**4. Results**

The likelihood of the refitted models declined with each step change in the controlled parameter away from its optimal value, as shown for New Zealand in Figure 6. The results for California were similar. The log-likelihood of the refitted model is plotted against the controlled spatial scaling factor in Figure 6a and against the temporal scaling factor in Figure 6b. An order of magnitude change in each scaling factor induced a modest reduction in the log-likelihood. The maximum reduction of about 34 units corresponded to an information loss per earthquake of about 0.2 relative to the overall optimal fit.

**Figure 6.** Log-likelihood of EEPAS model fitted with controlled values of (**a**) *σA* (Table 2) and (**b**) *aT* (Table 3) to the New Zealand earthquake catalogue.

The refitted mixing parameter *μ* tended to increase as the controlled parameter shifted further away from its optimal value, as shown for New Zealand in Figure 7. Again, the results were similar for California. The variation of *μ* with the spatial scaling factor is shown in Figure 7a and against the temporal scaling factor in Figure 7b. The values of *μ* increased from about 0.15 at the optimal fit to greater than 0.5 when the temporal or spatial scaling factors were changed by an order of magnitude. The *μ* value represents the proportional contribution of the background model to the total EEPAS model rate density. Higher *μ* values thus indicate a greater contribution of the background component and a smaller contribution of the time-varying component. In other words, higher *μ* values indicate that there were fewer target earthquakes with precursors matching the changed spatial and temporal distributions.

**Figure 7.** Fitted values of mixing parameter *μ* (0 ≤ *μ* ≤ 1) of the EEPAS model fitted with controlled values of (**a**) *σA* and (**b**) *aT* to the New Zealand earthquake catalogue.

As the controlled parameter was changed, the refitted values of the other parameters changed in a way that was consistent with the notion of a space–time trade-off. The results are shown for New Zealand in Figure 8a and for California in Figure 8b.

**Figure 8.** Trade-off of spatial and temporal scaling factors *<sup>σ</sup>A*<sup>2</sup> and 10*aT* , respectively, revealed by the fit of the EEPAS model with controlled values of *σA* (blue triangles) and *aT* (black squares). The straight line with a slope of −1 represents an even trade-off between space and time. (**a**) New Zealand. (**b**) California.

In each plot, the pairs of scaling factors resulting from controlling *σA* are shown as blue triangles, and those resulting from controlling *aT* are shown as black squares. The temporal scaling factor decreased as the controlled spatial scaling factor increased, and the spatial scaling factor decreased as the controlled temporal scaling factor increased. However, the curves had different slopes depending on whether *σA* or *aT* was the controlled variable. An even trade-off line with a slope of −1 is drawn through the intersection of the two curves (straight blue line in Figure 8a,b). Its slope lies between the average slopes of the two controlled fitting curves.
