## Validation Methodologies of the Proposed Method  
The proposed method learns multipath/direct-signal only features from training datasets; then, test datasets can be used to validate the trained detector. Note that the trained detector does not learn any multipath features from test datasets. Except for data used in parameter tuning (detail is presented in Section 5), all the other training and test datasets are collected in different satellite geometries with different multipath errors in observations. The proposed method has been validated with simulated and real data. The aim of testing with simulated data is to validate the proposed method in a well-controlled environment, where all the multipath/direct-only signals can be manually defined and clearly labelled. Real data is used to train and test detectors so that the feasibility of the proposed method can be examined in practice. Recall, rate of false detection, and accuracy are used to assess the performance of the trained detector, and their definitions are shown in Table 2. Recall is a metric used in machine learning to evaluate how many 'true positive' samples (i.e., multipath contaminated measurements in this research) can be detected by a classifier. A high recall means a classifier is sensitive to 'true positive' samples. The rate of false detection is the number of incorrect multipath detections as a percentage of the total number of multipath detections. Accuracy is the number of correct detections (for both multipath and direct-signal only) samples as a percentage of all samples. Training and testing with simulated data and real data are independent (i.e., no cross-validation between simulated and real data is performed) because their labels are determined with different methods (labelling of simulated data is explained in Section 4). A criterion is used to label multipath measurement on the real data based on double-differenced carrier phase residual error: if the double-differenced carrier phase residual error of a measurement at an epoch exceeds 99% (three-sigma) of measurement errors in a low multipath environment (i.e., three times of standard deviation (S.D.) of the double-differenced carrier phase errors), then that measurement is labelled as a multipath measurement. Because the proposed method process MP and CNR separately in feature extraction and 'true' carrier phase residuals are used to label multipath measurements, the orthogonality between CNR and MP with carrier phase has no effect on multipath detection. Multipath at the reference receiver is negligible. Real multipath errors may be small and they cannot be distinguished from noise in practice; they are labelled as a direct-signal only class. Therefore, real data will have a higher rate of \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Simulated/Real Data**} & \multicolumn{3}{c}{**Label**} \\ \cline{2-4}  & & **Multipath** & **Direct-Signal Only** \\ \hline \multirow{2}{*}{Detection} & Multipath & N\({}_{11}\) & N\({}_{12}\) \\  & Direct-signal only & N\({}_{21}\) & N\({}_{22}\) \\ \hline \multirow{2}{*}{Rate of false detection (on multipath) Accuracy} & Recall & N\({}_{11}\)/(N\({}_{11}\) + N\({}_{21}\)) \\  & & N\({}_{12}\)/(N\({}_{11}\) + N\({}_{12}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of recall, rate of false detection, and accuracy. false detection than simulated data. The S.D. of un-differenced random carrier phase errors are 1 mm (L1) and 1.2 mm (L2) for GPS determined in [32]. Detection results can be further validated in position domain by down-weighting the detected multipath measurements. However, developing linear or nonlinear down-weighting methods is not the main focus of this paper due to its complexity.