1 Introduction
2 Time series prediction
3 The improved neuro-endocrine model (INEM)
4 LDWPSO for the improved neuro-endocrine model
4.1 LDWPSO algorithm
4.2 Optimizing the parameters of the improved model neuro-endocrine model with LDWPSO
5 Applications and results
5.1 Experiments setting
5.2 Mackey–glass time series (MG)
Neural model | Neuro-endocrine model without interaction of glands | Improved model | |
---|---|---|---|
Training | |||
Mean | 0.9122 | 2.0758e-004 |
8.4494e-005
|
Std | 1.4680 | 1.3380e-004 |
3.0569e-005
|
Best | 5.4702e-005 | 5.2127e-005 |
4.0670e-005
|
CPU time (0.001) | 42.7660s |
36.9530s
| |
Successful ratio | 83.3 % | 100 % | 100 % |
Testing | |||
Mean | 1.0366 | 2.1038e-004 |
8.6815e-005
|
Std | 1.6627 | 1.4536e-004 |
3.3291e-005
|
Best | 5.2078e-005 | 5.5016e-005 |
4.1827e-005
|
Methods | RMSE |
---|---|
Auto-regressive model [31] | 0.19 |
Cascade correlation NN [31] | 0.06 |
Sixth-order polynomial | 0.04 |
Linear predication method | 0.55 |
Wang and Mendel [23] Product T-norm | 0.0907 |
GA and Fuzzy system [21] | 0.049 |
PG-RBF network [22] |
0.0028
|
WNN + hybrid [31] | 0.0059 |
Feed-forward neural model with PSO | 1.0181 |
Neuro-endocrine model without interaction of glands | 0.0145 |
Improved model | 0.0093 |
5.3 Box–Jenkins gas furnace time series (BJ)
Neural model | Neuro-endocrine model without interaction of glands | Improved model | |
---|---|---|---|
Training | |||
Mean | 2.0565e-004 | 2.4100e-004 |
1.4412e-004
|
Std | 7.0618e-005 |
1.1757e-006
| 2.7320e-005 |
Best | 1.0985e-004 | 1.0721e-004 |
8.6327e-005
|
CPU time (0.001) |
12.1710s
| 30.2618s | 38.2810s |
Successful ratio | 100 % | 100 % | 100 % |
Testing | |||
Mean | 2.3148e-004 | 2.6012e-004 |
1.6027e-004
|
Std | 7.2019e-005 |
1.3281e-006
| 3.0143e-005 |
Best | 1.2311e-004 | 1.1694e-004 |
9.0217e-005
|
Method name | Inputs | RMSE |
---|---|---|
ARMA [26] | 5 | 0.843 |
Tong’s model [29] | 2 | 0.685 |
Pedrycz’s model [30] | 2 | 0.566 |
Xu’s model [25] | 2 | 0.573 |
Surmann’s model [27] | 2 | 0.400 |
Lee’s model [32] | 2 | 0.638 |
ANFIS model [28] | 2 | 0.085 |
Neural tree model [31] | 2 | 0.026 |
WNN + hybrid [31] | 2 | 0.081 |
Feed-forward neural model with PSO | 10 | 0.0152 |
Neuro-endocrine model without interaction of glands | 10 | 0.0161 |
Improved model | 10 |
0.0127
|
5.4 Electroencephalogram (EEG) data
Neural model | Neuro-endocrine model without interaction of glands | Improved model | |
---|---|---|---|
Training | |||
Mean | 0.0078 |
0.0076
|
0.0076
|
Std | 4.6530e-004 | 1.1524e-004 |
1.0648e-004
|
Best | 0.0076 | 0.0075 |
0.0074
|
CPU time (0.001) |
0.2030s
| 6.8942s | 8.750s |
Successful ratio | 100 % | 100 % | 100 % |
Testing | |||
Mean | 0.0076 |
0.0075
|
0.0075
|
Std | 4.7310e-004 | 1.2712e-004 |
1.1086e-004
|
Best | 0.0075 | 0.0074 |
0.0070
|
Method name | MSE |
---|---|
