TimeInf: Time Series Data Contribution via Influence Functions
Authors: Yizi Zhang
Jingyan Shen
Xiaoxue Xiong
Yongchan Kwon
Venue: ICLR 2025 Poster (2025)
Keywords: Time Series Data Contribution
Time Series Anomaly Detection
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: Evaluating the contribution of individual data points to a model's prediction is critical for interpreting model predictions and improving model performance. Existing data contribution methods have been applied to various data types, including tabular data, images, and text; however, their primary focus has been on i.i.d. settings. Despite the pressing need for principled approaches tailored to time series datasets, the problem of estimating data contribution in such settings remains under-explored, possibly due to challenges associated with handling inherent temporal dependencies. This paper introduces TimeInf, a model-agnostic data contribution estimation method for time-series datasets. By leveraging influence scores, TimeInf attributes model predictions to individual time points while preserving temporal structures between the time points. Our empirical results show that TimeInf effectively detects time series anomalies and outperforms existing data attribution techniques as well as state-of-the-art anomaly detection methods. Moreover, TimeInf offers interpretable attributions of data values, allowing us to distinguish diverse anomalous patterns through visualizations. We also showcase a potential application of TimeInf in identifying mislabeled anomalies in the ground truth annotations.

Published as a conference paper at ICLR 2025
TIMEINF: TIME SERIES DATA CONTRIBUTION VIA INFLUENCE FUNCTIONS
Yizi Zhang Jingyan Shen
Xiaoxue Xiong Yongchan Kwon
Columbia University
ABSTRACT
Evaluating the contribution of individual data points to model's prediction is critical for interpreting model predictions and improving model performance. Ex- isting data contribution methods have been applied various data types, including tabular data, images, and text; however, their primary focus has been on i.i.d. settings. Despite the pressing need for principled approaches tailored time series datasets, the problem of estimating data contribution in such settings mains under-explored, possibly due to challenges associated with handling inherent temporal dependencies. This paper introduces TimeInf, model-agnostic data contribution estimation method for time-series datasets. By leveraging influence scores, TimeInf attributes model predictions to individual time points while pre- serving temporal structures between the time points. Our empirical results show that TimeInf effectively detects time series anomalies and outperforms existing data attribution techniques as well as state-of-the-ae anomaly detection methods. Moreover, TimeInf offers interpretable attributions of data values, allowing us distinguish diverse anomalous patterns through visualizations. We also showcase a potential application of TimeInf in identifying mislabeled anomalies in the ground truth annotations.
INTRODUCTION
Understanding how individual data points impact model outcomes is a crucial task, especially in fields like finance and healthcare, where the decision decision-making process depends heavily on model predictions. By evaluating the influence of training data points, we can identify outliers or low-quality samples that deviate from the normal behavior or significantly degrade the model performance. Moreover, the identification of influential data points allows researchers and practitioners focus on meaningful patterns in their datasets. From all these motivations, the attribution of value each data point-diitinguisshnng between beneficial and detrimental patterns-is of critical importance to various applications, including anomaly detection and the recognition of valuable data patterns (Kwon & Zou 2023 Sun et al. 2025).
A standard data attribution approach is the regular influence function, originally developed in robust statistics (Hampell 1974] [Cokk& & Weisberg 1980). It measures the change in model parameters or predictions in response infinitesimal adjustment in the weight of specific data point, effectively capturing the impact of individual data points. However, its underlying assumption of independent and identically distributed (i.i.d.) data limits its direct application to time series datasets, due the temporal dependency within data points. To address this, Kunsch (984) introduces a conditional influence measure for autoregressive (AR) models, which quantifies the impact of single time point given the values of its immediate predecessors. Although this approach extends and validates the use of the regular influence function in time series through some preprocessing, its dependency on past data limits its capability to express data point's 's influence within specific historical contexts and fails to fully capture the influence. Alternatively, Ghosh et al. (2020) employs approximate leave-one-out cross-valddatin to quantify the effect of removing single observed time point model performance while preserving temporal dependencies in the latent space. This method gives a rigorous attribution method for time series data, however, it is strictly limited to specific latent state-space models.
"Equal contribution
Corresponding author (e-mail: yz4123 @oolubbia.cdu; yk301 20000i/coommmacddo

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Published as a conference paper at ICLR 2025
We propose new approach called Timelnf that addresses the challenges of existing time series data attribution methods. Our method integrates the influences over multiple sampled time blocks (Hall
1985 Kunsch 1989 Bühlmann 2002) that include the particular time point of interest. Unlike the
conditional influence of Kunsch 1984), which evaluates a time point's impact hased on its immediate
past, distinctive integration considers various temporal patterns and accurately reflects the true effect of each data point. Moreover, unlike Ghosh et al. (2020), our method does not require specific model structures.
We demonstrate its effectiveness in time series anomaly detection. On multiple real-world datasets, our method outperforms existing approaches detecting harmful anomalies. Timelnf provides intuitive data value attributions, facilitating the identification of harmful anomalies and beneficial temporal patterns through visualizations the computed influence scores. We present a potentially important application of Timelnf in identifying mislabeled anomalies in the ground truth annotations. In Appendix, we demonstrate that Timelnf is model-agnostic and can distinguish between beneficial and detrimental training data for time series forecasting.
PRELIMINARIES
Notations. We denote a set of time series observations by )i where Ty E are drawn from stationary ergodic process (Edition al. 2002). We construct + (m n) overlapping time blocks of consecutive observations x;; == (I Ti-m+ E R" with the empirical m-
dimensional marginal distribution fm defined as
8 [m)) x
where daim) is the point mass distribution at [mi, and From E marginals stationary process.
Mm star where Mm is the mn-dimensional
stat
Influence Functions. One way to introduce influence functions for autoregressive models is through the concept of contaminated distribution function. For Fm E the contaminated distribution function is defined as follows.
MM
stat < 1, and ziml E Rm
F (mlml) (1 F) m + edgim).
Intuitively, PTT (2lmn) is contaminated distribution in which the original distribution Fm is contami- nated by a sample 2mm] with a probability of
Denote 0 E θ C R29 as the time series model parameters, and Ô as estimator. We introduce general definition of the influence function for time series models as follows.
Definition 2.1 (Influence Function for mn-Dimensional Marginal). The influence function of sample 2ml (or equivalently )gmm)) ) on an estimator M + Ro at a distribution Fmm is defined as follows.
8 0(FTT (2lma)) - ((F"))
Ia(8gtm) FT)
lim <00
That is, influence functions measure the infinitesimal effect of upweighting 2mm) by e to evaluated at Fm. We set Ig (6 Fmm (z(mll) when the second argument fr is an empirical distribution fm and simpler form Ia when the context is clear.
Example 1 (M-estimatorss. Equation 1 depends specific choice of Ô. Given that most machine learning estimators are based on optimizing a criterion function, we focus on broad class of these estimators, which is also known M -estimaatoss (Hampel 2001). Suppose loss function
We focus univariate data for notation simplicity, hut the concepts can be readily extended to multivariate settings.
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Table Comparative overview of time series data contribution estimation methods. Data Instance: the specific
instance whose influence on the model we aim to quantify. Quantity of Interest: the model output or characteristic that changes in response changes in the data instance. Contaminating Distribution: the distribution of z!"), introduced perturb the original distribution F Model Choice: the range of model types compatible with the
method.
Method
Dara Instance
Quantity of Interest
Contaminating Dismibution
Model Choice
Regular Inftuence (Kah & Lingg 2012
L.D.
Loss
Difterentiale Model
Conditional Iollueine
Parmetter
Autoregresivve Model
LWCV (Ghooh
2020
Time
Differontable Latent Suae-Space
Timelnt (Ouu))
Time Point
Loss
Mixture 6,omt
Model-Agnostic
p: R θ E is differentiable with respect to 9 and strongly convex. We define M M-estimators as the solution to the following minimization problem
/ ptal"n) ()dd(((!!m))
argmin AE
which can be expressed as the solution to a set of equations
f (xlmm) O)d (x:lml) = 0, (almm),0) 0): dp(alm))
d0
This class includes the maximum likelihood estimator in AR models (Davis et al. 1992 Bai & Wu 1997). In practice, with mild regularity conditions (Giddambe) 1960), the empirical influence function Cook & Weisberg
1980;
Koh & Liang 2017) of 2|m² on 0 at F is given as follows.
( di
Iz(z|ml)= -
d0 df df
((2lll,0)
(3)
See Appendix Section A for derivations of Equation
3 for detailed proof.
2.1 PRIOR WORKS IN DATA ATTRIBUTION
Equation (T) presents general definition of influence functions, encompassing many existing influence function- based methods as special cases. This section briefly summarizes the strengths and limitations of existing works, paving the way for new framework to assess data contribution in time series settings.
