Unifying Knowledge from Diverse Datasets to Enhance Spatial-Temporal Modeling: A Granularity-Adaptive Geographical Embedding Approach
Authors: Zhigaoyuan Wang
Ying Sun
Hengshu Zhu
Venue: ICML 2025 poster (2025)
Keywords: Geographical Modeling
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Spatio-temporal forecasting provides potential for discovering evolutionary patterns in geographical scientific data. However, geographical scientific datasets are often manually collected across studies, resulting in limited time spans and data scales. This hinders existing methods that rely on rich historical data for individual entities. In this paper, we argue that heterogeneous datasets from different studies can provide complementary insights into the same underlying system, helping improve predictions for geographical entities with limited historical data. To this end, we propose a Segment Quadtree Geographical Embedding Framework (SQGEF). SQGEF integrates knowledge from datasets with varied target entities, time spans, and observation variables to learn unified representations for multi-granularity entities—including those absent during training. Specifically, we propose a novel data structure, Segment Quadtree, that flexibly accommodates entities of varying granularities. SQGEF not only captures multi-level interactions from grid data but also extracts nested relationships and human-defined boundaries from diverse entities, enabling a comprehensive understanding of complex geographical structures. 
Experiments on real-world datasets demonstrate that SQGEF effectively represents unseen geographical entities and enhances performance for various models.

Unifying Knowledge from Diverse Datasets to Enhance Spatial-Temporal Modeling: A Granularity-Aaaptiv Geographical Embedding Approach
Zhigaoyuan Wang'
Ying Sun*!
Hengshu Zhu 23
Abstract
1. Introduction
Spatial-temporal data and its analysis are crucial across vari- ous fields. For instance, analyzing trends in spatio-temporal carbon emission data for geographic entities (Peters et al., 2012) is essential for developing effective environmental strategies. Additionally, examining changes in their global spatial distribution aids in improving climate modeling.
Spatio-temporal forecasting provides potential for discovering evolutionary patterns in geographi- cal scientific data. However, geographical scien- tific datasets are often manually collected across studies, resulting in limited time spans and data scales. This hinders existing methods that rely on rich historical data for individual entities. In this paper, we argue that heterogencous datasets from different studies can provide complemen- tary insights into the same underlying system, helping improve predictions for geographical en- tities with limited historical data. To this end, we propose a Segment Quadtree Geographical Em- hedding Framework (SQGEF). SQGEF integrates knowledge from datasets with varied target en- tities, time spans, and observation variables learn unified representations for multi-granularity entities including those absent during training. Specifically, we propose a novel data structure, Segment Quadtree, that flexibly accommodates entities of varying granularitics. SQGEF not only captures multi-level interactions from grid data but also extracts nested relationships and human- defined boundaries from diverse entities, enabling comprehensive understanding of complex geo- graphical structures. Experiments on real-world datasets demonstrate that SQGEF effectively rep- resents unseen geographical entities and enhances performance for various models.
In the past decade, extensive efforts have been made in spatio-temporal forecasting. In time series forecasting, cur- rent methods treat the observations of each target entity as time series, leveraging historical data to uncover tem- poral patterns, such as frequencies (Zhou et al., 2022) and trends (Wu et al., 2022), within individual entities for down- stream forecasting. contrast, spatio-temporal forecasting goes beyond using each entity's own history by further cap- turing interactions between multiple target entities. For example, ST ST-ResNet (Zhang et al., 2017) employs CNNs to model relationships between geographically nearby entities, while GNN-based methods (Wu et al., 2019b) extend this to capture both local and long-distance dependencies across entities. These approaches, grounded in target entities' his- tories, have proven effective in forecasting tasks such as electricity demand and traffic flow prediction.
However, traditional spatio-temporal methods heavily rely abundant historical data, which is often limited in scien- tific studies due to the high cost of data collection. Many sci- entific datasets are gathered through expensive specialized equipment, which leads to data being collected from only small number of targets. Additionally, time-consuming survey methods restrict the time span of the datasets. As a result, spatio-temporal methods struggle capture entity relationships from such sparse datasets.
In scientific research scenarios, although individual studies have limited data, numerous related studies exist, and their data are inherently connected. Different datasets collect dif- ferent entities across various granularities, yet they share in- herent geographical relationships. For example, provincial- level carbon emission data provides macro-level reflection of the emissions patterns observed at the city level, offering valuable prior information when city-level data is scarce. Likewise, neighboring cities often experience similar en-
Corresponding author: 'AI Thrust, HKUST(GZZ) Guangzhou, China CCmpuuter Network Information Center, Chinese Academy of Sciences, Beijing, China 'University of Chinese Academy of Sciences, Beijing, China. Correspondence Ying Sun <yings@hkuu hkust- gz.edu.cn>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Segment Quadtree:A Granularity-Adaptiv Geographical Embedding Approach
vironmental conditions, making one city's data useful for inferring patterns in the other. Second, different metrics within a unified industrial system are often correlated. For instance, carbon emissions and energy consumption, though sourced from different studies, all reflect energy usage dy- namics and can compensate for missing observations. Con- sequently, the information within these different datasets is inherently complementary. Leveraging these cross-datasse relationships enables the integration of entity interactions across datasets while capturing their hierarchical structure and geographical dependencies. Therefore, we argue that: heterogeneous datasets from different studies can pro- vide complementary insights into the same underlying system, thereby improving predictions for geographical entities with limited historical data.
France
Grid
(a) Grid
(b) Geographical Entity
Figure I: (a) Grid Gig in the European region. (b) Geo- graphical entity consisting of grids France at the country granularity and Madrid at the province granularity.
We conduct experiments on country, province, and city- granularity forecasting tasks across different regions. The results demonstrate that our method effectively models geo- graphical entities of varying granularities from different re- gions, benefiting both timeseries and spatial-temport mod- els. Further experiments prove the robustness of our model, and the embeddings we learn effectively reflect the relation- ships of entities in the real world.
However, integrating heterogeneous datasets to model un- seen entities presents several challenges. First, the target task dataset may contain entities that have not appear in the avaliable datasets, obtaining relevant information for unseen entities is challenging. Second, granularity of dif- ferent datasets can largely differ, each granularity offering unique interactions. For example, country-level interac- tions in national datasets differ from city- interactions in city datasets. Finding way to utilize these heteroge- neous datasets to model unified complex interactions is a significant challenge. Third, datasets can have different data structures, such as entity-based and grid-based. Entity- based datasets present interactions within human-defined boundaries with administrative knowledge, while grid- based datasets present interactions within naturally divided units of region with geographical information. Extracting and fusing relationships from different data structures to create an integrated model is another key challenge.
The key contributions of this work are as follows:
To our knowledge, we are among the first to propose a novel framework that can integrate different hetero- geneous datasets to address the data scarcity issue in scientific datasets.
We propose a novel embedding method that can repre- sent different granularity entities within a region, even if they are not present the training dataset.
We conduct comprehensive experiments demonstrat- ing that our method effectively represents unseen geo- graphical entities across all granularities and regions, benefiting range of models and showcasing its versa- tility and robustness.
To address these challenges, we propose Segment Quadtree Geographical Embedding Framework (SQGEF) that utilizes heterogeneous datasets achieve a unified representation of different granularity entities within a region, even if they are not present in the training dataset. This framework is built on a novel data structure, the Segment Quadtree, and two innovative learning methods tailored to train this structure on entity and grid datasets, respectively. (1) The Segment Quadtree provides a hierarchical representation of the region, in contrast to single-layer grid- partitioning that treats all regions uniformly. It offers the flexibility to represent entities and grids of any granularity while explic- itly capturing nested relationships between nodes, enabling multi-level information storage and aggregation. (2) Hierar- chical BriddBBaeddda Learning captures interactions at multiple levels from grid datasets, incorporating global interactions for child nodes within the Segment Quadtree. (3) Geographi- cal Entity- Learning aggregates grids within entities to extract nested relationships and numan-defined boundaries information across different granularity entities.
2. Preliminaries
2.1. Problem Formulation Definition 1: Grid: In this study, we partition the region containing all geographical entities into I x T grid map based on longitude and latitude, where each grid cell Gi,j represents specific area within the current region, as illus- trated in Figure ((a).
Definition 2: Geographical Entity: A geographical entity E, is represented as set of grid cells. Specifically, a grid cell Gr,j part of the geographical entity Ex, then GJ,j e & Therefore, geographical entity & can be defined as: & (G., (i,j) E Ia x J&} where Ia C {1...., Z} and Is S (1, (1 denote the indices of the grid cells that belong to the entity εx, as shown in Figure 1(b).
In this paper, we aim to forecast the multivariate time se- ries for a given set of geographical entities, denoted as
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Segment Quadtree:A Granularity-Adaptis Geographical Embedding Approach
N J. Let x; E R- rep- resent the values of Nieet geographical entities at time step t, where xa[i] E IR denotes the value of the i-th geographical entity at time step t. Given a sequence of historical observations over Lx time steps, X = (xt,; Xtt...,.....,..1 J. our objective is to predict the future values Y - (xtex 111 Xtxx 12 Xt2x Ly of these geo- graphical entities based on the previous Lx time steps and additional relevant information.
glest {gtest 22 3..... Steat
ing the model solely on the original granularity, which fo- cuses only on interactions specific level, our approach re-divides datasets into different granularities for training, By training child nodes together with their ancestor nodes on these re-divided datasets, our method incorporates global interactions, enabling each node to understand and capture interactions at various levels.
Geographical Entity-Based Learning. Various entities of different granularities from subregions within a region ex- hibit distinct boundaries and nested relationships that define meaningful groupings and interactions within the grids. Ig- noring these aspects would result in a loss of administrative knowledge regarding interactions within and between grids of human-defined entities. Focusing solely on the relation- ships between grids and their nearby grids overlooks the inherent group information among grids within the same en- tity. Our approach addresses this limitation by training grids within the same entity together. This helps the model learn both the nested relationships and the human-defined bound- aries within different entities. Each entity can be represented by nodes at different levels within the Segment Quadtree, facilitating communication across levels and parent nodes.
