Online Time Series Forecasting with Theoretical Guarantees

Dear Review CFrf, we sincerely appreciate your informative feedback and helpful suggestions that helped clarify our contributions, improve the readability of our theories, and the completeness of our experiments. Please see our point-to-point response below.

Q1: The authors are recommended to verify their citation formatting carefully, as it appears inconsistent with standard practice. Additionally, citations in the main text do not link correctly to the corresponding references listed.

A1: Thank you for pointing this out. We appreciate your careful reading. We have thoroughly checked all citations and corrected inconsistencies. Additionally, we have fixed the incorrect linking between in-text citations and the reference list in the revised manuscript.

Q2: The authors need to clarify the rationale for emphasizing "online time-series forecasting" explicitly. If there is no substantial distinction from general time-series forecasting, it would be beneficial to unify the terminology, enhancing clarity and facilitating progress within the field.

A2: Thank you for this constructive comment, which helps us better clarify the scope of our work.

The explicit emphasis on “online time-series forecasting” is important because the setting differs substantially from general time-series forecasting. In online forecasting:

• We cannot access the entire historical data; instead, we can only use sequentially arriving data streams and rely on the most recent observations within a moving window for forecasting.
• Handling nonstationarity in such a constrained setting is crucial. One of our key theoretical contributions (Theorem 2) proves that at least 4 consecutive observations are sufficient to identify the latent variables driving nonstationarity, which aligns directly with the requirements of online forecasting.

We agree that our methodology is general and also applicable to nonstationary time-series forecasting in offline settings. Following your suggestion, we have applied our method to standard nonstationary forecasting tasks, and the experiment results are shown as follows:

In light of your suggestions, we have revised the manuscript to make this distinction clearer while also emphasizing the broader applicability of our framework.

Q3: Please explicitly state the theoretical framework’s practical implications, detailing how it theoretically informs real-world applications and contributes to advancing the research area.

A3: Thank you for this question. We would like to respectfully clarify that we have already provided detailed discussions of the theoretical framework’s practical implications and real-world validity in the original paper (Lines 175–197) and Appendix C. Please kindly let us know if you cannot find it. Specifically:

• Assumption 1 (Bounded and continuous densities):
  This assumption only requires that the transition probability densities of observed and latent variables are bounded and continuous, which is easy to satisfy in practice. With sufficient data, the estimated probability density functions are naturally bounded. For instance, temperature records in climate data change continuously within a reasonable range.
• Assumption 2 (Injectivity of linear operators):
  This condition only requires sufficient variation in the density of one variable for different distributions of its parent variables. For example:

$$L_{x_{t}, x_{t+1} \mid x_{t-1}, x_{t-2}}$$

implies that historical observations have a strong influence on future observations—e.g., historical temperature values strongly influence future temperature values. Multiple practical examples of injectivity verification are provided in Appendix C.3.

• Assumption 3 (Spectral decomposition uniqueness):
  This assumption requires that changes in the latent variables $z_t$ produce a sufficiently large and distinguishable effect on the mapping $x_{t-1} \to x_t$. For example, if $z_t$ encodes seasonal information and $x_t$ denotes temperature, the day-to-day temperature change in winter versus summer differs markedly. Since this assumption only requires the existence of any two such distinguishable states, it is satisfied in many real-world scenarios with continuous $x_t$ (details in Appendix C.6).
• Assumption 4 (Sparse mixing procedure):
  This assumption is mild and common in real-world scenarios. For instance, in financial data, if $z_{t,i}$ denotes a policy and $x_t$ denotes stock prices, a single policy typically does not influence all stocks.
  This assumption can also be empirically checked by evaluating the Jacobian matrix of $x_t$ with respect to $z_t$ during reconstruction; if the estimated Jacobian is sparse, it validates the sparse mixing procedure.

In light of your comment, we have further emphasized these points in the revised version to make the practical implications of the theoretical framework clearer.

Q4: The experimental validation presented in the manuscript is insufficient, as it employs only two metrics (MSE and MAE). It is advised to repeat experiments multiple times, reporting both mean and standard deviation of the results to ensure robustness.

A4: Thank you for your suggestion. We would like to clarify that we have already addressed this concern in the current manuscript. Specifically, as stated in Lines 343–344, since some methods report the best results in their original papers, we show the best results in the main text. We also provide the experimental results with the mean and variance over three random seeds in Appendix D.2.4. Please kindly let us know if you cannot find it.

