## 4 Analysis of Field Test Results
### Machine Learning Residual Estimation
For the training set, all methods, except for our proposed algorithm, generated a Gaussian-shaped distribution, indicating the lack of consideration for multipath effects by the other models. In contrast, our machine learning-based SBDD model resulted in a bimodal shape because it accounted for a multipath effect, \(res_{LPF}\) in Equation (22), as discussed in Section 3.2. While the normalized HPF data, \(\frac{res_{HPF}}{\sqrt{(\sigma_{HPF}^{2}+\sigma_{LPF}^{2})}}\), follows a Cauchy distribution, the LPF magnitude, \(\frac{res_{LPF}}{\sqrt{(\sigma_{HPF}^{2}+\sigma_{LPF}^{2})}}\), is the dominant component of the mixed distribution, leading to a distribution that deviates from the standard normal distribution. As the effects of the multipath are non-zero, the normalized distribution has peaks near \(\pm 1\) when the multipath has been modeled correctly for the training dataset. However, if the modeling error is sufficiently large, the distribution will return to being of a Gaussian shape. On the contrary, the statistics within the \(1\sigma\) and \(2\sigma\) bounds reveal different results than those inferred from the discrepancies in the distribution shapes near the center. For fault monitoring purposes, our primary focus lies on the statistics within the \(1\sigma\) and \(2\sigma\) bounds. Figure 9: Probability Distribution Functions of the training data residuals normalized by the models. The model that exhibits the closest match to these bounded statistics is the one we need to utilize. Although the distribution shape is close to Gaussian, the ZBDD underestimated the residuals because it removes all the multipath and atmospheric noise, as previously explained. As a result, probabilities bound to \(\pm 1\) and \(\pm 2\) by ZBDD were only 0.073 and 0.153, respectively, instead of being 0.686 and 0.954 of a unit normal distribution. As summarized in Table 2, the elevation-based function also underestimated the residuals, while the C/No-based modeling method overestimated them. Despite the difference in the shape of the probability density function, the statistical properties in the central part of the SBDD are the closest to those of the unit normal distribution, with probabilities of 0.683 and 0.980 for the \(\pm 1\) and \(\pm 2\) bounding, respectively. Figure 10 and Table 3 show the results of residual modeling for the test data, using the models obtained from the training data. In contrast to Figure 9, where the SBDD distribution was concentrated at \(\pm 1\), the SBDD distribution exhibits a unimodal shape due to the increase in the modeling error. Using the same training dataset for both model generation and residual normalization resulted in a significant effect of the multipath, leading to a bimodal distribution. However, when the test dataset was used for the normalization, which differs from the sigma modeling, the model's uncertainty increased. As a result, the influence of \(\frac{res_{\textit{BDD}}}{\sqrt{(c_{\textit{BDD}}^{2}+c_{\textit{LFF}}^{2})}}\) in Equation (22) becomes more pronounced, causing the bimodal distribution to be less noticeable. Similar to the statistics in Table 2, other methods over- or under-estimated the actual residuals, but the proposed method produced a distribution that is statistically most similar to the standard normal distribution. The metrics for bounding are provided in Table 3, and it is evident that the proposed model surpasses its counterparts in performance.