Transparent Networks for Multivariate Time Series

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 14:37:45

Abstract: Transparent models, which are machine learning models that produce inherently interpretable predictions, are receiving significant attention in high-stakes domains. However, despite much real-world data being collected as time series, there is a lack of studies on transparent time series models. To address this gap, we propose a novel transparent neural network model for time series called Generalized Additive Time Series Model (GATSM). GATSM consists of two parts: 1) independent feature networks to learn feature representations, and 2) a transparent temporal module to learn temporal patterns across different time steps using the feature representations. This structure allows GATSM to effectively capture temporal patterns and handle dynamic-length time series while preserving transparency. Empirical experiments show that GATSM significantly outperforms existing generalized additive models and achieves comparable performance to black-box time series models, such as recurrent neural networks and Transformer. In addition, we demonstrate that GATSM finds interesting patterns in time series. The source code is available at https://anonymous.4open.science/r/GATSM-78F4/.

Main paper: 
 claims: 
 Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes]Justification: The main claims made in the abstract and introduction accurately reflect the paper's contributions and scope.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper.• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.• The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.• It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
related works: 
 Various XAI studies have been conducted over the past decade [7, 8, 9, 17, 18] ; however, they are less relevant to the transparent model that is the subject of this study. Therefore, we refer readers to [19, 20] for more detailed information on recent XAI research. In this section, we review existing transparent models closely related to our GATSM and discuss their limitations. The simple linear model is designed to fit the conditional expectation g (E (y| x)) = M i=1 x i w i ,where g(•) is a link function, M indicates the number of input features, y is the target value for the given input features x ∈ R M , and w i ∈ R is the learnable weight for x i . This model captures only linear relationships between the target y and the inputs x. To address this limitation, GAM [10] extends the simple linear model to the generalized form as follows:EQUATIONwhere each f i (•) is a function that models the effect of a single feature, referred as a feature function.Typically, f i (•) becomes a non-linear function such as a decision tree or neural network to capture non-linear relationships.Originally, GAMs were fitted via the backfitting algorithm using smooth splines [10, 21] . Later, Yin Lou et al. [22] and Harsha Nori et al. [23] have proposed boosted decision tree-based GAMs, which use boosted decision trees as feature functions. Spline-and tree-based GAMs have less flexibility and scalability. Thus, extending them to transfer or multi-task learning is challenging. To overcome this problem, various neural network-based GAMs have been proposed in recent years. Potts [24] introduced generalized additive neural network, which employs 2-layer neural networks as feature functions. Similarly, Rishabh Agarwal et al. [7] proposed neural additive model (NAM) that employs multi-layer neural networks. To improve the scalability of NAM, Chun-Hao Chang et al. [8] and Filip Radenovic et al. [9] proposed the neural oblivious tree-based GAM and the basis network-based GAM, respectively. Xu et al. [25] introduced a sparse version of NAM using the group LASSO. One disadvantage of GAMs is their limited predictive power, which stems from the fact that they only learn first-order feature interactions-i.e., relationships between the target value and individual features.To address this, various studies have been conducted to enhance the predictive powers of GAMs by incorporating higher-order feature interactions, while still maintaining transparency. GA 2 M [26] simply takes pairwise features as inputs to learn pairwise interactions. GAMI-Net [27] , a neural network-based GAM, consists of networks for main effects (i.e., first-order interactions) and pairwise interactions. To enhance the interpretability of GAMI-Net, the sparsity and heredity constraints are added, and trivial features are pruned in the training process. Sparse interaction additive network [28] 𝑥!,! ℎ# 𝑥!,! ℎ! 𝑥!,! ℎ$ 𝑥!,! is a 3-phase method for exploiting higher-order interactions. Initially, a black-box neural network is trained; subsequently, the top-k important features are identified using explainable feature attribution methods like LIME [1] and SHAP [2] , and finally, NAM is trained with these extracted features.… 𝑥!,# ℎ# 𝑥!,# ℎ! 𝑥!,# ℎ$ 𝑥!,# … 𝑥!,% ℎ# 𝑥!,% ℎ! 𝑥!,% ℎ$ 𝑥!,% … … 𝑤 !,! #$% 𝑤 !,& #$% 𝑤 !,' #$% 𝑤 !,! #$% 𝑤 !,& #$% 𝑤 !,' #$% 𝑤 !,! #$% 𝑤 !,& #$% 𝑤 !,' #$% 𝑥 #!,! 𝑥 #!,Dubey et al. [29] introduced scalable polynomial additive model, an end-to-end model that learns higher-order interactions via polynomials. Similarly, Kim et al. [15] proposed higher-order NAM that utilizes the feature crossing technique to capture higher-order interactions. Despite their capabilities, the aforementioned GAMs cannot process time series data, which limits their applicability in realworld scenarios. Recently, neural additive time series Model (NATM) [30] , a time-series adaptation of NAM, has been proposed. However, NATM handles each time step independently with separate feature networks. This approach cannot capture effective temporal patterns and only takes fixed-length time series as input. Our GATSM not only captures temporal patterns but also handles dynamic-length time series. Table 1 shows the advantages of our GATSM compared to existing GAMs.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: The limitations of our work are described in section 6.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.• The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.• The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.• The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.• The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.• If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.• While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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problem statement: 
