Future exogenous variables provide valuable information for forecasting endogenous time series. Existing covariate-aware methods primarily learn the direct influence of exogenous variables on endogenous variables. However, these effects can be complex and change with the pattern of the exogenous variables, making them difficult to capture. Beyond this perspective, we observe that a given exogenous pattern often co-occurs with only a small set of endogenous response patterns. These associations motivate a strategy that matches future and historical exogenous patterns and uses the corresponding endogenous patterns to enhance forecasting. However, in real-world forecasting scenarios with multiple exogenous variables, each exogenous variable provides a distinct dimension for matching, creating a dilemma for this strategy between precise matching and sufficient historical support. To bridge this gap, we propose XMatch (EXogenous MATCHing), a covariate-aware forecasting model that realizes the aforementioned strategy through a tree-structured matching process that adaptively adjusts the number of exogenous variables used as matching conditions. Specifically, we first introduce the ProtoTree Creator, which organizes historical correspondences between exogenous and endogenous patterns into a ProtoTree, whose deeper levels incorporate additional exogenous variables for matching. For forecasting, we then design the ProtoTree Matcher, which uses future exogenous variables to query the ProtoTree and adaptively determines how many exogenous variables to use for matching based on exogenous pattern similarity and historical support. Finally, the matched endogenous patterns are used as explicit historical evidence to enhance forecasting. Extensive experiments on 12 real-world datasets demonstrate that XMatch outperforms state-of-the-art baselines.
Figures & tables
Figure 1: Illustration of the frequency of co-occurrences between exogenous and endogenous patterns in real-world datasets. Rows denote exogenous patterns and columns denote endogenous response patterns, while darker cells indicate more co-occurrences. The darker cells in each row are often concentrated in a few columns, showing that a given exogenous pattern tends to co-occur with only a small set of endogenous patterns. We provide further evidence in Appendix B.3 .
Figure 2: The architecture of XMatch. (a) The ProtoTree Creator organizes correspondences between exogenous and endogenous patterns according to the descending discriminative power of exogenous variables. (b) The ProtoTree Matcher constructs matching queries from known future exogenous variables and uses Hierarchical Proto-Matching to match and aggregate endogenous response prototypes, thereby improving forecasting accuracy.
Models
XMatch
DAG
KITE
GCGNet
TimeXer
TFT
TiDE
DUET
CrossLinear
Amplifier
TimeKAN
Metrics
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
NP
0.282
0.299
0.362
0.344
0.325
0.323
0.370
0.348
0.418
0.371
0.379
0.375
0.443
0.400
0.411
0.408
0.371
0.387
0.420
0.418
0.405
0.419
PJM
0.085
0.178
0.093
0.180
0.096
0.179
0.095
0.187
0.108
0.198
0.114
0.207
0.142
0.246
0.102
0.197
0.112
0.223
0.137
0.246
0.139
0.262
BE
0.420
0.273
0.423
0.280
0.428
0.286
0.431
0.294
0.452
0.290
0.454
0.291
0.498
0.325
0.515
0.354
0.479
0.337
0.559
0.413
0.548
0.407
FR
0.410
0.216
0.414
0.219
0.387
0.225
0.415
0.234
0.427
0.241
0.504
0.257
0.484
0.281
0.496
0.327
0.483
0.298
0.554
0.408
0.547
0.374
DE
0.349
0.369
0.370
0.370
0.350
0.370
0.401
0.389
0.475
0.418
0.489
0.446
0.499
0.447
0.482
0.430
0.485
0.452
0.473
0.441
0.473
0.445
Table 1: Average results on 12 real-world datasets, where the inputs are Xendo , Xexo , and Yexo . Red : the best, Blue : the 2nd best. Full results with Yexo are reported in Table 5 .
Dataset
BE
DE
Sdwpfh1
Metrics
mse
mae
mse
mae
mse
mae
(a) w/o tree retrieval
0.440
0.298
0.371
0.373
0.442
0.477
(b) w/o variable ordering
0.438
0.289
0.410
0.396
0.457
0.481
(c) w/o search stopping
0.430
0.283
0.380
0.375
0.419
0.454
(d) endo history retrieval
0.433
0.299
0.356
0.373
0.478
0.491
(e) mix exo retrieval
0.425
0.285
0.363
0.377
0.470
0.487
Table 2: Average results of ablation studies for XMatch.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
#Num
Ex. Descriptions
En. Descriptions
Sampling Frequency
Lengths
Split
NP
2
Grid load, wind power
Nord Pool electricity price
1 Hour
52,416
7:1:2
PJM
2
System load, COMED zonal load
COMED zonal electricity price
1 Hour
52,416
7:1:2
BE
2
Belgian load, French generation
Belgian electricity price
1 Hour
52,416
7:1:2
FR
2
Generation, load
French electricity price
1 Hour
52,416
7:1:2
DE
2
Wind power, Amprion zonal load
German electricity price
1 Hour
52,416
7:1:2
Energy
5
Battery, geothermal, hydroelectric, solar, wind
Thermoelectric generation
1 Hour
13,064
7:1:2
Appendix
Table 3: Statistics of datasets. #Num denotes the number of exogenous variables. Ex. and En. are abbreviations for the exogenous variable and endogenous variable, respectively.
