Biased Temporal Convolution Graph Network for Time Series Forecasting with Missing Values
Authors: Xiaodan Chen
Xiucheng Li
Bo Liu
Zhijun Li
Venue: ICLR 2024 poster (2024)
Keywords: Spatial and temporal graph neural network
bias
missing values
time series forecasting
Primary area: general machine learning (i.e., none of the above)
TL;DR: We design a biased temporal convolution graph network for time series forecasting with missing values.

Abstract: Multivariate time series forecasting plays an important role in various applications ranging from meteorology study, traffic management to economics planning. In the past decades, many efforts have been made toward accurate and reliable forecasting methods development under the assumption of intact input data. However, the time series data from real-world scenarios is often partially observed due to device malfunction or costly data acquisition, which can seriously impede the performance of the existing approaches. A naive employment of imputation methods unavoidably involves error accumulation and leads to suboptimal solutions. Motivated by this, we propose a Biased Temporal Convolution Graph Network that jointly captures the temporal dependencies and spatial structure. In particular, we inject bias into the two carefully developed modules, the Multi-Scale Instance PartialTCN and Biased GCN, to account for missing patterns. The experimental results show that our proposed model is able to achieve up to $9.93$\% improvements over the existing methods on five real-world benchmark datasets. Our code is available at: https://github.com/chenxiaodanhit/BiTGraph.

Published as a conference paper at ICLR 2024
BIASED TEMPORAL CONVOLUTION GRAPH NETWORK FOR TIME SERIES FORECASTING WITH MISSING VAL-
UES
Xiaodan Chen', Xiucheng Li (Ez), Bo Liu', Zhijun Li (+4)
School of Computer Science and Technology, Harbin Institute of Technology
2 School of Computer Science and Technology. Harbin Institute of Technology (Shenzhen)
{2 (21B3030048stu., lixiuchenge, 2389360278stu. jun_ooe fhit en
ABSTRACT
Multivariate time series forecasting plays an important role in various applications ranging from meteorology study, traffic management to economics planning. In the past decades, many efforts have been made toward accurate and reliable forecasting methods development under the assumption of intact input data. However, the time series data from real- world scenarios is often partially observed due device mal- function or costly data acquisition, which can seriously impede the performance of the existing approaches. A naive employment of imputation methods unavoidably involves eтror accumulation and leads to suboptimal solutions. Motivated by this, we propose a Biased Temporal Convolution Graph Network that jointly captures the temporal dependencies and spatial structure. In particular, we inject bias into the two carefully developed modules- the Multi-Scale Instance PartialTCN and Biased GCN-to account for missing patterns. The experimental results show that our proposed model able to achieve up to 9.93% improvements over the existing methods on five real-world benchmark datasets. Our code is available https:: http:/ggitttu com/ wwwnxiaadaaniiic BeaddaniiiBBiTrraaalaa /BiTGraph.
INTRODUCTION
Multivariate time series forecasting finds its applications in a wide spectrum of domains such as meteorology, traffic, energy consumption, economics, etc. The real-world demand has spurred the development of various forecasting approaches in the literature. From the generative perspective, the multivariate time series data produced by collection of N instances (e.g., sensors) during period of time. Thus, an accurate characterization of the underlying dynamics requires faithfully modeling both the temporal dependencies (intra-instance correlation) and the spatial structure (inter- instance correlation). The statistical methods ARIMA (Nelson, 1998). VAR (Zivot & Wang, 2006) have made early attempts by building autoregressive models to capture the temporal dependencies. However, their linear dependency assumption often leads poor performance in practice. Inspired by their successes in Natural Language Processing, there has been an increasing trend in designing forecasting models based on RNNs and Transformers to explore their nonlinear modeling and complex pattern extraction capacity (Salinas et al., 2020; Zhou et al., 2021; Liu et al., 2021; Wu et al., 2021; Zhou et al., 2022). Especially, benefiting from the wide-range receptive fields enabled by the attention mechanism, the Transformer-base methods have exhibited excellent prediction performance on long-term forecasting tasks.
Apart from the methods dedicated to temporal dependencies modeling. there is another line of work toward exploiting the spatial correlation of multivariate time series. Many proposals (Salinas et al., 2020; Liu et al., 2021) model the spatial dependencies implicitly and simply rely on a hidden representation to capture the correlation. BRITS (Cao et al., 2018) proposes to use a dense connection layer to learn the correlation between every instance pair, which results in a high model complexity. The advent of graph neural networks (GNNs) (Kipf & Welling, 2016; Defferrard et al., 2016) enables us to effectively explore the Euclideen structure data. Indeed, the DCRNN (Li et al., 2018 1)) proposes to build graphs and conduct graph convolution operations to capture the spatial correlation explicitly in traffic flow forecasting, in which the graphs induced by spatial proximity. To apply

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GNNs to the more general forecasting scenarios, in which the graph structures are not available, the proposals (Bai et al., 2020; Wu et al., 2020) propose to learn the graphs adaptively by learning each node an embedding and building the graphs using the node embeddings, which achieves great progress in enhancing the prediction accuracy.
Despite the promising results achieved, the existing methods pay relatively less attention mul- tivariate time series forecasting with missing values. In practice, the collected time series data is often partially observed, caused by device malfunction, communication failure, or data acquisition difficulty. One commonly adopted solution is to employ the existing time series imputation meth- ods (Cao et al., 2018: Marisca al., 2022: Cini et al., 2022) and then build the forecasting models on the imputed data. However, this two-step process separates the forecasting from the imputation, and the accumulated errors may impede the model performance and lead to suboptimal solutions. GRU-D (Che et al., 2018) proposes a decayed-GRU mechanism to handle the missing values for time series classification, and a similar idea is also adopted by BRITS (Cao et al., 2018) and GRUI (Luo et al., 2018) for time series imputation. However, these methods are not dedicated to time series forecasting. Neural ODE (Chen et al., 2018) is capable of handling geegulaary-samplee time series data and many variants including LatentODE, NeuralCDE, and CRUs (Rubanova et al., 2019; Kidger et al., 2020; Schirmer et al., 2022) have been proposed. However, these methods often entail an ODE-solver computation for each iteration and have to align time steps of different time series, and thus cannot utilize the sparsity of the observations.
Motivated by the above observations, in this paper, we propose Biased Temporal Convolution Graph Network, dubbed BiTGraph, to jointly capture the temporal dependencies and spatial struc- ture by explicitly exploring the missing values in the model architecture design. We develop two core modules-thy the Multi-Scale Instance PartialTCN and Biased GCN. The Multi-Scale Instance PartialTCN performs instance- -anauncceiddeeddentt partial temporal convolution to capture the intra-instance temporal dependencies contaminated by the missing values. Furthermore, the Biased GCN module explores the spatial structure by constructing a biased graph to account for the missing patterns. Besides, we integrate the two modules with hierarchical architecture, in which the missing patterns will be updated progressively along the temporal and spatial dimensions to maximize information propagation and minimize the impacts of missing values. To summarize, our contributions are as follows.
We present BiTGraph to jointly capture the temporal dependencies and spatial structure for the time series forecasting with missing values, the proposed model explicitly considers the missing patterns in its model design.
We introduce Multi-Scale Instance PartialTCN to effectively model temporal dependencies destroyed by the missing values and present Biased GCN to propagate information among instances by building a biased graph in missing patterns aware manner.
BiTGraph achieves up to 9.93% improvements over the existing forecasting methods under various missing values scenarios as verified five real-world benchmark datasets.
2 RELATED WORK
Time series forecasting with complete data Due to its practical importance, lot of efforts have been devoted to developing accurate time series forecasting methods. The classic ARIMA (Nelson, 1998), VAR (Zivot& & Wang, 2006) build the autoregressive models based on linear dependency assumption. RNNs-based methods (Salinas et al., 2020) and (Zaremba et al., 2014) exploit the expressive power of recurrent neural networks to relax the linear assumption. Very recently, various Transformer- based methods have been proposed to exploit the wide-range receptive fields of attention mechanism for long-term forecasting. To reduce the quadratic complexity of vanilla attention, Informer (Zhou et al., 2021), Pyraformer (Liu et al., 2021), Autoformer (Wu et al., 2021), and FEDformer (Zhou et al., 2022) have been proposed successively. Non-stationary Transformer (Liu et al., 2022) aims to renovate the attention mechanism to account for the non-stationary property of time series data. PST (Nie et al., 2023) explores the patch and lhannel-independenn Apart from enhancing the temporal dynamics modeling capability, many proposals are dedicated to exploring spatial correlation. DCRNN (Li et al., 2018), AGCRN (Bai et al., 2020), MTGNN (Wu al., 2020), GTS (Shang al., 2021), and SAGDFN (Jiang et al., 2024) model the spatial structure
design.
