Spatio-temporal Partial Sensing Forecast of Long-term Traffic
Authors: Zibo Liu
Zhe Jiang
Zelin Xu
Tingsong Xiao
Zhengkun Xiao
Yupu Zhang
Haibo Wang
Shigang Chen
Venue: Accepted by TMLR ()

Abstract: Traffic forecasting uses recent measurements by sensors installed at chosen locations to forecast the future road traffic. Existing work either assumes all locations are equipped with sensors or focuses on short-term forecast. This paper studies partial sensing forecast of long-term traffic, assuming sensors are available only at some locations. The problem is challenging due to the unknown data distribution at unsensed locations, the intricate spatio-temporal correlation in long-term forecasting, as well as noise to traffic patterns. We propose a Spatio-temporal Long-term Partial sensing Forecast model (SLPF) for traffic prediction, with several novel contributions, including a rank-based embedding technique to reduce the impact of noise in data, a spatial transfer matrix to overcome the spatial distribution shift from sensed locations to unsensed locations, and a multi-step training process that utilizes all available data to successively refine the model parameters for better accuracy. Extensive experiments on several real-world traffic datasets demonstrate its superior performance. Our source code is at https://github.com/zbliu98/SLPF

Haibo Wang Published in Transactions on Machine Learning Research (08/2025)
Spatio-temporal Partial Sensing Forecast of Long-term Traf- fic
Zibo Liu"
ziboliu edu zhe.jiang jiang edu zelin.ra@uuj ru edu riaotings ong Buff. edu rifff edu y. z.zhang Wuff. edu
Zhe Jiang
Zelin Xu
Tingsong Xiao
Zhengkun Xiao
Yupu Zhang
Department of Computer & Information Science Engineering, University of Florida
haibo Queee.iieee.on
Department of Computer Science, University of Kentucky
Shigang Chen
sgchen edu
Deportment of Computer & Information Science & Engineering, University of Florida
Reviewed on OpenReview: https: openeevee......e...em net/ forum? id=Ff0BaPjV.j
Abstract
Traffic forecasting uses recent measurements by sensors installed at chosen locations to forecast the future road traffic. Existing work either assumes all locations are equipped with sensors or focuses on short-term forecast. This paper studies partial sensing forecast of long-term traffic, assuming sensors are available only some locations. The problem challenging due to the unknown data distribution at unsensed locations, the intricale spatio- temporal correlation in long-term forecasting, well noise to traffic patterns. We propose Spatio-temppral Long-term Partial sensing Forecast (SLPF) model for traffic prediction, with several novel contributions, including rank- based embedding technique to reduce the impact noise in data, a spatial transfer matrix to overcome the spatial distribution shift from sensed locations to unsensed locations, and a multi-step training process that utilizes all available data successively refine the model parameters for better accuracy. Extensive experiments on several real world traffic datasetss demonstrate its superior performance. Our source code is at https://gii httpg/gttttu...../ wwt.://cwmmmmmmmm1.11..1
1 Introduction
Background: The traffic forecast problem is to use the recent measurements by sensors installed at chosen locations to forecast the future road traffic. This problem has significant practical values in traffic management and route planning, considering the ever worsening traffic conditions and congestion in cities across the world. Most existing work considers full-sensing scenario, where permanent sensors installed at all locations to be forecast. We observe that lower cost, more flexible solution should support partial sensing, where only some locations have permanent sensors, yet supporting traffic forecast. at other locations currently without sensors. The partial sensing research is in its nascent stage, with limited recent work on short-term forecast. This paper investigates the long-term partial sensing forecast problem that aims to train prediction model to use recent measurements from currently sensed locations to forecast. traffic al currently unsensed locations deep into the future.
Challenges:
The long-term partial sensing forecast problem poses several non-trivial challenges. Fig. shows the traffic rates at locations 121, 44, and 33 over two days from the PEMS08 dataset. Using
* *Corresondingg author

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Published in Transactions on Machine Learning Research (08/2025)
Fluctuations
500
Flow Rate
400
300
Location 121
Location 44
Location
08 Sas alle
19:00
08/20 03,00
23:00
07:00
11:00
15:00
19:00
23:00
03:80
12:112:00
07:00
08/21 15:00
11:00
08/20
08/20
08/20
08/20
08/20
08/21
08/11
0821
Time (5 mins each interval)
Figure 1:
Flow rates of three locations for two days
past traffic measurements at one location to infer future traffic another location requires not only the adaptability of a model to transfer the knowledge from locations to location based on the limited input features but also learning the intrinsic, subtle spatio-temperal connections across the locations. Next, long- term forecasting requires the learning of informative embeddings capable of capturing more intricate spatial and temporal correlations than what is required for short-term forecasting. We also observe high-frequency traffic fluctuations at the locations of Fig. | which are noises against the underlying traflic patterns. We will show that the impact of noise on traffic forecast accuracy much larger under long-term partial sensing than under traditional short-term full sensing.
Related Work: Most existing work on traffic forecast focuses on full sensing forecast, where all locations of interest are deployed with permanent sensors, including recurrent neural networks (RNNs) (Zhang et al.
2017),
convolutional neural networks (CNNs) or Multiple Layer Perceptrons (MLPs) with graph neural
networks (GNNs) (Yu et al. 2017 Li et al.
2018 Wu et. al. 2021b Li & Zhu 2021 J Jiang et al. 2023b
Yi et al.
2023 neural differential equations (NDEs)
(Fang et al. 2021
Liu et al. 2023c Bayesian neural
network MMio et al. 2022). transformers (Liu et al.
2023a Jiang et al. 2023a). mixture--f-experts (MOE)
(Shazeer al., 2017) based spatio-temporal model
(Lee & Ko 2024), and large language model (LLM)
base sptiao-tempors
model (Liu et al. 2024: 2025c a by. A few recent works study partial sensing forecast,
where only some locations deployed with permanent sensors, such as Frigate (Gupta et al., 2023 STSM
(Su et al. 2024), and GinAR (Yu et al.] 2024). But these methods are designed for short-term forecasting. The most related work is GinAR, which performs much better than Frigate, and unlike STSM, it does not require additional knowledge about location environment (such as nearby malls). But GinAR mainly focuses on missing observations and does not have a mechanism to address the impact of noise, and is therefore not suitable for long-term forecast, as we will demonstrate.
Spatial transfer learning methods (Mallick et al.) 2021
Tang ct al. 2022 Ouyang et al. 2023 utilize traffic measurements from data-rich locations to forecast the traffic at data-scarce locations. Data imputation
2023 interpolate the missing measurements at some locations where sensors may sometimes fail. They still require sensors at all locations. Spatial extrapolation methods, such as non-negative matrix factorization methods (Lee & Scung. 2000). and GNN methods (Wu et al. 2021a Zheng et al., 2023 Hu et al. 2023 Appleby et al. 2020), could be used to estimate traffic at unsensed locations basod on data from sensed locations within the same time period, but they do not model the complex spatio-temporal dynamics in long-term traffic forecast, and hence are not suitable for traffic forecast deep into the future.
methods (Li et al. 2023 Cini et al.) 2021 Chen et al.