The additional information includes: 1. Grid Dataset: Let G. E RTx represent the values of the grid cells at time step 4, where G denotes the value of the grid cell Go at time step t. Given a sequence of historical obser- vations over Lx time steps, G = {Gr,, Gr....., Gizx 2. Geographical Entity Dataset: Let Errain represent the set of geographical entities in the training dataset, where gtrain = Eirain, Egraiin denotes the number of geographical entities in the training dataset. Each entity
Etrain and Ntrain
g Efrai has an associated time series X = [t2] (t),ai(t),........ atrain [xyx] for histor- ical observations.
It is important to note that different Geographical Entity datasets have entirely different sets of geographical enti- ties, i.e., grrain n great = 0. The entities in the target dataset grest will not appear in training datasets Errein emphasizing the challenge of transferring knowledge across datasets to unseen entities. By integrating these heteroge- neous datasets, our model effectively predict the multivariate time series for the given set of geographical entities.
3. Method
In this section, we present the details of SQGEF. 3.1. Segment Quadtree Embedding To effectively model geographical regions, it is essential to represent each grid within a defined area, as these grids serve as the fundamental units for understanding spatial interactions. By modeling each grid, we can capture lo- calized phenomena and their time series data, providing insights into how different geographical areas interact over time. However, treating each grid in isolation presents sig- nificant drawbacks. This approach neglects the hierarchical relationships that exist between different granularity lev- els of geographical entities, such as cities, provinces, and countries, leading to fragmented understanding of spatial dynamics.
2.2. Framework Overview
Our framework contains a data structure and two learning methods to train it, as shown in Figure 2. Before present- ing the details, we first introduce the pipeline of how to train the model and then elaborate on the motivations and advancements behind their design.
Segment Quadtree Embedding. To represent entities at different granularities, we use a hierarchical representation of region by dividing it into multiple levels, with each child node representing a quarter division of its parent. This ap- proach ensures that nodes represent all granularities within the region. Each node in the Segment Quadtree corresponds to specific granularity level, capturing the interactions and information relevant to that level.
To address these limitations, we employ a segment quadtree structure, which allows us to represent the region in hi- erarchical manner. Formally, let T denote the segment quadtree. The root node To,0 covers the entire I x J grid region, and each node Ti.j at level i can be divided into four child nodes T+41.4++1 T 1,5+22 T++1,44+3, and T.+1.4++4 at level i + 1, corresponding to the northwest, northeast, southwest, and southeast quadrants, respectively. Each node Ti+1,A4* (where & E (1, 2, 3, 4)) contains the grid cells Ga.s where and b correspond to the appropriate size of the region: [2 I 2.2 I - 1] or (2 1 I 2i+2 I 1] for 2
An entity's variation is influenced by higher-level variations, so the Segment Quadtree is trained from the top down. This process begins with a high-level representation and progres- sively models more fine-grained representations at lower levels. During this training, lower-level nodes are trained simultaneously with higher-level information, allowing each node to incorporate both the global context from its parent node and the local information specific to itself. Hierarchical Brsed Learning. tions across multiple levels in grid datasets, instead of train-
2. 24+1_ J 1] or [2++1 J 24,2+22.. 2 1]
a, and [2
23
for b. depending on the quadrant. This hierarchical struc- ture ensures that different granularity representations are learned, preserving the hierarchical nature of geographical entities. Extracting features from the segment quadtree and
To capture interac-
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Segment Quadtree:A Granularity-Adage Adaptiiv Geographical Embedding Approach
(a) Segment Quadtree Embedding
(b) Hierarchical Grid-Based Learning
Grid Query
Segment Quadtree T
Grid Interaction
Forecasting Model Set
1
2
Model
3
4
[eij]
Training Set:
Entity Query
Grid Dataset
Model
19
20
17
Model
69
(c) Geographical Entity-Based Learning
(d) Inference Process
Geographical Entity Interaction
Forecasting Model Set
Unseen Entity Interaction Forecasting Model Set
Model
Model 1
Training Set: Geographical Entity Dataset
Model 2
Training Set:
Model 2
Unseen Entity Dataset
Countrien
Provinces
Citiex
???
M
Model
Model
M
M
Figure 2: Overview of SQGEF
obtaining the embeddings allows to construct the rela- tions of any subregions or entities, enhancing time series forecasting methods. This approach captures interactions at multiple levels and provides a comprehensive understanding of spatial dynamics, benefiting forecasting on target datasets.
is calculated based on the cosine similarity between the mean embeddings of different nodes, For instance, the entry
A (F represents the relationship between node T and node Ty, and computed as:
A = mij m..
(2)
3.2. Hierarchical Grid-Baset Learning
m my
In time series methods, the forecast of multivariate time series for given set of geographical entities, denoted as = Efeest ctest ..... gtest can be enhanced by leveraging a graph structure G E N which captures the contextual relationships between variables. This graph-base approach provides additional context, improving the model's ability to capture interactions between variables. Here, we introduce the construction of between each grid.
where m,j ma.j jr is the dot product of the mean embed- dings and |mm and ||m,,,'l are their magnitudes. Several models mi, my, m3. are considered, each taking as input the data X and the graph 9 with shared weights. Let L, denote the loss for model m.-
optimize the performance across all models, we train each model in turn. For each model m,, the loss L, is defined using the loss between the output and the target:
To effectively train each node and its children together, we construct the graph for all nodes at the same level using the embeddings of each node and its children to calculate the adjacency matrix. For each node Tr, at level i, we compute the mean embedding m,j- which represents the node Tr,j and its children. Let C(T.,) be the of children of node Tej. The mean embedding m.. j is then given by:
L. (X, w) = -- l0:. Vi.1l,
where Dr. is predicted value and Vi. is the target value for model m, at time step and W represents the shared weights to construct the graph with adjacency matrix A.
The training process involves minimizing the loss for each model sequentially. Each model m, has its specific param- eters O; but they share the weights of the same graph G. The combined optimization objective, taking into account the shared weights of the graph, is:
Σ
m., HT.s) Th. (UTT
where (Ta,j contains the node Tr, and its children, and H(,s) U C(T.,) is the cardinality of this set. Using the mean embeddings m,,j, the adjacency matrix A for the graph G is constructed. The adjacency matrix
g" arg min Cx G; O;) for i 1,2...., (4)
C.
where (0:) represents the parameters of all models.

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Segment Quadtree:A chanularity-Adaptiin Geographical Embedding Approach
By iteratively training each model on multiple grid datasets, the shared graph 9 is refined with its level nodes and their child nodes from Segment Quadtree.
for initializing and optimizing the hierarchical segmentation within T. After training with the Grid Dataset, we proceed to train using the Geographical Entity Dataset. Each dataset contains entities at varying granularities, such as country, province, and city datasets, allowing the segment quadtree to extract hierarchical grouping information at different levels. This dataset enables the refinement of the segment quadtree T to capture specific hicrarchical representations tailored to geographical entities.
3.3. Geographical Entity-Based Learning
Geographical entities are defined by distinct administrative boundaries that dictate meaningful groupings and nested relations within the data. To effectively capture these bound- aries and interactions of grids within their boundaries, we train the Quadtree T Geographical Entity Datasets.
For each geographical entity En in the dataset Etrain the grid layout is represented by binary matrix Mr E {0, (0,1)TxJ, where indicates grid presence and O indicates absence. To extract the entity representation from the grid representation using the quadtree traversal, we traverse from the root node To,0 downward. During traversal, if a node Th overlaps with any 1 in MEn,, it is added to the set Qe and traversal of its child nodes is halted. This approach ensures that nodes Ti, E Çe, at various levels i capture different aspects of & 's spatial structure, effectively repre- senting the spatial layout of ED at different granularities.
3.4.2. INFERENCE PROCESS
Once the segment quadtree TT has been trained on both types of datasets, we use it to enhance any models for infer- ence on target datasets. Given the boundaries of the set of geographical entities appearing in the downstream dataset Erear = {Efest, E ,..., EN }, we query TT to get repre- sentations for all entities and build the interaction graph G. We train the model on historical observations X over Lx time steps. The interaction graph G and the inference model are jointly trained this historical data to fit the down- stream dataset with pretrained knowledge. After training, we perform inference on the test set Y, where Y consists of time steps following the historical data. The trained model leverages the pretrained representations from T and task- specific relations from the downstream dataset to predict results for these future time steps.
The embedding еεp for En is obtained by aggregating the embeddings of nodes Ti. E QeD. weighted by their level i. Specifically, the embedding is computed as:
CεD= =
-E
eij 2
(5)
TE.EERE
4. Theoretical Analysis
where e, is the embedding of node Tr. This hierarchical aggregation ensures that eEp captures information from different levels of 'T allowing nodes from various levels to effectively communicate their information. By leveraging the embeddings еg, obtained for each geographical entity Ep, we construct graph GE, that captures the contextual relationships between these entities. Each node in GED represents a geographical entity, and edges between nodes encode relationships based on their embeddings.
In this section, we briefly analyze how the segment quadtree representation regulates model complexity to better capture entity relationships, the complete proof is provided in Ap- pendix A.5. We prove that the SQGEF yields tighter gener- alization error bound compared to the traditional Grid-based Partitioning Embedding using Rademacher complexity. We compare two embedding methods for an entity with grids region: Grid-based Partitioning Embedding: Fuses k grid embeddings using i embeddings. The hypothesis class is de- noted Hgria. Segment Quadtree Embedding: Constructs a segment tree over the grids of region, fusing m = Oloo k) node embeddings to represent the entity. The hypothesis class is denoted Heeg-
Similar to Frid-Based Learning, we train sharing weights of the same graph 9 on several models M2, ma,-- -- in turn. Unlike Grid-Based Learning, which focuses single- level grid cells, geographical entity-based learning aggre- gates information across hierarchical nodes of the Segment Quadtree and captures longer-distanee interactions between nodes from different parent nodes. This approach enables the model to capture dependencies and interactions that span various geographical scales and political boundaries.
