Algorithmic Recourse in Abnormal Multivariate Time Series
Authors: Xiao Han
Lu Zhang
Yongkai Wu
Shuhan Yuan
Venue: Accepted by TMLR ()

Abstract: Algorithmic recourse provides actionable recommendations to alter unfavorable predictions of machine learning models, enhancing transparency through counterfactual explanations. While significant progress has been made in algorithmic recourse for static data, such as tabular and image data, limited research explores recourse for multivariate time series, particularly for reversing abnormal time series. This paper introduces Recourse in time series Anomaly Detection (RecAD), a framework for addressing anomalies in multivariate time series using backtracking counterfactual reasoning. By modeling the causes of anomalies as external interventions on exogenous variables, RecAD predicts recourse actions to restore normal status as counterfactual explanations, where the recourse function, responsible for generating actions based on observed data, is trained using an end-to-end approach. Experiments on synthetic and real-world datasets demonstrate its effectiveness.

Published in Transactions on Machine Learning Research (01/2026)
Algorithmic Recourse in Abnormal Multivariate Time Series
Xiao Han Utah State University
aiao.han&@usuu edu
Lu Zhang University of Arkansas
1z006@wark edu
Yongkai Wu Clemson University
yongkaws@cdemsonn.cde eduu
Shuhan Yuan Utah State University
shahhn,, yuon Wusuuuuuuu...mm edu
Reviewed on OpenReview: https: openreview. net/ forum? id= k
Abstract
Algorithmic recourse provides actionable recommendations to alter unfavorable predictions of machine learning models, enhancing transparency through counterfactual explanations. While significant progress has been made in algorithmic recourse for static data, such tabular and image data, limited research explores recourse for multivariate time series, particularly for reversing abnormal time series. This paper introduces Recourse in time series Anomaly Detection (RecAD), framework for addressing anomalies in multivariate time series using backtracking counterfactur reasoning. By modeling the causes of anomalies as external interventions on exogenous variables, RecAD predicts recourse actions to restore normal status as counterfactual explanations, where the recourse function, responsible for generating actions based on observed data, is trained using an end-to-end approach. Experiments on synthetic and real-world datasets demonstrate its effectiveness.
1 Introduction
Algorithmic recourse refers to the process of offering individuals or entitics actionable recommendations to change undesirable outcomes generated by predictive models (Karimi et al,, 2022). In the context of counterfactual explanations, algorithmic recourse aims identify minimal and feasible modifications
input features that can shift model's prediction (Verma et al. 2024), which enhances transparency and interpretability in machine learning systems.
In recent years, algorithmic recourse has been extensively studied in the context of tabular (Karimi et al. 2020: Chen et al. 2020 Creager & Zemel 2023
Gao & Lakkaraju 2023 De Toni et al. 2024) and image data Jung et al. 2022 Wang & Vasconcelos 2020 Von Küigelgen et al. 2023). Despite significant advancements in these areas, there is a noticeable lack research on algorithmic recourse for time series. Time series data, which consist of scquential obscrvations over time, are ubiquitous domains such as industrial systems (sensor readings), finance (slock prices), and healtheare (patient vital signs). The temporal dependencies and complex dynamics inherent in time series data present unique challenges that differ from those in static tabular or image data, as interventions made at one point in time can lead to cascading effects across subsequent periods, making it essential to accurately account for temporal and causal dependencies when designing recourse actions.
This paper studies algorithmic recourse for time series data. While our method could serve as general framework, in this paper, we focus on abnormal multivariate time series due to their practical significance. Abnormal time series characterized by anomalies or unexpected patterns across multiple variables. Such

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anomalies can indicate critical events such system failures, fraudulent activities, or health deterioration. Providing algorithmic recourse in this context means offering explanations and actionable strategies to mitigate or prevent undesirable outcomes reflected by the anomalies, advancing anomaly detection beyond mere identification toward practical, intervention-driven solutions. For example, Figure| presents the usage data of two control nodes, nodes 117 and 124, in an OpenStack testbed (Nedelkoski et al. 2020). Each node's 's performance metrics include CPU and memory usage. When an anomaly detection model triggers an anomaly alert (indicated by the red area in the top figure), recourse action recommended to free up memory usage node 124, aiming to correct the abnormal behavior. After implementing the recourse action (shown as the green area in the bottom figure), the system returns to normal, as indicated by the dashed lines in the bottom figure. This example demonstrates how algorithmic recourse can provide quick and cost-effective remediation of the issue when an anomaly detection model detects an abnormal status.
CPU.117 117
RAM_117
C2U,122
RA
Anemaly
Value
Recnarse incommnndation
Time
Figure 1: Recourse recommendations for flipping an abnormal status of a distribution system.
To recommend recourse abnormal time series, we treat anomalies as external interventions on exogenous variables in structural causal model. For instance, in distributed computing environment, sudden surge in incoming traffic to particular node, such as caused by an external request spike, may lead to abnormal CPU or memory usage. Such anomaly ariscs not from change in the system's 's internal dynamics but from an external factor influencing the exogenous inputs to the node. Methodologically, we align this formulation of anomalies with the concept of backtracking counterfactual reasoning (Von Kügelgen et al. 2023), which we adopt as the foundation our method. Backtracking counterfactuals involve reasoning backward from an observed outcome to infer the necessary changes in exogenous variables that could have led to different result, assuming that all causal relationships remain intact. This approach is particularly suitable for interpreting abnormal states in time scries, it allows anomalies to be traced back systematically to plausible external interventions, providing explanatory insights and actionable recourse strategies.
Specifically, we propose a neural network-based framework for algorithmic Recourse in time series Anomaly Detection (RecAD). Our method assumes that the causal relationships within the time scries have been modeled using Granger causality discovery methods, such as Generalized Vector Autoregression (GVAR) (Marcinkevičs & Vogt 2021). Given a time series with abnormal segments, RecAD addresses the question: What is the most likely ezternal intervention that explains these obnormal states? Leveraging backtracking counterfactual reasoning, we identify recourse actions that mitigate anomalies by restoring abnormal states to normal ones. We formulate this task as a constrained maximum likelihood problem, where the identified actions serve both as plausible counterfactual explanations and actionable interventions. To enable end-to-end learning of a recourse function, we parameterize the recourse function with neural networks, mapping observed data to recourse actions. We adopt the Abbduction-Acttoo-Prelee Prediction procedure to model the downstream effects of interventions, deriving the counterfactual postcrior through cross-world abduction. A differentiable loss function is then defined to guide the learning of the recourse function. Furthermore, we discuss practical considerations, such handling sequence-level anomalies and addressing anomalies caused by interventions on structural equations rather than on exogenous variables. Empirical studies on two synthetic and one real-world datasets demonstrate the effectiveness of RecAD in recommending recourse actions for restoring abnormal time series.
The contribution of this paper can be summarized as follows. 1) We propose Recourse in time series Anomaly Detection (RecAD), a novel framework that generates recourse actions to correct anomalies multivariate time series data; 2) by treating anomalies due to external interventions on exogenous variables, we apply
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the concept of backtracking counterfactuals to time series analysis and formulate algorithmic recourse as constrained maximum likelihood problem using backtracking counterfactual reasoning; and 3) we addressing practical considerations and demonstrating effectiveness through empirical studies using both synthetic and real-world datasets.
