decoder structure bution of exogenous variables in the normal status. When an exogenous intervention occurs, the derived exogenous variables should significantly differ from the normal ones. Meanwhile, because explicitly derive the exogenous variables at each time step, even if the time series is still ab- normal due to error propagation through time, AERCA can distinguish the root cause from the downstream impact. Figure I shows the framework of AERCA. In the following, we explain each components of the framework.
the normal time series, the model can capture the distri-
Time Series
Granger Causal Discovery
Normal
Abductive Reasoning
+i
Encader 69g
Go
Decoder 5i
Desoder oig
(4)
Deductive Reasoning
Abnormal
Root Cause Analysis
Root Cause Score
Encoder
olo
Figure 1: The overview of AERCA.
4.2 GRANGER CAUSAL DISCOVERY
Motivation. To model the data generation process, i.e., the causal relationships as well as the distributions of exogenous variables, we adopt the encoder-decoder structure to simulate both the abductive and deductive reasoning processes. Abductive reasoning to seek the most plausible explanations, i.e., to infer the most likely exogenous variables (causes) that could have generated the observed time series data. As shown in Eq. [[] based on the Granger causality, the value of the time series at step ? is function of past time series plus an exogenous term at the current step, ie., X₁ := fx + uy, with simplified notations. To simulate abductive reasoning, the encoder derives the exogenous variables based on the observed data by rewriting Eq. Tas
X, f(x5+-1)-
On the other hand, deductive reasoning derives effects from known causes, i.e., reconstructing the observed data from exogenous variables. By recursively resolving each previous time such as expressing X;-1 terms of its predecessor X;-2, and continuing this process backward the first time step-we can rewrite Eq. iin different way as function of the exogenous variables:
X, = =(( Ue
which shows that the observed data at step t is represented as function f(-) of all preceding ogenous variables. Within the encoder-decoder framework, this function acts as the decoder for reconstructing the observed data directly from the exogenous variables.
Based on the above analysis, we develop an encoder-decoder structure, where the encoder learss Granger causal relationships f(-) by using past time series values input compute the exogenous
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variables, simulating Eq. The decoder f() then takes these exogenous variables from the encoder as imput to reconstruct the value the current time step Xt, simulating Eq. 4
Encoder-decorder Structure. Given normal multivariate time series X (X), . .T.......)).. we define a window with length as W = (xt-K+1s-r +.., Xx) and convert a time series X to a sequence of sliding windows W (W K: W K+1,-.. Wr). We first aim learn the Granger causality of time series in a window, i.e., a window causal graph (Assaad et al.) 2022).
Given time series window, first parameterize the Granger causality in time series defined in Eq. as
K X; We. (xx=k)xx-k + ut
(5)
where wa, (x:-x) indicates the k-th neural network to predict the Granger causal relationship be- tween X and X,. The output of waz (xx-k) can be reshaped as a d d coefficient matrix, where the entry element (i,j) indicates the influence of x on (3) As shown in Eq, 5 K neural net- works are used to predict the weights of past K time legs on deriving x,. Therefore, relationships between d time series over K time lags can be explored by inspecting coefficient matrices. Fol-
lowing Eq. 3 rewrite the Eq. 5
K Σ
X
wan (xI-))x1-k-
(6)
Then, given time series window Wt, t. apply the encoder K times to derive the exogenous variables in a window, denoted as U, (u.-K+1,...... Ut).
To enforce independence between the derived exogenous variables, ensure that the distribution of U, adheres to an isotropic standard Gaussian distribution Q. By assuming that the exogenous variables follow multivariate Gaussian distribution and applying the KL divergence to quantify the distribution difference, we formulate the independence constraint
(P(U.))I)))))) 2 tr( tr( Ex) (nQ -1:) (nQ m1) -dd -d= llog det EQ det E,
2 (tr(2c) + pf pr d - log det 2.)
where lQQ O and Eq represent the and covariance matrix of the isotropic standard Gaussian distribution Qi and Σn are the mean and covariance matrix of U,.
The decoder to reconstruct the input x, based on the exogenous variables U, One challenge is that theoretically, the value x; at the current time step is computed by the exogenous variables of all the previous time steps. However, considering the potential infinite length of the time series, is impractical to reconstruct by using all the previous time steps. To tackle this challenge, we iteratively replace the x; with X, (1++) for subsequence with length and derive the following proposition.
Proposition Consider a basic autoregressive model where wis = Wg. (x) as a framework for analyzing Granger causality. The value the current time step x; can be derived by the exogenous variables from a previous window [u,-1.... ut-k] K] and the observed time series from a previous window [x,- K 11.,. Xx- (222]] with the following equation:
K
K
K
Σ
X, ak- (K + PKRIKK
+
aK+1-m
w k k- IK+1m
(8)
km
where Wk indicates the parameter of Granger causality, and On Σm Wn 1 1 K, is
recursive equation with og
We provide proof of the proposition in the Appendix A.I Inspired by Proposition]] 1 we propose a decoder structure that combines both observed time series and exogenous variables. Specifically, parameterize the impact of exogenous variable u on by a neural network Wa, and the impact of observed time series X on X, by another neural network w/ Then, the decoder computes x, based on the following equation.
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& و (u-
+
Leg K- (
Up
k=1
where & indicates the reconstructed value at time step and u computed by encoder defined in Eq.
The whole encoder-decoder structure can be defined as x, A B. (X<x). Given a time series with length T. the objective function to train the encoder neural netework we, and decoder neural networks Wg, 4Bg is defined as:
C
| xxll2 BDKL Ren + Rdel(( Adc
k=
(10)
T-1 (Ten S(e41 + Sa.5( (4) + Yde (5)
1=++1
where D.T. indicates the independence constraint U, defined in Eq. 7 S4 == wwox (x,- R)
wa, (x,-1)] indicates the concatenation of coefficient matrices over the past time steps; similarly, have 92 [wo (ut- k) :...: wa, (u,-1) and S2 ( (X--2k) :.... : Wee (X--x- R(-) indicates the LI and L2 norm penalty for sparsity of the coefficient matrices from the encoder and decoder; the -). is smoothness penalty, defined as S(1++1 14) [[0 S4ll22 À and hyperparameters.
Granger Causal Discovery. As the encoder-decoder is proposed to simulate the data generation process governed by Granger causality, we expect the function wa, can capture the causal relation- ships in time series. To further summarize the Granger causal relationships between variables summary causal graph, similar to (Marcinkeviõs & Vogt 2021), we aggregate the output from wa, into a summarized coefficient matrix as
Stj = max [median (xx (x:=))); for 1 S i,j S d, ISKSK
where Sty indicates the strength of the Granger causal effect from (() on x(i). To further derive the adjacency matrix A, we set a threshold r, if the value S T, then Ar, 1. In experiments, the threshold is set based on the quantile of the coefficient matrix S.
4.3 ROOT CAUSE LOCALIZATION
After training on the normal time series, we expect that the exogenous variables can be approximated by the encoder. When deploying the model for root cause localization, assume the time series arrived streaming manner. When a new time step 1* is arrived, we first adopt the encoder to
derive the exogenous variables u,- based on Eq. 6 Then, for each time series, ut- ()) we compute the
7-score as the root cause score 2 (.))) -2 .Gi where yنر and (5) indicate the mean and standard
deviation of the exogenous variable for the j-th time series in normal data. We then adopt streaming
beaks-over-thresholle (SPOT) (Siffer et al., (017)) to dynamically determine the threshold of labeling the potential root cause.