Regular Influence. The regular influence function is originally proposed by Hampell (9744) to measure data instance's impact model parameters and later repurposed by Koh & Liang (2017) to assess the impact on test predictions. It is a special case of the general influence function formulation in Equation (1) when m 1. It can be applied any differentiable models, providing rigorous and intuitive explanations for identifying which data points are helpful or harmful. However, its limitation to 1 does not extend general time series cases. For instance, the objective function for an AR model of order requires two consecutive data points, x! (z,, 1;-),, in a loss function, i.e., p(z(") .) (z; x;-10) . This is not feasible when = 1 because loss function is evaluated at individual data points. We generalize this framework by considering any m and construct rigorous influence functions for time series data.
Conditional Influence. There has been an approach to apply the regular influence function to a block of contiguous time series data points. While this seems a naive extension of the regular influence function, it has not been clear until Kunsch (1984) establishes its theoretical support for AR models. Specifically, Kunsch ( (1984) considers a conditional log likelihood (e.g., log p(z+++ T..... ".....,,,__.......... ff-m++1) for loss function / in Equation (2) and defines the conditional influence of ar given its immediate predecessors {æ,,... X4-m+11 by leveraging the regular influence function. Note that this method is also special case of Equation ! when m 1.
II is noteworthy that Equation 2 is general version of the standard M-estimator Hampel
2001 in
sense that each loss function term p(.) (mn () is evaluated multiple data points s.!"m) (xt. (.4...., m+))+ When Lee,, individual loss evaluated at data point, Equation yields the regular M-sstimator.
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Although Kunsch (1984)ss method measures time point's impact model parameters, it constrains the influence to a specific temporal arrangement as assigns point mass to its observed predecessors. In other words, it only considers the local historical context of the time point of interest, and result, it cannot account for the diverse contamination patterns that (#i,... ,, [iim++1) } can exhibit. We address this problem by considering more informative contaminating distribution.
LWCV. The leave-withing within-structurre-e outt cross- valsiattai (LWCV) method, introduced by Ghosh al.] (2020), measures the impact of removing a single time point on model loss without breaking the temporal dependency within the data. In contrast to existing influence function variants, which measure the impact on model parameters, LWCV captures the impact on model loss, providing additional insights into how data are influencing models. However, this approach is specifically designed for state-space models such as autoregressive hidden Markov models (AR-HMM) to maintain temporal dependencies through latent variables. This underlying assumption latent structures restricts LWCV's applicability to models with explicit latent states, excluding more complex models like transformers.
TIMEINF: TIME SERIES DATA CONTRIBUTION VIA INFLUENCE FUNCTIONS
We provide unified framework for time series data contribution that addresses the limitations of the aforementioned approaches. Throughout this section, we focus on differentiable models to ease the presentation of our methods, but the proposed method can be applied to non-differentiable models. For the treatment of non-differentiable models, see Appendix Section B.
The general influence function Iğ defined in Equation (1) captures how data contamination time block impacts model parameters. To understand a specific time block's influence on model predictions instead, we can use the chain rule to derive the influence function for a time block z!mm)
Thleek (2 (m)m), imll - (b(estt [mm]] Freest
0) T) (z(mml)
- +b (2/mm) Ø)T ( (e)(almm. d0
d.Fmm (2)m())
(52lm) 0),
which measures the impact of overweighting 2[m] on model prediction based on test sample [ml] This can be viewed as either straightforward extension of the regular influence function (Koh & Liang, (017)) or that of the conditional influence function (Kunsch) 1984). critical limitation of this approach is its inability to quantify influence scores for individual data points, as it is defined for block of contiguous data points. Moreover, the evaluation the influence function constraints to a specific temporal arrangement 2(ml because only considers point mass to the observed 2(m]
To address these challenges, we propose consider all contiguous time blocks |!ml that contain specific data point and aggregate them into a single value. This idea is formalized into an integration of block influence functions over an empirical contaminating process:
Definition 3.1 (TimeInt). Let S S2.Fm be the set of overlapping time blocks 2/mm] ~ fm E M
containing time point 2. For Gmm
[S_.fml Ex zm 8g(m), we define Timelnf the integration
(ml Istoc ml Freat with respect to ziml, Cm:
Ieime (z, 2les)) ;= /
"bbloc mm)
Zrest
ddGm (2lml)
(5)
(fiiicomm) dpm a dg
-1 (b(2mm) ()
(=(mm 0)T
(6)
TimeInf Timee (z, Prest aggregates block influence functions Tolack over various contaminating blocks
[m]
defined in S $², fmmm We find this integration be critically important as multiple temporal arrangements
considered in S_ lead to more robust influence measure that less dependent on any single block
Derivaations of Equation
are included in Appendix Section A.
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dmmm unlike Kunsch (1984) 's method. This advantage becomes clearer in an alternative expression
of TimeInf. A simple algebra gives:
E plaast 0( ((Frm()lm)))) -p00pp(at p(=nout A ((Fm))
Timee (2; Frest
lim ~0
(7)
where p is the loss function and the expectation is taken over Gm This implies that Timelnf can be interpreted the limit of expected infinitesimal change when a perturbation Ó_(=) is introduced randomly with contaminating distribution ÕTM. Compared existing influence function variations, the expectation in Equation (7) effeetively captures the real impact of a per-time data point, avoiding the randomness of particular block.
In summary, Timelnf is designed for time series data contribution and improves upon existing methods in the following ways:
Unlike the regular influence function, Timelnf can capture per-time-poin influence on specific model predictions while preserving the dependency structure between time points.
In contrast the conditional influence, TimeInf extends beyond the observed local historical context of the time point of interest.
Unlike LWCV, TimeInf is model-agnostic, extending its applicability to complex models without explicit latent states.
3.1 VARIANTS OF TIMEINF
Depending the task, different TimeInf variants can be used for data contribution estimation. When the goal is to estimate data quality, particularly anomaly detection, self-influence provides a means quantify the impact of specific time point its own prediction:
[zar(z)
Intock Bnock(2lmm)) (z!ml, g!mm))
| pm g(m) _S, p,
The key idea is that anomalies, being rare and distinct, hard predict using normal data but easier using anomaly data itself. Thus, anomaly points are expected to be most influential for their own predictions.
For time series forecasting, to understand how changing time point affects test predictions, we can average Time from Equation (6) across all test set time blocks:
(z)
Σ
Zume (z, Ziest )
S frmm
Fres. E.0
where See,, Fm. is the set of time blocks in the test set.
Example 2 (Timelnf for AR(m) model). Although any differentiable models can be used compute TimeInf, our primary focus this paper is on linear AR models due to their powerful performance
and computational efficiency (Zeng et al. 2023 Toner & Darlow 2024). For linear AR models of
[m++]]
x [m] 0)2. the Meesimattor
order m and the mean squared error (MSE) loss p(x,+1
(21+1
i corresponds the least squares estimator. In this case, Ij in Equation have dlosed-form solutions. Specifically, the influence overweighting on the model parameters given by:
3) and Totock in Equation (
[m]
-1
Ig(2
[mll
11
[m]] [mm]] x xi
mm) (2L41 - [mm]] 0).
n
1=1
The influence of overweighting 2 [m]]
the test prediction is:
-1
[2lock [m]] - = (al)))=((==ntttttta -Ztest,
0)
[im]
[m]] [m]]
2 Jml
[mm]] 0). (21+1
Frest.
5
1=1
which can be plugged into Equation
5 obtain Timelnf.

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3.2 IMPLEMENTATION OF TIMEINF
The challenge in computing Timelnf lies in the inversion of the Hessian in Equation expressed as I de(all)) d.Fmm (z.lm)). With n training points and 0 E R2, this involves ( + q²)) operations.
d0
While it is feasible to directly compute the inverse of the Hessian for linear AR models with a small q. for deep neural networks with millions of parameters, this computation becomes prohibitively expensive. To approximate this computation efficiently, we resort to the conjugate gradient (CG) method, as suggested by Koh & Liang (0117) and the Hessian-free free approach introduced by Pruthi et al. (2020). The CG method converts the matrix inversion into optimization problem solvable in (ng) time. The Hessian-free approach replaces the inverse Hessian matrix with an identity matrix, reducing the main computation bottleneck of the influence function to only the dol product of gradients. Once equipped with the influence of a time block (Ioloc)), we can compute TimeInf' (Tume), self-influence (Ieet), and the influence of a time point on the test prediction (Zusa), according to Equations 6). and
4 USE CASES OF TIMEINF
This section investigates the practical effectiveness of TimeInf real-world anomaly detection tasks. Our method outperforms existing time series data attribution methods as well state-of-the-at anomaly detection methods. In addition, we illustrate how Timelnf facilitates the detection of various anomaly patterns through interpretable visualizations. Furthermore, we highlight the capability of Timelnf to identify mislabeled anomalies within ground truth annotations. We include additional experiments in Appendices C and D to demonstrate the model-agnostic nature of TimeInf, and how different hyperparameters (e.g., block length and model size) affect the downstream task performance.