Model
Code repository
Model
Code repository
DAG
decisionintelligence/DAG
TiDE
thuml/Time-Series-Library
KITE
decisionintelligence/KITE
DUET
decisionintelligence/DUET
GCGNet
decisionintelligence/GCGNet
CrossLinear
mumiao2000/CrossLinear
TimeXer
thuml/TimeXer
Amplifier
aikunyi/amplifier
TFT
google-research/…/tft
TimeKAN
huangst21/TimeKAN
Appendix
Table 4: Code repositories for the baseline models.
Models
XMatch
DAG
KITE
GCGNet
TimeXer
TFT
TiDE
DUET
CrossLinear
Amplifier
TimeKAN
Metrics
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
mse
mae
NP
24
0.185
0.221
0.202
0.237
0.179
0.223
0.208
0.237
0.236
0.266
0.219
0.249
0.284
0.301
0.246
0.287
0.210
0.266
0.252
0.303
0.273
0.310
360
0.380
0.377
0.521
0.451
0.471
0.422
0.531
0.459
0.600
0.475
0.539
0.501
0.601
0.498
0.576
0.528
0.531
0.508
0.587
0.534
0.538
0.529
Avg
0.282
0.299
0.362
0.344
0.325
0.323
0.370
0.348
0.418
0.371
0.379
0.375
0.443
0.400
0.411
0.408
0.371
0.387
0.420
0.418
0.405
0.419
PJM
24
0.060
0.147
0.057
0.143
0.056
0.143
0.060
0.150
0.075
0.166
0.095
0.195
0.106
0.214
0.072
0.166
0.088
0.191
0.096
0.208
0.115
0.244
360
0.110
0.209
0.130
0.218
0.135
0.214
0.129
0.223
0.140
0.231
0.133
0.219
0.177
0.279
0.131
0.228
0.135
0.254
0.177
0.285
0.162
0.281
Appendix
Table 5: Full results on forecasting with historical and future exogenous variables across 12 real-world datasets, where the inputs are ( Xendo,Xexo , and Yexo ). Red : the best, Blue : the 2nd best. Avg means the average results from two forecasting horizons.
Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.
Kuiye Ding, Yifan Hu, Hanchen Wang +1
University of Technology Sydney · Tsinghua University · The Hong Kong University of Science and Technology (Guangzhou)
Covariate effects vary across contexts and shift over time, requiring forecasters to assess how to use them for each forecasting context. As forecasting proceeds, observations for earlier forecasts become available, providing feedback on past covariate use for subsequent forecasts. However, when multiple covariates act together, the forecast error reveals the numerical discrepancy from the observation but not how the covariates should have been used. We introduce JudgeCast, an experience-based framework for time series forecasting with covariates. Following the judgmental adjustment practice, a frozen TSFM provides the base forecast, while a frozen LLM uses the current context and relevant experience to adjust it. Within the adjustment, assessing covariate effects and determining the numerical adjustment serve distinct roles, so JudgeCast first forms explicit covariate-wise judgments and then determines the adjustment. After observation, JudgeCast uses the observed residual of the base forecast to reconstruct alternative judgments and evaluates the original and alternatives through their resulting adjustments. The best-performing decision is selected and retained as validated experience for subsequent forecasts. Across diverse real-world datasets, JudgeCast outperforms strong baselines. Ablations show that explicit covariate-wise judgment can improve forecast-time adjustment, while residual-guided experience construction yields more reliable forecasting gains than retaining raw decisions as experience.
Forecasting multiple time-series with high-dimensional covariates presents a core challenge: unifying common temporal patterns while retaining meaningful series-specific information. We introduce Hopformer (Homogeneity-Pursuit Transformer), a two-stage framework that addresses this challenge. In the first stage, we perform a Sparsity Pattern Aggregation (SPA) scheme extracting a common low-variance trend that incorporates the covariates. This acts as a homogenization layer. In the second stage, a LoRA-fine-tuned Transformer models the remaining complex dependencies in the residual. Our method is theoretically grounded. We prove that SPA achieves a near-optimal bias-variance trade-off via an oracle inequality. We also provide generalization bounds for the second stage under dependent time series data. Hopformer sets a new state of the art, improving MASE by an average of 6.56% across synthetic and real-world forecasting benchmarks.
Wan Zhang, Qinjie Lin, Chan Lee +3
The AMSS Center of Forecasting Science, Chinese Academy of Sciences, Beijing, China · Department of Computer Science, Northwestern University, Evanston, IL · Department of Statistics and Data Science, Northwestern University, Evanston, IL +1