5 EXPERIMENTS
5.1 EXPERIMENTAL. SETUP
We conduct experiments on four synthetic and two real-world datasets. By using the synthetic datasets, we have the ground truth about the structural causal models as well as the root cause of anomalies. For the real-world datasets, we only have information about root cause variables. Therefore, the real-world dataset only to evaluate the root cause identification.
Datasets.
Synthetic Datasets: Linear Dataset (Marcinkevičs Vogt 2021) is dataset with linear interaction dynamics. Nonlinear Dataset (bsar et 2023) is nonlinear time series dataset. Lotka- Volterra

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[Marcinkevičs & Vogt 2021) is a nonlinear model that simulates a prairie ecosystem with multiple species. Lorenz 96 (Marcinkevičs Vogt 2021) a non-linear time series data. We describe the data generation equation and abnormal behavior injection in Appendix A.2.1
Real-world Datasets: SWaT (Maturr & Tippenhauer 2016) is a dataset collected from a testbed that simulates real-world water treatment plant. 'The dataset consists of both normal operations and attack scenarios within the water treatment process. MSDS (Multi-Source Distributed System) (Nedelkoski al., 2020) is developed on OpenStack testbed. Instances of fault injections are identified as anomalies.
Table shows the statistics of datasets for training and evaluation. The number in the parenthe- sis indicates the dimensions of multivariate time series. The training set only has the normal time series. In the test set, for each sequence in the synthetic datasets, we conduct one or multiple exoge- nous interventions randomly to generate the anomalies. The last column shows the average number of variables receiving the exogenous interventions. Note that because Lorenz 96 Marcinkevičs & Vogt (2021) has the most complicated interdependencies between time series, we use more training samples to train AERCA for causal discovery.
Table Statistics of Datasets
Training
Test
Dataset
Stop
Sequences
Avg. (1)
Avg.
Linear (4)
5,000
100
500
3.75
Nonlineur (6)
5,000
100
500
5.25
Lotka- Volterra (40)
40,000
100
2,000
30.75
Lorenz 96 (20)
200,000
100
2,000
13.75
SWa' (51)
49,5001
20
51
13.35
MSDS (IO)
29,268
4,255
21
3.05
Evaluation Metrics. Because root cause identification is achieved based on the learned causal models, we evaluate AERCA in both causal discovery and root cause identification.
Causal Discovery. We adopt the commonly used metrics to evaluate the performance of causal discovery (Moraffah al.) [222] Hasan al. [203] Assaad et al. al [2022] [Sun et al.] Nauta al.] 2019 Pamfil et 2020), including FI, AUC-ROC, AUC-PR, and Hamming distance (HD). In the context of causal discovery, F1, AUC-ROC, and AUC-PR quantify the correctness of edge discovery, while Hamming distance calculates the proportion of disagreeing edges between the learned causal graph and the ground truth causal graph.
Root Cause Identification. Following the existing work (Ikram et al. 2022 Li et al. 2022 Yu al. 2021 Ma et al., 22020), we adopt the "recall at top-k" to evaluate the performance of root cause identification, denoted AC@K. This metric quantifies the probability of identifying the correct root cause in the list of variables with the top-k highest root cause scores. Given a set of time series X, the definition of AC@K is shown below.
(RC) n n[ 1,2,
ACOK
1 A
Σ XEX
min(K, IV (RC)))
where Rx [k] indicates the time series at the k-th rank for the sequence X, and (<(C) indicates a set of root cause variables over the whole time series X. Note that if time series receives multiple ex- ogenous interventions, it only counts as one root cause time series in V (RCC), We further compute the overall performance by computing the average AC@K, denoted Avg@K
x
Σk_AC@s AC@k.
AC@K K quantifies the performance of root cause analysis as long as the approach finds the root cause time series. However, in some cases, time series can receive exogenous interventions for multiple time steps. We also evaluate the effectiveness of approaches for locating the root cause time series specific time steps. The experimental results shown in Appendix A.2.3
The experimental results are reported as the average of five independent runs. Our code is publicly available at http:////tuub.oo//aaaaiao0000//1
Baselines. We choose two sets of baselines to compare the performance of causal discovery and root cause identification.
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Causal Discovery. We compare AERCA for the causal discovery with the following baselines. 1) VAR (Vector AutoRegressive) @Ltteephhl 2005) is a linear model to analyze and predict the temporal interdependencies between multple time series datasets; 2) cMLP (Tank al. 202I in- dicates structured multilayer perceptrons (MLPs) combined with sparsity penalties on the weights for Granger causal discovery; 3) cLSTM (Tank et al. (021) leverages recurrent neural networks (RNNs) for Granger causal discovery; 4) TCDF (Temporal Causal Discovery Framework) (Nauta) ct al. 2019) uses iitention-based convolutional neural networks for causal discovery from time series data; 5) eSRU ecconomy-Statistical Recurrent Units) (Khanna & Tan 2019) leverages spe- cial type of RNN called the statistical recurrent unit (SRU) for inferring the Granger causality; 6) PCMCI (Runge et al.) 2019) combines linear or nonlinear conditional independence tests with causal discovery algorithm to estimate causal networks; 7) PCMCI+ (Rungel (020) extends PCMCI to include discovery of contemporaneous links. 8) GVAR (Marcinkevičs & Vogt (021) is a vector autoregression with generalized coefficient matrices predicted by neural networks; 9) CUTS (Cheng ct al. 2023) is neural Granger causal discovery algorithm building causal adjacency matrix with imputed data under sparse penalty.
Root Cause Identification. We compare AERCA for root cause identification with the following baselines. 1) c-Diagnosis (Shan et al. 2019) assumes that the root cause nodes have significantly changed between the abnormal and normal periods and conducts pair-wise significant tests to locate the root cause; 2) RCD (Root Cause Diseovery) (Ikram et al.] 2022) learns the partial causal graph related the root cause and locate the root cause as the interventional targets; 3) CIRCA (Causal inference-based Root Cause Analysis) (Li et al. (022) builds structural causal graph via domain knowledge and locates the root cause in anomalies as the nodes with significant distribution changes
given its parents. All baselines are implemented by PyRCD (Liu et al. 2023).
5.2 EXPERIMENTAL RESULTS
Table 2: Overall performance (stanttstd.) of causal discovery.
Lineur
Nonlineur
Model
AUC-PR
AUC-ROC
HD
F1
AUC-PR
AUC-ROC
HD
VAR
1.9691
1.90
0.011 go.00a
0:178 10.44
0.529 10.1at
0.676 0.6761a.:00
0.258 1.0
cMLP
0. 0.74620.029
0.586=0.058
0. 0.529 25
20.29
0.419 194
0.327 =.307
0.60 0.609=0.00
0.3-10=0.117
cLTTM
0.681+0.042
0.522
0.7666200077
0.312 0622
0.378 000
0.233..0.00
0.50022..00
0.767=0.00
TCDF
0.94820.000
0.033=0.051
0. 2000
0.033
0.473
0.343==0.072
0. 0.656400077
0.307 20,07
eSRU
0.96420.coo ono
0.958
0.9
0,021 041
0.408 162
0.33220.07
0.61
0.267
PCMCI
0.969 can
0.981 11331_...om
0.025g6.nm 23.
0.607 2: one
0.456_...17
0.742gn 147
0,273 175
PCMC++
I. 1.00040.000 000
1.000 oco
1.000g0.000
0.000 .000
0.505 144
0.410
0.669-+0.84
0,233 100
GVAR
0. 0.86220.co ooe
0.981 200000
0.131 006
0.421
10 one
0.562_...15
0.6883ga.07
0.487 20 1o0
CUTS
0.81010.078
0.792
0.841
0.164 (n.