4.1 TIME SERIES ANOMALY DETECTION
When time series model is trained with outliers or anomalous patterns, the quality of its predictions is expected to be negatively affected by these samples. In such situations, evaluating the impact of individual data points can be potentially useful in detecting anomalies. Motivated by this, we conduct time series anomaly detection experiments and assess the detection ability of Timelnf.
Experimental Settings. We consider realistic and practical experimental settings where the training dataset is not necessarily anomaly-free, and the proportion of anomalies is unknown (Jiang et al., 2022, [Schmidl et al., 2022). We use the contaminated test set as training data for all methods and evaluate their performance on the same dataset. It is noteworthy that our setting differs from conventional time series anomaly detection settings, which often focus on detecting anomalies in a contaminated test dataset after training models
well-curated clean training dataset (Alnegheimish (et al. 2023 Si et al.) 2024). While we recognize that each setting has its advantages, datasets with anomalous data points have been considered to investigate the detection ability of data attribution methods. That is, we want to know which training data points are anomalies and attempt to detect them using data contribution methods. We provide implementation details in Appendix Section F.
We extensively evaluate TimeInf on five benchmark datasets: UCR (Wu & Keogh, 2021] SMAP (Hundman al., 2018). MSL, NAB-Traffic and SMD (Su et al. 2019). Detailed descriptions of these datasets and results on additional datasets are provided in Appendix Section F.
Datasets.
TimeInf Usage. We use self-influences Zueir for anomaly detection. For univariate time series, we compute the self-influence directly for each time point, normalizing the results to a range between 0 and 1 obtain anomaly scores. In the case of multivariate time series, we first apply self-influence to each dimension separately and then average the anomaly scores across all dimensions to obtain the final anomaly scores for individual time points. Although we can apply self-influences to all dimensions of the multivariate time series and obtain one set of anomaly scores, empirical results suggest that it is less effective than taking the average of anomaly scores obtained separately from each dimension. We provide detailed discussion in Appendix Section G.
Baselines. We compare TimeInf to three time series data contribution methods: block leave-one out- cross-validation (LOOCV), conditional influence (Kunsch) 1984), and LWCV (Ghosh et al. 2020). Conditional influence and LWCV are described in Section 2.1 We have proposed Block LOOCV as straightforward approach for calculating time point's influence. This method removes all time

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Table 2: Anomaly detection performance and time cost of Timelnf and baseline methods, For both and AUC
metrics, a higher value indicates a better performance. Timelnf outperforms sfate-of state-of-the-art time series anomaly detection methods (shown in white) and other data contribution techniques (shown in grey) across five real-world datasets, while being computationally efficient. Methods marked by failed to complete experiments on certain datasets due to computational inefficiency.
UCR
SMAP
MSL
NAR-Trffic
SMD
AUC
Time (5)
AUC
Fi
(5)
Time
AUC
F1
(S]
(S)
Isolation Forest
0.57
0.02
0.57
0.67
0.33
0.14
0.68
0.31
0.11
0.57
0.20
0.22
0.82
0.37
0.30
LSTM
0.60
0.04
803.27
0.55
0.31
245.53
0.70
0.34
8.77
0.57
0.19
49.13
0.79
0.30
307.08
ARIMA/VA /AAR
0.54
0.10
2596.32 32
0.58
0.26
20.43
0.82
0.24
6.85
0.55
0.20
17.52
0.71
0.16
6.38
TraAD
0,50
0.22
2,87
1.55
0.21
02.44
0.40
0.06
0.24
0.60
0.36
1.44
0.65
0.05
3.54
DCdetoctor
0.74
0.34
151 08
0.67
0.35
194.52
0.69
0.34
58:13
0.61
0.32
27.96
0.70
0.25
106 70
Ancuraly Tranformerr
0.04
20.13
0.05
246
Block LOOCV
30.93
0.62
Conditional Inffuence
0.65
9
0.70
0.37
7.78
4.85
0.61
0.29
6.50
1WCV
0.55
1540.47
0.63
0.36
0.32
109.00 84
0.59
0.37
67.10
0.79
Timelaf (Ours)
0.79
0.23
23.61
0.73
0.34
10.59
0.72
0.35
6.17
0.64
0.39
2.79
0.87
0.25
20.85
blocks containing the time point of interest, and then measures the impact of this removal on model losses; see Appendix Section F for implementation details about each baseline.
We additionally compare TimeInf to four commonly used anomaly detectors, covering classic approaches like Isolation Forest (Liu al.) 2008), Long Short-Term Memory (LSTM)-based detector (Hundman et al., 20118) AR integrated moving average model (ARIMA) and vector AR model (VAR), and state-of of-te-t transformer- based methods including Anomaly Transformer (Xu
et al. 202ID. TranAD (Tuli et al. (022) and DCdetector (Yang et al., (023). For AR-based models, we use ARIMA for univariate time series, and VAR for multivariate time series.
Forest and LWCV directly to score each time point. Since the other methods assume an AR model, we use time blocks of length 100, as empirical results show optimal model performance at this length with minimal gains beyond; see Appendix Section D. TimeInf, block LOOCV, and conditional influence provide per-time-point scores directly. For LSTM, ARIMA/VAR, and Anomaly Transformer, average scores across time blocks containing the target point to obtain its anomaly score.
Evaluation. We apply Isolation
Following the practice (Jiang et al., 2023), we apply the k-Means clustering algorithm to the calculated anomaly scores in our method, using a cluster of two to partition the scores into clusters of normal and anomaly time points. For other baselines, we consider two sets of anomaly predictions: (i) the k-Means clustering-base
method used for TimeInf and (ii) the top-k anomaly score-based method, where k is the true number of anomalies. When computing performance metrics, we select the highest score between these two classification approaches. In practice, the exact number of anomalies is unknown, making this scoring procedure favorable to the other baselines. Evaluation metrics include Fl and AUC for detecting accuracy, along with the computation time cost for each method. We calculate the AUC by comparing the ground truth anomaly labels (0 for normal and for anomalous data) with the anomaly scores produced by each baseline.
Main Results. As shown in Table 2 our method outperforms the other baselines in the UCR, SMAP, MSL, and NAB-Traffic datasets for anomaly detection. On the SMD dataset, our method achieves higher AUC than other methods. Notably, TimeInf outperforms other methods significantly on the UCR dataset, achieving a 0.79 AUC compared the sub-0.65 AUC of its competitors (a 20% improvemen).. Compared to sfate-of-the-ar transformer- TranAD by 7-30% and DCdetector by 3-17% in AUC across the datasets. Although we maintain consistent block length of 100 across all datasets for a fair comparison, using a block length optimized for each dataset could potentially yield even better results, as shown by the ablation study in the Appendix (Fi Additionally, TimeInf requires substantially less computation time than most baseline methods except Isolation Forest and conditional influence, enabling efficient data contribution assessment for large datasets.
approaches, Timelnf outperforms
4.2 INTERPRETABILITY OF TIMEINF
We investigate reasons behind the performance of TimeInf with a qualitative analysis using the UCR dataset (Wu & Keogh (02T). Figure compares the anomaly scores of TimeInf with other baselines across various types of time series anomalies, namely point anomaly, noisy data, local contextual,
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and global contextual anomalies. Our method better captures challenging local and global contextual anomalies, whereas other baselines tend to identify only large deviations in data values, struggling to distinguish abrupt changes in local or global context. For point anomalies and noisy data, while other baselines can detect them, our method produces very low anomaly scores for the normal points, effectively distinguishing them from anomalies.
Inf outperforms Isolation Forest due to its consideration of temporal structure, allowing it to better handle contextual anomaly patterns. Unlike deep learning-baset approaches such LSTM and Anomaly Transformer, our method does not rely on training on an anomaly-free subset before being applied to detect anomalies the contaminated subset. Due to this difference, TimeInf generally performs better than other prediction-base methods in identifying anomalies within the training dataset.
Among time series data contribution methods, Block LOOCV is strong competitor to Timelnf. It directly calculates the effect of removing all time blocks containing the time point of interest by retraining the model following each data removal. However, Block LOOCV has several limitations. Calculating leave-one-out values requires computationally expensive model retraining for each time point, as influence functions cannot be used to approximate this process. It is also less sample-effiient and potentially less accurate. For instance, with 1000 training time blocks and a window size of 100, removing one time point eliminates 10% of the training data. We observe that TimeInf outperforms Block LOOCV in anomaly detection, as Block LOOCV's removal of large portions of training data compromises parameter estimation accuracy. Figure I of TimeInf and Block LOOCV, as both methods aim to capture similar quantities. However, IimeInf more efficient by approximating the inefficient model retraining procedure used in Block LOOCV,
reveals comparable patterns in the data values
TimeInf surpasses conditional influence in anomaly detection by considering multiple temporal arrangements from various time blocks containing the target point, rather than focusing on a single specific arrangement of preceding points. This approach enables TimeInf to identify anomalies across diverse contexts. TimeInf outperforms LWCV, likely because LWCV is constrained to latent state-space models, specifically an HMM with Gaussian emissions. This imposes a restrictive Markov assumption, considering only the immediate past. In contrast, TimeInf employs an AR model of order 100, allowing it to account for a much longer historical context. Figure [ illustrates the superiority of TimeInf over conditional influence and LWCV, TimeInf produces less noisy data values by considering a broader range of temporal contexts in which the target time point may appear.
