Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Authors: Yicheng Luo
Bowen Zhang
Zhen Liu
Qianli Ma
Venue: ICML 2025 poster (2025)
Keywords: irregular multivariate time series
time series modeling
graph neural network
Primary area: applications->time_series

Abstract: Multi-scale information is crucial for multivariate time series modeling. However, most existing time series multi-scale analysis methods treat all variables in the same manner, making them unsuitable for Irregular Multivariate Time Series (IMTS), where variables have distinct origin scales/sampling rates. To fill this gap, we propose Hi-Patch, a hierarchical patch graph network. Hi-Patch encodes each observation as a node, represents and captures local temporal and inter-variable dependencies of densely sampled variables through an intra-patch graph layer, and obtains patch-level nodes through aggregation. These nodes are then updated and re-aggregated through a stack of inter-patch graph layers, where several scale-specific graph networks progressively extract more global temporal and inter-variable features of both sparsely and densely sampled variables under specific scales. The output of the last layer is fed into task-specific decoders to adapt to different downstream tasks. Experiments on 8 datasets demonstrate that Hi-Patch outperforms state-of-the-art models in IMTS forecasting and classification tasks.

Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Yicheng Luo' Bowen Zhang'
Zhen Liu'
Qianli Ma
Abstract
Due sensor malfunctions, varying sampling sources, or human factors, the final sampled time series often exhibits irregularities in real-world applications, resulting in an Ir- regular Multivariate Time Series (IMTS). The peculiarity of IMTS lies in two aspects: firstly, the sampling interval within each variable is uneven. Secondly, different variables are asynchronously sampled and have distinct sampling rates. The subgraph in the top left corner of Figure illus- trates an IMTS example with two variables.
Multi-scale information is crucial for multivari- ate time series modeling. However, most ex- isting time series multi-scale analysis methods treat all variables in the same manner, making them unsuitable for Irregular Multivariate Time Series (IMTS), where variables have distinct ori- gin scales/sampling rates. To fill this gap, we propose Hi-Patch, hierarchical patch graph net- work. Hi-Patch encodes each observation a node, represents and captures local temporal and inter-variable dependencies of densely sampled variables through an autra-patch graph layer, and obtains patch-level nodes through aggregation. These nodes are then updated and re-aggregated through stack of inter-patcl graph layers, where several scale-specific graph networks progres- sively extract more global temporal and inter- variable features of both sparsely and densely sam- pled variables under specific scales. The output of the last layer is fed into task ssecispei decoders to adapt to different downstream tasks. Exper- iments on 8 datasets demonstrate that Hi-Patch outperforms state-of-the-art models in IMTS fore- casting and classification tasks. Code is available at: https://gtthhbbcomm/aaaaiaaaaaaa
As unique type of multivariate time series, IMTS also ex- hibits the inherent multi-scale characteristies of time series. Unfortunately, most existing multi-scale analysis methods assume that the input series are regularly sampled and have limitations in handling IMTS. The mainstream approach of these methods is to downsample or segment the origi- nal series based several fixed time steps (Challu et al., 2023; Shabani et al., 2023), thereby forming several views that reflect features at different scale levels, as shown in the bottom row of Figure 1. On this basis, global temporal and varialle dependencies are extracted from coarse- grained views, while local dependencies are extracted from fine-grained views. This approach assumes that all variables have consistent original scale that each downsampled view consistently reflects the features of all variables at the same granularity level, thereby achieving consistent separa- tion of multi-scale patterns for all variables.
However, in IMTS, some variables have only a few obser- vations and do not possess fine-grained features, making their downsampled view meaningless (e.g., the variable Vn in Figure 1). Some other variables are densely sampled and require multiple downsampling steps to reflect global pat- terns (e.g., the variable V12 in Figure 1). In such cases, it is difficult to determine a consistently suitable downsampling level for all variables, making existing methods infeasible from the beginning. Furthermore, IMTS, downsampled view can contain mixed granularity features, as shown in the top row of Figure 1, but existing methods tend to extract and analyze single-level features at a specific scale, which are difficult to handle mixed features. Additionally, despite the distinct scales/sampling rates occurring among variables in IMTS, they are not entirely independent but exhibit cor- relations (Luo et al., 2024). Specifically, variables MTS may display different inter- intervvarable correlations different time scales (Cai et al., 2024). How to extract multi-scale
1. Introduction
Time series analysis has important applications in various fields such as healthcare (Lan et al., 2024), climate fore- casting (Verma et al., 2024) and traffic planning (Li et al., 2024b). Due to the complexity and non-stationarity of real- world systems, multivariate time series often exhibit dif- ferent variations and fluctuations different scales (Chen al., 2024a). Previous studies (Cai et al., 2024: Wang et al., 2024a) have demonstrated that effectively capturing multi-scale features is essential for time series modeling.
'School of Computer Science and Engineering, South China University Technology, Guangzhou, China. Correspondence to: Qianli Ma <qianlima @ scut.edu.cn?..
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
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Figure 1. Comparison of downsampling results of two variables for an irregular (top row) a regular (bottom row) multivariate time series with step sizes of (left column), 2 (middle column) and 4 (right column). Irregular time series contain mixed feature levels at specific downsampling scale, while regular time series exhibit single feature level at a specific downsampling scale.
temporal dependencies while considering inter-variable cor- relations under different scales IMTS, where variables have distinct original scales and asynchronous observations, remains a major challenge in IMTS modeling.
to flexibly represent and fully extract dependencies of specific variables IMTS at specific scales.
We propose Hi-Patch, using a hierarchical architecture to effectively achieve multi-scale modeling IMTS from fine-grained to coarse-grained.
To fill this gap, we propose Hi-Patch, a hierarchical patch graph network. Hi-Patch is based on the concept of patching, which has been recently proven to be effective in capturing local temporal dependencies (Nie et al., 2023; Chen et al., 2024a; Zhang et al., 2024). Hi-Patch first divides the orig- inal observation set into multiple patches based on a short time span and encodes each observation as node. Then, an intra-patch graph layer flexibly represents and fully captures local temporal and inter-variable dependencies of densely sampled variables at the original scale by employing fully connected graph network within each patch. Subsequently, to capture dependencies at a larger scale, the observation nodes are aggregated to obtain patch-level feature nodes as inputs for several inter-patch graph layers. Each inter-patch graph layer receives specific patch-level nodes, which are obtained through node aggregation of the previous layer, thereby forming a hierarchical architecture. These inter- patch graph layers gradually extract more global tempo- ral and inter-variable dependencies that both densely and sparsely sampled variables possess through scale-specific graph networks. The output of the last layer finally fed into specific task decoders to adapt to different downstream tasks. During this process, each intra/inter-pal -atth graph layer handles single-level features at specific scale. Fine-grained features of densely sampled variables are extracted in the lower layers, while coarse-grained features of all variables are extracted in the upper layers, achieving complete ex- traction of mixed multiple granularities temporal and inter- variable correlations in IMTS. Our main contributions are summarized follows:
We conduct experiments IMTS forecasting and classification on 8 datasets, and the evaluation results demonstrate that Hi-Patch outperforms existing meth- ods in most cases.
2. Related Work
Irregular Multivariate Time Series Modeling Existing methods can be generally divided into interpolation-basel and raw-data-based methods. The former, employing meth- ods such as kernel-based approaches (Shukla & Marlin, 2019; Wu et al., 2021b), hourly aggregation (Ma et al., 2020) or gaussian process (Tan et al., 202 aims to regu- larize sampling intervals. However, interpolation may dis- rupt the sampling patterns of the original series. Raw-data- based methods learn directly from IMTS. (Che al., 2018) enhances RNNs for uneven time intervals, while (Horn et al., 2020; Shukla & Marlin, 2021) introduce time em- beddings for arbitrary timestamps. (Rubanova et al., 2019; De Brouwer et al., 2019; Biloš et al., 2021; Schirmer et al., 2022; Chen et al., 2024b) use neural ODEs to address irreg- ularities. Recent research also integrates attention mecha- nisms (Jhin et al., 2021) or graph neural networks (Zhang et al., 2022; Yalavarthi et al., 2024; Zhang al., 2024; Luo et al., 2024) to capture inter-variable correlations in IMTS. Despite these advancements, most of these methods focus only on single-scale characteristics of IMTS, and how to comprehensively capture multi-scale features within IMTS remains challenge.
We introduce the hatra-patch/inter-pate graph layers
2

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Multi-scale Modeling for Time Series Multi-scale infor- mation is essential for time series modeling. (Chen al., 2021) leverages different frequencies to update sub-hidden states, while (Liu et al., 2022) employs pyramid attention for multi-scale feature extraction. (Challu et al., 2023) uses multi-rate sampling and hierarchical interpolation, and (Sha- bani et al., 2023) assigns forecasting models across tem- poral scales. (Chen et al., 2024a) introduces a multi-scale Transformer with adaptive paths, and (Wang al., 2024a) disentangles variations across scales for complementary pre- dictions. However, these methods are designed for regular MTS and are not well adaptive for IMTS. Although Warp- former (Zhang et al., 2023) presents a multi-scale approach for IMTS, its original goal is to balance the differences in sampling densities across variables and involves upsam- pling interpolation for sparse variables, which may distort the original data distribution.
1, ,,,,,,,,, < [و] and forecasting window 3 ([()),
1,... M,t; > ft). Elements ε and " of j-th observation tuple in the set of forecasting windows are combined into a forecasting query qj E Q, The prob- lem is to accurately predict the corresponding observation values Z, correspondence to forecasting query O based on the historical window X;:
F(X, 21) - 2,
(3)
where F()) denotes the forecasting model we aim to learn.
4. Methodology
In this section, we will take the i-th sample S; with Ma historical observations as an example for introducing our method. We introduce the observation encoder in Section 4.1, followed by the introduction of the single intra-patch graph layer and inter-patch graph layer in Sections 4.2 and 4.3, respectively. Section 4.4 describes how the hierarchical architecture is implemented, and Section 4.5 covers the task decoder. Figure 2 presents the overall architecture of our model.
Graph Neural Networks for Multivariate Time Series Recent studies have integrated GNNs with time series mod- eling frameworks to effectively capture inter-variable depen- dencies in MTS (Jin et al., 2024), achieving success in do- mains such as transportation (Rahmani et al., 2023), health- care (Wang et al., 2022), and economics (Wang et al., 2021). However, many of these methods are primarily designed for modeling synchronous correlations among variables, lacking sufficient extraction of asynchronous dependencies widely present in IMTS.
4.1. Observation Encoder
First, we introduce how to encode each observation into an dmode dimensiimmnsio graph node embedding. The j-th histori- cal observation og of S; corresponds to the tuple (t,, fy, 2z), representing the observation value 2j of variable vy at times- tamp Lj. We encode these three parts separately.
3. Problem Definition
For time encoding, we adopt continuous time embedding (Shukla & Marlin, 2021), which is designed specifically for IMTS. The d-th dimension of dmoder-diment -iieensioaa time embedding Φ(t,) for tý is calculated by:
Definition 1 (Irregular Multivariate Time Series) We con- sider a dataset D consisting of IMTS sample. Each sample of 'D is tuple, i.e., ' == ((S1, v1),.... ... (Sn, yn)}, }. where S, denotes the i-th time series sample and y; E {l,..., (a....,C C} is the class label (C is the number of categories). We de- scribe i-th sample S, as a set of AM = |S,] observations such that S, := for..... : ..3) ). Each observation 0j is a tuple (lz, Zz, U2), consisting of a timestamp (j E R+ observed value 2j- E R and variable indicator uy {1...., V}. where V represents the total number of variables. An IMTS sample can thus be represented as:
<tt wn 1 + 0z, sin(wa - L a.), if < d < dmodel
if dl = 0
(4)
where Wo, aep, Wa and an are Iearnable parameters. For vari- able encoding, we randomly initialize a learnable variable embedding for each variable, forming a variable embedding matrix E E TR. Vxdunell the embedding corresponding to variable vj is Evg. For value encoding, a linear layer f(-) is used to map the observation value 2j into dmodel mamnee-diimesional embedding.
S = {(t,, 2,, 0;)lı = M)..,.M,
(1)
Problem 1 (Irregular Multivariate Time Series Classi- fication). Given an IMTS sample S, the problem is to correctly predict its class label yı:
With these three embeddings, the graph node embedding for j-th observation 0j is calculated as:
hg = a (b(()) + E.,, + f(z;)) E dmmaee
(5)
C(S.) + yt:
where (t:) is ReLU activation function.
where C(.) denotes the classification model we aim to learn.
4.2. Intra-Patch Graph Layer
Problem 2 (Irregular Multivariate Time Series Forecast- ing). Given a split timestamp ts. each sample S; is seg- mented into a historical window X, := (lt, 2j, v,)lī =
In this section, we will introduce how to extract fine- -ne-grained features of densely sampled variables through an intra-patch
3

