Addressing Spatial-Temporal Heterogeneity: General Mixed Time Series Analysis via Latent Continuity Recovery and Alignment

We deeply appreciate Reviewer nZaK's positive acknowledgment of our work's originality, clarity, and quality. We are especially grateful for the detailed and insightful feedback provided. Rest assured, we are dedicated to addressing your concerns and enhancing our work.

W1.1&Q1: Smoothness constraints and K-Lipschitz continuity

Thanks for your insightful comments. These are all great points. We would like to address them with the following aspects:

• Purpose of the Smoothness Constraint: Consider an ideal continuous variable should be be equipped with interpretable autocorrelation or smooth variations, we design $\mathcal{L} _{\mathrm{Smooth}}$ to promote the recovered LCVs to achieve this. Also, we implement $\mathcal{L} _{\mathrm{Smooth}}$ with an adjustable coefficient $\lambda _1$. If a DV is known to commonly undergo sudden changes, $\lambda _1$ can be reduced accordingly.
• Collaboration and Mutual Restraint with Other Losses: Essentially, the smoothness loss works synergistically with other losses, such as reconstruction and task losses. While the smoothness loss aims to promote smooth changes, it is balanced by the other losses to prevent over-smoothing, which could negatively impact the performance of reconstruction and downstream tasks. That is to say, our model can adaptively learn a proper smoothness degree. We elaborated this point in $\underline{\text{Section 3.5 on Page 6}}$ of our paper, which may be of interest to you. Also, the sensitivity analysis of $\lambda _{1}$ presented in $\underline{\text{Figure 17 on Page 22}}$ verifies its robustness.
• K-Lipschitz Continuity as Constraints: We greatly appreciate your invaluable suggestion of using K-Lipschitz continuity, which is a wonderful idea and allows us to use CVs as guidance to determine the smoothness degree. We can formulate it as $\mathcal{L}_{\mathrm{smooth}}^{K\text{-Lipschitz}}$ =

$\sum_{t=1}^{T-1} \max( 0,|x_{t}^{LCV}-x_{t+1}^{LCV}|-K)$,

This term penalizes the consecutive points that exceed the bound $K$, which is determined by CVs. Empirically, we conducted experiments on the ETTh1&h2 dataset to verify its effectiveness as in $\underline{\text{Table 4 in global response PDF}}$, which shows comparable performance to our original formulation. In light of your suggestions, we promise to add the relevant results discussions to our paper.

W1.2&Q2: About the adversarial discrimination framework

Thank you for raising these excellent points. We are glad to address your concerns as follows:

• Discriminating Features in Time Series: For the recovered LCVs, we believe that if they present similar temporal variation features (e.g., autocorrelation, periodicity) and statistical features (e.g., distributions) like those in CVs, they can trick the discriminator. Also, we have provided visualization evidence in $\underline{\text{Figure 1 in global response PDF}}$, showing the LCVs accurately recover their actual temporal variations and distributions.

• Learning to Align Distributions: Actually, due to the difference in information granularity and distributions between CVs and DVs, directly modeling mixed variables would inevitably cause errors. The objective of our adversarial framework is to promote aligning the distributions of LCVs and CVs, akin to Domain Adaptation [1]. This alignment facilitates to model correlation across DVs (represented by LCVs) and CVs, which is crucial for downstream tasks.

• About Anomaly Samples: We agree that time series may experience abnormal variations, specifically for anomaly detection tasks. Here we would like to clarify that during the training phase, we focus on leveraging normal samples to capture the typical distributions and temporal variations, which is in line with previous works [2]. In the testing phase when anomaly may occur, the discriminator is discarded and irrelevant to the LCV recovery process.

• Training Stability: We ensure the overall training stability by incorporating other supervision signals, e.g., smoothness loss and task loss, whose synergy and mutual restraint have been discussed in $\underline{\text{Section 3.5 on Page 6}}$.

W2: Presentation issues

Thanks for your scientific rigor. Actually, we explain $z$ in lines 194~195 of our paper and we will further emphasize it. Also, in light of your suggestion, the error bars of MiTSformer will be included in our paper. You can also refer to $\underline{\text{Table 4 in global response PDF}}$ for relevant results.