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with the graph neural networks. In addition, CoST (Woo et al., 2022) and TS2Vec (Yue et al., 2022)
approach the time series forecasting from the self--uper learning perspective.
Modeling time series with missing values Caused by device malfunction, communication failure, or costly data acquisition, the world collected time series data is often incomplete and partially observed. To fill missing entries, many time series imputation methods BRITS (Cao et al., 201 8), GRIN (Cini et al., 2022), CSDI (Tashiro et al., 2021), SPIN (Marisca et al., 2022), GRIN (Cini et al., 2022), and TIDER (Liu et al., 2023) been presented in the machine learning community. To deal with the partially observed time series, one may attempt to build the forecasting models with the imputed results produced by the imputation methods. However, the imputation is disparate from the forecasting in this two-step process, and thus the accumulated errors may seriously degrade the forecasting performance. GRU-D (Che et al., 2018) presents a decayed-GRU to handle the missing values for time series classification without resorting to the imputation. Tang et al. (2020); Zuo et al. (2023) attempt to capture local dependencies based on global statistic characteristics for the missing value forecasting. The neural ODE-based models NeuralCDE, LatentODE, and CRUs (Chen et al., 2018; Rubanova et al., 2019; Schirmer al., 2022) are capable of handling irreularly-samplle time series data. Nonetheless, they have align time steps of different time series and cannot utilize the sparsity of the samples.
3 PRELIMINARIES
In this paper, we consider the multivariate time series X E NXTxD consisting of N univariate time series ( x (N) collecting over T time steps with D-dimension observation. Due to the malfunction of devices, communication failure, costly data acquisition, there may exist missing values in X, and we use a mask matrix M E R NxT to represent the missing patterns, which is defined as follows.
[1, if Xnx is observed, MnS = {o. otherwise.
(1)
(n) where Xna denotes the value of n-th instance (or channel) at time step we alternatively use z; Xnt to represent the same entry. Similarly, we use m!!) E RT to denote the n-th row of the mask matrix M. and both Mnt and m(") represent the n-row, t-column element of M. addition, the slice notation X, x F1
x denotes the values in a time window of size R H or X, E x
H from time step - to 1, i.e., the time interval [t H,t). the subsequent discussion, we will also refer to the mask as the missing pattern.
Multivariate time series forecasting with missing values Given the partial observed multivariate time series X and the corresponding mask matrix M, the multivariate time series forecasting with missing values problem aims to build a forecasting model é to predict the future F-step values Y X by taking inputs the historical observation X₃-H: and its mask M,-H:, that is, Y o(x:HHis ..) In the training phase, we only resort the observed values to provide the learning signals. More formally, the loss function L of the model can be described as follows.
N F+F L(Y, Y, Mc+FF) = -==1 r= m.
(2)
Σ
1+F-1 (n)
1-=
m.
which measures the mean absolute error between the predicted values and ground truths.
METHODOLOGY
The framework of our proposed BiTGraph (Biased Temporal Convolution Graph Network) shown in Figure 1-(a). It comprises L. identical blocks, dubbed Biased TCGBlock (Biased Temporal Convolution Graph Block), which is the basic building block of our proposed method. The Biased TCGBlock consists of two key modules: the Multi-Scale Instance PartialTCN module and the Biased GCN module. The two modules are responsible for fusing the information along the temporal dimension and spatial dimension, respectively. In contrast to the existing time series forecasting methods, we explicitly consider the missing values in the model design and inject bias to account for the different missing patterns, and the model also progressively updates the missing patterns
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RiR. 23
X E
x
M
M
Int
X galtatoo
B
A'
Biased TCG Block 1
Scale PurtaTTON
Graph
Blased TCG Block
X
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B.Bj Ba
X 6 Motthon
be R
MRIPT
Busedd GCN Modale
X R0
Bissed The itlook
Graph Diflision
Transforraatos
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Figure 1: (a) The framework of our proposed BiTGraph, and (b) the illustration of the Biased GCN module.
as the information diffusion proceeds. As the Figure 1-(a) shows, the Ê-th block takes as inputs ((-1)) IR X and the missing patterns ((t-1) E NxH, and produces two transformed
tensors (( E RNXHXD(0))) and (( E RNKH where ((0) is the feature dimension of the f-th block.
4.1 MULTI-SCALE INSTANCE PARTIALTCN MODULE
In this paper, we opt for Temporal Convolution Network (TCN) our backbone to capture the temporal dynamics for two main reasons: 1) it has been shown empirically that the TCN exhibits more favorable sequence modeling abilities in comparison to RNNs in variety of tasks (Bai et al., 2018), 2) the convolution operation permits a simple modification to account for partial observations evidenced in computer vision (Liu et al., 2018). Different from the vanilla partial convolution (PartialCNN), we propose to apply the partial temporal convolution (PartialTCN) within each time series (instance). i.e., the parameters of PartialTCN benefits are twofold: 1) we decouple the temporal dependency modeling from spatial correlation modeling, which enables exploring the temporal patterns invariant instances and enhancing the statistical strengths; 2) the PartialTCN is shareable across instances and this will lead to a more parameter-efficient model, we refer the resulting approach as Instance ParatialTCN. In addition, we further adopt the multi-scale strategy to develop Multi-Scale Instance ParitalTCN,, which can handle missing patterns from different time scales more effectively. Next, we illustrate the module by focusing on particular instance (")) R and its mask m (n) E R To keep the notation uncluttered, we drop the upper script temporally.
shareable across different instances. The
Instance PartialTCN Given the kernel size K, the TCN applies the same linear transformation into different time windows under the time translation invariance assumption, i.c.
X: W + b,
where Kit E RxD, is time window sequence with D; input features, x' RD is the output feature map at location t - 1, and W E RRxDx and b E RD, are convolution parameters. Motivated by the success of partial convolutions vision tasks, we introduce Instance PartialTCN to model the temporal dependencies of partially-observed time series to account for missing values as,
K
(x Kit
M Ka) W + b, if sum(m;;Ka)) > 0,
otherwise.
where @ denotes the Hadamard product. The Instance PartialTCN only attends to the time steps with observations compute the new feature maps and the factor K/ sum summm;- (m--k:t KRt) rescales the
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computation result to the same magnitude of convolutions on complete observations. In such manner, the missing patterns are integrated into the temporal dynamics modeling. As the temporal convolution proceeds, the time steps with missing values will have chances gather sufficient information from their surrounding neighhors. To account for this, the missing pattern m is updated as,
1, if sum(m; 0,
14 0, otherwise.
In other words, the time step / is considered filled if we could collect values from the present time window [t- - K. () The missing pattern m will be progressively filled as the convolution proceeds.
Multi-Scale Instance PartialTCN
To capture the multi-scale temporal dependencies of the time series, we propose to integrate the multi-scale convolution with different kernel sizes into the Instance PartialTCN, specifically, we adopt 1 3, 1 x 5, and 1 in this paper. Consequently, different kernels
will yield multiple different updated missing patterns m (n) E TR for each instance n, S i S Nker and Nker is the number of kernels. We propose aggregate these missing patterns generated by different kernels by max pooling as,
max(m!)) ("), 1 ISiS S Nicr-
(6)
The aggregated (n) will then be used in the subsequent graph convolution module to diffuse information along spatial dimensions. By applying the Multi-Scale Instance PartialTCN to each instance (2) (n 11.......NV N). we transform the input feature map ((=-1) E Nx 1)
and missing pattern ( E R into ((-11) E NxHxx and updated missing pattern (C E N respectively.
4.2 BIASED GCN MODULE
The Multi-Scale Instance PartialTCN focuses on capturing the temporal dynamics hidden in each instance without considering the inter-instance correlation. However, it is equally important to model both the spatial correlation and temporal dependencies for accurate multivariate time series forecasting. In this paper, we propose to use graph convolution networks to explore the spatial structure of the temporally fused feature map (t-1) and updated M produced by the Multi- Scale Instance PartialTCN. The graph neural networks have been exploited to model the spatial correlation for time series forecasting in the literature either by using the predefined (Li et al., 2018) or adaptively- leaaned graph structures (Bai al., 2020; Wu al., 2020; Shang et al., 2021), in which each time series is treated as graph node. In contrast to the existing approaches, we explicitly consider and incorporate a bias term (i.e., prior knowledge) into graph structure learning to account for the missing values, leading to Biased GCN. It is therefore able to deliver promising performance in the missing value scenarios. The Biased GCN module is shown in Figure 1-(b).