2 Related Work
A time scries anomaly is defined as sequence of data points that deviates from frequent patterns in the time series (Schmidl et al., 2022). Recently, large number of approaches have been developed for time series anomaly detection (Schmidl et al. 2022 Blazuuz-García et al. 2021). However, explaining the
detection results is under-exploited (Jacob et al. 2020 Kieu et al. 2022). Algorithmic recourse can provide counteractual explanations by recommending actions to reverse unfavorable outcomes from an automated decision-making system (Karimi al. 2022). ). Specifically, given a predictive model and a sample having an unfavorable prediction from the model, algorithmic recourse to identify the minimal consoquential recommendation that leads to favorable prediction from the model. The key challenge of identifying the minimal consequential recommendation is to consider the causal relationships governing the data. Any recommended actions on a sample should be carried out via structural interventions loading to counterfactual instance. Multiple algorithmic recourse algorithms on binary classification models have been developed (Krimi et. al.) 2020 2021 von Kügelgen et. al. 2022 Dominguez-Olmedo et al. (022).. Recently, algorithmic recourse for anomaly detection on tabular data has also been discussed (Datta et al. 2022). However, this study does not consider causal relationships when generating counterfactuals.
In this work, we focus on addressing the algorithmic recourse for anomaly detection in multivariate time series, considering causal relationships. Specifically, assune that anomalies result from external interventions, distinguishing our approach from typical counterfactual reasoning. We align our approach with backtracking counterfactuals (Von Kügelgen et al., 2023), which suggests a backtracking interpretation of counterfactuals where causal laws remain unchanged in the counterfactual world, and differences from the factual world are attributed to changes in exogenous variables that alter the initial conditions. Different from a recent study Kladny et al., 2024p, which also leverages backtracking counterfactual reasoning to generate counterfactual explanations for static high-dimensional data by developing a deep generative model, our work focuses on providing backtracking counterfactual explanations in abnormal multivariate time series.
3 Preliminary
Granger Causality. Granger causality (Granger 1969 Dahlhaus & Eichler
2003 is a standard approach
for capturing causal relationships in multivariate time series. Formally, let a stationary time-series be . .x............... , where X; Id is a d-dimensional vector (e.g., d-dimensional time series from d sensors) a specific time t. Define the true data generation mechanism in the form of (1)
X =
2 = (x
x + u"), for 1 i
(1)
where x x = [ 2 2-1 denotes the present and past of series j: v0 indicates exogenous variable of time series j at time step t; F= {f())..,, f(1) is a set unknown functions, and f(i) E F is the function for time series j that captures how the past values impact the future values of xl)) Then, the time series
()) (
Granger causes j, if f(f) depends on x St- i.c., (t) (6) 21-1 x (x (1-1 x 51-1 x x(_1)
# S1-1 (x (1)
11-1 11-1).
Granger causal discovery, i.e., learning Granger causal relationships from the observational data, has been extensively studied (Nauta et al.] 2019 Tank et al.] 2021 Marcinkevičs & Vogt, 2021). While multiple
methods are available, in this paper, we utilize a generalized vector autoregression (GVAR) approach that
can model nonlinear Granger causality (Marcinkevičs & Vogt 2021). GVAR models the Granger causality of the t-th time step given the past K lags by
K
X; =
Σ
ge (x
(2)
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where 9x(+): R -> Idd is feedforward neural network predicting a coefficient matrix at time step t- k; u is the exogenous variable for time step 1. The element (i,j) of the coefficient matrix from (x(xx)) indicates the inftuence of = on =²() Mcanwhile, K neural networks are used predict x,- Therefore, relationships between d variables over K time lags can be explored by inspecting K coefficient matrices. The neural
(i)
networks trained by the objective function:
C=
Σ k=
X
& +
Σ R(M)) T- I=
A
Σ f=++1
| -
K
T K
where &; Ek=z 91 (x)=))x=<+ indicates the predicted value by GVAR; M, == [x 9K IX, 91 (x:-1)) indicates the concatenation of generalized coefficient matrices over the past the K time steps; R(-) is the penalty term for sparsity, such as L1 or L2 norm; the third term is smoothness penalty; À and y are hyperparameters. After training, the generalized coefficient predicted by I& (xtxx+)) indicates the causal relationships between time series at the time step t - k.
Vi (m,,,,,
U uu U
(Ue-i++1 M(+z)
Recourse action
F u
11
K +
t M t+L
W. (Xx-K=11m..X))
V (ex+i))
(a) Backtracking counterfactual.
(b) Algorithmic recourse multivariate time series.
Figure 2: Illustration of backtracking counterfactual and algorithmic recourse.
Counterfactual Reasoning. Counterfactual inference provides a framework for understanding causal relationships by exploring counterfactual scenarios, often framed as "what-if" questions. For example, "would the CPU utilization stay within safe operating limits the memory usage followed its default profile?" In causal inference, counterfactuals are typically analyzed using two main approaches: interventional counterfactuals Pearl 2009) and backtracking counterfactuals (Von Kügelgen et al. 2023). In interventional counterfactuals, causal laws are modified while the factual and counterfactual worlds share the same background conditions. In backtracking counterfactuals, on the other hand, as shown in Figure 2a causal laws remain consistent, but the counterfact ual explanation adjusts exogenous variables to account for changes observed in the outcome. Both types of counterfactual reasoning are valuable for understanding causal mechanisms; however, backtracking counterfactuals are especially useful for explaining observed anomalies by inferring the most plausible prior conditions that led to the current state. This insight forms the basis of our method.
4 Methodology
In this section, we propose the algorithmic Recourse in time series Anomaly Detection (RecAD) method compute algorithmic recourse actions for anomalics multivariate time scries, assuming that the causal relationships have been captured using a Generalized Vector Autoregression (GVAR) model shown in Eq. (2. We first provide the general framework of the problem formulation based on the backtracking counterfactual. Then, we explain the detailed implementation of each component of the framework.
External intervention vs. structural intervention. Depending on how anomalies occur within a causal mechanism, they can be divided into two categories: those due to changes in exogenous variables (external interventions) and those due to changes in the structural equations (structural interventions). In this paper, we focus on anomalies caused by external interventions and briefly discuss in Section 4.5.2 how to handle anomalies resulting from structural interventions.