W3: The effectiveness of LCV recovery

Thank you for the opportunity to clarify these points and improve our paper. We would like to address your concerns with the following aspects:

1. Visualization Evidence: As suggested, we have visualized the recovered LCVs and the actual LCVs, together with quantitative deviation analysis. Both results are included in $\underline{\text{Figure 1 and Table 1 in global response PDF}}$, showing that MiTSformer achieves accurate recovery of LCVs for DVs, which re-establishs their latent fine-grained and informative temporal variations.
2. Necessity of LCV Recovery: By recovering LCVs, we reduce the information granularity gap between DVs and CVs and align their distributions, enabling us to flexibly and reliably model correlations across CVs and DVs for various tasks.
3. Directly Modeling DVs: We conducted ablation experiments ($\underline{\text{Row 5 in Table 4 on Page 9}}$) where we directly fused the embeddings of CVs and DVs without LCVs recovery. As presented, the performance in both forecasting ($\mathrm{MAE}: 0.433 \rightarrow 0.529$ ) and classification tasks ($\mathrm{Acc}: 71.9 \rightarrow 69.3$ ) significantly decreased, demonstrating the necessity and effectiveness of LCV recovery.

Reference:

[1]. Unsupervised domain adaptation by backpropagation. ICML, 2015

[2]. Anomaly transformer: Time series anomaly detection with association discrepancy. ICLR, 2022.

Dear Reviewer nZaK：

We are thrilled to hear that our responses have made a positive impact on the paper. We sincerely apologize for any misunderstanding that may have occurred regarding your question and we appreciate your patience and time in rephrasing these questions. Rest assured, we are delighted to resolve your remaining concerns:

A1. About Smoothness Constraints:

Thanks for your detailed and constructive feedback. We promise to highlight the issues of addressing DVs with inherent sudden changes in our revised paper. Here we would like to address your concerns as follows:

• Handling DVs with Inherent Sudden Changes: Our strategy for managing DVs with inherent sudden changes involves two aspects:

  1. Adjusting the Weights of Smoothness Loss. We implement the smoothness loss with an adjustable coefficient $\lambda_1$ as $\lambda_1\mathcal{L} _{\mathrm{Smooth}}$, allowing for flexibility based on the specific characteristics of the DVs. If a DV is known to undergo inherent sudden changes, we can set a relatively small $\lambda_1$ accordingly to reduce the potential impact.
  2. Balancing the Smoothness Loss with Restraint from Other Losses. While the smoothness loss aims to promote smooth changes, it is restrained by the other losses to prevent LCVs from being too smooth to be their actual nature. Because over-smoothing for DVs with inherent sudden changes could detrimentally affect other critical loss objectives, such as can not accurately reconstruct the original DVs (i.e., impacting the reconstruction loss $\mathcal{L} _{\mathrm{Rec}}$ in Equation 5) and affect the downstream tasks (i.e., impacting the task loss $\mathcal{L} _{\mathrm{Task}}$ such as classification accuracy). Thereby, these loss terms can balance and constrain $\mathcal{L} _{\mathrm{Smooth}}$ to some extent when encountering DVs with inherent sudden changes.

• Limitations: We acknowledge that our framework, despite being applicable in most time series scenarios, may still have limitations, especially for some extreme cases that involve drastic sudden changes. As suggested, we will explicitly highlight the applicable scopes and limitations as follows:
  "The smoothness loss in our method is suitable for DVs with sudden changes that are caused by inherent smooth variations. However, it may not adequately account for DVs with inherent sudden changes that are essential characteristics of the dataset. For such cases, we can further leverage less restrictive constraints such as K-Lipschitz continuity to determine the proper smoothness level."

