4.2. Nonlinear Modeling and Prediction of Dst Index
 Geomagnetic storms dominantly originate from solar wind flows. On the basis of physical research and our studies about geomagnetic storm and solar terrestrial physics, it is obvious that Dst index modeling based on solar wind conditions is nonlinear (may be a local dynamical model). It is reasonable to utilize a locally linear model for Dst index modeling such that several local lines are fit for each region of data set. In addition, soft switching between each local linear model with a membership function causes avoiding from discontinuity. The hourly omni-directional data are downloaded from the National Space Science Data Center and are used here. History of the Dst index, history of Bs, the Dst derivative, and the square root of dynamic pressure past values are used as geoeffective inputs to LOLIMOT for modeling and prediction of the next Dst values. The solar wind dynamic pressure is computed as follows:
where n and V are the density and velocity of solar wind, respectively. For the modeling and prediction of geomagnetic storms that is studied here, inputs to the LOLIMOT model depend on physical backgrounds mentioned before, and they are selected as follows:
For selecting the appropriate number of LLMs for prediction of storms in order to obtain the maximum generalization, the error criteria with respect to number of LLMs must be computed and plotted; the appropriate number of LLMs is that where the error criteria in the validation set start to increase. Some other criteria that are computed for predictions studied here are the average relative variance (ARV), normalized mean square error (NMSE), mean square error (MSE), and root mean square error (RMSE), which are computed as follows:
where y, , and are observed, predicted, and average observed values, respectively. The average accuracy of the forecasts in the sample relative to the accuracy of forecasts produced by a reference method is called skill, which can be measured by any number of so-called skill scores (SS). On the basis of MSE and RMSE, the skill scores are defined as
where MSE and RMSE are the mean square error of the sample forecasts and MSEr and RMSEr are the mean square error of the reference forecasts.
 The data before storms are predicted are used for model selection. About 80% of this data is used for training the model, and about 20% is used for validating (selecting the number of neurons) in order to predict the test data set. The test set differs from both the training and the validating sets. The data set from 1998 to 6 April 2000 is applied for model selection to predict the storm of 6–8 April 2000. Three LLMs are applied for prediction of this storm, and a correlation coefficient of 0.982 is achieved in this case. The data set from 1998 to 20 June 2002 is used for model selection for predicting the storms from 21 June 2002 to the end of 2002. In order to achieve maximum generalization, in Figure 3, NMSE is plotted with respect to the number of LLMs for both training and validation data; the LLNF model for best structure must have 11 LLMs. A correlation coefficient of 0.988 between observed Dst and predicted Dst for storms from 1 September 2002 to the end of 2002 is achieved that implies almost perfect prediction of Dst index with the LOLIMOT model on the basis of geoeffective solar wind conditions and Dst past values. In Figure 4, observed versus predicted values are depicted for prediction of these storms that shows a compact error bar and a little deviation between predicted values and observed values. In Figures 5 and 6, two storms are shown with higher timescale resolution for two models: in model 1, only the Dst past values are used in LOLIMOT and in model 2, both Dst past values and solar wind data are applied. As depicted in Figures 5 and 6, main and recovery phases except in commencement of the storm are predicted almost perfectly and have no time delay between predictions and observations, owing to dynamic modeling of the geomagnetic disturbance Dst index on the basis of geoeffective data and Dst dynamic data by using LOLIMOT. In model 1 (using only Dst past values), predictions in most phases are approximately the same as in model 2, but the prediction in the main phase has delay time. Correlation coefficients between predicted and observed Dst index for the storms in Figures 5 and 6 by model 2 are 0.99 and 0.987, and the ARVs of these predictions are 0.008 and 0.025, respectively, implying that 99% and 97.5%, respectively, of these storm variances are predictable. The correlation coefficients of model 1 in Figures 5 and 6 are 0.985 and 0.960, and the ARVs are 0.083 and 0.028, respectively. As found from this result, by using physical background for prediction of these storms in the LOLIMOT model, not only does the prediction accuracy increase but also the time delay for the main phase is eliminated, but both models have delay for prediction of storm commencement. Kugblenu et al.  employed solar wind data as external inputs to a multilayer perceptron (MLP) neural network for Dst prediction. Their network has three layers. They have used intense storms from years 1972, 1973, 1974, 1975, 1977, 1978, 1979, 1981, and 1982 for training the MLP network to predict the geomagnetic storm on 19 December 1980, and their method achieved a correlation coefficient of 0.96, ARV of 0.04, and RMSE equal to 11 nT. Although we have not used their data set, we have trained a similar MLP model with our data set with the same input parameters to both MLP and LOLIMOT models for comparison. The data sets from 1978 and 1979 and also part of 1980 prior to this event are used for model selection in order to predict this storm. By using the MLP neural network, the best number of hidden neurons is seven, and for the LOLIMOT model it is two. The LOLIMOT method achieved a correlation coefficient of 0.987 for prediction of this storm, and two LLMs was the best number of neurons to achieve maximum generalization. The prediction result is depicted in Figure 7. With the LOLIMOT method, the ARV is 0.024, which means that 97.54% of observed Dst variance is predictable from both Dst and the other geoeffective parameters. RMSE with the LOLIMOT model is 8.61, which is equivalent to SSRMSE = 0.217 with respect to the MLP model [Kugblenu et al., 1999]. The correlation of 0.985, ARV = 0.030, and RMSE = 9.61 nT are obtained with the MLP network, and SSRMSE of LOLIMOT with respect to the MLP model is 0.104.