Recent works have addressed related challenges from different perspectives, including inductive forccast- ing (ST-FiT [Lei et al., (025)), OOD generalization (STONE (Wang et al. 2024)), continual learning (URCL Miao et al.J 2024), and Expand-and Sraad--ad-Compress Compress (Chen & Liang] 2024)), ) and test-time adaptation (Learning with Calibration (Chen & Liang 2025)). ST-FiT assumes that the target sensors have no training data but all sensors are observed at inference, and focuses on short-term forecasting: our setting instead uses temporary sensors during training and assumes permanent missingness at test time, with long-term
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prediction. STONE models spatial and temporal distribution shifts via dynamic node sets, while set- ting assumes a fixed node set with partial sensor deployment. Fxpand-and-Cooppess introduces continual adaptation over time, whereas our framework considers a static deployment without continual evolution. Learning with Calibration performs online test-time adaptation, while our model remains frozen during in- lerence and aims for robust generalization without lest-lime updates. In contrast to these works, we are the first to study long-term forecasting under permanent partial sensing, where the spatial distribution shift is both large-scale and fixed deployment.
In summary, this is the first work specifically on the partial sensing long-term traffic forecast problem.
Contributions: We propose Spatial-temporal Long-term Partial sensing Forecast model (SLPF) for long-term trallic forecasting. SLPF has three main technical contributions. First, we design a rank-based node embedding, which makes the model more robust against noises in learning intricate spatio-temporal correlations for long-term forccast. Second, we propose a spatial transfer module, which aims to enhance our model'ss adaptability by extracting the dynamic traffic patterns from the transfer weights based on rank-based embedding in addition to spatial adjacency, allowing for more muanced and accurate predictions. Third, we use a multi-step training process to fully utilize the available training data. This approach enables successive refinement of the model parameters. Extensive experiments on several real-world traffic datasets demonstrate that our model out outperforms the state-of-the-arl and achieves superior accuracy in partial sensing long-term forecasting, Our source code is at bttps://githube com 2tt.1///b19
2 Preliminaries
2.1 Problem Definitions
Definition 1: Road Topology and Traffic Flow Rates. Consider a road system, and let N be set of chosen locations where traffic statistics are of interest. Let NN. The road topology of these locations is represented as a graph g = (N, A). A [Ai.]. i.j E N] E R=** is an adjacency matrix. Ar, that there is a road between location i and location j; otherwise A,,- = O. A traffic flow consists of all the vehicles that pass a location in N; the flow rale is defined as the number of vehicles passing through the location during a preset time interval. It a discrete function of time if we partition the time into a series of time intervals of a certain length, c.g.; 5 min. For simplicity, we normalize each time interval as one unit of time. Let M be subset of locations, i.e., M C N, and T be series of time intervals. As an example, T could be {t- [t-(+1, t 1,t) of intervals, where is the current time. The traffic matriz over locations M and times 7 is defined as XMrT i E M,j E 2'I, where Xi is the flow rate at location i during time j. Further denote the rate vector at location i as X, = [Kg..J E T). Hence, XAAT = [K, M).
indicates
Definition 3: Problem of Partial-Sensing Traffic Forecast. Suppose a subset M of locations are equipped with permanent sensors to continuously measure their flow rates. The subset of locations without permanent sensors is denoted as M' N M. Let m |M] and m' = [M'. The problem is forecast future traffic recent traffic matrix XM,, that has
matrix XM Th over the locations currently without sensors, based been just measured at the locations with sensors, where T" = {t + 1, ..., t + ('), T = {t - 1+ I,..., t t), and l is the current time. It is called partial-sensing traffic forecast if M C N and M o 0; it is full-sensinn traffic forecast if M- = N. In most existing work (Fang et al. 2021 Song et al. 2020 Jin et al.) 2022 Shao ct al. 2022a) Jia & Benson 2020 Ji et al. 2022), both and are set to hour for short-term forecast. Note that 7 should not be too large to avoid excessively large models and computation costs that come with them. Moreover, research has shown that too large may actually degrade forecast accuracy (Zeng et al. 2023). For long-term forecast, which is the focus of this work, is set much larger than l.
2.2 Embeddings
Embeddings are a common technique to enhance feature expression. We adopt the embedding setting in (Shao al. 2022a Liu al., (223aa), where the timo-of-day embedding and the day-of-weck embedding capture the temporal information, and node embedding captures the spatial property of locations. We use Flod IRmx Edow Rmxxd and Ey, E mmmx to denote them respectively, where d is the embedding
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dimension. A day is modeled as Drad 288 discrete units of 5 minutes each. A week has Wdow 7 days. The time-of-day is i/Diod for the ith unit in the day, and the day-of-week is i/Wdow for the ith day in the week. Trainable vectors Biad E R2 i E [0, Dlud), and B, E R2, i E [0, Wdow), are the time-of- day embodding bank and the day-of- week bank. In practice, for the input flow rate data AM.T E R T (1 I,..,, 1,4), we only consider the time feature of flow rate data XMA--411 E Rm al 1-1+1, yielding FM E Rmxdd EXP E Rmxdd For the node embedding, given any location E M, the trainable node embedding bank is denoted as B E IRd For the flow rate data XMr,T, we obtain E E Rmxd from the node embedding banks By = {B i M). Similar notations are defined over N (or M'), with M and in the above notations replaced by N and n (or M' and m').
3 Long-term Partial Sensing Forecast with Rank-based Node Embedding
3.1 Impact of noise
GinAR short-term full-sensing
30
GinAR short-term partial-sensing
GinAR long-term full-sensing
29
GinAR long-term pertial-sensing
MAE
28
26
Added Noise
Figure 2: Noise has the greatest impact on GinAR for long-term partial sensing than for long-term full sensing, which is in turn more sensitive to noise than traditional short-term full sensing.
We find that, compared to the traditional short-term full sensing forecast, the impact of noise is exaggerated for long-term partial sensing forecast, which therefore requires special attention in the model design to handle noise impact. The "noise" in this paper refers to high-frequency fluctuations in traffic, overlaid on the underlying daily patterns as shown in Fig. 1, which may be the result from natural traffic fluctuations from one lime interval to the next or sometimes from sensor misreadings. larger fluctuations, are harder forecast accurately. To demonstrate the impact of noise, we have performed traco-bascd experiments with the most related work, GinAR (Yu al.] (202)), on dataset PEMS08, where the detailed settings for all our experiments and description of the datasets can be found in Section 5.1 and the details on more noise-related studies can be found in Section 5.4 Below we only give the most relevant information for understanding the experiment results.
intuitive that noiser data, with
Among the 170 locations PEMS08, we randomly select half them for the sensed locations M and the other half for the unsensed locations M'. As the baseline for comparison, we use GinAR perform the traditional short-term full sensing traffic forecast amongst the locations in M only, i.e., forecasting XMr.T from XM,T, where both T and T' are one hour. And also conduct GinAR short-term partial sensing traffic forecast, ie., forecasting XM from XM,T. To quantitatively investigate the impact of noise, we add controlled amount of noise from zero-mean i.i.d. Gaussian distribution, N (000 ?2), to the datasct, where n² E (0, 10, 50, 100}, and 72 = 0 represents the scenario without additional noise. The experimental results are presented as the blue line (GinAR short-term full sensing) and the orange line (GinAR short-term partial sensing) Fig. 2 where MAE (Mean Absolute Error) is the average forccast error among all locations M. demonstrate that when the noise increase, the problem of short-term partial sensing forecasting is more challenging than short-term full sensing forecasting.
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Next we move to long-term full sensing traffic forecast, which still forecasts XM,r" from XM,T, but T is eight hours, while T remains one hour. The experimental results are presented as the green line (GinAR long-term full sensing) in Fig. 2 which shows that noise has a larger impact on forecast accuracy in the long-term case because the crror increases at a much faster rate as we increase noise. Finally we investigate the impact of noise on our new problem of long-term parlial sensing truffic forecusl, which forecasts XMr TT from XM,T, where M' is the unsensed locations, T" is eight hours and T is one hour. The experimental results are shown as the red line (GinAR long-term partial sensing), where the impact of noise on forecast accuracy is the largest and the error increases at the fastest rate when we increase noise. The results imply that long-tern teg-term partial sensing traffic. forecast is more sensitive to noise than long-term full sensing, which is in turn more sensitive to noise than traditional short-term full sensing.