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Output
26 Embheeklings
Nade of V,
Task Decader
Node of
H1 (3)
Inter-Patcih Graph Layer
Node of V. Aggregation
[logz Mpace
Y
Inter -aatch Graph Layer
Update
20
Aggregation
2((2)
time Prediction
(4. (0())
Update
Classifier
Scale oe(3)
Clesification
Intra Patth Graph Layer
history
fature
Aggregation
Update
Observation Encoder
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20 time
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Figure 2. The model framework of Hi-Patch. The example input is an IMTS with three variables and historical window of 0-40s which is divided into N = 4 patches with a patch size / = 10s. First, each observation is encoded into a graph node through an observation encoder. Then, fine-grained features of densely sampled variables within each patch extracted via an intra-pater graph layer and nodes are aggregated into patch-level nodes as inputs for multiple stacked inter-patch graph layers. After passing through [log, N = 2 inter-patch graph layers, coarse-grained features of all variables are progressively extracted and single node embedding for each variable is obtained. Finally, the task decoder computes the downstream task output based on these node embeddings.
graph layer. First, we divide the historical observation nodes into several non-overlapping patches based on short time span P. Given a total historical time span of T, it can be divided into N = - patches. The n-th patch includes observation nodes of all variables within the period from (n 1)- P to n P. the index set of observation nodes in n-th patch is:
We use graph attention network (GAT) (Veličković et al., 2018) to update the node states within each intra-patch graph for sufficiently extracting these dependencies. The formula for updating the state of node v is:
h by +
Σ MHA Bay hy), nEA((e)
(8)
I, úi = 1,2, .... Mn. - (1- Sn PJ. (6)
where N(w) denotes the set of neighboring nodes of node v and MHA denotes the multi-head attention mechanism (Vaswani et al., 2017):
The corresponding set of node states is H,, = (h₂ 1 I,), initialized by node embeddings. Each patch, divided based on a short time span, primarily consists of observation nodes from densely sampled variables, with few observation nodes from sparsely sampled variables, thus forming fine-gras tne-grained view of densely sampled variables.
(Wahh)) (Wghh)))W)))))))
(Wolhe))
MHA (1yps Auy (ww) = Softmax
Wohu
dimodeel
(9)
Due to the existence of three different types of dependen- cies, we adopted three sets of { g. WR. W.} parameters to handle them separately. Compared to conventional methods that separately extract features along the time and variable dimensions, our intra-patch graph more comprehensively ac- counts for the asynchronous characteristics of IMTS. After the I--layer state update with (8), the local dependencies of the densely sampled variables the original scale are thoroughly extracted.
4.2.1. UPDATE
After obtaining several patches as fine-gran lne-ggrined views of the densely sampled variables, each pair of nodes within a patch connected by an edge to form fully connected graph (intra-patch graph) for flexible representation of three types of local dependencies in IMTS in unified manner, namely: 1) the same variable at different times, 2) different variables at the same time (synchronous) and 3) different variables at different times (asynchronous). The set of edges in n-th patch is:
4.2.2. AGGREGATION
To further extract features at a larger scale, we aggregate nodes of the same variables within each patch to create the
En = ((u,) u In, Inj.
4

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
overall feature nodes of variables during the period of each patch. Specifically, for the nodes of the w-th variable in the n-th patch, the index set of nodes is In.; - j e In. Vy = = We first calculate the average observation timestamp as a reference time 12 = T... EjEz,,. 4, and then use multi-time attention (Shukla & Marlin, 2021) to aggregate the nodes to 7(0).
we calculate the average timestamp between two adjacent nodes as the reference time ((2))
12 2n
where 1. +
10)
man
n 1, N/2, m m2n-1. mm. O, and then we aggregate the adjacent nodes to this reference time through multi-time attention to obtain feature nodes of 2P scale:
(0)
(0)
Σ JE(2n-1,20)
MHA MHA(6())) (o() ( (,0)
of D( R.D j.0
(12)
100 = Σ (T.)
(10)
JEI
The node state set after aggregating is H22) ( I
When a variable has no observation within a patch, no ag- gregation node is created, thus preserving the irregularities of the original series. We use a mask indicator mn," to indicate whether the aggregation node exists of the w-th variable at the n-th patch. If | = 0, ma,e is set O, (0) else mm,1" = 1. The new set of node states for all N patches of all V variables after aggregation is 7((0) (1 | 1, N,,,,.,, N,
1, 2, N/2, v = 2, ..... V, m = 1}. which is served as input for the next inter-patch graph layer (2P-scale).
(0)
4.4. Hierarchical Architecture
(0)
Section 4.3 describes the first inter-patch graph layer (P- scale). This layer takes O0 as input, updates the node states, and aggregates them to 22) as the input of next inter- patch graph layer (2P scale). Generalizing to the general case, the formula for the I-th inter-patch graph layer is:
1,2,..., V, m = =1}.
4.3. Inter-Patch Graph Layer
H EegreggationUUpaatee (Update(H( (H(())))
(13)
After passing through an intra-patch graph layer, fine- grained features with time scales less than P of the original series have been extracted. In this section, we aim to extract features with time scale P based on the aggregated node set H(() through an inter-patch graph layer.
where H(( = (no, n = 1,2, N/22,a
(!) 13 and HD-11 (t-1) = 1}. The recur-
(1-1)
1, 2, ... V. ma,"
{
n
1, 1,2 N/2 sive form of (13) allows us to build a hierarchical archi- tecture by stacking inter-patch graph layers, thereby further extracting features at scales of 2P, 4P, 8P, 16P, .... Un- til l = [ogg N7. the output of this layer is H(ooo,, { [nn ([es, N1) where a single node embedding is obtained for each variable which has at least one observation. This process involves a total of [log, N] inter-patch graph layers, and tneinnaieed features of relatively densely sampled variables are extracted in the lower layers, while coarse- -rauine features of both densely and sparsely sampled variables are extracted in the upper layers. The node embedding of the e-th variable is H (llog, N)) Rddaia
= 1, 2.... V.m..o
1. UPDATE
Each node in the set ()) represents the overall feature within time span P, and the reference time interval between nodes in adjacent patches is approximately equal to P. We connect the nodes located in adjacent patches pairwise in node set (0) by an edge to form a P-scale graph (inter- patch graph). This phte-pptchh graph flexibly represents the temporal and asynchronous variable dependencies of densely sampled variables and some sparsely sampled variables with an origin sampling interval less than P at scale P in a unified manner. We continue to update the node states in this P-scale graph using GAT to extract the dependencies at scale P. The formula for updating is:
1, = 1, V, V.m.l.e
m[logg M/) = 1},
4.5. Task Decoder
V
h = (0) +
Σ
Σ
MHA
(0) (0) n. ne
4.5.1. CLASSIFICATION
w'E(nnl.n1)eed 1'=1
We calculate the sum of dmodel channels for each variable's node embedding to get V-dimensional vector C and use this to predict the final classification probabilities: Softmax(W** 6²). The training objective is to minimize the cross-entropy loss between I and the ground truth y.
(11)
4.3.2. AGGREGATION
After the node state update, the features at scale P are extracted. Subsequently, to extract features at scale 2P, we aggregate every two nodes located on adjacent patches of the same variable by multi-time attention again to obtain the input of the next inter-patch graph layer. Specifically,
4.5.2. FORECASTING
Given a query q, = (t,, vy), we follow (Zhang et al., 2024) by concatenating the node embedding of the query variable
5

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
by with the embedding of the query time 1y, and then pass it through MLP projection layer to generate the forecasting result: 2, = MLP(H (Dogg N1) (t,)))). The training ob-
5.1.2. EVALUATION SETUP
For the forecasting task, we follow the data pre-processing method described in (Zhang al., 2024) and randomly divide all the instances among each dataset into training, validation, and test sets according to ratios of 6:2:2. We use Mean Square Error (MSE) and Mean Absolute Error (MAE) to evaluate forecasting performance.
jective is to minimize the mean squared error loss between 2j and the ground truth 2z. The pseudo-code for Hi-Patch is presented in Appendix A (Algorithm 1).
5. Experiment
For the classification task, we follow the method described in (Harutyunyan et al., 2019) and divide the dataset into three parts for training, validation, and testing with the ratio of 70%.15%.155 on the MIMIC-III dataset. For the re- maining three datasets, we adhered to (Zhang et al., 2022)'s approaches, and the ratio of training, validation, and test- ing set is 8:1:1. We measure the classification performance with the Area Under the Receiver Operating Characteris- tic Curve (AUROC) and Area Under the Precision Recaal Curve (AUPRC) since all the four datasets are binary classi- fication datasets with highly imbalanced class distribution. More details of metrics can be found in Appendix D.
In this section, we present forecasting and classification experiments using a range of models and 8 datasets.
5.1. Experimental Setting
5.1.1. DATASETS AND BASELINES
For the forecasting task, we follow (Zhang et al., 2024) and use four datasets: PhysioNet (Silva et al., 2012),
MIMIC-III (Johnson al., 2016). Human Activity, and USHCN (Menne et al., 2015), covering the fields of health- care, biomechanies, and climate science. We compare our method with seventeen relevant baselines, covering the SOTA models from (1) MTS forecasting: iTransformer (Liu et al., 2024). ModernTCN(LLo& & Wang, 2024), TimesNet (Wu et al., 2023), PatchTST (Nie et al., 2023), (2) multi- scale MTS forecasting: Pathformer (Chen et al., 2024a) TimeMixer (Wang et al., 2024b), MSGNet (Cai et al., 2024), MICN (Wang al., 2023), (3) IMTS classification: Warpformer (Zhang et al., 2023), Raindrop (Zhang et al., 2022), GRU-D (Che et al., 2018), (4) IMTS forecasting: IPatchGNN (Zhang et al., 2024), GraFITi ( (Yalavarthi et al.,
5.1.3. IMPLEMENTATIO DETAILS
We adopt the Adam (Kingma& & Ba, 2014) optimizer with a learning rate of 0.001, stopping when the validation loss doesn't 't decrease over 10 epochs. All experiments are con- ducted with five random seeds, and the average and standard deviation are reported. All the models are experimented with using the PyTorch library on 2 GeForce 24G GPUs. The detailed settings of hyperparameters can be found in Appendix E.
2024), CRU (Schirmer et al., 2022), mTAND (Shukla &
Marlin, 2021), Neural Flows (Biloš et al., 2021), Latent ODEs (Rubanova et al., 2019).
5.2. Main Results
For the forecasting task, we test the model's performance under 3 varying observations and forecast horizons on each dataset. Table shows the results of the default horizon, and the complete results are presented in Appendix F.1. Table reports the models' classification performance on the other four datasets. In addition, to demonstrate the robustness of our method, we test whether Hi-Patch can achieve good classification performance when a subset of variables is completely missing. Table 9 (Appendix F.2) reports the results under different missing ratios. In summary, Hi-Patch achieves the best performance in 62 out of 72 metrics across 8 datasets for both classification and forecasting tasks.
For the classification task, we conduct experiments four datasets in medical field where IMTS is most widely used, namely P19 (Reyna et al., 2020), PhysioNet (Silva et al., 2012), MIMIC-III (Johnson et al., 2016) and P12 (Gold- berger et al., 2000) where PhysioNet is a reduced version of P12 considered by prior work (Shukla & Marlin, 2021). We compare our method with the state-of-the-art methods for irregular time series classification, including GRU-D (Che et al., 201 ODE-RNN (Rubanova et al., 2019), IP- Net (Shukla& & Marlin, 2019). SeFT (Horn et al., 2020), mTANN (Shukla & Marlin, 2021), Raindrop (Zhang et al., 2022), StraTS (Tipimeni & Reddy, 2022), DuETT (Labach et al., 2023), VITST (Li et al., 2024a) and Warpformer (Zhang et al., 2023). In addition, we also compare our method with two approaches initially designed for forecast- ing tasks, namely DGM2-O (Wu et al., 2021b) and MTGNN (Wu et al., 2020). The implementation and hyperparameter settings of these baselines are consistent with those of the original paper. More details of datasets and baselines can be found in Appendix and C.
Among these baselines, regular MTS forecasting models do not demonstrate competitive performance in IMTS forecast- ing since they have limited capability to handle the irreg- ularities within and among variables in IMTS. In contrast, although our Hi-Patch is initially designed for IMTS, it also demonstrates competitive performance in regular MTS fore- casting, shown in Appendix F.3. As for the baselines specifically designed for IMTS, most of them focus ex-
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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Table Method benchmarking on IMTS forecasting. The best results highlighted in bold, and the second-best results in underlined. The results in the table are presented in the form of (Mean + Std). ' indicates numerical overflow error.
Methodk
Duman Activity (3000ms + 1000ms)
USIICN (21monihs + Imonth)
PhysioNet (2th + 24h)
MIMIC-III (2th 2-1b)
MSEX10-3
MAEX10-2
10
MAEX10-1
MSEX10-3
MAEX 10-2
MSEX 10-2
MABx 10
Transformer
3,97 1 0.10
4,30 2.08
6.17 0.07
53.55 1 19.59
16.87 2:551
7.24
21.24 0.97
ModeraTINN
4.32
0.13
0.08
28.99 11.96
0.24
10.07
10.76 1.30
TimesNer
3.79 0.05
4.78
0.12
9.30
2.34 0.54
8.09
PatchTST
5.21 0.33
5.10 0.20
5.88=0.10
3.66 0.13
25.56 4.42
10.00 1.08
7.24 0.65
19.50 0.87
Pathfommer
0.16
3.65 0.08
6,75 + 0.41
4.61 = 0.20
TimeMiser
4.97 0.31
5.02 0.16
0.10
3.59 0.07
13.98 0.31
6.88 0.00
4.78 0.09
14.29 0.96
MSGNEet
6.12
6.06
0.05
=
0.18
MICN
6,93 0.12
6.04 0.06
0.19
0.08
10.34 0.24
0.14
2.36 0.06
8.43
Warpformer
5,04
3:72
1.76
1.27
Raindrop
4,42 0.25
4.65 +
0.11
3.29 0003
10.65 0.29
6,02 0.11
2.31 0.07
8.61 + 0.12
GRU-D
3,94 0.29
4.37 0.21
a.06
3.21 0.05
5,76 0.34
4.53 0.15
2.35 0.06
8.34 0.22
IFaccGNN
2.79 0.09
3.21 0.06
3.07 0.05
5,06 0.10
3.75 a 0.07
1.97 2: 0.05
1,76 a 0.22
GraFITi
3.83
3.45
5,07 0.05
0004
0.11
3.06
1,76 0.04
7.28
CRU
3,83
3,60 0.04
5.15 0.50
6.43
4.51
2.23 0.03
7.99 0:22
ITANN
3.14 0.09
3.71 0.06
5.03 0.05
3,00 0.06
6.18 0.31
4.44-0..19
2.15 0.05
8.00 0:06
NeurilFlow
4.29 0.63
4.61 0.43
5.41 0.05
3.35 0.06
7,68 0.37
4.54 0.19
2.34 0.05
8.09 0.09
3.32
0.0
2.11
Hi-Patch
2.57 0.02
3,11 0.00
4.94
0.04
4.86 0.03
3.62
1.75
7.34
Table Method benchmarking on IMTS classification. The best results are highlighted in bold, and the second-best results are in underlined, The results in the table presented in the form of (Mean Std %).
P19
PhysioNet
MIMIC-II III
P12
Methods
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRO
GRU-D
88.7 + 1.2
57.6 2.3
79.1 6.9
42.7 + 7.2
82.2 1.8
43.3
79.6 0.6
41.7 + 1.8
ODE-RNN
87.1 1.0
5.26
75.5 2.8
33.7+4.1
81.
42.3
78.8
37.4 2.6
IP-Net
9.2
58.6
0.6
15
84.1
4.
83.7
46.3
ScFT
8.30
0.5
0.2
2.9
67.9
2..2
7.5
3,.9
MTGNN
1.0
15
4.41
7.3
78.5
3..2
82
4..
mTANND
8.99
1.5
86.8 1.3
52.5 1.3
4.3
46.6 1.61...
0.3
49.3 1.0
DGM2-0
0.56
1.3
0.7
5.24
8.5
3.80
8.1
0.7
Raindrop
2.76
1.
0.9
2.0
1.3 79.8
35.2 1.1
8.60
42.7 = 1.7
StraTS
9.3
58.4 1.4
84.9 1.5
4..
84.4
46.4
8.7
$2.1 1.5
DuETT
8.5
5..0
81.3 14
4..9
78.8 0.8
34.3 1.0
83.4 1.2
45.4 1.5
VITST
9.1
0.7
81.3 1.9
2.
39.6
0.1
50.8
Warpformer
0.4
2..6
0.7
2.3
8.36
46.6
8..4
50.4 1.5
Hi-Patch
92.1 a 0.4
6.11
86.8 0.9
57.3 1.9
84.8 a 0.2
47.2
86.9 : 0.7
53.3 + 0.9
tracting correlations at the origin scale, failing to achieve comprehensive multi-scale feature extraction and resulting in suboptimal performance. Additionally, conduct periments with a fixed prediction horizon while varying the observation horizons to demonstrate our model's ability to learn more temporal dependencies from long-term data in Appendix F.4. Furthermore. we provide the computational efficiency analysis in Appendix G and comprehensive dis- cussions on the technical details of Hi-Patch and existing methods in Appendix H.
Table 3. Ablation results of Hi-Patch two datasets. The results in the table are presented in the form of (Mean + Std).
Methods
Hunam Activity
USHEN
MSEx 10
MAEX 10
MSEx10
MARX 10
HI-Patch
2.57 a 0.02
3.11 0.03
4,94 0.05 0.05
2.96 1.9.200.00
wio Hie
2:70 0.04
3.13 0.03
0.15
0.25
wio DVDT
2.68 0.04
3.12 0.01
5.21 00,05
3.11 1.11
3W
2.00
0.03
wio TEAGG
3,22 0.05
5.23 0.04
3.10
of the entire historical window, only extracting features at the original scale. (2) w/o DVDT: We removed the asyn- chronous edges of different variables at different times in intra/inter patch graph layers, retaining only the edges be- tween nodes of the same variable at different times and different variables at the same time; (3) w/o 3W: We use only one set of attention parameter matrices { We. W.} for three types of edges; (4) w/o TEAGG: We aggregate
5.3. Ablation Study
In this section, we investigate the performance benefits gen- erated by each key component of the proposed method the forecasting task. We compare the Hi-Patch with its four variants: (I) w/o Hie: We remove the hierarchical multi- scale architecture and set the patch size to the time span