The Grid-based Partitioning Embedding method employs an MLP to map & embeddings, each in Rdd into a single d-dimensional embedding. The input is concatenation of k vectors (dimension kd), with L hidden layers and parameter count p scaling with kd. The Rademacher complexity is bounded as (Bartlett et al., 2017):
3.4. Model Training and Inference
Having defined two distinct learning approaches, we now proceed with the overall training process for the segment quadtree and inference on downstream datasets.
| 4-1 /VV VpVkd kd
( S C₁
(6)
3.4.1. 1. TRAINING PROCESS
The Grid Dataset serves as the initial training data for con- structing the segment quadtree T. This dataset provides
where C > 0 is a constant, ||]] is the maximum spectral norm of weight matrices, and p x kd due to input layer weights.
spatial information organized into a grid structure, essential
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Segment Quadtree:A Granularity Geographical Embedding Approach
Similarly, the Segment Quadtree Embedding fuses m Ollog k) embeddings, each in TRd, into single d- dimensional embedding using MLP. The input dimension is md, with L hidden layers and parameter count p' scaling with md. The Rademacher complexity is bounded as:
Entity-Based Learning contribute to improved model per- formance? Q4: How do various hyperparameter settings influence the performance of the model? Q5: Does the model accurately learn relationships that reflect the actual interactions between entities in real world? Q6: Can our method generalize effectively across datasets from different scientific domains?
V/l md ||]]
R.s(Heeg) S C2-
5.1. Data Description
T
We set three downstream forecasting tasks with different granularities from two regions for carbon emissions: China Province, Europe Country, and China City. The China City dataset suffers from extreme data scarcity, making it unsuit- able for some baseline models. The datasets used for each task are summarized in Table 1, with detailed descriptions provided in the appendix.
where C2 > 0 is constant, and p' x md.
Assume I. and [W] > are the same for both methods, and the constants C1 as C2 are approximately equal, with p xx kid and p' x md. The Rademacher complexities are for Grid-based Partition- ing Embedding and R.s(H x || vn for Segment Quadtree Embedding. Since m = Ollg k) < k. and p' < P. we have Vmdvp' < Vkdyp, thus R.s( (Hsegy) < RRs(Hggaa) the generalization bound for the Segment Quadtree Embedding is tighter.
Rs(Hggia) x [[]]
5.2. Experimental Setup
5.2.1. METRICS
We used mean absolute error (MAE) and mean squared error (MSE) to assess the performance of our model and baselines. For both of these metrics, smaller values indicate better performance.
5. Experiments
Table 1: Experimental Datasets for Training and Testing
5.2.2. BASELINES
Dataset Name
Data Type
Function
We selected seven representative baseline methods, falling into two categories: time series models and spatio-temporal models. For the first category, we compare the three most widely utilized time series models: Informer (Zhou et al., 2021), FEDformer (Zhou et al., 2022), Autoformer (Wu et al., 2021), as well as the recent state-of-the-an method TimesNet (Wu et al., 2022). For the second category, we compare the three most widely utilized spatial-temporal models; AGCRN (Bai et al., 2020), MTGNN (Wu et al., 2020), and GWNet (Wu et al., 2019a). Detailed descriptions are provided in the appendix.
ODIAC Ikm China
Grid
Train
ODIAC ddeg China
Grid
Train
MEICModel Region
Entity
Train
Carbonmonitor China
Entity
Test
ODIAC Ikm BU
Grid
Train
ODIAC Ideg EU
Grid
Train
Carbonmonitor EU
Entity
Test
ODIAC 1km China
Grid
Train
ODIAC Ideg China
Grid
Train
Carbonmonitor China
Entity
Train
5.3. Overall Performance(Q1)
MEICModel City
Entity
Test
As shown in Table 2.w/ Seg denotes the enhancement of baseline models by SQGEF SQGEF achieves the best per- formance across all datasets and generally improves the performance of all baseline methods. This indicates that SQGEF effectively represents entities of different granulari- ties, from countries to cities in various regions, and benefits various kinds of models.
To validate that SQGEF can represent unseen differentt gran- ularity geographical entities from various regions, we apply it to CO2 emission forecasting tasks for entities at three dif- ferent granularities: country, province, and city, across two distinct regions-China and Europe. For each region, we utilize multiple grid or geographical entity datasets to train the model and subsequently test it on a geographical entity dataset not included in the training phase. We then analyze the experimental results, demonstrating improvements in accuracy by comparing with various baselines. Additionally, we design different experiments to address the following questions: Q1: Can method effectively model unseen geographical entities of different granularities? How do factors such as the granularity, region of entities, and ex- treme data scarcity impact performance? Q2: Is Segment Quadtree necessary for effective modeling, as opposed to modeling each grid individually? Q3: Does Geographical
On all datasets, SQGEF shows greater improvements for time series methods compared to spatio-temporal meth- ods. This is because SQGEF helps the time series methods to leam spatial relations, which they were unable to cap- ture before. Additionally, SQGEF can also enhance the performance of spatio-temporal baselines. This indicates that SQGEF learns better representations from the training dataset compared to a completely data-driven approach.
However, the China City dataset unique because all the cities contains are obscure and less representative in the
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Segment Quadtree:A chaaulaaity-Adaptiii Geographical Embedding Approach
Table 2: Overall performance comparison. The best results among all the models are highlighted in bold.
China Province
Europe
Country
China City
Method
MSE
MAE
Method
MSE
MAE
Method
MSE
MAE
Informer
0.8890 =.890000003
0.7329 # 0.0006
Informer
2.0979 = +00027
1.0842
0.0003
Informer
7.6327 #0.155
1.7926 * 0.0185
w/Seg
0.7638 0.0011
0.6815 * 0.0001
w/Seg
2.0373 + 0.0004
1.0658
0.0001
w/Seg
7.3984 0.0239
1.7488 * 0.0069
FEDformer
133588200000
0,4753 1 0.0000
FEDformer
2.7053 0.0000
1.2411
0.0000
FEDformer
6.4412
1.3191 = 0.0005
w/Seg
0.2941 00,0000
0.3971 * 0.0000
w/Seg
1.3831=0.0000
1.1772 0.0000
w/Seg
6.2043 0.3618
1.1631 # 0.0006
Autoformer
0.3242 = 0.0003
0,4490 +: 0.0001
Autoformer
0.0002
1.2554 = 0.0000
Autoformer
7.3596 0.6210
1.3848 = 0.0103
w/Seg
0.2934 0.0000
0.4008 1 0.0000
w/Seg
1.1806 =.0000
0.0000
w/Seg
7.1760 0.0390
1.2261 = 0.0005
TimesNet
+45003=0.0000
0.5257 1 0.0004
TimesNet
1.2406 2.0000
0.0001
TimesNet
w/Seg
0.2743 0.0001
0.3900 +: 0.0001
w/Seg
2.3866
1.1767 0.0002
w/Seg
AGCRN
0.9951 00.007
0.7359 + 0,0002
AGCRN
2.2722 #0,0001
1.1458 + 0.0000
AGCRN
16546
1.8930 = 0.0034
w/Seg
0.7061 26
0.6383 + 0.0028
w/Seg
2.0247 +.00049
1.0763 0.0007
w/Seg
7.4233 00.0422
1.8644 + 0.0034
MTGNN
0.5926=
0.5539 + 0.0008
MTGNN
2.1919 =0.0011
1.1175 0.0001
MTGNN
7.1258
1.6903 = 0.0063
w/Seg
0.4978 0.0109
0,5465 # 0.0047
w/Seg
2.1395 #0.0012
1.1048 = 0.0001
w/Segg
7.2951=0.000
1.8278 # 0.0005
GWNett
0.3953 = 0.0000
0,4573 1 0.0000
GWNet
2.2498
0.0000
1.1328
0.0000
GWNeet
1.5994 1: 0.0039
w/Seg
0.3732 0.0039
0.4691 = 0.0015
w/Seg
2.0856 0.0013
1.0981
0.0011
w/Seg
7.1551 00.061
1.7891 = 0.0014
TimeXer
0.4027 0.0000
0.4894 1: 0.0000
TimeXer
2.3852 a: 0.006
1.1684 0.0005
TimeXer
8.1542 0.7333
1.3078 1: 0.0022
w/Seg
0.3569 0.0002
0.4596 a: 0.0001
w/Seg
2.2058 + 0.0022
1.1266
0.0002
w/Seg
7.3635 a 0.3326
1.2838 a: 0.0026
Transforment
0.4772 = 0.0064
0,5004 # 0.0043
Transformet
0,0000
1. 1.1808 0.0000
Transformert
8. 5,5086 0.4485
1.3474 # 0.0018
w/Seg
0.4319 0.0001
0.4790 a: 0.0001
w/Seg
2.2671
0.0001
1.1454 0.0000
w/Seg
7.8494 0.0765
1.3240 a 0.0015
Table 3: Ablation Study of each module China Province dataset.
