Going Beyond Static: Understanding Shifts with Time-Series Attribution

Thank you so much for your advice to improve our paper. We have addressed your concerns as follows, and we will incorporate these into our final version.

Q1. Discussion with previous works about shift in time-series data

There’s been lots of prior works on detecting shifts in time-series data, and we discuss them in Appendix E. In time-series anomaly detection, they focus primarily on detecting breakpoints. However,

• Our goal is to understand why a model’s performance declines in a prediction task by attributing this drop to specific properties of interest.
• The shifts considered in time-series anomaly detection are typically associated with breakpoints, whereas our work covers a broader range of temporal properties
• Our method can also be used to diagnose performance drops in anomaly detection methods as well.

Thanks for your advice, and we will move the related work into the main body to improve the clarity.

Q2. Novelty

We would like to clarify that our contributions are significant at the problem-solving level, methodology level, and theory level:

• Setting level: We focus on time-series settings, where we mainly consider the shift in non-stationary properties. This setting is much more complicated, and we are the first to deal with this. Previous methods cannot deal with this effectively. As shown in Appendix D.3 (Table 2), static region analysis cannot produce reliable safe/risk regions.
• Methodology level: Our attribution method is novel and different from previous ones. We formulate the attribution problem as an average treatment effect estimation problem, marking, to the best of our knowledge, the first instance of this approach. And we derive the doubly robust estimator to estimate the attribution score.
• Theory level: Our attribution has a solid theoretical foundation. Rather than simply incorporate average treatment effect estimation, we rigorously prove the unconfoundedness assumption within our framework—a substantial advancement over traditional causal inference literature, where this assumption is typically taken for granted. Based on this, we prove the double robustness of our estimator. Our framework thus paves the way for a novel perspective on performance attribution, creating opportunities for further exploration in this domain.

Q3. Utility of Sufficiency measure

Thank you for your valuable suggestion. We would like to clarify the utility of the sufficiency measure by addressing it from two aspects:

• Experiment: we select top-K features with $K \in \\{10,20,30,all\\}$, calculate the sufficiency measure as well as the worst-accuracy (among $\mu_{\mathbb P}$, $\mu_{\mathbb Q}$, and $\pi$) of our attribution model. The results show a clear trend: higher sufficiency scores correspond to improved prediction performance and more precise attribution, thereby highlighting the practical utility of this measure.

• Theory: As demonstrated in Proposition (2), the attribution estimation error is directly controlled by the sufficiency measure. This theoretical result further underscores the importance and relevance of this metric in ensuring robust and reliable attribution.

Q4. Clarity in case studies

Thank you for your suggestions. We add more details and demonstrations as follows:

• Experimental details:
  • Original model (under evaluation) architecture: we use Transformer model (n_head:4, n_layer:3, hidden_dim:32), learning rate is $1e^{-3}$, the total epoch number is 200, batch size is 256, and the early stop is used during training (according to last 10 epoch);
  • Attribution model: The model architecture is shown in Figure 2, where we use two-layer MLP with hidden size 32, learning rate $1e^{-3}$, and batch size 64.
• Splitting criterion in Case Study 3: The setting is to simulate the early diagnosis [1], where we hope the model can predict the outcome as early as possible. This problem is important in real-world health-care applications.

We will add these into the Appendix for better clarity.

[1] Wang, Lulu. "Early diagnosis of breast cancer." Sensors 17.7 (2017): 1572.

[2] Yuan, Kuo-Ching, et al. "The development an artificial intelligence algorithm for early sepsis diagnosis in the intensive care unit." International journal of medical informatics 141 (2020): 104176.

	Attr.(S) = \mathbb{E}[R(X_{-S}, T=1) - R(X_{-S}, T=0)],

Thank you for addressing my points and incorporating the revisions in red text. I appreciate the updates and clarifications, as they have improved my understanding. I have updated my score.

Regarding case scenario 3, I recognize that the purpose is to evaluate a model in a temporally shifted scenario rather than predicting the past. My concern was about the potential implications if the training, validation, and test patient cohorts overlapped. However, the paper clarifies that the training set is a separate subset of patients, which resolves this concern.

I believe the paper could still benefit from greater clarity and organization, particularly with the new updates. For example, the sentences, “Additional demonstrations on the relationship between our TSSA approach and average treatment effect (ATE) estimation can be found in Section 6. And Appendix E illustrates how our approach can be integrated with Shapley Value,” appear at the end of the methods section, even though ATE and Shapley Value are not discussed earlier in that section. I recommend repositioning this information to better align with where these topics are first introduced in the paper. For instance, it may be more effective to consolidate all mentions of Shapley Value (e.g., related work, why Shapley Value is not used, and compatibility with Shapley Value) into a single section in the appendix, while referencing this section when the topic is mentioned in the related work. Similarly, when treatment infection estimation is first referenced in line 228, it would be helpful to immediately include the sentence, “Additional demonstrations on the relationship between our TSSA approach and average treatment effect (ATE) estimation can be found in Section 6.” This restructuring would improve cohesion and readability for readers.