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4.1 Problem Formulation: Algorithmic Recourse based on Backtracking Counterfactual
Following the common setting for time series anomaly detection (Audibert et al. 2020 Tuli et al 2022 given a multivariate time series X, we consider local window with length K as W, - x2) and convert a time series X' to sequence of sliding windows W = (Wx K, W K++,.... Wr). Consider score-based anomaly detection function s(-) that takes a time window W. as the input. If s( (W) > then the time step will be labeled as in an abnormal state. Assume that an anomaly occurs at time t, which causes abnormal states in time window Ve+L= (St....,(++2)) (assume that L < K) due to the ripple effect. By treating the anomaly as an external intervention on the exogenous variable ut, our objective to find the recourse action 0, at the time step t to reverse the abnormal states in the time window V(+L via counterfactual reasoning. Such action can be viewed as the most plausible counterfactual explanation for the abnormal states. To simplify our discussions, we mainly focus on point anomalies. Extending our approach to sequence anomalies is straightforward and will be addressed in Section 4.5.1
Specifically, the backtracking counterfactual computes the probability the exogenous variables in a counter- factual world, which may differ from their factual counterparts, given both the factual and counterfactual endogenous variables. As shown in Figure 2b in our context, the factual endogenous variables are W,, i..,, the observational data. We denote the backtracking counterfactual exogenous variable at time by uy, which is obtained by performing the recourse action, i.e, U, u, + Oz, where is the factual exogenous variable. We then denote the counterfactual endogenous variables by Vi+L = (x.,,..,,;+ i.e., the counterfactual variants of Vi+L obtained by performing the recourse action, with the requirement that their states are restored to normal. As a result, the problem of algorithmic recourse can be formulated to maximize the likelihood of uf conditioning on Wr, subject to the constraints that s s(Vū.) < for t' [t, L].
Problem Formulation 1 Given local windows of time series W, -X, x..... and a score function s(-) with time step t labeled as abnormal, i.e., s(W.) > the algorithmic recourse aims to find the recourse action at time 1, i.e., _º = u, + 0., to restore normal states in a future time window Vi+L = (X,,.., x++1), by solving the following constrained marimization likelihood problem:
arg max P(u W. s.t. s(V))< <T, ttE(ttt
0.
By introducing penalty terms to the negative log-likelihood to enforce constraints, derive the revised problem formulation as follows.
Problem Formulation The algorithmic recourse aims to solve the optimization problem:
L+L
arg min L(O.) max (s(Vp) - r, 0} - Aoo P(u W.).
(3)
Next, we discuss the implementation of each component of the problem formulation to enable end-to-end learning of recourse function.
4.2 Implementing Recourse Function
To generate effoctive recourse actions, we learn a recourse function ha(-) parameterized by & that maps from the observed time series perturbation of the exogenous variables, i.e., 0, (Wr,1,, Ar), where O, is the proposed intervention at time step
The design of h is guided by two key considerations. 1) Temporal context is essential for accurate intervention, as the causal dependencies in multivariate time series often span multiple time lags. Therefore, we encode the preceding time window W using a Long Short Term Memory (LSTM) network to capture latent temporal features and long-range dependencics. 2) Deviation from expected dynamics another critical signal for recourse. We define the deviation term A, 2, &,, where & is the expected value time based GVAR. Intuitively, A, quantifies how much the current. state deviates from its normal trajectory
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Concretely, our implementation is where @ denotes vector concatenation. The final leedforward network combines the latent temporal state and the deviation signal to produce the recourse action.
and serves as proxy for anomalous inffuence from exogenous variables. structured as follows:
4 = LW
2A 2A = =A=F(N(AA))
0, = FFNN 2A),
(4)
We empirically validate the necessity of each component through ablation studies, showing that removing either the LSTM or the deviation encoder significantly degrades recourse effectiveness.
4.3 Inferring Downstream Effect
We then infer the downstream effect Vū because, based Granger causality, the recourse action at time leads to the counterfactual variants of the following time steps, i.e, Vi+L counterfactual variants, according to the additive noise assumption presented in Eq. ① given backtracking counterfactual u we directly have x = X4 u, + i x, + 0,- However, since we perform the recourse action only at time t, remains necessary to infer the factual exogenous variables for all t' Thus, we proceed to utilize the Abduction-Actiin on the causal relationships learned by GVAR shown in Eq. ( the
x; To compute the
(Pearl 2009) procedure compute x Formally, based Action-Prediction (AAP)
procodure to compute x for t' = t 1, ", L can be described below.
Step 1 (Abduction):
K
(xe -
(5)
Step 2 (Action): Not required
Step 3 (Prediction):
K Σ k=l'-(1
x
Σ 1=1
x
9k (xx x)xe + uy
(6)
In essence, the abduction is to derive the exogenous variable uy at step 1' based on the observed value, then the prediction to predict the counterfactual value x at step after conducting action O, at step t.
4.4 Deriving Counterfactual Posterior
According to Problem Formulation the next step is to derive the counterfactual posterior P(u; W.).
Following the backtracking counterfactual (Von Kügelgen et al. 2023), we apply cross-world abduction obtain:
Plu: u&) (()PP(WW))) ut)
P(u W.) Σ
P(u u/ (W))== Σ
P(W.)
u
up
Σ
P(ui, P(W,(u, = Σ P(u P().(W))
Σ P(u,)u))P(W,, -(W W.),
where the second equality is based the Bayes rule, the third equality is due to the fact that už and W, are conditionally independent given u, (obtained using d-scparation in Figure 2a), I(-) is the indicator function, u, is the factual exogenous variable obtained by abduction, and P(u; Ue) is the backtracking conditional. Following the suggestion in (Von Kügelgen et al.) 2023), we construct P(u uz) based on the distance between oy and u,, defined as P(u u,) 1/Z exp du,, u.) where Z a normalization constant. By using the squared Mahalanobis under isotropic covariance, a result, we have that P(u; W.) exp(-llu; u exp{-110.113.

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Combining all the components above, obtain our final loss function:
(()) Σ max (s(V)) - 7, + 1110.112
where @ are the parameters of the recourse function defined in Eq. and hyperparameter.
4.5
Other Considerations
4.5.1 From Point Anomalies to Sequence Anomalies
So far, have introduced our framework in the setting point anomalies, where single abnormal state is explained by one recourse action uz- This allows us to present the core intuition and mechanics of our approach in a straightforward manner. In practice, however, abnormal states in time series data are often caused by sequence of anomalies, which requires exlending our formulation. To handle this situation, we generalize the single recourse action i, in Problem Formulations and to a sequence of recourse actions
U = (uz, 42415 1 M(+M).
Correspondingly, the recourse function is modified to adopt a sequence-to-sequenee network architecture. The subsequent procedures, including the Abduction-Actee Action-Preiiiiiiiioo pipeline and backtracking counterfactual reasoning, remain conceptually the same and can be applied in this extended setting. key challenge here is that the length of the anomaly sequence is not always known. As a practical solution, we propose computing the shortest sequence of recourse actions that adequately explains the abnormal states, i.e., the minimal sequence that restores normal states for all subsequent time steps. We describe the pseudo-code of the training process for recourse predictions with multiple abnormal time steps in Appendix A
4.5.2 Dealing with Anomalies due to Structural Intervention
We have assumed that anomalies are caused by external interventions on exogenous variables, without altering the Granger causal relationships. This type of anomaly occurs, for example, when a sensor is attacked. However, there are other types of anomalies that cannot be treated as external interventions but must be considered structural interventions. In other words, they caused by replacing the normal equations with abnormal ones. For instance, when system experiences a structural change due to component degradation or failure. Despite the intrinsic differences between external interventions and structural interventions, it still possible to explain anomalies caused by structural interventions using recourse actions. This is because (x(() (x---1''
rearrange the structural equation as follows: (.] f() 5t-1'" x + u()
(00
((-11) is an abnormal function (d) "00---, xx-1) 4.th 4,)), for S j d, where f(t)) (x()) (1-11" (1)
xx (d)
for the time series j at time and c. (()) (x (I) can describe the anomalies as
51-1" x x((1)) -xx)(xx))) (1-1111 x xx1)). As result, we
2 = (x)
(1)
(0) x
al + 9 C for 1 siidd
where anomaly term &t can be due to a structural intervention. However, the significance of using recourse actions to explain structural intervention is worthy of future study.