Method
MSE
MAE
|Mthod
MSE
MAR
|Mttho
MSE
MAP
Method
MSE
MAE
Method
MSE
MAE
Method
MSE
MAE
Methed
MSE
MAB
Informer
0,8800
0.7329
FEDformer
0.3588
0.4753
Autaformer
0.3042
0.4490
TimesNett
0.4503
0.5257
AGCRN
0.9911
0.7359
MTGNN
0.5926
0.5539
IWWNe
0.3053
0.4573
w/ Seg
0.7638
0.6815
m/ Seg
0.29411 0.3971
m/ Seg
0.2914
0.4008
m/ Seg
0,2743
0.3900
eil Seg
0.7061
0.6383
w/Seg
0.4978 0.5465
Seg
0.3732
0.4691
w/o GB
0,7805 0.6879
l/fo GB
0,3550 0.4574
Iolo GB
0,3155
0.4198
wio GEB
0.3661 0,4885
inoo GE
0,7054 0.6210
GE
0.5357
0,5585
udo GB
0,3876
0.4822
w/o SE
0.7391
julo SE
0.5526
jw/o SE
0.3044
0.4343
julo
0,5188
julo SE
10187
0.7452
SE
0,5175 0.5318
jul SE
0.6250
0.5576
training dataset. Despite our model achieving the best per- formance on this dataset, it decreases the performance of two spatio-temporal baselines. This is because the infor- mation of these under-represented cities is overwhelmed by nearby larger cities, causing SQGEF to learn representations that include both the obscure city and its surrounding areas. Addressing the noise when modeling obscure geographical entities should be focus for future work.
5.4.2. GEOGRAPHICAL ENTITY-BASED LEARNING (Q3) As shown in Table removing Geographical Entity-Based Learning decreases performance for all methods. This demonstrates the importance of capturing boundaries in the data. MTGNN shows the largest performance decrease because it uses top-K relations for each entity. Without cap- turing boundaries, the relationships become too smooth, and the differences between values are smaller. Other methods that include all relationships between entities are less af- fected, but the top-K approach used by MTGNN can cause the model to miss important relationships.
5.4. Ablation Stu and Q3)
To delve into the contributions of each module, we perform an ablation study using the China Province dataset. In this study, w/o SE denotes models that use only grid embedding solely without hierarchical Segment Quadtree and w/o GE denotes models trained exclusively on grid datasets without the Geographical Entity- Learning component.
5.5. Parameter Sensitivity(Q44)
To assess the robustness of SQGEF, we evaluate how vary- ing parameter choices influence the model's performance. All parameters are set to their default values except for the one being tested. The embedding size denotes the number of dimensions for each node in the segment quadtree, while the temperature factor regulates the smoothness of the con- structed graph in the softmax function. Our findings indicate that the method is not sensitive to minor variations in these parameters across all three experiments. It maintains high performance with cost-effective parameter settings, demon- strating both its effectiveness and robustness. Specifically, embedding size of 18 yields optimal performance for the China Province and China City datasets, whereas size of 20 is more suitable for the Europe Country dataset. This suggests that the Europe Country dataset exhibits greater variability and thus requires a larger embedding size to cap-
5.4.1. 1. SEGMENT QUADTREE EMBEDDING (Q2)
As shown in Table 3, the results show that removing Seg- ment Quadtree Embedding significantly reduces perfor- mance. Without this embedding, the model can only capture interactions at the grid level and fails to capture higher-level interactions. This causes the model to perform poorly on province-level data. Spatio-tempore methods AGCRN and GWNet show a significant drop in performance without Seg- ment Quadtree Embedding because they rely on relational graphs. Unlike other spatio-temporal methods like MTGNN, which focus on top-K relations for each entity, AGCRN and GWNet are more affected by noise in the relationships.
7

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Segment Quadtree:A chanularity-Adaptiii Geographical Embedding Approach
ture it effectively. For the temperature factor, the best per- formance is observed 0.95 for the China Province dataset, which differs from the optimal settings for the other two datasets. This implies the China Province dataset contains many similar provinces, necessitating a lower temperature factor to accurately differentiate between them.
you madar A33g Hubel Torader HABAl
Welluan redhereet Mrite
25
- 200 255
the c5 xie o.
ab the
(a) China Province
(b) Europe Country
(c) China City
Figure 5: Embeddings learned from Segment Quadtree com- pared to MTGNN on all datasets
6:30
4.35
630
5.50
5.7. Cross-Domain Generalization Performance(Q6)
(a) TimesNet w/Seg on(b) AGCRN w/Seg on (c) FEDformer w/Seg China Province
Europe Country
China City
To further demonstrate our framework's 's adaptability across domains, we conducted additional experiments using air pollution datasets. The results confirm the model's 's versatil- ity and effectiveness across different domains, with detailed results and settings provided in the appendix.
Figure 3: Parameter sensitivity of Segment Quadtree with Embedding Size in terms of MSE and MAE
6.20
2.00
6. Related Work
24.3
Our related work can be summarized into two categories: Spatio-temporal forecasting and Scientific computing...
6.08
6.5
Wesseste Ieder
115
Ierpersuurs Fitt
Sumper sture Facke
(a) TimesNet w/Seg on(b) AGCRN w/Seg on (c) FEDformer w/Seg on China City
6.1.
Forecasting
China Province
Europe Country
Spatio-temporal forecasting has applications across various granularity geographical granularities. At the country level, Jiao er al. leverage spatial dependence and heterogeneity to predict tourist arrivals across 37 European countries (Jiao et al., 2020). At the provincial or state level, carlier stud- ies have examined the demand for inbound tourism within different provinces of a single country (Yang & Zhang, 2019). At the city level, previous studies examined the composition and spatio-tempore variations of municipal solid waste (Mushtaq et al., 2020) and job mobility (Zhang et al., 2021). At the street level, RDAT (Liu al., 2023a) fo- cuses on making Intelligent Transportation Systems resilient to adversarial attacks, while MM-DAG (Lan et al., 2023) introduces multi-task learning algorithm for managing multi-modal traffic at intersections. It is common to utilize graphs to model the relationships between entities, thereby enhancing forecasting. The graph extracts relevant informa- tion from similar nodes, improving the accuracy (Liu et al., 2022b) and robustness (Li et al., 2022; Liu et al., 2023d; Li et al., 2024) of predictions. However, these methods primar- ily focus on utilizing historical data of forecasting targets, overlooking the potential of other heterogencous datasets. To address this, we propose Segment Quadtree Embedding to extract information from heterogeneous datasets with different granularities and data structures, providing a com- prehensive representation of entities within a region.
Figure 4: Parameter sensitivity of Segment Quadtree with Temperature Factor in terms of MSE and MAE
5.6. Case SQu)y(Q5)
In this section, we compare the quality of embeddings gen- erated by our method with those that adaptively construct the relation graph from the downstream dataset, such as MTGNN. The entities selected for comparison are based on their similarity in economic or industrial indicators, en- suring meaningful evaluation of embedding quality. The results highlight method's 's superior ability to represent geographical entities. First, we compare the embeddings of Jiangsu and Hubei. Both provinces exhibit similar carbon emission patterns due to their high concentration of energy- intensive industries. Figure 5(a) shows that embeddings learned from our method are much closer each other com- pared to those from MTGNN. Similarly, Figure 5(b) com- pares the embeddings of France and Italy. Both countries are notable for their positive efforts in expanding renewable energy and improving energy efficiency, setting them apart from other EU countries. Our method yields embeddings that are closer to each other than those produced by MT- GNN. Finally, we compare the embeddings of Yichuan and Xining, two inland, semi-arid cities in China that rely heav- ily fossil fuels. Our method provides embeddings that are closer to each other than those from MTGNN. These experiments demonstrate that our method achieves better representations of geographical entities compared to the em- beddings adaptively learned from the downstream dataset.
6.2. Scientific Computing
Scientific computing leverages advanced computational ca- pabilities to analyze complex natural problems. the cli-
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Segment Quadtree:A Granularity-Adaptis Geographical Embedding Approach
mate field, understanding the correlation between different geographical entities is erucial for effective environmen- tal strategies, such forecasting carbon emissions (Peters et al., 2012) and earthquake (Zhu et al., 2020). Astsatryan et al. utilize deep learning models to predict city tempera- tures for weather forecasting (Astsatryan et al., 2021). In geology, Pao et al. employ finite element methods (FEM) to simulate the complex interactions across regions to forecast earthquakes (Paolucci et al., 2018), while Taborda et al. use numerical methods to predict earthquake effects in urban areas to aid urban planning (Taborda & Roten, 2015). In social science, AI techniques are used to detect financial fraud (Liu et al., 2021; Huang et al., 2022: Gong & Sun, 2024) and model recruitment variations (Sun et al., 2021; 2024; Qin et al., 2025b). In biology, Al used to identify protein and gene functions (Bao & Yang, 2024; Wang et al., 2024). Furthermore, AI is increasingly applied to solve scientific problems (Qin et al., 2025a), including mathemat- ical problem-solving (Liu et al., 2023c;b; 2022a; Lin et al., 2021) and causal attribution analysis (Ji et al., 2025). How- ever, scientific computing often faces data scarcity. The high cost of collecting comprehensive data limits analy- task-specific datasets, resulting in scientific datasets with few observation points and short time spans, making it difficult to provide sufficient historical information for model training. To address this, we propose SQGEF to unify knowledge from diverse datasets, enhancing forecasting on target datasets.
structures. Analyses and experiments with datasets from var- ious regions and granularities demonstrate that our method effectively represents different unseen geographical enti- ties and improves the performance of various models. By integrating heterogeneous datasets and providing relation- ships for unseen entities, SQGEF offers valuable support for emerging scientific tasks facing data scarcity.
Impact Statement
This work aims to enhance spatio-temporal forecasting for geographical scientific data by integrating heterogeneous datasets. Our approach has potential applications in envi- ronmental monitoring and resource management. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
A Acknovledgements
This work is partly supported by the National Natural Sci- ence Foundation of China (No. 62306255, 92370204), the National Key Research and Development Program of China (No. 2023YFF0725000,, the Guangdong Basic and Ap- plied Basic Research Foundation (No. 2024A151501 1839), the Fundamental Research Project of Guangzhou (No. 2024A04J4233), and the Education Bureau of Guangzhou Municipality.