---

Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
nodes with the same variables within patch using mean ag-
Ap-1
5.04
5,00
gregation rather than multi-time attention aggregation. The results on the Human Activity dataset and USHCN dataset are presented in Table 3, while the results for the remaining datasets are presented in Table 11 (Appendix l). The results show that all components of Hi-Patch are necessary.
5.02
5.00
MSE
4.95
4.98
4.90
4.06
4.91
1.250
1.375 1,500 1.625 1.750
4.85
5
7
Patch Size (months)
Patch Size (hours)
5 3.00 10-3
(a) USHCN
(b) PhysioNet
5.3
MSE
5.2-
MSE
4.95
Figure 5. Effect of different patch size
5.1
5.5. Effect of Patch Size
2
3
5
6
4.85
3
5
Number of Scales
Number of Scales
The impact of patch size on the performance of our model in two real-world datasets, USHCN (climate) and PhysioNet (clinical), is illustrated in Figure 5. For the USHCN dataset, the optimal model performance is achieved at a patch size of 1.5 months. At this patch size, our hierarchical architecture extracts features at scales of 1.5 months, 3 months, 6 months, and 12 months, which precisely cover several important ob- servational scales in climatology, including the seasonal (Doblas-Reyes et al., 2013) (3 months), monsoonal (Clift & Plumb, 2008) (6 months) and annual (Almazroui et al., 2012) cycle (12 months). For the PhysioNet dataset, the optimal patch size is 6 hours. At this patch size, our model extracts features at the 6-hour, 12-hour, and 24-hour scales, which also align with actual cycles in clinical medicine. Among them, 6 hours is common clinical monitoring pe- riod used in medical practice (Seymour et al., 2017), while 12-hour and 24-hour cycles reflect circadian rhythms and daily cycles, which are crucial for assessing patients" phys- iological changes and disease fluctuations (Klerman et al., 2022). This alignment with real-world periodicity enables it to better capture the inherent temporal dynamics in di- verse applications. By aligning with critical observational and diagnostic timeframes, our model enhances its predic- tive power and interpretability, making it highly adaptable and effective across practical scenarios that require nuanced temporal understanding.
(a) USHCN
(b) PhysioNet
Figure 3. Effect of different scale quantities on two datascts.
Time Span (hours) 3
020
Span
42.5
40.0
Didributtoon curee
Sample Faint
Dake bution
sample Part
00 0.005
0.030
10 000
202
Langth
220
300
ido Length
150
(a) USHCN
(b) PhysioNet
Figure 4. Distribution of sample length and time span on two datasets. Each blue dot represents a sample, with its x-coordinate indicating the sample length and the left y-axis representing its time span. The red curve the distribution curve of sample length.
5.4. Effeet of Scale Quantity
Figures 3 and 8 illustrate the impact of different scale quan- tities on four datasets. Generally, increasing the number of scales improves performance within a certain range. How- ever, beyond a certain point, further increasing the number of scales has a negative effect. This is because the number of scales is inversely related to patch size in our method: as the number of scales increases, the patch size decreases. A patch that is too small may not contain enough observations to extract local patterns effectively. Additionally, we observe that on the PhysioNet and MIMIC-III datasets, performance with too many scales is even worse than with single scale. To further investigate, we visualize the distribution of sam- ple lengths and time spans across the four datasets in Figures 4 and 9. We find that samples in PhysioNet and MIMIC-III primarily exhibit a 'short length, long span' characteristic, which indicates that samples in these two datasets predomi- nantly show coarse -rarine patterns with few fine-grained local features. In such cases, using too many scales becomes redundant and can significantly degrade performance. We provide visualizations of the multi-scale views at different layers in our model Appendix K.
6. Limitations
Although our proposed method introduces a multi-scale framework for sparse IMTS, it exhibits certain limitations in terms of scalability. As detailed in Appendix G, the pri- mary computational cost of our approach stems from the intra-patch graph layer, whose complexity scales quadrati- cally with the number of observed points within each patch. While this cost is acceptable for sparsely sampled IMTS data, it becomes prohibitive when applied to large-scale datasets with dense observations. Consequently, a promis- ing direction for future work is to reduce the computational complexity of the intra-patch graph layer in Hi-Patch to improve its scalability. Appendix G.3 provides further dis-
8

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
cussion this topic.
Cristian Challu, Kin G Olivares, Boris N Oreshkin, Fed- erico Garza Ramirez, Max Mergenthaler Canseco, and Artur Dubrawski. Nhits: Neural hierarchical interpola- tion for time series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 37, pp. 6989-6997, 2023.
7. Conclusion
In this paper, we propose Hi-Patch for modeling IMTS. The proposed method leverages the patraaapccc/inte-patell graph neural network to flexibly represent and fully extract features specific scales in IMTS. Based on this, the hierar- chical architecture is used effectively achieve multi-scale modeling of IMTS in bottom- up manner (from local to global). Experimental results demonstrate that Hi-Patch outperforms existing methods in both IMTS forecasting and classification tasks. Our future work will focus on adaptive multi-scale modeling of IMTS, which selects the most suit- able scales based on the specific temporal characteristies and dynamics of each sample to further improve performance.
Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):6085, 2018.
Peng Chen, Yingying Zhang, Yunyao Cheng, Yang Shu, Yihang Wang, Qingsong Wen, Bin Yang, and Chenjuan Guo. Pathformer: Multi-scale transformers with adaptive pathways for time series forecasting. In International Conference on Learning Representations, 2024a.
Impact Statement
Yuqi Chen, Kan Ren, Yansen Wang, Yuchen Fang, Weiwei Sun, and Dongsheng Li. Contiformer: Continuous time transformer for irregular time series modeling. Advances in Neural Information Processing Systems, 36, 2024b.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Zipeng Chen, Qianli Ma. and Zhenxi Lin. Time-aware multi-scale rnns for time series modeling. In IJCAI, pp. 2285-2291, 2021.
Acknowledgements
Peter D Clift and R Alan Plumb. The Asian monsoon: causes, history and effects, volume 288. Cambridge Uni- versity Press Cambridge, 2008
The work described in this paper was partially funded by the National Natural Science Foundation of China (Grant No. 62272173), the Natural Science Foundation of Guangdong Province (Grant Nos. 2024A1515010099 2022A1515010179), the Science and Technology Planning Project of Guangdong Province (Grant No. 2023A0505000100 and the National Key R&D Program of China (Grant No. 2023YFA1011601).
Jesse Davis and Mark Goadrich. The relationship between precision-recall and roc curves. volume 06, 06 2006. doi: 1 143844. 143874.
Edward De Brouwer, Jaak Simm, Adam Arany, and Yves Moreau. Gru-ode-bayes: Continuous modeling of sporadically-obsereed time series. Advances in Neural Information Processing Systems, 32, 2019.
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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
A. Algorithm of Hi-Patch
Algorithm The pseudo-code of Hi- Pi-Patch for forecasting
Input: An IMTS sample S, with M observations, split time is historical window X; := {(G), 2/ By)lj I..,,,,, M, 2 ) forecasting query Q; == #y)li = 1.... M,t, >> tt); patch size P, total time span T
Output: Predicted value set 2,
1: b Observation Encoder
2: for j 1, 2, ..., | [ do
3: encode observation tuple o, 2z, as graph node embedding h using Eq.(S)
4: end for
5: Intra-Patch Graph Layer
6: forp= - 1,2, | do
7: construct intra-pate patch graph in patch using (q) and (7)
8: update states of intra-patch graph nodes patch p through GAT using (8. and Eq.(9)
9: aggregate nodes of the same variable in patch through multi-time attention using (1,( get 7(
end for
Inter-Patch Graph Layers
12: for / = 1,2, .... [log2 [ do
13:
construct /P-seale inter-patch graph
14:
update states of LP-sca inter-patch graph nodes through GAT using Eq.(11)
15: aggregate nodes of every two adjacent nodes of the same variable in (P-scale inter-patch graph through multi-time
attention using E12( get 72(1)
end for Task Decoder
18: for 1,2, ..., 1 Q do
(ooga # ((1)))
19:
; = MMHP((He
20:
Z: + Z, U {
21: end for
22: return 2
B. Datasets
We use irregularly multivariate time series datasets to evaluate the performance of our model and baseline models. The dataset statistics are summarized in Table 4.
Table 4. Dataset statisties.
Task
Datasets
# Samples
Variables
Avg Sample Length
Missing Ratio
Human Activity
5400
12
120
75%
USHCN
26736
5
163
77.9%
Forecasting
PhysioNet
11988
36
74
88.4%
MIMIC-III
23457
96
96.7%
P19
38803
34
36
94.9%
P12
11988
36
74
88.4%
Classification
MIMIC-III
21107
16
78
65.5%
PhysioNet
3997
36
74
84.9%
B.I. Forecasting
For the forecasting task, we use four datasets and follow (Zhang et al., 2024)'s data preprocessing program. Here the detailed information of these datasets. We use three historicalfforecasting horizons on each dataset.
13