We have added more visualizations in Figures 10 and IT in the Appendix to further compare the performance of different anomaly detectors TimeInf. Morcover, we have explored other datasets, which can be found in Appendix Section F.
4.3 IDENTIFYING MISLABELED ANNOTATIONS IN ANOMALY DETECTION
Recent research by Wu & Keogh (02T) has exposed significant flaws in many anomaly detection datasets, casting doubt the reliability of numerous published comparisons of anomaly detection algorithms. These benchmark datasets frequently contain mislabeled ground truth annotations. For example, Figures 2 and 3 illustrate two types of problematic annotations: segments with distinet anomalous patterns that lack anomaly labels, and segments labeled as anomalies that appear normal. Additional instances of mislabeling can be found in Figure in the Appendix, as well as in Wu & Keogh 22021).
While our method excels in many real-world datasets, Appendix Table 4 shows that TimeInf is only comparable or slightly poor compared to Isolation Forest on problematic datasets such as PSM (Abdulaal al. 2021). NAB-Tweets (Ahmad et al. 2017). SWaT (Mathur & Tippenhauer 2016). and KDD-Cup99 (Stolfo et al. 2000). However, we believe that this poor performance is due to the low-quality problem of datasets, rather than the methods themselves. In fact, TimeInf provides highly intuitive scores in identifying real anomalies, rather than adhering to potentially mislabeled annotations, in many qualitative analyses. For instance, in Figure Segment 3, TimeInf assigns higher anomaly scores to time points within segments not officially marked as anomalies, raising questions about the reliability of these labels. Figures 2 and 3 provide more examples of problematic ground truth annotations, showing how TimeInf can expose potential mislabeling in data, while Isolation Forest adheres to mislabeled annotations. This highlights a novel application of TimeInf to debugging and improving the quality of anomaly detection datasets.
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Obsered
Contantual
Anooall
Noday Data
2080
Data Contribution
NIC 100T1:301
AUE 3413
AUC: 1A0
ABC 1.0 677
Triakf
(Oung
Blak
LOCOV
Cundioonal Infuence
MIC 4571-115
AUD 0721320.00
AUC: 03477:228
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Figure 1: Qualitative example of Timeluf and baseline methods in the UCR dataset. The first row displays the observed time series, while the other rows show the anomaly scores generated by each method. For better visualizations, we normalize the anomaly scores from each method to range between 0 and The ground truth anomaly intervals are marked in red segments. TimeInf outperforms state-of-the of-the-arr art time series anomaly detection methods and other data contribution techniques in identifying diverse anomaly patterns in terms of AUC and F1, and it also provides interpretable attributions.
5 CONCLUDING REMARKS
We propose TimeInf, efficient method for measuring the contribution of time series data model predictions. This approach serves as a unified framework, addressing limitations of existing time series data contribution methods. Our experiments demonstrate that TimeInf effectively identifies harmful time points through anomaly detection and provides intuitive data values, enabling easier distinction of diverse anomaly patterns. Moreover, Timelnf can expose mislabeled ground truth annotations in anomaly detection datasets, potentially improving training data quality. Its applications extend beyond anomaly detection; in Appendix Section E, we illustrate Timelnf's ability identify helpful time points for time series forecasting through a data pruming experiment. Our research also shows that TimeInf is model-agnostic and robust against hyperparameter changes, as detailed in Appendix Sections C and D.
Limitations. Although TimeInf can used to measure the influence of a data point on various metrics, this work focuses the validation loss function. This choice highlights TimeInf's capability in time series anomaly detection and its potential applications (e.g., identifying low-quality datasets in Section 4.3). We believe there many promising applications for influence functions-ddtecciio detecting the most representative patterns under distribution shifts- but we leave this as an intriguing direction for future work. Another important challenge is that many data attribution methods, including TimeInf, often show unsatisfactory downstream task performance, especially when data are high- dimensional. We find that Timelnf's performance gains over other data contribution methods are marginal for nigh-dimensional data, which poses the question of whether this is due to low-quality data or methodological differences. We believe requires a thorough investigation using high-quality multivariate time series datasets with reliable annotations.

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Treets Sogment
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ا اللل باا اللا
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ation Forest AUC: 0.45
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Islation Forss AUG, 0:67
A qualitative example of mislabeled anomaly points in the NAB-Tweets dataset. Each panel displays segment from the observed data the first row, with anomaly scores from Timelnf and Isolation Forest in the following rows, Anomaly scores are normalized a 0-1 range for clarity, The ground truth anomaly is marked in red. In NAB-Tweets segments and 2. a large anomalous section is incorrectly labeled; most of it is normal, with only a small anomalous segment. Segment 3 shows a clear anomaly between time points 5000 and 6000 but lacks an anomaly label. Isolation Forest adheres to mislabeled annotations and obtains a high AUC, while TimeInf accurately identifies the anomalies but receives low AUC.
Figure 2:
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Figure 3: A qualitative example of mislabeled anomaly points in the PSM dataset. The first three rows display selected dimensions of the time series, while the remaining rows show normalized anomaly scores from TimeInf and Isolation Forest (range 1). The ground truth anomaly is marked red. Although a point anomaly labeled near time point 600, it appears normal, while the preceding segment seems more suspicious. TimeInf does not flag the labeled point as anomalous. Although TimeInf and Isolation Forest have similar anomaly scores near the labeled anomaly, Isolation Forest obtains higher AUC, highlighting the problematic annotations.
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ACKNOWLEDGEMENTS
We acknowledge computing resources from Columbia University's Shared Research Computing Facility project, which is supported by NIH Research Facility Improvement Grant IG20RR030893- 01, and associated funds from the New York State Empire State Development, Division of Science Technology and Innovation (NYSTAR) Contract C090171, both awarded April 15, 2010.
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APPENDIX
A. DERIVING INFLUENCE FUNCTIONS FOR M-ESTIMATORS
Let (X.) denote the random process. Denote Fr the distribution of X!!" (X....... X An
M-estimator ê for model parameters the solution to a minimization problem over the data:
i = argmin Elp( 0)]
AEe
We consider class of M-estimators for which p is differentiable and convex (Planiden & Wang 2014 Bertsekas et al. 2003), enabling us to express the estimate as the solution to set of equations:
[m] (xx!m, 0) (p(x!ml..) 0)
EL X [m]] 0)] O,
d0
Influence functions compute the infinitesimal change in the estimator if the original distribution 1r is contaminated by =!m] from Gm 811m) by some small amount, giving us a contaminated
estimator & = argmin E(p( (X. !") 2mml, 0.)]. For the contaminated distribution F 0 must satisfy:
8.00
[ (x!!) 0.) (m)( 0.)1 (1 - c) (xx 0.)] + (2lm ( = 0.
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We differentiate determine the effect of changing A: mm)
d de
d <F(ab(2!!)) ammm,00 0..3]
(1 - [Xx [c)l
de
[m] 0.)1 + (1 e) dx. [im) .b.). d0, Bl 0.)] =IEi dr de |,llll,, 0.) de
d0. de
(X
d0
d0
o. thus the influence function for model Around O, the estimator must satisfy ( !ml 00) parameters in Equation 33 is
dô -
- 0)
dyp (X.
( 0)].
de c=0
Applying the chain rule, the effect of changing the loss evaluated at test point Zest in Equation is then
dp(zrest, Oc) de
dp(zrest, 0) dd=
dxp
0) 111[(//) 0)]
= -1- -b D)
c== d0 de l==
d0
Algorithm 1 Timelnf Estimator
Require: Time point of interest training time blocks {x]mm] Irel ..... _ }. test time block =|sal index of time blocks that contain the time point of interest Sg, model parameters 0, loss function 4, and block length m.
Ensure: Timelnf (z,e (mm))
(Optional if is ready) Learn the parameter by training a model on {0] ..... [mi }.
1. Compute the gradients at the training time block va(a!"" [m)] 0) oe _a 0) for all i
{1...., n and the Hessian Hg.
2. Compute the gradient at the test time block (= 0) = d ((=lenl for i E S2 do
Iime (2, 210ss
1.
(22(21 0) H. "b,""" (m] 0) + Izime (z, 24m2 )).