A2. About the Discrimination:

We are delighted that you can agree with our ideas. Also, we greatly agree that real data examples can help to improve the interpretability. We are keen to provide the comparison showcases of the CVs and LCVs recovered from DVs for you in our anonymous Github:https://anonymous.4open.science/r/MiTSformer/Visualization_showcases/Supplementary_Figures_for%20Reviewer_nZaK.pdf. By the way, you can also refer to $\underline{\text{Figure 1 in global response PDF}}$, which plots the recovered LCVs and the actual LCVs, showing the recovered LCVs are indeed equipped with key temporal variation features like autocorrelations, trends, and periodic patterns, like those in CVs.

A3. About LCV and CV:

Thanks for kindly raising this concern. Our original statements may cause some misunderstandings. Here we would like to kindly clarify that the results you requested might be included in $\underline{\text{Figure 1 in Global Response PDF}}$ ( you can download it at the bottom of Summary of Revisions and Global Response ). Specifically, the results are based on ETTh1 and ETThh2 benchmarks [1][2], which are real-world electric transformer datasets. The original ETT datasets consist of purely continuous variables. To meet our mixed variable setting, we processed these datasets by randomly selecting half of the variables discretizing them into discrete variables (DVs). Such operation also allows us to visualize the recovered LCVs and the actual LCVs. As depicted in the figures, our MiTSformer achieves accurate recovery of LCVs for DVs, which re-establishes their actual fine-grained and informative temporal variations. Also, we conduct quantitative error analysis in $\underline{\text{Table 1 in Global Response PDF}}$, further validating the accuracy and necessity of LCV recovery.

We hope these revised explanations have addressed your concerns. Please do not hesitate to contact us if we have not answered your question completely. In addition, we would like to express our gratitude for requesting clarification. Your feedback has been immensely valuable in enhancing the overall quality of our work!

Reference

[1]. Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting. AAAI, 2021.

[2]. Film: Frequency improved legendre memory model for long-term time series forecasting. NeurIPS, 2022.

Dear Reviewer nZaK:

We deeply appreciate the time and effort you have invested in reviewing our paper and providing such invaluable comments. We are also truly grateful for your timely and insightful feedback during the discussion period. In the days following your feedback, we have dedicated ourselves to thoroughly studying your comments. This reflection has significantly deepened our understanding of the critical issues you highlighted. In response, we are eager to offer additional discussions and clarifications to better illustrate our work and effectively address the concerns raised.

Supplement to A1: About Smoothness Constraints:

• Purpose of the Smoothness Constraint: we design $\mathcal{L} _{\mathrm{Smooth}}$ to promote the recovered LCVs to be equipped with interpretable autocorrelation or smoothly varying trend properties, which are suitable for DVs with apparent sudden changes that is potentially caused by inherent smooth variations. Intuitively, consider a sine wave $\mathrm{sin}(t)$ with a mean of 0 that is discretized based on a threshold of 0, transforming it into a binary discrete time series as

$$x^{DV}_t = 1 \text{ if } \sin(t) > 0, \quad 0 \text{ if } \sin(t) \leq 0$$

Around the threshold $0$, even small variations would lead to sudden changes of $x^{DV}_t$ to the opposite state, while the underlying variation process of $\mathrm{sin}(t)$ is smooth and continuous. Our smoothness constraint can help to recover its inherent actual continuous nature.

• Limitations to handle DVs with Inherent Sudden Changes: We agree that in some cases there are DVs with inherent sudden changes that are essential characteristics of the dataset. Our strategy for managing them involve adjusting the weights $\lambda _1$ of $\mathcal{L} _{\mathrm{Smooth}}$ and balancing the $\mathcal{L} _{\mathrm{Smooth}}$ with restraint from other losses, e.g., $\mathcal{L} _{\mathrm{Rec}}$ and $\mathcal{L} _{\mathrm{Task}}$, as discussed in our previous reply. Also, we acknowledge that such strategies may still have limitations, especially for some extreme cases that involve drastic sudden changes, and we committed to discussing this issue in our revised paper.