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Segment Quadtree:A Granularit-AAdaptiv Geographical Embedding Approach
A. Technical Appendix
A.1. Data Description
ODIAC
The ODIAC (oda, 2011) provides high-resolution global data on fossil fuel CO2 emissions. combines country-level estimates from the Carbon Dioxide Information Analysis Center (CDIAC) with spatial proxies such as point source locations and satellite nightlight data, covering the years 2000-2021. We employ two different resolutions: Ikm x Ikm and Ideg deeg grid subsets China and Europe locations.
MEIC
The MEIC (Tong et al., 2020; Cheng et al., 2021; 2023) is a platform to model atmospheric emissions from human activities. generates detailed emission inventories for China, supporting scientific research and policy evaluation from 1997 to 2017. The region subset is regional-scale, split by industrial zones and cities, including some obscure cities.
CARBON MONITOR
Carbon Monitor (Liu al., 2020) offers daily, up-to-date estimates of CO2 emissions from fossil fuels and cement production. This international initiative tracks crucial data for climate change studies from 2019 to 2023. We choose the China and Europe regions for experiments.
A.2. Baselines
Informer (Zhou et al., 2021) proposes the ProbSparse self-attention mechanism replace inner product self-attention, thereby reducing time and space complexity.
FEDformer (Zhou et al., 2022) designs two attention modules that utilize the Fourier transform and wavelet transform to perform the attention operation in the frequency domain.
Autoformer (Wu et al., 2021) a transformer-style model that introduces the Auto-Correlation Mechanism to replace the dot-product attention mechanism. It also presents the Decomposition Architecture, which extracts more predictable components from complex temporal patterns.
TimesNet (Wu et al., 2022) models two types of temporal changes: within modeling period (Intraperiod) and between periods (Interperiod). It expands one me-dimensional temporal data into two-diwenne vro-dimensional space for analysis.
AGCRN (Bai et al., 2020) combines graph convolutional networks with recurrent neural networks to capture both spatial and temporal dependencies time series data.
MTGNN (Wu et al., 2020) extracts relationships among variables using graph learning module. It integrates external knowledge and captures spatial and temporal dependencies from the data.
GWNet (Wu et al., 2019a) integrates graph convolutional networks with WaveNet architecture to effectively model spatial-temporal correlations in sequential data.
A.3. Implementation Details
We implemented our model using PyTorch 2.1.0, based on Python 3.8.18, and utilized Tesla A6000 GPU for training, The dimensions of node embeddings are set to 20. We used the Adam optimizer (Kingma & Ba, 2015) for parameter optimization. The learning rate was set to 0.0001 and the dropout rate to O.1. We employ Gaussian initialization for the node embeddings to provide robust starting point for training. The graph is constructed by calculating the interaction strengths between nodes through the product of their embeddings. Specifically, we compute the pairwise dot product of node embeddings form the adjacency matrix. This is followed by softmax normalization with temperature parameter applied to each row of the adjacency matrix to control the sharpness of the distribution, resulting in the final interaction probabilities. For each experiment, we trained the model in the order listed in Table 1.
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Segment Quadtree:A Gramularity-Adaptin Geographical Embedding Approach
A.4. Cross-Domain Generalization Performance
To better demonstrate our framework's adaptability across various domains, have included additional experiments using air pollution datasets. We trained our model with PM2.5 and PMI0 grid datasets and tested it using O3 data at the provincial level. The results affirm the model's versatility and effectiveness actoss different domains.
Method
MSE
MAE
Informer
2.0194
1.1581
w/Seg
1.9953
1.1510
FEDformer
2.0272
1.1584
w/Seg
1.7034
1.0619
Autoformer
2.1276
1.1355
w/Seg
2.0754
1.1215
AGCRN
2.7700
1.3346
w/Seg
2.2893
1.2133
MTGNN
2.5206
1.2658
w/Seg 2.3239
1.2139
GWNet 2.5746
1.2803
w/Seg
2.5782
1.2796
Table 4: Performance on air pollution datasets.
A.5. Detailed Proof of Generalization Bound for SQGEF
We prove that the SQGEF yields tighter generalization error bound compared to the traditional Grid-based Partitioning Embedding using Rademacher complexity.
1. PROBLEM SETUP AND NOTATION
Let 5 = ((11,1))..... vi)...., (In, yo)} be a training set of n samples drawn i.idd from an unknown distribution D over X y. For a hypothesis h E H. define the true error as L(h) = B(..))=((((((z).)) v)] and the empirical error as (h) = E ((h(r;), v.), where É y [0, 1] is bounded loss function.
We compare two embedding methods for entity with k grids region: Grid-based Partitioning Embedding: Fuses k grid embeddings using & embeddings. The hypothesis class is denoted Hgria. Segment Quadtree Embedding: Constructs segment tree over the grids of region, fusing m = Ollog k) node embeddings to represent the entity. The hypothesis class denoted Heeg-
A.5.2. GENERALIZATION BOUND WITH RADEMACHER COMPLEXITY
We rely standard result from statistical learning theory:
Theorem A.1 (Rademacher Complexity Bound (Bartlett & Mendelson, 2002)). For any hypothesis class H, with probability at least - å over the draw of S, for all h e H.
In (2/88))
L(h) < (s(h) 2R) + +3/
2n
where Rs(H)= = - (suрлe E (:E()();), v:)) is the empirical Rademacher complexity, and a; Unif[ (+1,-(+1])) -1)) are Rademacher variables.
The generalization gap L(h) (h) is controlled by Rs(H), which measures the complexity of H. We now bound
R. (HHag))) and R& (Hog).
A.5.3. COMPLEXITY ANALYSIS
For clarity, we consider an MLP to fuse l embeddings without loss of generality. The Grid-based Partitioning Embedding method employs an ML.P to map & embeddings, each in Rd into single d-dimensional embedding. The input is
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Segment Quadtree:A Gramularity-Adaptix Geographical Embedding Approach
concatenation of vectors (dimension kd), with L hidden layers and parameter count p scaling with kd. The Rademacher complexity is bounded as (Bartlett et al., 2017):
Rs (Hgrs) S SC [| VDV kd
where C > 0 is a constant, [W] is the maximum spectral norm of weight matrices, and p oc kd due to input layer weights. Similarly, the Segment Quadtree Embedding fuses Ollog k) embeddings, each in R., into single d-dimensional embedding using an MLP. The input dimension is md, with L hidden layers and parameter count p' scaling with md. The Rademacher complexity is bounded as:
| [+1
vp /md
Rs( 5 C2-
vn
where C2 > 0 a constant, and p' x md.
A.5.4. COMPARISON BOUNDS
Assume L and [W] are the same for both methods, and the constants C1 R C2 are approximately equal, with p x kd and p' md. The Rademacher complexities are RHHHgna (rr] for Grid-based Partitioning Embedding / for Segment Quadtree Embedding. Since m = O(log k) < k, and p' < p. we have and RH(H) x [[W]]-1 I Vmd < /kd.T. thus (HHog) < RRs(Hgidd
Applying Theorem A.I, for hgrid E Hgria.
In (2/6))
L (hhgria) S (isgiaii)) +?Rr(H) + 3/
2n
For heog E Haeg
ln(2/6)
L(hsez) S (lses)))) + (Rsc() + =3/
2n
Given that (Hooo)) < (Hgria) the generalization bound for the Segment Quadtree Embedding is tighter.
A.6. Data Scarcity
Data scarcity primarily refers to the fact that all forecasting targets the test set are unseen during pretraining, meaning their historical data is entirely absent from the pretraining stage. Consequently, the data available to directly model relationships between these targets is extremely limited. To address this, SQGEF leverages large volume of heterogeneous datasets from related regions or entities, even though these datasets do not contain the historical records of test targets. For example, in the China Province experiment, we aim to forecast future carbon emissions for several Chinese provinces. During pretraining, we utilize two distinct grid-based datasets and historical records of various regions, but none of these include the historical data of the target provinces themselves. This approach allows us infer patterns indirectly.
To quantify this scarcity, compare the data volume between the pretraining and test stages across three metries: total data points, number of temporal points, and temporal duration. The table below summarizes these statistics for our three experimental settings:
From this table, it is evident that the test set contains significantly fewer data points than the pretraining set across all experiments. For China Province and Europe Country, the test set's temporal duration is notably short (2 years). In the China City, test set has an extremely limited number of temporal records (37 points), indicating sparse sampling. Collectively, these statisties highlight the severe data scarcity in the test set, both in terms of quantity and coverage.
A.7. Complexity Analysis
Space Complexity: Both SQGEF and the naive grid embedding method have a space complexity of O(N) (where N is the number of grid cells). SQGEF introduces only a constant-level overhead store multi-level interaction information.
14

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Segment Quadtree:A Granularty-AAaaptiv Geographical Embedding Approach
Table 5: Data scarcity comparison between pretraining and test sets
Pretrain Set
Test Set
Pretrain Set
Test Set
Experiment
Data Points
Data Points
Time Points (Span)
Time Points (Span)
China Province
8,651,704
37,062
581 (22 years)
1,278 (2 years)
Europe Country
8,650,752
39,618
528 (22 years)
1,278 (2 years)
China City
8,687,814
962
1,806 (22 years)
37 (37 years)
In our Segment Quadtree, the total number of nodes is N (root level) + (1/4) (next level) +... + (1/4)* N. where [log, N is the tree height. This geometric series sums to ((44/3)NV) which remains linear in N. Thus, our approach scales efficiently even with higher-resolution grids, incurring minimal additional storage costs.