In this paper, we choose the adaptive graph structure learning approach since it is more flexible and applics to cases where the graph structures are unavailable. In particular, we learn the graph structure or adjacency matrix A by using two learnable embedding matrices E,, E₂ Nx Dnode as follows.
A ReLU E.)
The i-th row of E1 (resp. E2), denoted by e, (resp. e, 2 is the embedding of i-th time series and cl (1)T d(2) quantifies the correlation strength from node to node j. The reason we choose two embeddings instead of one E and computing A = ReL ReLU(tanh( [ET)) is that the spatial correlations very likely to be asymmetric in practice. This learned adjacency matrix A will be used by the subsequent graph convolution operation to aggregate information and aid in the eventual forecasting task, and thus the embedding matrices E, and E2 be learned end-to-end.
However, Eq. 7 fails to account for missing patterns. Intuitively, the information propagation intensity should vary against the missing patterns and we choose the inner product quantify it as
A = ReLU E, E, )) + $softmax MM,
(8)
where the first term denotes global spatial correlation strength indicating the global message passing strengths among nodes, the second term is specific to particular time window [t H, () and can be
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considered as time-windowe weree--iidowseeeeee bias that corrects the global message passing strength according to the current missing pattern in graph diffusion process, 3 denotes a learnable global parameter that controls the intensity of the correctness. By intuition, the information propagation should also be directed and more information should flow from nodes with fewer missing values to the ones with more missing values, but the second term is symmetric matrix and cannot mirror this intuition. To correct this, we assign each node a learnable scalar bias by and use b; - by to adjust towards the asymmetries. Let b E R be the learnable bias term, we propose to learn the graph structure as,
B = (o,tmax(M,-HaNN) Hit + b1 - -bb)
A ReLU (2,E2 BB,
where 1 is length-N all-one vector. As shown in Figure I-(b), global message passing strengths between node 4 and node 5 are corrected by the time- fime-wiidow-speciff. bias.
To ensure the structure sparsity, we clamp the small entries of to zeros by only preserving the neighbors of node i with the top-A correlation strengths and use the clamped A in the graph convolution operation to aggregate information (as will be shown shortly). Being analogous to the Instance PartialTCN, propose to update the missing patterns of node i after aggregating the information from its spatial neighbors as follows,
m (4) = m)) JE (i) UM,
(10)
where N; indicates the neighbors of node in the graph. The missing pattern updating process is illustrated with the node 1 in Figure 1-(b).
Now considering the 1-th block of the model, it performs the graph convolution to diffuse information as follows.
(
( + b(4)
(11)
where (((1-) is the output of the Multi-Scale Instance PartialTCN in the f-th block, D, and D. are the in-degree and out-degree matrix of A, respectively, and ( and (()) are the graph convolution parameters of the f-th block. (() and (()) will then be fed to the next block as the inputs.
4.3 HIERARCHICAL ARCHITECTURE
By stacking layers of Biased TCGBlock, we could enhance both the spatial and temporal receptive fields of the model. We initialize (0) and ( with the original partial observation X e xxxXXx and its corresponding missing pattern M E N and the outputs of the L-th block are X e x x and (L)) e NxH. (L.) fuses both the spatial and temporal features, which will be used produce the multi-step prediction Y simultaneously by linear transformation. The mask of (-th layer (()) is updated progressively as information flows from bottom to up, and the model parameters are learned by optimizing the prediction loss in Eq. 2.
5 EXPERIMENTS
We evaluate BiTGraph against the state-of-the-art forecasting methods under different missing rates on five real-world benchmark datasets. We first assess the forecasting performance of different methods in terms of three commonly used metrics, and then we verify the efficacy of our proposed modules by ablation study.
5.1 EXPERIMENT SETTINGS
Datasets We select five most commonly used time series forecasting datasets: Metr-LA, Electricity, PEMS, ETThI, and BeijingAir, whose statistics are summarized Table 1. The five datasets are collected from different domains and cover diverse magnitude ranges, sampling frequencies, and statistics. We randomly drop the data according to the missing rate r ranging from 0.1 to 0.8, including 0. 0.2, 0.4, 0.6, and 0.8.
Baseline methods We compare our proposed BiTGraph with the latest state-of-the-ee forecasting methods well several classic methods. BRITS (Cao et al., 2018), SPIN (Marisca et al., 2022),

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Table 1: Dataset description.
Metr-LA
Electricity
PEMS
ETTh1
BeijingAir
#Samples (T")
34272
26304
52116
17420
8759
#Instances (N)
207
321
325
36
Frequeney
5 min
1 h
5 min
1
1 h
Mean
53.72
2538.79
62.62
4.58
72.01
Variance
410.53
2.26 10
92.05
42.68
79.07
GRIN (Cini et al., 2022), GCN-M (Zuo et al., 2023), CRUs (Schirmer et al., 2022) are representative forecasting methods designed specifically for time series with missing values. Mcanwhile, we also include three Transformer based methods, vanilla Transformer (Zerveas et al., 2021), STWA (Cirstea GNNs-base
al., 2022), and FEDformer (Zhou et al., 2022), as well as two methods, AGCRN (Bai et al., 2020) and MTGNN (Wu et al., 2020). Since these five methods require complete input to perform prediction, we study their two variants, namely, filling the missing entries with zeros and the values imputed by TimesNet (Wu et al., 2023), the state-of-the-arr time series imputation approach. We denote the corresponding variants as Modelo, and Model, respectively. The missing masks are fed as covariates to guide the forecasting for the latter five baseline methods. The details of baseline methods are presented in Appendix A.
Implementation details The number of blocks L of BiTGraph is set to 3, the number of top-k nearest neighbors is set to 10 in all our experiments. The batch size is 32, the learning rate is 0.001. We split the datasets into training. validation, and test datasets with the ratio 0.6/0.2/0.2 chronologically. The future window size F is set to 24 for all methods, and the history window size H for our proposed method is 24. We select the best history window size from the set {24, 48, 96) for the baseline methods and report their best results. All methods are trained on Nvidia V100 GPUs. Our method is implemented with PyTorch 2.0 and we use the source codes released by the authors for all baseline methods. We adjust the hyperparameters of baseline methods to obtain the best performance on each dataset, and evaluate the performance of different methods in terms of Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE).
5.2 OVERALL PERFORMANCE
Table 2 presents the forecasting performance the datasets (Metr-LA and Electricity) of different methods under the missing rates of 0.2, 0.4, 0.6, and 0.8, the results are averaged over five funs We move the results of the PEMS, ETTh1, and BeijingAir datasets and the results under the missing rate of 1 to Appendix B to save space. It can be seen from the table that our proposed BiTGraph is able to achieve the best results most cases terms of all three metrics. Its performance gains become more evident when the missing rate grows to 0.8, which benefits from the ability of the Multi-Scale Instance PartialTCCN module and the Biased GCN module handling the missing patterns adaptively. It is worth noting that SPIN and GRIN, both of which are explicitly designed to address missing values, demonstrate a marked superiority. However, their practical applicability is constrained by the necessity of pre-defined graphs. SPIN, MTGNN, and STWA, achieves the best results among all baseline methods under different cases. In comparison, our proposed BiTGraph is able to deliver the best results consistently. Notably, it achieves up 9.93% improvement over the best baseline in terms of RMSE on the Electricity dataset.
5.3 ABLATION STUDY
In this section, we conduct ablation studies to evaluate the effectiveness of our proposed modules, Multi-Scale Instance PartialTCN (MSIPT) and Biased GCN (BGCN) modules. The results are shown Table We divide the MSIPT module or BGCN module into two distinct procedures. The first part (Eq. 6 or Eq. 10) relates to the mask updating process (MUP), whereas the second part (Eq. 4 or Eq. 9) regarding the information aggregation process (IAP). Firstly, we carry out ablation studies (w/o MSIPT, w/o. BGCN, and BiTGraph) to assess the joint significance of UID and MUP across temporal
The model is trained with five different random seeds.

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Table 2:
The forecasting performance of different methods.