Supplement to A2. About the Discrimination & A3. About LCV and CV:

• Visualization Evidence: We would like to kindly remind you that the results you requested, i.e., visualization plots of the recovered LCVs and the actual LCVs, might be included in $\underline{\text{Figure 1 in Global Response PDF}}$ (in the bottom of Summary of Revisions and Global Response). For your convenience, we have also prepared an anonymous link for you as https://anonymous.4open.science/r/MiTSformer/Visualization_showcases/Supplementary_Figures_for%20Reviewer_nZaK.pdf, which plots the actual/recovered LCVs (Figure 1) and other observed CVs (Figure 2). The visualization suggests that the LCVs are accurately recovered and are indeed equipped with key temporal variation features like autocorrelations, trends, and periodic patterns, like those in CVs.
• Necessity and Effectiveness of LCV recovery: Due to the difference in information granularity and distributions between CVs and DVs, directly modeling mixed variables would inevitably cause errors. Thereby, in our work, we propose to recover the latent continuous variables (LCVs) behind DVs to reduce the information granularity gap between CVs and DVs and align the distributions. The recovered LCVs can further facilitate the spatial-temporal correlation modeling among mixed variables, and benefit various downstream tasks.

We sincerely thank Reviewer ZwAv for the positive evaluation of our work's innovativeness, creativity, and technical quality. Your detailed and insightful comments have helped us improve our work substantially. We hope your concerns are addressed.

W1&Q3: Mathematical formulas and the LCV Recovery

Thanks for your kind valuable suggestion. We are glad to provide a more detailed and explicit explanation:

1. LCV Recovery Network: Our LCV recovery is based on a recovery network composed of residual dilated convolutional neural networks, which hierarchically aggregate multi-scale temporal context information to transform DVs into LCVs. Also, we provided the detailed network structure in $\underline{\text{Appendix A.3 on Page 13}}$.
2. Mathematical Formulation: The recovery network receives a DV as input and outputs its LCV, as depicted in $\underline{\text{Figure 4 on Page 5}}$. This process can be mathematically described as

$$x^{LC} = \mathrm{Rec}\text{-}\mathrm{Net}(x^{D}) = h_{n}$$

where the residual dilated convolutional network is defined by the iterative process as

$$h_{i} = \mathrm{Conv}^{d_{i}}(h_{i-1}) + h_{i-1}, \quad \text{for} \; i = 1, 2, \ldots, n$$

In this formulation, $h_{i}$ represents the output of the $i$-th residual block, $\mathrm{Conv}^{d_{i}}$ denotes the convolution operation along the temporal axis to aggregate contextual information with dilation rate $d_{i}$. The final output, $x^{LC}$, is the result after $n$ residual blocks. Additionally, we perform z-score normalization for each $x^{LC} \in \mathbb{R}^{1 \times T}$ to ensure training stability as

$$x^{LC} = \frac{x^{LC}-\mathrm{Mean}(x^{LC})}{\mathrm{Std}(x^{LC})}$$

where $\mathrm{Mean}$ and $\mathrm{Std}$ denote the mean and the standard deviation along the time axis, respectively. According to your suggestion, we promise to include these detailed formulations in our submission to enhance clarity and comprehensiveness.

W2: Better explaining the smoothness regularization term

We appreciate your constructive feedback. We would like to clarify this issue in detail:

Inspired by the temporal autocorrelation nature of continuous variables, we devise the smoothness constraint loss $\mathcal{L} _{\mathrm{smooth}}$ to regularize the LCV recovery process. Mathematically, we aim to enforce the smoothness of adjacent points of LCVs as

$$\sum_{t=1}^{T-1}{\mathrm{Abs}( x_{t+1}^{D}-x_{t}^{D}) \cdot (x_{t+1}^{LC}-x_{t}^{LC}) ^2}$$

where $\mathrm{Abs}\left( x_{t-1}^{D}-x_{t}^{D} \right)$ denotes the absolute difference between two adjacent time points of DVs, whose value can be $\\{0, 1 \\}$ ( $1$ indicates "sudden change" and $0$ indicates "no sudden change"). For computational efficiency, we introduce a constant-valued smoothness matrix $\boldsymbol{S}$ to achieve the above objective by multiplying it with $x^D$ and $x^{LC}$ as

$$\mathcal{L} _{\mathrm{smooth}}=\left\| \mathrm{Abs}\left( \boldsymbol{S}x^D \right) \otimes \left( \boldsymbol{S}x^{LC} \right) \right\| _{2}^{2}$$

where $\otimes$ denotes the Hadamard product operation.
Thanks for highlighting this issue, we promise to better explain the smoothness regularization term in our paper.