0.357 10.000
0.249 (o.03a
0.536 Lavos
0.513 10.121
AERCA
1.000_0.om
1. 1.000.0.000 000
1.00000.000
0.000.a 000
0.826 057
0.996_.0001
0.998 1.998.0.00
0.027 20.014
Model
Lotka- Volterra
Lorenz 96
AUC-PR
AUC-ROC
F1
AUC-PR
AUC-ROC
HD
VAR
1. 1.00
1,000
0.101 na2
0.562 10.377
O. 0.761gaauu
0.360
eMLP
0.511
0.005
0.508
0.049 20,00
0.472 oss
0.202=0.027
a.509
0.193
cLTM
0.3
0.052-0.001
0.500=00.00
0.100=0.428
0.158
0.191=0.021
0.572=n.041
0.232
TCDF
0.853
0.740==0.000
2.000
20.002
0.422
0.290=0..oc0
2.646
0.260=0..111
eSRU
0.422+0.089
0.323=2.030
D. 0.634g0.16
0.055
0. 0.19
0.225
0. 0.530=a.com
0.215
PCMCI
0. 0.465g..com ons
0.201_00.0l
0.906ga007
0.109 20 one
0.368
0,227 207 oor
0.540 021
PCMCH+
0. 0.77
10.11
0.851 0.051=20022
0.024 D0G
0.502
0.320
0. 0.709
0.163
GVAR
0. 0.787200013
00.08
0.027 20 ooe
0.56% 330
0.582 ae1
0. 0.776g0.10
0.3422g..18 101
CUTS
0. 0.877a
0.791_...07
D. 0.892 Laves
0.011 101l,0.002
0.341 Lo
0,2066 o.002
0.621 Laxos
0o.02
AERCA
0.857
1.000g0.000
1.0000n..on
0.026 woo
0,800 oo0
0.999 3399....00 001
0.105 one
Performance of Causal Discovery. Table 2 shows the results of AERCA and baselines for causal discovery. For baselines, different approuches can achieve good performance on different datasets. For example, VAR can achieve high F1, AUC-PR, AUC-ROC, and low HD on the Linear dataset, but the performance of VAR on other nonlinear datasets is poor, which is expected as VAR linear model. Some other advanced approaches, such as TCDF, GVAR, and CUTS, can achieve good performance on the Lotka-Volterra dataset. However, none of the baseline approaches can achieve satisfactory performance on both Nonlinear and Loren296 datasets. In contrast, AERCA achieves the perfect performance on the Linear dataset with 1 F1, AUC-PR, AUC-ROC scores, and 0 HD, indicating AERCA can learn the causal graph without any error. For the three more challenging nonlinear datasets, AERCA can achieve high FI, AUC-PR, and AUC-ROC scores, as well as very low HD, showing the capability of AERCA discover nonlinear causal relationships.
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Table 3: Overall performance (td) of root cause analysis.
Dataset
Model
AC I
AC@5
AC TO
Avger Td
Diagnosis
0.900
0.850
1. 1.000=0.000
1. 1.000
0..50+
RCD
0.500
0.817=0.189
1.000
1.000 000
0.907 +0.076
Linear
CIRCA
0.600
0.800
1. 1.000-0.000
1. 1.000:0.m
0.910 106
AFRCA
1.000 000
1.000 000
1. 1.000 1.0000
1 1.000 10.000
1.000 0.000
Diagnosis
0.400
0.667 #0.525
0.880
1.000 000
0.837 18.180
Nonlincar
RCD
490 0.600
0.750 10.344
0.880 10 165
1. 1.000
1o 000
0.878
CIRCA
0. +00
0.717 +0
0. 0.835
1. 1.000
0 0.863+o.
AERCA
1.000 000
1. 1.000
1.000
1.000 000
1.00000000
<- -Diagnosis
0.100 300
0.133
0.13840.149
0.247 181188
0. 0.158
Lotka-
RCD
0.100 10. 300
0.133
01 0.138 10.1100..9
0.247 10. 188
0. 1..
Volterra
CIRCA
0.120 925
0.107 +007700.100
0.120
0.225 230
U. 0.14
AERCA
1.000 o80
1. 1.000 40.000
1. 1.000=0.000 0..000
1 1.000 000
0000 =.00=00000
e- -Diagnosis
0.100 100300
0.200 2.221
1.220
0.450 330
1.314
Lorenz96
RCD
0,200 400
0.333 533
0.400
0.556 537
0. 20.278
CIRCA
0.360000.500 480
0.330+0.244
0.346+0.24p
24.
0.539 263
0. 40.20
AFRCA
0.996 009
0.996 10.00
0.997 10.00e
008 0.996
0.990 10.011
e- DDiagnosis
0.075
0. 2..
0. 0.12570.217
0.375
0. +0.
RCD
0.000
0.000
0.000)
0.300
U. 0.10
SWaT
CIRCA
0.000
0.000
0.000
2.300
0. 100
AERCA
0. 0.22010.11 111
0. 0.290 088
1.333_00.0l 0.048
0.45510 044
0.342 20.022
DiDignosis
0.004
0.266
0.4
1. 1,000 000
0. 0.49200.001
MSDS
RCD
0.412 10
0.573
0.984
001
1.000 Lo 000
0.821 10.012
CIRCA
0.454
0.860
0.9 0.91740.084
1 1.000
0.809+0.035
AERCA
0.381 40.408
0.90840.000
0.974
1.000
0.896
presents the results of AERCA and baseline models for root cause identification across synthetic and real-world datasets. Although baselines show promising results terms of AC@5 and AC@ 10 on both Linear and Nonlinear datasets, it's important to note that these datasets only have few time series (low dimensionality). In contrast, AERCA shows exceptional performance across all datasets, excluding the SWaT dataset, even at the AC@1 metric, indicating its capability accurately identify the time series with the highest root cause score. For the SWaT dataset, we observe that the performance of all methods declines, likely due violations of assumptions such as independence and the presence of complex causal rela- tionships, including hidden confounders and instantaneous effects. However, our AERCA method consistently identifies root cause time series with significantly higher accuracy compared to the baseline methods.
Performance of Root Cause Identification. Table 3
AERCA
AERCA w/o
0.91
0.040
0,807
0.00
0.29
0.7
0.0
LOW
(a) AUC-PR
(b) AUC-ROC
Figure 2: Impact of the independent constraint on exogenous variables (Eq. 7 for causal discovery.
Ablation Study. To properly learn the exogenous variables, is critical to ensure the exogenous variables of different time series are independent of each other. Therefore, we have an independent constraint on exogenous variables defined in Eq. 7 To show the importance of the independent constraint for causal discovery, we conduct the ablation study compare the performance of causal discovery when AERCA trained with and without the independent constraint in the objective function. As shown in Figure 2 on Linear, Nonlinear, and Voltera datasets, without the

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independent constraint exogenous variables, the performance of AERCA for causal discovery is much worse. On Lorenz96, the impact is minor, but without the independent constraint, the performance of AERCA is still lower. The experimental results demonstrate the importance of the independent constraint on exogenous variables for causal discovery.
We show more experimental results in Appendix A.2 including root cause identification specific time steps, sensitivity analysis on the continuous erogenous interventions, and case study.
