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where k is the number of PSNN blocks, o is a non-linear transfer function, m is the input vec- tor dimension size, w is a trainable weight, x is the input and y(f) is network output at a previ- ous time step.

Figure 3 where k is the number of PSNN blocks, o is a non-linear transfer function, m is the input vec- tor dimension size, w is a trainable weight, x is the input and y(f) is network output at a previ- ous time step.

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Fig 1. Ridge Polynomial Neural Network. PSNN stands for Pi-Sigma Neural Network.
Fig 1. Ridge Polynomial Neural Network. PSNN stands for Pi-Sigma Neural Network.
Fig 2. Dynamic Ridge Polynomial Neural Network. PSNN stands for Pi-Sigma Neural Network and Z ‘ denotes the time delay operator
Fig 2. Dynamic Ridge Polynomial Neural Network. PSNN stands for Pi-Sigma Neural Network and Z ‘ denotes the time delay operator
Fig 3. Ridge Polynomial Neural Network with Error Feedback. PSNN stands for Pi-Sigma Neural Network, a(t+ 1) is the desired output at time t+ 1 and Zl daanatac thatima:dalayw anoraicr 1dded PSNN block, P,(t + 1) is P(t + 1) after its weights are frozen, ois the sigmoid activation
Fig 3. Ridge Polynomial Neural Network with Error Feedback. PSNN stands for Pi-Sigma Neural Network, a(t+ 1) is the desired output at time t+ 1 and Zl daanatac thatima:dalayw anoraicr 1dded PSNN block, P,(t + 1) is P(t + 1) after its weights are frozen, ois the sigmoid activation
doi:10.1371/journal.pone.0167248.t001 Table 1. Time series information. exchange rate. The data set contains 781 observations covering the period from January 3, 2005 to December 31, 2007 [44]. The data was collected from [45, 46]. The last time series was generated from the Mackey-Glass time-delay differential equation which is defined as follows:
doi:10.1371/journal.pone.0167248.t001 Table 1. Time series information. exchange rate. The data set contains 781 observations covering the period from January 3, 2005 to December 31, 2007 [44]. The data was collected from [45, 46]. The last time series was generated from the Mackey-Glass time-delay differential equation which is defined as follows:
doi:10.1371/journal.pone.0167248.g004 The settings used in this study for these series are shown in Table 1. These settings were also used in the studies of [42-44, 48-59]. The used intervals for training and out-of-sample sets are shown in Figs 4-7.
doi:10.1371/journal.pone.0167248.g004 The settings used in this study for these series are shown in Table 1. These settings were also used in the studies of [42-44, 48-59]. The used intervals for training and out-of-sample sets are shown in Figs 4-7.
Fig 5. Monthly smoothed sunspot numbers time series from November 1834 to June 2001.
Fig 5. Monthly smoothed sunspot numbers time series from November 1834 to June 2001.
Fig 7. Mackey-Glass time-delay differential equation time series.
Fig 7. Mackey-Glass time-delay differential equation time series.
Table 2. Network topology and training. t-tests with a significance level of 0.05 to highlight significant performance. The equation for these metrics are given by: Root Mean Squared Frror (RMSE):
Table 2. Network topology and training. t-tests with a significance level of 0.05 to highlight significant performance. The equation for these metrics are given by: Root Mean Squared Frror (RMSE):
Table 3. Root Mean Squared Error (RMSE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. E means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; 7 means the RPNN has significant performance using t-test.
Table 3. Root Mean Squared Error (RMSE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. E means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; 7 means the RPNN has significant performance using t-test.
Table 4. Normalized Mean Squared Error (NMSE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. E means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; means the RPNN has significant performance using t-test.
Table 4. Normalized Mean Squared Error (NMSE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. E means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; means the RPNN has significant performance using t-test.
Table 5. Mean Absolute Error (MAE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; 7 means the RPNN has significant performance using t-test.
Table 5. Mean Absolute Error (MAE) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; 7 means the RPNN has significant performance using t-test.
Table 6. Signal to Noise Ratio (SNR) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; means the RPNN has significant performance using t-test.
Table 6. Signal to Noise Ratio (SNR) improvement of RPNN-EF to RPNN and DRPNN. These are the de-normalized results for 30 simulations. means the RPNN-EF has significant performance using t-test; P means the DRPNN has significant performance using t-test; means the RPNN has significant performance using t-test.
Table 7. Best Single Simulation Results. These are the de-normalized results. doi:10.1371/journal.pone.0167248.t007
Table 7. Best Single Simulation Results. These are the de-normalized results. doi:10.1371/journal.pone.0167248.t007
Fig 9. Correlation function between forecast and observed values for StarBrightness time series based on the best RPNN-EF simulation. doi:10.1371/journal.pone.0167248.g009
Fig 9. Correlation function between forecast and observed values for StarBrightness time series based on the best RPNN-EF simulation. doi:10.1371/journal.pone.0167248.g009
Fig 10. Correlation function between forecast and observed values for Sunspot time series based on the best RPNN-EF simulation.