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
PhysioNet (Silva al., 2012) This dataset includes 12000 IMTS from different patients, each with 41 clinical signals collected irregularly during the first 48 hours of ICU admission. We use the first 24/36/12 hours as the observed data to predict the queried values in the subsequent 24/12/36 hours.
(Johnson al., 2016) MIMIC-III is a clinical database containing IMTS data from 23457 patients, each with 96 variables recorded during the first 48 hours of ICU admission. We use the first 24/36/12 //2 hours as the observed data to predict the queried values in the subsequent 24/12/30 hours.
MIMIC-III
Human Activity This dataset consists of 12 irregularly measured 3D positional variables from sensors worn on the ankles, belts, and chests of five individuals performing various activities. To better align with realistic forecasting scenarios, the original time series is chunked into 5400 IMTS, each spanning 4000 milliseconds. The first 2000/2000) /1000 milliseconds are used as observed data to predict the sensor positions for the next 1000 22000/200000000 milliseconds.
USHCN (Menne et al., 2015) The USHCN dataset includes over 150 years of climate data from multiple U.S. stations, covering climate variables. Following established preprocessing methods, we focus on data from 1114 stations between 1996 and 2000, resulting in 26736 IMTS. Each instance uses data from the previous 24 months to predict the next 1 16//12 month's ss climate conditions.
B.2. Classification
P19 (Reyna al., 2020) The PhysioNet Sepsis Early Prediction Challenge 2019 dataset contains medical records of 38,803 patients. Each record includes 34 variables and static vector detailing attributes such as age, gender, time between hospital and ICU admission, ICU type, and ICU length of stay in days. Each patient also receives a binary label indicating whether sepsis occurs within the next 6 hours. We exclude samples with excessively short or long time series following (Zhang et al., 2022). Available at http:://oyyiionntoooc//mmmnttttt
P12 (Goldberger et al., 2000) The P12 dataset comprises data from 11,988 patients after removing 12 inappropriate samples identified by (Horn et al., 2020). Each record includes multivariate time series data from the first 48 hours of ICU stay, consisting of 36 sensor measurements (excluding weight) and static vector with 9 elements, including age and gender. Patients labeled based on ICU stay duration: a negative label indicates three days less, and positive label indicates more than three days. Available at
MIMIC-III (Johnson al., 2016) MIMIC-III is widely used dataset containing de-identified EHRs of ICU patients admitted to Beth Israel Deaconess Medical Center from 2001 to 2012, originally with around 57,000 records covering variables such as medications and vital signs. We focus on the io-hospital mortality prediction task, using subset established by (Harutyunyan et al., 2019). After preprocessing, our dataset includes 16 features and 20,107 data points. Available at http:://hhyyiine...ooo/oommmmmmm
PhysioNet (Silva et al., 2012) Physionet contains the data from the first 48 hours of ICU patients, which is a reduced version of P12 considered by prior work. Therefore, we follow the same preprocessing methods used for the P12 dataset. The processed data set includes 3997 labeled instances. We focus on predicting in mosospital mortality.
C. Baselines
C.1. Forecasting
C.1.1. METHODS FOR REGULAR MTS
For methods for regular MTS, we organize each sample into a V x Tr history matrix and V x T forecasting matrix, where V represents the maximum number of variables and Ty/Ty represents the historical/forecasting length of the sample. Unobserved positions are filled with zeros, the sequence length is set to the maximum historical length across all samples, and the forecasting length set to the maximum forecasting length across all samples.
iTransformer (Liu et al., 2024) uses inverted Transformers for time series forecasting. We use the following setting in our experiment: Clayers = 3, duayers 1, factor 3, d'motee = 512, da = 512.
14

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
TimesNet (Wu al., 2023) analyses general time series through temporal DD-Variation, We use the following setting in experiment: Clayers = duyers 1, factor 3, dmndee 256, di 512, top-k 5.
PatchTST (Nie et al., 2023) is a Transformer-basee model using patch and channel independence for long-term time series forecasting. We use the following setting in experiment:
Clayers dlayers 1, factor 3, patch_len 16,
stride = 8.
(Wang et al., 2023) achieves multi-scale local and global context modeling for long-term time series forecasting. We use the following setting in experiment: Clayers = dayeass 1, factor 3, dmodel 256, du 512, top k 5, decomp_kernel = 32, conv_ conv_kernel = 24, isometric_kernel = [18, 6].
MICN
TimeMixer (Wang et al., 2024b) achieves complementary predictive capabilities by disentangling variations in multi-scale
series. We the following setting in experiment: Elayers = 3, duyars 1, factor 3, dmodel down www.ssampinng..ccmmmmmmmmmmm..n. layers 3, down downn sampling window =
16, da = 32,
For the above methods, we use the implementation in Lime--eries-
ModernTCN (Luo & Wang, 2024) is modern pure convolution structure designed for general time series analysis. We use the implementation in https://github.com/// com/uuuchhh//Mooomm and the following setting in our experiment: ratio = 8, aatch_size = 8, strie 4, num blocks = 1, large_size = 51, small_size = 5, dims = 64
Pathformer (Chen et al., 2024a) multi-scale transformer architecture with adaptive pathways for time series forecasting. and the following setting in our experiment:
We use the implementation in http://hiuuu.ooooo//c/coonnnnnnn 2, nums 3, dmodel = 16, dr = 64.
MSGNet (Cai et al., 2024) learns multi-scale inter-series correlations for multivariate time series forecasting. We use the implementation in
and the following setting our experiment: Clayers = 2, dmodel 512, du = 64, n heads = 8, top_k k = 5, dropout = 0.1, nums kernels = conv channel = 32, skip ship_channel = 32, gen.depth 2, propalpha 0.3, node_dim 10, gen-dropout 10.
C.1.2. METHODS FOR IMTS
IMTS forecasting methods can directly make predictions. For IMTS classification methods, we use them as encoders to extract variable-level representations for each sample, followed by forecasting using the decoder described in Section 4.5.2.
Warpformer (Zhang et al., 2023) A transformer-bast based network that captures features at different scales in IMTS using
warping modules and dual attention mechanisms. We use three scales with normalized length [()) = 0, Σ()) 0.2 and 7,(2) = 1. The dimension of representations and 2, respectively. We use the implementation at http://giih..oommmmmaaaa//a////a
set as 32. The attention heads and the layers of the warpformer are set as
Raindrop (Zhang et al., 2022) A graph neural network that embeds IMTS while learning the dynamics of sensors purely from observation data. We use the following setting in our experiment: dos 4, p' = 16, r. = 16, L = 2, d2 = 20, da = We use the implementation
GRU-D (Che et al., 2018) GRU-D takes two representations of missing patterns, i.e., masking and time interval, and effectively incorporates them into a deep model architecture. The number of hidden states of GRU-D is set as 49. We use the implementation from https://github.com https///gihhub..o//Haaa.... com/Han-JD//RU..
IPatchGNN (Zhang et al., 2024) is a transformable patching graph neural network for IMTS forecasting. We use the implementation https://gihub..ommusiiilttt husssllllhusstt-PaachhNt and use hyperparameters specified in their scripts.
GraFITi (Yalavarthi et al., 2024) use bipartite graph for representing and forecasting IMTS. We use the fol- lowing setting in our experiment: latent.dim = 128, nalayers = 4, Theads = 1. We use the implementation at http://ggghbbbcoomyyaaaaittaaaa
15