6 CONCLUSIONS
In this paper, we have developed AERCA for the root cause analysis of anomalies in multivariate time series through Granger causal discovery. AERCA assumes that the anomalies caused by external interventions on exogenous variables. To achieve root cause analysis. AERCA explicitly considers the exogenous variables when simulating the data generation process. After training on the normal time series data, AERCA can learn the causal relationships among time series as well as derive the exogenous variables. During deployment, exogenous variables that deviate from normal values will be assigned high root cause scores. Experimental results on multiple datasets demon- strate that AERCA achieves state-of-the-art performance in both causal discovery and root cause identification.
ACKNOWLEDGGENTT
This work was supported in part by NSF 1910284, 2142725, and 2103829.
REFERENCES
Saima Absar, Yongkai Wu, and Lu Zhang. Neural time-invariant causal discovery from time series data. In 2023 International Joint Conference on Neural Networks (IJCNN) pp. 1 1-8. IEEE, 2023.
Charles K Assaad, Emilie Devijver, and Eric Gaussier. Survey and evaluation of causal discovery methods for time series. Journal of Artificial Intelligence Reseurch, 73:767-819, 2022.
Charles K Assaad, Imad Ez-Zejjari, and Lei Zan. Root cause identification for collective anomalies in time series given an acyclic summary causal graph with loops. In International Conference on Artificial Intelligence and Statistics, pp. 8395-88- 8404. PMLR, 2023a.
Charles K. Assaad, Imad Ez-zejjari, and Lei Zan. Root Cause Identification for Collective Anoma- lies in Time Series given an Acyelic Summary Causal Graph with Loops. In Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (AISTATS). arXiv, 2023b.
Kailash Budhathoki, Lenon Minorics, Patrick Blöbaum, and Dominik Janzing. Causal structure- based root cause analysis of outliers. In International Conference on Machine Learning, pp- 2357-2369, PMLR, 2022.
Yuxiao Cheng, Runzhao Yang, Tingxiong Xiao, Zongren Li, Jinli Suo, Kunlun He, and Qiong- hai Dai. Cuts: Neural causal discovery from irregular time-series data. arXiv preprint arXi022302.07458, 2023.
Rainer Dahlhaus and Michael Eichler. Causality and graphical models in time series analysis. Oxford Statistical Science Series, pp. 115-137, 2003.
Clive WJ Granger. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, pp. 424 438, 1969.
Uzma Hasan, Emam Hossain, and Md Osman Gani. A survey on causal discovery methods for iid and time series data. arXiv preprint arXiv:2303. /5027, 2023.
Azam Ikram, Sarthak Chakraborty, Subrata Mitra, Shiv Saini, Saurabh Bagchi, and Murat Kocaoglu. Root cause analysis of failures in microservices through causal discovery. Advances in Neural Information Processing Systems, 35:31158-31177 2022.
10

---

Published as a conference paper at ICLR 2025
Hiranya Jayathilaka, Chandra Krintz, and Rich Wolski. Performance monitoring and root cause analysis for cloud-hosted web applications. In Proceedings of the 26th International Conference World Wide Web, pp. 469-478. 2017.
Vimalkumar Jeyakumar, Omid Madani, Ali Parandeh, Ashutosh Kulshreshtha, Weifei Zeng, and Navindra Yadav. Explainit!-a declarative root-cause analysis engine for time series data. In Proceedings of the 2019 International Conference on Management of Data, pp. 333-348, 2019.
Saurabh Khanna and Vincent YF Tan. Economy statistical recurrent units for inferring nonlinear granger causality. arXiv preprint 1rXiv:1911.09873 2019.
Myunghwan Kim, Roshan Sumbaly, and Sam Shah. Root cause detection in a service-orientee architecture. ACM SIGMETRICS Performance Evaluation Review, 41(1)933-104 2013.
Mingjie Li, Zeyan Li, Kanglin Yin, Xiaohui Nie, Wenchi Zhang, Kaixin Sui, and Dan Pei. Causal inference-based root cause analysis for online service systems with intervention recognition. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 3230-3240 2022.
Chenghao Liu, Wenzhuo Yang, Himanshu Mittal, Manpreet Singh, Doyen Sahoo, and Steven CH Hoi. Pyrca: A library for metric-based root cause analysis. arXiv preprint arXiv:2306. 2023.
Helmut Lütkepohl. New introduction multiple time series analysis. Springer Science & Business Media, 2005.
Meng Ma, Jingmin Xu, Yuan Wang, Pengfei Chen, Zonghua Zhang, and Ping Wang. Automap: Diagnose your microservice-based web applications automatically, In Proceedings of The Web Conference 2020, pp. 246-258, 2020.
Ričards Marcinkevičs and Julia E Vogt. Interpretable models for granger causality using self- explaining neural networks. arXiv preprint arXiv:2101.07600, 2021.
Aditya Mathur and Nils Ole Tippenhauer. Swat: A water treatment testbed for research and training ics security. In 2016 international workshop on cyber-physical systems for smart water networks (CySWater), pp. 31-36. IEEE, 2016.
Yuan Meng, Shenglin Zhang, Yongqian Sun, Ruru Zhang, Zhilong Hu, Yiyin Zhang, Chenyang Jia, Zhaogang Wang, and Dan Pei. Localizing Failure Root Causes in Microservice through Causality Inference. In 2020 IEEE/ACM 28th International Symposium on Quality of Service (IWQoS), pp. 1-10, 2020.
Raha Moraffah, Paras Sheth, Mansooreh Karami, Anchit Bhattacharya, Qianuu Wang, Anique Tahir, Adrienne Raglin, and Huan Liu. Causal inference for time series analysis: Problems, methods and evaluation. Knowledge and Information Systems, 63:3041-3085, 2021.
Meike Nauta, Doina Bucur, and Christin Seifert. Causal discovery with attention-based convolu- tional neural networks. Machine Learning and Knowledge Extraction, 1(1):19, 2019.
Sasho Nedelkoski, Jasmin Bogatinovski, Ajay Kumar Mandapati, Soeren Becker, Jorge Cardoso, and Odej Kao. Multi-source distributed system data for ai ai-powered analytics. In Orivice-Oriented and Cloud Computing: 8th IFIP WG 2. 14 European Conference, ESOCC 2020, Heraklion, Crete, Greece, September 28-30, 2020, Proceedings pp. 161-176. Springer, 2020.
Mark EJ Newman. Estimating network structure from unreliable measurements. Physical Review E, 98(6):06331,, 2018.
Nastaran Okati, Sergio Hernan Garrido Mejia, William Roy Orchard, Patrick Blöbaum, and Do- minik Janzing. Root cause analysis of outliers with missing structural knowledge. arXiv preprint 0451:24406.00114.. 2024.
Roxana Pamfil, Nisara Sriwattanaworachai, Shaan Desai, Philip Pilgerstorfer, Konstantinos Geor- gatzis, Paul Beaumont, and Bryon Aragam. Dynotears: Structure learning from time-series data. In International Conference on Artificial Intelligence and Statistics, pp. 1595-1605. Pmlr, 2020.

---

Published as a conference paper at ICLR 2025
Jakob Runge. Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets. In Conference on Uncertainty in Artificial Intelligence, pp. 1388-1397. Pmlr, 2020.
Jakob Runge, Peer Nowack, Marlene Kretschmer, Seth Flaxman, and Dino Sejdinovic. Detecting and quantifyyng causal associations in large nonlinear time series datasets. Science advances, (11) :eeaau49996 2019.
Huasong Shan, Yuan Chen, Haifeng Liu, Yunpeng Zhang, Xiao Xiao, Xiaofeng He, Min Li, and Wei Ding. 2-diagnosis: Unsupervised and real-time diagnosis of small-window long-tail latency in large-scale microservice platforms. In The World Wide Web Conference, pp. 3215-3222, 2019.