 We tackle the problem of the existing GAMs on time series. Equation (1) outlines the GAM framework for tabular data, which fails to capture the interactions between current and previous observations in time series. A straightforward method to extend GAM to time series, adopted in NATM, is applying distinct feature functions to each time step and summing them to produce predictions:EQUATIONwhere X ∈ R T ×M is a time series with T time steps and M features, and t is the current time step.This method can handle time series data as input but fails to capture effective temporal patterns because the function f i,j (•) still does not interact with previous time steps. To overcome this problem, we suggest a new form of GAM for time series defined as follows:EQUATIONDefinition 3.1 GAMs for time series, which capture temporal patterns hold the form of Equation 3.In Equation ( 3), the function f (•, •) can capture interactions between current and previous time steps.Therefore, GAMs adhering to Definition 3.1 are capable of capturing temporal patterns. However, implementing such a model while maintaining transparency poses challenges. In the following section, we will describe our approach to implementing a GAM that holds Definition 3.1. To the best of our knowledge, no existing literature addresses Definition 3.1.4 Our Method: Generalized Additive Time Series Model To overcome this problem, we extend neural basis model (NBM) [9] to time series as:EQUATIONWe refer to this extended version of NBM as time-sharing NBM. Time-sharing NBM has B basis functions, with each basis h k (•) taking a feature x i,j as input. The feature-specific weight w nbm j,k then projects the basis to the transformed feature xi,j . As depicted in Equation 4, the basis functions are shared across all features and time steps, drastically reducing the number of required feature functions T × M to B. We use B = 100 and implement h k (•) using multi-layer perceptron (MLP).Masked MHA: GATSM employs multi-head attention (MHA) to learn temporal patterns. Although the dot product attention [16] is popular, simple dot operation has low expressive power [31] .Therefore, we adopt the 2-layer attention mechanism proposed by [31] to GATSM. We first transform 4as follows:xi = [x i,1 , xi,2 , • • • , xi,M ] ∈ R M produced by EquationEQUATIONwhereZ ∈ R M ×D is a learnable weight, pe i = [pe i,1 , pe i,2 , • • • , pe i,D ] ∈ R D is the positionalencoding for i-th step, and D indicates the hidden size. The positional encoding is defined as follows:EQUATION)The positional encoding helps GATSM effectively capture temporal patterns. While learnable position embedding also works in GATSM, we recommend positional encoding because position embedding requires knowledge of the maximum number of time steps, which is often unknown in real-world settings. After computing v i , we calculate the attention scores as follows:EQUATIONEQUATION)where k is attention head index, σ (•) is an activation function, w attn k ∈ R 2D , and m i,j ∈ R is the mask value used to block future information. The time mask is defined as follows:EQUATION)Inference: The prediction of GATSM is produced by combining the transformed features from time-sharing NBM with the attention scores from masked MHA.EQUATION)where K is the number of attention heads,a k,t = [a k,i,1 , a k,i,2 , • • • , a k,i,T ] ∈ R T is the attention map in Equation 8, X = [x 1 , x2 , • • • , xT ] ∈ R T ×Mis the transformed features in Equation 4, andw out k ∈ R M is the learnable output weight.Interpretability: We can rewrite Equation 10 as the following scalar form:EQUATIONEquation 11 shows that GATSM satisfying Definition 3.1. We can derive three types of interpretations from GATSM: 1) a k,t,u indicates the importance of time step u at time step t, 2)h b (x t,m ) w nbm m,b w out k,mrepresents the time-independent contribution of feature m, and 3) a k,t,u h b (x t,m ) w nbm m,b w out k,m represents the time-dependent contribution of feature m at time step t.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [NA]Answer: [Yes] Justification: We used public datasets and opened our code.Guidelines:• The answer NA means that paper does not include experiments requiring code.• Please see the NeurIPS code and data submission guidelines (https://nips.cc/ public/guides/CodeSubmissionPolicy) for more details.• While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. 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experimental setup: 
 Datasets: We conducted our experiments using eight publicly available real-world time series datasets. From the Monash repository [32] , we sourced three datasets: Energy, Rainfall, and AirQuality. Another three datasets, Heartbeat, LSST, and NATOPS, were downloaded from the UCR repository [33] . The remaining two datasets, Mortality and Sepsis, were downloaded from the PhysioNet [34] . We perform ordinal encoding for categorical features and standardize features to have zero-mean and unit-variance. For forecasting tasks, target value y is also standardized to zero-mean and unit-variance. If the dataset contains missing values, we impute categorical features with their modes and numerical features with their means. The dataset is split into a 60%/20%/20% ratio for training, validation, and testing, respectively. Table 2 shows the statistics of the experimental datasets. Further details of the experimental datasets can be found in Appendix B. Baselines: We compare our GATSM with 12 baselines, which can be categorized into four groups: 1)Black-box tabular models include extreme gradient boosting (XGBoost) [35] and MLP. 2) Black-box time series models include simple recurrent neural network (RNN), gated recurrent unit (GRU), long short-term memory (LSTM), and Transformer [16] . 3) Transparent tabular models are simple linear model (Linear), explainable boosting machine (EBM) [23] , NAM [7] , NodeGAM [8] , and NBM [9] . 4) NATM [30] is a transparent time series model.We implement XGBoost and EBM models using the xgboost and interpretml libraries, respectively. For NodeGAM, we employ the official implementation provided by its authors [8] . The remaining models are developed using PyTorch [36] . All models undergo hyperparameter tuning via Optuna [37] . The pytorch-based models are optimized with the Adam with decoupled weight decay (AdamW) [38] optimizer on an NVIDIA A100 GPU. Model training is halted if the validation loss does not decrease over 20 epochs. We use mean squared error for the forecasting tasks, and for classification tasks, we use cross-entropy loss. Further details of the model implementations and hyper-parameters are provided in Appendix C.Table 3 shows the predictive performances of the experimental models. We report mean scores and standard deviations over five different random seeds. For the forecasting datasets, we evaluate R 2 scores. For the binary classification datasets, we assess the area under the receiver operating characteristic curve (AUROC). For the multi-class classification datasets, we measure accuracy. We highlight the best-performing model in bold and underline the second-best model. Since the tabular models cannot handle time series, they only take x t to produce y t .On the Energy and Heartbeat datasets, which are small in size, our GATSM demonstrates the best performance, indicating strong generalization ability. EBM, XGBoost, and Transformer struggle with overfitting on the Energy dataset. For the Mortality and Sepsis datasets, there is no significant performance difference between tabular and time series models, nor between black-box and transparent models. This suggests that these two healthcare datasets lack significant temporal patterns and feature interactions. It is likely that seasonal patterns are hard to detect in medical data, and the patient's current condition already encapsulates previous conditions, making historical data less crucial. Since these datasets contain variable-length time series, the performance of NATM, which can only handle fixed-length time series, is not available. On the Rainfall, AirQuality, LSST, and NATOPS datasets, the time series models significantly outperform the tabular models, indicating that these datasets contain important temporal patterns that tabular models cannot capture. Additionally, the black-box