5
Experiments
5.1
Experimental Setups
Datasets. We conduct experiments on two semi-synthetic datasets and one real-world dataset. The purposes of using semi-synthetic datasets are as follows. 1) We can derive the ground truth downstream time series after the intervention on the abnormal time step based on the data generation equations. 2) We can evaluate the fine-grained performance of RecAD by injecting different types of anomalies.
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Semi-synthetic datasets: 1) Linear Dataset (Marcinkeviés & Vogt 2021 is time series dataset with linear interaction dynamics. 2) Volterr is a nonlinear time series model that simulates prairie ecosystem with multiple species. For both datasets, we adopt the structural equations defined in (Marcinkevičs & Vogt 2021 which are included in Appendix We also describe the anomaly injection strategics in Appendix B
Real-world dalaset: Multi-Source Distributed System (MSDS) (Nedelkoski et al. 2020) consists of 10-dimensional time series. The first half of MSDS without anomalies is used as a training set, while the second half, including 5.37% abnormal time steps, is used as test set. As real-world datasct, cannot observe the downstream time series after the intervention. In the test phase, we use GVAR and AAP to generate the counterfactual time scries for evaluation.
Table 1: Statistics of three datasets for anomaly detection.
Test (Anomalies %)
Dataset
Dim.
Train
Point (External)
Seq. (External)
Seq. (Structural)
Linear
50,000
250,000 (2%)
250,000 (6%)
250,000 (6%)
Lotka- Volterra
20
100,000
500,000 (1%)
500,000 (3%)
500,000 (3%)
MSDS
146,340
146,340 (5.37%)
Table shows the statistics of three datasets. Training datasets only consist of normal time series. Note that the test sets listed in Table [ are used for evaluating the performance of anomaly detection. After detecting the abnormal time scries in the test sct, for the synthetic datasets, we use 50% of abnormal time series for training RecAD and another 50% for evaluating the performance of RecAD on recourse prediction, while for the MSDS dataset, we use 80% of the abnormal time series for training RecAD and the rest 20% for evaluation.
Baselines. To our best knowledge, there is no causal algorithmic recourse approach in time series anomaly detection. We compare RecAD) with the following baselines: 1) Multilayer perceptron (MLP), which is trained with the normal flattened sliding windows to predict the normal values for the next step; 2) LSTM, which can learn complex temporal dependencies in time series to make predictions for the next step; 3) Vector Autoregression (VAR) is a statistical model that used to analyze GC within multivariate time series data and prodict future values; 4) Generalised Vector Autoregrossion (GVAR) (Marcinkevičs & Vogt 2021 an extension of self-explaining neural network that can infer nonlinear multivariate GC and predict values of the next step.
For all the basclines, in the training phase, train them to predict the last value in time window on the normal time series. In the testing phase, when a time window is delected as abnormal by an anomaly detection model, indicating the last time step X₁ is abnormal, we use baselines to predict the expected normal value in the last time step & Then, the recourse action values can be derived as 0, = X, Xx. For the sequence anomalies, we keep using the baselines to predict the expected normal values and derive the action values by comparing them with the observed values.
Evaluation Metrics. We evaluate the performance of algorithmic recourse in terms of effectiveness, feasibility, and efficiency.
1) Flipping Ratio (t) (Effectivenoss). The Flipping Ratio measures the proportion of detected anomalous time steps that are successfully reverted to normal after applying the recourse actions. Formally, let Tanom be the set of detected anomalous time steps, let 0, be the recourse actions, & the counterfactual sequence produced by the recourse algorithm, and s(-) the anomaly score with threshold т. Then
EteTamn I(s(x,) <T}
Flipping Ratio
[Tanoml
A higher Flipping Ratio indicates more effective recourse, as a larger fraction of anomalies is eliminated after intervention.
2) Action Cost (1) (Feasibility). The Action Cost quantifics the magnitude of all interventions required to flip anomalics. For each time step t, let 0, E Rd denote the perturbation applied the d-dimensional system state. The cost at time is measured using the Faclidean norm ||0: [2, and the average cost per abnormal
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sequence is computed
Action Cost
||0:1.2.
Taaom
EE TE7anom
Lower Action Cost reflects more parsimonious interventions that achieve anomaly flipping with minimal total perturbation.
3) Action Step (4) (Efficiency). The Action Step metric counts the number of discrete time steps which interventions are applied. For each anomalous sequence, let st = if 0g (i.e., an action is taken) and 82 0 otherwisc. The average number of steps per sequence is
Action Step
Σ
TTaoo
IETanom
A smaller value implies that fewer intervention points are required, indicating temporally efficient recourse.
5.2 Experimental Results
To implement the score function s(-), in our experiments, we leverage the UnSupervised Anomaly Delection for multivariate time series (USAD) (Audibert et al. 2020) as well as a transformer-based anomaly detection model (TranAD) (Tuli et al. 2022) as base anomaly detection models that output an anomaly score for multivariate time series. We include the implementation details as well the performance of USAD and TranAD for anomaly detection in the Appendix. For all experimental results, report the mean and standard deviation over 10 runs. Our code is available online"
5.2.1 Evaluation Results on Synthetic Datasets
The performance of recourse prediction on anomalies caused by external interventions. Table shows the performance of RecAD for recourse prediction anomalies caused by external interventions. Note that the anomalies consist of both point and sequential anomalies. First, in all settings, RecAD can achieve the highest flipping ratios, which shows the effectiveness of RecAD in flipping abnormal behavior. Mcanwhile, RecAD can achiove low action costs and action steps compared with other baselines. Because no baseline considers the downstream impact of recourse actions, they usually require more action steps to flip the anomalics.
The performance of recourse prediction on anomalies caused by structural interventions. We examine the performance of RecAD anomalies caused by structural intervention. As shown in Table 3 RecAD can achieve the highest flipping ratio compared with basclines on both Lincar and Lolte-Volterra datasets, indicating that the majority of anomalies caused by structural interventions can be successfully flipped. Meanwhile, RecAD can also achieve low action costs and action steps with high flipping ratios. Overall, RecAD meets the requirement of algorithmic recourse, ie., flipping the abnormal outcome with minimum costs on anomalies caused by structural interventions.
Therefore, based Tables and 3
demonstrate that RecAD can provide recourse prediction on different types of anomalies in multivariate time series.
5.2.2 Evaluation Results on the MSDS Dataset
The performance of recourse prediction. Because the types of anomalics in the real-world datasct are unknown, report the performance of recourse prediction on any detected anomalies. As shown in Table 4 RecAD achieves much higher flipping ratios than all baselines. Regarding the average action cost per time series and the average action step, RecAD also outperforms the baselines by having the lowest values. This suggests that, by incorporating Granger causality, RecAD can identify recourse actions that minimize both cost and the number of steps.
httpe:///gghhub..cm//aaaaiinee
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Table 2: The performance of recourse prediction on anomalies caused by external interventions.
(a) USAD as the anomaly detection model
Point
Seq.