Figure 3. Error criteria for selecting the appropriate number of LLMs for prediction of storms in year 2002: data from first of year 1998 to 20 June 2002 are used for model selection and from September 2002 to end of 2002 for validation; appropriate number of LLMs to achieve maximum generalization is 11 with RMSE = 4.7 nT.
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Figure 4. One-step predicted versus observed Dst index with LOLIMOT method with 11 LLMs, based on solar wind conditions for the first of September to the end of December 2002. Data from year 1998 prior to September 2002 are used for model selection. Correlation coefficient = 0.988.
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Figure 5. One-step prediction of geomagnetic storm on 2–3 September 2002 by LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.985, ARV = 0.03, NMSE = 0.022, and RMSE = 8.43 nT) and model 2, both Dst past values and solar wind parameters (correlation = 0.995, ARV = 0.008, NMSE = 0.006, and RMSE = 4.55 nT).
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Figure 6. One-step prediction of geomagnetic storm on 8–9 September 2002 by LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.960, ARV = 0.083, NMSE = 0.017, and RMSE = 12.04 nT) and model 2, both Dst past values and solar wind parameters (correlation = 0.986, ARV = 0.029, NMSE = 0.006, and RMSE = 7.10 nT).
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Figure 7. One-step prediction of storm on 19 December 1980 with LOLIMOT based on Dst past values and geoeffective parameters. Correlation coefficient = 0.987 and ARV = 0.024.
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 A question may arise whether the complex modeling and extrapolation method (locally linear with LOLIMOT) could perform better than linear extrapolation method. As mentioned in section 2, first, the model fits a global linear model, and if the performance is good, then the algorithm stops, and if not, the algorithm continues for constructing a locally linear model. In Figure 3, an error criterion is plotted as a function of LLM number. LLM = 1 indicates a global linear extrapolation model that is less efficient with respect to a locally linear model with LLM = 11 for this data set (see caption for Figure 3). In some other data sets, one linear model might be the best model, but such nonlinear processing is essential to find how many locally linear models have the best structure for applied data for modeling.
 Gleisner et al.  have shown that the solar wind history for 18–24 hours is needed as input to a trained network in order to reproduce all phases of a geomagnetic storm. They developed a time delay feed forward neural network based on a temporal sequence of solar wind data. Their networks showed better performance with larger temporal size of the input data sequence and were able to reproduce 85% of the Dst variance, which is quite an improvement on the earlier work of Lundstedt and Wintoft . Wu and Lundstedt , attempting to reproduce the recovery phase that was difficult to model with the networks of Lundstedt and Wintoft , used Elman recurrent networks. The Elman network is an extension of the multilayer back propagation networks with an addition of a feedback connection from hidden to input layers which allows the network to detect and generate time-varying patterns. Their Elman network model obtained a relatively high correlation coefficient, that is, 0.91, for a very long period of Dst (900 hours). For each storm period, however, their model does not always give a good fit to the observations, but the LOLIMOT method with geoeffective data shows very good performance for prediction of several intense geomagnetic storms. In Table 1, correlation coefficient, RMSE, NMSE, and ARV for prediction of several intense geomagnetic storms with the LOLIMOT method on the basis of solar wind data and Dst past values are reported. According to Table 1, prediction of several storms has a high correlation coefficient with observed values, and ARV for all of the case studies is less than 0.082, which shows that even in the worst case, 92% of storm variance is predictable, and in the best case, 98% of storm variance is predictable. In addition, the best number of LLMs for achieving maximum generalization is at least two for prediction of storms on 15–18 July 2000, and the other geomagnetic storms have different numbers of LLMs as reported in Table 1; this implies that the geomagnetic disturbance Dst index based on solar wind data is a nonlinear dynamic system, which could be considered as an extension of Burton et al.'s  empirical linear model.