W3: Strengthening the analytical explanation of LCVs

Thanks for your kind valuable suggestion. For a better investigation of LCV recovery, we have provided both visualization and quantitative analysis of the recovered LCVs and their actual LCVs. You can refer to $\underline{\text{Figure 1 and Table 1 of global response PDF}}$ for the results.

As presented, MiTSformer achieves accurate recovery of LCVs for DVs, which re-establishs their continuous fine-grained and informative temporal variation patterns. By recovering LCVs, we reduce the information granularity gap between DVs and CVs and align their distributions, which facilitates sufficient and balanced correlation modeling of mixed variables and further enhances the performance of various tasks. We promise to add relevant results and discussions to our paper to further validate the effectiveness and necessity of LCV recovery.

Q1: Scale with large datasets or in real-time applications

Thanks for your insightful questions. We would like to address your concerns from the following aspects.

• For Scaling with Large Datasets: First, MiTSformer exhibits great computational efficiency as demonstrated in $\underline{\text{Appendix B on Page 20}}$. Secondly, we can empower MiTSformer with efficient linear self-attention techniques, e.g., Flowformer [1] to further reduce the computational burdens. Furthermore, we can leverage efficient pre-training and fine-tuning frameworks like LoRA [2] for scalability at deployment.
• For Real-Time Applications: MiTSformer can process real-time mixed time series data in a similar way to most time series models. In real-time applications, well-trained MiTSformer can be deployed in online systems. When new data points arrive, they can be appended to the historical data to form a full sequence and then fed into MiTSformer to generate online predictions.

According to your feedback, we will supplement the above discussions in our paper.

Q2: Computational complexity of MiTSformer

Thanks for your invaluable feedback. We highly agree that computational complexity is a key issue for investigating the MiTSformer's practicality, and we analyzed it in terms of training time and memory cost, which are included in $\underline{\text{Appendix B on Page 20}}$. In general, MiTSformer maintains great performance and efficiency, especially for the training time. Enlightened by your feedback, we promise to further emphasize computational complexity analysis in the main text of our paper.

Reference:

[1]. Flowformer: Linearizing transformers with conservation flows. ICML, 2022.

[2]. LoRA: Low-Rank Adaptation of Large Language Models. ICLR, 2022.

We deeply appreciate Reviewer 3NoE's positive acknowledgment of our methodology's innovation, comprehensiveness, effectiveness, and experimental robustness. We are especially grateful for the constructive and insightful feedback provided. Rest assured, we are dedicated to addressing your concerns.

W1: Implementation and usability

Thanks for kindly raising this concern and we would like to address it for you below:

• Necessity of Each Module: We would like to emphasize that mixed time series modeling differs significantly from typical time series tasks due to variable heterogeneity. Thereby, each component in MiTSformer is essential to address it via latent continuity recovery and alignment. Our experiments, particularly the ablation studies displayed in $\underline{\text{Table 4 on Page 9}}$ further justify this point.
• User Accessibility: Like most deep time series models, MiTSformer can be trained and deployed in an end-to-end manner. Also, we have tried our best to make our framework accessible to practitioners by providing open-source and well-organized codes and scripts with sufficient implementation details ($\underline{\text{Appendix A on Pages } 13\sim 19}$) to facilitate ease of use.

In light of your feedback, we will add relevant clarifications to our paper.

W2: Data dependency

You are right that the performance of MiTSformer, like most deep learning models, benefits from high-quality training data. However, we can empower MiTSformer's robustness under data limitations with advanced techniques such as Data Augmentation [1] and Self-supervised Pre-Training [2]. We promise to update these discussions in our paper.