Ali Shojaie and Emily B Fox. Granger causality: A review and recent advances. Annual Review of Statistics and Its Application, 9(1):289-319) 2022.
Alban Siffer, Pierre-Alain Fouque, Alexandre Termier, and Christine Largouet. Anomaly detection in streams with extreme value theory. In Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1067-1075, 2017.
Jacopo Soldani and Antonio Brogi. Anomaly detection and failure root cause analysis in (micro) service-based cloud applications: A survey. ACM Computing Surveys (CSUR), 55(3):1-39, 2022.
Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, prediction, and search. MIT press, 2001.
Xiangyu Sun, Oliver Schulte, Guiliang Liu, and Pascal Poupart. Nts-notears: Learning nonparamet- ric dbns with prior knowledge. arXiv preprint arXiv:2109.04288 2021.
Alex Tank, Ian Covert, Nicholas Foti, Ali Shojaie, and Emily B Fox. Neural granger causality. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(8):4267-4279, 2021.
S Tuli, Casale, and NR Jennings. Tranad: Deep transformer networks for anomaly detection in multivariate time series data. arxiv 2022. arXiv preprint arXiv:2201.07284, 2022.
Dongjie Wang. Zhengzhang Chen, Yanjie Fu, Yanchi Liu, and Haifeng Chen. Incremental causal graph learning for online root cause analysis. In Proceedings of the 29th ACM SIGKDD Confer- ence on Knowledge Discovery and Data Mining, pp. 2269-2226222 2278, 2023a.
Dongjie Wang, Zhengzhang Chen, Jingehao Ni, Liang Tong, Zheng Wang, Yanjie Fu, and Haifeng Chen. Interdependent causal networks for root cause localization. Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 5051-5060, 2023b.
Ping Wang, Jingmin Xu, Meng Ma, Weilan Lin, Disheng Pan, Yuan Wang, and Pengfei Chen. Cloudranger: Root cause identification for cloud native systems. In 2018 18th IEEE/ACM In- ternational Symposium on Cluster, Cloud and Grid Computing (CCGRID), pp. 492-502. IEEE, 2018a.
Ping Wang, Jingmin Xu, Meng Ma. Weilan Lin, Disheng Pan, Yuan Wang, and Pengfei Chen. CloudRanger: Root Cause Identification for Cloud Native Systems. In 2018 18th IEED/ACM In- lernational Symposium on Cluster, Cloud and Grid Computing (CCGRID), pp. 492-502, 2018b.
Jianping Weng, Jessie Hui Wang, Jiahai Yang, and Yang Yang. Root cause analysis of anomalies of multitier services public clouds. IEEE/ACM Transactions on Networking, 2018.
Wenzhuo Yang, Kun Zhang, and Steven C. H. Hoi. A Causal Approach to Detecting Multivariate Time-series Anomalies and Root Causes, 2022.
Guangba Yu, Pengfei Chen, Hongyang Chen, Zijie Guan, Zicheng Huang, Linxiao Jing, Tianjun Weng, Xinmeng Sun, and Xiaoyun Li. Microrank: End-to-end latency issue localization with extended spectrum analysis in microservice environments. In Proceedings of the Web Conference 202/, pp. 3087-3098, 2021.
12

---

Published as a conference paper at ICLR 2025
Chuxu Zhang, Dongjin Song, Yuncong Chen, Xinyang Feng, Cristian Lumezanu, Wei Cheng, Jingchao Ni, Bo Zong, Haifeng Chen, and Nitesh Chawla. A deep neural network for un- supervised anomaly detection and diagnosis in multivariate time series data. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 1409-1416, 2019.
Yingying Zhang, Zhengxiong Guan, Huajie Qian, Leili Xu, Hengbo Liu, Qingsong Wen, Liang Sun, Junwei Jiang, Lunting Fan, and Min Ke. Cloudrca: A root cause analysis framework for cloud computing platforms. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 4373-332,, 2021.
Hang Zhao, Yujing Wang, Juanyong Duan, Congrui Huang, Defu Cao, Yunhai Tong, Bixiong Xu, Jing Bai, Jie Tong, and Qi Zhang. Multivariate time-series anomaly detection via graph attention network. In 2020 IEEE international conference on data mining (ICDM), pp. 841-850. IEEE, 2020.
A APPENDIX
A.1 APPROACH
Proposition 2. Consider basic autoregressive model where wi (Xt-k) as framework for analyzing Granger causality. The value at the current time step x, can be derived by the exogenous variables from previous window [Me-1:... Mt-K] and the observed time series from the immediate previous window [x-K-1- Xt-2K] with the following equation:
K-1
X1 OK- UK FOKX
Σ aK+l-m
Σ
W
(11)
m=
kmm
where wk indicates the parameter of Grunger causality, and On Σi= an 1 1sn S K, is recursive equation with 0ug =1.
Proof. According to the definition of Granger causality, we have
WiXii Mr-
(12)
Then, recursively replace variables in the previous window with their autoregressive equations. By substituting X with Σx=1 Wk + Pc-1, have
K
= 1IX
+
WKXi
2=2
Wk W + MC-1) + ( Σ wix + 14)
wi (
By further substituting X we have
K Σ
K
X, = =<[w + w2) (wiXr-3+ +
wh - Mo-2) + wi (2(22x--3 +
Σ W + pe-11
2=2
K
+ (Wixx-3 Σ
WkXXe + 14)
k=4
By defining an En Wn S n K, ag = 1. the above expression can be rewritten as
Xx =02 (IXX +
WhX: 2- 2 ar w2X1-3 + Σ
Wh 1-k + Pt-
K
+ au W3X4-3 +
Σ
WkXi-k + 14)
km4
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By following this pattern, for any < K, we have that
x, On-1 WiXxin
Σ
+ me-+t) + on (00-2(w220 (2xx-n +
Σ
- + Mt-n+2
k=2
k=3
K
+ +... + ao WnXien +
Σ
w + 11)
km=+1
By rearranging the above expression, we have that
n+1
K
Σ m=2
an-mi (n-mmm) + Xc-nn
an+1
WkX,, k- 6-(x+-mm
m=1
kmm
Finally, as we focus on representing X, by the exogenous variables from previous window and the observed time series from the immediate previous window, we assign K. Then, we have that
K+1
X Σ
ak-
+ FOxX-KK K +
Σ
ak+1-m
Σ
W k- (K+ 1-
A.2 EXPERIMENTS
A.2.1 SYNTHETIC DATASETS
Linear Mataase/Mareinkeviie & Vogt (2221 is synthetic time series dataset with linear interaction dynamics. The structural equations are defined as:
(1)
(1)
(1) (1)
St a1 + u4 + 6
2 (2) 0
+ 03
(1)
u +
(2) (2)
4 a4d
+ 05
(2)
u
(3)
+
(1) (1)
(3) (4) (1) + u
= ag
+ 0:
agæ
+ +
where coefficients a; ~ (1[1 0.2] U |0.2, 0.8)). additive innovation terms uf (1) (00,0.1)) and anomaly term e
Nonlinear Dataset Absar et al. (2023) is synthetic time series dataset with non-linear interaction dynamics, of which the structural equation is defined as:
X, AT
Bm cos X + +1) +
where 8 is the regression coefficient, and e represents standard Gaussian noise. The noise scale is kept below and proportional the value of d. The non-linear relationship between time series is introduced through the cosine function. The adjacency matrix A of the underlying causal graph is generated using the Erdös Erdös-Rényi model Newman 2018).