Dataset
Model
Flipping Ratio f
Action Cost
Action Step
Flipping Ratio t
Action Cost. +
Action Step +
MLP
0.778+0.0r,
8,406+0.257
1.188
0.867
22.394+0
2.261
LSTM
0.807 065
8,383
1.170+0.010
0.8784a.001
22. 20.4
2.2 2.248
Linear
VAR
0.6764o 003
1.31 1.311+0.0
23. 690 511
2.434g000s
GVAR
0.775 053
8.446:00249
1.207
0.848
22.4152o ser
2.287 2187200020
RecAD
0.901 ::
8.201
1.104 022
0.944.0.041
21.264 war
2.193 046
MLP
0. 120
277.580g117.100
1.237
0.65820:259
761.550
2. 2.195
LSTM
898 057
313.510
1.096
0.88020.007
1500.483gss r26
1.330
Lotka-Volterra
VAR
0.558 to 196
326.967 112003222
1,57 20.4
0.504go.151
1445.084. 189
2.570
GVAR
0.498
270.335 1117:22
2.02011.049
0.6606.0.00
149.792,
2.547 10:900
RecAD
0.915
242.759_0000
1.085
0.972:0.016
1374.1 177.1122344.447
1.329
(b) TranAD as the anomaly detection model
Point
Seq.
Dataset
Model
Flipping Ratio
Action Cost +
Action Step
Flipping Ratio f
Action Cost.
Action Step
MLP
0.657
5. 5.260.2211
1.145go.11
0.73220.228
14.006 048
2.238-20.22
LSTM
0.671
6.271
144
0.
14.228 988
2.189
Linear
VAR
0.557
5.349
1.55 20.
20:27
14.147 2012
2.373
GVAR
0.590
5.326
1.549
0.673 2. 20777
14.211
2.335
RocAD
0.8841o.121
7.021
1. 1.11210.000
0.924_.o.10 1.16
16.793.5.000
1.905 10.00000000
MLP
0. 0.796
253.537
1.230=0.246
0.735
326.481=s80
2.251
LSTM
0.839
274.916;:119.66
1.146=00173
0. 0.833
1457.3144344
1.866=0.214
Lotka-Volterra
VAR
0.636
166.372::116.002
1,554g0.56
0.42020.00s
1261.513
2. 2.833
GVAR
0.72020339
244.942.1144.
1.664_1.052
0,550
680.18 1 329
2.501
RecAD
0.902
278.432
1.099
0.925 034
1254.6646
1.856 138
Table 3: The performance of recourse prediction on anomalics caused by structural interventions.
(a) USAD as the anomaly detection model
Datasel
Model
Flipping Ratio f
Action Cost. $
Action Step 4
MLP
0.884=0.023
41.387 11.387=2.151
2.359
LSTM
0.903=0.021
42.412=2.00
2.398
Linear
VAR
0.782=0.035
59.346-3.228
2.883=0.0622
GVAR
0.874_0.024
39.474_2.116
2.415-0.041
RecAD
0.919
38.917
2.16911 206
MLP
0.665=0.277
1578.597
2.247
LSTM
0.89020.099
3159 3159.333
2.310=0.100
Lotka-Volterre
VAR
0.48820.178
3088. 713387888435.000
2.630=0..440
GVAR
0.384
1946.415=3344 144
2.618
RecAD
0.970
2767.819 .488
1. 1.386 120
(b) TranAD as the anomaly detection model
Dataset
Model
Flipping Ratio f
Action Cost
Action Step 1
ML.P
0.83410.117
52.812 404
2.722
LSTM
0.92144.031
$2.759-431.20
2..48
Linear
VAR
0.77820.083
61.716137.610
2.908
GVAR
0.785410.00
59.3031135.30
2.84022.430
RecAD
0.942 048
48 48.227:131 905
2.260:1.447
MLP
0.72740.167
1541.431 460.0a6
2.228
LSTM
0.853 091
2783 639 121.1121.60 .66
7.735 176
Lotka- Volterra Lotka-Volterr
VAR
0.44440.10
2441.456::44
2.791 100
GVAR
0.572.1.177
1387.077 118
2. 2.495
RecAD
0.929
2276.269.
1. 1.809
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Table 4: The performance of recourse prediction in MSDS datasets.
(a) USAD as anomaly detection model
(b) TranAD as anomaly detection model
Model
Flipping Ratio
Action Cost
Action Step &
Model
Flipping Ratio 1
Action Cost
Action Step
MLP
0.687
6.84842.500
1.443=0.68
MLP
0.66420.337
4.999=3.136
1.47040.891
LSTM
0.83010.211
6.798
1.279_0
LSTM
0.749
5.44323.011
1.301
VAR.
0..24
6.759
1.432_0.500
VAR
0.643 366
5.197
1.525
GVAR
0..12
8.92343.258
1.425=0.4e
GVAR
0.739
6,443-3.270
1.32740.448
RecAD
0.841:0.080
6.747
1. 1.24910.008
RecAD
0.837
4.91811.28
1.273_0.455
5.3
Case Study
We further conduct case studies to show how to use the recourse action predicted by RecAD as an explanation for anomaly detection in multivariate time scries.
Antslope
Hare
Gray
Anomaly
Recourse recommendation
Value
Time
Figure 3: Recourse recommendatioss for intervening in an imbalanced ecosystem to restore balance.
Case study on the Lotka- Volterra dataset. Figure shows a simulation of a prairie ecosystem that contains antelope, hare, fox, and gray wolf based on the Lotka- Volterra model (Bacaër 2011). where each time series indicates the population of a species. As shown in the top figure, in most of the time steps, the numbers of carnivores (fox and gray wolf) and herbivores (antelope and hare) keep stable in balanced ecosystem, say 0. 1.1k-1k 1k antelopes, 1k-10k hares, 0.1k-1k foxes, and 0. 0.1kkkk gray wolves. After detecting abnormal behavior at specific time step (red area in the top figure), the algorithmic recourse aims to provide recourse actions to flip the abnormal outcome. In this case, the algorithmic recourse model recommends the intervention of reducing the populations of harcs, foxes, and gray wolves by 100.1k, 9.3k, and 7.5k, respectively. After applying the recourse actions (green area in the boltom figure), we can notice the populations of four species become stable again (the dashed line in the bottom figure). Therefore, the recourse actions can provide recommendations to restore the balance of the prairic ecosystem.
Lotka-Volterra
Case study on the MSDS dataset. Figure ] depicts case study on MSDS with control nodes 117 and 124. USAD detects a subsequence of anomaly consisting of two abnormal time steps from two-time series (CPU and RAM usages on node 117), highlighted in the red area of the top figure.
When the first abnormal time step detected, RecAD suggests releasing the CPU usage by 6.7 on node 117 (the green area the middle figure). In other words, it also mcans the anomaly here is due to the higher CPU usage than normal with a value of 6.7. After taking this action, the following time steps are affected by this action. A counterfactual time series is then generated using the AAP process, which is shown the dashed lines in the middle of Figure RecAD continues to monitor subsequent time steps for any abnormalities.