Thus, the robustness of our method has been verified, and the corresponding results are reported in the appendix for completeness.

Q5: From the current numerical results, the proposed method does not exhibit significant improvements in forecasting performance. Additional experimental evidence should be provided to clearly demonstrate the advantages of the proposed approach.

A5: Thank you for your comment. We respectfully disagree with the statement that our method does not exhibit significant improvements in forecasting performance.

As summarized below, we have computed the average relative improvements of our method over different backbones and datasets:

These results clearly show that our method consistently improves forecasting performance across all tested backbones. Notably, the improvement is larger for relatively older or simpler backbones, demonstrating that our framework is general and complementary to different model architectures.

Following your suggestion, we further evaluated additional backbone networks, including ER[1] and DERpp[2]. Experiment results are shown as follows:

[1] Chaudhry,et al. On tiny episodic memories in continual learning.
[2] Buzzega et al. Dark experience for general continual learning: a strong, simple baseline.

Q6: Visualization Results.

A6: Thank you for this helpful suggestion. We fully agree that visualization is important for improving interpretability in time-series forecasting.

Due to policy restrictions, we are unable to include the figures in the manuscript. However, to address this concern, we have presented the forecasting trends in a tabular format, which conveys the same information as line plots. Additionally, we have included corresponding analyses to highlight the interpretability of the results.

The visualization results from the above table show that our method achieves a better performance.

Dear Reviewer CFrf,

Thank you for dedicating time to review and provide feedback on our submission. We hope our responses effectively address your concerns. If there are additional matters you'd like us to consider, we eagerly await the opportunity to respond.

Best regards,

Authors of submission #8290

Dear Reviewer 2i9r, we highly appreciate the valuable comments and helpful suggestions on our paper and the time dedicated to reviewing it. Below, please see our point-to-point responses.

Q1&Q7: As for the explanation of assumptions.

A1&Q7: Thank you for your question. We would like to respectfully clarify that these conditions are standards in the field of latent variables identifiability [1,2,3]. Moreover, we have already provided detailed discussions of the practical implications and real-world validity of these assumptions in the original paper (Lines 175–197) and Appendix C. Specifically:

• Bound and Continuous Density: This assumption only requires that the transition probability densities of are bounded and continuous. It is easy to satisfy in practice because, with sufficient data, the estimated probability density functions are naturally bounded. For instance, temperature records in climate data change continuously within a reasonable range. This assumption is naturally satisfied since we use VAE-based model.

• Injectivity: The injectivity condition only requires that there is sufficient variation in the density of one variable for different distributions of its parent variables. For example, $L_{x_{t},x_{t+1}\mid x_{t-1},x_{t-2}}$ implies that historical observations have enough influence on future observations—e.g., historical temperature values strongly influence future temperature values. We further provide multiple examples of the injectivity of linear operators in Appendix C.3. However, this assumption is not testable. In a word, when the mapping from latent variables to observations is not fully injective, our model can only recover the injective components of the latent variables.

• Uniqueness of Spectral Decomposition: This assumption asks that changes in the latent variables $z_t$ produce a sufficiently large and distinguishable effect on the mapping $x_{t-1} \to x_t$. For example, if $z_t$ encodes seasonal information and $x_t$ denotes temperature, the day-to-day temperature jump in winter versus summer differs markedly. Since A3 only requires the existence of any two such distinguishable states, it is satisfied in many real-world scenarios with continuous $x_t$. More details are provided in Appendix C.6. Importantly, this decomposition uniqueness condition is empirically testable. After identifying the latent variables, we can use the observed variables to directly test whether the decomposition is unique by using tensor unique decomposition tools.

• Sparse Mixing Procedure: The Sparse mixing procedure assumption is also common in real-world scenarios. Specifically, it denotes that the Markov network over $z_t$ and $x_t$ is sparse. For example, in the field of financial data, consider $z_{t,i}$ and $x_t$ as one policy and stocks, and this policy may not have an influence on all the stocks. In principle, this assumption is not testable. When it does not hold, we may no longer achieve component-wise identifiability. However, our method can still ensure block-wise identifiability of $z_t$, which is sufficient to model how $z_t$ influences the mapping from $x_{t-1}$ to $x_t$. This allows the framework to remain effective for online forecasting, even when full sparsity is violated.