Thank you so much for your positive review! We would like to address your concerns as follows:

Q1. Baseline Comparisons

Thank you for your suggestions. We have (in Appendix) and add more baselines to further validate our effectiveness:

• For safe/risk region analysis: In Appendix D.3, we compare our methods with “static” methods to demonstrate the need to consider non-stationary properties.
• For sample-efficiency or sufficiency measure: In Appendix D.4, we compare our model with XGB (SOTA for tabular prediction and does not share representations) to show that our shared representation can indeed improve the performance (with a better sample complexity).
• For model intervention in Case Study 2: We add the results of finetuning, where we use 50% samples (3200) from the original test set for finetuning, and test on the remaining 50% test samples. For our model intervention, we mix the original training set with the finetuning samples, and then retrain a classifier with balanced weights according to the identified features (age, ICU stay time, etc.). The results are as follows, where we can see that our intervention can significantly outperform simple finetuning.

According to your suggestions, we will move these results into the main body.

Q2. Additional theoretical results

The convergence results align closely with traditional findings on SGD, and we do not present them as a novel contribution. Additionally, CLT results can be further expanded upon our consistency results.

Q3. Font size & Error bars

Thank you very much for your advice! We have updated the paper (Figure 5 and Figure 7) to address these typos.

Q4. Equation (21)

Yes, it is derived from the tower property.

Q5. Frequency

The frequency metric used in our paper is in $\mathbb R$.

Thank you so much for your reply! We would like to address your concerns as follows:

1. For the frequency:

   • The output of FFT is in $\mathbb C$, denoted as $FFT[k]=a_k+ib_k$.
   • We then calculate the amplitudes as: $\sqrt{a_k^2+b_k^2}$.
   • Finally we extract the dominant frequency (with the maximal amplitude) from positive frequencies, where the positive frequency values are real numbers, calculated as $f_k = \frac{k}{NT_s}$ (in our experiments we set $T_s=1$).

2. For the error bar: The error bar represents the 2 times of the standard error (i.e. $\pm$2xSE), calculated as $2*\frac{s}{\sqrt{n}}$, where $s=\sqrt{\frac{1}{n-1}\sum_{I=1}^n(x_i-\overline{x})^2}$ denotes the sample standard deviation, and $n$ denotes the time of repeated experiments (in our experiments $n=10$).

3. For the hyper-parameters:

   • for the Transformer model under evaluation: We select a set of standard hyperparameters to obtain a transformer model, which remains fixed during evaluation. Our evaluation results are based on this model. Since our method focuses on evaluating and diagnosing (any) model's failures, it is adaptable to various hyperparameter configurations. Different hyperparameters can be applied, and the evaluation results will correspond to the model configured with those settings.
   • for the attribution model: we choose the hidden size in $\\{16, 32, 64\\}$ and select the best parameter (32) according to the cross-validation results (accuracy of $\mu_{\mathbb P}(\cdot), \mu_{\mathbb Q}(\cdot), \pi(\cdot)$). As for the optimizer, we use Adam optimizer with the default (standard) learning rate ($1e^{-3}$). We will clarify this in the final version.

4. For the tower property: We add more detailed illustrations below:

   • Tower Property: The tower property states that $\mathbb{E}[\mathbb{E}[X | \mathcal{G}]] = \mathbb{E}[X]$, where $\mathcal{G}$ is a sub-$\sigma$-algebra. This means that when taking the conditional expectation of a random variable $X$, and then the result is equal to the total expectation of $X$.
   • For our equation (21), the inner conditional expectation $\mathbb{E}_{\mathbb P}[\ell(X, Y) | g(X)]$ can be viewed as a function of $g(X)$, denoted as $h(g(X))$. And the outer expectation considers the expectation of $h(g(X))$ over some event $A \in \sigma(g(X))$.
   • Here by tower property, the result of LHS equals to $\mathbb{E}[\ell(X, Y) \cdot \mathbb{1}_A]$ because $\mathbb{1}_A$ is measurable with respect to $\sigma(g(X))$, the $\sigma$-algebra generated by $g(X)$. And this ensures that $\mathbb{E}_P[\ell(X, Y) | g(X)]$ is well-defined and depends only on $g(X)$.

We will add this illustration into Appendix in the final version. As for the reference, we will add [1] to our proof.

Thank you so much for your advice!

[1] Billingsley, Patrick. Probability and measure. John Wiley & Sons, 2017.

Thank you so much for your suggestions to improve the paper's clarity! We would like to address your concerns as follows:

Q1. Ill-defined/Unclear concepts:

Thank you for your thoughtful suggestions to improve the clarity of our paper. Below, we clarify key definitions and claims:

• Shift within shift: This term distinguishes the type of distribution shifts we address from the natural temporal shifts inherent in time-series data (e.g., changes in features over time). Specifically, the distribution shifts discussed in our work extend beyond static settings to include dynamic changes in the non-stationary properties of time-series data, such as trends, seasonality, and frequency. To improve the clarity, we will change the term to shift in non-stationary properties.
• Application-specific: The key idea of this work is to understand the specific distribution shift patterns first when facing performance drop in real applications. This concept focuses on the nature of the problem itself, rather than being tied to a specific data modality (e.g., time-series). To enhance clarity and avoid potential misunderstanding, we will revise the term from "application-specific" to "problem-specific".