The following time step is still detected as abnormal in the time series of memory usage of node 117. RecAD recommends releasing the RAM usage by 13.39 on node 117 (the green area in the bottom figure), meaning that the abnormal time step here due to high memory usage in a margin of 13.39. After taking the recourse action, the counterfactual time series then generated (the dashed lines in the bottom figure). We can then observe that the entire time series returns to normal.
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CAU 117
CPU 124
RAM_124
Anomaly
Recourse recommendation
Va luee
Figure Recourse recommendations for restoring the abnormal CPU and RAM usages in MSDS.
In summary, recourse actions recommended by RecAD can effectively flip the outcome and lead to normal counterfactual time series. Mcanwhile, based on the recourse actions, the domain expert can understand why a time step is abnormal.
Table 5: The performance of recourse prediction using different components of RecAD.
Dataset
Metric
RecAD w/o PFNN
RecAD w/o LSTM
RecAD
Flipping Ratio f
0.340+0.191
0.676=0.085,
0.922 0.040
Linear
Action Cost
119. 19,2866114.172
23. 23.589423.44 23.4433.43
22.794
Action Step
2.90010.88
2.27620.999
1.822:0.55 521
Lotka-
Flipping Ratio f
0..231
0.523 0.000
0.952 o 054
0.0054
Action Cost
876. (6.,107=811 181000000
1035. 1135.192481..00
1468. 230 1086 .90
Volterra
Action Step
2.706
2.304
1.266 180
Flipping Ratio f
0.228.10.14
0.697 1..977
0.841 1 080
MSDS
Action Cost
3.494 22.4:2.660
4. 4.13640..27
6.747 + 1.544
Action Step
2.04840.607
10.51
1 1.249 068
5.4 Ablation Study
We evaluate the performance of using different parts of RecAD (i.e., FFNN and LSTM) for recourse prediction. In this experiment, we adopt USAD as the base anomaly detection model. As RecAD contains an LSTM to catch the previous K time lags and a feedforward neural network (FFNN) to include the time lag exclusion lerm Å,, we then test. the performance of these two parts separately. Table 3 shows the average flipping ratio, action cost, and action step for three types of anomalies for the syuthetic datasets and results for the real-world dataset MSDS. We observe that RecAD achicves higher flipping ratios while requiring fewer or comparable action steps compared to methods that utilize only parts of RecAD. In cases where other methods exhibit lower action costs, they achieve very low flipping ratios, indicating their inability to successfully flip abnormal samples, which is undesired. This result shows the importance of considering both information for reasonable action value prediction.
5.5 Sensitivity Analysis
The objective function defined in Eq. 7 for training RecAD employs the hyperparameter À to balance the flipping ratio and action cost. A large À indicates a large penalty for high action costs, which could potentially hurt the performance of flipping abnormal time stops as small action costs may not be sufficient flip the anomalies. As shown in Figure 3 on both synthetic datasets, we have similar observations that with the increase of X, both action cost and flipping ratio decrease. Specifically, smaller values of yield higher
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1.0
1.0
20.0
1200
0.8
0.8
Flipping Ratio
17.5
Flipping Ratio
1000
Action C68
Action Cost
0.6
15.0
0.6
800
12.5
600
0.4
0.4
10.0
400
0.2
Flipping Ratio
7.5
0.2
Flipping Ratio
Action Cost
5.0
Action Cost
200
0.0
0.0
le-4
1e-3
1e-2
le-1
1
1e-4
1e-3
1e-2
1e-1
1
A
(a) Linear
(b) Voltera
Figure Effects of the hyperparameter À Eq.
action costs but higher flipping ratios, whereas larger values produce lower costs and lower flipping ratios. By varying X, the model generates multiple distinct recourse trajectories with different eosteeffectivesess trade-offs, introducing diversity among the counterfactuals.
6 Conclusions
In this work, have developed RecAD, novel framework for algorithmic recourse in abnormal multivariate time series. RecAD suggests actionable interventions to restore the multivariate time series to normal status, offering counterfactual explanations for abnormal patterns. By leveraging backtracking counterfactual reasoning, we formulated the problem of learning the recourse function from data as a constrained maximum likelihood problem to enable end-to-end training. The empirical studies have demonst.rted the effectiveness of RecAD for recommending explainable and practical recourse actions in abnormal time series.
7 Limitations and Future Work
Our current framework adopts GVAR to infer Granger causal relationships from multivariate time scries. While GVAR provides principled and computationally tractable approach, several limitations remain: (i) GVAR may be sensitive to latent confounding and hyperparameter choices governing sparsity and temporal smoothness. (ii) While GVAR model nonlinearity and time variation, its reliability still depends regularization design and stability selection thresholds when aggregating coefficients into summary graph. (iii) GVAR does not explicitly handle interventions or invariance across environments, which can improve robustness to spurious associations. In future work, we plan to integrate recent advances in causal discovery to complement GVAR's learned coefficients, including invariance- baasdd or multi-envirrnment selection to reduce spurious edges, procedures that address latent confounders and non-stationarity, and hybrid models that combine constraint-based, score-based, and neural approaches.
Another limitation of the current work lies in handling anomalies caused by structural interventions. In Section 4.5.2, we reformulated structural interventions as external interventions and addressed them using a similar method. However, this approach does not directly capture or explain the structural changes themselves. As future work, we plan to develop methods that can learn a modified causal structure, which, if implemented, would restore the system normal behavior. In particular, we aim design an optimization procedure that enables the encoder-decoder architecture to adapt rapidly local causal mechanism changes present in abnormal sequencos.
Acknowledgment
This work supported in part by NSF 1910284, 2103829, and 2142725.
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Appendix
A Algorithm of RecAD
Algorithm shows the pscudo-code of the training process for recourse predictions. Given abnormal time window W. with the abnormal time step l and subsequence window Vi+r of length L, we first predict the action value on to reverse the abnormal status. After conducting the recourse actions, we also need to ensure that the following time steps are also normal. Therefore, we then derive the counterfactual subsequence Vi+r. step by step. For each following step, denoted as Vi++> the subsequence Vi+t still abnormal, we further conduct recourse actions flip the abnormal status. Afler going through time sleps, update the action prediction function h-(+) based on the objective function defined in Eq. 7
Algorithm 1: Pseudo code of training RecAD
Pretrained GVAR ge(+), anomaly detector a()), an abnormal window W, with the abnormal time step E, and the following window Vi+L
Input:
Output: Updated ha(+)
1: x < Action Prediction (Wn, g)("), k)(-))
2: V |2+r = x
3: for / +++1 to L do
4: Derive Vi+ by Eqs. 5 and
if s(Vi+t) > r then
Xi41 + Action Prediction inliictiinniiiui (Vi+z) gh((),, h-)
Vi+1!!] =x#
end if
end for
10: Update h-(-) based on the objective function [()) in Eq. 7
11: Function Action Prediction( We, (x(:),, (-))
12:
W + We {x:}
13: Compute &, = Ek=l Ih xx(xx=x)xx with W
14:
Compute A. = Xt &t
15: Compute 0, ho (W,.. A.)