=
18a
end for
B. BLACK-BOX MODELS AND NON-DIFERENTIABLE LOSSES
Although we focus on linear AR models in the main paper due to their computational efficiency, the computation of TimeInf is model-agnostic, allowing to explain any models of interest under our proposed framework. For "black-box" models with differentiable losses, like RNN and LSTM,
we compute the Hessian E( dv(x." m") m)))))
())
in Equation (4) using automatic differentiation packages. The primary challenge lies in computing the inverse cessiann-vecton products, and various solutions proposed by Koh & Liang (2017), Grosse al. (2023), Kwon et al. (2023). Kim et al. (2023). Pruthi ctt al. (2020), and others have scaled this computation to deep neural networks. For linear AR models, we simply compute the inverse of the Hessian. However, for black-box models, this computation becomes intractable. Therefore, we employ the CG method in Koh & Liang (017) and TracIn (Pruthi ct al. 2020) to approximate the inverse of the Hessian.
da
For non-differentiable models such as tree-based models and ensemble models, we compute non- parametric influence functions (Feldman & Zhang 2020). The idea is to calculate Leave-One- Out (LOO) data value by excluding a data point from the training set, retraining the model, and evaluating the impact of this removal prediction performance. Given its computational costs, Feldman & Zhang (2000) employs data subsampling to approximate LOO. The nonparametric counterpart of Tolock Equation is:
Inomparametnie (2(m) ((m)) = [x. [p(
[mm] (s.))
[m]] (s.)) 2!!l] & Sy). | [m)) E St] - [xx-(K][[((e))
where P denotes the loss function, Å represents the learned parameters of the model under consider- ation. The notation 0s, is used for the parameters of the model fitted subset St. The process
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Figure 4: The performance and computation time of TimeInf across different model choices on the MSL and SMAP datasets. Panels (A) and (B) show the mean AUC and mean runtime in seconds for each model choice. Linear AR models provide the fastest computation for Timelnf. with negligible time costs not visible in panel (B) due to their small computation time. Choosing more complex model improves performance TimeInf' but at the expense of increased computation time.
involves K subsets denoted as Sk. which are randomly sampled from the training data of size n with smaller sample size sabset < n. Subsequently, the models of interest are trained separately on each sampled subset. Once equipped with mumparametrie,, we can compute the influence of a time point (Tume), self-influence (Ieer), and the influence of a time point on the test prediction (Iwea), according to Equations 6 and
C. MODEL AGNOSTICITY
Although we used linear AR models to compute TimeInf for anomaly detection, TimeInf is model- agnostic; see details in Appendix Section B. To understand the variation performance and compu- tation time across different models, we compare TimeInf computed using linear AR models against an RNN (1,153 params), L.STM (4,513 params), transformer-base. PatchTST (Nie al. 2022)
(299.073 params), and a gradient boosting regressor with 100 trees of max depth 3. All models use MSE as the loss function. For RNN and LSTM, we apply the CG used in Koh & Liang (0017) to compute TimeInf for differentiable black-box models. For the large nonlincar PatchTST model, we use the scalable "Hessian-feee free" (Pruthi et al. 2020) method to compute Timelnf. For gradient boosting, we compute nonparametric Timelnf according to Feldman & Zhang (2020); see details in Appendix Section B. To ensure fair comparisons, all models are thoroughly trained and perform well on held-out validation sets.
We evaluate Timelnf computed with different model choices on the SMAP and MSL. datasets, as shown in Figure Panel (A) shows that the performance of TimeInf in anomaly detection improves with the transition from a linear AR model to nonlinear models such as LSTM, RNN and PatchTST. In Panel (B), there is a slight increase in computation time when nonlinear model is used. The computation of nonparametric TimeInf is time-consuming due to the need for data subsampling and model traiiinn
In summary, using higher dimensional or more complex models improves TimeInf performance but increases computation time. The choice between a complex model for better performance or simple linear AR for efficiency depends on the user's needs. Moreover, these findings suggest that the results in Table could be further improved using flexible deep neural network models. We highlight that the computation time for TimeInf is small, even for moderately sized transformers (300K parameters). Therefore, TimeInf can potentially scale to large over-parameterized models, enabling interpretability of time series foundation models. See the ablation study in Appendix Section (Figure for detailed performance comparisons across model sizes.
D. ABLATION STUDY
This section explores how hyperparameters of TimeInf - block length used for constructing the empirical contaminating distribution (m in Equation 6). and the model size affect the downstream task performance through an ablation study.
Experiments on Different Block Lengths. The performance of Timelnf in anomaly detection can be affected by the block length, which determines the order of the AR model used in its computation.
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The performance and computation time of TomeInf and baseline methods across different block lengths. Panels (A) and (B) show the mean AUC and mean runtime in seconds at the block lengths of 25, 50, 100, and 150. Compared to other methods, TimeInf achieves a higher AUC while requiring significantly lower computation time across various datasets.
Figure 5:
Table 3: The performance of Timelnf across different time strides. Ablation studies the SMAP and SMD datasets show that the anomaly detection AUC decreases as the time stride increases.
Time Stride
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Figure 5 shows how AUC and computation time vary across block lengths 25, 50, 100, and 150 for each anomaly detector. In Figure (A), we observe that the AUC Timelnf improves when block length becomes larger across most datasets, as larger block length enables Timelnf to better capture contextual anomalies that persist for an extended duration. Panel (B) indicates an increase in computation time when block length increases for most methods. This is because extended block length increases the dimension of the model parameters, and thus increases the time needed for computing the inverse Hessian. In summary. TimeInf consistently outperforms other anomaly detectors across wide range of block lengths, while requiring significantly less time.
Experiments on Time Stride. The time stride, which determines the number of time blocks time point appears in, affects the anomaly detection performance of Timelnf. Table ssows that as the time stride increases, the AUC decreases, with the threshold varying across datasets. Larger strides provide fewer time blocks containing the point, limiting the sampling of temporal configurations around Consequently, with fewer samples, TimeInf estimate becomes less reliable, which is why we choose stride in our experiments.
Experiments on Model Size. Experiments are conducted to examine the effect of model size on Timelnf's performance. The anomaly detection results the MSL and SMAP datasets using TimeInf computed with models of different sizes are compared. As depicted in Figure panel (A), the anomaly detection performance of TimeInf improves with increasing model size the MSL dataset, with the cansformer-basee PatchTST model scoring the highest. For the SMAP dataset, although PatchTST performs worse, the nonlinear RNN model achieves the best performance. Despite the computation time increasing with model size, PST with over 300 thousand parameters remains fast, enabling scaling to larger deep learning models.
E. DATA PRUNING
Identifying the most influential time patterns in the training dataset can improve forecasting perfor- mance by prioritizing patterns that contribute significantly to accurate predictions. In this section,
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Figure 6: The performance and computation time of Timelnf across different model sizes on the SMAP and MSL datasets. Panels (A) and (B) show the mean AUC and mean runtime in seconds for each model size. Linear AR models are the smallest and provide the fastest TimeInf computation, with negligible time costs not visible panel (B) due their small computation time. Choosing larger model improves performance TimeInf but at the expense of increased computation time.
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Figure 7: Data pruning results for Timelnf and other data contribution methods. We estimate the contribution of each time block the training data using various baselines. For each method, we iteratively remove the most influential time block from the training set and track the resulting performance degradation, measured by I22 and RMSE. The better the data contribution method, the lower the I2 curve and the higher the RMSE curve. TimeInf shows steeper decrease in R and steeper increase in RMSE than state-of-the of-e----rrrtttt data contribution methods, demonstrating superior performance in identifying helpful data patterns for time series forecasting across four real- wellwworld datasets.
demonstrate TimeInf's ability to identify helpful time patterns for forecasting through a data pruning experiment.
Experimental Settings. For each dataset, we conduct 100 independent experiments, each time randomly sampling 5,000 consecutive data points. These points are then sequentially partitioned into a training set, a validation set, and a test set, with fixed sizes of 3000, 1000, and 1000 points, respectively.
Contrary to the point removal benchmark commonly used data valuations (Jiang al. 2023), our experiment employs block-wise data pruning approach, which is more in line with common practices in time series forecasting. We divide the time series into equally sized blocks. Fach block within the training dataset is assigned value indicating its contribution to future time series forecasting. Subsequently, the blocks are removed in descending order of the values. After each block is removed, train linear model the remaining dataset and assess its performance
held-out test set.
Datasets. We consider four datasets commonly used in time series forecasting literature (Lai et al. 2018): Traffic, Solar Energy, Electricity (Trindade) 2015), and Exchange Rate (Lai et al. 2018). The Traffic da aggregates hourly road occupancy rates over 48 months from 862 sensors across California. The Solar Energy d comprises records from 137 photovoltaic power plants in
dot www.ddtt..c..ooooo
https: hwtp://wwwwwttm prel http://w......../oooooooeeeeeeee
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Figure 8: Quantitative results for Timelnf and other data contribution methods from a data pruning experiment across four real-world datasets. We estimate the contribution of each time block in the training data using various baselines. For each method, we iteratively remove the least helpful time block from the training set and track the resulting performance degradation, measured by R², aiming for less rapid decrease even an increase in performance.
from
Alabama State. The Electricity dataset includes the electricity consumption patterns of 321 clients from 2012 to 2014 (Trindade 2013). The Exchange Rate dataset includes daily exchange rates for countries, covering the period from 1990 to 2016 (Lai et al. 2018).