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
CRU (Schirmer et al., 2022) models IMTS with continuous recurrent units. We use the following setting in our ex- periment: latent state dim 20, hidden units 50, bandwidth 10, num basis 20, trans covar 0.1. We the implementation at
TTAND (Shukla & Marlin, 2021) A deep learning framework for IMTS data that learns an embedding of continuous time values and uses an attention mechanism to produce fixed-length representation. We the latent dimension and the hidden size of GRU to 32. The number of reference points and the dimension of time embedding is 128. We use the implementation at https///gihuub.co///mmll///i/i
NeuralFlows (Biloš et al., 2021) use neural networks model ODE solution curves to mitigate the expensive numerical and the following setting in our experiment: flow wwwdemooddooooooe "CouplingFlow'' hidden_dim 50, Jayess 3, latents 20, rec_dims 40.
solvers in neural ODEs. We the implementation
(Rubanova al., 2019) is an ODE-based model that improves RNNs with continuous-time hidden state dynamics specified by neural ODEs. We use the implementation in https:/github..com www.m//laaauaaaaaaaaaaaaeeeaea .ode and the following setting in our experiment: latents 20, units 50, gen layers rec_ dims = 40. rec rec_layers gruunits 50.
Latsnt-ODEs
C.2. Classification
ODE-RNN (Rubanova al., 2019) ODE-RNN uses neural ODEs to model hidden state dynamics and an RNN to update the hidden state in the presence of observation. The latent dimension is set 40, and the ODE function has 3 layers with 50 units. We use the implementation at https:/github..om//llaaa YuliaRubanova/laten ode
SEFT (Horn et al.. 2020) set function approach where all the observations are modeled individually before pooling them together using attention-based approach. We constant architecture for the attention network f' with 2 layers, 4 heads and dimensionality of the dot product space d of 128. In addition, the attention network f' was always set to use mean aggregation. We use the implementation from https:/giituu.coo//Brggaaaa aab/SeFT. Fab
IP-Net (Shukla Marlin, 2019) A model architecture for IMTS data based on several cami-parametric interpolation layers organized into interpolation network followed by prediction network GRU. The number of reference points is set as 192. The hidden size of GRU is 100. We take the source code https://github. https://gihbb..co/mmdsalabba
- (Wu et al., 2021b) A generative model, which tracks the transition of latent clusters instead of isolated feature representations, achieves robust sparse time series modeling. We use the DGM and set both the hidden dimension and the cluster 10. We use the source code at https://tthbb.com//uwwwyyjjjjnn
MTGNN (Wu et al., 2020) A general graph neural network framework designed for MTS data. We use 5 graph convolution and temporal convolution modules with the dilation exponential factor 2. The graph convolution and temporal convolution modules have 16 output channels. The skip connection layers all have 32 output channels. The first layer of the output module has 64 output channels, and the second layer has 1 output channel. We use the implementation from:
StraTS (Tipimeni & Reddy, 2022) is self-supervised transformer for sparse IMTS. We use the implementation at https//ggthhb.coomm/innarr/////7 and the following setting in our experiment: hidden dim 64, num layers heads 16, dropout = 0.2.
DuETT (Labach et al., 2023) is a dual event time transformer for Electronic Health Records (EHRs). We use the implementation at wtt.:ghtww..ooooooaeaaaaaaaaiiia and the default settings of the model declaration in this repository.
VITST (Li et al., 2024a): transforms IMTS into line graph images and adapts powerful vision transformers to perform time series classification in the same way as image classification. We use the implementation at http://gthubb.coom.eeeeee//////
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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Please refer C.1.2 for details of GRU-D (Che al., 2018), m'TAND (Shukla & Marlin, 2021). Raindrop (Zhang et al., 2022) and Warpformer (Zhang al., 2023).
D. Performance Metries
MSE MSE (Mean Squared Error) measures the average the squared differences between predicted and actual values. The calculation formula
MSE
Σ 1=1
(Us D2)
(14)
where Y: represents the actual value, Y: represents the predicted value, and is the number of observations. A smaller MSE indicates better model performance. Since errors are squared, MSE is sensitive to large errors or outliers.
MAE MAE (Mean Absolute Error) measures the average of the absolute differences between predicted and actual values. The calculation formula is:
1
MAE
114
Uel.
(15)
21
where yi represents the actual value, O represents the predicted value, and is the number of observations. smaller MAE indicates better model performance. Compared MSE, MAE is less sensitive to outliers and provides straightforward average measure of error.
AUROC AUROC is commonly employed in binary classification tasks, where one class is designated positive and the other as negative. It represents the area under the Receiver Operating Characteristic (ROC) curve, constructed by plotting the True Positive Rate (TPR) against the False Positive Rate (FPR). AUROC ranges from 0 to 1, with higher value indicating better model performance in accurately discriminating between positive and negative instances. An AUROC equal to 0.5 indicates a model's 's performance equivalent to random guessing, while an AUROC greater than 0.5 signifies superiority over random guessing.
The Area Under the Precision-Resd Recall Curve is widely used as performance metric for imbalanced binary classification tasks. provides comprehensive assessment of model's precision-recall trade-off. The Precision-Recall curve is constructed by plotting recall on the x-axis and precision the y-axis. AUPRC ranges from 0 to and higher value indicates better model performance in achieving high precision and recall simultaneously. It has been suggested as good criterion for unevenly distributed classification problems (Davis & Goadrich, 2006).
AUPRC
E. Hyperparameters Settings
We search all hyperparameters in the grid for our proposed model Hi-Patch. Specifically, our model has total of hyperparameters: patch size P. dimension of node state dmodel, number of multi-head attention heads Rheads: number of GAT layers L.. Since the number of patches N = p the dataset-specific maximum time span, we search N over the range {2, 4, 8, 16, 32} to maintain the number of inter-patch graph layers as an integer. Thus the patch size P of each dataset determined by P = [ Additionally, we search dmotel in {16, 32, 64, 128), Theads in {1, 2, 4, 8} and L in 2,2.3.. The best hyperparameters for each dataset are reported the code.
T
and the number of inter-patch graph layers equals to logg N where 7 is
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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
F. Additional Experiments
F.1. Varying Observation and Forecast Horizons
Table Performance of varying observation and forecast horizons on Human Activity dataset. The best results highlighted in bold, and the second-best results are in underlined. The results in the table are presented in the form of (Mean Std).
Horizon
2000ms +
2000ms
1000ms + 3000ms
Metric
MSEx10
MAEx -
MSE 10
MAE 10
iTransformer
7.49 4 4.72
6.08
5.58
5.13
ModemaTN
5.26 + 0.06
4.96 0.04
12.12 = 1.01
5.72 2
TimesNet
5.38 0.30
5.32
9.90 0.42
7.34 1.81
PatchTST
7.25 0.29
6.17
8.97 1.96
6.94 0.94
Pathformer
4.67 + 0.22
4.58
5,49 + 0.12
5,05
TimeMixer
5.39 0.54
5,05 0.38
5,96 0.19
5.36
MSGNet
7.90 0.36
6,85 0.13
4.29
7.02 0.12
MICN
7.57 # 0.05
6.43
8.16 0.12
6.78 # 0.05
Warpformer
3.60 + 0.08
3.81 0.03
4.26
4.26 0.04
Raindrop
5.57 + 0.34
5.15
5.75 0.33
5.37 # 0.22
GRU-D
5.93 0.10
5.66 0.66
6.14
5.75 0.49
IPatchGNN
3.71 + 0.20
3.89
4.56
4.32
GraFITi
4.59 + 0.04
4.45
4.01
4.62
CRU
4.12 0.08
0.06
4.85
4.86
mTAAN
4.38 0.37
4.29
5.39
5.22
NeuralFlow
5,47 + 0.49
+.35
6.01 0.91
5.66 # 0.60
Latent-ODE
5.04 + 0.46
5.11
5.48 0.21
5.33 0.14
Hi-Patch
3.29 + 0.04
3.70 + 0.04
4.21 # 0.08
4.25 = 0.07
Table Performance of varying observation and forecast horizons on MIMIC-III dataset. The best results are highlighted in bold, and the second- best results are underlined. results in table are presented the form (Mean Std).
Horizon
36h + 12h
12h 36h
Metric
MSEX 10
MAEX10
MSE
MAEX10
iTransformer
7.27 + 0.04
2..52
7.24
21.47 + 0.45
MCdemTTN
5.12 0.59
11.41 # 0.30
3.96 0.23
10.79 0.39
TimesNet
2.02 + 0.09
8.18 : 0.22
2,39 0.06
8.35 + 0.20
PatchTST
8.81 # 0.96
22,97 # 1.32
7.11 # 0.65
19.96 + 1.24
Pathformer
2.45 0.02
8.65 0.09
TimeMixer
4.31 + 0.29
13.44 + 01.28
6.31 0..4
18.41 + 0.06
MSGNet
2.48 + 0.12
9.42 # 0.32
2.57 0.03
9.02 0.06
MICN
2.15
8.66 0.26
2.39
8.58
Warpformer
1.45 0.10
6.74 0.08
2.32 0.04
8.14 0.07
Raindrop
2.21 0.37
9.17 0.49
2.36 0.03
8.63 0.11
GRU-D
2.03 0.13
8.14 0.26
2.39
8.43 +: 0.13
(PatchGNN
1.44 0.08
6.78 0.14
2.35
8.23 0.08 8.13
GraFITi
1.61 + 0.27
7.16 0.36
2.22
CRU
2.00 + 0.13
8.16 0.26
2.34
8.32 + 0.13
iTTAND
2.01 + 0.09
8.13
2.29 0.03
8.38 0.08
NeuralFlow
1.97 4 0.12
8.39 0.25
8.29 + 0.10
Latent-ODE
1.90 + 0.03
7.92
2.38 0.05
8.35 + 0.113
Hi-Patch
1.56 0.10
6.71 0.16
2.32 0.02
8.11 0.08
18

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
Table Performance of varying observation and forecast horizons USHCN dataset. The best results highlighted in bold, and the second- best results are in underlined. The results in the table are presented in the form of (Mean = Std).
Horizon
24months
+ 6months
24months + 12months
Metric
MSFX -1
MAEX 10
MSEx 10
MARx 10
Trannsoormer
6.05 = 0.03
3.89 = 0.14
6.19 0.12
4.01 0.12
MC
6.03 0.10
3.68 = 0.06
6.19 + 0.07
3.75 1 0.03
TimesNet
5.68 0.05
3 0.06
5.84 0.06
3,78 0.10
PatchT'T
6.12 0.03
4.01 0.08
6.55 0.12
4.20 +20
TimeMixer
5.89 + 0.07
3.65 0.05
6.06 0.27
3.76 0.04
MSGNet
5.76 0.02
3.74 = 0.08
5.93 # 0.05
3.77 # 0.03
MICN
6.02 0.03
3.85 0.04
6.00 0.03
3.85 0.03
Warpformer
5.12 0.03
3.13 0.08
5.10 0.07
3.13 0.12
Raindrop
7.01 = 0.49
4.24 =244:.33 0.33
7.61 # 0.02
4.61 # 0.05
GRU-D
5.29 0.09
3.34 0.09 0.09
5.36 0.12
3.25 # 0.07
IPatchGNN
5.23 0.02
3.24
6.23 = 0.10
3.83 0.60
GraFITi
5
0.10
5.01 # 0.03
3.14
CRU
6.77 1.04
4.61
6.64
0.51
mTAND
3.1600.10
3.00
5.07
3.09 + 0.03
NeuralFlow
5.52 0.05
3.05
5,48
3.56
Latent-ODE
5.18 0.04
3.36 = 0.04
5.23 0.04
3.35 # 0.02
Hi-Patch
5.07 0.17
3.06 0.06
5.02 + 0.05
3.07 = 0.08
Table 8, Performance of varying observation and forecast horizons PhysioNet dataset. The best results are highlighted in bold, and the second-best results are underlined. results in table are presented
(Mean Std).
Horizon
36h > 12h
12h + 36h
Metric
MSE 10
3
MAEX 10 -
MSE 10
3
MAEX 10 2
iTranffrmmrr
56.83 21.17
17.05 5.21
54.17 = 17.91
17.03 4.18
MCdernTTNN
24.87 = 9.44
6.58 # 0.58
31.48 + 3.96
6.73 a 0.53
TimesNet
9.43 0.53
5.55 0.20
9.26 0.18
5.58 # 0.10
PatchTST
26.13 + 1.75
11.25 + 0.29
25.63 + 1.51
10.70 0.38
Pathformer
6,85 0.42
4.61 + 0.18
8.19 + 0.28
5.03 # 0.07
TimeMixer
12.52 0.45
6.47 0.21
19.86 0.17
8.38 0.18
MSGNet
10.44 = 0.44
5.94 = 0.08
9.91 + 0.12
5.74 0.08
MICN
10.98 0.31
6.01 0.11
10.24 0.14
5.85
Warpformer
4.17
3.38 0.08
6.51 4 0.12
4
Raindrop
10.67 + 0.33
5.87 = 0.20
10.24 + 0.18
5,83 4 0.10
GRU-D
6.85 0.37
4.88 0.18
7.80 + 0.22
5.13
(PatchGNN
4.22 0.09
3.38 0.04
6.45 0.11
4.24
GraFITi
4.58
3.65
6.30 0.14
4.38 + 0.12
CRU
6.74
0.12
7.66 0.14
4.97 4.97
mTAND
5.61 0.31
4.05
7.46
4.85 +.05
NeuralFlow
8.87 1.00
5.43
7.98 0.57
5.08+0.24
Latent-ODE
6.99 0.24
0.11
7.28 + 0.13
4.83 + 0.07
Hi-Patch
4.16 0.08
3.31 0.06
6.30 + 0.06
4.12 + 0.05
19

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
F.2. Leave-variables-out Classification
Table Classification performance on samples with fixed set of left-out variables. The best results are highlighted bold and the second best results underlined.
Discurd ratio
Dataset
Methods
10%
20%
30%
40%
50%
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
GRU-D
6
2.2
68.2 2.1
34.5 2.9
66.8 3.3
4,6
65.8 13.84.0
512
65.1 +.1
30.4 5.5
mTAND
0.6
0.6
74.0
36.5
7.84
34.1
7.66
3372
#35
3.6
DGM--0
P12
MTGNN
7.12+
3.3
29.7 + 2.8
6.2
28.5
68.1
23.
6.26
3.8
Raindop
73.2
73.0 1.6
31.7 + 1.1
2.62
31.1
71.5
30.6
70.8 +
2.3
SmaTS
8.48
42.4
8.7
41.8
1.6
40.
78.7+119
38.4+42
7
4.4
DuETT
a.7
35.8 2.3
74.7 1.8
35.3
2.2
34.1
72.8
3.73
72.3
32.6
Warpformer
0.7
2.2
0.8
36.7 2.3
73.8 2.9
3.1
72.8 12.2344
4.6
72.1
32.2 4.7
Hi-Patch
79.7 1.2
392
3.0
GRU-D
2.3
$4.6 3.7
88.8
3.4
8 2.5
5..4
4.96
86.4
8.6
mTAND
1.8
28.6 1.9
79.2
2.11
2.4
2.9
7.72
2.9
76.2 +
2.8
DGMF-O
0.6
53.4 1.5
87.3 0.8
53.2 1.7
86.6 1.6
4.1
85.8
4.9
85.2 2.2
6.7
P19
MTGNN
84.5
48.9
84.8
84.0
4.2
83.3
4.5
82.5
4,2
Raindrop
1.5
59.7
88.1 1.3
59.8 1.4
1.2
1.7
87.6
18
87.1 1.5
2.37
StaTS
9.9
56.4 3.0
91.0 0.9
56.3
9.9
56.0
90.8
55.1 +30
90.4
$4.4
DuETT
53.7
Warpformer
9..3
55.2 5.6
0.8
55.1 * 5.6
0.8
5..0
0.7
5..4
91.2
56.2 3,9
Hi-Patch
9.4
60.7 + 2,0
92.0 = 0.4
60.6 + 1.9
91.9 a 0.5
60.3 1.8
=
60.0 + 1.7
91.6 0.8
59.5 1,9
GRU-D
70.0
32.1 &
69.5 2.6
31.1 3.6
69.2 3.0
31.0
68.3 3.6
30.1 5.3
68.1 3.7
29.8 1.3
mTAND
2.1
42.8 = 4.0
3.42
76.3 4.7
3.77
7..6
5.6
5.1
DGM--
0.9
38.6 2.8
8.9
28
1.9
3..1
7.77
4.4
5.00
3..0
PhysioNet
MTGNN
6.19
25.8 a: 4,8
69.3 4.3
26.6 2..
4.8
2.2
6.23
2.8
67.2 a
2.84
Raindrop
76.5
33.4
1.
3.3
2.0
3..8
74.7
29.7
3.2
28.8
StaTS
2.44
2.6
80.8
4.2
2.85
3.7
3.0
3.8
78.1+34
3.7
DGETT
7.82
3.5
78.3 3.0
3.7
76.7
3.59
7..9
37.0
74.9
35.9
Warpformer
78.2
33.3 2.1
77.7 1.6
33.6
75.8
31.8 3.0
73.8
3.12
72.7
29.2 4.2
Hi-Patch
5.5
GRU-D
0.6
42.1
mTAND
8.2
804 1.1
41.9 1.2
79.7 1.4
4.70
79.3
40.4
78.8 + 1.6
39.8 2.3
DGMF-0
7.5
3.92
0.8
33.9 1.1
77.6 1.2
33.4
77.3
33.1
76.8 + 1.5
32.6 1.4
MIMIC-III
MTGNN
34.5
76.3
2.4
Raindrop
1.12
33.7
1.3
3.9
76.4
3.4
7..0
32.5
75.7 1 2.0
32.3 1.4
StrTTS
8.74
2.
82.1 0.6
43.7 2.1
81.7
43.1
81.5
4..6
81.0
4.49
DuETT
7.50
7 0..0
33.7
7..6
1.0
7.44
1..0
32.61
Warpformer
8.55
43.1
0.9
42.5 1.2
81.2 1.1
4.2
80.6
41.8
80.0 1.9
41.3 1.6
Hi-Patch
8.38
4.23
82.3 + 0.6
44.1 + 1.3
8..
43.6 1.4
8
42.8
80.5 4: 1.9
4
20