Method
Mete-LA RMSE
Electricity
(r = 0.2)
MAE
МАРE
MAE
RMSE
MAPE
BRITS
8.32 0.02
13.18
0.10
18.26 = 0.71
1029.30
1.10
10126.
175 30.57
47.73 + 0.35
SPIN
6.46
11.21
0.05
0.03
GRIN
6.80 0.02
0.112
16.18 + 0.24
GCN-M
6.78 +: 0.03
11.12
13.50 1 0.02
CRUs
10.80 0.02
12.49 0.15
19.66 + 0.34
464.66 4.14
5276.49
53.36
25.64 + 0.53
AGCRN,
14.88
4.53
26.81
0.22
Transformero
0.04
13.08 0.08
17.07 1 0.09
296.03
5.77
2432.09
22.15
29.14 0.27
FEDformero
7.09 +: 0.03
12.76
16.73 0.19
368,29 + 3.71
2574.37 + 25.89
31.29 1 0.31
STWA,
6.04
12.89 = 0.13
272.60 7.35
2263.
24.10
28
0.26
MTGNN,
6.34
1.061.0100
12.51 +: 0.19
274.68 5.56
2016.44
28.54
AGCRN,
1.00
0.21
27.06 0.18
1049.23
12.06
11751.49 20.67
57.76 0.18
Transformer:
6.90 0.08
12.98
16.49 0.21
280. 6.78
2274.28 25.11 25.18
28.74 0.35
FEDlormer,
6.89 0.06
11.76 0.17
16.01 = 0.09
313.59
4.96
2000. 93 26.31
32.83
0.23
STWA,
6.20 + 0.02
10.71 0.11
12.26 + 0.18
261.92 4.65
2089. + 19.35
27.37 + 0.26
MTGNN,
6.13 0.02
10.76 0.07
12.11
269.25 6.27
2176.21 12.49
27.71
0.78
BiTGraph
6.04 0.02
10.69
0.02
11.69
243.23 2.12
1834.18 15.36
27.38 0.46
Method
Metr-LA RMSE
Electricity RMSE
(r = 0.4)
MAE
MAPE
MAE
MAPE
BRITS
8.38 = 0.08
12.97 + 0.11
18.39 1:.99:000 0.28
1029.73 73 = 1.48
10136.39 + 63.63
47.96 + 0.56
SPIN
6.52 = 0.07
11.94
13.22 1.00
GRIN
6,91
12.60
16.59
GCN-M
7.09 + 0.01
12.42+
17.06 0.04
CRUs
1..04
1..1
20.13
496.95
6.03
5397.31
52.52
27.94 1: 0.33
AGCRN,
0.00
14.30
0.00
1526.90
13.77
14823.39
21.68 2586.69
68.73
0.41
Transformer,
0.04
12.97
17.72 0.08
310.88
4.67
22.73 3606.49
31.79
0.15
FEDformero
7.15
0.12
406.17
27.73
33.14
STWA,
6.37
1.01
0.11
292.47 1.666
2764.34
20.06
29.07
1: 0.16
MTGNN,
0.05
11.10
1..79
305.46 165.46:100 6.77
2576.44 25.51
23.15 1 0.37
AGCRN,
12.73
12.49 0.14
0.16
1283.27 8.49
13743.42 49.38
58.62
0.36
Transtormer,
6.09
16.45
300.43 10.17
19.14 2529.26
28.86 0.20
FEDformer,
330.90
STWA,
6.28 + 0.03
10.93 + 0.14
12.68
0.07
289.59
6.13
2355.34
17.67
28.29 + 0.31
MTGNN,
6.26 1 0.05
10.90 1 0.10
12.49
0.04
281 32 6.82
2236. 74
16.81
28, 1: 0.19
BiTGraph
6.13 + 0.01
10.76 + 0.02
12.41
0.12
270.14 3.77
2091.88 .8 30.49
22.04 + 0.36
Method
Metr-LA RMSE
Electricity RMSE
0.6)
MAE
MAPE
MAPE
BRITS
0.02
12.94
0.08
18.66 = 0.22
1029.38 1.84
10118.18 + 33.04
48.25 = 0.29
SPIN
6.61 0.02
0.17 11.35
13.31
0.12
GRIN
7.04 0.04
1.17
17.04 0.03
GCN-M
7.27 = 0,02
11.55 0.02
16.42 0.03
CRUs
11.02 0.02
13.38
13.38 0.24
20.40 0.04
664.07 = 9.88
8126.82 000.8222 59.42
31.44 + 0.45
AGCRN,
14.87 0.04
14.30
29.92 0.06
1945.61 1: 6.38
13891.03 000100311 17.38
75.20 1 0.20
Transformer,
0.00
1..03
17.09
346.43 5.59
25.54 2952.28
28.96
0.37
FEDformer
7.50
12.32
17.31 0.07
535.72 7.67
5329.18
STWA,
6.82 = 0.02
11.72 0.10
13.66 +.00
325.47 6.62
2479.75
21.17
30.06 + 0.22
MTGNN,
6.95 0,03
12.09
12.09110 0.02
0.09 13.87
329.18 4.61
2490. 0500.44-2 23.38
28.20 - 0.27
AGCRN,
12.73 0.02
0.14
21.13
1374.64 7.11
12069.56 19.73
61.92 = 0.28
Transformer,
7.22 = 0.08
2.11
16.75 0.06
327.17 8.08
2506. 23.17
29.27
0.40
FEDformer.
7.26 100--0000 0.04
1 0.07 13.08
17.16 0.03
341.66
5.25
2682.73 73 24.97
1 29.87
STWA,
6.55
11.28
13.57
0.03
312.25
5.36
2407 39 23.05
MTGNN,
0.63 0.02
11.10 0.04
13.48 + 0.05
309.59
4.73
2399.5 51 = 20.09
25.37
0.13
BiTGraph
6.32 0.01
10.93
0.03
12.67 + 0.11
295.23
2.75
2239.06 26.39
27.38
0..9
Method
Mete-LA RMSE
Electricity RMSE
(r = 0.8)
MAE
MAPE
MAE
MAPE
BRITS
8.56 0.09
13.03
0.18
18.92 + 0.09
1027. 28 0.50
10150.54
31.05
48.04 0.02
SPIN
6.68 0.31
11.42
0.35
14.41
1.20
GRIN
8.00 0.02
0.09 12.68
18.35 + 0.05
GCN-M
7.75 +: 0.03
11.65 0.04
17.94 + 0.02
CRUs
14.06 14.06 0.70
22.08 + 0.22
623.
13.07
7033.29
17.85
33.29 0.74
AGCRN,
2351.41
26.79
16824.28
207.77 0.56
Transformer
8.06 0.02
12.82
18.37 1: 0.11
398.99
6.62
24.19 3612.37
30.07 0.18
FEDformer
7.83
12.97 0.14
17.93 1 0.06
676.93
5.62
7839.76 31.13
61.7940..3
STWA.
7.57
12.15 0.07
17.31 0.12
376.26 5.36
3512.37 22.09
31.15 0.008
MTGNN,
0.00
17.22 + 0.09
383.89
3539.74 15.22
30.29
AGCRN,
1..00
14.20
29.92 0.10
1841.76 76117616.88
17376.51 44.79
70.38 1
Transformer.
7.32
12.96 0.08
16.87 + 0.05
391.83 4.17
3451.33 5.66
32.26 4 0.17
FEDlormer,
3.06
13.17 0.06
16.71 0.04
350.06 3.39 3.39
3335.18 20.10
31.56 0.13
STWA,
6.90 0.03
11.30 0.05
13.69 0.07
362.25 3.21
3156.68 24.41
29.22 0.15
MTGNN,
6.79 0.04
0.07 11.05
13.54
355.68 5.11
3023.30 11.46
28.7
0.31
BiTGraph
6.63 0.01
11.20
13.44
0.02
347.35 1.76
2839.79
25.49
27.97 0.27
8

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Published as a conference paper at ICLR 2024
Table 3: The results of ablation studies Metr, Electricity, and PEMS datasets under the missing rates of 0.2, 0.4, and 0.6.
Metr
Electricity
PEMS
Missing Rate
Model
MAB
RMSE
MAE
RMSE
MAE
RMSE
TCGNet
6,25 + 0.01
11.01 0.02
255.46 - 2.86
2026.43
35.79
1.94 0.00
4.35 = 0.02
w/a. MSIPT
6.34 +: 0.04
0.00
279.73
4.91
2159.16
28.74
1.97 0.02
4.40 + 0.04
w/o. BGCN
6.26
0.03
11.03
263.
