A comparative study of wavelet-based ANN and classical techniques for geophysical time-series forecasting

Time-series modeling forms an important area of research in geophysics. Time-series models can be linear, like linear state-space models or non-linear, like artificial neural networks. One way to judge the goodness of different models associated with a given time-series is to assess the prediction capabilities of these models. Some of the important techniques used in time-series forecasting are: (i) Minimum mean squared error (MMSE) forecast obtained using conditional means of ARIMA(p,d,q) models, (ii) Kalman filter approach and (iii) Artificial neural networks (ANN) approach. However, the wavelet-based versions of these techniques, respectively denoted as W-MMSE, W-Kalman and W-ANN, rather than the original techniques themselves, have been found to possess better capabilities in forecasting the highly nonlinear geophysical data. Using the original and predicted data, these observations have been validated by determining the RMSE (root mean squared error) and correlation coefficients between them. The prediction capabilities of both versions of the above techniques are tested on (i) the ionospheric total electron content (TEC) data, (ii) the daily average rainfall data and (iii) the gamma-ray log data from an offshore oil well, off the west coast of India. The TEC and daily average rainfall data sets designate as examples of data with very high correlations pertaining over very large lags. They also have a strong seasonality component associated with them. However, the gamma-ray log data sets show no seasonality component and have no trends associated with them. Therefore, the choice of different nonlinear data sets having diverse sources of their origins are apt to test the forecasting capabilities of these techniques. It has been observed that W-ANN gives the best prediction, when compared with the other algorithms discussed in this paper. This is believed to be due to the use of non-linear activation function by ANNs to produce regression that results in capturing the inherent non-linear dynamics of the process effectively. The results also show the usefulness of discrete wavelet transform (DWT) coefficients as training features for both linear and non-linear forecasting approaches. The better performance of the wavelet-based forecasting algorithms can be attributed to the fact that DWT coefficients are wide-sense stationary sequences. Thus the wavelet-based versions of these models like W-MMSE and W-Kalman provide better fit to these coefficients, compared to the original time-series data itself.