---

Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
E33 Regular Multivariate Time Series Forecasting
We Hi-Patch on traditional datasets with regular sampling frequency enhance the model's generality. Specifically, we add forecasting experiments four regular MTS datasets: ETTm1, ETTm2, Exchange and Weather. The lengths of the historical and prediction windows are both set to 96. The experimental pipeline is based that provided by iTransformer (Liu et al., 2024), and we compared our model with five SOTA regular time series forecasting methods in recent years: PatchTST (Nie et al., 2023), Crossformer (Zhang & Yan, 2023), FEDformer (Zhou et al., 2022) and Autoformer (Wu et al., 2021a).
Table 10 shows the results. From the experimental results, our Hi-Patch achieved an average ranking of 2.25 terms of MSE and MAE metrics across the four datasets, notably outperforming the latest iTransformer the Exchange and Weather datasets, Although
method is initially designed for IMTS, its ability extract multi-scale temporal dependencies and inter-variable dependencies is general for time series modeling. Consequently, it can also achieve competitive performance in regular time series forecasting tasks, further demonstrating the generality of our method.
Table 10. Forecasting performance of Hi- Ili-Pach four regular time series datasets.
ETTml
ETTm2
Exchange
Weather
Methods
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
iTransformer
0.334
0.180
0.264
0.086
0.206
0.174
0.214
PatchTST
0.329
0.367
0.175
0.259
0.088
0.205
0.177
0.218
Crossformer
0.404
0.426
0.287
0.366
0.256
0.367
0.158
0.230
FEDformer
0.379
0.419
0.203
0.287
0.148
0.278
0.217
0.296
Autoformer
0.505
0.475
0.203
0.287
0.197
0.323
0,266
0.336
Hi-Patch
0.380(4th)
0.405(3rd)
0. 184(3rd)
0.269(3rd)
0.085(1 st)
0,206(2nd)
0.158(
0.208(1st)
Human Activity
USHCN
PhysioNet
3.3
Warpformer
Warpformer
GraFITi
5.6-
GraFITi
tPatchGNN
tPatchGNN
Ours
MSE (x1e-3)
Ours
3.0
5.4
2.9
6.0
2.8
52-
Warpformer GraFITi
5.5-
2.7
tPatchGNN
5.0
Ours
5.0
1000
1500
2000
2500
3000
12
18
24
12
18
24
Observation Horizon (ms)
Observation Horizon (months)
Observation Horizon (hours)
Figure 6. Forecasting performance with varing observation horizons and fixed forecast horizon.
F.4. Varying Observation Horizons and Fixed Forecast Horizon
We conduct additional experiments to evaluate the model's ability capture more temporal dependencies from long-term data on datasets. Specifically:
Human Activity Fixed forecast length at 1000ms, varied observation horizons {1000, 2000, 3000)
USHCN Fixed forecast length at month, varied observation horizons as (6, 12, 24 months.
PhysioNet Fixed forecast length at 24hours, varied observation horizons as {6, 12, 24 hhors.
Figure 6 shows the results of our model and the three most competitive baseline methods: Warpformer, GraFITi and IPatchGNN. It evident that our method consistently improves MSE as the historical length increases. This demonstrates
21

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
the model's capability learn more temporal dependencies from longer-term data. Conversely, some existing IMTS methods do not have this ability, such Warpformer on the USHCN and Human Activity datasets, and GraFITi on the Human Activity dataset, both showing an increase in MSE with increasing observation horizons. These supplementary experiments further highlight the superiority of our Hi-Patch approach compared to prior methods.
G. Computational Complexity Analysis
G.1. Theoretical Analysis
For a dataset with V variables, maximum series length Lmax. and missing rate M, suppose the data is divided into N patches using fixed time span. Hi-Patch consists of one Intra-Patch Graph Layer and log, N Inter-Patch Graph Layers. We analyze the computational complexity of these two components separately.
Intra-Patch Graph Layer Complexity For each patch, assuming missing points are uniformly distributed, the number of nodes per patch is V Lmxx (1 - M) //N. The complexity of attention operations within a patch is ( Lmxx (M))N/ ))) For N patches computed in parallel, the total complexity is O(V2 I. (1 M)² / Additionally, aggregating nodes of the same variable the reference time point has a complexity of L Lma M)). Thus, the overall complexity of the Intra-Patch Graph Layer is O(V2 1 max (1 M) + VT Tmmax ((1 M)) = ( 1 (1 M)² /NN).
Inter-Patch Graph Layers Complexity In the first Inter-Patch Graph Layer, the maximum number of nodes is NV. Each node computes graph attention with all 2V variable nodes in adjacent patches, resulting in complexity of ( Aggregation, where each node interacts with only one adjacent node of the same variable, has a complexity of ((V)). Thus, the first Inter-Patch Graph Layer has a total complexity of O(2NV2 + NV) = (NV?). For the second layer, the number of nodes is halved compared to the first layer, leading complexity of O /2). Summing over log, N layers, log, NV2 O(V 2)).
the total complexity of Inter-Patch Graph Layers is O Σ
From the above, the primary complexity of our model lies in the first Patra-Patchhae Graph Layer. Since each observation is treated as a node for attention operations to capture asynchronous dependencies IMTS, this layer has quadratic complexity with respect to both the number of variables and the sequence length. Fortunately, IMTS datasets are often highly sparse (missing rate M > 80% as shown in Table 4), and the coefficient (1 M)² reduces the practical complexity by 1-2 orders of magnitude. Additionally, increasing the number of patches N further reduces computational overhead in practice. Therefore, the practical computation time of our method is acceptable, as shown in Appendix G.2. In summary, our method achieves a more comprehensive extraction of multi-scale asynchronous correlations in IMTS at the cost of an acceptable increase in computational complexity.
G.2. Empirical Analysis
We compared 8 leading IMTS models: Warpformer, GRU-D, GraFITi, (PatchGNN, CRU, mTAND, NeuralFlow, and Latent-ODE. All models are evaluated using the same batch size (32 for Human Activity, 128 for USHCN, 64 for PhysioNet, and 8 for MIMIC-III) to assess their training time per epoch and MSE. The results shown in Figure indicate that the training time of our method ranks 5-th on average across the four datasets. It is worth noting that our model leverages a hierarchical graph structure extract asynchronous correlations and multi-scale features in IMTS that other methods fail to capture, which inevitably leads to increase in computational overhead. Nonetheless, our model's 's training time remains in the same order of magnitude as the fastest models. In this context, We believe that the trade-off of sacrificing some training time to extract richer features in IMTS is worthwhile.
G.3. Future Scalability on Dense Data
Should the need arise to extend our approach to large-scale dense datasets, existing scalability techniques can be integrated to reduce the overhead of the intra-patch graph. For instance, edge pruning-such as connecting each node only to its nearest l observation points- can reduce the complexity to Ok:N), or alternatively, leveraging improved attention mechanisms like the ProbSparse Self- Attention from Informer (Zhou et al., 2021) can lower the complexity to ON log N). In these cases, the orade-off between a slight reduction feature extraction and enhanced scalability would be worthwhile.
22

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Hii-Path:: Hierarchical Patch GNN for Irregular Multivariate Time Series
10a
Warpformer
GRU-D
GrafiTi
Warptormer
10a
TPatchGNN
GrafiTi
mTAND
time (c)
IPatchONN
[x]
NeuralFlow
CRU ITTAND Neurall low Latent-ODF Ours
Latent-ODE
Ours
107
X
X
10a
2.50
3.25
3.50
4.25
5.0
5.2
5.3
5.4
Meen Squared Error
Meen Squared Error
(a) Human Activity
(b) USHCN
X Werpformer
Werpformer
GRU-D
GRU-D
tPatchENN
102
tPatchGNN
ITAAD
mTTAND
NeuralFlow
tima (5)
NouralFlow
20
Latent-O0D
Latent-ODE
10a
Mean Squared Error
Moan Squared Error
(c) PhysioNet
(d) MIMIC-III
Figure 7. Comparison of IMTS models terms of efficiency: training time per epoch against error metric.
H. Comprehensive Discussion the Technical Details of Hi-Patch and Existing Methods
Differences from patch-based methods for regular multivariate time series Representative methods include PatchTST (Nie al., 2023) and Pathformer (Chen et al., 2024a). These approaches encode observations within fixed-length time steps of univariate time series into patches and use attention mechanisms extract temporal correlations between patches. However, these methods apply uniform time step all variables, which makes them unsuitable for IMTS. In IMTS, some variables may only have a few observations, resulting in patches with long time spans, while other variables with frequent observations will result in patches covering very short time spans. These methods cannot account for differences in time spans across patches, and the patches for different variables are often misaligned temporally, complicating the extraction of vatiall correlations. In contrast, Hi-Patch segments patches using fixed time spans and each patch contains observations for all variables. This design guarantees consistent time spans across patches and facilitates the modeling of inter-variable relationships.
Differences from GNN-based methods for regular multivariate time series A latest typical example is MSGNet (Cai et al., 2024), which extracts correlations between variables at the same timestamp using variable graphs. However, such methods struggle to capture asynchronous correlations that are prevalent in IMTS. Our method uses both intra-patch and inter-patch graphs to simultaneously extract: 1) Correlations of the same variable across different time points (SVDT), 2) Correlations different variables the same time point (DVST) and 3) Correlations of different variables across different time points (DVDT).
Differences from patch-based methods for irregular multivariate time series Pioneering work is (PatchGNN (Zhang et al., 2024), which also segments patches based on fixed time spans and models intervarriale correlations via GNNs. However, tPatchGNN considers only one variable per patch and fails to capture fine ame-grained inter-variable correlations within a patch, Moreover, it extracts features separately temporal and variable dimensions, limiting ability to model
23

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
asynchronous correlations. In contrast, Hi-Patch extracts fine-grained dependencies within patches and coarse-grained dependencies across patches while modeling both synchronous and asynchronous inter-variable correlations. Additionally, Hi-Patch can extract multi-scale information from IMTS, which is absent in (PatchGNN.
Differences from GNN-based methods for irregular multivariate time series Representative methods include Raindrop (Zhang et al., 2022) and GraFITi (Yalavarthi et al., 2024). Raindrop models variable interactions at each timestamp using GNNs but cannot capture asynchronous correlations. GraFITi represents IMTS using variable-time bipartite graph but, as acknowledged in their paper, has limited ability to extract asynchronous correlations. Moreover, these methods lack the capability to extract multi-scale information from IMTS. Hi-Patch stands out by capturing both synchronous and asynchronous dependencies at multiple scales, which provides significant advantage over these methods.
Differences from multi-scale methods for irregular multivariate time series Among methods specifically designed for IMTS, Warpformer (Zhang et al., 2023) is the only method that considers the multi-scall characteristics of IMTS and achieves the most competitive performance. However, Warpformer involves interpolation of sparse variables, which may distort the original distribution of IMTS. In addition, Warpformer employs attention separately in the time and variable dimensions, limiting its ability to capture asynchronous dependencies. In contrast, our Hi-Patch flexibly represents and extracts both synchronous and asynchronous dependencies IMTS under different scales through an intra-patch graph layer and several inter- inter-patch graph layers. Moreover, Hi-Patch only handles actually observed points, avoiding the accumulation of imputation errors, particularly cases of higher missing ratios, exhibiting a degree of robustness.
More Results for Ablation Study
Table 11. Ablation results of Hi-Patch on two datasets. The results in the table are presented the form of (Mean + Std).
PhysioNet
MIMIC-III
Methods
MSE x -0-3
MAEX10
MSEX 10 10-2
MAEx10-2
Hi-Patch
4.86 + 0.03
3.62 + 0.07
1.75 + 0.26
7.24 0.18
w/o Hie
4.96 + 0.07
3.68 = 0.08
1.77 + 0.30
7.30 0.13
w/o DVDT
4.88 + 0.05
3.67 # 0.04
1.80 + 0.04
7.41 0.18
w/o 3W
4.98 = 0.07
3.74
1.79 + 0.04
7.38 # 0.09
w/o TEAGG
6.34 + 0.60
4.44
1.92 + 0.07
7.63 # 0.26
J. More Results for Effect of Scale Quantity
1e-3
1.85 1e-2
2.70
2.65
MSE
MSE
1.80
2.60
1.75
2
5
2
3
5
Number of Scales
Number of Scales
(a) Human Activity
(b) MIMIC-III
Figure 8, Effect of different scale quantities on Human Activity and MIMIC-III datascts.
24