The most related work, GinAR (Yu et al. 2024), does not have a component that is explicitly designed to moderate the impact of noise, which reflected in the experimental results above. This is where our work will come in, by incorporating a new component that helps reduce the impact of noise.
3.2 Rank-based Node Embedding
To handle the noise impact on long-term partial sensing traflic forecast, we introduce a new rank-based node embedding. Consider an arbitrary time interval j E T in the input XM,T. We assign a rank value to each location & E M as follows: Sort Xzj, Vi E M, in ascending order (with ties broken arbitrarily), and the rank of location k at time is the index number of Xa,y's position in the ordered list.
Consider the example of Fig. [ location 44 (middle red curve) exhibits a daily traffic pattern with high- frequency noises as the traffic fluctuates normally or sometimes due to inaccurate sensor rcadings. Suppose location 44 is a sensed location, whereas location 121 (top blue curve) is unsensed. Traffic fluctuations at location 44 and location 121 blur their patterns, making it harder to use a blurred pattern at location 44 to predict the noisy traffic at location 121, especially for long-term prediction where the intrinsic connection between the two locations becomes weaker and thus the impact of noise will be bigger due to a smaller "signal-to-noise" ratio in data. Comparing to the actual, fluctuating flow rates, the ranks are discretized values that tend to smooth out noise fluctuation and be more stable. Especially for partial sensing, the fewer the number of sensed locations is, the more stable the rank-based embedding becomes. The stability of the ranks could also enhance the model's adaptability from one spatial area (say, with sensor data) to a new area (without sensor data), whereby their actual flow rates can differ greatly but their relative ranks follow stable patterns. In other words, the mank-based embedding is more robust to the distribution shift from the sensed locations to the unsensed locations.
3.3 Long-term Partial Sensing Forecast Model
We present our Long-term Partial sensing Forecast model (LPF) with rank-based node embedding. At each time interval, we rank the flow rates at the locations in M. Given the i-th rank of the locations, vector BE E Rd denotes the rank-based node embedding bank for rank i. For the input traffic data. XM,T, we perform ranking for each time interval in T over M locations, yielding the rank-based node embedding Ey E mmxixd
A Multi-Layer Perceptron (MLP) processes the input data XM, to produce a feature embedding Fiy, E ma We then concatenate this with the other four embeddings: time-of-day embedding Eted, wac- embedding Eyow, node embedding Ei, rank-based node embedding EM: to form an aggregated feature E Rmxxdd This aggregated feature then passes through another MLP to produce a digh-dimensional repre- sentation H E Rmx6d.
Ey = MLP(XM.T)
(1)
HM. = || [E (Edg")
(2)
Next, we want to capture the latent correlations between sensed locations and unsensed locations and transfer the feature embedding from the sensed locations in M to the unsensed locations in M'. For that,
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introduce the node embedding enhanced spatial transfer matria, M', follows:
AM.M = AM.M + (FiM +
(3)
where T is the transpose of the matrix, and AM.M = [Ar.5, E M,j E M'] E Rmmxmm the partial adjacency matrix between locations in M and locations in M'. We enhance our model's spatial correlation by incor- porating the node embedding banks B& and the rank-based node embedding EYr, thereby producing more robust embeddings. Besides, the node embedding banks B% makes the enhanced spatial transfer matrix aware of the location properties. The rank-based node embedding Fi makes the matrix sensitive to the changes in traffic patterns, and thus enhances the model's adaptability. With this enhanced spatial trans- fer matrix, an MLP is then used to map the high-dimensional representation, A HMT,, to the traffic forecast output Ry
R = MLP((M,, - (HMT)
(4)
4 Additional Training Data
4.1 Inference v.s. Training
After a forecast model is trained and deployed, the permanent sensors at the locations in M will measure flow rates XM,T, which will be used input to the model to inference (forecast) future rates of the locations in M', ie., N
Before model deployment, when we learn such a model, we will need the ground truth of XA in the training data, which is unavoidable also for the related work including GinAR (Yu et al., 2024).. One way is to use mobile sensors that are temporarily deployed at the locations in M' for a period of time to collect the training data. This is reasonable because mobile sensors can be re-depolyed at future new locations or different cities for collecting the needed training data, i.e., the ground truth for the model output, offering a lower overall cost than implementing permanent sensors at all locations of all citics that need traffic forecast now or in the future.
With mobile sensors temporarily deployed to collect the traffic data at M', say, for three months, we naturally have both X and XM for that period of time. So, the training datasct will naturally contain XM,T and XM,T from the permanent sensors, and XMr,T and X r from the mobile sensors. During the model training phase, even though the input to the model must still be XMrT and the output be XA,,,, we should fully utilize the side information of XMO, T and XM,T which is available in the training data, to refine the model parameters.
4.2 Spatio-Tempont Temporal Long- Term Partial Sensing Forecast Model
We propose the Spatio-temporal Long-term Partial sensing Forecast model (SLPF) to utilize the additional training data, illustraatdd in Fig. 3 where the training phase (left plot) consists of three sequentially steps: 1. the dynamic adaptive step, which builds a module with XM.T input and X T expected output; 2. the long-lerm forecasling slep, which builds another module with XM.T and the previous module's oulput as input, and with XM,T as expected output; and 3. the aggregation step, which builds yet another module with XMrr and the previous two modules' output as input, and with XMr.T as expected output. The above three modules are trained sequentially, and we refer to them as the dynamic adaplive module, the long-term forecasting module, and the aggregation module, respectively, which together form the proposed SLPF.
4.3 Dynamic Adaption Step
The main objective this module to capture the latent correlations between the locations in M and the locations in M' and transfer the fcature embedding from M to M'. We first use XM through Eq. 12 to obtain the representation HMT.. Then, the node embedding enhanced spatial transfer matrix A as stated in Fq. 3 is used to transfer the knowledge from M to M where the node embedding banks B%
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Locations
M
M'
T
M,T
XM M'
Time
T"
X
AM' ,T
1: Dynamic Adaption Step from XM,T to NM
2: Long-Term Forecasting Step from XM,T XM &M,T Aggregation Step from XM,T, X XM, to N
Figure 3: The model Training phase consists of three sleps above. After the model trained and deployed, only XM is collected for input, and XA Tr output.
Bir, and the rank-bace based node embedding Eig are trained during this step. Finally, an MLP is used to map the high-dimensional representation, A'r,M M,T, an output &M T,,,, which the module output for the ground truth AMrT by the dynamic adaption step.
4.4 Long- Term Forecasting Step
The main objective of this module is to enhance SLPF's long-term forecasting power, by learning informative embeddings capable of capturing more intricate spatial and temporal correlations than what's needed for short-term forccasting. The input consists of AMæ and &M which is the output of the previous step. We first produce rank embedding enhanced spatial transfer matrix, similar to Eq. 3 as follows:
(5) where AN,M = [AL,, E N,J E M] E Rexm is the partial adjacency matrix between locations in N and locations in M, the rank embedding E% E Raxd is obtained from X [ &Mr,T, using the method
AN.. = AN,M + (EK + (xx)
described in Section 3.2 and B% and By are defined in Section 2.2
The representation HN, E gaxdd is
obtained from X'v.r E IR nx similar to Eq. 12 as follows:
E MLP(X
(6)
HN. = MI Ex (B
(7)
Finally, an MLP is used to produce EM of length 1', as the module output for the ground truth XMr.