Single multiplicative neuron model with BP [8] | 0.0142 |
Single multiplicative neuron model with PSO [8] | 0.0080 |
Single multiplicative neuron model with GA [8] | 0.0081 |
Single multiplicative neuron model with CRPSO [8] | 0.0081 |
Feed-forward neural model with PSO | 0.0076 |
Neuro-endocrine model without interaction of glands |
0.0075
|
Improved model |
0.0075
|
5.5 IBM common stock closing prices model (IBMCSCP)
Neural model | Neuro-endocrine model without interaction of glands | Improved model | |
---|---|---|---|
Training | |||
Mean | 2.4115e-004 | 2.4106e-004 |
2.4101e-004
|
Std | 1.3324e-006 | 1.3554e-006 |
1.1075e-006
|
Best | 2.3956e-004 | 2.3851e-004 |
2.3820e-004
|
CPU time (0.001) |
2.1034s
| 4.3866s | 5.2190s |
Successful ratio | 100 % | 100 % | 100 % |
Testing | |||
Mean | 2.5618e-004 | 2.5433e-004 |
2.4712e-004
|
Std | 1.5541e-006 | 1.4718e-006 |
1.0176e-006
|
Best | 2.4819e-004 | 2.4781e-004 |
2.4022e-004
|
Models | Inputs | RMSE |
---|---|---|
SVM [19] | 4 |
0.0129
|
ANN [19] | 6 | 0.0158 |
Feed-forward neural model with PSO | 2 | 0.0160 |
Neuro-endocrine model without interaction of glands | 2 | 0.0159 |
Improved model | 2 | 0.0157 |
5.6 Canadian Lynx data (CLYNX)
Neural model | Neuro-endocrine model without interaction of glands | Improved model | |
---|---|---|---|
Training | |||
Mean | 0.0360 | 0.0357 |
0.0334
|
Std | 0.0025 | 0.0022 |
0.0019
|
Best | 0.0315 | 0.0316 |
0.0290
|
CPU time (0.04) |
30.8600s
| 42.6538s | 48.5780s |
Successful ratio | 100 % | 100 % | 100 % |
Testing | |||
Mean | 0.0379 | 0.0368 |
0.0354
|
Std |
0.0057
| 0.0068 | 0.0074 |
Best | 0.0361 | 0.0388 |
0.0358
|
Models | MSE |
---|---|
PADD [20] | 0.046 |
SETAR1 [34] | 0.042 |
FAR [20] | 0.036 |
SAR [20] | 0.038 |
SETAR2 [38] | 0.042 |
SBL [20] |
0.022
|
GP [20] | 0.028 |
Feed-forward neural model with PSO | 0.0360 |
Neuro-endocrine model without interaction of glands | 0.0357 |
Improved model | 0.0334 |
5.7 Comparisons using t test
Datasheet | ANN | Neuro-endocrine model without interaction of glands | ||
---|---|---|---|---|
MG | Training |
t value |
3.4033
|
4.9121
|
P value |
0.001213
|
0.000008
| ||
Testing |
t value |
3.4145
|
4.5385
| |
P value |
0.001172
|
0.000029
| ||
BJ | Training |
t value |
4.4509
|
19.4049
|
P value |
0.000039
|
0.00000
| ||
Testing |
t value |
4.9958
|
18.1260
| |
P value |
0.000006
|
0.00000
| ||
EEG | Training |
t value |
2.2950
| 0.0000 |
P value |
0.02537
| 1.00000 | ||
Testing |
t value | 1.1272 | 0.0000 | |
P value | 0.2643 | 1.00000 | ||
IBMCSCP | Training |
t value | 0.4426 | 0.1565 |
P value | 0.659701 | 0.876182 | ||
Testing |
t value |
26.7136
|
22.0701
| |
P value |
0.0000
|
0.000000
| ||
Lynx | Training |
t value |
4.5352
|
4.3337
|
P value |
0.000029
|
0.000059
| ||
Testing |
t value | 1.4659 | 0.7630 | |
P value | 0.148076 | 0.448556 | ||
Better | 7 | 6 | ||
Same | 3 | 4 | ||
Worse | 0 | 0 |