ProCAST: A Projection Framework for Coupled
Aggregation Constrained Multivariate Time Series
Forecasting
Jiaqi Xue, Hongji Dong, Yucen Gao, Xiaofeng Gao
∗
, Guihai Chen
Shanghai Key Laboratory of Scalable Computing and Systems, School of Computer
Science, Shanghai Jiao Tong University, China
∗
Corresponding Author
Introduction
In aggregation structures, we often focus on top-level forecasts,
as they are most relevant for sales trends, supply chain planning,
and industry outlooks. When aggregation is coupled, simple non
negativity constraints on underlying sequences can induce highly
complex constraints on top-level sequences.
Challenges ① constraints are intractable to derive; ② unconstrained
forecasts rarely feasible; ③ underlying value often noisy/missing.
ProCAST integrates projectionbased constraint enforcement into
any base forecasting to deliver accurate and feasible toplevel
predictions with theoretical guarantees, while remaining effective
without access to underlying sequences.
Hierarchical Forecasting: Top-level, Underlying Sequences & Aggregation Structure
Category A:
Fashion Sytle
Category B:
Retro Style
Category C:
Furniture Decoration
Multivariate Forecasting: Top-level Sequences Only
Proposed ProCAST: Top-level Sequences & Aggregation Structure
Butterfly Mirror
ba
Sandalwood Fan
c
Retro Tissue Box
d
Retro Cup
Underlying Sequences
Top-level Sequences
Aggregation
Structure
×
=
Aggregat
e
Overview of ProCAST, including two optional projection methods: orthogonal and oblique
Top-Level Sequence Prediction
Multivariate Time Series Base Model
✗
✓
✗
Violation of Coupled Aggregation Constraints
Projection
Orthogonal Projection
OR
Oblique Projection
→
⋮
Virtual
Underlying
Sequence
Reconciliated Result
✓
Coupled Aggregation Constraints
Projection Reconciliation
Theoretical Guarantees
orthogonal
oblique
Orthogonal projection minimizes the Euclidean distance to the feasible set,
guaranteeing error reduction under distancebased loss function.
Oblique projection incorporate a data driven approach that learn a “projec
tion matrix” from historical data, which enables the projected points to lie
within the convex cone. Assuming stationarity over time , oblique
projection reduces the error in expectation.
Experiment Results
Best results are bolded; second-best underlined; cells shaded in gray indicate predictions that violate Coupled-Coherence.
Dataset E-Commerce Dataset RH Dataset
Metric MAE RMSE MAE RMSE
Method Raw BU MinT Ortho. Obliq. Raw BU MinT Ortho. Obliq. Raw Ortho. Obliq. Raw Ortho. Obliq.
NBEATS 1092 1149
1083
1092
1024
1983 2052
1975
1983
1877
827.3
827.3 752.7
1262
1262 1153
NBEATSx 1092 1149
1083
1092
1024
1983 2052
1975
1983
1877
827.3
827.3 752.7
1262
1262 1153
NHITS 1103 1194
1100
1103
1032 2001
2102 2002
2001 1901
867.5
867.5 807.0
1331
1331 1240
TimesNet
1050
1175 1082
1050 1038 1949
2087 1989
1949 1887 584.1
584.6
562.9
880.4
879.9 842.4
TCN
1067
1258 1086
1067 1054 1959
2155 1985
1959 1912 674.7
674.7
638.2
1013
1013 951.2
BiTCN
1073
1174 1075
1073 1069 1978
2084 1979
1978 1896
3382
3379 3046
5674
5655 5306
DeepNPTS
1058
1209 1111
1058 1039 1944
2115 2015
1944 1886
664.0
664.0 630.3
1023
1022 961.7
TFT
1065
1236 1074
1065 1048 1957
2133 1969
1957 1877 608.7
610.2
582.7
913.7
909.3 860.9
TiDE
1080
1122 1120
1080 1032 1985
2033 2030
1985 1886 1258
1258
1133
1949
1949 1762
DLinear 1456 1549 1496
1414 1319
2411 2500 2406
2330 2226
3690
3311 3283
9510
7620 7539
Informer 3053
2890
3048 3053
1406
3702
3560
3698 3702
2347
2.2e4
2.2e4 2.1e4
2.7e4
2.7e4 2.6e4
Autoformer
1050
1114 1058
1050 1052 1950
2032 1963
1950 1880
1025
1025 910.4
1554
1554 1388
FEDformer 1053 1115 1059
1053 1045
1949 2029 1958
1949 1874
595.5
595.5 571.9
901.5
901.4 854.7
PatchTST
1041
1122
1054 1041 1057 1924
2030 1945
1924 1881
589.7
589.6 561.8
864.3
864.3 817.9
TimeXer 1074 1169 1082
1074 1061
1973 2075 1981
1972 1897 799.3
799.6
741.3
1214
1213 1115
TimeMixer
1063
1149 1067
1063 1051 1966
2062 1972
1966 1887 608.3
608.8
585.9
920.9
920.5 878.1
TSMixer
1057
1140 1065
1057 1048 1959
2054 1968
1959 1877 619.0
619.0
589.0
948.6
948.6 888.6
TSMixerx
1062
1134 1067
1062 1057 1963
2047 1966
1963 1897 731.6
733.5
704.8
1123
1118 1061
iTransformer 1097 1173
1091
1097
1076
1983 2075
1981
1982
1928 612.7
612.8
595.2
917.8
916.4 883.7
RMoK
1072
1162 1076
1072 1047 1962
2069 1969
1962 1901 682.1
682.6
634.6
1037
1037 957.5
SOFTS 1106 1174
1093
1106
1091
1984 2075
1981
1984
1921 669.7
670.5
634.9
986.8
986.3 927.8
StemGNN
1059
1236 1083
1059 1055 1950
2136 1981
1950 1886 890.7
894.0
826.0
1396
1367 1258
RMSE
dierent base models
Results: Avg. RMSE reduction 5.15%
(ECommerce) and 7.10% (RH); MAE
reduction 4.33% and 6.19% respec
tively. ProCAST removes all coupled
coherence violations and outperforms
hierarchical baselines in practice.
Stationarity is consistently validated
across multiple base forecast models.
Contact:
Advanced Network Laboratory
Shanghai Jiao Tong University
xuejiaqi@sjtu.edu.cn Repo:
https://github.com/HellOwhatAs/ProCAST/