= MAT==MPP'(AA) (N.T)
(8)
4.5 Aggregation Step
So far, the model has learned how to transfer knowledge from M to M' during the dynamic adaption step and how to perform long-term forecasting in the long-term forecasting step. This aggregation step aggregates them together for long-term forecast of XMr,T. Its input comprises three parts, XM.T, &M from the dynamic adaption step, and &MTr from the long-term forecasting step. Let &y be the concatenation of XMAT and &M We first produce two rank embedding enhanced spatial transfer matrices, similar to Eq. 3 as follows:
AN = AN.M + (E + (B 7)
(9)
NM = AM.M + (E + B
(10)

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Published in Transactions on Machine Learning Research (08/2025)
Table 1: Basic statistics of the datasets used in our experiments.
Dataset
PEMS03
PEMS04
PEMS0S PEMS-BAY
METR-LA
Area
in CA, USA
Time Span
9/1/2018
1/1/2018
7/1/2016
1/1/2017
3/1/20112
11/30/2018
2/28/2018
8/31/2006
5/31/2017
6/50/20012
Time Interval
5 min
Number of Locusions,
358
3077
170
325
207
Number Time Intervals
17,856
52,116
34,272
where AN,M' and AM,M are adjacency matrices, while Ey and Esr the rank-based embeddings. The representations, HN, Ruxdd and HMrr Rmxxdd are obtained from &N, and RMrT, respectively, similar to Eq. 2 Finally, an MLP is used to produce Rai of length I,, as the output for the ground truth Na
RM =0 MTP MLP' MA Hx.) (1 a)*
MLP'(AM HMT))
4.6 Complexity Analysis
We provide the parameter complexity and the time below.
Paramcter complexity: Let n be the number of locations (and ranks), d the embodding dimension size, D D= 5d the aggregated dimension size, and the number of MLP layers. The parameter complexity is:
O ((2n + Ndow + +Nod) d + n² + a D²)
Time complexity: SLPF has three stages: adaptation, forecast, and aggregation. The dominant terms are:
Ondp
m.I.D
+ m. D
m D
+ +mm D.I final MLP
imput emubedding
MLP encader
transfer matrix
O fore
D.I-D
D2
n n.m- n.m.D.
+ m' D-L'
imput embedding
MLP encoder
transfer matrix
final MLP
O Oagg
(n 1 m 1')-11 D embedding from previous predictions
(n m) D2 + (n + m) m D +' ml D. - transfer matrix
MLP encoder
final MLP
5 Experiment and Evaluation
5.1
Settings and Datasets
5.1.1
Datasets
We use five widely used public traffic flow datasets, METRA-LA, PEMSBAY, PEMS03, PEMS04 and PEMS08 ! The details of data statistics shown in Table | All datasets measure the flow rates, i.c., number of passing vehicles at each location during each 5-min interval. We split each dataset into three subsets in 3:1:1 ratio for training, validation, and testing. A traffic forecast model uses hour of flow-rate data to predict the future 8 hours of flow-rate data. Because the time interval for flow-rate measurement is five minutes, each hour has 12 intervals. We pre-process the flow rates by the z-score normalization beforc using them as model input. The z-score normalization subtracts the mean rate from each flow rate a dataset and then divides the result by the rates' standard deviation.
The datasets provided in the STSGCN github repository Sttps://githhn http:////ittata.ccaaaaaaaaa//s// wtt..//aviidaann///Ssee/ github repository htttts:/ggithub..... com/ hatm//lyyguang/DeeY
DCRNN

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Table 2: Performance comparison over five datasets, where n is the total number of locations and m' the number of unsensed locations. bascline model with suffix * is the model adapted for the partial sensing Lask. Our models are in bold at the boLtom.
PEMS0S
PEMS04
PEMS08
PEMSBAY
METRLA
Modall
039
51%
150,
58, 58.2%
250,
70.9%
150, (nn
72-4%
MAE
RMSEE
MAPE
MAM
RMSE:
MAPE
MAE
RSISE
MAPE
MAE
RMSE
MAPE
MAR
RMSE
MAPE
Maceix Factoriaacion
Semng
2000
110.17
105.41
01.08
150.00
122.65
76.01
182.64
6.25
15.63
20.21
15.72
34.21
PandiTST
3022
57.71
86.05
108.28
86.15
54.00
63.00
64.68
4.08
8.66
12.60
11.67
TTansfomee
20050
67.31
103.07
127.88
126.04
90.1
00.81
91.03
01.24
5.18
9.60
15.02
13.65
22.58
27.51
D28TGNK
Stao wl
2002g
$7.31
01.1
15.60
47.29
75.41
45.05
07.57
07.4
35.2
6.02
10.94
15.52
14.85
27.25
42.33
RoorieGNN
vi
35.55
52.00
60.03
43:46
60.47
45.09
64.07
02:06
91.13
5.85
7.50
10.87
7.80
14.51
36 11
STSGON
Sing m al
3000
28.57
15:00
39.17
32:70
09.07
97.03
31:74
40.00
21:47
5.12
0.48
1.87
0.77
9:57
15:76
STGODE
Bang ct
2001
29.78
38.00
32.00
27:88
42.45
25.03
33.01
01.20
24.00
5.15
0..21
7.78
6.48
11.40
19.70
ST SSL*
ct
XX61
24.58
42.61
36.15
27.71
44.28
22:07
30.992
47.65
22,23
5.1.
6.31
7.86
5.96
12.98
14.80
DyLISL*
2hoo 0.
20020
25.12
45.22
30.39
29:17
46.44
22.03
29.30
45.59
21.87
3.68
7.87
10.05
9.14
36.15
25.85
STID*
Stso el x
11774
21.70
39.07
21:36
24:17
40.70
17:14
21.00
12:07
15.42
5.00
5.98
7:43
00.00
14.00
19.68
MegaC RX
Jian ml
21.00
40:32
28:30
94:15
398.
18:15
29.41
47:57
20.58
5.09
0.20
7:42
5.50
10.30
18:42
STEP*
Shan et al
21.54
39:01
25.53
35.50
39:14
17:40
28.33
40:51
17:30
1.84
9:13
13:01
7.1
12.51
25.75
TESTAM
Les & Ko
2004
21.54
43:12
27.00
24.11
39.13
17:11
26.47
44.43
16.43
3.155
0.31
7.40
4:28
8.85
14.00
STLLM*
Lat ct m
2024
27,02
41.19
38.00
33.995
58.12
26.85
30.15
45.13
20,14
3.03
6,30
7.51
4.30
9.25
10.04
STLLM **
L2a ct.ctil
20200
27.50
44.98
20.01
31.992
48.20
26.93
30,003
45.19
21,76
3.02
6,28
7.47
4.59
9.30
10.10
STAEEFomee
Liu ml.
30230
22.47
41.60
26,00
23.11
38.32
16.35
23.31
41.20
14.63
2.72
5,38
6.50
4.07
8.43
13.00
STAEForass + 1GNNK Wa
2021.
21.01
42.12
27.32
23:07
30.21
17.09
23.50
43.15
16,32
2:80
5,50
6.89
4.32
8.78
14.21
FTAFFormer STONP
al.