---

Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
4000
Time Span (hours)
10.06
Ratios samples
Time Span (ms)
0:100
40-
3800-
Sample
3600
Distribution
0.075
30
0,04
20
Sample Paint Distribution curve
0.002
3200
0.025
10
3000
0.000
0.00
0
100
200
400
600
Length
Length
(a) Human Activity
(b) MIMIC-III
Figure 9. Distribution of sample length and time span on two datasets. Each blue dot represents a sample, with its x-coordinate indicating the sample length, and the left y-axis representing its time span. The red curve is the distribution curve of sample length.
K. Visualization of Multi-scale Views
2.0
25
2.
1.5
1.6
0.5
0.5
6.0
500
1000
1500
2000
2303
Tinic (ms)
(mc)
Time (mS)
(a) Scale (Origin)
(b) Scale
(c) Scale
Figure 10. Visualization of views three scales on Human Activity dataset.
1000
150
400
50
15
20
25
Time (h)
Time th!
(a) Scale (Origin)
(b) Scale 2
Figure 11. Visualization of views on two scales on MIMIC-III dataset.
25

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Hi-Patch: Hierarchical Patch GNN for Irregular Multivariate Time Series
20
1.5
1.5
Vplue
Value
Velee
0.0
-03
-0.
10
15
20
10
15
30
20
(a) Scale (Origin)
(b) Scale
(c) Scale
0.8
0.8
0.5
0.2
0.0
0.0
Time (man))h)
Time
(d) Scale
(c) Scale 5
Figure 12. Visualization of views five scales USHCN dataset.
300
volve
200
159
Value
200
100
20
2.5
5.0
7,5
10.0
15.0
135
200
2255
Time (h)
Time 1h)
Time
1h1
(a) Scale (Origin)
(b) Scale
Scale
Figure 13, Visualization of views on three scales PhysioNet dataset.
26