Training Stage Time Complexity: While both SQGEF and the naive grid embedding method have a computational complexity of ( (where N is the number of grid cells). SQGEF adds only constant overhead to process multi- granularity data. While SQGEF models interactions across all granularity levels, the additional computational cost is minor relative to the substantial performance gains it delivers. The complexity comprises two parts: (1) constructing a graph to capture spatial relations between grids, and (2) computing temporal relations across timestamps. For the graph, the naive method's s complexity is O(N2). In SQGEF, this becomes O(N2 + (1/16) + .... + (1/16)" across tree levels, summing O((16/ (1)) a slight increase over O Temporal relations are treated as constant T, as no additional temporal computation is required beyond the input sequence. This modest overhead enables rich multi-level interactions, significantly enhancing forecasting accuracy, evidenced by experimental results.
Inference Stage Time Complexity: During inference, SQGEF outperforms the naive grid embedding method in efficiency. Using the segment tree query method [1] for region [4 [x2] x [aY1, (Y2), we recursively check overlaps with the current node's region [X1, X2] x [Yi, Y2)- With a tree height of O N). and up to 4 nodes visited per level (if the query spans quadrants), the total nodes visited is bounded by 4 Olog, N) = (log N) (since log, N = (1/2) N). This contrasts with the naive grid method's (NN complexity, making our approach far more efficient.
During training, SQGEF maintains O(N) space complexity and O(N2) computational complexity, identical to the naive method, enabling the capture of multi-granularity relationships that yield significant performance improvements. In inference, our method achieves (log N) complexity. versus O(N) for the naive approach, ensuring superior scalability and efficiency. These trade-offs make SQGEF highly practical and advantageous, especially for numerous downstream spatio-temporal forecasting tasks, where its fast inference speed delivers substantial benefits across a wide range of applications.
Summary:
15


Reviews:
[{"review_id": "K9aoyqZH3l", "reviewer": "Reviewer_cJrD", "summary": "This paper aims to integrate diverse heterogeneous datasets to address the data scarcity problem in scientific research. The authors propose the Segment Quadtree Geographical Embedding Framework with a novel data structure called the Segment Quadtree. The framework employs two learning strategies: capture interactions at multiple levels from grid datasets, and extract nested relationships and human-defined boundary information from entity datasets. As a result, SQGEF enhances spatio-temporal forecasting for geographical entities across various regions and granularities, even for entities unseen in the training set.\n\n## Rebuttal summary\n\nData scarcity has been a fundamental issue in scientific research. To resolve this, this paper proposes integrating diverse heterogeneous datasets with a novel Segment Quadtree Geographical Embedding Framework. This framework is verified with comprehensive experiments. My initial concerns about experimental designs have been reasonably addressed after the rebuttal, so I have raised my score from 3 to 4.", "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": "Thank you for your thorough review! Your suggestion to include additional experiments and analysis will further strengthen our paper.\n\n# outdated baselines\n\nThank you for your suggestion! We have updated our experiments to include two state-of-the-art baselines: TimeXer (NeurIPS 2024) and iTransformer (ICLR 2024). The results demonstrate that our framework significantly enhances the performance of these cutting-edge baselines, underscoring its robustness and adaptability.\n\n|China Province||||Europe Country||||China City|||\n|---|---|---|---|---|---|---|---|---|---|---|\n|Method|MSE|MAE|Method|MSE|MAE|Method|MSE|MAE|\n|TimeXer|0.4027±0.0000|0.4894±0.0000|TimeXer|2.3852±0.0066|1.1684±0.0005|TimeXer|8.1542±0.7333|1.3078±0.0022|\n|w/Seg|0.3569±0.0002|0.4596±0.0001|w/Seg|2.2058±0.0022|1.1266±0.0002|w/Seg|7.3635±0.3326|1.2838±0.0026|\n|iTransformer|0.4772±0.0064|0.5004±0.0043|iTransformer|2.5433±0.0000|1.1808±0.0000|iTransformer|8.5086±0.4185|1.3474±0.0018|\n|w/Seg|0.4319±0.0001|0.4790±0.0001|w/Seg|2.2671±0.0001|1.1454±0.0000|w/Seg|7.8494±0.0765|1.3240±0.0015|\n\n# complexity analysis\n\nThank you for your insightful review! We agree that including an efficiency analysis strengthens our paper, and hightlights the effectiveness of our method that we only introduces a constant-level additional complexity to achieve significant performance increase.\n\n**Space Complexity:** Both SQGEF and naive grid embedding method have a space complexity of O(N) (where N is the number of grid cells), SQGEF introduces only a constant-level overhead to store multi-level interaction information. In our Segment Quadtree, the total number of nodes is N (root level) + (1/4)N (next level) + ... + (1/4)^h N, where h = ⌈log₄N⌉ is the tree height. This geometric series sums to O((4/3)N), which remains linear in N. Thus, our approach scales efficiently even with higher-resolution grids, incurring minimal additional storage costs.\n\n**Training Stage Time Complexity:** While both SQGEF and the naive grid embedding method have a computational complexity of *O(N²)* (where *N* is the number of grid cells), SQGEF adds only a constant overhead to process multi-granularity data. While SQGEF models interactions across all granularity levels, the additional computational cost is minor relative to the substantial performance gains it delivers. The complexity comprises two parts: (1) constructing a graph to capture spatial relations between grids, and (2) computing temporal relations across timestamps. For the graph, the naive method’s complexity is O(N²). In SQGEF, this becomes O(N² + (1/16)N² + ... + (1/16)^h N²) across tree levels, summing to O((16/15)N²), a slight increase over O(N²). Temporal relations are treated as a constant T, as no additional temporal computation is required beyond the input sequence. This modest overhead enables rich multi-level interactions, significantly enhancing forecasting accuracy, as evidenced by our experimental results.\n\n**Inference Stage Time Complexity**: During inference, SQGEF outperforms the naive grid embedding method in efficiency. Using the segment tree query method [1]for a region [qX₁, qX₂] × [qY₁, qY₂], we recursively check overlaps with the current node’s region [X₁, X₂] × [Y₁, Y₂]. With a tree height of O(log₄N), and up to 4 nodes visited per level (if the query spans quadrants), the total nodes visited is bounded by 4·O(log₄N) = O(log N) (since log₄N = (1/2)log₂N). This contrasts with the naive grid method’s O(N) complexity, making our approach far more efficient.\n\n**Summary:** During training, SQGEF maintains *O(N)* space complexity and *O(N²)* computational complexity, identical to the naive method, enabling the capture of multi-granularity relationships that yield significant performance improvements. In inference, our method achieves O(log N ) complexity, versus O(N) for the naive approach, ensuring superior scalability and efficiency. These trade-offs make SQGEF highly practical and advantageous, especially for numerous downstream spatio-temporal forecasting tasks, where its fast inference speed delivers substantial benefits across a wide range of applications.\n\n[1] Lee, D. T., & Preparata, F. P. (1984). *Computational Geometry: A Survey*. IEEE Transactions on Computers.", "claims_and_evidence": "The paper makes two main claims: 1. Heterogeneous datasets from different studies can provide complementary insights into the same underlying system. 2. SQGEF is designed to learn unified representations for multi-granularity entities, including those absent during training.\n\nThe authors support these claims by using datasets from different sources within the same region. The improved performance on forecasting targets that were absent in the training data validates these claims. Overall, the claims are well-supported and convincing.", "methods_and_evaluation_criteria": "Yes, they make sense.", "theoretical_claims": "I reviewed the proof in Section 4, and it is correct.", "experimental_designs_or_analyses": "The overall setting is comprehensive. The authors mainly support the claims from different perspectives. However, the baseline is outdated. The paper should include some more advanced baselines.", "relation_to_broader_scientific_literature": "This paper addresses the data scarcity problem in the scientific data domain, which has broad applications in fields such as satellite data analysis, and remote sensing.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n1. The paper addresses a meaningful and challenging problem: integrating heterogeneous datasets to tackle data scarcity in scientific research.\n2. The methodology is well-motivated, and the Segment Quadtree data structure is innovative and interesting.\n3. The paper provides a comprehensive set of experiments, addressing multiple research questions and demonstrating the framework's effectiveness across various scenarios.\n\nWeakness:\n1. The baselines used in the experiments are really outdated. There are many more advanced baselines available.\n\n2. The paper lacks a thorough analysis of the framework's efficiency and scalability. For example, there is no discussion of computational complexity, training time, or memory usage, which are critical for real-world applications.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. What is the performance of the framework on more advanced baselines, both for time series models and spatio-temporal models?\n\n2. What is the computational complexity of the model? Can it generalize to larger datasets or higher-resolution grids?", "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": "8CLNzn6Dyr", "reviewer": "Reviewer_pADa", "summary": "This paper tackles the challenge of data scarcity in geographical scientific datasets for spatio-temporal forecasting by introducing a novel framework: the Segment Quadtree Geographical Embedding Framework (SQGEF). SQGEF integrates knowledge from diverse datasets that vary in granularity, time span, and observation variables. It learns unified representations for multi-granularity entities, including those not present during training. The key contributions of this work are threefold: (1) a novel method for integrating heterogeneous datasets; (2) a unique embedding approach for entities of different granularities; and (3) comprehensive experiments that demonstrate the method's effectiveness and robustness across various regions and granularities.\n\n## update after rebuttal\n\nThe author's detailed response has largely addressed my initial concerns, so I have updated my assessment to positive.", "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": "Thank you for your valuable comments on our work. We greatly appreciate your feedback, including the detailed suggestions for analysis and clarification. However, due to space constraints, we cannot include all the content in the main text. Since URLs are not permitted except for figures, we have included the updated PDF with detailed proofs in the supplementary materials (uploaded to an anonymous repository). All referenced appendices are provided. We sincerely hope you will review these materials and reconsider our score. \n\n# Benefits of Multi-granularity\nThe ablation study w/o SE empirically shows that combining multi-granularity datasets is effective, as performance drops noticeably when this component is removed. To shed more light, we explain below the intuition behind why this approach boosts forecasting performance.