43:15
16:39
13:37
41:30
14.90
2.73
5.00
0:35
6.19
13:20
DiTGGraph Chen
44.80
17.00
27,02
06.85
5.20
6.50
7.83
4.43
10.75
CmaAR Yu
44.05
24.95
19.00
26.91
47:13
3.01
7.50
15.10
LPF
21.24
37.10
25.01
22.01
38.99
17.01
23.91
11.19
15.01
2.64
5.34
6.35
3.71
7.20
11.03
SLPF
18.41
33.51
20.69
21.19
37.07
15.49
22.18
37.96
14.14
2.31
4.71
5.59
3.31
6.61
9.45
5.1.2 Hyperparameters
The numbers of sensed locations m and unsensed locations m' vary in our experiments, with m + m' = n. The number of time intervals in model input is 12, and the number I' of time intervals in model output is 96. For the embedding parameters, Ndoow 7, Aotod 288, and the dimension is d = 64. We use two layers of CNN with residual connection as the MLP structure in each of the three steps in Fig. 3 The input passes through one layer of CNN, the relu activation function (Agarap) 2018), the dropout layer (Srivastava et al., 2014h with 0.15 dropout rate, and then the second layer of CNN. A residual connection (Szegedy et al, 2017 of the original input then is added to the result for the final output. a in the aggregation step is 0.5. During training, we set the batch size at 64, the learning rate at 10-3, and the weight decay at 10-3 for all datasets. The optimizer is AdamW (Lohhhhlovv & Hutter 2019 the loss function.
We use Mean Absolute Error (MAE) as
5.1.3 Baselines
The baselines for comparison can be found in Table 2 for partial sensing forecast by adding fully connected layers to map the feature dimensions from the sensed locations to the unsensed locations. For the short-term forecast models, the output length is increased from hour to hours. These adapted models are identified with a suffix * For the spatial extrapolation models, in order to use them in the context of partial sensing traffic forecast, we need to first apply a full sensing forecast model (such as STAEFormer (Liu et al. 2023a)) that uses XM,T to forecast XM,T' and then use the spatial extrapolation models to estimate Xas T from XM.T.. We have excluded from our comparison the models that either performed very poorly (such as Frigate (Gupta et al. 2023) which performed worse than all above baselines) or could not be adapled in our experimental setting (such as STSM (Su et al., 2024) which uses the environmental knowledge of each location such as nearby shopping malls). For all baselines, we exccuted their original codes and adhered their recommended configurations. The reported results represent the average outcomes from four separate experiments with four random sceds, where under the same random seed.
We adapt the existing full-sensing forecast models
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Table 3: Different ablation methods under weighted selection and m' condition on PFMS08 dataset over the average horizon.
Method
MAE
RMSE
MAPE
SLPF
16.58
29.22
17.85
LPF step, see Fig. 3 from Xay. to Maryee)
18.47
32.08
18.83
Step (Irom Astr Xar-r. rr
Nuc., Aap-or to Xan.r)))
17.21
Plinn Spatial Transfer Matrix
17.80
32.35
17.87
No Spatiel Transfer Matrix
17.78
32.35
17.85
hase- Embeedding
5.1.4
Location Selection
We use two methods for the selection of sensed/unsensed unsensed locations: random selection and weighted selec- lion. In random selection, the sensed/ gensed/unsensed /unsessedd locations are randomly chosen. In weighted selection, the probability of each location to be sensed is proportion to its flow rate.
5.1.5 Accuracy Metrics
We evaluate the traffic forecast accuracy of the proposed work and the bascline models by Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE). Given
m
the ground truth Kiy and the forecast result Rs.j, E M', T MAE at time j mm Σ
Xi.
Ras
m Ki
R Ris
) * 100%, and RMSE at time j = N
m Ki. - R15)
m' Σ 1=1
MAPE at time j == mn
5.2 Traffic Forecast Accuracy
Table 2 compares methods, LPF and SLPF, with the baselines traffic forecast accuracy in terms of average MAE, MAPE, and RMSE over 96 future intervals (i.c., 8 hours). All experiments were conducted using the weighted selection method. The number of unsensed locations is set to 250 for larger datasets in- cluding PFMS03, PEMS04 and PEMSBAY, and 150 for PEMS08 and METRLA, which have fewer locations. The percentage of locations that are unsensed ranges from 69% to 88.2%. The table shows that our model outperforms all bascline models in traffic forecast accuracy, for example, achieving 15.5% improvement by LPF and 28% improvement by SLPF in MAPE against. the best. baseline result over the METRLA dataset.
The long-term traffic forecast models, PatchTST* and iTTrassfrmmr** do not sufficiently model spatial corre- lation patterns, thus resulting in relatively poor performance. Among all the baseline models, STAEFormer* - which is a spatio-temporal short-term forecast model - achieves the best overall performance, benefiting from its sophisticated embodding techniques (c.g., node embedding, time-of-day and day-of-wock embed- ding), coupled with a robust transformer structure. TESTAM* performs close to STAEFormer*, thanks to its informative MOE architecture. The spatial extrapolation models, IGNNK and STGNP, were originally designed for extrapolation to relatively modest extent, such 30,, in contrast to experimental setting of extrapolation 69-88.29 88% of locations that are unsensed from only 12.8-31% of locations that are sensed. In addition, they use STrrFrrmer for extrapolating in the temporal dimension from one hour data to eight hours data. In comparison, our model LPF performs better than all baselines as expected because this is the first work designed specifically for the problem of long-term partial sensing traffic forocast with a noise miligating component of bank-based embedding. SLPF performs betier than LPF with novel three-steps model design to take advantage of additional training data.
5.3 Ablation Study
To investigate the effect of different components of SLPF, we conduct ablation experiments on PEMS08 with several different variants of our model. LPF: It does not use the additional training data. in the training
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phase. 2-Steps: This variant of SLPF removes the long-term forecasting step but keeps the other two steps. It first produces an estimate of Aw,, in the dynamic adaption step and then integrates XM, and the estimate of XM in the aggregation step to forecast XM,,. Plain Spatial Transfer Matrix: We retain SLPF's 3-steps structure, but use a learnable matrix to add up with the partial adjacency matrix, instead of the node embedding bank B and the rank-based node embedding E". No Spatial Transfer Matrix: This variant excludes the spatial transfer matrix of SLPF. Instead, it employs a single linear layer of MLP with shape of MMM to map the input sensor dimensions to the desired output dimensions, akin to the method used for adapting the full sensing models for the partial sensing task. No Rank-based Node Embedding: We remove the rank-based node embedding, while keeping all other embeddings as well as the 3-steps structure.
Table 3 compares SLPF, LPF, and other variants above in terms of average MAE, MAPE and RMSE over 8-hour traffic forecast on dataset PEMS08, under the weighted selection of unsensed locations with m' = 50. Comparing SLPF (3 steps) with LPF (1 step) and 2-Steps, we can see that our 3-steps training process keeps improving forecast accuracy as more supervised information is used. Comparing SL.P with No Rank-based Node Embedding, we show that the rank-based node embedding improves the average forecast accuracy, The significance of the ranked-based embedding will be further revealed in the next subscction when we take deeper look at its impact. Comparing SLPF with Plain Spatial Transfer Matrix and No Spatial Transfer Matrix, we show that our node embedding enhanced spatial transfer matrix is more effective than simple or no spatial transfer matrix. From Table 3 as each variant disables one design component of SLPF, we observe about 1-2% MAE degradation, which non-negligible in this line of research, as seen from Table 3. Hence, from all above variants, the ablation study justifies the importance of all those design components, which contribute to SLPF in different ways: rank improves robustness against noise, transfer matrix enhances spatial generalization, and temporal embeddings capture periodic patterns, etc.
