## 5 Discussion  
As seen in Figure 8, the controlled fits produced two curves which did not lie on the even trade-off line but instead had higher or lower slopes. This result can be explained by the limitations on the length of the catalogue and the size of the search region. The fitted parameters could only adjust to the precursors that were contained in the catalogue and not to those that were screened out by such limitations. We now consider in detail the trend of the fitted \(\sigma_{A}\) value away from the even trade-off line for the controlled values of \(a_{T}\). The trend of the other curve can be explained similarly. As \(a_{T}\) was stepped down to lower values (i.e., the time scale was shortened), fitting the trade-off required earthquakes at increasingly longer distances from the target earthquakes. However, at longer distances, more precursory events were screened out by the spatial limitation on the input catalogue. The precursors of the largest earthquakes in the target magnitude range would be most affected by the spatial limitation because they had larger precursory areas (Figure 2). The spatial limitation at small \(a_{T}\) values forced the fitted values of \(\sigma_{A}\) to increasingly fall below the even trade-off line. On the other hand, as \(a_{T}\) was stepped up to higher values, the precursory time scale became longer and exceeded the available lead time. This temporal limitation most affected the largest earthquakes in the target magnitude range, which had the longest precursor times (Figure 2). Thus, more and more precursory earthquakes on the specified time scale were screened out by the limited time span of the catalogue. The remaining precursors for fitting \(\sigma_{A}\) would be those at the lower end of the time distribution. Because of the space-time trade-off, these remaining precursors tended to be at longer distances than the screened-out events. This forced the fitted \(\sigma_{A}\) to increasingly exceed the even trade-off line. The space-time trade-off in the EEPAS model shows that as the mean of \(f(t|m)\) in Equation (4) increased, the area of the fitted \(h(x,y|m)\) in Equation (5) decreased and vice versa. This phenomenon can be interpreted in terms of the predictive scaling relations on which the EEPAS model is based (Figure 2). Figure 2 shows that \(T_{P}\) and \(A_{P}\) both increased with the precursory earthquake magnitude \(M_{P}\). Similarly, in the EEPAS model, the mean of the time distribution \(f(t|m)\) and the area of the location distribution \(h(x,y|m)\) both increased with \(m\). Now, the space-time trade-off observed in the EEPAS model can be interpreted in terms of the space-time distribution of precursors to an individual major earthquake; that is, the earliest precursors tend to occur very close to the source, and the later precursors to occupy a wide area around the source. This interpretation only applies to precursors occurring more than 50 days before the mainshock because of the time lag applied for EEPAS model fitting here. The existence of this trade-off raises the question of how it can be exploited to improve the performance of the EEPAS model. The EEPAS model treats the time and location as independent variables, but the trade-off implies that they are correlated. We will illustrate how to improve forecasting by forming hybrid models. The hybrid models are mixtures of three EEPAS models with the values of \(a_{T}\) and \(\sigma_{A}\) chosen from points on the even trade-off line with a slope of \(-1\). We constructed two models, Hybrid_1F and Hybrid_1R, starting from two different EEPAS models: EEPAS_1F and EEPAS_1R, respectively. EEPAS_1R was similar to EEPAS_1F in nearly all aspects, apart from having fewer optimized parameters. Its fixed and optimized parameters are given in Table 4. An important difference between the two models was that EEPAS_1F (Table 1) had a larger value of \(\sigma_{T}\) than EEPAS_1R(Table 4). The parameter \(\sigma_{T}\) was optimized in the fitting of EEPAS_1F but not in EEPAS_1R.