We have incorporated this into the main body, and please feel free to point out other points that you’d specifically like us to fix for better clarity.

Q2. Criteria for selecting metrics in each category

For each category, we collect metrics from various domains including statistics, analytical chemistry, signal processing, finance, etc. Since this is the first work to deal with this, we collect as many as possible metrics to improve the attribution power. We acknowledge that these metrics are not exhaustive, and more metrics can be added (as said in line 188).

Q3. Contribution is incremental

We would like to clarify that our contributions are significant in problem setting level, methodology level, and theory level:

• Setting level: We focus on time-series settings, where we mainly consider the shift in non-stationary properties. This setting is much more complicated, and we are the first to deal with this. Previous methods cannot deal with this effectively. As shown in Appendix D.3 (Table 2), static region analysis cannot produce reliable safe/risk regions.
• Methodology level: Our attribution method is novel and different from previous ones. We formulate the attribution problem as an average treatment effect estimation problem, marking, to the best of our knowledge, the first instance of this approach. And we derive the doubly robust estimator to estimate the attribution score. Tiffany’s work [1] uses inverse propensity score reweighting to estimate the score, while our estimator is doubly robust (contain hers). That is, when their method works, our estimator works too. Jean’s work [2] uses Shapley value for the attribution, which estimates the average effect of one feature on all other possible feature subsets. However, as discussed in Appendix C, we focus on the impact of one single data property on the model’s performance degradation, while keeping all other properties (or features) constant. As a result, a full calculation of Shapley values may be unnecessary and less suitable for our objectives.
• Theory level: Our attribution has a solid theoretical foundation. Rather than simply incorporate average treatment effect estimation, we rigorously prove the unconfoundedness assumption within our framework—a substantial advancement over traditional causal inference literature, where this assumption is typically taken for granted. Based on this, we prove the double robustness of our estimator. Our framework thus paves the way for a novel perspective on performance attribution, creating opportunities for further exploration in this domain.

[1]Tiffany Tianhui Cai, Hongseok Namkoong, and Steve Yadlowsky. Diagnosing model performance under distribution shift. arXiv preprint arXiv:2303.02011, 2023

[2]Jean Feng, Harvineet Singh, Fan Xia, Adarsh Subbaswamy, and Alexej Gossmann. A hierarchical decomposition for explaining ml performance discrepancies. arXiv preprint arXiv:2402.14254, 2024.

Thank you so much for your positive review! We would like to address your concerns as follows:

Q1. Code release

Thank you very much for your advice. The code is based on PyTorch and our algorithm is easy to implement. We promise to release our code after acceptance.

Q2. Double robustness network architecture

Thank you for your thoughtful suggestions. We would like to demonstrate more on the double robustness and our model architecture:

• Double robustness: Our attribution approach is doubly robust, meaning that as long as either the outcome prediction or the propensity score estimation is accurate, the overall estimation remains reliable. This robustness is a key strength of our method.
• Model architecture: The shared representation space utilized in our model is a widely adopted approach in treatment effect estimation and multi-task learning [1,2,3]. By employing multi-head structures and multi-task loss functions, the model is able to capture all necessary information within the representation space. One significant advantage of this design is improved sample efficiency: while we may lack sufficient samples from the target distribution, we have ample samples from the training distribution. This allows the model to learn better representations from the available data.
• Experiments: Furthermore, as illustrated in Appendix D.4 (Figure 7), our model demonstrates superior prediction accuracy for both outcome prediction and propensity score estimation compared to single-network architectures.

[1] Nonparametric Estimation of Heterogeneous Treatment Effects: From Theory to Learning Algorithms. Alicia Curth, Mihaela van der Schaar. AISTATS 2021

[2] Adapting Neural Networks for the Estimation of Treatment Effects. Claudia Shi, David M. Blei, and Victor Veitch. NeurIPS 2019.

[3] Learning Shared Representations for Value Functions in Multi-task Reinforcement Learning. Diana Borsa, Thore Graepel, John Shawe-Taylor.

Q3. How much risk assessment would be to clinicians

Thank you for raising this thoughtful question. We deeply appreciate your consideration of the practical implications of our work.

• Our attribution results are designed to provide a comprehensive framework for both data scientists and clinicians to understand the distribution shifts responsible for performance drops. Specifically, the interpretability of both static features and extracted non-stationary properties allows clinicians to identify the underlying reasons behind model failures. This can facilitate actionable insights in medical practice.
• Moreover, our safe and risk region analysis further enhances practical utility by enabling informed decision-making in deployment scenarios. Even in cases where the model cannot be modified, this analysis supports smarter and safer deployment strategies in real-world settings.

Q4. Illustration of Equation (39)

Equation (39) relies on the assumption that the samples are independent, which is a standard assumption in similar settings. In our work, each sample represents the non-stationary properties extracted from the time-series data of a single patient, and we assume that these samples are independent.

To ensure clarity, we will explicitly state this assumption in the main body of the paper. We appreciate your suggestion and will make the necessary adjustments to improve transparency.