5.4 Rank-based Node Embedding
Location 25
Location 117
400
Flow Rate
Flow Rate
250
200
150
SLPF wo. Rank
wo. Rank
SLPF
SLPF
50
Ground Truth
Ground Truth
100
08/19 14:00
18:00
22:00
02.00
06:00
10:00
14:00
18:00
22:00
08/21 02:00
88/21 06:00
08/19 14:00
18:00
22:00
02.00
06:00
10:00
14:00
18:00
22:00
08/21 02:00
08/21 06:00
08/19
08/19
08/20
08/20
08/20
08/20 08/20
05/20
08/19
08/19
08/20
08/20
08/20
06/20
08/20
08/20
Time (5 mins)
Time (5 mins)
Figure 4: Visualization on different locations when our model with without ranking embedding.
We begin with two case studies to visualize the impact of rank-based node embedding. Fig. 4 shows the actual flow rates (blue), the forecast rates of SL.PF (red), and the forecast rate of SLPF without. rank embedding (gray) at Location rank embedding allows the red line of SLPF to track the underlying pattern by smoothing out the noise (traffic fluctuations) of the actual rate curve, outperforming the gray line of SLPF without rank embedding. We hypothesized in Section 3.2 that node ranks, which are essentially the normalized, discrete flow rates at the nodes, are more stable input to the model than the actual rates, which fluctuate. Hence, rank embedding is more resistant. against noise in traffic, and thus helps capture the underlying patterns better. Below, we to verify our hypothesis.
and Location 117 in the PEMS08 datasct. It clearly demonstrates that
use experiments (Fig.
To quantitatively study the robustness of amounts of zero-mean i.i.d. noise to the training dataset, considering three commonly used distributions:
node embedding against noise, we add controlled
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m 50
m'= 100
m'= 150
SLPF
SLPF
42.5
42.5
Without Rank
42.5
40.0
40.0
40.0
RMSE
RMSE
RMSE
35.0
32.5
32.5
32.5
SLPF
30.0
30.0
30.0
Without Rank
100
100
50
100
Gaussian noise
Gaussian noise
Gaussian noise
Figure 5: Performance comparison with varying y under Gaussian noise on PFMS08 dataset.
m's 50
m's
100
m'= 150
45.0
45.0
45.0
42.5
Without Rank
42.5
Without Rank
42.5
40.0
RMSE
37.5
37.5
RMSE
37.5
35.0
35.0
35.0
32.5
32.5
32.5
SLPF
Without
0
50
100
0
50
100
0
50
100
Uniform noise
Uniform noise
Uniform noise
Figure 6: Performance comparison with varying y under Uniform noise on PEMSO8 dataset.
== 50
m'= 100
m'= 150
45.0
45.0
45.0
SLPF
42.5
Without Rank
42.5
42.5
40.0
40.0
40.0
37.5
37.5
RMSE
37.5
35.0
35.0
35.0
32.5
32.5
32.5
SLPF Without Rank
SLPF
30.0
30.0
30.0
Without Rank
50
100
100
100
Laplace noise
Laplace noise
Laplace noise
Figure 7: Performance comparison with varying y under Laplace noise PEMS08 dataset.
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Gaussian noise 22);
x²2
p(x)
exp
2
27²
Uniform noise (-),))):
i E 7-
p(a) =
27
0,
otherwise
Laplace noise Laplace(0,
p(x) 27 exp
For all three distributions, the noise magnitude is controlled by a shared scale parameter Y where y E {0, 10,50, 100). The case 7 = corresponds to the scenario without additional noise.
The experimental results under these noise types are presented in Fig, Fig. and Fig, Across all cases, SLPF without rank-based node embedding suffers significant performance degradation the noise level increases. In contrast, SLPF with rank-based node embedding consistently demonstrates strong robustness, exhibiting much smaller performance drops under increasing noise levels, regardless of the noise distribution.
5.5 Efficiency Study
Accuracy and Efficiency Trade-off D2STGNN
FourierGNN*
RMSE
STSGNN
STGODE
MegaCRN* TESTAM-
DyH5L* BiTGraph
STEP PDFormrr*
STAEForarriGGNIK GİNAR
STAEFormer STÄEFOrmer+STONP
LPF
SLPF
30
0
50
100
150
200
250
Training Time (mins)
Figure 8: Comparing our model SLPF with the baselines in training efficient, in the context of accuracy- efficiency tradeoff, on dataset PEMS08, with weighted selection and m' 50.
Fig. S compares our models, LPF and SLPF, with other models in the context of efficiency-accuracy tradeoff. Experiments were conducted on server with AMD EPYC 7742 64-Core Processor @ 2.25 GHz, 500 GB of RAM, and NVIDIA A100 GPU with 80 GB memory. The batch size uniformly set to 16. We record the total training times and the average RMSE values over hours of traffic forecast on dataset PEMS08 with m' - 50. Our models achieve the best RMSE among all models with relatively low computational cost. This dual advantage of high accuracy and efficiency sets models apart in the realm of partial sensing traffic fore- cast. On the one hand, STID*, FourierGNN STGODE* and STSSL* exhibit commendable efficiency, yet they fall short in achicving the accuracy level demonstrated by our models. On the other hand, MegaCRN* GinAR, BiTGraph, STSGCN*
TESTAM* STAFFormer* STAEFormer+IGNNKS STA STAEForner+STGN DyHSL* * D2STGNN* STEP*, and PDFormer* show slower training speed and worse accuracy than our models. The balance of efficiency and accuracy makes our models attractive for real-world applications intelligent traffic management. Between LPF and SLPF, the former less accurate but more efficient, while the latter is more accurate but less efficient, representing accuracy-efficiency tradeoff.
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35
Improvement (%)
25
15
5
0-5%
5-10%
10-15%
15-20%
20-25%
25-30%
30-35%
35.40%
40 40-45%
5% forecasts with descending MAE
Figure 9: Improvement of SLPF over its variant without rank in bins of 5% forecasts with descending MAE, on dataset PEMS08 datasct with weighted selection and m' 50.
5.6 Worst-case Improvement
With rank-based embedding to handle the impact of noise, particularly, large noise, we expect to observe more worst-case accuracy improvement than the average-case improvement in Table g To verify this hypothesis, we sort. all hours of traffic forecasts the descending order of MAE, and then group them into bins, each with 5% of all forecasts. We compute the average MAE in each bin for SLPF, do the same for its variant without rank, and then compute the percentage improvement in average MAE of SLPF over its variant without rank. The results are presented in Fig. g where SLPF consistently improves over its variant without rank over all bins shown. The improvement amongst the first 5% percent forecasts (i.e., worse case) is close to 35%, where the largest irregularities are suspected to reside.
5.7 Number of Unsensed Locations and Random Selection vs Weighted Selection
Full Sensing Benchmark
STAEFormer* FourierGNN
DyHSL* SLPF
STID*
PDFormer*
ST-SSL*
Random Selection
Weighted Selection
45.0
56.68
45.0
46.50
62.66
42.5
42.5
RMSE
37.5
37.5
35.0
35.0
32.5
32.5
27.5
27.5
50
100
150
50
100
150
Number Unsensed Locations
Number of Unsensed Locations
Figure 10: Accuracy comparison in terms of RMSE with respect to the number of unsensed locations under different selection methods on PEMS08 dataset.
Fig. 10 compares our model SLPF with the baselines on forecast accuracy in terms of RMSE over dataset PEMS08 with respect to a varying number of unsensed locations from 50 to 100 to 150. The left plot uses the random selection method, and the right plot uses the weighted selection method. In cach plot, for each number of unsensed locations on the horizontal axis, a group of bars present the average RMSE values of the Full Sensing Benchmark (leftmost), STID* and our method SLPP (rightmost)), respectively.