models outperform the transparent models, suggesting that these datasets have higher-order feature interactions that transparent models cannot capture. Nevertheless, GATSM is the Choice of feature function: We evaluate the performance of GATSM by changing the feature functions using three models: Linear, NAM, and NBM. Table 4 presents the results of this experiment.The simple linear function performs poorly because it lacks the capability to capture non-linear relationships. In contrast, NAM, which can capture non-linearity, shows improved performance over the linear function. However, NBM stands out by achieving the best performance in six out of eight datasets. This indicates that the basis strategy of NBM is highly effective for time series data.We evaluate the performance of GATSM by modifying the design of the temporal module. The results are presented in Table 5 . GATSM without the temporal module (Base) fails to learn temporal patterns and shows poor performance in the experiment. GATSM with only positional encoding (Base + PE) also shows similar performance to the Base, indicating that positional encoding alone is insufficient for capturing effective temporal patterns. GATSM with only multi-head attention (Base + MHA) outperforms the previous two methods, demonstrating that the MHA mechanism is beneficial for capturing temporal patterns. Finally, our full GATSM (Base + PE + MHA) significantly outperforms the other methods, suggesting that the combination of PE and MHA creates a synergistic effect. Consistent with our previous findings in section 5.2, all four methods show similar performances on the Mortality and Sepsis datasets, which lack significant temporal patterns.In this section, we visualize four interpretations of GATSM's predictions on the AirQuality dataset.In addition, interpretations for the Rainfall dataset can be found in Appendix E. Time-step importance: We plot the average attention scores at the last time step T in Figure 2 .The process for extracting the average attention score of time step u at time step t is formalized as K k=1 a k,t,u . This process is repeated over all data samples, and the results are averaged. Based on Figure 2 , it seems that GATSM pays more attention to the initial and last states than to the intermediate states. This indicates that the current concentration of particulate matter depends on the initial state.Global feature contribution:  
future works & conclusion: 
 Although GATSM achieved state-of-the-art performance within the transparent model category, it has several limitations. This section discusses these limitations and suggests future work to address them. GAMs have relatively slower computational times and larger model sizes compared to black-box models because they require the same number of feature functions as input features. To address this problem, methods such as the basis strategy can be proposed to reduce the number of feature functions, or entirely new methods for transparent models can be developed. The attention mechanism in GATSM may be a bottleneck. Fast attention mechanisms proposed in the literature [39, 40, 41, 42, 43] , or the recently proposed Mamba [44] , can help overcome this limitation. Existing time series models, including GATSM, only handle discrete time series and have limited length generalization ability, resulting in significantly reduced performance when very long sequences, unseen during training, are input. Extending GATSM to continuous models using NeuralODE [45] or HiPPO [46] could address this issue. GATSM still cannot learn higher-order feature interactions internally and shows low performance on complex datasets. Feature interaction methods proposed for transparent models may help address this problem [29, 15] .In this papre, we proposed a novel transparent model for time series named GATSM. GATSM consists of time-sharing NBM and the temporal module to effectively learn feature representations and temporal patterns while maintaining transparency. The experimental results demonstrated that GATSM has superior generalization ability and is the only transparent model with performance comparable to Transformer. We provided various visual interpretations of GATSM, demonstrated that GATSM capture interesting patterns in time series data. We anticipate that GATSM will be widely adopted in various fields and demonstrate strong performance. The broader impacts of GATSM across various fields can be found in Appendix A.NeurIPS Paper ChecklistQuestion: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes]Justification: We described the experimental setting in section 5.1.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.• The full details can be provided either with the code, in appendix, or as supplemental material. 
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 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes]Justification: We provided standard deviations with experimental results.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).• The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)• The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean.• It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.• For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).• If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
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 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes]Justification: We provided information on the computational resource used in the experiments.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.• The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.• The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.• The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
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 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [Yes] Justification: We discussed the potential impacts of GATSM in Appendix A.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.• Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.• The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.• If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
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: 
 Since employing distinct networks across different time steps requires a massive amount of learnable parameters, the feature networks in GATSM share the weights across all time steps, while the temporal module independently learns temporal patterns. GATSM then generates final predictions by integrating the feature representations with the temporal information from the temporal module. This strategy allows GATSM to effectively capture temporal patterns and handle dynamic-length time series while preserving transparency. Additionally, this approach facilitates the separate extraction of time-independent feature contributions, the importance of individual time steps, and time-dependent feature contributions through the feature functions, temporal module, and final prediction. To demonstrate the effectiveness of GATSM, we conducted empirical experiments on various time series datasets. The experimental results show that GATSM significantly outperforms existing GAMs and achieves comparable performances to black-box time series models, such as recurrent neural networks and Transformer [16] . In addition, we provide visualizations of GATSM's predictions to demonstrate that GATSM finds interesting patterns in time series. 
 