Lotka- Volterra Marcinkeviës & Vogt (2021) is synthetic time series model that simulates prairie ecosystem with multiple species. The structural equations are defined as:
dx(()
di (1) B
Σ
ylm) ())x for 1 Sijsp
)EPa(x(0)
8y(i) Σ
x(1 (4)
()) for 1 5 S
di
kEPa(y())
(6)
+ (1) for 1 sisp. yi
It
Et
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where x(9 and yL0)) denote the population sizes of prey and predator, respectively, a, a,3, n. 6, parameters that decide the strengths of interactions, (a(x(1) and Pa(y()) correspond the Granger Causality between prey and predators for x(6) and y(i) respectively, and cl the abnormal term. We simulate 20 prey species and 20 predator species.
Lorenz 96 Mareinkeviis & Vogt (2021) a synthetic time series data, where the i-th variable is defined by the following nonlinear differential equations:
(x(()) (x.(+ x (i) x0() FF, for Sisd.
dt
where (0) (d),, x(-1) = z(d-1), and x (d++l) = x²: and is a constant controlling the nonlinearity of the data.
Abnormal behavior injection the synthetic datasets. For point anomalies, the anomaly term is single or multiple extreme values for randomly selected time series variables at specific time step /. For example, point anomaly at time step can be generated with an abnormal term 6, = [0, 2, 4, 0). which means the second and third time series have extreme values.
For sequential anomalies, the anomaly terms are cunction-generaln values in a given time range. (1) = 0.1 x i, for 0 S S will cause a trend anomaly for time series For instance, setting Cz+
variable (²( setting 'tti ~ N (V,00.16)) for 0 < i < n, will cause shapelet anomaly; and setting
(1)
(1) = (a1 + (442)) + (a13 + (1) 4 (a) (a1 + u4 2 for 0 S i S n, will C+t
(1)
cause scasonal anomaly.
A.2.2 IMPLEMENTATION DETAILS
We implement distinct neural network configurations tailored to the complexity of the dataset at hand. Specifically, for synthetic datasets, we employ two-layer feedforward neural network archi- tecture with a hidden dimension of 50, whereas for real-world datasets, the architecture is expanded to eight layers, each boasting a hidden dimension of 1000. Preprocessing of data is standardized across datasets using MinMax scaler, with further efficiency measures including downsampling of the SWaT dataset at intervals of every 10 seconds and the MSDS dataset every 5 steps. The training framework is anchored by learning rate of 1 10- with the Adam optimizer facilitating parame- ter optimization. Hyperparameters 3, X, and 7 are initially set to 1, ensuring a balanced approach to regularization and loss function adjustment. The maximum training epochs porating an early stopping criterion that halts training if no improvement in loss is observed for 20 consecutive epochs. All experiments were conducted on an Ubuntu 20.0 20.04 server equipped with an AMD Ryzen 3960X 24-Core processor at 3,&GHz, dual GeForce RTX 3090 GPUs, and 128 GB of RAM. The implementation uses Python 3.9.7 and PyTorch 1.11.0.
5000, incor-
A.2.3 PERFORMANCE OF ROOT CAUSE ANALYSIS AT SPECIFIC TIME STEPS
To quantify the effectiveness of approaches for locating the root cause time series at specific time steps, we further develop the metric "recall at top-k over all time steps" below.
U V2l (RC) n R [&] = 2, KH
AC
X
min (K, ULET V(((C1))
XEX
where V2(C)) indicates the of root cause time series the t-th time step; R [k] indicates the time series at the k-th rank over all time steps. Similarly, we also compute the overall performance by computing the average AC*@ @K, denoted Aug @K 1/4 _x_ @C
For baselines that cannot identify the root cause at specific time steps, we consider the root cause predicted by the baselines indicating the abnormal time series the last time step in sliding win- dow.
As shown in Table for the more challenging metric AC @K, AERCA achieve high AC @ scores on most datasets except the SWaT dataset, meaning that AERCA can successfully detect the root cause at specific time steps. Furthermore, AERCA achieves near-perfect AC* @ 10 scores
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most datasets. Considering that there are thousands even tens of thousands of candidates (T dd) on each dataset when trying to highlight the root cause at specific time steps, the performance of AERCA is promising.
Table Overall performance (stan=-std.) of root cause analysis at specific time steps.
Dataset
Model
AC @
AC*
@
AC @ 100
AC* 550
Avg' @ @500
--Dianosis
0.000+0,000
0.000=0.co0
0.000 000
0.250+0.316
0.086++0.125
RCD
0.000
0.000=0.000
0.000 000
0. 0.150 .00
0.0644+0.16
Linear
CIRCA
0.000
0.000
0.025 075
0.233 .22
0,088
AERCA
O. o..
0.990+a.018
1.000 000
1.000 0.000
0.998
Diagnosis
4..000
0.000
0.000 00.000
0.092
0.056
RCD
+0.000
00.0000.000
0.080
+0,0040.240 240
0.2633
0.11640.215
Nonlinear
CIRCA
+0.000
0.000 00000
0.01 7.01740.00 +0.050
0.160
0.0661+0.075
AERCA
O. 0.3::.1122
0.830+0.00 094
0.9944:.00
0.995
0.987 =0.094 094
D-iagnosis
00.00000000
0.000=0.000
0.000
0. 0,0000+0.00 .000
0.000
Lotka-
RCD
0.000
0.000
0,000
0,000 000
0,000
Volterra
CIRCA
0.000
0.000
0.000
0.000 0000
0.000
AERCA
00.00
0. +0.044
1.000
1.000=0.000
1.000000.000
Diagnosis
0.000
0.000
0.000
0.000
0.000
RCD
00.00000000
0.000=0.000
0.000 000000
0.000
0.000
Lorenz96
CIRCA
0.000
0.000 0.000
0.000 000
0.0001 0.000
0.000+0.000
AERCA
0.84220.01e
0.970+0.0m
0 009
0.996=2.000
0.987 1987=z =0.000
Diagmosis
0.075+0.179
0.125
O. 0.125+0.211
1. 1.000
0.633
RCD
0.000 +0.000
0.025
0.21 4.4......60
0.799
0.416+0.02x
SWaT
CIRCA
=000000.000
0.025
0.214 0.0.0
0.799
0.416-10.028
AERCA
40.006
0.32010.026
1.000 0.000
1.000=0.000
00.02
D-Daagnosis
0.000
0.389=o.110
0.706 310
1.000=0.000
0.880
RCD
0.025
0.216
0.806 205
1.000
0.908
MSDS
CIRCA
0.000
0.102=0.108
0.741 40.11
1.000=0.0on
0,884+0.083
AERCA
0.230 00.04
1. 1.000
1.000
000
1.000=0..00
0.997
10.000
1.00
1.00
0.90
0.90
0.80
Avg@11
0.80
Avg W5500
0.70
0.70
Linear
0.60
Linear
0.60
Nonlinear
0.50
Nonlinear
VolterVoltera
Lotka-Volterra
0.50
Lorenz96
Lorenz96
0.30
1
2
6
8
9
5
6
7
8
Continue Time Steps
Continue Time Steps
(a) Avg 10
(b) Avg" @500
Figure 3: Performance of root cause identification with various numbers of continuous exogenous interventions.