Baselines. For each block in the training dataset, we assess its impact by averaging the score Zalock from Equation 4 across all blocks in the validation set. We fix the block length 100. For anomaly detection, standard methods are available, but for data pruning, no established method exists. Therefore, employ the state-of- the-art data valuation methods as baselines: data valuation methods, namely KNN Shapley (Jia et al. 2019), and out-of-bag estimates of data contribution, such Data-OOB (K won & Zou 2023). We exclude Data Shapley (Ghorbani & Zou 2019) from consideration due to its computational inefficiency, making it difficult to apply to our datasets.
saalley-valuebaate
For fair comparison, we consistently employ a linear AR model across all baseline data valuation methods. We assign a data value to each time block in the training dataset and iteratively remove time blocks, starting from the most helpful to the least helpful for forecasting on the validation set. After each removal, we train a linear AR model on remaining dataset and assess the model predictions on the held-out test set.
Evaluation. We use R2 and root mean squared error (RMSE) as evaluation metrics for model predictions. A larger R² and smaller RMSE correspond to better prediction performance.
Main Results. Figure 7 iilluttrate the effectiveness of Timelnf in identifying the most valuable patterns for forecasting in four real-world datasets. We found that predictive performance declines more rapidly with TimeInf, suggesting its superior efficacy compared to other baseline methods. For instance, in the Traffic dataset, we observe a more than 10% drop in R² after removing 30% of the blocks, whereas other methods result in less than 2% decrease. Figure SoureePlllsse illustrates qualitative example from the Electricity, which demonstrates how TimeInf successfully identifies influential periodic patterns for test forecasting.
Additionally. TimeInf compared to baseline methods in identifying the least helpful (or most harmful) patterns for forecasting. Time blocks are iteratively removed from least helpful to most helpful based on data values assigned by each method. After each removal, a linear AR model is trained and evaluated. TimeInf can correctly identify the least helpful temporal patterns in most datasets except the Solar dataset, as shown in Figure g
F. ANOMALY DETECTION EXPERIMENT DETAILS
Dataset Details. In the main paper, we extensively evaluate TimeInf on five benchmark datasets.
The univariate datasets include UCR (Wu & Keogh| 202 featuring 250 time series from diverse fields, and -raffffffic which contains real-time traffic data from the Twin Cities Metro area. For multivariate datasets, MSL (Mars Science Laboratory) and SMAP (Soil Moisture Active Passive satellite) are NASA datasets (Hundman et al., 2018) monitoring spacecraft telemetry anomalies. SMD (Server Machine Dataset) is dataset from Internet company (Su et al. 2019).
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Time
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Timettt 37.70
Timalit 12,82
Timsant 11.57
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Timaent: 11
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Figure 9: Qualitative results of the data pruming experiment for Timeluf on the Electricity dataset. The first row shows the raw test time series, with subsequent rows depicting the most influential (green) and detrimental (red) time blocks in the training data linked to the example test block (black) based on Timelnf. Timelaf effectively captures the periodic pattern in the time series, offering valuable insights for forecasting.
Table 4: Quantitative results for Timelnf and other anomaly detectors across seven real-worla datasets. For both Fl and AUC metries, higher value indicates better performance. TimeInf faces challenges in these datascts due issues with the reliability of ground truth anomaly labels; see Appendix Section F for details.
Twe-Tweets
NAB-Taxi
PSM
SWaT
WADI
COD-Cup59
NAB- Exchange
AUC
F1
AUC
F1
AUC
F1
AUC
F1
ALC
FI
AUC
F1
AUC
Fl
Isolation Forest
0.55
0.18
0.64
0.13
0.71
0.51
0.87
0.69
0.74
0.34
0.74
0.41
0.50
0.11
LSTM
0.55
0.17
0.33
0.04
0.62
0.38
0.30
0.16
0.49
0.08
0.98
0.88
0.56
0.12
ARIMA/ VAR
0.56
0.17
0.49
0.05
0.55
0.33
0.16
0.11
0.11
0.01
0.87
0.52
0.51
0.14
Anomaly Transformer
0.50
0.08
0.51
0.06
0.48
0.15
0.00
0.22
0.50
0.06
0.54
0.17
0.44
0.07
Timelnf (Ours)
0.52
0.16
0.52
0.02
0.63
0.02
0.61
0.08
0.63
0.14
0.79
0.43
0.54
0.34
In Table we include additional datasets: (i) NAB-Tweets (Ahmad et al. 2017) comprises Twitter mentions of publicly-traded companies such as Google and IBM. (ii) NAB-Taxi (Ahmad et al. 2017) records NYC taxi passenger numbers with anomalies during events like the NYC marathon and holidays. (iii) The Pooled Server Metrics (PSM) (Abdulaal et al. 2021 dataset from eBay captures anomalies in application server nodes. (iv) The Secure Water Treatment (SWaT) (Mathur & Tippenhauer 2016) dataset features data collected from sewage water treatment facility, where the anomalies are caused due to cyberattacks. (v) The Water Distribution (WADI) (Ahmed et al.)
2017) dataset is a distribution system consisting of large number of water distribution pipelines. (vi) KDD-Cup99 (Stolfo et al., 2000) contains network connection data for intrusion detection in a collects online advertisement clicking rates. Details of all datasets are summarized in Table 5 Although our method performs well in many real-world datasets, TimeInf faces challenges in these datasets due to issues with the reliability of ground truth anomaly labels. Visual examples of problematic anomaly annotations are in Figure 12
military network environment. (vii) NAB-AdExchane
3 and illustrating how Timelnf can expose potential mislabeling in the data.
The UCR, SMAP, MSL, SMD, PSM, and SWaT datasets have pre-partitioned structure: a clean training set without anomalies and a contaminated test set with anomalies. In these datasets, we exclusively train and evaluate the baseline anomaly detection methods on the contaminated test set. Conversely, WADI, KDD-Cupp99, -AB-Traffic, -weetss and NAB-Tas have a single contaminated dataset with dispersed anomalies. For these datasets, baseline methods are trained and evaluated on the entire dataset.
Baseline Details. For block LOOCV, we employ time block length of 100 and utilize linear AR model for prediction. The conditional influence Kunsch (1984) method is implemented following the approach of Grosse al. (2023), using a linear AR model with an order of 100. For LWCV, we
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Table Details of the anomaly detection datasets. Note that discrete- valued dimensions are excluded from each dataset. The term "Average Length" refers to the average length across all time series in the dataset.
Dataset
Dimensions
Num. Time Series
Average Length
Average Anomaly Ratio
UCR
1
250
56,205
0.8%
SMAP
25
55
8,068
12.8%
MSL
55
27
2,730
10,5%
NAB-Trafie
1
7
2238
10.0%
SMD
38
28
25,300
4.2%
NAB-AdExchange
1
6
1602
10.0%
NAB-Tweets
1
15863
9.9%
NAB-Taxi
1
1
10320
5.2%
PSM
25
I
87841
27.8%
SWaT
51
449919
12.1%
WADI
123
172,751
5.7%
KDD-CUP99
42
494,021
19.7%
employ HMM with Gaussian observations (Ghosh et al. 2020), incorporating two discrete states: one hidden state representing normal conditions, and another representing abnormal conditions.
G. APPLYING TIMEINF TO MULTIVARIATE TIME SERIES
For anomaly detection in multivariate time series, we first apply self-influence to each dimension individually, and then average the anomaly scores across all dimensions to derive the final anomaly scores for individual time points. We refer to this approach as "Seplnf" for simplicity. We empirically find that "SepInf" demonstrates better performance than directly applying self-influence to the multivariate time series, which we refer to as "Multilnf". In this section, we explore potential challenges associated with using MultiInf and illustrate why it may not be suitable for anomaly detection.