 Appendix: We discuss the expected impacts of GATSM across various fields.• Time series adaptation: GATSM extends existing GAMs to time series, enabling tasks that traditional GAMs could not perform in this context -e.g., better performance on time series and finding temporal patterns.• Improved decision-making system: GATSM can show users their exact decision-making process, providing trust and confidence in its predictions to users. This enables decision-makers to make more informed choices, crucial in high-stakes domains such as healthcare.• Ethical AI: GATSM can examine that their outcomes are biased or discriminatory by displaying the shape of feature functions. This is particularly important in ethically sensitive domains, such as recidivism prediction.• Scientific discovery: Transparent models have already been used in various research fields for scientific discovery [47, 48] . GATSM also can be applied to these domains to obtain novel scientific insights.Despite these advantages, it is important to remember that the interpretations of transparent models do not necessarily reflect exact causal relationships. While transparent models provide clear and faithful interpretations, they are still not capable of identifying causal relationships. Causal discovery is a complex task that requires further research.We use eight publicly available datasets for our experiments. Three datasets -Energy, Rainfall, and AirQuality -can be downloaded from the Monash repository [32] . Another three datasets -Heartbeat, LSST, and NATOPS -are available from the UCR repository [33] . The remaining two datasets can be downloaded from the PhysioNet [34] . Details of the datasets are provided below:• Energy [49] : This dataset consists of 24 features related to temperature and humidity from sensors and weather conditions. These features are measured every 10 minutes. The goal of this dataset is to predict total energy usage.• Rainfall [50] : This dataset consists of temperatures measured hourly. The goal of this dataset is to predict total daily rainfall in Australia.• AirQuality [51] : This dataset consists of features related to air pollutants and meteorological data.The goal of this dataset is to predict the PM10 level in Beijing.• Heartbeat [52] : This dataset consists of heart sounds collected from various locations on the body.Each sound was truncated to five seconds, and a spectrogram of each instance was created with a window size of 0.061 seconds with a 70% overlap. The goal of this dataset is to classify the sounds as either normal or abnormal.• Mortality [53] This dataset consists of records of adult patients admitted to the ICU. The input features include the patient demographics, vital signs, and lab results. The goal of this dataset is to predict the in-hospital death of patients.• Sepsis We used get_UCR_data() and get_Monash_regression_data() functions in the tsai library [57] to load the UCR and Monash datasets. We use 13 models, including GATSM, for our experiments. We implement XGBoost and EBM using the xgboost [35] and interpretml [23] libraries, respectively. For NodeGAM, we employ the official implementation provided by its authors [8] . The remaining models are developed using PyTorch [36] . In addition, we implement the feature functions in NAM and NBM using grouped convolutions [58, 59] to enhance their efficiency. XGBoost and EBM are trained on two AMD EPYC 7513 CPUs, while the other models are trained on an NVIDIA A100 GPU with 80GB VRAM. All models undergo hyperparameter tuning via Optuna [37] with the Tree-structured Parzen Estimator (TPE) algorithm [60] in 100 trials. The hyperparameter search space and the optimal hyperparameters for the models are provided below:• XGBoost: We tune the n_estimators in the integer interval • RNN, GRU and LSTM: We tune the hidden_size in the integer interval [8, 128] , dropout in the continuous interval [0, 0.9], learning_rate in the continuous interval [1e-3, 1e-2], and weight_decay in the continuous interval [1e-6, 1e-1] on a log scale.• Transformer: We tune the n_layers in the integer interval [1, 4] , emb_size in the integer interval [8, 32] , hidden_size in the integer interval [8, 128] , n_heads in the integer interval [1, 8] , dropout in the continuous interval [0, 0.9], learning_rate in the continuous interval [1e-3, 1e-2], and weight_decay in the continuous interval [1e-6, 1e-1] on a log scale.• Linear: We tune the learning_rate in the continuous interval [1e-3, 1e-2], and weight_decay in the continuous interval [1e-6, 1e-1] on a log scale.• EBM: We tune max_bins in the integer interval [8, 512] , min_samples_leaf and max_leaves in the integer interval [1, 50] , inner_bags and outer_bags in the integer interval [1, 128] ,learning_rate in the continuous interval [1e-6, 100] on a log scale, and max_rounds in the integer interval [1000, 10000].• NodeGAM: We tune n_trees in the integer interval [1, 256] , n_layers and depth in the integer intervals [1, 4] , dropout in the continuous interval [0, 0.9], learning_rate in the continuous interval [1e-3, 1e-2], and weight_decay in the continuous interval [1e-6, 1e-1] on a log scale.• GATSM: We tune nbm_batchnorm in the descret set {False, True}, nbm_dropout in the continuous interval [0, 0.9], attn_emb_size in the integer interval [8, 128] , attn_n_heads in the integer interval [1, 8] , attn_dropout in the continuous interval [0, 0.9], learning_rate in the continuous interval [1e-3, 1e-2], and weight_decay in the continuous interval [1e-6, 1e-1] on a log scale. The optimal hyper-parameters for GATSM across all experimental datasets are provided in Table 6 .The inference speed of machine learning models is a crucial metric for real-world systems. We evaluate the throughput of various models. The results are presented in Table 7 . Since the datasets have fewer features than the number of basis functions in NBM, NAM achieves higher throughput than NBM. Transparent tabular models typically exhibit fast speeds. However, their throughput significantly decreases in datasets with many features, such as Heartbeat, Mortality, and Sepsis, because they require the same number of feature functions as the number of input features. Transformer shows higher throughput than the transparent time series models because it does not require feature functions, which are the main bottleneck of transparent models. Additionally, the PyTorch implementation of Transformer uses the flash attention mechanism [61] to enhance its efficiency.NATM has slightly higher throughput than GATSM, as it does not require the attention mechanism and has fewer feature functions compared to the number of basis functions in GATSM. We evaluate GATSM by varying the number of basis functions in the time-sharing NBM. The results for forecasting, binary classification, and multi-class classification datasets are presented in Figure 6 .For the Sepsis dataset, using 200 and 300 basis functions causes the out-of-memory error. For the Energy and Heartbeat datasets, performance improves up to 100 basis functions but shows no further benefit when the number of bases exceeds 100. In other datasets, performance changes are not significant with different numbers of basis functions. In addition, there is a trade-off between the number of basis functions and computational speed. Therefore, we recommend generally setting the number of basis functions to 100. Note that the performance of GATSM with this hyper-parameter depends on the dataset size and complexity. Hence, a larger number of basis functions may benefit more complex datasets. In addition to the interpretations on the AirQuality dataset in section 5.4, we present another interesting interpretations of GATSM on the Rainfall dataset.Time-step importance:  