+: 3.32
2029.16
1.00
4.37 0.03
w/o. Eq. 9
6.14 0.02
10.72++0.02
250.75
5.01
2020.13
1.94 + 0.02
4.33 + 0.03
wio, Eq.
6.12 0.01
10.91
246. 0.03
2.89
2003.22 = 35.09
+.00+:000
4.30 0.02
BiTGraph
6,04 0.02
0.02 10.69
243.23 + 2.12
1834.18 + 15.36
0.01 1.90
0.01 4.28
TCGNet
6.41 = 0.03
1.01
281.39
+ 5.26
2323.03 = 10.15
1.99
0.01
4.17 0.03
wio. MSIPT
6.48 +: 0.02
11.20 0.08
299.34 +: 4.17
2361 779 37.82
2.02 0.02
4.50 0.03
w/o. BGCN
6.40
0.02
2337.69
31.98
1.98
0.02
w/o. Eq. 9
6.18 + 0.01
10.81 + 0.04
282. 12 2.88
2236.82 + 30.26
1.98 0.01
4.32 + 0.02
w/o. Eq.
6,25 0.00
10.87 0.02
280.30
2.73
2277.50 28,
1.97 0.01
4.34 0.01
BiTGraph
6.13 0.01
1.00
270.14
1: 3.77
2091.88 :: 30.49
1,96 0.00
1.3 0.02
TCGNet
6.48 +: 0.02
313.60
+: 3.29
2372.36 36 36.19
2.04 0.02
4.53 :: 0.02
w/o. MSIPT
6.65 + 0.03
11.50+0.00 0.04
332.39
3.82
2469.15
33.63
2.09 0.02
4.62 + 0.04
w/o. BGCN
6.65 0.02
1.004
322
2487.22
25.39
2.03
0.6
w/o. Eq.
6.35 + 0.03
11.06
306. 59 + 3.97
2366.30 + 32.16
2.03 + 0.01
1.52 = 0.02
w/o. Eq
6.38 + 0.02
10.84 + 0.02
301 25 + 2.05
2312.39
22 22.46
2.03 + 0.02
4.54 0.00
BITGraph
6.32 0.01
10.93 + 0.03
295.23 :: 2.75
2239.06 +: 26.39
1,99 0.01
4.47
and spatial dimensions. The results reveal that when we adopt MSIPT or BGCN the performance drops significantly, which can be explained by that the MUP builds complete -aaaainngn path between the spatial and temporal dimensions and the absence of any module will cut off the information flow between the two dimensions.
To further validate the effectiveness of IAP and MUP, we conduct ablation studies by modifying the IAP. First, we replace the temporal convolution in the MSIPT module with the standard convolution operation (e.g.. Eq. 3). Next, we alter the generation of adjacent matrix A by using Eq. 7. The results shown in the fourth and fifth rows of the table. As we can see, the replacement of either spatial or temporal operations within IAP leads to a notable performance drop, which further verifies the effectiveness of our proposed modules in handling missing values. We also conduct the corresponding ablation study under the block missing scenarios and the results are shown in Appendix C. The results of parameter sensitivity including window size H1, the number of blocks and the number of nearest neighbors k are shown in Appendix D. We analyze the role of 8 in Appendix E. We visualize the prediction curves in Appendix The model complexity analysis is given in Appendix G.
6 CONCLUSIONS
In this paper, we present BiTGraph for the time series forecasting with missing values. BiTGraph jointly captures the temporal dynamics and spatial structure by explicitly taking the missing values into consideration. We inject bias into the two carefully designed modules, the Multi-Scale Instance PartialTCN and Biased GCN, to account for the missing patterns. The experimental results on five real-world benchmark datasets verify its superiority under various missing value scenarios. The ablation studies also show that its excellent performance stems from the two carefully designed the future, we would like to
Multi-Scale Instance PartialTCNN and Biased GCN components. explore the Transformer architecture as the backbone of our temporal module to further enhance its long-term forecasting performance for partially observed time series data.
ACKNOWLEDGEMENNTS
This work is supported by the National Natural Science Foundation of China under Grant No. 62206074 and Grant No. 62072137, Shenzhen College Stability Support Plan under Grant No. HWWWD20220811132233000 and the National Key R&D Program of China under Grant No. 2023YYB4503300

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Published as a conference paper at ICLR 2024
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A EXPERIMENTAL DETAILS
A.1 DETAILS OF BASELINE MODELS
The details of the baseline models are briefly summarized as follows. For BRITS, SPIN, GRIN, CRUs, AGCRN, MTGNN, FEDformer, and STWA, we use the source codes released by their the version implemented in FEDformer.
authors. While for Transformer,
B It is time series imputation model that combines Bidirectional Recurrent Neural Network with the time decay mechanism to establish the relationship between missing values and observed data.
SPIN:: : It handles missing values by constructing a sparsely connected graph in both spatial and temporal dimensions.
GRIN:: It incorporates recurrent neural network (RNN) and graph neural network (GNN) to capture inter- and intra-series dependencies to build the relationship between missing values and observed ones.
GCN-M*: considers local spatiotemporal features and global historical patterns in an attention-based memory network.
CRUs It combines the Kalman filter and encoder-decoder frameworks update the continuous hidden states.
AGCRN': The adaptive graph and node-specific patterns learned by node embeddings and matrix factorization, respectively.
The approach constructs a skew-symmetrical spatial correlation matrix and employs the Temporal Convolutional Network (TCN) and GCN to capture the intra- and inter-series dependencies.
MTGNN
The classic sequential model that uses a stack of self-attention blocks to capture the temporal dependencies in time series.
Transformer:
FEDformer": The method decomposes the time series into seasonal and trend components and employs the self-attention mechanism in the frequency domain.
STWA 10: It constructs spatial-temporal aware embeddings within the self-attention mecha- nism and introduces window attention reduce complexity.
A.2 DETAILS OF METRICS
The metrics of mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) adopted in the paper are defined as follows.
MAE-
Eijen Vij
Dig
RMSE
Eisen (Jig dú)
MAPE
Σ gen
Wi
dig
(12)
[2
[
[ig
where Ω represents the index set of observed values.
13

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Published as a conference paper at ICLR 2024
Table The results forecasting error on Metr-LA and Electricity datasets with the missing rate
0.1.
Method
Metr-LA RMSE
Electricity RMSE
0.1)
MAE
МАРЕ
MAE
MAPE
BRITS
8.21 = 0.01
12.63 + 0.10
18.04
1027.47 3.92
10157.39
64.012
47.54 0.01
SPIN
6.36 = 0.00
11.07 + 0.01
12.00 = 0.00
GRIN
6.69 + 0.02
12.27 + 0.00
15.88
0.01
GCN-M
6.72 + 0.01
12.33+
13.06
0.02
CRU
1..00
1..00
19.71
334.40
2923.44 39.73
AGCRN
7.07 0.06
1.00
1361
8.39
12569
30.09
62.54
Transformer,
12.97
0.113
289.65 3.39
2296.17
27.77
27.77
25.08 - 0.32
FEDlormero
12.37 0.008
16.22
337
5.25
2713.72 33.37
30.24 0.28
STWA,
6.22 0.04
1..64
12.71 0.05
269.81
+ 5.95
2039.64
24.06 1974.50
22.39 + 0.33
MTGNN,
6.25
AGCRN,
13.72
0.223
0.118
1109.57 3.95
10794.08
34.26
57.95 1: 0.18
Transformer,
6,90 0.07
12.81 0.04
16.49
19.49-000 0.06
266. 76 6.03
206-1.82 1 23.51
20.06
1 0.27
FEDformer,
6.61 + 0.03
11.09
0.06 13.23
283.63 2.15
2269.11 + 19.86
23.23 + 0.18
STWA,
6.17 1: 0.02
10.82 1 0.09
0.04
248.8 1: 3.79
1945. 16 23:20
18.20
1: 0.16
MTGNN,
0.02
10.69 1 0.03
12,02
254.67
1: 3.22
1994.07 +: 25.90
22.41
1
BiTGraph
5.96
10.71 1: 0.00
231.70 1.76
1823.18
25.49
0.27
Table 5: The results of forecasting error on PEMS, ETTh1, and BeijingAir datasets with the missing
rate r r 0.1 and 0.8.