Suppose have T training time blocks of size m. {a; ml 1T J1=1 x; (Zt, It-1.... Tr-m+1)
Rm<p,, where is the dimension of the multivariate time series. To compute MultiInf, we fit a vector AR model, i.c., multivariate linear regression, as follows
xz0 Er+11
(10)
[m] where, 2141 R2 represents the one-step ahead prediction, while x E mmx denotes the past time blocks used for prediction. Additionally, u E E. and 6141 E RR' correspond to the intercept and error terms, respectively. The parameters of the AR model, denoted as 0 = (0; [0,}/=1 E IRxxxxp, are capable of capturing correlations across each dimension of the multivariate time series, between timesteps within time block, and between each timestep in time block and each dimension. This expressed as follows:
θ 11
0
f51p
0
0,22
0,2p
0j
E Rpp
(1I)
A fm1
0jme
0 pmp
There are two practical challenges associated with computing the influence functions and over- weighting time block on the parameters 0: (1) Anomalies may not occur simultaneously in all dimensions of the time series. Consequently, capturing correlations among feature dimensions could result poor anomaly scores because the ground truth anomaly labels may depend only on anomalies in some dimensions but not the others; see Figures 11 and 12 for examples. (2) There may not be enough training samples to accurately estimate the high-dimensional parameters Inaccurately estimated 0 can lead to the computation of inaccurate influence functions. For instance, when fitting the model to a 100-dimensional time series with block length of 100, we have to learn 1 million model parameters, demanding large amount of training data. (3) Additionally, using Multilnf causes issues computationally because the inversion of the Hessian matrix scales with the number of model parameters.
size p X m X p.
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In contrast, SepInf is computed by initially applying self-influence each dimension separately and then averaging the anomaly scores (scaled self-infleences) across all dimensions of the multivariate time series. This is equivalent to fitting multiple linear regression in the same form as Equation (IO) under the assumption that each dimension is independent. This requires learning {0z {0z} with
0g = 10 032
jjml R"",
since the off-diagonal entries in Equation (II) are zero. This approach effectively reduces the number of parameters to learn. Furthermore, since anomaly patterns are often vastly different between dimensions, and the ground truth anomaly labels may be influenced by only some feature dimensions, computing SepInf effectively separates the influences of time points from different dimensions, thereby mitigating potential issues stemming from multicollinearity. Moreover, computing SepInf is more efficient because it involves fewer parameters when constructing the inverse Hessian matrix.
H. USING TIMEINF FOR VARIOUS DOWNSTREAM TASKS
In the paper, we use anomaly detection and time series forecasting as representative examples to demonstrate that TimeInf can identify both harmful and helpful data points in the data. In this section, we outline the setup for time series forecasting and anomaly detection, as well as how to apply TimeInf to both tasks.
HI. TIME SERIES FORECASTING
Let (mt)T=1 represent a univariate time series, with the objective of predicting future values based on past observations. We define input window of size sr!"T) (xi, -.. f2-m++1). Let fo be a forecasting model with parameters 0, such as an autoregressive model or a neural network. The forecasted value time + 1 is given by fo (m)
To evaluate the model performance, we use loss function 1 [o(x) (1))" such as the mean squared error (MSE). For time series forecasting, we particularly interested in understanding the impact of a training data point on model performance at test time. Therefore, we partition the original time series into training and test sets.
First, we train the model the training data to obtain the parameters i and the Hessian matrix H8-
We can then compute the influence of any time window [mm] in the training data on any time window
[m] in the test data as follows:
Festtt
Telock [m]]
(=t"m) -ima)) d
[ml d0 ff
d H.y de ((++1 (((+))ff(())" (())))))).
To measure the impact specific time point 2, in the training data on the entire test set, we define the following quantity:
Inca(Z2)
1
Σ !!
I2lock
[mm] -
SSest
[md "loa.
where Sz, is the set all time windows in the training data that include the time point 24 and Siss is the set of all time windows in the test set. The quantity I measures the influence of upweighting a training time point on the test loss.
The interpretation of the influence of this time point varies depending on the chosen loss function. For instance, if we use MSE as the loss function, negative value (2,)(2;) indicates that upweighting will lower the MSE, suggesting that is important for good forecasting performance. On the contrary, Z 2x) is positive, upweighting z, will increase the MSE, implying that z, is detrimental to forecasting accuracy.
H2. TIME SERIES ANOMALY DETECTION
For anomaly detection, the objective is to identify anomalies by reconstructing the time series and comparing the reconstructions with the actual observations. Let ga be reconstruction model
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with input window of size m, defined as iy. [m]] ( The reconstructed time
window given by (x.""")). The anomaly score is based on the reconstruction error, expressed as ((x]") (o(=x(")) )). The underlying assumption is that if time point anomalous, reconstructing using its own values should be casier than using normal points, as normal time points show distinctly different behaviors from anomalous ones.
Thus, we focus on measuring the reconstruction error without needing to partition the time series into training and test sets; we treat the entire series as single dataset. First, we train the model on this data to obtain the parameters & and the Hessian matrix H3. We can then compute the influence of any time window [mm on itself as follows:
Tplcck (zt mm] mi) ,
dø ((= -g((=("))
d
H TH d do ((z .9g(z"")))
Next, we compute self-influence, which measures the impact of upweighting time point on its own reconstruction loss, as our anomaly detection metric:
Intack (z(m) 2(ml)
S
Using MSE as the reconstruction error, negative ( suggests an anomaly, as upweighting this point improves its own reconstruction. This is based on the rationale that reconstructing an anomaly is easier with abnormal points than with normal ones. We take the absolute value of self-influence to compute the anomaly scores, with higher scores indicating higher likelihood of being an anomaly.
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SPAP charnel 6-1
SHAP chernel Gx
SMAP channel S-1
0.25
WVVVVV
TrGe 6.99
Tinchf AUC 0300
Timchhf AUC: 0.69
Iudetion Formet AUC
INcadoon Frwes car
ALC. 0.76
0.0
Anemady burn
Aramsdy 630
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Tiniltr 5.80
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OwN
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Estation Forist AUC: 0.79
Edlation Aurestuc 0.0
1STH 8.C
ASTHAUG:000 0.74
LSTH AUC. 0.5
ATTMANII AUC: 0.50
ARIMANAI AUC: 75
Anomall AUC
arrfome 0.6.
Tarsfore 0.09
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Figure 10: Qualitative results for Timeluf and other baseline anomaly detectors on selected channels from the SMAP and MSL datasets. The first row of each panel displays the first dimension of the time series, which captures the major anomaly pattern in the data. The remaining rows of each panel show the anomaly scores generated by each detector. For better visualizations, we normalize the anomaly scores from each method to range between 0 and 1. The ground truth anomaly intervals are marked in red segments. TimeInf outperforms state-of of -fee--of-hhhrrrr art time series anomaly detection methods in identifying both single and multiple anomaly patterns in the data.
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SMD Dimension
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ARIMWYAR AUC: 0.59
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Figure 11: Qualitative results for Timelıf and other baseline anomaly detectors on a selected channel from the multivariate SMD dataset. The first seven rows show selected dimensions of the time series, while the remaining rows show the anomaly scores generated by each detector. For better visualizations, scores from each method range between 0 and The ground truth anomaly intervals marked in red segments. TimeInf outperforms state-of oate-of-the-arr patterns in multivariate time series data.
normalize the anomaly
series anomaly detection methods in identifying anomaly
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KDD Cupés Dimenson?
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Figure 12: A qualitative example of mislabeled anomaly points in the KDD- KDD-Cup99 dataset. The first seven rows showcase selected dimensions of the time series, while the last displays the anomaly scores generated by TimeInf. For better visualizations, we normalize the anomaly scores from each method to range between and 1. The ground truth anomaly intervals are marked red segments. While not explicitly marked as anomalies, segments within the ranges 0 to 400, 1250 to 1500, and 3500 to 3600 exhibit distinct patterns that set them apart from others. TimeInf yields elevated anomaly scores for time points within these segments, raising questions about the reliability of the labels in these instances.