Table 1. Real-Time Prediction of Some Geomagnetic Storms With LOLIMOT Models Based on Solar Wind Data
|Geomagnetic Storm Date||Correlation||RMSE, nT||NMSE||ARV||Best Number of LLMs|
|6–8 Apr 2000||0.982||13.458||0.011||0.035||3|
|15–18 Jul 2000||0.960||27.446||0.028||0.081||2|
|11–13 Aug 2000||0.984||8.691||0.007||0.032||4|
|31 Mar to 2 Apr 2001||0.989||15.852||0.009||0.021||3|
|1–2 Oct 2000||0.981||9.451||0.006||0.041||11|
 Figure 8 shows the one-step prediction of a very intense geomagnetic storm from 1200 UT on 31 March to 1200 UT on 2 April 2001. The data set from 1998 prior to this event is utilized for model selection to predict the event. In training and validating the data set, the value of events does not exceed −301 nT, but in the test set, a geomagnetic storm with value −387 nT, is predicted perfectly. The minimum value of the event is −387 nT, and the prediction of this value is −372.51 nT. A correlation coefficient of 0.989 and an ARV of 0.021 are achieved for prediction of this extreme storm, which indicates that almost 98% of Dst variance of this storm is predictable; these results imply near-perfect prediction of this event without delay, which is depicted in Figure 8. As found from Figure 8, by using the solar wind parameters, the main phase of the storm is not only predicted without time delay owing to utilization of the southward IMF, but also the prediction of the recovery phase of the storm has an almost perfect match with observed values owing to the use of dynamic pressure history as an external input to the novel nonlinear LOLIMOT model. By using only Dst past values, the prediction in the main phase not only has a 1 hour delay but also in the recovery phase has less accuracy with respect to a situation using the solar wind parameters as external physical parameters. The data set from 1995 to 1999 is fitted with three LLMs, and a linear correlation coefficient of 0.984 and a RMSE of 4.31 nT are gained. In addition, the model is tested for the first half of the year 2000, and a correlation coefficient of 0.983 and a RMSE of 4.38 nT are obtained for this interval. It is worth mentioning that we do not compare these results with Temerin and Li's  model through (15), since we have used a different set of inputs, which makes the comparison unfair.
Figure 8. One-step prediction of geomagnetic storm from 1200 UT on 31 March to 1200 UT on 2 April 2001 with LOLIMOT with two sets of input parameters: model 1, only Dst past values (correlation = 0.981, RMSE = 22.68 nT, and ARV = 0.041) and model 2, both Dst past values and solar wind parameters (correlation = 0.989, RMSE = 15.852 nT, and ARV = 0.021).
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 To highlight the contribution of physical parameters for Dst prediction with the LOLIMOT model, first, the LOLIMOT model is trained and validated with both solar wind and past values of measured Dst, then the model is applied for prediction of further values of Dst by using only the observed solar wind data and predicted values of Dst from previous steps of the LOLIMOT model instead of using the measured Dst. In other words, the input to the LOLIMOT model in prediction mode for predicting the further time steps of Dst index is as follows:
where st(t − 1), st(t − 2), and st(t − 3) are the Dst predictions at times t − 1, t − 2, and t − 3, respectively. Figure 9 shows the prediction result of the storm on 19 December 1980. It can be seen from Figure 8 that the peak of the storm and its vicinity are predicted well. More significantly, we see that before the commencement of the storm and decreasing of the measured Dst index, the prediction starts to decrease, and this indicates that the magnetosphere and Dst index are controlled by solar wind. This result is compatible with the result of recent physical models [Temerin and Li, 2002]. The result of Figure 9 also implies that the LOLIMOT model could correlate the dynamic of solar wind data to Dst index and have a correlation coefficient of 0.88 when prediction is done only with observation of solar wind condition. However, prediction with LOLIMOT for a long time by using the predicted Dst instead of measured Dst as input to LOLIMOT is not possible, and after a time, prediction diverges to infinity. An important point is that when solar wind is southward for several hours, then the entry of plasma to magnetosphere starts, and the change in the ring current leads to the occurrence of a geomagnetic storm. We can predict the Dst index that in this situation assumes an abnormal negative value by using solar wind parameters. The large-amplitude oscillation around the mean Dst behavior, however, indicates that even with the inclusion of the solar wind data, the model loses forecasting value significantly when predicted past values of Dst are used instead of the observed past values. The model predicts Dst of −100 nT, indicating a major geomagnetic storm, about 8 hours before the actual commencement of the storm. But this essentially renders the LOLIMOT prediction without using the observed data worthless since 8-hour error is well beyond any tolerance level of usable Dst forecast. It can therefore be concluded that although physical data is very important for achieving good prediction and alerting capability, the identification of the internal dynamics of the Dst time series via advanced nonlinear identification and prediction techniques is a crucial factor, and without the use of observed data, one has absolutely no faith in the changes of the modeled Dst that could signal, for example, storm commencement or the beginning of the recovery phase.
Figure 9. Prediction of storm on 19 December 1980 by using predicted Dst (not the measured Dst) for prediction of consequent Dst values. Correlation coefficient = 0.88.
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 Then using the solar wind parameters in the LOLIMOT model causes that model to detect the decrease in the Dst index more promptly, and as a result, the contribution of physics or solar wind parameters is in eliminating the time delay in the main phase of the storm and also increasing the prediction accuracy. If only the past values of the Dst index are utilized for the LOLIMOT model, then the prediction, especially in the main phase of the storm, has longer time delay, and this kind of prediction has less worth, and the model may in some crucial cases perform slightly more than attributing the present observed Dst value as the prediction of the next hour.