# Prediction Equation for Horizontal Ground Motion Peak Values in the Loess Plateau Region
## 6 Residual Analysis of Horizontal Peak Parameter  
### Attenuation Relationships  
In this section, the residuals refer to the logarithmic differences (Base 10) between the observed and predicted values from the strong motion records, which are the logarithms of the observed values minus the logarithms of the predicted values. The observed values are from the attenuation research dataset while the predicted values are calculated based on the peak prediction equations presented in this paper. Figure 20 depicts the distribution of PGA fitting residuals based on epicentral distance and Figure 21 illustrates the distribution of PGA fitting residuals based on magnitude. Figure 21: Distribution of PGA fitting residuals by magnitude. Figure 20: Distribution of PGA fitting residuals by epicentral distance. The left chart in Figure 20 displays the distribution of PGA residuals for Model I, for a residual range of \(-1.30\) to \(2.08\) and an average of all residuals of \(0.196\). The right chart displays the PGA residuals for Model II, with a residual range of \(-1.21\) to \(1.02\) and an average of all residuals of \(-0.0657\). Comparing the two charts reveals that the fitting residuals for Model II are smaller than those for Model I, indicating that Model II provides a better fit than Model I. In Figure 21, the left and right charts represent the distribution of residuals for Model I and Model II, with pentagrams marking the arithmetic mean of residuals for different magnitude brackets. For Model I, the average residuals for each magnitude bracket are as follows: \(M_{\rm S}\) 3.2 at \(0.44\), \(M_{\rm S}\) 4.2 at \(0.50\), \(M_{\rm S}\) 4.5 at \(0.26\), \(M_{\rm S}\) 5.0 at \(0.12\), \(M_{\rm S}\) 5.5 at \(0.09\), \(M_{\rm S}\) 6.2 at \(0.13\), and \(M_{\rm S}\) 6.3 at \(0.04\). For Model II, the corresponding values are \(M_{\rm S}\) 3.2 at \(0.12\), \(M_{\rm S}\) 4.2 at \(-0.04\), \(M_{\rm S}\) 4.5 at -0.15, \(M_{\rm S}\) 5.0 at \(-0.30\), \(M_{\rm S}\) 5.5 at \(-0.22\), \(M_{\rm S}\) 6.2 at \(0.09\), and \(M_{\rm S}\) 6.3 at \(-0.09\). Hence, there are differences in Model I and Model II performance under various magnitude conditions. Model I underestimates the PGA values for magnitudes less than \(M_{\rm S}\) 4.5 and Model II slightly overestimates the PGA values for magnitudes between \(M_{\rm S}\) 4.5 and \(M_{\rm S}\) 5.5. For magnitudes less than \(M_{\rm S}\) 4.5, the predictions from Model II are more accurate, whereas, for magnitudes between \(M_{\rm S}\) 4.5 and \(M_{\rm S}\) 5.5, the predictions from Model I are more favorable. When the magnitude exceeds \(M_{\rm S}\) 5.5, the prediction accuracy of the two models is comparable. Figure 22 displays the distribution of fitting residuals for EPA, where the left chart refers to Model I, ranging from \(-1.28\) to \(2.17\) with an average of \(0.23\), and the right chart refers to Model II, ranging from \(-1.24\) to \(1.09\) with an average of -0.08. Comparing the two charts indicates that Model II has smaller residuals and a superior fitting result than Model I. Figure 23 presents the EPA residuals for different magnitude brackets, where the left chart refers to Model I and the right for Model II, with pentagrams indicating the arithmetic mean of residuals for each magnitude. The trend in the distribution of the EPA residuals is roughly consistent with that of PGA. The average residuals for Model I are \(M_{\rm S}\) 3.2 at \(0.46\), \(M_{\rm S}\) 4.2 at \(0.57\), \(M_{\rm S}\) 4.5 at \(0.32\), \(M_{\rm S}\) 5.0 at \(0.20\), \(M_{\rm S}\) 5.5 at \(0.15\), \(M_{\rm S}\) 6.2 at \(0.10\), and \(M_{\rm S}\) 6.3 at \(0.06\); for Model II, they are \(M_{\rm S}\  
magnitudes while Model II's predictions are more accurate, particularly for magnitudes below \(M_{\rm S}\) 4.5. Figure 24 illustrates the distribution of fitting residuals for PGV based on epicentral distance.