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MethodComplexityper layerMaximum path length
CNN (Munir et al., 2018)O(L)O(L)
RNN (Salinas et al.,2020)O(L)O(L)
Full-Attention (Vaswani et al.,2017)O(L2)0(1)
ETC (Ainslie et al., 2020)O(GL)0(1)
Longformer (Beltagy et al., 2020)O(L)O(L)
LogTrans (Li et al., 2019)O(L log L)O(log L)
PyraformerO(L)0(1)
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MethodsParametersDatasetsNRMSENDQ-K pairs
Full-attentionO(N(HDDK+DDF))Electricity0.3280.041456976
Wind0.1750.082589824
App Flow0.4070.080589824
LogTransO(N(HDDK+DDF))Electricity0.3330.04150138
Wind0.1730.08158272
App Flow0.3870.07358272
ReformerO(N(HDDK+DDF))Electricity0.3590.047677376
Wind0.1830.086884736
AppFlow0.4630.095884736
ETCO(N(HDDK+DDF))Electricity0.3240.04179536
Wind0.1670.074102144
App Flow0.3970.069102144
LongformerO(N(HDDK+DDF))Electricity0.3300.04141360
Wind0.1660.07552608
AppFlow0.3770.0752608
PyraformerO(N(HDDK+DDF) +(S-1)CD²)Electricity0.3240.04117648
Wind0.1610.07220176
App Flow0.3660.06720176
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MethodsMetricsETTh1ETTm1Electricity
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InformerMSE1.0751.3291.3840.5560.8410.9210.7451.5794.365
MAE0.8010.9110.9500.5370.7050.7530.2660.3230.371
Q-K pairs188040188040423360276480560640560640188040188040423360
LogTransMSE0.9831.1001.4110.5540.7861.1690.7911.5844.362
MAE0.7660.8390.9910.4990.6760.8680.3400.3360.366
Q-K pairs74664746642167442547606487686487687466474664216744
LongformerMSE0.8600.9751.0910.5260.7671.0210.7661.5914.361
MAE0.7100.7690.8320.5070.6630.7880.3110.3430.368
Q-K pairs6364863648249120329760100713610071366364863648249120
ReformerMSE0.9581.0441.4580.5430.9240.9810.7831.5844.374
MAE0.7410.7870.9870.5280.7220.7780.3320.3340.374
Q-K pairs10160641016064270950453084161445068814450688101606410160642709504
ETCMSE1.0251.0841.1370.7621.2271.2720.7771.5864.361
MAE0.7710.8110.8660.6530.8800.9080.3260.3400.368
Q-K pairs125280125280288720331344836952836952125280125280288720
PyraformerMSE0.8080.9451.0220.4800.7540.8570.7191.5334.312
MAE0.6830.7660.8060.4860.6590.7070.2560.2910.346
Q-K pairs264722647274280572649638496384264722647274280
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NotationSizeMeaning
LConstantThe length of historical sequence.
GConstantThe number of global tokens in ETC.
MConstantThe length of future sequence to be predicted.
BConstantBatch size.
DConstantThe dimension of each node.
DKConstantThe dimension of a key.
XB×L×DInput of a single attention head.
YB×L×DOutput of a single attention head.
QB×L×DkThe query.
KB×L×DkThe key.
VB×L×DkThe value.
WQD×DkThe weight matrix of the query.
WKD ×DkThe weight matrix of the key.
WvD×DkThe weight matrix of the value.
SConstantNumber of scales.
AConstantNumber of adjacent nodes at the same scale that a node can attend to.
CConstantNumber of finer scale nodes that a coarser scale node can summarize.
NConstantNumber of attention layers.
n(DThe l-th node at scale s.
Nlen(N)) × D
Alen(A()) × D The adjacent A nodes at the same scale with n(s).
Clen(C()) × D
Plen(P(s) × D The parent node of n(s).
FpB×M×DThe prediction tokens.
FeB ×Ltot ×D The output of the encoder. Ltot represents the output length of the encod
Fd1B×M×DThe output of the first attention-based decoder layer.
HConstantThe number of attention heads.
DF
ConstantThe maximum dimension of the feed-forward layer.
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Datasetprediction lengthNSHAChistorical length
ETTh116844634168
33644634168
72044654336
ETTm19644635384
28844655672
67244636672
Elect16844634168
33644634168
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MethodMSEMAE
Full attention3.5501.477
LogTrans3.0071.366
ETC4.7425.509
Informer7.5462.092
Longformer2.0321.116
Reformer1.5383.069
Pyraformer6,6,61.2580.877
Pyraformer12,7,41.1760.849
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A=3A=9A = 13
MSEMAEQ-K pairsMSEMAEQ-K pairsMSE MAEQ-K pairs
C=21.0350.811735121.0290.8151626481.0030.807221112
C=31.0290.817589921.0090.7981289761.0560.805174672
C=41.0010.802532081.0280.8061158481.0270.804156696
C=50.9990.796499921.0050.7961087441.0170.797147192
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MethodMetrics96288672
CSCM OnlyMSE0.5760.7820.883
MAE0.5440.6830.752
PyraformerMSE0.4800.7540.857
MAE0.4860.6590.707
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History LengthMSEMAE
841.2340.856
1681.2260.868
3361.1080.835
6721.0570.806
13441.0620.806
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CSCMMSEMAEParameters
Max-pooling0.8420.7000
Average-pooling0.8330.6930
Conv.0.7960.6793147264
Conv.w/bottleneck0.8080.683328704
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