16:
x X, 0e
17:
Return x;
B
Experiments
B.1 Datasets
Linear Dataset (Marcinkevičs & Vogt 2021 is synthetic time series dataset with linear interaction
dynamics. We follow the structure from the original paper (Marcinkevičs & Vogt, 2021 where the linear interaction contains no instantaneous effects between time series and can be defined as:
z (1) a + u + +
(1)
(2) a2=1=1
(2) + 03 (1)
(2) (2)
u +
(9)
2 a,
(3)
(3)
05
22)
(3) (3)
+
(4) d
+ 07
ag
(3)
+
(1) + c
where coefficients ~ u([-0.8, 0.2| U [0.2, 0.8),, additive innovation terms ut -N(000.16) and anomaly term E
Abnormal behavior injection. For point anomalies, the anomaly terms are single or multiple extreme values for randomly selected time series variables specific time step For example, point anomaly at time
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step t can be generated with an abnormal term e, 20, which means the second and third time series have extreme values.
For non-causal sequence anomalies, the anomaly terms are function-generated values for a given time range (1)
from & to ++ n. For instance, setting fi+s 0.1 x i, for 0 S i S will cause trend anomaly for time series variable setting 444 ~ (0, 0.16), for 0 S i S n, will cause a shapelet anomaly for time series variable (2)
(1) ; and setting '++r = (a) 2 + "(+2)) + (a1 142--2 11-211 - (a1 + uitt), for 0 < isss S will cause seasonal anomaly for time series variable z!
For causal sequence anomalies, we consider two scenarios: 1) changing the coefficients A= = {ai, 42, ... as from a normal one to a different one in a time range to t + n; 2) changing generative functions from the original Equation to a different one.
Lotka-Volterya Volterra (Bacaér 2011 is another synthetic time series model that simulates a prairie ecosystem with multiple species. We follow the structure from (Marcinkevičs & Vogt 2021). which defines as:
dx(())
ax
B
E
yb0 - )x for 1 s S
dt
jEPa(x(0)
dy()
8y(0)
Σ
(() py(i) for 1 sssp.
(10)
dt
kEPaly(i)
( (i)
+ 9 (1)
Ft It
for S s
(1) ( d.) for 1 م sssp
V1 vt
where (0) and y(f) denote the population sizes of prey and predator, respectively; a, 8, 8, p are parameters that decide the strengths of interactions, Pa(x(1) and Pa(yl)) correspond the Granger Causality between prey and predators for x(1) and y( respectively, and e is the abnormal term. We adopt 10 prey species and 10 predator species.
Abnormal behavior injection. For point anomalies and non-causal sequence anomalies, we perform similar procedure as the linear dataset, i.e. randomly select time series variables at specific time step t and assign single or multiple extreme values as point anomalics, and assign function-generated abnormal terms for a time range from & to + n as sequence anomalies.
For causal sequence anomalies, we still consider two scenarios: 1) changing the coefficients a, B, n, p different values than the normal ones; 2) changing Pa(x()) and Pa(y()) to different ones from the original generative functions Equation 110
B.2 Implementation Details
Similar to (Audibert et al. 2020), we adopt a sliding window with sizes 5, 5, and 10 for the Linear, Lotka- Volterra, and MSDS datasets, respectively. We set the hyperparameters for GVAR by following (Marcinkevičs & Vogt 2021). When training (.), we set in the objective function as L = 1, which is to ensure the following one-time step should be normal. The cost vector с can be changed according to the requirements or models that cannot take the cost into
prior knowledge. Because the bascline models are prediction- account, to be fair, we as the cost. vector. The implementation details of neural norwsrks in experiments are described in the Appendix.
For baselines, MLP is forwafo neural network with structure of ((K - 1) * 10-00-100----- that the input is the flattened vector of K - time steps with d dimensions and the output the predicted value of the next time step. The LSTM model consists of one hidden layer with 100 dimensions and is connected with a fully connected layer with a structure of 100-d. We use statsmodel,[] to implement the VAR model. The baseline GVAR model the same as GVAR within our framework. To implement ha(-) in RecAD, we utilize
httpp::/www/
wtwp://tttttmmee...m http:://www.etttmmdel..oo
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LSTM that consists of one hidden layer with 100 dimensions and feed-forward network with structure d-100. Then we use another leed-forward network with structure of 200-d to predict the intervention values.
All experiments were conducted on an Ubuntu 20.04 server equipped with an AMD Ryzen 3960X 24-Core processor at 3.8GHz, dual GeForce RTX 3090 GPUs, and 128 GB RAM. The implementation uses Python 3.9.7 and PyTorch 1.11.0.
B.3 Performance of anomaly detection.
Table 6: Performance of Anomaly Detection: TranAD and USAD Synthetic Datasets
Linear
Lotka Volterra
Anomaly Types
Metrics
TranAD
USAD
TranAD
USAD
F1
0.7 0.736 0.
0.857 40 020
0.787 100
Non-causal
AUC-PR
0.602 40.060
0.619=0.024
0.742+0.0sa
0.840
Point
AUC-ROC
0.8 0.816
0.950 o02
0.851
0.829 40.093
0.878
0.783
0.677 +0.001
Non-causal
AUC-PR
0.730+0.123 0.708=0.015
0.63340.054
0.519+0.026
Seq.
AUC-ROC
+0.066
0.91420.010
0.879
0.794+0.089
0.740
0.756 003
0.747 .25
0.71440.020
Causal Seq.
AUC-PR
0.595 065
0.604 0044
0.581 045
0.659
AUC-ROC
0.853
0.877=0.002
0.860
0.82440.078
Table 7: Performance of Anomaly Detection the Real-Worle MSDS Dataset.
Metrics
TranAD
USAD
F1
0.863
0.888 00097
AUC-PR
10.003
0.001
AUC-ROC
0.981
0.985
We evaluate the performance of USAD for anomaly detection in terms of the Fl score, the area under the precision-recall curve (AUC-PR), and the area under the receiver operating characteristic (AUC-ROC) on two synthetic datasets. Table G shows the evaluation results on synthetic datasets, while Table shows the evaluation results on the real-world MSDS dataset.
Overall, both USAD and TranAD achieve promising performance different types of anomalies and both synthetic and real-world datasets, which lays solid foundation for recourse prediction.