Fig 10. Correlation function between forecast and observed values for Sunspot time series based on the best RPNN-EF simulation.
Fig 11. Correlation function between forecast and observed values for EUR/USD time series based on the best RPNN-EF simulation.
Fig 11. Correlation function between forecast and observed values for EUR/USD time series based on the best RPNN-EF simulation.
Fig 12. Correlation function between forecast and observed values for Mackey—Glass time series based on the best RPNN-EF simulation. doi:10.1371/journal.pone.0167248.g012
Fig 12. Correlation function between forecast and observed values for Mackey—Glass time series based on the best RPNN-EF simulation. doi:10.1371/journal.pone.0167248.g012
Fig 13. Out-of-sample forecasting for StarBrightness time series based on the best RPNN-EF simulation.
Fig 13. Out-of-sample forecasting for StarBrightness time series based on the best RPNN-EF simulation.
Fig 14. Out-of-sample forecasting for Sunspot time series based on the best RPNN-EF simulation.
Fig 14. Out-of-sample forecasting for Sunspot time series based on the best RPNN-EF simulation.
Fig 15. Out-of-sample forecasting for EUR/USD time series based on the best RPNN-EF simulation.
Fig 15. Out-of-sample forecasting for EUR/USD time series based on the best RPNN-EF simulation.
Fig 16. Out-of-sample forecasting for Mackey—Glass time series based on the best RPNN-EF simulation.
Fig 16. Out-of-sample forecasting for Mackey—Glass time series based on the best RPNN-EF simulation.
Fig 17. The histogram of the forecasting error for StarBrightness time series based on the best RPNN-EF simulation.
Fig 17. The histogram of the forecasting error for StarBrightness time series based on the best RPNN-EF simulation.
Fig 18. The histogram of the forecasting error for Sunspot time series based on the best RPNN-EF simulation.
Fig 18. The histogram of the forecasting error for Sunspot time series based on the best RPNN-EF simulation.
Fig 19. The histogram of the forecasting error for EUR/USD time series based on the best RPNN-EF simulation.
Fig 19. The histogram of the forecasting error for EUR/USD time series based on the best RPNN-EF simulation.
Fig 20. The histogram of the forecasting error for Mackey—Glass time series based on the best RPNN-EF simulation.
Fig 20. The histogram of the forecasting error for Mackey—Glass time series based on the best RPNN-EF simulation.
Table 8. Comparison of the performance of various existing models on Sunspot series. ART, adaBoost.regression and threshold; FNN, fuzzy neural networks; DRPNN, dynamic ridge polynomial neural network; RFS, recurrent fuzzy system; RPNN, ridge polynomial neural network; RPNN-EF, ridge polynomial neural network with error feedback.
Table 8. Comparison of the performance of various existing models on Sunspot series. ART, adaBoost.regression and threshold; FNN, fuzzy neural networks; DRPNN, dynamic ridge polynomial neural network; RFS, recurrent fuzzy system; RPNN, ridge polynomial neural network; RPNN-EF, ridge polynomial neural network with error feedback.
Table 9. Comparison of the performance of various existing models on Mackey-Glass series. DE, differential evolution; DRPNN, dynamic ridge polynomial neural network; GD, gradient descent; KFA, kalman filtering algorithm; MLMVN, multilayer neural network with the multi-valued neurons; PSO, particle swarm optimization; RPNN, ridge polynomial neural network; RPNN-EF, ridge polynomial neural network with error feedback; SVD, singular value decomposition; WNN, wavelet neural network
Table 9. Comparison of the performance of various existing models on Mackey-Glass series. DE, differential evolution; DRPNN, dynamic ridge polynomial neural network; GD, gradient descent; KFA, kalman filtering algorithm; MLMVN, multilayer neural network with the multi-valued neurons; PSO, particle swarm optimization; RPNN, ridge polynomial neural network; RPNN-EF, ridge polynomial neural network with error feedback; SVD, singular value decomposition; WNN, wavelet neural network
Fig 23. Learning curves for Daily Euro/Dollar (EUR/USD) exchange rate time series forecasting based on the best RPNN-EF simulation.
Fig 23. Learning curves for Daily Euro/Dollar (EUR/USD) exchange rate time series forecasting based on the best RPNN-EF simulation.
Related topics:
Recurrent Neural NetworkFeedback-Error-LearningLyapunov functionError feedbackHigher order neural networkRidge Polynomial Neural Network
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