25


Reviews:
[{"review_id": "ZvRitVvXqR", "reviewer": "Reviewer_rAep", "summary": "This paper presents an interesting approach to time series interpretability research—by using noise perturbations to measure the contribution of data to the model. However, the method designed by the authors is a theoretical extension of existing approaches and lacks more extensive experimental evidence to demonstrate its effectiveness.", "questions": "See above.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The idea of inferring interpretability and anomaly analysis by measuring the contribution of time series data to the model's output is quite novel. The mathematical derivation is well-developed, and the details are thorough.", "weaknesses": "1. Compared to existing methods, the advantages of this approach are not fully emphasized, and the innovative aspects are not clearly defined. The main contribution of the paper can be seen as an extension of existing theory, but the method primarily involves deriving formulas based on Equation (3) and considering the Mixture of \\( \\delta z^{[m]} \\). This operation has limited significance for theoretical extension and does not effectively demonstrate the contribution claimed. I do not believe it addresses the shortcomings of existing methods.\n2. The paper conducts experiments on anomaly detection, interpretability, and mislabeled error in datasets. However, the paper does not discuss current state-of-the-art baselines in anomaly detection, such as methods based on LLMs or DCdetector. Moreover, the AUC and F1 results differ significantly from the original work. While the authors' skepticism about dataset quality is commendable, more evidence is needed to support this claim, as only partial visualizations of the data are presented. Additionally, the ablation experiments in the appendix are limited to parameter experiments. Based on my experience, complex theoretical methods often perform unsatisfactorily in neural network training, and more evidence is required to validate the effectiveness of the theory.\n3. Compared to traditional machine learning papers, there is no clear and complete process to present the designed algorithm, such as the input, output, and objective function. The majority of the paper is focused on theoretical formulas and related comparisons, making it difficult to understand", "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": "nzIPNJbtAC", "reviewer": "Reviewer_xCEM", "summary": "This paper presents a new method for measuring the contribution of different timestamp to the final predictions. The major idea is to leverage influence scores to assign model predictions to individual time points while preserving temporal structures between the time points. Experiments in anomaly detection can verify the effectiveness of the proposed method.", "questions": "Please reply to the above weaknesses if I have any misunderstandings. Thank you.\n\nAnother question: Can the proposed model process multivariate time series?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n2. The presented solution is technically solid.\n3. The experiments on anomaly detection can demonstrate the efficacy of the proposed method.", "weaknesses": "1. The relationship between the proposed model and unsupervised anomaly detection is unclear and should be further explained.\n\n2. The model's generalization ability to other time series tasks (e.g., forecasting, imputation) is uncertain. Specifically, how would the contribution score of historical timestamps be measured for future time points?\n\n3. There is no discussion on scalability. How does the method perform as the number of historical time points increases?", "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": "nj6ygUbLSi", "reviewer": "Reviewer_kJ9U", "summary": "This paper introduces TimeInf, a novel method for estimating data contribution in time series datasets using influence functions. Unlike previous data contribution methods that primarily focus on i.i.d. settings, TimeInf specifically addresses the challenges of temporal dependencies in time series data. The method works by analyzing overlapping blocks of consecutive time points to preserve temporal structure, then uses influence functions to measure how individual time points impact model predictions. TimeInf's key innovation is its ability to integrate data values across consecutive time series observations while accounting for different temporal arrangements of data points. The authors demonstrate TimeInf's effectiveness through extensive experiments on real-world datasets, showing it outperforms existing methods in both identifying harmful anomalies and helpful time points for forecasting. The method is particularly noteworthy for its computational efficiency, interpretable results through visualizations, and strong performance in practical applications, while being theoretically grounded in robust statistics and autoregressive models.", "questions": "See weakness.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The primary technical strength of TimeInf lies in its novel and theoretically sound approach to data contribution estimation in time series. It successfully extends influence functions to handle temporal dependencies, addressing a significant gap in time series analysis that previous methods overlooked. The method is built on a strong mathematical foundation combining robust statistics with innovative use of overlapping blocks to preserve temporal structure. This theoretical rigor is balanced with practical efficiency, as TimeInf proves to be computationally faster than deep learning alternatives while remaining scalable to large datasets with millions of time points. The method provides clear, quantifiable influence scores and intuitive visualizations that help users understand both anomaly patterns and forecasting behaviors. Its comprehensive evaluation across multiple real-world datasets demonstrates consistent improvement over existing approaches in both effectiveness and efficiency.", "weaknesses": "1. Methodological Limitations:\nThe paper’s core claim about their “distinctive integration” considering various temporal patterns seems to overlap significantly with existing attention mechanisms, without adequately differentiating itself. The theoretical justification for TimeInf lacks rigorous analysis of its properties, such as consistency or asymptotic behavior. Additionally, the paper doesn’t thoroughly explore the sensitivity of TimeInf to various hyperparameters, like block length or model architecture choices.\n\t2.\tExperimental Issues:\nThe experimental setup, particularly for anomaly detection tasks, is not clearly described, making result reproduction challenging. The reported results differ significantly from those in original papers for baseline methods, raising questions about comparison fairness. Important baselines in time series anomaly detection such as MTAD-GAT[1], TransAD[2] are missing, limiting the ability to fully assess TimeInf’s performance relative to state-of-the-art methods.  [1] and [2] are using the unspervised learning to learn the distribution and don't use the \"clean data assumption\" as well. The paper also lacks a comprehensive evaluation of TimeInf’s interpretability claims compared to other approaches.\n\t3.\tLimited Scope and Analysis:\nWhile the paper focuses primarily on anomaly detection tasks, it’s not clear how well TimeInf generalizes to other time series tasks or domains. The computational complexity of TimeInf compared to existing approaches is not thoroughly analyzed, despite mentioning efficiency improvements through conjugate gradient and Hessian-free approaches. The paper lacks a thorough discussion of TimeInf’s limitations or potential failure cases, which is crucial for understanding when and where the method is most applicable. Lastly, the evaluation metrics used are standard (AUC and F1 score), but don’t consider other potentially relevant metrics for time series anomaly detection, such as detection delay or false positive rate at a fixed detection threshold.\n\n\n[1] Zhao, Hang, et al. \"Multivariate time-series anomaly detection via graph attention network.\" 2020 IEEE international conference on data mining (ICDM). IEEE, 2020.\n[2] Tuli, Shreshth, Giuliano Casale, and Nicholas R. Jennings. \"TranAD: deep transformer networks for anomaly detection in multivariate time series data.\" Proceedings of the VLDB Endowment 15.6 (2022): 1201-1214.", "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": "0QXjq2bvvs", "reviewer": "Reviewer_kAP5", "summary": "This work presents TimeInf, a data contribution estimation model for time series. TimeInf overcomes the traditional i.i.d. setting of influence functions, making it more suitable for time series data. By considering multiple time blocks, TimeInf captures contextual information within time series data. Additionally, its model-agnostic design enhances its versatility and flexibility.", "questions": "+ Experimentally, TimeInf can effectively identify anomalies in time series. However, in the Data Pruning experiment, the authors claim that TimeInf can identify the most influential time patterns, which appears different from detecting anomalies (removing anomalies should not necessarily lead to a significant drop in model performance). Why is TimeInf able to identify influential time patterns at times, while it detects anomalies at other times? This seems somewhat contradictory.\n\n[1]. Tuli S, Casale G, Jennings N R. TranAD: deep transformer networks for anomaly detection in multivariate time series data[J]. Proceedings of the VLDB Endowment, 2022, 15(6): 1201-1214.\n\n[2]. Xiao C, Gou Z, Tai W, et al. Imputation-based time-series anomaly detection with conditional weight-incremental diffusion models[C]//Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2023: 2742-2751.\n\n[3]. Xu H, Wang Y, Jian S, et al. Calibrated one-class classification for unsupervised time series anomaly detection[J]. IEEE Transactions on Knowledge and Data Engineering, 2024.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "+ TimeInf addresses the gap between existing data contribution methods and the unique requirements of time series data.\n+ The proposed influence function is highly versatile, with multiple variants supporting diverse anomaly detection settings. Its model-agnostic nature further enhances its applicability across various models.\n+ TimeInf is computationally efficient, leading to significant resource savings.", "weaknesses": "+ Some key mathematical terms in the paper could benefit from clearer explanations, such as $\\delta_{x}[m]$ and $z^[m]$.\n+ The paper lacks a clear mathematical definition of the task, especially regarding the tasks targeted by the variants of TimeInf, which could cause confusion.\n+ Although the authors describe TimeInf as a data contribution estimation method, its task definition and benchmark choices make it appear primarily as an anomaly detection model. While data pruning experiments are provided in Appendix Section E, more direct evidence would help demonstrate that TimeInf goes beyond anomaly detection.\n+ The comparisons would benefit from including more advanced anomaly detection algorithms [1], [2], [3].\n+ In Section 4.3, the authors attribute TimeInf's relatively poor performance in Table 4 to mislabeled ground truth annotations. However, they need to clarify why other benchmark models are unaffected by these mislabeled annotations. Demonstrating that other models do adhere to potentially mislabeled annotations would help validate this claim.", "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": "K3Uixel7QN", "reviewer": "Reviewer_Zos6", "summary": "The paper introduces TimeInf, a model-agnostic data contribution estimation method for time-series datasets. It uses influence scores to attribute predictions to individual time points, preserving temporal dependencies. TimeInf shows effectiveness in detecting anomalies, providing interpretable attributions, and identifying mislabeled anomalies in ground truth.", "questions": "1.\tCan you elaborate on how TimeInf extends beyond the local historical context to capture a more global perspective as mentioned in Section 3?\n2.\tFollowing Q1, regarding Table 3, can you provide baseline comparisons to show the model’s advantages in capturing long-term dependencies?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper is well-organized with clear problem formulation and concise visualization. It effectively demonstrates the utility of TimeInf across multiple tasks, providing robust results on time-series datasets.", "weaknesses": "1.\tMy main concern is that while TimeInf is presented as a general model-agnostic method, the experiments in the main text focus solely on anomaly detection. It would be better to include more different tasks to support the advantage of this general method rather than only elaborate on one task.\n2.\tTable 2 contains misformatted bold values that do not reflect the best performance (UCR and SMAP’s F1).\n3.\tThe data pruning experiments use non-standard time-series forecasting metrics; it will be better if more common metrics like RMSE and MAE can be considered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)