A.2.4 SENSITIVITY ANALYSIS
We evaluate the performance of AERCA on root cause identification when the anomalies are caused by continuous exogenous interventions. We tune the number of time steps having the exogenous interventions and check the performance change. Recall that to make the task more challenging, at each time step, the exogenous intervention is conducted on different time series. Figure 3 shows the evaluation results. We can observe that in terms of Avg @ 10, the performance of AERCA remains stable on both Linear and Nonlinear datasets and slightly decreases on the Lotka- Volterra dataset when increasing the time steps receiving the exogenous interventions. It shows that AERCA can identify the root cause correctly with continuous interventions on different time series. Meanwhile, in terms of Avg* @ 500, AERCA still achieve reasonable performance Linear, Nonlinear,
16

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Published as a conference paper at ICLR 2025
and Lorenz96 datasets, indicating that in most cases, AERCA can identify the time series receiving exogenous interventions at specific time steps with high root cause scores. The main reason that the performance of AERCA on Lotka- Voltera significantly decreases is because Lotka-Volterra Volterra has more variables, which makes the candidates of root cause significantly larger, especially for computing the metric AC* @ K. In summary, AERCA achieve promising performance root cause identification for continuous interventions when the number of dimensions in multivariate time series is moderate.
A.2.5 CASE STUDY
Figure4ssows short snippet of multivariate time series the Nonlinear dataset, where conduct four exogenous interventions on four different time series at different time steps. We highlight the predicted root cause with the top highest root can notice that AERCA correctly detects the root cause time series when exogenous interventions are conducted at specific time steps. Meanwhile, shown the bottom of each time series, the distribution of root cause score (z-score) matches the exogenous variables, especially when the time series receives the exogenous interventions.
scores (AC*@5) @5) via purple bars. We
Sround Tethh
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Figure 4: Visualization of multivariate time series, exogenous variables, and predicted root cause scores on the Nonlinear dataset (6 dimensions) with the ground truth and predicted root cause.
17


Reviews:
[{"review_id": "HGtKoG9GvD", "reviewer": "Reviewer_pxU9", "summary": "The paper proposes the AERCA model, an encoder-decoder framework integrating Granger causal discovery with root cause analysis to address anomaly detection in multivariate time series. AERCA identifies causal relationships by defining anomalies as interventions on exogenous variables, capturing deviations that point to root causes. By explicitly modeling the distributions of exogenous variables under normal conditions, AERCA can differentiate abnormal time series data when anomalies arise. The approach achieves promising results on synthetic and real-world datasets, demonstrating its efficacy in both causal discovery and root cause identification.", "questions": "- Why was Granger causality chosen over more robust causal methods like Structural Causal Models (SCMs) or Dynamic Causal Models (DCMs), given its limitations in capturing true causation?\n- How does the model handle scenarios where exogenous variables are interdependent, as this independence assumption may not hold in real-world systems?\n- Could the authors provide insights into the model’s sensitivity to key parameter settings, particularly the thresholds for anomaly and root cause detection?\n- Given the complexity of the encoder-decoder structure, how do the authors intend to make the model’s reasoning more interpretable for practitioners?\n- How does the model address the potential influence of latent confounders, given that Granger causality does not inherently account for unobserved variables?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The integration of Granger causal discovery with root cause analysis via an encoder-decoder structure is a novel approach. This fusion provides a comprehensive method to identify both causal links and root causes of anomalies.\n- Modeling exogenous variables under normal conditions adds robustness to anomaly detection, enabling the model to distinguish between normal variations and true anomalies.\n- Experiments conducted across multiple datasets, including synthetic and real-world time series, demonstrate AERCA’s high accuracy and adaptability, especially in distinguishing nonlinear causal relationships.\n- The model is versatile, as it handles both point and sequence anomalies, making it applicable across various domains requiring real-time anomaly detection.", "weaknesses": "- The paper relies on Granger causality to identify causal relationships, which only capture temporal predictability rather than true causation. This can result in spurious relationships, especially in systems with latent confounders or feedback loops, limiting the robustness of causal inference.\n- AERCA assumes exogenous variables are mutually independent, which might not hold in real-world scenarios where dependencies can exist between exogenous factors. This assumption could reduce model robustness when applied to systems with interdependencies among exogenous factors.\n- While the model structure is powerful, its complexity may hinder interpretability for users, especially when deployed in high-stakes industries like finance or healthcare, where transparent causal explanations are crucial.\n- The paper does not extensively discuss the sensitivity of the model to hyperparameters, especially the thresholds for identifying root causes. Further analysis of parameter robustness would help ensure consistent performance across datasets.\n- Granger causality is observational and cannot account for unobserved confounders, leading to potential biases in causal inference. Without mechanisms to address these confounders, causal relationships may be inaccurately represented\n- One significant limitation of this paper is the omission of several proven and state-of-the-art causal discovery algorithms in its comparison. For example, while PCMCI was used as a baseline, the more efficient PCMCI+ variant (http://auai.org/uai2020/proceedings/579_main_paper.pdf) , which enhances computational performance and accuracy for high-dimensional time series, was not included. Additionally, more recent methods like Kernel-based Causal Discovery (KCD) (https://ieeexplore.ieee.org/iel7/9873850/9873995/09874142.pdf) offer advanced capabilities in handling non-linear dependencies and would have provided a stronger benchmark for evaluation. Conversely, some included approaches, such as Vector Autoregression (VAR) and Temporal Causal Discovery Framework (TCDF), are relatively outdated and come with inherent limitations, particularly in capturing complex non-linear causal relationships. Including these more robust and recent algorithms would enhance the comparative rigor and provide a clearer understanding of the proposed approach’s strengths and weaknesses relative to modern causal discovery methods.", "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": "VPBd5uWL1F", "reviewer": "Reviewer_ofk8", "summary": "To identify root causes of anomalies in time series data, this paper models anomalies as interventions on exogenous variables. It then proposes a methodology (AERCA) to detect anomalies by explicitly modeling the mutually independent exogenous variables in the underlying autoregressive model, using encoders and decoders. A root cause is then identified by performing statistical tests on each of the associated exogenous variables. Experiments are performed for causal discovery and root cause identification on various synthetic datasets. Root cause identification is also assessed on two real datasets.", "questions": "The importance of mutually independent exogenous variables is demonstrated in the ablation study, such that the method performs significantly worse if the independence constraint is not invoked. Have the authors considered datasets where the exogenous variables are inherently correlated (due to latent confounders)? It would be interesting to see what would happen for root cause identification in this case, since this is a realistic scenario, which would likely be significantly more challenging. It would also be interesting to discuss model misspecification in this case (imposing independence constraint when it is not true).\n\nThe method seems to perform poorly on the SWaT dataset. Limited analysis/discussion is offered. Why do the authors think that this is the case? Is it due to higher numbers of root causes, latent confounders, and more complicated causal relationships? \n\nSimilarly, is the ground truth causal graph known for the real datasets? It seems likely that the causal graph is not being learnt by AERCA given the performance on root cause identification, which suggests that the exogenous variables are not precisely modeled. \n\nWhat is the importance of assuming stationarity of the underlying process (L111)?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem is well motivated, since root cause identifiaction is prevalent, and existing methods are impractical.\n\nThe given definition for root cause, as well as the proposed method are intuitive.\n\nA broad range of system dynamics (ex. linear, nonlinear, lotka-volterra) are considered for empirical evaluation on synthetic datasets.