W3: Scalability to very large datasets

This is an interesting question. We would like to elaborate it with the following points:

1. Computational Efficiency: As demonstrated in the computational efficiency analysis in $\underline{\text{Appendix B on Page 20}}$, MiTSformer exhibits great efficiency concerning memory footprint and training time, which ensures the feasibility for very large datasets.
2. Fast Attention: MiTSformer adopts self- and cross-attention to model variable correlations, which is the primary source of computational load. When encountering datasets with a large number of variables, we can adopt efficient linear attentions, e.g., Flowformer [3] to further reduce the computational burdens.
3. Efficient Pre-training and Fine-tuning: We can empower MiTSformer with efficient pre-training and fine-tuning frameworks like LoRA [4] to further enhance the scalability.

Enlightened by your feedback, we will include these discussions in our paper.

W4: Explanation of Hyperparameters

We highly agree that hyperparameters are key issues for MiTSformer. Owing to page constraints, we had to provide the details of the hyperparameter setting in $\underline{\text{Appendix A on Pages 13-15}}$, and hyperparameter sensitivity analysis in $\underline{\text{Appendix C on Pages 20-22}}$. In summary, we find that MiTSformer is relatively stable in the selection of $d_{model}$ and $l_{layer}$. Also, MiTSformer is quite robust to the weights of loss items (i.e., $\lambda_1$, $\lambda_2$, and $\lambda_3$), and moderate weights (e.g., $0.3 \sim 1.0$) would bring optimal performance. Also, we promise to further emphasize hyperparameter analysis in the main text.

Q1: Handle non-linear patterns

MiTSformer effectively handles highly non-linear temporal patterns in both CVs and DVs via Spatial-temporal attention blocks as depicted in $\underline{\text{Figure 5 on Page 6}}$, where self-attention and cross-attention can model the nonlinear spatial-temporal correlations within LCVs and CVs and across LVs and CVs, respectively.

Q2: Handle non-binary discrete variables

This is an insightful and interesting question. A quick answer is "yes". Actually, DVs can take on multiple states ($\geq 2$) that reflect the magnitude and can be directly input into the recovery network to obtain LCVs outputs. It is noted that more states in a DV imply richer information granularity. We chose the binary scenarios in our experimental settings as they are the most challenging and commonly encountered in the real world.

Also, we conducted experiments on classification datasets with various numbers of discrete states in $\underline{\text{Table 2 of global response PDF}}$, showing that as the number of states increases, accuracy improves due to the richer information. The above results and discussions will be included in our paper.

Q3： Computational efficiency

We highly agree that computational complexity is a key issue for MiTSformer's practicality, and we analyzed it in terms of training time and memory cost, which are included in $\underline{\text{Appendix B on Page 20}}$. In general, MiTSformer maintains great performance and efficiency. Enlightened by your feedback, we promise to further emphasize the analysis of computational efficiency in the main text.

Q4: Real-time applications

Thanks for your valuable question. MiTSformer can process sequentially arrived data in a similar way to most time series models. For example, well-trained MiTSformer can be deployed in real-time systems. When new data sequentially arrive, they are appended to the historical data to form a full sequence, which is then fed into the well-trained model to generate online predictions.

Reference：

[1]. Self-supervised contrastive representation learning for semi-supervised time-series classification. IEEE TPAMI, 2023.

[2]. TimeSiam: A Pre-Training Framework for Siamese Time-Series Modeling. ICML, 2024.

[3]. Flowformer: Linearizing transformers with conservation flows. ICML, 2022.

[4]. LoRA: Low-Rank Adaptation of Large Language Models. ICLR, 2022.

We sincerely thank Reviewer GdPs for the favorable recognition of our work's problem setting, effectiveness, and invaluable contribution to designing robust time series foundation models. We deeply appreciate your detailed and perceptive feedback. Rest assured, we are committed to addressing your concerns and improving our work.

