CNN-LSTM based incremental attention mechanism enabled phase-space reconstruction for chaotic time series prediction

Algorithm: Parameter estimation algorithm
Input: $\rm{all} \; (m{\mathrm{,}}\; \tau) $
Output: $\rm{best} \; (m{\mathrm{,}}\; \tau) $
1: Initialize: Set $m_0=1{\mathrm{,}}\; \tau_0=0 $
2: for $ m=1{\mathrm{,}}\; 2{\mathrm{,}}\; \cdots{\mathrm{,}}\; m_{{\mathrm{max}} } $ do
3: 　for $\tau=0{\mathrm{,}}\;1{\mathrm{,}}\; \cdots{\mathrm{,}}\; \tau_{{\mathrm{max}} } $ do
4: 　if $(m{\mathrm{,}}\; \tau) $ fit DWC then save;
5: 　　else continue;
6: 　　end if
7: 　end for
8: end for

Table 2. Phase space reconstruction parameters of logistic system.
View table
View in Article
Table 2. Phase space reconstruction parameters of logistic system.
Method Embedding dimension Delay
IAN 2 5
CAO 5 9
FNN 4 15
C-C 12 3

Table 3. Dimension contribution weight of the logistic system.
View table
View in Article
Table 3. Dimension contribution weight of the logistic system.
Dimension Contribution weight Incremental
1 [1.0] Yes
2 [0.3808, 0.6192] Yes
3 [0.2507, 0.4120, 0.3384 ] No
4 [0.2430, 0.1731, 0.3057, 0.2782] No

Table 4. Predicted RMSE error table of the logistic system.
View table
View in Article
Table 4. Predicted RMSE error table of the logistic system.
Model IAN CAO FNN C-C
CNN-LSTM 0.0061 0.0084 0.0087 0.0072
LSTM 0.0069 0.0188 0.0084 0.0103
CNN 0.0087 0.0138 0.0115 0.0907
SVR 0.0499 0.0664 0.0516 0.0873
RNN 0.0146 0.0277 0.0437 0.0298

Table 5. Predicted MAE error table of the logistic system.
View table
View in Article
Table 5. Predicted MAE error table of the logistic system.
Model IAN CAO FNN C-C
CNN-LSTM 0.00025 0.00034 0.00036 0.00029
LSTM 0.00028 0.00077 0.00034 0.00042
CNN 0.00036 0.00056 0.00047 0.00370
SVR 0.00204 0.00271 0.00211 0.00356
RNN 0.00060 0.00113 0.00178 0.00122

Table 6. Phase space reconstruction parameters of the Lorenz system.
View table
View in Article
Table 6. Phase space reconstruction parameters of the Lorenz system.
Method Embedding dimension Delay
IAN 3 6
CAO 7 17
FNN 5 17
C-C 6 10

Table 7. Dimension contribution weight of the Lorenz system.
View table
View in Article
Table 7. Dimension contribution weight of the Lorenz system.
Dimension Contribution weight Incremental
1 [1.0] Yes
2 [0.01051, 0.98949] Yes
3 [0.1748, 0.2845, 0.5427] Yes
4 [0.1911, 0.2078, 0.3186, 0.2825] No
5 [0.1689, 0.2367, 0.1999, 0.2808, 0.1136] No

Table 8. Predicted RMSE error table of the Lorenz system.
View table
View in Article
Table 8. Predicted RMSE error table of the Lorenz system.
Model IAN CAO FNN C-C
CNN-LSTM 0.1264 0.3306 0.2801 0.2393
LSTMs 0.1331 0.2279 0.2511 0.1857
CNN 0.1491 0.3062 0.4627 0.6170
SVR 1.7839 2.1778 2.1305 1.9869
RNN 0.1352 0.3709 0.3106 0.3161

Table 9. Predicted MAE error table of the Lorenz system.
View table
View in Article
Table 9. Predicted MAE error table of the Lorenz system.
Model IAN CAO FNN C-C
CNN-LSTM 0.00516 0.01350 0.01144 0.00977
LSTM 0.00543 0.00930 0.01025 0.00758
CNN 0.00609 0.01250 0.01889 0.02519
SVR 0.07283 0.08891 0.08698 0.08111
RNN 0.00552 0.01514 0.01268 0.01290

Table 10. Phase space reconstruction parameters of the sunspot series.
View table
View in Article
Table 10. Phase space reconstruction parameters of the sunspot series.
Method Embedding dimension Delay (month)
IAN 5 3
CAO 12 14
FNN 5 14
C-C 3 27

Table 11. Dimension contribution weight of the sunspot series.

View table

View in Article

Table 11. Dimension contribution weight of the sunspot series.

Dimension	Contribution weight	Incremental
1	[1.0]	No
2	[0.4175, 0.5825]	Yes
3	[0.2307, 0.2431, 0.5263]	Yes
4	[0.1751, 0.2042, 0.2757, 0.3450]	Yes
5	[0.0180, 0.0587, 0.1143, 0.2405, 0.5685]	Yes
6	[0.2109, 0.1194, 0.1376, 0.1675, 0.1984, 0.1662]	No
7	[0.1173, 0.2157, 0.1516, 0.1115, 0.1474, 0.1319, 0.1245]	No

Table 12. Predicted RMSE error table of the sunspot series.
View table
View in Article
Table 12. Predicted RMSE error table of the sunspot series.
Model IAN CAO FNN C-C
CNN-LSTM 25.6003 28.7535 33.9891 30.2471
LSTMs 25.9045 25.9927 26.5143 26.8429
CNN 25.7932 28.1824 26.2913 26.5868
SVR 30.2663 32.5171 32.1185 30.5299
RNN 25.7662 25.8984 26.5871 26.6672

Table 13. Predicted MAE error table of the sunspot series.
View table
View in Article
Table 13. Predicted MAE error table of the sunspot series.
Model IAN CAO FNN C-C
CNN-LSTM 1.04513 1.17386 1.38760 1.23483
LSTM 1.05755 1.06115 1.08244 1.09586
CNN 1.05300 1.15054 1.07334 1.08540
SVR 1.23562 1.32751 1.31123 1.24638
RNN 1.05190 1.05730 1.08541 1.08868

Tools

Get Citation

Copy Citation Text

Xiao-Qian Lu, Jun Tian, Qiang Liao, Zheng-Wu Xu, Lu Gan. CNN-LSTM based incremental attention mechanism enabled phase-space reconstruction for chaotic time series prediction[J]. Journal of Electronic Science and Technology, 2024, 22(2): 100256

Download Citation

EndNote(RIS)BibTex Plain Text

Set citation alerts for article

Save article for my favorites