In light of your comment, we have provided a further discussion in the updated version.

[1] Fu, et al. "Identification of Nonparametric Dynamic Causal Structure and Latent Process in Climate System.
[2] Lachapelle et al. "Discovering Latent Causal Variables via Mechanism Sparsity: A New Principle for Nonlinear ICA."
[3] Hu et al. "Instrumental variable treatment of nonclassical measurement error models."

Q2: As for the contribution.

A2: Thank you for your comment, which helps us highlight our contributions. We would like to clarify that our contribution lies in addressing a fundamentally different and practically motivated real-world problem. Specifically, we aim to address the online time series forecasting, which has been rarely handled in the field of time series forecasting. To address this problem, we have to formulate the specific model and then tailor the techniques, like eigenvalue–eigenfunction decomposition, and incorporate further constraints to achieve our identifiability results.

Moreover, we would like to highlight the contributions of Theorems 1 and 3, which are, to the best of our knowledge, entirely new:

1. Theorem 1 formally establishes how latent variables affect predictive performance under nonstationarity.
2. Theorem 3 introduces the sparse mixing procedure into temporal causal representation learning and proves that latent variables are identifiable under this mild and practically meaningful assumption.

Q4: As for the work's scope.

A4: Thank you. We would like to clarify that we have already explicitly acknowledged this limitation in the manuscript (see Lines 370-372). Our current theoretical analysis and experiments focus on regularly sampled time series, as it allows for a clear formulation of the identifiability conditions.

Importantly, within this scope, our framework is general; it accommodates a wide range of time series data with a discrete time step. And we believe it is unrealistic to expect a single theoretical framework to cover all possible data types, especially in the early stages of developing identifiability theory for latent-variable models. Extending our framework to continuous or irregularly sampled time series is indeed an interesting direction, and we have emphasized this as future work in the revised manuscript.

Q5: The paper mentions that "latent variables introduce unknown distribution shifts", can you elaborate? Is the paper also interested in shifts in the data space?

A5: Thanks a lot. By "latent variables introduce unknown distribution shifts" we refer to the fact that the evolution of latent variables influences the causal process of observed variables, which in turn changes their marginal distribution over time. Specifically, in the data generation process

\\begin{array}{ccc} z_{t} & \\rightarrow & z_{t+1} \\\\ \\downarrow & & \\downarrow \\\\ x_{t} & \\rightarrow & x_{t+1} \\end{array}

the transition $x_t \rightarrow x_{t+1}$ is governed by $z_t$. As $z_t$ evolves, the marginal distribution $p(x_t)$ also changes over time, leading to distribution shifts. Because $z_t$ is unobserved, these distribution shifts are "unknown".

Regarding the second part of your question, our work focuses on distribution shifts in the data space (observed variables). We have explicitly highlighted how latent variables have influence on the evolution of the observed distribution in the updated version.

Q6: As for time-delayed and instantaneus parents

A6: Thank you for your question. Let us consider the following causal graph:

\\begin{array}{ccc} z_{t,1} & \\rightarrow & z_{t+1,1} \\\\ \\downarrow & & \\downarrow \\\\ z_{t,2} & \\rightarrow & z_{t+1,2} \\end{array}

In this example:

• Time-delayed parents:
  The historical latent variables, such as $z_{t,1}$ and $z_{t,2}$, are time-delayed parents of $z_{t+1,1}$ and $z_{t+1,2}$, respectively, because they influence these variables through temporal transitions $z_{t,1} \to z_{t+1,1}$

• Instantaneous parents:
  At the current time step $t$, $z_{t,1}$ is an instantaneous parent of $z_{t,2}$, as it directly influences $z_{t,2}$ within the same timestep. Similarly, $z_{t+1,1}$ is an instantaneous parent of $z_{t+1,2}$.

Q8: As for computational requirements compared to baseline methods.

A8: Thanks! We have provided a detailed clarification regarding computational requirements, scalability with sequence length and dimensionality, and runtime comparisons.