Method (0.1)
PEMS
ETTh1
BeijingAir
MAE
RMSE
MAE
RMSE
MAE
RMSE
BRITS
3.06 + 0.01
6.39 0.02
1.76 0.03
3.17 + 0.03
45.79 0.24
67.83
0.21
SPIN
2.03 0.00
4.62+ 0.00
44.93 0.03
68.13 0.75
GRIN
2.63 2.63
0.06
45.96 0.25
67.06
1.30
GCNM
2.13
0.07
47.68 0.11
68.29 0.09
CRUs
3.21
0.03
2.80 - 0.03
4.82 + 0.05
56.92 0.72
0.87
AGCRN,
5.00
0.00
2.39 0.04
4.76:0000
55.50 = 0.12
81.31 0.11
Transformer,
0.07
0.02
1.88
3.25 0.07
48.58 = 0.07
69.50 0.21
FEDformer,
2.61 0.U5
5.10
1.09
3.22 0.0
49.65 0.09
72.77 + 0.13
STWA,
2.01
4.57 + 0.03
1.00
3.15 0.00
46.71 = 0.14
70.10 0.07
MTGNN,
0.00
4.52 0.04
1.001
2.99 + 0.02
0.02
44.37 0.05
65.92 + 0.0G
AGCRN,
5.00
10.05 0.00
0.03
4.29 + 0.05
47.08 = 0.26
69.62 - 0.37
Transformer;
0.02
6.05
0.02
3.25 + 0.13
47.43 =7.3=0011 0.16
69.69 0.08
FEDformer;
2.45 0.03
1.67 0.02
3.22 1 0.04
44.87 0.16
0.16
66.54 0.07
STWA,
1.98 0.02
3.51 0.04
0.04
3.04 + 0.02
45.28 0.13
68.93
0.04
MTGNN,
1.93
0.01
3.35
0.03
1.54
2.96
43.32
65.81
0.07
BiTGraph
1.56 + 0.02
2.97 + 0.02
1.51 0.01
2.92 + 0.02
42.11 = 0.11
65.53 0.23
Method
PEMS
ETTh1
BeijingAir
(0.8)
MAE
RMSE
MAE
RMSE
MAE
RMSE
BRITS
3.26 + 0.10
7.02 0.03
2.14 + 0.01
3.81 0.09
46.75 0.59
68.17 0.87
SPIN
2.26 - 0.01
5.03
44.94 0.13
66.58 0.25
GRIN
2.96+0.02 0.02
6.10
52.07 0.92
74.60 1.85
GCN-M
4.54+0.02
0.07
52.57 + 0.08
73.71 = 0.14
CRUs
3.15
6.07
3.15 + 0.11 0.11
5.31
57.10
73.18 = 0.14
AGCRN,
5.09
10.07
3.18 + 0.12
6.02
55.59 0.03
81.58 = 0.10
Transformero
2.97
0.08
2.81 0.07
5.00
52.47
73.10 = 0.16
FEDformer
2.89 0.04
0.06
2.59 0.03
5.78 0.02
56.94
81.88 = 0.09
STWA,
2.34
5.16+0.02
2.56 0.02
5.10
74.95 0.08
MTGNN,
2.39
0.09
0.04
5.30 0.02
$3.05
73.57 0.10
AGCRN,
5.10
10.07=0.00
0.09
5.25 0.12
55.48 0.05
78.86 0.13
Transformer,
2.85 0.07
0.11
2.01
4.98
50.09
72.00 0.07
FEDformer,
2.76
0.02
2.55 0.00
4.69
49.83
71.22 = 0.08
STWA,
2.27+0.00 0.02
0.00
2.11 0.02
3.73 0.M4
46.97
72.93 0.07
MTGNN.
2.211
4.89 0.001
2.01 4 0.01
3.73 0.02
45.93
68.17 = 0.11
BiTGraph
2.15
4.73 0.02
1.91 + 0.01
3.54 = 0.01
45.47
66.98 = 0.29
14

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Published as a conference paper at ICLR 2024
Table 6: The results of forecasting error on PEMS, ETThI, and Beijing Air datasets with the missing rate y 0.2, 0.4, and 0.6.
Method
PEMS
ETTh1
BeijingAir
(0.2)
MAE
RMSE
MAE
RMSE
MAE
RMSE
BRITS
3.07 + 0.02
6.34 + 0.00
1.81 + 0.05
3.24 + 0.00
47.81 + 0.16
70.66
2.41
SPIN
2.08 + 0.05
4.74 + 0.16
44.03 + 0.12
66.34 0.75
GRIN
2.69 0.09
0.22
46.58 = 0.03
68.81 1.37
GCN-M
2.08
5.07
50.35 0.03
70.51 0.08
CRUs
0.01
5.60
2.87 + 0.05
4.92 + 0.05
57.81 0.55
75.49 0..1
AGCRN,
5.00+0.03
+0.0
2.57 0.02
4.93 0..5
53.47
81.06 0.12
Transformer
2.79
6.00
2.31 0.07
3.79
50.37 0.06
73.15 0.07
FEDformer
2.70=0000
0.11
+.00
3.00
50.03 0.09
71.81 0.14
STWA,
2.07
+.03
0.01
0.23
45.16 0.05
67.96 0.13
MTGNN,
2.00
0.02
L.65
0.02
44.71
66.11 0.06
AGCRN,
0.00
0.01
1355:000
0.04
0.08
78.55 0.12
Transformer,
0.01
5.42
1.75
0.06
47.72 0.04
69.17 0.16
FEDformer,
2.56
5.77
0.011
0.03
45.70 0.09
68.91 0.02
STWA,
2.03 0.02
2.03
4.61 0.07
1.67 + 0.02
0.01
45.08 0.02
67.57
0.10
MTGNN,
1.98 0.01
4.51 0.06
1.56 - 0.01
3.05
0.01
44.40 - 0.06
66.86 0.11
BiTGraph
1.90
0.01
4.28 + 0.01
1.56 0.01
2.97
0.02
42.94 + 0.13
66.36 0.25
Method
PEMS
ETTh1
BeijingAir
(0.4)
MAR
RMSE
MAR
RMSE
MAR
RMSE
BRITS
3.08 + 0.01
6.31 + 0.03
1.82 0.07
3.26 0.06
46.20 0.35
67.95 0.06
SPIN
0.24
45.37 0.55
67.38 0.77
GRIN
2.81 0.03
6.76 0.14
46.79 0.25
68.22
GCN-M
2.26+0.02
5.51 + 0.03
50.61 0.04
71.61 + 0.17
CRUs
2.85
5.65 1 5.65
2.04
5.01 1.110.04 0.04
0.46
76.41 1.16
AGCRN,
5.09
0.02
3.19
0.07
0.00
81.27 0.03
Transformero
0.02
6.38 0.07
2.36
4.79
0.06
77.03 0.17
FEDformer,
0.05
5.81
0.00
3.61
52.18
74.78 0.16
STWA,
2.02
4.67 0.03
0.00
3.00
47.38 1138+00 0.07
71.16 0.06
MTGNN,
2.02
4.50 + 0.02
1.92
3.03
45.17
66.73 +.0000..00 0.02
AGCRN,
5.10
10.06 + 0.01
2.87 0.04
0.04
5.00
82.67 0.10
Transformer;
0.00
6.09 0.07
1.88 0.02
0.08
0.12
0.000
FEDformer,
2.74
5.78 + 0.03
1.86 + 0.02
0.00
46.96
68.17 0.08
STWA.
2.07 = 0.03
4.80 + 0.02
1.81 + 0.04
0.07
45.69 0.13
70.56 0.19
MTGNN,
2.05 0.02
4.61 + 0.07
1.67 0.01
3.12 0.01
44.29
0.02
66.39 + 0.08
BiTGraph
1.96 = 0.00
4.34 + 0.02
1.64
3.07 0.02
43.13
0.22
65.55 0.24
Method
PEMS
ETTh1
BeijingAir
(0.6)
MAE
RMSE
MAE
RMSE
RMSE
BRITS
3.14 0.03
6.22 0.01
1.87 1 0.11
3.24 + 0.02
46.71 + 0.17
67.93 1 0.04
SPIN
2.18 0.05
4.91
44.28 + 0.45
65.68 + 0.08
GRIN
2.84
6.11
49.10 + 1.01
71.43 1.63
GCN-M
2.39 2.02
0.02
51.47 0.05
7.241
CRUs
2.97 0.31
0.09
3.05
0.03
56.63
75.42
AGCRN,
+.01
10.06
3.43
6.81+0.00
55.67 0.06
82.10
Transformero
2.91
0.04
2.03
0.00
50.79
7.09
FEDformer
0.04
6.06
2.06
0.02
55.32 0.13
79.83
STWA,
0.02
0.06
2.16
0.09
48.22
70.09
MTGNN,
0.01
0.02
2.36
0.04
4
67.80
AGCRN,
0.00
1.00
3.35
0.09
55.45
8.07
Transformer,
0.04
5.22
2.19 0.01
4.29+
49.23
72.38 0.122
FEDformer,
2.79+0..