\n\nIn SQGEF, combining multi-granularity datasets enhances forecasting by allowing each Quadtree node to capture info from its own and higher granularity levels. This hierarchical integration occurs during training, where a node’s queried embedding is computed by **fusing its own embedding with those of all child nodes** stored in the tree, as shown in Equation 1. When a high-level node gets supervision from a coarse-granularity dataset, its child nodes are updated simultaneously. This top-down approach ensures lower-level nodes benefit from the richer, global info in higher-level datasets.\n\nThis mechanism provides key benefits. First, it lets lower-level nodes gain global interactions beyond their limited local data. Second, it supports cross-scale training: lower-level nodes are improved using higher-level dataset supervision, allowing the model to capture nested relationships across granularities. Thus, SQGEF builds a richer representation of geographical entities, enhancing generalization to unseen entities or data-scarce regions.\n\n# W1\nSQGEF introduces only constant additional computational complexity. During training, it maintains the same space and computational complexity as the naive method while capturing multi-granularity relationships for significant performance gains. During inference, SQGEF achieves lower complexity than the naive approach, ensuring superior scalability and efficiency.\n\nDue to space constraints, the detailed analysis is available in the complexity analysis (Rebuttal, Reviewer cJrD) or Appendix A.8.\n\n# W2\nWe have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here.\nFrom the Rademacher Complexity Theorem, for any $\\mathcal{H}$, the generalization error bound is $L(h) \\leq L^S(h) + 2\\hat{\\mathcal{R}}_S(\\mathcal{H}) + 3\\sqrt{\\frac{2\\ln(2/\\delta)}{n}}$, where $\\hat{\\mathcal{R}}_S(\\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\\hat{\\mathcal{R}}S(\\mathcal{H}{\\text{seg}}) \\leq C_2 \\cdot \\frac{|W|^{L+1} \\sqrt{p'} \\sqrt{m d}}{\\sqrt{n}}$, with $m = O(\\log k)$ and $p' \\propto m d$, giving a tighter bound for Segment Quadtree Embedding.\n\n# W3\nWe first clarify what data scarcity means in our work and then provide statistical evidence to support it. Due to space constraints, we have added detailed description in Appendix A.6.\n\nIn our paper, data scarcity refers to forecasting targets in the test set being unseen during pretraining, with no historical data available at that stage. This severely limits direct modeling of their relationships. To tackle this, SQGEF uses a large volume of heterogeneous datasets from related regions or entities, despite lacking test target historical records.To quantify this scarcity, we compare the data volume between the pretraining and test stages across three metrics: total data points, number of temporal points, and temporal duration. \n\n\n|Experiment|Pretrain Set Data Points|Test Set Data Points|Pretrain Set Time Points (Span)|Test Set Time Points (Span)|\n|-|-|-|-|-|\n|China Province|8,651,704|37,062|581 (22 years)|1,278 (2 years)|\n|Europe Country|8,650,752|39,618|528 (22 years)|1,278 (2 years)|\n|China City|8,687,814|962|1,806 (22 years)|37 (37 years)|\n\nFrom this table, it is evident that the test set contains significantly fewer data points than the pretraining set across all experiments. For China Province and Europe Country, the test set’s temporal duration is notably short(2 years). In the China City, test set has an extremely limited number of temporal records(37 points), indicating sparse sampling. Collectively, these statistics highlight the severe data scarcity in the test set, both in terms of quantity and coverage.\n\nMoreover, Table 2 in our paper shows that this scarcity goes beyond data volume to informational richness. Models using only the scarce historical data of forecasting targets consistently lag behind SQGEF, which combines this limited target data with richer, more plentiful data from related grids and entities during pretraining.", "claims_and_evidence": "The claims are partially supported. How combining multi-granularity datasets benefits forecasting performance seems to be missed.", "methods_and_evaluation_criteria": "The method is theoretically effective and feasible.", "theoretical_claims": "A rough theoretical analysis is given, while missing some details about calculation procedure.", "experimental_designs_or_analyses": "The experimental design is reasonable but could be improved.", "relation_to_broader_scientific_literature": "The idea of segment quadtree is novel to me. I haven't seen similar technology in existing literature.", "essential_references_not_discussed": "The coverage of references is comprehensive.", "other_strengths_and_weaknesses": "**Strengths:**\n\n- The idea of segment quadtree is novel. And the whole framework is technically sound.\n- The experiment covers different regions, data scarcity, and backbones. \n\n**Weaknesses:**\n\n- Additional time consumption of applying the proposed framework should be provided.\n- The theoretical analysis is sketchy. More details should be provided, e.g., the calculation of bounds.\n- The scarcity of different datasets should be given.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "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": "ztI6WD7ZNc", "reviewer": "Reviewer_y3tt", "summary": "This paper addresses the challenge of data scarcity in geographical scientific datasets for spatio-temporal forecasting by proposing a novel framework called Segment Quadtree Geographical Embedding Framework (SQGEF). The framework integrates knowledge from diverse datasets with varying granularities, time spans, and observation variables to learn unified representations for multi-granularity entities, including those absent during training. The key contributions include a novel method for integrating heterogeneous datasets, a unique embedding approach for different granularity entities, and comprehensive experiments showcasing the method's effectiveness and robustness across various regions and granularities.\n\n## Update after rebuttal: Thanks for author rebuttal and I have read the rebuttal. I think my concerns are addressed and will keep my score this time (for weak acceptance).", "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": "Thank you for your valuable feedback on our work. We greatly appreciate your insightful comments, including your detailed suggestions regarding the calculation of bounds, clarifications on data scarcity, and additional experiments. Due to space constraints, we are unable to incorporate all of this content into the main text. Since URLs are not permitted (except for figures), we have instead included an updated PDF with detailed proofs in the supplementary materials, which has been uploaded to an anonymous repository. All referenced appendices are provided in full. We sincerely hope you will review these additional materials and reconsider our score.\n\n# the calculation of bounds\n\nWe have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here.\nFrom the Rademacher Complexity Theorem, for any $\\mathcal{H}$, the generalization error bound is $L(h) \\leq L^S(h) + 2\\hat{\\mathcal{R}}_S(\\mathcal{H}) + 3\\sqrt{\\frac{2\\ln(2/\\delta)}{n}}$, where $\\hat{\\mathcal{R}}_S(\\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\\hat{\\mathcal{R}}S(\\mathcal{H}{\\text{seg}}) \\leq C_2 \\cdot \\frac{|W|^{L+1} \\sqrt{p'} \\sqrt{m d}}{\\sqrt{n}}$, with $m = O(\\log k)$ and $p' \\propto m d$, giving a tighter bound for Segment Quadtree Embedding.\n\n# W1\n\nThank you for your valuable suggestion! We first clarify what data scarcity means in our work and then provide statistical evidence to support it. Due to space constraints, we have added detailed description in Appendix A.6.\n\nIn our paper, data scarcity refers to forecasting targets in the test set being unseen during pretraining, with no historical data available at that stage. This severely limits direct modeling of their relationships. To tackle this, SQGEF uses a large volume of heterogeneous datasets from related regions or entities, despite lacking test target historical records.To quantify this scarcity, we compare the data volume between the pretraining and test stages across three metrics: total data points, number of temporal points, and temporal duration. \n\n|Experiment|Pretrain Set Data Points|Test Set Data Points|Pretrain Set Time Points (Span)|Test Set Time Points (Span)|\n|-|-|-|-|-|\n|China Province|8,651,704|37,062|581 (22 years)|1,278 (2 years)|\n|Europe Country|8,650,752|39,618|528 (22 years)|1,278 (2 years)|\n|China City|8,687,814|962|1,806 (22 years)|37 (37 years)|\n\nFrom this table, it is evident that the test set contains significantly fewer data points than the pretraining set across all experiments. For China Province and Europe Country, the test set’s temporal duration is notably short(2 years). In the China City, test set has an extremely limited number of temporal records(37 points), indicating sparse sampling. Collectively, these statistics highlight the severe data scarcity in the test set, both in terms of quantity and coverage.\n\nMoreover, Table 2 in our paper shows that this scarcity goes beyond data volume to informational richness. Models using only the scarce historical data of forecasting targets consistently lag behind SQGEF, which combines this limited target data with richer, more plentiful data from related grids and entities during pretraining.\n\n# W2\n\nThank you for your suggestion! This will help clarify the effect of different granularity datasets. We use only one grid dataset to train the model. As shown in the following table, using one single dataset significantly drops the performance, which verifies the validity of unifying various datasets.\n\n||MSE|MAE|\n|-|-|-|\n|Informer|0.7983|0.7036|\n|FEDformer|0.3628|0.4674|\n|Autoformer|0.3241|0.4518|\n|TimesNet|0.3985|0.5162|\n|AGCRN|0.8112|0.7142|\n|MTGNN|0.5518|0.5753|\n|GWNet|0.3984|0.4954|", "claims_and_evidence": "The claims are supported by experimental result.", "methods_and_evaluation_criteria": "The proposed method is technically sound.", "theoretical_claims": "The authors try to give theoretical analysis of the method, but more details are needed to make it clear and convincing, e.g., the calculation of bounds should be provided.", "experimental_designs_or_analyses": "The overall designs of experiment are acceptable.", "relation_to_broader_scientific_literature": "The Segment Quadtree is novel in the literature. I appreciate the interesting idea.", "essential_references_not_discussed": "The coverage of references is good.", "other_strengths_and_weaknesses": "**Strengths:**\n- Novel and interesting idea. The Segment Quadtree data structure provides a hierarchical representation of geographical regions, capturing multi-level interactions and geographical entity knowledge\n- The method shows improvements across various models, including time series and spatio-temporal models, and has potential applications in different scientific domains.\n\n**Weaknesses:**\n- Authors should provide the information about data scarcity of utilized datasets.