W1: Experimental settings of DVs

Thanks for raising this insightful point. Actually, in the real world, many DVs originate from latent CVs due to constraints like measurement and storage limitations. We elaborate on this in $\underline{\text{Line } 47 \sim 58 \text{ and Figure 2}}$ in our paper, which may interest you. Our conversion process simulates the generation process of DVs and discretizes variables while preserving their inherent coupling relationships and properties. This operation also allows us to compare the recovered LCVs with the actual ones, which are investigated in the responses to W3&Q2 for you. In light of your feedback, we will add relevant clarifications to our paper.

W2: Practical implementation and scalability

We appreciate your valuable insights and are glad to address your concerns below.

• Necessity: We would like to emphasize that mixed time series modeling differs largely from typical time series tasks due to the variable heterogeneity. Each module in our MiTSformer is essential to address this problem. Specifically, the recovery network is a lightweight module but is indispensable for recovering LCVs. The ablation study displayed in $\underline{\text{Table 4 on Page 9}}$ also justifies its effectiveness.
• Computational Efficiency: We analyzed computational effort in $\underline{\text{Appendix B on Page 20}}$, demonstrating the great efficiency of MiTSformer. Additionally, we conducted an experiment for the recovery network based on ETTh1, showing that the computational costs brought by the recovery network are minimal, while the performance improvements are significant.

• Pre-training for Real-world Applications: We can employ self-supervised pre-training for the backbone of MiTSformer, whose parameters can be fixed during deployment and we can fine-tune the task head if needed, ensuring both efficiency and scalability.

Thanks for the opportunity to clarify these issues and we will add them to our paper.

W3&Q2: Verification of LCV recovery

This is an invaluable and interesting suggestion. As suggested, we have included the relevant visualizations and quantitative evaluations in $\underline{\text{Figure 1 and Table 1 of global response PDF}}$, showing MiTSformer achieves accurate recovery of LCVs for DVs, which re-establishs their continuous and fine-grained temporal variations. In light of your suggestion, we will add relevant results and discussions to our paper.

W4: Effectiveness of the discrimination loss

Thanks for your rigorous concerns. we design the variable modality discrimination loss $\mathcal{L} _{\mathrm{Dis}}$ to align the feature distributions of CVs and LCVs, facilitating their correlation modeling. The discrimination loss works collaboratively with other loss terms and we discussed their synergy in $\underline{\text{Section 3.5 on Page 6}}$. The performance decrease without the discrimination loss is not substantial because smoothness loss $\mathcal{L} _{\mathrm{Smooth}}$ and reconstruction loss $\mathcal{L} _{\mathrm{Rec}}$ can promote the recovery of LCVs to some extent. However, incorporating $\mathcal{L} _{\mathrm{Dis}}$ can further provide a consistent performance gain throughout all datasets for various tasks.

Q1: Comparison with mixed naive Bayes (NB) and variational inference (VI)-based methods

Thanks for your constructive feedback. We would like to resolve your concerns regarding the following aspects:

• Theoretically, mixed NB and VI methods are typically designed for tabular data [1][2]. They struggle with time series, and often rely on certain assumptions like conditional independence [3], limiting their ability to model the correlations of DVs and CVs [2]. In contrast, MiTSformer leverages the temporal adjacencies to achieve latent continuity recovery, making it capable of handling time series data and effectively capturing inherent nonlinear correlations.

• Empirically, we compare MiTSformer against typical mixed NB-based models HVM [1] and VI-based methods VAMDA [2] on mixed time series classification datasets and summarized the results in $\underline{\text{Table 1 of global response PDF}}$, showing MiTSformer consistently outperforms HVM and VAMDA. We promise to include these results and discussions in our paper.

About the limitations:

Thanks for highlighting this. The original expressions in Appendix D may have caused some misunderstanding. We would like to clarify that our framework is designed for DVs with numerical magnitude by effectively recovering their LCVs. Actually, categorical variables can be divided into those with numerical magnitude (e.g., ordinal categories) and those without numerical magnitude (i.e., nominal categories such as gender). Our framework is effective for the former, which is frequently encountered in real-world time series. For the event data and sales demand that you have mentioned, if categorical data represent varying levels of a quantity (e.g., sales volume), MiTSformer can also effectively handle it. We will revise the relevant clarifications to avoid any confusion.

