# Application of Machine Learning Models to Multi-Parameter Maximum Magnitude Prediction
## 2 Data and Preprocessing
### Multi-Parameter Selection  
This study draws extensively on the research by [PERSON] and [PERSON] (2019) and [PERSON] et al. (2019), calculating 11 seismicity indicators as characteristic parameters from the earthquake catalog data (Table 2). These parameters not only combine the Gutenberg-Richter (G-R) inverse power law and the characteristic distribution of earthquake magnitude but also cover a variety of prediction factors that can comprehensively reflect the relationship of earthquake spatiotemporal intensity, such as the \(b\)-value. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Magnitude Range** & **Region A** & **Region B** \\ \hline
5 \(<M_{\rm L}\leq 6\) & 127 & 297 \\ \(M_{\rm L}>6\) & 23 & 71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Earthquakes of magnitude five or greater in the study area. Figure 1: Schematic diagram of the study area. This index represents the mean magnitude of the most recent \(n\) earthquakes within the time window. The following formula gives the definition:  
\[M_{mean}=\frac{\sum M_{i}}{n} \tag{1}\]  
where \(M_{i}\) is the magnitude value of the \(i\) event. 2. \(E^{1/2}\)  
Most seismic zones can be approximated as open physical systems where energy gradually accumulates through the motion of lithospheric plates. When the stored energy reaches a certain threshold, it is suddenly released in the form of a large earthquake event [29]. The following formula gives the definition:  
\[E^{1/2}=\left(\sum 10^{12+1.8M_{i}}\right)^{1/2} \tag{2}\]  
3. \(b_{1}\) and \(a\)  
In seismology, the \(b\)-value and a-value are two important parameters derived from actual earthquake observation data by fitting the G-R relationship [31]. The G-R relationship is defined as follows:  
\[log_{10}N=a-bM \tag{3}\]  
In this equation, \(N\) represents the number of earthquake events with a magnitude of \(M\) or greater within a given time window. The parameter a indicates the intensity of seismic activity, while the parameter \(b\) represents the ratio of small events to large events. The \(b\)-value determines the frequency of smaller earthquakes compared to larger ones: the higher the \(b\)-value, the more frequent the smaller earthquakes. The \(b\)-value and \(a\)-value obtained through linear least squares regression analysis are as follows:  
\[b_{1}=\frac{n\sum\left(M_{i}log_{10}N_{i}\right)-\sum M_{i}\sum log_{10}N_{i}} {\left(\left(\sum M_{i}\right)^{2}-n\sum M_{i}^{2}\right)} \tag{4}\]  
\[a=\frac{\sum(log_{10}N_{i}+bM_{i})}{n} \tag{5}\]  
4. \(b_{2}\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Seismicity Indicator** & **Description** \\ \hline
1 & \(M_{\text{mean}}\) & Mean magnitude \\
2 & \(E^{1/2}\) & Square root of total earthquake energy \\
3 & \(b_{1}\) & Least squares to obtain the value of \(b\) \\
4 & \(a\) & Least squares to obtain the value of \(a\) \\
5 & \(b_{2}\) & Maximum likelihood method \\  & & to obtain the value of \(b\) \\
6 & \(\sigma\) & Sum of the mean square deviations of \\  & & the G-R curve \\
7 & \(\Delta M\) & Deficit of maximum magnitude \\
8 & \(Lon_{mean}\) & Mean longitude \\
9 & \(Lat_{mean}\) & Mean latitude \\
10 & \(Lon_{E}\) & Energy-weighted mean longitude \\
11 & \(Lat_{E}\) & Energy-weighted mean latitude \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Seismicity parameters. When calculating the \(b\)-value, the least squares method is simple, easy to apply, and widely used. However, it has limitations in practical applications, such as inconsistent weighting between large and small earthquakes and issues with magnitude binning when the sample size is small. Therefore, under specific conditions, the maximum likelihood method can serve as a more practical alternative or supplement [32]. The formula is defined as follows:  
\[b_{2}=\frac{loge}{M_{mean}-M_{c}} \tag{6}\]  
5. \(\sigma\)  
This metric is the sum of the mean square deviations of the G-R curve. A smaller mean square deviation indicates a smaller discrepancy between the G-R curve and the observed data, reflecting a better fit.