\n- It will be better to compare the performance gains from combining data of different granularity compared to using a single data set.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "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": {"ethical_review_concerns": "N/A"}}, {"review_id": "AJUYRwl1iR", "reviewer": "Reviewer_1nAg", "summary": "Spatio-temporal datasets are often heterogeneous, each with its own granularity and representation of entities. Given the data scarcity issue in such datasets, it is crucial to integrate them effectively and learn unified representations for multi-granularity entities. The authors propose a framework to achieve this, called the Segment Quadtree Geographical Embedding Framework (SQGEF). Their approach leverages a quadtree data structure to embed entities of varying granularities. They evaluate SQGEF on three different datasets for spatio-temporal forecasting of carbon emissions and demonstrate that their method generally improves baseline performance for this task.", "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": "Thank you for your detailed and thorough feedback on our work. We greatly appreciate your insightful comments. Due to space constraints, we are unable to incorporate all of this content into the main text. Since URLs are not permitted (except for figures), we have instead included an updated PDF with detailed proofs in the supplementary materials, which has been uploaded to an anonymous repository. All referenced appendices are provided in full. We sincerely hope you will review these additional materials and reconsider our score.\n\n# W1\nThe idea of an indextree in references you given is to efficiently index spatial entities by hierarchically partitioning a region into proper granularity levels but coarser than entity level (containing multiple cities), implemented as a quadtree where child nodes separate the region of parent nodes, bring no additional information. Our work, neither motivated by nor following indextree, addresses the problem of using heterogeneous datasets to extrapolate knowledge to model unseen entities, a challenge indextree cannot solve due to its focus on summarizing known entities without extrapolation capabilities. To tackle this, we use finer granularity nodes (grids below city level), stores distinct information across parent-child levels from different granularity datasets (detailed discussion in Appendix A.9), and employs a segmenttree query method to fuse mixed-level embeddings, capturing different-level interactions. In summary, our Segment Quadtree is a new data structure designed for knowledge extrapolation, distinct from indextree’s indexing paradigm. \n\nWe listed more detailed foundational differences between indextree and Segment Quadtree are shown in Appendix A.7.\n\nThe \"segment\" in Segment Quadtree reflects its query method, inspired by segment trees, a data structure for efficiently computing range sums over continuous arrays. For example, in a segment tree with values (a1,a2, a3,a4), querying the sum of (a1, a2,a3) uses precomputed nodes like ([a1, a2]) and (a3), reducing computation. Similarly, in our Segment Quadtree in Figure 2(a), querying an yellow entity’s boundary fuses embeddings from mixed levels: one high-level node (Node3), one mid-level node (Node17), and two low-level nodes (Nodes69,71), rather than aggregating all lowest-level grids, as done for France in Figure 1(b). This multi-level fusion accelerates queries and leverages hierarchical information, distinguishing it from standard quadtrees like indextrees, where fusing across levels is less meaningful due to uniform node content.\n\n\n# W2\nDue to constraints in a paper, we cannot conduct experiments across a wider range of scenarios. Therefore, we focused on two key cases to allow for a more thorough analysis. We acknowledge this limitation and will explore additional applications of our method in future work. \n\nTo address the air pollution datasets description concern, we have improved the dataset description for air pollution in Appendix A.1.\n\n# W3\nWe have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here.\nFrom the Rademacher Complexity Theorem, for any $\\mathcal{H}$, the generalization error bound is $L(h) \\leq L^S(h) + 2\\hat{\\mathcal{R}}_S(\\mathcal{H}) + 3\\sqrt{\\frac{2\\ln(2/\\delta)}{n}}$, where $\\hat{\\mathcal{R}}_S(\\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\\hat{\\mathcal{R}}S(\\mathcal{H}{\\text{seg}}) \\leq C_2 \\cdot \\frac{|W|^{L+1} \\sqrt{p'} \\sqrt{m d}}{\\sqrt{n}}$, with $m = O(\\log k)$ and $p' \\propto m d$, giving a tighter bound for Segment Quadtree Embedding.\n\n# W4\nFollowing your recommendation, we shifted from visualization-based analysis to a quantitative evaluation of embedding distances on all embeddings. Specifically, we computed distances between all pairs of embeddings and compared them with the corresponding time series distances of carbon emission amounts. For the carbon emission time series, we calculated the L2 norm between every pair of provinces. Similarly, for each model, we computed the L2 distances between all pairs of embeddings. We then assessed the correlation between these embedding distances and the real carbon emission distances using the Pearson correlation coefficient. The results demonstrate that the Pearson score for TimesNet w/Seg embeddings is significantly higher than that of MTGNN.\n\n|Exp|TimesNet w/Seg|MTGNN|\n|-|-|-|\n|China Province|0.377|0.121|\n|Europe Country|0.480|0.343|\n|China City|0.247|0.226|\nResults indicate that our method better captures the structure of entities compared to MTGNN.\n\n# W5\nWe have written an related work section about Quadtree Applications in Spatial Data Management in Section 6.3.\n\n# W6\nWe have clarified the differences between the Index Tree and Segment Quadtree, added air pollution experiments, provided a detailed theoretical and complexity analysis, and quantative data scarcity in the original paper.", "claims_and_evidence": "The approach proposed by the authors is particularly useful for spatio-temporal datasets and could be applied to various applications. Their method appears to improve existing approaches for spatio-temporal forecasting. However, the authors claim to \"propose a novel data structure\" for spatial representation—specifically, segment quadtrees (see abstract). Quadtrees have long been used for spatial data representation, as evidenced by works such as Kothuri et al. (2002). Yet, the authors do not reference any prior work related to quadtrees.\nI would like to understand the rationale behind omitting such an important body of work. Additionally, what distinguishes a segment quadtree from a standard quadtree? I believe the authors should clarify this distinction in the text.\n\nKothuri, Ravi Kanth V., Siva Ravada, and Daniel Abugov. \"Quadtree and R-tree indexes in oracle spatial: a comparison using GIS data.\" In Proceedings of the 2002 ACM SIGMOD international conference on Management of data, pp. 546-557. 2002.", "methods_and_evaluation_criteria": "The method performs well on the three proposed downstream tasks. However, in my opinion, the experimental datasets are somewhat limited for evaluating this framework for two main reasons:\n- It focuses on a single spatio-temporal application—carbon emissions.\n- The three tasks are closely related, with two of them differing only by an inversion of the training and testing datasets, and the other two using different regions. \n\nIdeally, I would like to see an additional experiment with a different setup to better assess whether the method can be generally applied to various spatio-temporal modeling scenarios, as suggested by the title.\n\nAdditionally, the datasets and their spatial distribution are barely discussed in the main text and are only briefly covered in the supplementary materials.\n\nEdit: After reading the supplementary material, I noticed the experiment on the air pollution dataset, which is not mentioned in the main text. I believe this experiment should be given more emphasis and better described, including relevant references.", "theoretical_claims": "I find the theoretical analysis unclear. For example, it is not clear for me where the generalization error bound comes from or what the terms $C$ and $n$ represent. I believe more details should be provided, either through additional references or in the appendix.", "experimental_designs_or_analyses": "I believe Figure 5 is misleading and cannot be interpreted as the authors suggest. What do the two axes represent? Did the authors perform dimensionality reduction on the embedding space? If so, this should be clearly explained. Moreover, if dimensionality reduction was applied, I do not think it is appropriate to compare the embedding spaces of two models in this way, as distances between points, especially after reduction, may not be directly comparable. There should be more than two points per model to begin to understand the structure of the embedding space.", "relation_to_broader_scientific_literature": "As mentioned earlier, there is a lack of references and connections to previous works on using quadtrees for spatial representations. Additionally, the paper has a limited number of references (only 24), with no citations from the location encoding field.", "essential_references_not_discussed": "References to quadtrees for spatial representations are missing, such as:\n\nKothuri, Ravi Kanth V., Siva Ravada, and Daniel Abugov. \"Quadtree and R-tree indexes in oracle spatial: a comparison using GIS data.\" In Proceedings of the 2002 ACM SIGMOD international conference on Management of data, pp. 546-557. 2002.\n\nYin, Xiang, Ivo Düntsch, and Günther Gediga. Quadtree representation and compression of spatial data. Springer Berlin Heidelberg, 2011.", "other_strengths_and_weaknesses": "I believe the paper lacks clarity due to missing details. For example, in Equation (1), $e_{k,l}$ are introduced for the first time without any explanation of what they are and how they are obtained. Later, they are referred to as embeddings, but it seems to be never explained how they are derived or trained.\n\nHowever, the method may have significant value, and the setup considered is interesting to explore. It could also benefit many other applications.", "other_comments_or_suggestions": "Line 107: $N^{test}$ instead of $N$", "questions_for_authors": "I would like the authors to address the following points raised earlier:\n- Clarification and references to quadtrees\n- Lack of diversity in datasets considered and a more detailed dataset description\n- Improvement to the theoretical analysis section\n- Addressing potential issues with Figure 5\n- More references\n- General clarification of the text\n\nI am willing to increase my recommendation if the authors address these points, as the method has potential.", "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: The paper proposed a novel framework, the Segment Quadtree Geographical Embedding Framework (SQGEF), to address the challenge of data scarcity in geographical scientific datasets. SQGEF integrates knowledge from diverse datasets that vary in granularity, time span, and observation variables. The reviewers agree that the proposed ideas are interesting and useful. Some concerns related to experimental evaluations and paper presentations are addressed during rebuttal. The authors are suggested to incorporate these revisions in the manuscript.
Decision: Accept (poster)