18


Reviews:
[{"review_id": "ZlIcLPY2ch", "reviewer": "Reviewer_P2a8", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses algorithmic recourse in the context of time series data, i.e., identifying minimal changes that would alter the output of a predictive model. Unlike prior work that mainly focused on static data (e.g., images, tabular records), the authors emphasize that time series carry inherent temporal correlations: an intervention at time $t$ not only affects predictions at that time but may also propagate into the future. This motivates the need for specialized methods rather than direct adaptation from other modalities.\n\n# Strengths\n\n- The focus on time series recourse is well-motivated and timely: temporal propagation makes interventions fundamentally different from static data domains.\n- The paper is generally well-written, structured, and easy to follow, with relevant references to prior work.\n- Experimental validation demonstrates both feasibility and practical relevance.\n\n# Weaknesses\n\n- Notation and figures are sometimes confusing or inconsistent (see below). This may reduce accessibility to readers.\n- The sensitivity analysis (Sec. 5.5) is somewhat unclear (see below).", "claims_and_evidence_yn": "Yes", "claims_explanation": "Experiments are convincing, even if I do not understand why the sensitivity analysis is performed on synthetic data alone (see below).", "audience_yn": "Yes", "audience_explanation": "The paper tackles an important problem and clearly formulates how this problem is under-addressed in the current litterature.", "requested_changes": "# Critical\n\n## Improve notation and figures:\n\n* Fig 2b is misleading: if I understood correctly, the $\\star$ notation is used for the counterfactual variables, whereas it seems in Fig 2b that $u$ and $u^\\star$ just correspond to different timestamps. Maybe the figure could be refactored such that on one side of the axis we have the factual variables and on the other side the counterfactual ones (this is just a suggestion but at least the Figure should be made clearer)\n* Also, the notation is not consistent: in Fig 2b and Sec 4.1, $W_t$ is the window that ends in $x_t$ and $V_t$, $u^\\star$ etc start with $x_t$ so there is an overlap between them whereas in Sec 4.2 the past window is denoted $W_{t-1}$ from which $\\theta_t$ is computed (and you stated above that you wanted to optimize $P(u^\\star_t|W_t)$)\n* In Eq. 7, do you actually optimize on $\\|\\theta_t\\|_2$ or do you rather use $\\|\\theta_t\\|_2^2$ that seems better suited for optimization purposes (especially when using gradient based optimization). Note that in (Von Kügelgen et al., 2023) they suggest to use squared Mahalanobis distance.\n\n# Non-critical (but recommended)\n\n## Clarify the sensitivity analysis in Sec. 5.5:\n\nI don't understand the sensitivity analysis presented in Section 5.5: here, the authors study the impact of hyperparameter $\\lambda$ on both the flipping ratio and the \"Action Value\". If I understand well, the \"action value\" is the same as the action cost defined in Sec. 5.1. If so, it seems very difficult to reach an interesting trade-off between flipping ratio and action cost since both metrics seem to be highly correlated. Why isn't the same experiment presented on the real-world MSDS dataset?\n\n## Improve presentation:\n\n* Putting the GVAR loss function inline makes it difficult to read, please make some space for an actual isolated equation\n* USAD acronym is used (Sec 5.1) before being properly introduced (Sec 5.2)", "broader_impact_concerns": "--", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "kLaVrXzHbp", "reviewer": "Reviewer_SqwT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In order to address the lack of methods for algorithmic recourse in multivariate time series, RecAD proposed by the authors achieves effective and cost-efficient reversal of anomalies through framing recourse as a constrained maximum likelihood problem based on backtracking counterfactuals, but its effectiveness is contingent upon a pre-computed and accurate causal graph and is primarily designed for anomalies arising from external rather than structural interventions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Clear methodology: Well-defined problem formulation, the Abduction-Action-Prediction (AAP) process, and counterfactual posterior derivation, ensuring transparency. Causal reasoning integration: Treating anomalies as external interventions and using counterfactual reasoning enhances interpretability and effectiveness. Component validation: Ablation studies confirm the necessity of key components like the retracing function with temporal context and anomaly dynamics. Comprehensive experiments: Results on synthetic and real datasets consistently show RecAD outperforming baselines across multiple anomaly types and metrics. Case studies and sensitivity analysis: Demonstrations in ecological and distributed systems plus hyperparameter studies improve practical understanding. While limitations remain in causal graph dependency and structural intervention handling, the evidence convincingly supports the claims.", "audience_yn": "Yes", "audience_explanation": "Novel Problem Formulation: The paper pioneers the application of algorithmic recourse to multivariate time series, a domain with significant practical importance. The formulation using backtracking counterfactuals is theoretically sound and well-motivated for explaining anomalies as external shocks. Methodological Rigor: The end-to-end framework is elegant, combining a sophisticated recourse function with a causally-grounded procedure (AAP) for modeling downstream effects. This moves beyond naive prediction-based approaches. Comprehensive Evaluation: The experiments are thorough, using a mix of synthetic data for controlled analysis (point vs. sequence, external vs. structural anomalies) and real-world data for demonstrating practical relevance. The inclusion of ablation and sensitivity analyses further strengthens the empirical claims.\n\nIt fills a research gap in multivariate time-series anomaly retracing. It has high practical relevance in fields like healthcare, finance, and industry. The methodology is innovative, combining causal reasoning with counterfactual inference in an end-to-end framework.", "requested_changes": "Strengthen the robustness of causal relationship modeling, particularly by providing a more in-depth discussion or potential improvements regarding the dependency on the predefined causal graph. Clarify and unify the problem formulation for sequence-level attribution actions to ensure better consistency with the algorithm description. Enhance the treatment mechanism for structural interventions by offering more concrete solutions for such anomalies or outlining a more comprehensive future research outlook. Expand the experimental analysis on baseline comparisons and cost functions to improve the completeness and persuasiveness of the experimental results.", "broader_impact_concerns": "Limited Handling of Structural Interventions: The paper's primary assumption is that anomalies are caused by external interventions. While a method for handling structural interventions is proposed by reframing them as an additive anomaly term (Section 4.5.2, page 7), the authors themselves state that its significance \"is worthy of future study\" (page 7). This remains a key limitation for anomalies caused by fundamental system changes (e.g., component failure). Assumption of Stationarity: The use of Granger causality models like GVAR typically assumes the underlying causal relationships are stationary over time. The framework may not perform well in systems with non-stationary or dynamically changing causal structures.", "additional_comments": "Dependency on Causal Discovery: The entire framework's performance is predicated on having an accurate Granger causal graph, which is learned beforehand using GVAR. If the GVAR model fails to capture the true causal dynamics, the recourse actions generated by RecAD could be suboptimal or incorrect. The paper does not analyze the model's sensitivity to errors in the causal graph.", "extra_scores": {}, "extra_text": {}}, {"review_id": "feVZaufZBa", "reviewer": "Reviewer_mPHQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a new approach for algorithmic recourse, specifically for Time Series anomaly detection; which is yet unexplored and has practical application. The authors do a good job of motivating and describing the problem, and in  describing a clear framework. \nThe Granger Causality based approach is straightforward conceptually, and aligns with the general understanding of multivariate time series, along with GVAR that fits well into this framework. However, the assumptions are strong for applying to real world systems that may not be isolated, with additional factors.\nThe backtracking counterfactual argument is sensible in the problem formulation, as a the key motivation is often mitigation. The authors could better explain their take on structural interventions ( or at least clarify delegation to future work in a crisper manner).  The methodology is clearly presented, and technically sound. In the experimental section, however , the baselines seem not sufficiently meaningful. Also, it would be interesting to see how the system would work under presence of complex anomalous patterns such as Change Points, and distribution shifts. Moreover how does RecAD behave with different scoring mechanisms ( including paradigms such as Point Adjustment ) would be interesting to see as well. some of  metrics provided do not tie back to the traditional Algorithmic Recourse (or Algorithmic Recourse for Anomaly Detection). The counterfactuals generated here also do not have a diversity aspect, which is often important.\nThe paper's writing presents a clear narrative, however some doubts on the practical applicability remains.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The manuscript clearly explains the technical contributions.", "audience_yn": "Yes", "audience_explanation": "Algorithmic Recourse is an active research area, and Algorithmic Recourse for anomaly detection is nascent. The work is novel in exploring a new territory.", "requested_changes": "- Revise the section on metrics\n- Address diversity of counterfactuals\n- {optional) how different AD systems affect the performance of RecAD", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)