Reference:

[1]. Hybrid variable monitoring: An unsupervised process monitoring framework with binary and continuous variables. Automatica, 2023.

[2]. A flexible probabilistic framework with concurrent analysis of continuous and categorical data. IEEE TII, 2023.

[3]. Naive Bayesian classification of structured data. Machine learning, 2004

Dear Reviewer GdPs：

Thanks sincerely for acknowledging the motivation of our work. We sincerely apologize for any misunderstanding that may have occurred regarding your question and we appreciate your patience and time in rephrasing these questions. Rest assured, we are glad to resolve your concerns.

Q1. The necessity and benefit of converting some of CVs to DVs:

Here we would like to further clarify our experimental settings for generating DVs from the following aspects：

• Modeling Mixed Time Series: Different from existing methods that primarily focus on time series composing purely continuous variables, our research focuses on modeling mixed time series that are composed of both continuous variables (CVs) and discrete variables (DVs), which are frequently encountered in real-world scenarios. Essentially, due to measurement limitations or storage requirements, many intrinsically continuous-valued signals are often recorded with discrete-valued forms as DVs. Given an example in industrial sensing systems, some temperature variables are recorded as binary alarm signals based on whether they exceed the control limits.

• Necessity and Benefits: Despite being a fundamental issue, modeling mixed time series remains underexplored in academia, with a lack of specialized benchmark datasets for mixed time series. To meet the tasks for mixed time series, we employed benchmark time-series datasets and implemented discretization to convert some CVs into DVs. We would like to emphasize that such discretization strictly follows our problem formulation and aligns with the aforementioned generation mechanisms of DVs, which does not alter the intrinsic relationships of the variables but presents them in discrete forms. Also, the original CVs selected for discretization can be considered as the ground truth of the latent continuous variables (LCVs), which allows for comparisons between the actual LCVs and the recovered LCVs. Thereby, the discretized datasets can be viewed as benchmarks for mixed time series to facilitate future research.

• Application to datasets with natural DVs: In addition, our model is also applicable to datasets with natural DVs. We are carrying out experiments on a real thermal power plant dataset that contains alarm discrete signals without the corresponding original continuous signals. Also, we compare the differences between these two ways as follows:

Owing to time limitations, we would like to explain the following questions in the current reply first and we are committed to including the relevant results in our revised paper.

Q2. The necessity of recovering LCVs from DVs & About simply observe the relationship between CVs

In our experimental setting, the LCVs are unobservable, and we can only observe the DVs and other CVs. Thereby, we can not directly observe relationships between CVs and LCVs. Also, due to the difference in information granularity and distributions between CVs and DVs, directly modeling mixed variables would inevitably cause errors. Thereby, in our work, we focus on recovering the LCVs for DVs to reduce the information granularity gap and align the distributions, which can facilitate the spatial-temporal correlation modeling among mixed variables and benefit various tasks.

Q3. How much continuous and fine-grained temporal variations can be recovered when recovering LCVs for DVs:

We have included showcases based on ETTh1&h2 datasets to visualize the (i). observed DVs, (ii) recovered LCVs, and (iii) actual LCVs (i.e., DVs before discretization) in $\underline{\text{Figure 1 in Global Response PDF}}$. These visualization plots show our method accurately re-establishes the continuous fine-grained and informative temporal variation patterns of actual LCVs for the observed DVs. Specifically, the recovered LCVs are indeed equipped with key temporal variation features such as autocorrelations, trends, and periodic patterns, like those in CVs. For your convenience, we have also prepared an anonymous link for you (https://anonymous.4open.science/r/MiTSformer/Visualization_showcases/Supplementary_Figures_for_Reviewer_GdPs.pdf), which involves the visualization plots of the recovered/actual LCVs and observed CVs.

We hope these revised explanations have addressed your concerns. Please do not hesitate to contact us if we have not answered your question completely. In addition, we would like to express our gratitude for requesting clarification. Your feedback has been immensely valuable in enhancing the overall quality of our work!