Decision: This paper proposes a Generalized Additive model for transparent and interpretable modeling of time series data. 
The reviewers generally agree about the importance of the motivation, and the breadth of the datasets used in the empirical validation.
However, several concerns have been raised about the originality of the work, the clarity of the presentation, and the limitations of the model.
Specifically, reviewers expressed some doubts regarding the novelty of the approach and how transparency is being quantified concretely. These doubts are compounded by the sometime unclear exposition, in particular regarding the evaluation metrics and the presentations of the empirical results that fail to elucidate and quantify the gains in interpretability.
In addition, the reviewers have raised concerns about the benchmarking against black-box baselines, as the number of state-of-the-art methods that have been compared against the proposed method has been indicated as limited. The addition of new experimental results in the rebuttal is appreciated, but it is still not capturing the full spectrum of the state-of-the-art baselines currently available for time-series data, including more recent transformer-based architectures and foundation models.
Overall, the reviewers pointed out several strength of the paper, but not enough to counterbalance the concerns that were raised.

Reviews: review: paper_summary: paper_summary: This work aims to build transparent models for the time series domain for better interpretability. Specifically, they proposed a work called Generalized Additive Time Series Model (GATSM) that consists of independent feature networks as well as a temporal attention module to learn temporal patterns. The corresponding model can be written into a scalar form to ensure interpretability/transparency. The authors applied their model to several datasets, and showed that GATSM can outperform existing generalized additive models. The model can also be used to interpret the features in original dataset.
strength_weakness: strength_weakness: While this work is not of my direct expertise, I think the following contents have room for improvement:

1. Experimental results are weak.
- Black-box Time Series Models seem out-of-dated. The authors should consider better models such as TimesNet, PatchTST, FreqTransformer, or Informer for commonly used black-box models.
- Forecasting tasks use R2 score for evaluation, but an R2 score of 0.07 (or in general, below 0.5) seems very low. The authors should show some visualization examples to ensure the model is functioning.
- Figure 4/5 are not self-explainable, the authors should try to explain what is happening in those figures, and how the interpretability is quantified.
- The work could benefit from synthetic dataset, where casual relationships are manually crafted and thus can be evaluated.
questions: questions: - Line 123: How one should understand the proposed basis functions, are they similar to quantized vectors in works like VQVAE?
- Eq 5: How would positional embeddings affect the interpretability of the model?
- How does the interpretability results differ in datasets with very long length (e.g. Mortality) v.s. that of shorter length?
limitations: limitations: Yes
paper_summary: paper_summary: The paper introduces a Generalized Additive Model for time series, combining feature embedding and attention layer. The proposed solution is evaluated on forecasting, binary and multiclass classification, over 8 datasets, against black box and transparent models. Global, local, time-focused and feature focused interpretability methods are provided.
strength_weakness: strength_weakness: I am not sure if the work is original. The idea of using DNN first on the time axis without covariate interaction is not new, but wether there is a model similar to the proposed solution, I do not know. The review of previous works focuses on Generalized Additive Models, but a similar neural structure may have been presented without being positioned as a GAM.