1. Experimental environment:
   All experiments were conducted on a single NVIDIA GTX 3090 (24GB) GPU and an Intel i7-7700k CPU with 64GB memory.
2. Scalability of the architecture:
   As discussed in Line 252, our framework is model-agnostic and can be combined with different backbone networks. The computational scaling thus mainly depends on the chosen backbone:
   • Encoder and forecaster: Their scalability with sequence length and input dimensionality follows the standard complexity of the selected backbone (e.g., LSTD or OneNet).
   • Decoder: The decoder is implemented as a simple MLP and is independent of sequence length; it only scales with the output dimensionality.
   • Noise estimators: Each latent variable dimension corresponds to two noise estimators. Thus, the complexity of these components grows linearly with the latent dimensionality.
3. Runtime comparisons:
   We have conducted runtime evaluations across different datasets and backbones to provide a fair comparison with baseline methods. The table below shows the average runtime (seconds) per training epoch for LSTD and Proceed backbones.

Q9: 4 consecutive observations for identifiability.

A9: Thank you for this insightful question. As discussed in Lines 152–159, Theorem 2 implies that at least 4 consecutive observations are theoretically required to describe the variation of the latent variables $z_t$. This is the theoretical minimum for identifiability in our framework. Specifically, with only 3 consecutive observations, the distributional changes are insufficient to uniquely recover the latent transitions, making $z_t$ unidentifiable under the assumptions in Theorem 2. And using more than 4 observations (e.g., 5) is allowed and does not violate the theoretical framework. Following your suggestion, we conducted additional experiments by varying the number of observations used for identification. Experiment results are shown as follows:

Dear Reviewer 2i9r,

Thank you very much for taking the time to review our work. We are sincerely grateful for your constructive comments.

We feel delighted that your concerns have been fully addressed. We would be grateful if you might consider updating your rating accordingly. Thank you again for your thoughtful and invaluable feedback!

With best wishes,

Authors of submission #8290

Dear Reviewer hiQT, we sincerely appreciate your encouraging and insightful feedback, which has significantly helped us clarify the boundary of our method and make our experiment solid. Please find our point-by-point responses below.

Q1: The theoretical results rely on assumptions such as bounded continuous densities (A1) and injective linear operators (A2), which may not hold in highly nonstationary or noisy real-world scenarios. For example, abrupt distribution shifts (e.g., sudden policy changes) could violate the continuity assumption, potentially limiting the framework’s robustness in extreme cases.

A1: Thank you for raising this important concern. We would like to clarify that the continuity assumption in (A1) refers specifically to the probability density functions of the latent and observed variables, not to the continuity of the data generation process. Therefore, even in the presence of abrupt distribution shifts, such as sudden policy changes, our theoretical results remain valid as long as the latent and observed variables admit bounded continuous densities. Moreover, such conditions still hold in many realistic nonstationary or noisy settings, where the the distributions do not become degenerate.

Q2: Will the sampling frequency affect this module, thus causing the assumption conditions to be violated and making four observation points insufficient for identifying the latent variables?

A2: Thank you for the thoughtful question. The sampling frequency can indeed affect the identifiability of latent variables, particularly when it induces a form of subsampling in which some observation variables become unobservable. In such cases, the corresponding latent variables may no longer be fully identifiable unless additional observations are incorporated.

For example, consider the following generative structure:

\\begin{array}{ccccccccccc} z_1 & \\rightarrow & z_2 & \\rightarrow & z_3 & \\rightarrow & z_4 \\\\ \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow \\\\ x_1 & \\rightarrow & x_2 & \\rightarrow & x_3 & \\rightarrow & x_4 \\end{array}

If $x_2$ becomes unobservable due to subsampling or sensor failure, then the identifiability of $z_1$ and $z_3$ may require additional observable variables beyond the original four observed variables. However, for online forecasting tasks, even if some latent variables are only partially identified, our method can still leverage the identifiable components to make effective predictions of future observations.

Q3: This framework seems very universal. Can it be used for offline model training?

A3: Thank you for the great suggestion, which extends the boundary of our contributions. Indeed, our methodology is universal and can naturally be applied to offline model training, where the distributions of historical time series change over time. In light of your questions, we conducted additional experiments on the offline training scenarios with non-stationary time series. Specifically, we follow the setting of [1], and consider Koopa [1], FITS [2], SOILD [3] as baselines. Moreover, we evaluate our method on ECL, ETTh2, Exchange, ILI, and Traffic datasets. Experiment results are shown as follows:

We can find that our method can still achieve the ideal performance in offline scenarios.