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The Full Sensing Benchmark tries to establish some sort of accuracy bound that partial sensing model such as SLPF can possibly achieve. We reason that partial sensing model without the recent. knowledge of unsensed locations, i.e,, AM,,, should not beat a full sensing model with the knowledge of XMr. This knowledge of unsensed locations is hypothotical, and thus the Full Sensing Benchmark is not implementable, but it gives us an accuracy bound for partial sensing model. Table | shows the forecast. accuracy of the full-sensing baselines on dataset PEMS08, assuming the knowledge of AM such that they can use XN, infer AMrT, , where N M M'. Among the full-sensing baselines, PDFormer performs the best in RMSE and thus its results are used as the Full Sensing Benchmark in Fig. 10
Table 4: Performance comparison on PEMS08 dataset among the full-sensing baselines with the hypothetical assumption that they have the knowledge of the unsensed locations. The best results are in bold.
PEMS08
Models
MAE
RMSE
MAPE
STID Shao sl.
2022A
19.39
32,20
13.05
PDPurmner Jiang al.
2023a
18.58
31.86
12.64
STAEFormer
Liu et al 202M
18.77
31273
12. 10
TESTAM
Loe & Ko
2024
18.90
32.81
12.45
MegaCRN Jaang ct al.
20233
19.99
33.67
13.09
STEP
Shao es al.,
2022h
18.97
32.98
12.66
D2SIGNN Shao
16.13
GourlerGNN
Yi al
2023
62.66
38.70
ST-SSL Ji al. 2012
24.35
37.31
17.59
DyHSL Zhao et sl.
2023
26.01
39.28
20.09
Note that (1) the vertical axis of Fig. 10 RMSE, begins from 27.5, not zero, and (2) ends at 45.0, while the higher bars (which go beyond the figure) have their heights shown at the top of the figure.
First, the figure shows that the choice between the random selection method and the weighted selection method does not cause significant difference in forecast accuracy. It suggests that preference of installing sensors at locations of larger flow rates does not improve performance significantly over random selection of sensor locations.
Second, our model SLPF outperforms the basclines under different numbers of unsensed locations, consistent with the results in Table 2 Although the performance of SLPF is worse than the Full Sensing Benchmark (as is expected), when the ratio of unsensed locations to all locations is relatively low, e.g., m' 50 and m' /n- 50/170- = 29.4%, the accuracy of SLPF is very close to the Full Sensing Benchmark. Even when the ratio of unsensed locations to all locations is high, e.g., m' = 150 and m' /n 150 = 88.2% SLPF is about 10% less accurate than the Full Sensing Benchmark. This comparison demonstrates the validity of the partial sensing approach in traffic forecasting: By installing a significantly smaller number of permanent sensors (e.g., m = n m' = 170 - 150 = 20), we can achieve long-term traffic forecasting accuracy within about 10% of what the full sensing models can achieve under much more expensive deployment of installing sensors at all locations.
5.8 Forecast Length
Our partial sensing long-term traffic forocast experiments output traffic estimates for 8 hours (i.c., 96 time intervals) into the future. The forecust length refers a specific time interval in the output. For example, when we consider forecast length of 1 hour (i..,, the 12th interval), we refer to the traffic estimates for the last interval of the first hour into the future.
Fig. compares our method with the baselines and the hypothetical Full Sensing Benchmark on forecast accuracy in terms of RMSE over dataset PEMS0S with respect to a varying forecast length of 1, 2, 4, and 8 hours. The figure shows that our method significantly outperforms all basclines at different forecast lengths. This result complements the average accuracy comparison in Table by providing more detailed comparison in forecast accuracy over time. Interestingly, the accuracy of method converges towards its accuracy
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Study of Forecast Length
50
Full Sensing Benchmark
STID
STEP*
45
MegaCRN*
PDFormer*
RMSE
40
STAEFormer"
35
ST.SSL"
DyHSL*
TESTAM
30
SLPF
25
2
Hours into the Future
Accuracy comparison in terms of RMSE with respect different forecasting lengths on PEMS08 dataset with the random selection method and the number of unsensed location being m' = 50.
Figure 11:
bound of the hypothetical Full Sensing Benchmark we stretch the forecast length from hour to hours, suggesting the suitability of SLPF long-term traffic estimator.
5.9 Effectiveness of Node Embedding
44
SLPF
Without Rank
40
RMSE
36
32
50
75
100
125
150
number of unsensed locations
Figure 12: Impact of number of unsensed locations on forecast accuracy (RMSE) of SLPF when it with or without based node embedding, on PEMS08 dataset with weighted selection.
The Choice of
Choice of dimension
38.5
38.4
38.3
RMSE
RMSE
38.3
38.2
38.2
38.1
38.1
38.0
38.0
0.4
size of dimension d
Figure 13: Parameter sensitivity of the proposed SLPF on the average horizon of PEMS08 with weighted selection and m' 150.
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Extending the comparison between SLPF and No Rank-based Node Embedding (i.e., SLPF without rank) in Table 3 we present additional RMSE results in Fig. 12 with varying number m' of unsensed locations from 50 to 150. It shows that the performance gap between SLPF and its variant without rank increases as the value of m' increases. This demonstrates that rank-based node embodding is more beneficial when the number of unsensed locations high.
5.10 Parameter Sensitivity
We study the parameter sensitivity of our model SLPF with two experiments on the average RMSE over 8 hours of traffic forecast on dataset PEMS08 with m' 150. One parameter is a, the ratio factor of the two parts in the aggregation step; the other the embedding dimension d. From Fig. 13 the RMSE at 0.5 is slightly better than the RMSEs at other values. The RMSE at d 64 is slightly better than the RMSEs at other d values. The reason is that when the dimension d is smaller than 64, SLPF may not fully capture the complex spatio-tempore patterns; when d is larger than 64, SLPF may suffer from overfitting.
6 Conclusion
This paper proposes a novel long-term partial sensing forecast model (SLPF). It has a multi-step train- ing/ ing/inference structure that enables progressive refinement of model parameters. It introduces a new rank- basod node embedding to handle the impact of traffic noises. uses the node embedding enhanced spatial transfer matrix to enhance data representation across varying phases by shared parameters.