6.31 0.05
4.26+
49.21 +
68.27 0.07
STWA.
2,14
0.03
0.02
0.00
46.06
69.92
MTGNN.
2.11
0.00
1.79
0.02
3.29 - 0.03
44.85 0.02
67.78 0.05
BiTGraph
1.99 + 0.01
4.47 + 0.01
1.74 + 0.00
3.21 + 0.01
44.23 + 0.15
64.20 + 0.32
15

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Published as a conference paper at ICLR 2024
Table 7: The results of forecasting error on Metr-LA and ETTh1 datasets with the block missing rate 0 0.0015 and 0.002.
Method
Metr-LA RMSE
ETTh1 RMSE
(r 0.0015)
MAE
MАPE
MAE
MAPE
BRITS
8.66 1 0.02
12.24 + 0.02
18.49 + 0.03
2.12 0.01
4.55
0.04
96.73
0.09
SPIN
6.58
11.02
0.03
0.03
GRIN
0.02
11.27 0.03
0.04
GCN-M
6.07
1..0
0.00
CRU
0.02
12.04
0.02
1.92 0.00
4.08 + 0.02
81.09 : 0.06
0.06
AGCRN,
14.88
1.02
2.79 0.02
5.02
116.64
Transformero
7.59
12.87 +: 0.03
1.00
2.01
4.20
0.005
FEDformer,
7.50
12.55
0.00
3.96 2.00
8.028
STWA,
7.02
14.84 0.03
0.00
1.90
3.69
79.29
MTGNN,
7.08
15.19 0.04
18.31
0.02
0.01
8.06
AGCRN,
14.88
14.30 + 0.01
29.94
1.002
2.000
4.92
107.79
Transformer:
7.08
11.75 + 0.03
1.04
1.02
3.39 0.01
84.67 0.06
FEDformer,
6,81 + 0.02
11.56 + 0.03
13.15 0.03
1.77 1.700
3.43 0.01
73.58 - 0.07
STWA,
6.56 0.01
10.57 = 0.04
12.90 0.03
1.79 0.00
3.21 0.02
74.59 - 0.06
MTGNN,
6.41 + 0.01
10.99 = 0.02
12.74 0.04
1.61 0.00
3.09 + 0.02
68.31 0.07
BiTGraph
6.22
10.97 +: 0.03
12.70
0.02
1.56
3.03
0.01
66.20
Method
Metr-LA RMSE
ETTh1 RMSE
(r 0.002)
MAE
MAPE
MAE
MAPE
BRITS
9.09 + 0.02
0.02
11.94 0.04
18.39 0.11
2.37 + 0.02
4.66 + 0.03
107 07.38 + 0.06
SPIN
6,68 108:0000 0.03
11.19 0.02
12.88 0.004
GRIN
6.87
12.18 0.06
0.04
GCN-M
6.85
11.74 0.03
13.35
0.00
CRU
7.91 +.002
13.25 + 0.03
1.05
2.97 0.01
5.05 + 0.00
97.58 0.11
AGCRN,
0.00
14.26 0.02
0.09
2.01
5.03
124.61 0.13
Transformero
0.02
13.67 0.02
1.00
0.00
4.24
103.02 1.02:0000
FEDformer
0.01
13.36 0.03
0.09
2.13
4.20
8..07
STWA,
0.00
16.76 0.07
0.03
2.00
3.76
79.43
MTGNN,
0.00
11.51 0.09
0.03
0.00
4.08
79.84
AGCRN,
14.88
14.26 + 0.02
0.02
0.00
4.80
99.99 0.16
Transformer;
0.00
12.34 0.04
0.03
0.00
0.00
75.32 0.08
FEDformer.
0.00
12.11 0.06
0.03
1.001
3.001
STWA.
0.00
10.92 + 0.03
0.03
1.000
3.33
76.30
MTGNN,
6.49
11.20 1 0.01
1..00
0.00
3.27 0.02
71.41
BITGraph
6,31 + 0.01
10.84 +: 0.02
12.49 + 0.02
1.60 +: 0.01
3.18 +: 0.02
68.11 + 0.04
B
MORE COMPARISON
Table 4 presents the results of different methods for the missing rate 0.1 Metr-LA and Electricity datasets. Table and Table demonstrates the results on ETTh1, PEMS and BeijingAir datasets. Our proposed BiTGraph demonstrates significant superiority, especially in the Electricity dataset with the highest variance. The performance of STWA, very close to that of MTGNN, and both models outperform the other baseline models. Moreover, the BiTGraph improves the MAE by 5.12 percent under the largest mask ratio on the PEMS dataset, which further demonstrates the effectiveness of our proposed method.
C BLOCK MISSING
To further substantiate the effectiveness of our method, we assess its performance in block-missing scenarios. We follow the method of GRIN to generate block-missing masks, and the results are presented Table We can observe that the proposed BiTGraph also demonstrates its great superiority, achieving notable 5% improvement the best baseline.
16

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Published as a conference paper at ICLR 2024
MAZ easter agannd
HMSC vantra segaieed on MetriA
versee againd Matr
11.2
11.1
100
12.0
0.1
10.0
24
Figure 2: The performance under different window sizes with H=24, 48, 96, and 168
MBC wile mL Mi
HAsSC nawe aganet on LA
MAPE vot agiietl MMl
11.5
130
120
Figure 3: The performance under I.=3, 5, 7, and 9
D HYPERPARAMETER SENSITIVITY
In this section, we evaluate the hyperparameter sensitivity of our method with respect to the history window size H1, the number of blocks L, well as the number of nearest neighbors k on the Metr-LA dataset.
Figure shows the performance changes over the history window size H. No evident performance improvements are observed as H increases from 24 to 168. We then evaluate the sensitivity of L. As shown in Figure BiT BiTGraph achieves the best performance when L=5. However, the performance does not improve further when I. continues growing. We hypothesize that this is due to the over-smoothing issue of GNNs that is, the node representations become indistinguishable when the graph convolutional layer reaches 5. We then further assess the impact of the k on the model performance. In particular, We study the metrics change as k varies from 5, 10, 15, to 20. As observed from Figure 4. the performance gains by incorporating spatial information from more neighboring nodes. However, once enough spatial correlations are obtained (i.e., k=10), further increasing the number of neighbors will no longer yield performance improvement. This suggests that only few neighborhood nodes have a significant impact on a given node.
E
THE ROLE OF 3
Table 8 shows the change of learnable 3 against missing rates on three representative datasets (traffic, solar energy, and air quality). We found that demonstrates similar values with different missing rates, indicating that 8 is primarily responsible for adjusting the strength of correctness and is
MMA MeriLA
1500
12.0
60
Figure 4: The performance under k =5, 10, 15, and 20
17

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Published as a conference paper at ICLR 2024
Table 8: The learned value of / on three datasets.
Dataset
0.2
0.4
0.6
0.8
Metr-LA
0.035
0,037
0.034
0.035
BeijingAir
0.475
0,463
0.471
0.467
ETThl
0.397
0.391
0.419
0.419
Pel
Grouss Tum BrGGmphh
Ground Truth
ETTNE
12.00
Crocan OrTSsrpph
Ground Truth
13000
1000
Grouns Turn BTGrruph
Ground Truth MITOON
Eelinglkr
Brgal
18TGraph
7//
Wan
300
Figure 5: The forecasting curves produced by our BiTGraph on four datasets.
independent of missing rates. Additionally, the possible reason for the value variation of across different datasets may caused by the unique characteristics of different datasets.
F PREDICTION VISUALIZATION
In this section, demonstrate the forecasting curves of our method four datasets with missing ratio r = 0.8. The time series down- wown-sampled in the temporal domain, and it is randomly selected in the spatial domain. As shown in the left part of Figure 5, our proposed method is able to produce results that well match the ground truth trends on all five datasets, which demonstrates the effectiveness of BiTGraph. In addition, we present the forecasting results of representative method MTGNN, which are displayed in the right part of Figure 5. We can observe that the curves produced by MTGNN, struggle to match the ground truths when the missing rate is high. Especially, on the BeijingAir dataset, the forecasting curves largely deviate from the ground truths.