One improvement to be done would be to add more clarity on figure captions. In its present form, it requires back and forth to the text to understand both what is plotted and what conclusions to draw from it.
questions: questions: In forecasting tasks, there are often long horizons to be considered. Is it possible to build a multioutput model for forecasting using GAM formalism? In which case, would the interpretation be able to relate future timestamps together, as series are often autocorrelated?

Can correlation among dimensions be detected with GAM? This would be especially relevant in high dimensional MTS.
limitations: limitations: Several limitations are identified: possible overfitting of the model due to overparameterization in NBM part, slow attention mechanisms that do not benefit from state of the art methods, and the fact that it was not evaluated for long sequences (and might not be suited for them).
paper_summary: paper_summary: This paper introduces GATSM(Generalized Additive Time Series Model), designed for handling multivariate time series data with a focus on transparency and interpretability. Using independent networks to learn feature representations and transparent temporal modules to learn cross-time step dynamics, GATSM effectively learns temporal patterns and maintains interpretability.  It achieves comparable results achieves comparable performance to black-box time series models on various datasets, and proves the transparent predictions with cases.
strength_weakness: strength_weakness: 1.  The need for a large number of feature functions can limit scalability (even if it is reduced from TxM to B), particularly with high-dimensional data.
2. The dataset tasks encompass both 1-step forecasting and classification, each requiring distinct evaluation metrics. Presenting all results in a single table without proper clarification leads to confusion. 
3. The selected baseline black-box models are relatively simple. Consider including one or two state-of-the-art methods for a more comprehensive comparison.
questions: questions: Can this model be applied to a multi-step prediction explanation since one-step forecasting is simple to real-world applications.
limitations: limitations: Yes
paper_summary: paper_summary: GATSM effectively captures temporal patterns and handles dynamic-length time series while preserving transparency, outperforming existing GAMs and matching the performance of black-box models like RNNs and Transformers.
strength_weakness: strength_weakness: * It is not clear how multi-head attention works in Definition 3.1 to learn temporal patterns. My understanding is that the global feature interacts with the current feature in Eq.3, so why is it that the input to the attention is not $x$ but the transformed $\tilde_x$. And then, are temporal patterns captured from attention? 

* The addition of attention to NBM is under-motivated, so why not just replace $w^{nbm}$ to attention weight, so that you can learn one less set of parameters $w^{nbm}$.

* The experiment lacks DLinear [1], a strong baseline and providing interpretability. I think it's highly relevant.
1. Are Transformers Effective for Time Series Forecasting? AAAI 2023.