Reviews:
[{"review_id": "rE7dgunV6k", "reviewer": "Reviewer_ZR6y", "summary": "This paper introduces a Hierarchical Patch Graph Neural Network (Hi-Patch) for Irregular Multivariate Time Series (IMTS) modeling, where variables have distinct sampling rates and exhibit multi-scale dependencies. Existing multi-scale analysis methods struggle with IMTS due to their assumption of regular sampling, making them ineffective in handling asynchronous and mixed-granularity data. Hi-Patch addresses this by first encoding each observation as a node, capturing local temporal and inter-variable dependencies through an intra-patch graph layer, and then progressively aggregating these nodes into higher-level patch representations using inter-patch graph layers. This hierarchical structure enables multi-scale feature extraction, preserving both fine-grained and coarse-grained patterns across different variable scales. The model is evaluated on forecasting and classification tasks across eight datasets, outperforming state-of-the-art methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Responses to Reviewer ZR6y\n\n**W1. Hi-Patch appears computationally expensive, which may limit its scalability.**\n\n**A1.** We address this concern from both sparse and dense data perspectives.\n\n1. **Trade-off in Sparse IMTS**\n\n   As detailed in Appendix G.1, the primary computational cost of our method arises from the intra-patch graph layer, whose complexity scales quadratically with the number of observation points (N) per patch. \n\n   Our focus is on irregular multivariate time series, which are typically characterized by extremely sparse sampling. As shown in Table 4, all our datasets exhibit a missing rate exceeding **75%**. \n\n   In these scenarios, the computational cost decreases sharply due to the limited number of available data points. We contend that the additional overhead is justified by the improved extraction of scarce data patterns. Moreover, our empirical study in Appendix G.2 indicates that our overhead is moderate—**ranking 5th out of 9 methods**—and remains within an acceptable range.\n\n2. **Future Scalability on Dense Data**\n\n   Should the need arise to extend our approach to large-scale dense datasets, existing scalability techniques can be integrated to reduce the overhead of the intra-patch graph. \n\n   For instance, **edge pruning**—such as connecting each node only to its nearest k observation points—can reduce the complexity to **O(kN)**, or alternatively, leveraging improved attention mechanisms like the **ProbSparse Self-Attention** from Informer can lower the complexity to **O(N log N)**. \n\n   In these cases, the trade-off between a slight reduction in feature extraction and enhanced scalability would be worthwhile. This extension is among our planned future works. For the current paper, our core contribution remains the introduction of a multi-scale framework for sparse IMTS. We will also add a Limitations section discussing model scalability.\n\n------\n\n**W2. The performance improvement in the classification task seems marginal. The PhysioNet and P12 datasets are redundant.**\n\n**A2.**\n\n1. **Classification Performance Improvement**\n\n   Although our method only shows marginal improvements over the best baseline on each individual dataset, it is important to note that **no single baseline consistently outperforms the others across all datasets**. The table below presents the average AUROC and AUPRC for our method and the three most competitive baselines across four datasets. **Our method achieves an average AUROC improvement of 1% and an average AUPRC increase of 2.7%**, which is noteworthy given that the AUROC values are already near 90%.\n\n   |            | Avg AUROC  | Avg AUPRC  |\n   | ---------- | ---------- | ---------- |\n   | IP-Net     | 86.2       | 52.0 (2nd) |\n   | StraTS     | 86.7 (2nd) | 51.1       |\n   | Warpformer | 86.3       | 50.3       |\n   | Ours       | 87.7 (1st) | 54.7 (1st) |\n\n2. **PhysioNet and P12 Datasets**\n\n   The P12 dataset comprises three subsets—set-a, set-b, and set-c—each containing 4,000 samples (totaling 12,000 samples). Previous studies [1, 2] have utilized only the set-a subset (namely PhysioNet dataset in our paper), whereas others [3, 4] employed the full P12 dataset. \n\n   **By evaluating both configurations, we aim to test the robustness of our method with respect to sample size and distribution.** The consistent performance across these configurations further validates the stability of our approach.\n\n   [1]. Multi-Time Attention Networks for Irregularly Sampled Time Series, ICLR, 2021\n\n    [2]. Warpformer: A Multi-scale Modeling Approach for Irregular Clinical Time Series, KDD, 2023\n\n    [3]. Graph-Guided Network for Irregularly Sampled Multivariate Time Series, ICLR, 2022\n\n    [4]. Time Series as Images: Vision Transformer for Irregularly Sampled Time Series, NeurIPS, 2023\n\n------\n\n**W3. The motivation for studying multi-scale information in IMTS.**\n\n**A3.**\n\n1. **Foundational Evidence**\n\n   Prior studies—such as those on Pathformer, TimeMixer, and MSGNet—have demonstrated that multi-scale information is crucial for general time series analysis. Although irregular multivariate time series (IMTS) are characterized by uneven sampling intervals, they inherently retain multi-scale properties.\n\n2. **Practical Example**\n\n   For instance, the USHCN weather dataset exhibits monthly, quarterly, and yearly patterns (see Figure 12). Each temporal scale shows distinct patterns that are essential for accurate prediction. This example underlines the importance of multi-scale information in IMTS.\n\n3. **Prior Empirical Validation**\n\n   Empirical evidence from previous work Warpformer [2], supports the benefits of incorporating multi-scale information into IMTS modeling. While Warpformer has its own limitations, its success further motivates our deeper exploration into a multi-scale modeling approach for IMTS.\n\nCollectively, these points underscore the significance of multi-scale information in IMTS modeling and reinforce the motivation behind our work.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "No proofs in the paper", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "The proposed Hi-Patch might be applied to IMTS modeling across various domains.", "essential_references_not_discussed": "The key related works are included.", "other_strengths_and_weaknesses": "Strengths:\n1. The studied classification and forecasting problems of Irregular Multivariate Time Series (IMTS) have significant applications in various crucial domains, such as healthcare, finance, climate science, and astronomy.\n2. The proposed Hi-Patch aims to address several unique challenges posed by IMTS modeling, including irregularity, asynchrony, and mixed granularity. These challenges may not be present and addressed in the widely studied Regular Multivariate Time Series.\n3. The experimental results exhibit promising performance in IMTS forecasting task.\n\nWeaknesses:\n1. The Hi-Patch model appears computationally expensive due to its use of graph attention operations on fully connected graphs of all observation nodes across variables. This may limit its scalability for large-scale variables with dense observations.\n2. The performance improvement in the classification task seems marginal. Additionally, the PhysioNet and P12 datasets both originate from The PhysioNet/Computing in Cardiology Challenge 2012 [1], making them somewhat redundant.\n3. As a crucial driving factor for this study, the motivation of studying multi-scale information in IMTS should be further justified. Why is it an important factor in IMTS modeling?\n\n[1] https://physionet.org/content/challenge-2012/1.0.0/.", "other_comments_or_suggestions": "NA", "questions_for_authors": "see weakness", "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": "YnQWaI8Blg", "reviewer": "Reviewer_2e2R", "summary": "The paper introduces a graph-based framework called **Hi-Patch** to handle irregularly sampled multivariate time series. The approach divides the time axis into patches of short intervals, capturing local (fine-grained) temporal patterns for densely sampled variables in each patch. It then progressively aggregates and propagates patch-level information through multiple “inter-patch” graph layers, forming a hierarchical structure that extracts coarse-grained temporal and inter-variable correlations for densely and sparsely sampled variables. This bottom-up, multi-scale pipeline allows for comprehensive feature extraction spanning local to global time scales. Empirical results show that Hi-Patch outperforms a variety of baselines in both classification (area under ROC/PR curves) and forecasting (MSE/MAE) tasks. The method’s key contributions include a patch-based intra-graph for local context, a stack of inter-patch graphs for higher-level features, and a final decoding layer that uses these learned representations for downstream predictions.\n\n## update after rebuttal\nThe authors covered most of my concerns in the rebuttal, so I kept the original positive rating.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Responses to Reviewer 2e2R\n\n**W1 & Q1. The increased complexity of the hierarchical design may affect scalability in massive datasets. How does it scale computationally when applied to highly long time series or datasets with significantly higher sampling densities?**\n\n**A1.**\n\n1. **Complexity in Sparse IMTS**\n\n   As detailed in Appendix G.1, **the primary computational cost of our method stems from the intra-patch graph layer**, whose complexity is quadratic in the number of observation points (N) within each patch rather than hierarchical design.\n\n2. **Future Scalability on Dense Data**\n\n   In scenarios involving highly long time series or datasets with much higher sampling densities, we can readily adopt existing scalability techniques to mitigate the overhead of the intra-patch graph. \n\n   For instance, **edge pruning**—where each node connects only to its nearest k observation points—can reduce the complexity to **O(kN)**. Alternatively, one could improve the attention mechanism (e.g., by incorporating the **ProbSparse Self-Attention** from Informer, which reduces complexity to **O(N log N))**. \n\n   In these cases, the trade-off between a slight reduction in feature extraction capability and improved scalability would be justified. This extension is part of our planned future work; the current paper focuses on the initial proposal of a multi-scale framework for sparse IMTS. In the revised version, we will add a Limitations section to discuss scalability aspects.\n\n------\n\n**W2 & Q2. More discussions or experiments on the interpretability of the learned hierarchical multi-scale features.**\n\n**A2.** Taking the USHCN dataset as an example—comprising climate data from 1,218 centers across the United States for five variables (precipitation, snowfall, snow depth, maximum temperature, and minimum temperature)—we use the first two years (24 months) as historical input to predict future changes.\n\n1. **Effect of Different Scales on Final Prediction Results**\n\n   As illustrated in Figure 3(a) in Section 5.4, increasing the number of scales from 1 to 5 (by incorporating scales of 12, 6, 3, and 1.5 months) results in a continuous decrease in MSE, demonstrating the benefit of multi-scale features. Notably, when a 0.75-month scale is added (resulting in 6 scales), the MSE increases, indicating that this extra scale introduces redundancy rather than additional useful information. \n\n   **This experiment not only identifies the scales that are beneficial for prediction, but also reveals which scales most strongly influence the final predictions through the slope changes.** For instance, the largest drop occurs when the 6-month scale is introduced, highlighting it is the most important scale for weather prediction.\n\n2. **Discussion on the Interpretability of Learned Multi-Scale Features**\n\n   As shown in Figure 5(a) in Section 5.5, our model achieves optimal performance with a patch size of 1.5 months, which enables feature extraction at scales of 1.5, 3, 6, and 12 months. **These scales correspond well with key climatological cycles and are largely interpretable.** Figure 12 in Appendix K provides visualizations of the time series at these scales based on our patching approach:\n\n   - **Scale 1 (Original Scale):** The original scale view retains the full sequence data, exhibiting high local variability. \n   - **Scale 2 (1.5-Month Aggregation):** By aggregating data every 1.5 months, the resulting view smooths short-term fluctuations, highlighting mid-term climate trends. \n   - **Scale 3 (3-Month Cycle):** Aggregating every two observations from Scale 2 yields a 3-month cycle view. Seasonal trends become more evident in this view, with red and purple temperature curves show an upward, downward, upward and downward trend, broadly corresponding to the four seasons. \n   - **Scale 4 (6-Month Cycle):** Further aggregation at 6-month intervals reveals monsoonal climate trends, where the red and purple temperature curves first rise and then fall, reflecting the impact of summer and winter monsoons. \n   - **Scale 5 (12-Month Cycle):** At a 12-month aggregation, the view reveals annual macro-climate trends, such as overall increases, decreases, or stabilization in climate patterns. \n\n   We will include this analysis in the appendix in future revisions.\n\n------\n\n**Comment1. Minor typographical errors and grammatical slips.**\n\n**A3.** Thank you for your careful review. We will thoroughly proofread and correct these issues in the revised version of the paper.\n\n------\n\n**Comment2. A short discussion on potential limitations or failure cases.**\n\n**A4.** We will incorporate a discussion on the scalability limitations of Hi-Patch in light of our response to W1 & Q1.", "claims_and_evidence": "The paper’s primary claims—namely that Hi-Patch achieves state-of-the-art results on irregular multivariate time series forecasting and classification through a novel hierarchical patch-based architecture—are substantiated mainly. The experiments cover multiple datasets of varied nature (e.g., clinical, climate, activity), comparing against both specialized irregular-series baselines and prominent time-series models and typically show consistent improvements in performance metrics (e.g., MSE/MAE, AUROC/AUPRC).  \n\nOne minor point to note is that while the method’s patch-based hierarchical graph structure is well-tested under moderate sequence lengths, claims about scalability to massive datasets or exceedingly fine-grained irregularities could benefit from deeper empirical exploration. Overall, the evidence presented—including comparative results, ablation studies, and multi-dataset analysis—convincingly supports the paper’s central claims.", "methods_and_evaluation_criteria": "Yes. The authors use benchmark datasets covering domains where irregular multivariate time series naturally arise (e.g., healthcare, climate, motion). These datasets and tasks (forecasting and classification) are appropriate for evaluating the benefits of the proposed hierarchical patch-based method. Comparisons to both general-purpose time-series models and specialized irregular-series frameworks further demonstrate the paper’s relevance to the problem setting and the robustness of its approach.", "theoretical_claims": "The submission does not appear to present formal proofs or a rigorous theoretical framework beyond outlining model architectures and derivations (e.g., attention mechanisms, graph-update formulas). Consequently, there are no explicit proofs to check for correctness. The authors support their modeling choices with empirical evidence rather than theoretical guarantees.", "experimental_designs_or_analyses": "The paper’s experimental design appears methodologically sound—particularly its multi-dataset, multi-task approach. The authors follow established splits (e.g., 6:2:2) and compare against well-chosen baselines across domains. They include ablation studies and variance reporting, further bolstering validity. No critical flaws or inconsistencies in the experimental design or analyses were observed.", "relation_to_broader_scientific_literature": "The paper extends patch-based approaches like PatchTST—which typically handles regular time series—by leveraging graph-based links to account for the irregular sampling patterns seen in many real-world datasets. This builds on prior GNN research (e.g., tPatchGNN, MTGNN) that models asynchronous inter-variable relationships; however, it is combined with a hierarchical, multi-scale design. The work synthesizes established patching methods in time series analysis with graph architectures that capture local (intra-patch) and global (inter-patch) dependencies, advancing existing literature on irregular time series modeling and multi-scale feature extraction.", "essential_references_not_discussed": "The paper already cites a range of prior works on patch-based modeling (e.g., PatchTST), continuous-time or irregular methods (e.g., Latent ODEs, GRU-D), and multi-scale networks (e.g., MSGNet). From the perspective of irregular time-series GNNs and multi-scale forecasting, no critical, well-known work appears omitted. The paper’s references collectively offer sufficient background for its primary contributions.", "other_strengths_and_weaknesses": "The paper exhibits notable originality by creatively combining patch-based time series segmentation with hierarchical graph neural networks. This approach effectively addresses the challenges of irregular sampling and multi-scale feature extraction. This innovative synthesis bridges ideas from patch representations and continuous-time graph models, filling an essential gap in the literature. The empirical evaluation is comprehensive, spanning diverse datasets and tasks, which underscores the significance of the method across various real-world applications. Additionally, the clarity in the presentation of the model architecture and the experimental setup enhances the paper's overall quality.\n\nOn the downside, the increased complexity of the hierarchical design introduces additional hyperparameters and potential computational overhead, which may affect scalability in massive datasets. Furthermore, while the paper demonstrates substantial performance improvements, more discussion on the interpretability of the learned multi-scale features and a deeper analysis of the metamethod's sensitivity to hyperparameter choices could further strengthen the contribution.", "other_comments_or_suggestions": "Other Comments and Suggestions:  \n- Overall, the paper is well-written and clearly describes the methodology and experiments.  \n- Minor typographical errors were observed, such as occasional inconsistencies in notation formatting (e.g., varying use of boldface for vectors) and a few grammatical slips that can be easily corrected during revision.  \n- A short discussion on potential limitations or failure cases could further balance the presentation.", "questions_for_authors": "1. How does the proposed hierarchical framework scale computationally when applied to highly long time series or datasets with significantly higher sampling densities? \n2. Can the authors elaborate on any techniques or experiments aimed at interpreting the hierarchical features learned by the model—for instance, identifying which scales or graph connections most strongly influence the final predictions?", "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": "omrKlcvTGI", "reviewer": "Reviewer_Ko9v", "summary": "The paper introduces Hi-Patch, a hierarchical patch graph network designed for IMTS, where variables have different sampling rates. Hi-Patch models both local and global dependencies across different scales. It represents observations as nodes, captures short-term dependencies using intra-patch graphs, and progressively learns global features through inter-patch layers. The final representations are fed into task-specific decoders. Experiments on eight datasets show that Hi-Patch outperforms SOTA models in IMTS forecasting and classification.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Responses to Reviewer Ko9v\n\n**Q1. The motivation and experimental validation of IMTS multi-scale modeling.**\n\n**A1.** \n\n1. **Motivation and Significance of Multi-scale Information**\n\n   Our work is grounded in the premise that multi-scale information is essential for general time series analysis—as demonstrated by previous studies (e.g., Pathformer, TimeMixer, MSGNet). \n\n   Since irregular multivariate time series (IMTS) inherently exhibit multi-scale features (for instance, monthly, quarterly, and yearly patterns in the USHCN weather dataset), it is natural to assume that multi-scale information is also critical for IMTS. \n\n   Moreover, prior work such as Warpformer has empirically verified the benefits of multi-scale modeling for IMTS, while also revealing certain limitations. \n\n   These intuitive and empirical insights jointly motivate our further exploration into multi-scale modeling for IMTS.\n\n2. **Experimental Validation and Ablation Studies**\n\n   In our ablation experiments (the “w/o Hie” setting), we set the patch size to span the entire historical window, thereby removing the hierarchical structure so that the model processes only single-scale raw information(Table 3 and Table 11 in Appendix I). \n\n   The significant performance drop observed (3.90%↓ on MSE and 3.07%↓ on MAE on four datasets) directly confirms the pivotal role of the hierarchical design in extracting multi-scale features. \n\n   Additionally, Figures 11–13 in Appendix K visually demonstrate the diverse patterns that Hi-Patch can \"see\" at different layers—a capability that distinguishes our model from many existing IMTS methods.\n\n------\n\n**Q2. Analysis of computational complexity.**\n\n**A2.** As detailed in Appendix G.1, the primary computational cost of our method arises from the intra-patch graph layer, whose complexity scales quadratically with the number of observation points (N) per patch. \n\nOur focus is on irregular multivariate time series, which are typically characterized by extremely sparse sampling. As shown in Table 4, all our datasets exhibit a missing rate exceeding **75%**. In these scenarios, the computational cost decreases sharply due to the limited number of available data points. \n\nOur empirical study in Appendix G.2 indicates that our overhead is moderate—**ranking 5th out of 9 methods**—and remains within an acceptable range. Consequently, We contend that the additional overhead is justified by the improved extraction of scarce data patterns.\n\n------\n\n**Q3. Clarification of parameter tuning.**\n\n**A3.** \n\n1. **Intuitive Parameter Choice**\n   Key parameters—such as the number of layers and patch size—have clear physical interpretations. Layers correspond to different scales of feature extraction, while the patch size reflects the local time window. These parameters can be set according to the specific task requirements and data characteristics. Our sensitivity analyses (Sections 5.4 and 5.5) demonstrate that the optimal parameter ranges is related to the sample distributions and inherent periodicities. Detailed search ranges are provided in Appendix E to guide practical parameter selection.\n2. **Practicality and Generalizability**\n   Although some hyperparameter tuning is required—common in deep learning methods[1, 2, 3]—our approach shows competitive or superior performance compared to state-of-the-art methods. The tuning process further adds flexibility, allowing the model to adapt to a variety of applications.\n\n[1]. Random Search for Hyper-Parameter Optimization*. Journal of Machine Learning Research, 13:281–305, 2012.\n\n[2]. Practical Bayesian Optimization of Machine Learning Algorithms. NeurIPS, 2012\n\n\n\n**Q4. The advantages of designing hierarchy at the feature level.**\n\n**A4.** \n\n1. **Carrying Multiple Semantic Cues**\n   A hierarchy built in the raw value space merely propagates numerical value information. In contrast, a feature-level hierarchy embeds additional semantic details (e.g., timestamps, variable identifiers, and sampling density). This enriched representation enables the model to capture not only numerical variations but also temporal dynamics and inter-variable heterogeneity—an essential aspect for effectively modeling IMTS.\n2. **Efficient Single-time Feature Encoding**\n   With a feature-level hierarchy, a single feature encoding is performed at the outset, and subsequent layers focus solely on dependency extraction and fusion. Constructing the hierarchy at the raw value level would require repeated encoding at every layer, thus incurring higher computational cost.\n3. **Enhanced Information Transmission and Control**\n   We employ multi-head and multi-time attention mechanisms at the feature level to ensure appropriate weighting and filtering during the aggregation process, thereby mitigating inappropriate information passing. Our ablation experiments (w/o TEAGG) confirm that this design substantially improves high-level feature learning (11.11% on MSE↑ and 5.53% on MAE↑ on four datasets).", "claims_and_evidence": "There are no problematic main claims.", "methods_and_evaluation_criteria": "The model designed in the paper is both reasonable and effective for addressing the IMTS modeling problem. The dataset used is a commonly employed real-world dataset in IMTS modeling, which also provides practical guidance for solving real-world problems.", "theoretical_claims": "There are no theoretical claims in this paper.", "experimental_designs_or_analyses": "The experiments in this paper are relatively comprehensive and demonstrate the effectiveness of the model. However, the ablation study does not sufficiently highlight the importance of hierarchical information. Additionally, the hierarchical illustrations in the appendix (Figures 11–13) fail to clearly show that the proposed model effectively captures better hierarchical information.", "relation_to_broader_scientific_literature": "This paper details efforts to advance IMTS modeling across various scientific domains. The paper provides a new perspective on the modeling of ISMTS to some extent and effectively improves downstream task performance.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "1. The motivation of this paper is unclear. While it emphasizes the importance of multi-scale information in multivariate time series modeling, there is no direct evidence supporting this claim. Even in the ablation study, no dedicated experiments demonstrate the significance of hierarchical information. Additionally, the hierarchical illustrations in the appendix (Figures 11–13) do not convincingly show that the proposed model captures better hierarchical structures.\n2. The proposed model involves multiple graph layers, which may impact its time and space complexity. Therefore, it would be helpful if the authors could provide an analysis of Hi-Patch’s computational complexity.\n3. The paper requires tuning of several important parameters, including but not limited to the number of layers and patch size.", "other_comments_or_suggestions": "No.", "questions_for_authors": "What are the advantages of designing hierarchy at the feature level compared to designing it at the raw value level? Specifically, as shown in the paper, the hierarchical structure obtained through aggregation at the feature level may lead to inappropriate information being passed to the next layer, potentially affecting the learning process in higher layers.", "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:

Meta-review: This paper introduces Hi-Patch, a hierarchical patch-based GNN framework for modeling Irregular Multivariate Time Series (IMTS). The approach effectively captures multi-scale temporal and inter-variable dependencies through intra- and inter-patch graph layers. The reviewers appreciated the paper's novelty, solid empirical results across diverse datasets, and relevance to real-world IMTS applications. Concerns were raised about scalability, clarity of multi-scale motivation, and interpretability of learned features. The authors provided thorough and satisfactory rebuttals, addressing computational complexity, validating the importance of hierarchy via ablations, and adding insights into scale interpretability and practical impact. Multiple reviewers raised or confirmed their scores post-rebuttal.
Decision: Accept (poster)