\n\nWriting and overall presentation are quite clear.", "weaknesses": "I don't think that this should impact the methodology or results, but the provided definition of Granger causality seems more like actual causality, since the structural equation model is assumed to match the true data-generating process (L112). Although I checked the two references provided, I think that Granger causality is typically understood to be about aiding predictive power given an imperfect model, rather than the true model. Indeed, the causal graph (which is recovered in experiments) is typically defined from the true SEM rather than Granger causality. \n\nThe method assumes that the exogenous variables are mutually independent, and this implicitly assumes causal sufficiency (no latent confounders). The importance of this assumption is demonstrated in the ablation experiment. I think that this assumption should be emphasized, since it does not seem like the proposed method is equipped to handle latent confounders.\n\nIt is not obvious that the method performs significantly better than alternative methods for root cause identification on real data. On the two offered datasets, performance was poor across all methods for the SwAT dataset, and besides epsilon-diagnosis, results are comprable across methods on the MSDS dataset. This may suggest that there are latent confounders, or other complications in the real datasets, which prevent AERCA from properly modeling the exogenous variables.\n\nDiscussion and analysis is limited in the experiments section.", "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": "TyNgMyIMaY", "reviewer": "Reviewer_QZnY", "summary": "This paper introduces AERCA, an autoencoder-based model for root cause analysis in anomaly detection that integrates Granger causal discovery and root cause analysis using an encoder for abductive reasoning to identify exogenous variables and a decoder for deductive reasoning to reconstruct observations. This structure enables AERCA to learn causal relationships and detect anomalies by identifying deviations in exogenous variables. \nThis approach is evaluated empirically on both synthetic and real-world multivariate time series data. Synthetic datasets include Lotka-Volterra and Lorenz 96, while real-world data comes from SWaT and MSDS. \nAERCA outperforms baselines in identifying causal structures and root causes, achieving high F1, AUC-PR, and AUC-ROC scores across four synthetic and two real-world datasets, with strong accuracy in both synthetic and real-world data.", "questions": "Can you describe what is the strategy that will need to be followed by practitioners interested in applying your framework in real world applications?\n\nWhat assumptions and conditions will an interventional problem need to fulfill for your approach to appropriately model the problem?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper has several strengths. It combines Granger causality with anomaly detection, making it suitable for complex systems with multiple interacting variables. This approach identifies not only the affected time series but also the specific time steps, offering precise localization for root cause analysis. The evaluations consider various datasets and baseline models over a range of performance metrics including F1 score and AUC-PR.", "weaknesses": "The most noticeable weakness of this approach, in addition to the obvious decreased interpretability due to the reliance on the encoder-decoder autoregressive architecture, is that the model relies strongly on the correct estimation of exogenous variables, which may vary widely in real-world applications.", "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": "MZ1Jo7YvoU", "reviewer": "Reviewer_J7CE", "summary": "This paper proposed a method to identify the exogenous variable in multiple time series, such that if the deviation from normal state of this exogenous variable is detected, the cause relation is established. The paper built a Encode-Decode pipeline to implement their design, in which the root cause is defined from Granger causality.", "questions": "At line 120-121, the definition of Granger causalitysed in the literature is too sensitive. Could the authors adopt the standard definition, and make the cause effect more statistically significant but not so sensitive?\n\nIn Proposition 1, the form of the model is not aggreable with ARMA since in the latter case the width of windows, i.e., the order of model, are determined by some criterions such as AIC. But in current work, the authors didn't give any explaination. Could the authors give some ideas or explanations?\n\nCould the authors describe the statistical property of the exogenrous variable such as distribution, the independence, and the power (variance)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Root cause analysis is one of the most challenging problem in multivariate time series analysis. This paper proposed an algorithm to identify the significant deviation of exogenous variable from its normal state, from which one can identify the root cause among multiple time series. Clearly, this is a very interesting approach.", "weaknesses": "Due to the assumption that the series are stationary, and the model is based on splitting window with fixed length $K$, thus we may regard the model from neural network as optimization processing in building ARMA model. However, if we regard it as ARMA, the content in (8) of Proposition 1 must be re-checked, since it lacks any criterion of choosing a reasonable window width $K$ (the order of model), and further more, we can not measure the effect of required parameters in the model. Actually, the coefficients of the model in this paper are trained by neural networks, which might be a nonlinear function, thus it might involve serious sensitivity effect.\n\nSecondly, the presentation of this paper is partially confused with standard ARMA model.  I hope the authors could rigrously distinct it with standard ARMA model, and indicates the difference. There actually exist many practical algorithms in time series theory to build an ARMA model as well as detailed performance analysis, which might help to consider the current formulation in more reasonable manner. \n\nIn equations (1)(2), the authors lead a description on the potential exogenous intervention on a specific time series at a time step. Then the problem is to detect the abnormal exogenous variable ASAP (Def 1). To my understanding, this exogenous variable is essentially the noise in ARMA model, so the philosophy of this paper is not attractive. That is, I do not agree that the author offers a method to search the root cause which is actually as the change point that affect the trend of time series.", "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": "JHC8jdmCrk", "reviewer": "Reviewer_sPkL", "summary": "This paper proposes AERCA, a novel method for root cause analysis (RCA) based on Granger causality. \nAERCA uses an encoder-decoder scheme to iteratively approximate the time series with an approximation of the exogenous variables (noises). \nThe root causes are modeled as variables that are added to the noises but have significantly higher variance.  \nBy training on normal data (no root causes) AERCA is able to detect the root causes on the test-data.\nThis was demonstrated empirically on known synthetic (or simulated) and real-world datasets both", "questions": "I would appreciate if you could provide some feedback on the comments on the weaknesses section. \n\nAlso, I believe it would be helpful to provide your code as supplementary material for reproduction.\n\nI noted two mistakes/typos:\n1. In Eq. (4) you should use $g$ instead of $f$ as this should be a different function depending only on $\\mathbf{u}_{\\leq t- 1}$\n2. In Proposition 1. I would not characterize the window $\\mathbf{W}_{t-K}$ as *immediately* previous, rather just previous. Because, before you had sliding windows, each moving by one time-step.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper is very well written and the contribution is clear and novel. \nThe proposed method is simple and efficient.\nIn particular, I would like to highlight the ability to handle multiple root causes at once and at multiple time steps, where typically existing work in RCA detects a single root cause. \nMoreover, it analytically computes the estimated root causes, rather than backtracking on the computed causal graph (as in existing work in RCA).\nI highly appreciate the computational analysis and the result of Proposition 1.\nThe empirical evaluation is extensive and shows the method's efficiency over prior work in synthetic and real scenarios.", "weaknesses": "It is hard for me to point out obvious weakness points. I will however suggest some points for potential improvement, but these do not affect my opinion negatively on the paper.\n\n1. I think it is quite restrictive to assume that normalize data always exist or that we can distinguish them from the faulty ones. How are you sure that the data that you are training on do not contain any root cause (this could affect your training). I am referring to a real scenario where the ground truth is unknown. \n\n2. I would also suggest trying out a bit more intuitive real world experiment where you can show with an example the discovered root cause and its meaning (what was the fault at that node and what did it cause).\n\n3. For the proposed setting of Eq. (5), together with the anomaly term from Eq. (2) are there any theoretical results regarding identifiability? This is well known for some cases ([1] equal variance, [2] imposing sparsity in root causes for static data). Would be interesting to see such a result and it would enhance your contribution. \n\n4. How big time-series can your method handle (how many variables)?\n\nYou can discuss the above by adding a limitation section.\n\n[1] Peters, Jonas, and Peter Bühlmann. \"Identifiability of Gaussian structural equation models with equal error variances.\" Biometrika 101.1 (2014): 219-228.\n[2] Misiakos, Panagiotis, Chris Wendler, and Markus Püschel. \"Learning DAGs from data with few root causes.\" Advances in Neural Information Processing Systems 36 (2023).", "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 (Oral)