G MODEL COMPLEXITY ANALYSIS
In this section, we assess the complexity of different methods in terms of memory usage and number of parameters. Table shows the model complexity of different methods. proposed BiTGraph has relatively small memory usage as well as a small number of parameters, which implies the model can trained with much less energy consumption.
be observed that our
18

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Published as a conference paper at ICLR 2024
Table 9: The model complexity of different methods.
Method
Memory Usage
#Parameters
BRITS
85.48M
173.72K
SPIN
10.85G
1.31K
GRIN
2,00G
12.76K
GCN-M
15.39G
396.50K
CRUs
41.97M
54.75K
AGCRN
10.17M
1.56K
MTGNN
250.55M
9.184K
Transformer
14.85G
15.46M
FEDformer
15.57G
16,08M
STWA
1.96G
256.50K
BiTGraph
194.17M
11.82K
19


Reviews:
[{"review_id": "0AvhdeY5Q6", "reviewer": "Reviewer_sYzZ", "summary": "The paper presented a multivariate time series forecasting model that applies temporal convolution and graph convolution. Such model is shown to handle missing values in inputs without additional pre-processing effort and outperforms the tested approaches.", "questions": "Introducing missing masks to feature map, motivated from vision tasks, makes sense; yet it'd be good to compare against, introducing the missing masks as input feature directly so the model can be trained with the knowledge what inputs are missing.\n\nIt was implied that the proposed approach could capture missing pattern; yet the experiments seem to only tested the scenario where values are missing at random. In actual application, it's rare that values are missing at random. It'd be good to also test different scenarios to show the efficacy of the proposed methods.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "The idea of introducing masked temporal convolution from vision tasks to time series is quite interesting and novel. \nThe paper conducted intensive experiments including comparison against multiple methods and ablation studies.\nThe paper is well written in the technical details.", "weaknesses": "The experiment setup is not clear to me whether it's sound.\nSpecifically, (1) It is not clear what loss function is used in both the proposed method and the tested methods. The paper compared MAE, MAPE and RMSE; it's the authors' choice what loss function to be used. Hence, if the task is to optimize for RMSE, the loss function for all the tested methods should be RMSE because the output would be the optimal mean estimator. Without such clarity, it's hard to derive where the accuracy difference comes from; is it from the matching loss function and evaluation metric, or is it from architecture innovation?\n(2) for MAE and MAPE, both are the evaluation metric for the median estimate as the denominator from MAPE, without clarifying, is assumed to be the same for all methods? But I saw for example in Table 2, under these two metrics, the ranking of methods can differ. This made me worry whether the experiment setup or the metric calculation is different from what I assumed.\n\nAnother minor note: an interval is implied for all the reported results and only in caption from Table 2, it was mentioned that 'the results are averaged over 5 runs'. What are these 5 runs referring to? Do they refer to different random seeds for training only? I assumed so given that the paper mentioned the train/test/validate is splitter based on ratio chronically which will fix the dataset so no variation from the data.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jkIwNBPsJP", "reviewer": "Reviewer_naqz", "summary": "This paper introduces a Biased Temporal Convolution Graph Network (BiaTCGNet) for forecasting from partially observed time series. BiaTCGNet is designed to jointly capture the temporal dependencies within and spatial structure of time series, while accounting for missing patterns by injecting bias into its components (MultiScale Instance PartialTCN and Biased GCN). Experiments conducted on several real-world benchmarks demonstrate the effectiveness of BiaTCGNet over alternative approaches.", "questions": "My questions and suggestions for the authors are included along with the weaknesses of this work (in the “Weaknesses” section of this review).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "* The proposed Biased Temporal Convolution Graph Network (BiaTCGNet) is designed specifically to account for both the temporal and spatial aspects of multivariate time series. Namely, this is achieved through BiaTCGNet’s two constituent modules: a Multi-Scale Instance PartialTCN and a Biased GCN. The former is capable of performing instance-independent temporal convolution to capture temporal (intra-instance) correlations within each individual time series, while the latter constructs a graph and diffuses information over it to capture the spatial correlations between the time series instances (channels).\n\n* In contrast to other existing time series forecasting methods, BiaTCGNet explicitly considers missing values in its model design through bias injection in the Multi-Scale Instance PartialTCN module that helps account for the different missing patterns; while progressively updating the missing patterns during Biased GCN’s information diffusion process.\n\n* Experiments on five real-world benchmark datasets have been conducted, the results of which suggest that BiaTCGNet achieves improvements of up to 11% over the existing forecasting methods under various scenarios involving missing values.\n\n* The paper is technically sound, well written and organized in a reasonably clear and comprehensive manner. The notation used throughout the paper is clear and consistent.\n\nUpdate: Thanks to the authors for detailed responses, after reading them, as well as comments from all reviewers, I am updating my rating from 5 to 6.", "weaknesses": "* Generally speaking, BiaTCGNet appears to be a result of (1) an almost direct application of a TCN with a straightforward modification to account for partial observations (Liu et al., 2018), and (2) leveraging a conventional GCN for learning of an adjacency matrix relying on two node embedding sets so as to capture asymmetric spatial correlations. In that regard, the novelty of this work may be considered incremental. Therefore, I would encourage the authors to further clarify and/or elaborate on the novelty of the two modules within BiaTCGNet.\n\n* In Eq. (8), the authors introduce $\\beta$, a learnable weight that is aimed to serve as a time-window-specific bias that corrects the global spatial correlation strength in accordance with the present missing patterns. Nevertheless, the role of this weight has not been discussed further. I believe that this work would benefit from including the learned values of $\\beta$ for each individual dataset used in the experiments. If possible, I would also suggest that the authors consider including a brief discussion on the interpretation of those values.\n\n* There seems to be a fairly recent work [C1] on attention-based memory networks for joint modeling of local spatio-temporal features and global historical patterns in multivariate time series with missing values. Moreover, the forecasting problem formulation in [C1] seems to be consistent with the one considered in this work. Therefore, I would suggest that the authors consider comparing the GCN-M method from [C1] with BiaTCGNet. Alternatively, I would ask the authors to provide the specific reason as to why this method has not been included among the baselines?\n\n[C1] Zuo, J., Zeitouni, K., Taher, Y., & Garcia-Rodriguez, S. (2023). Graph convolutional networks for traffic forecasting with missing values. Data Mining and Knowledge Discovery, 37(2), 913-947.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1DFcN2EI8M", "reviewer": "Reviewer_dZwn", "summary": "To handle missing values in time series, the paper employs\npartial convolutions from computer vision. The authors then\nuse a graph neural network with channels as nodes and\na induced adjacency matrix for forecasting. They evaluate\ntheir model on several datasets and show that it ourperforms\nseveral baselines from the literature.", "questions": "Using partial convolutions is a simple and plausible approach to\ntreat missing values in time series (s2). The experiments show a\nconsistent decrease in error over five datasets (s3) and the ablation\nstudy shows the effect of the two different components (s4).\n\nTwo points one could discuss:\nw1. limited methodological contribution: the paper merely combines\n  two existing methods.\n \nw2.  missing principled  baseline: forecast based on the time series\n  **and the imputation mask**.\nIn the main experiment in table 2 the authors compare several\nforecasting models for completely observed time series after\nimputing zeros (\"Transformer_0\") and after imputing with a\nspecific imputation method (\"Transformer_t\"). However, this way\nthe information which observations have been missing gets lost,\nand this might be relevant (\"informed missingness\").  The default\napproach is to impute a zero **and** to add a channel carrying\nthe imputation mask (i.e., as \"1\" if the value was observed and\na \"0\" if it originally was missing and now has been imputed).\nThis way, the information about missingness is not lost. -- It\nwould be really important to add these principled baselines\nto make sure that the specific way the proposed models deals\nwith the missing values really is causing the observed differences.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "s1. handling missing values is an important problem,\n   esp. in time series.\ns2. treating missing values with partial convolutions is interesting.\ns3. the method consistently reduces errors in the experiments.\ns4. ablation study shows effect of the two different components\n  of the method.", "weaknesses": "w1. limited methodological contribution: the paper merely combines\n  two existing methods. \nw2.  missing principled  baseline: forecast based on the time series\n  **and the imputation mask**.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)