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20


Reviews:
[{"review_id": "dyYIJtdjUB", "reviewer": "Reviewer_yA4H", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n+ This work presents SLPF, a long-term traffic forecasting model with partial sensor coverage, where only a subset of locations is equipped with sensors.\n+ Extensive case studies.\n\nWeaknesses:\n- Different noise models need to be considered to further verify the performance of SLPF.\n- The time complexity and space complexity analysis of the algorithm need to be supplemented.\n- The impact of different numbers of missing sensors on the performance of SLPF needs to be further explored.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the challenging problem of long-term traffic forecasting with partial sensor coverage, where only a subset of locations are equipped with sensors. To tackle issues like unknown data at unsensed locations, spatio-temporal complexity, and noisy traffic patterns, the authors propose a novel model called Spatio-temporal Long-term Partial sensing Forecast. Key innovations include a rank-based embedding to mitigate noise, a spatial transfer matrix to handle distribution shifts between sensed and unsensed locations, and a multi-step training strategy that progressively refines model parameters using all available data. Extensive experiments on real-world traffic datasets show that the proposed model significantly outperforms existing methods in forecasting accuracy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Comments:\n\n- From Table 3, I do not seem to see that different modules in SLPF have a significant impact on SLPF performance, because their experimental results are relatively close. Could the authors provide more explanation?\n\n- From Figure 4, it seems that SLPF performs worse as the number of m' increases. So the authors need to explain at least two things: 1) how the performance of other baselines changes with the change of m'; 2) this paper considers the problem of missing sensors, but the results do not seem to solve this problem well.\n\n- This paper only uses the Gaussian noise model to verify the robustness of SLPF. I would like to know how SLPF performs under other noise models.\n\n- Considering the complexity of the SLPF structure, it is necessary to analyze the algorithm complexity theoretically.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "nAdISoiccb", "reviewer": "Reviewer_Cwax", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths \n\n1. **Comprehensive Experiments**: Extensive experiments are conducted on multiple real datasets, including datasets of different scales and geographical locations, verifying the universality and effectiveness of the model.\n\n2. **Good Robustness**: The stability of the model under different noise levels is demonstrated through comparative experiments, proving its potential in practical applications.\n\n---\n## Weaknesses\n\n1. **Lack of Related Work**: The motivation of this paper actually comes from the challenge of spatiotemporal OOD generalization, so the author lacks discussion and comparison of some related works. I list some key works for the authors to read. [1] gives an inductive setting for spatiotemporal learning. [2] systematically explains the OOD generalization setting for spatiotemporal learning, [3] systematically explains the continual learning setting for long-term dynamic evolution of spatiotemporal learning, and [4] further systematically summarizes these works and provides another perspective. In addition to these works, some baseline related works in their paper need to be discussed.\n\n[1] Lei, et al. ST-FiT: Inductive Spatial-Temporal Forecasting with Limited Training Data. AAAI, 2025.\n\n[2] Ma, et al. STONE: A Spatio-temporal OOD Learning Framework Kills Both Structural and Temporal Shifts. KDD, 2024.\n\n[3] Chen, et al. Expand and Compress: Exploring Tuning Principles for Continual Spatio-Temporal Graph Forecasting. ICLR, 2025.\n\n[4] Chen, et al. Learning with Calibration: Exploring Test-Time Computing of Spatio-Temporal Forecasting. arXiv, 2025.\n\n2. **High computational complexity**: Multi-step training and multi-module design may lead to high consumption of computing resources, limiting its application in large-scale real-time systems.\n\n3. **Lack of theoretical analysis**: Although ranking embedding is proposed, the explanation of how it specifically affects spatial transfer and long-term prediction is not in-depth enough.\n\n4. **Lack of analysis of long-term stability of the model**: The possibility of parameter drift or performance degradation of the model during long-term operation and the coping strategies are not discussed in detail.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a **Spatio-temporal Long-term Partial Sensing Forecast (SLPF)** model for long-term traffic flow forecasting. The main contributions include: a ranking-based node embedding technique is proposed. Combining node embedding and ranking-based embedding, the model's knowledge transfer ability from sensor-based areas to sensor-free areas is enhanced, effectively solving the problem of spatial distribution offset. Using three steps of dynamic adaptation, long-term prediction and aggregation, the model parameters are gradually refined, the training data is fully utilized, and the prediction accuracy is improved.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. **Model complexity optimization**: It is recommended to explore how to reduce the computational complexity while maintaining model performance in subsequent work, such as through model compression or more efficient network structure design.\n\n2. **Enhanced interpretability**: It is necessary to further analyze the specific impact mechanism of ranking embedding on spatial transfer matrix and long-term prediction, and increase the transparency of the model.\n\n3. **Expand the scope of experiments**: It is recommended to conduct experiments on more diverse traffic datasets (such as LargeST Datasets) to verify the universality of the model.\n\n4. **Discussion and comparison of related work**: Supplement the comparison of competing work and the discussion of related work.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "pJYawDDWmu", "reviewer": "Reviewer_PP6v", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "+  The paper defines a practical and challenging problem. Most prior work either assumes full sensor coverage or focuses only on short-term predictions.  Addressing long-term forecasting with partial sensing is a valuable and timely contribution, given the high cost of deploying and maintaining permanent sensors everywhere.\n+ The three-step training process for the SLPF model is a smart way to utilize all available data during the training phase, where ground truth for unsensed locations might be temporarily available.\n+ The experimental evaluation is comprehensive. The authors test on five standard datasets , compare against a large number of appropriately adapted baselines , and provide detailed ablation studies to validate each component of their model.\n\n- The SLPF model's superior performance hinges on a multi-step training process that requires ground truth data for the unsensed locations during training.  The authors suggest using temporary mobile sensors for this purpose.  While they argue this is cheaper than permanent installation, this assumption represents a significant logistical and financial hurdle for real-world deployment.\n- The specific formulation of the \"node embedding enhanced spatial transfer matrix\" in Equation 3 feels somewhat ad-hoc. While the ablation study shows it works better than simpler alternatives, the theoretical or intuitive justification for this exact structure is not explored.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces and tackles the problem of \"long-term partial sensing traffic forecast\", which aims to predict future traffic flow for many time steps ahead at locations without sensors, using only recent data from a sparse set of sensored locations.  The authors propose two models: LPF, and its more advanced version, SLPF. The key contributions include a rank-based node embedding designed to mitigate the impact of high-frequency noise in traffic data , a \"node embedding enhanced spatial transfer matrix\" to capture correlations between sensed and unsensed locations , and a multi-step training process for SLPF that leverages additional data available only during training (e.g., from temporary sensors) to progressively refine the model.  Extensive experiments on five real-world datasets show that the proposed models, particularly SLPF, outperform a wide array of adapted baselines in terms of accuracy and offer a good trade-off between accuracy and training efficiency", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The description of the experiment in Section 3.1 appears to contradict the legend in Figure 2. For instance, the text states, \"we use GinAR to perform the traditional short-term full sensing traffic forecast... The experimental results are presented as the blue line in Fig. 2\", but the figure legend labels the blue line as \"GinAR short-term partial-sensing\". \n2. The high-level description of the three-step training in Section 4.2 could be clearer. The paper states that \"Each step leverages the module parameters derived from its previous step(s) ...\". It is unclear how this is done. \n3. For the \"No Spatial Transfer Matrix\" ablation variant, the paper mentions using \"an MLP to map the input sensor dimensions to the desired output dimensions\". It would be helpful to specify the structure of this MLP e.g., a single linear layer?\n4. typo? In sec 5.8 \"Fig. 10 compares our method with the baselines...\"", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "A0J5StVJeI", "reviewer": "Reviewer_7cV9", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper studies an important problem of traffic prediction. The authors propose a spatio-temporal long-term partial sensing model with extensive experiments. Generally, the paper is well-written and easy to follow. The proposed method is sound. However, it is suggested to improve the paper as follows.\n1. More recent baselines are required as follows.\n\n[1]. ST-LLM+: Graph Enhanced Spatio-Temporal Large Language Models for Traffic Prediction, TKDE 2025.\n\n[2]. Spatial-Temporal Large Language Model for Traffic Prediction, MDM 2024.\n\n2. It is suggested to compare the efficiency of the proposed method and baselines. In addition, theoretical computational analysis, i.e., time and space complexities, is required.\n3. It would be better to include a case study to intuitively show the effectiveness of the proposed method.\n4. More recent related work is required.\n\n[3]. MBA-STNet: Bayes-enhanced discriminative multi-task learning for flow prediction, TKDE 2023.\n\n[4]. A unified replay-based continuous learning framework for spatio-temporal prediction on streaming data, ICDE 2024.\n\n[5]. TimeCMA: Towards LLM-Empowered Multivariate Time Series Forecasting via Cross-Modality Alignment, AAAI 2025.\n\n[6]. Towards Cross-Modality Modeling for Time Series Analytics: A Survey in the LLM Era, IJCAI 2025.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies an important problem of long-term traffic forecasting. A spatio-temporal partial sensing forecast model is proposed. Experiments show the effectiveness of the proposed method.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weaknesses.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)