* Given the emphasis on interpretation or white-box modeling, qualitative experiments of the contributions/explanations need to be compared rather than visualization. If there is no ground-truth of the contribution, occlusion experiments in post-hoc methods [2, 3] can also be designed to explore the trade-off between performance and additive features.
2. Encoding time-series explanations through self-supervised model behavior consistency, NeurIPS 2023.
3. TimeX++: learning time-series explanations with information bottleneck, ICML 2024.

* In Table 7, why is the throughput of GATSM so much lower than NBM? Does NBM add up as features at the time level without sharing?
questions: questions: See the weaknesses.
limitations: limitations: It's hard to deal with the higher-order interactions, e.g. GA^2M/GA^NM in time series, since time series often have long time points and the high complexity of gam-based techniques. In addition, the temporal-level causal interrelationships can be further explored.

Comments: title: Response to Reviewer JRxn
Thanks
comment: Thank you for your feedback. We agree that the experiments you suggested will improve the quality of our paper. Thus, we have added the following new experimental results:
- In the multi-step forecasting experiments, our PatchTST and DLinear outperform our GATSM. However, our primary contribution is the development of a novel, interpretable model for multivariate time series, GATSM, which offers human-understandable interpretations with a slight trade-off in accuracy.
- We trained PatchTST and DLinear on the Energy, Rainfall, and AirQuality datasets, reporting their $R^2$ scores and standard deviations across five random seeds. However, performance results for the remaining five datasets are unavailable for the following reasons: the Mortality and Sepsis datasets have dynamic-length sequences, which are incompatible with forecasting models that require fixed-length observations. Additionally, the Heartbeat, LSST, and NATOPS datasets involve many-to-one classification tasks (binary or multi-class), a setting not supported by the publicly available implementations of PatchTST and DLinear.

||Energy($\uparrow R^2$)|Rainfall($\uparrow R^2$)|AirQuality($\uparrow R^2$)|
|---------------|--------------------|---------------|-------------------|
|DLinear|0.486(+-0.102)|0.086(+-0.014)|0.685(+-0.024)|
|PatchTST|0.413(+-0.236)|0.098(+-0.019)|0.638(+-0.045)|
|GATSM|0.493(+-0.173)|0.073(+-0.027)|0.583(+-0.026)|

- We conducted an experiment using a synthetic tumor growth dataset[1] that simulates changes in tumor size over time. The tumor size at any given time step is influenced by three factors: prior tumor size, chemotherapy, and radiotherapy. We set the chemotherapy coefficient to 10, that is chemotherapy significantly reduce tumor. The experimental results showed that while GATSM underperforms compared to more complex black-box models, it still achieves strong accuracy ($R^2$ > 0.9). Additionally, we visualized the contributions of the three factors, revealing that recent time steps are notably more influential than earlier time steps due to the direct impact of prior tumor size. GATSM also effectively captured the effects of chemotherapy and radiotherapy on reducing tumor size, with chemotherapy having a significantly greater impact than radiotherapy.

||SyntheticTumorGrowth ($\uparrow R^2$)|
|-|-|
|Transformer|0.965(+-0.002)|
|DLinear|0.956(+-0.009)|
|PatchTST|0.953(+-0.006)|
|GATSM|0.906(+-0.016)|

[1] Geng, C., Paganetti, H., & Grassberger, C. (2017). Prediction of treatment response for combined chemo-and radiation therapy for non-small cell lung cancer patients using a bio-mathematical model. Scientific reports, 7(1), 13542.
Thank you for the response. The new results show that GATSM performs worse on SOTA black-box networks on new datasets, and the authors did not provide results showing the performance of SOTA black-box networks on datasets presented in the paper. Additionally, the authors did not provide new synthetic results as I requested. Thus, I'd keep my score to be the same.
Thank you for the answer. I believe that the answer to Q3 especially would improve the paper, paired with interpretation plots attending temporal patterns in multioutput models. I will pass my rating to weak accept, but I will keep my review confidence as 1, as proposing new neural structure isn't my domain.
Thanks for the detailed response. My major concerns have been well addressed, and I believe they will be addressed in the final versions thus I decided to raise my rating to 5.
Thanks for the detailed response. My major concerns have been well addressed, and I have decided to raise my rating to 5.