[1] Liu, Yong, et al. "Koopa: Learning non-stationary time series dynamics with Koopman predictors." NeurIPS 2023
[2] Xu, Zhijian, Ailing Zeng, and Qiang Xu. "FITS: Modeling time series with $10 k$ parameters."ICLR2024
[3] Chen, Mouxiang, et al. "Calibration of time-series forecasting: Detecting and adapting context-driven distribution shift." KDD2024.

Dear Reviewer uFSc, we sincerely appreciate your informative feedback and helpful suggestions that make our paper more complete. We have added the new experiments following your suggestions, provided more discussions about the sparse mixing procedure and online forecasting, as well as modified the paper and appendix accordingly. Please see our point-to-point response below.

Q1. The authors should spend a bit more discussing a simple experiment within the confines of the main text. No details are given on the data generation process, e.g., time series length, the type of sparsity, and so on. This makes it hard to develop an intuition for how the method works.

A1: Thanks for your insightful suggestions, which make our experiment more solid. In light of your suggestion, we have added more simulation experiments. Specifically, we have considered four datasets with different types of sparsity: (1). the sparse latent process but the dense mixing procedure; (2). the dense latent process but the sparse mixing procedure; (3). the sparse latent process and the sparse mixing procedure; and (4). The dense latent process and the dense mixing procedure. All of these datasets contain 10000 5-dimensional time series samples with a length of 7. Experiment results are shown as follows:

We can find that the latent variables are hard to identify when both the mixing and latent dynamics are sparse.

Q2: The "sparsity of the mixing procedure" should also be made more clear. The way I understood it, the sparsity is in having each z_i only influence a few of the x_i's. Is this indeed the case? This would mean that each z_i must be "active" but at the same time there is a limitation on how many x_i's it can be connected to?

A2: Thank you for your question, which indeed helps us improve the readability of our paper. You are correct that the sparse mixing procedure means each $z_{t,i}$ only influences a limited number of $x_{t,j}$. However, we would like to clarify that our sparsity requirement is mild. Specifically, we only assume that the Markov network over $z_t, x_t$ is sparse, rather than imposing strong structural sparsity across the entire latent process. Given a toy example as follows

\\begin{array}{ccc} z_{t-1,1} & \\leftarrow & z_{t-1,2} \\\\ \\downarrow & \\searrow & \\downarrow \\\\ z_{t,1} & \\leftarrow & z_{t,2} \\\\ \\downarrow & \\searrow & \\downarrow \\\\ x_{t,1} & \\leftarrow & x_{t,2} \\end{array}

,the sparsity assumption is already satisfied although $z_{t,1}$ is fully connected to $x_{t,1}$ and $x_{t,2}$. This is because the corresponding Markov network is shown as follows

\\begin{array}{ccc} z_{t-1,1} &—& z_{t-1,2} \\\\ | & \\diagdown & | \\\\ z_{t,1} &—& z_{t,2} \\\\ | & \\diagdown & | \\\\ x_{t,1} &—& x_{t,2} \\end{array}

is also sparse. And the intimate neighbor set (defined in Line 202-204) of latent variables $z_{t,1}$ and $z_{t,2}$ are empty sets, which implies identifiability by using sufficient changes from historical information.

This mild assumption is sufficient to guarantee the component-wise identifiability of latent variables, as shown in Theorem 3. In light of your insightful question, we have provided a detailed explanation with the aforementioned graphical example in the updated version.

Q3: So even if the distribution changes in a way that some x_i becomes highly correlated with a block of x_i's driven by a latent variable, the sparsity constraint can prevent this x_i from being associated with that z_i?

A3: Thanks for your question! We would like to highlight that the sparsity constraint does not mean to prevent some observations from being associated with others.

Let us using the same example in A2, $x_{t,1}$ is indirectly influenced by $z_{t,2}$ through $x_{t,2}$. In this case, $x_{t,1}$ is associated with $z_{t,2}$ even if the data generation process is sparse.

Q4: It would be nice to have a cleaner description of the "online" component: how are updates done once a single new data point comes in?

A4: Thank you for this helpful suggestion. We follow the definition of online time series forecasting from [1,2], where the forecasting model generates future predictions using a moving window and updates the model immediately after the corresponding ground truth time series data arrive.

For a better understanding, we assume a first-order Markov process for both observations and latent variables as follows:

\\begin{aligned} & z_1 \\rightarrow z_2 \\rightarrow z_3 \\rightarrow z_4 \\rightarrow z_5 \\rightarrow z_6 \\\\ & \\downarrow\\quad\\;\\;\\;\\downarrow\\\;\\;\\;\\quad\downarrow\\quad\\\;\\;\\;\\downarrow\\quad\\;\\;\\;\\downarrow\\quad\\;\\;\\;\\;\\downarrow \\\\ & x_1 \\rightarrow x_2 \\rightarrow x_3 \\rightarrow x_4 \\rightarrow x_5 \\rightarrow x_6 \\end{aligned}

where we use the new arrived $(x_5)$ and the historical observations in moving window, e.g. $(x_1,x_2,x_3,x_4)$ to predict the values of $x_6$. Based on this example, the update procedure proceeds as follows:

1. Latent Variables Identification: When new data arrive, we use only the most recent adjacent observations to identify the latent variables. For example, given $(x_1, x_2, x_3, x_4, x_5)$, we can estimate $\hat{z}_3$ and $\hat{z}_4$ from $(x_1, x_2, x_3, x_4)$ and $(x_2, x_3, x_4, x_5)$, respectively (leverage our consecutive four observed variables).
2. Latent Variables Estimation: Online transition update: With the newly identified latent states, we learn the latent transition $p(z_t|z_{t-1})$ incrementally (e.g., updating from $(z_3,z_4)$ to infer $z_5, z_6$) as the transition of latent variables is stationary.
3. Prediction with Latent Variables: Finally, we combine the estimated latent variables $z_6$ with the latest available observations, i.e., $(x_1, x_2, x_3, x_4, x_5)$ to forecast the future observation $x_6$.

Thanks for your insightful question. We have provided a detailed example and an algorithm flow chart in the updated version.

[1] Lau, Ying-yee Ava, Zhiwen Shao, and Dit-Yan Yeung. "Fast and Slow Streams for Online Time Series Forecasting Without Information Leakage." ICLR 2025.
[2] Zhao, Lifan, and Yanyan Shen. "Proactive model adaptation against concept drift for online time series forecasting." KDD 2025.

Q5: Following up on the above. Can we try online learning in non-stationary environments? This methodology seems well-suitable for this kind of approach. And, do multi-step-ahead forecasting in such environments would be the ultimate robustness test of this methodology.

A5: Thank you for this insightful suggestion. Indeed, our methodology is general and naturally applicable to non-stationary environments. However, we would like to clarify the difference between our online forecasting setting and conventional non-stationary forecasting. In the nonstationary time series forecasting, we can leverage more historical time series data. In online forecasting, we cannot leverage the entire historical data; instead, we only have access to sequentially arriving data streams and must rely on the most recent observations for updating. Our theoretical contribution lies in showing how to identify the latent variables that drive non-stationarity with minimal adjacent observations, which is critical for real-time online updates.

We appreciate your suggestion to further evaluate our method on non-stationary time series forecasting. Specifically, we follow the experimental setting of [1] and use Koopa[1], FITS[2], and SOILD[3] as baselines. We evaluate our method on the ECL, ETTh2, Exchange, ILI, and Traffic datasets. The results are presented below.

We can find that our method can still achieve ideal performance in non-stationary environments.

[1] Liu. et al. "Koopa: Learning non-stationary time series dynamics with koopman predictors." NeurIPS 2023
[2] Xu. et al. "FITS: Modeling time series with $10 k$ parameters."ICLR2024
[3] Chen et al. "Calibration of time-series forecasting: Detecting and adapting context-driven distribution shift."KDD2024.

Q6: Do we explicitly care about the dynamics of the $z_t$'s at any point? Or only about which $z_t$'s are relevant at t?

A6: Thank you for this thoughtful question. We do care about the dynamics of $z_t$ and we explicitly model the dynamics of $z_t$ in Section 4.1. Specifically, when identifying the latent variables, we explicitly estimate the independent noise terms across different time steps, i.e., $\epsilon_{t,i} = r(z_{t,i}, z_{t-1})$. This procedure essentially models the temporal dynamics $z_{t-1}\rightarrow z_t$. In light of your questions, we have highlighted it in the final version.