## 3 Z-Score Method Using the Proposed Scale Estimator
### Detecting Outliers with the MAD Scale Estimator  
The conventional Z-score method based on Equation (2) is highly influenced by outliers. An outlier can skew the results of an observation, causing the mean to no longer be representative of the observation. The approach to detecting outliers using the modified Z-score is the same as using the conventional Z-score method; however, the only difference is that the mean of the distribution is replaced by the median of the distribution, and the SD is replaced with the MAD.  
Given the time-series RSS observations shown in Equation (1), the modified Z-score value of an \(\mathbf{rss}(\mathbf{n})\) is obtained using Equation (5).  
\[z_{MAD}(rss(n))=\frac{\left|\mathit{rss}(n)-\mathrm{MED}_{\mathbf{rss}_{i}} \right|}{\alpha\mathrm{MAD}(\mathit{rss}(n))} \tag{5}\]  
where \(\alpha=1.4285\) is called the correction factor and is used to make the MAD scale estimator consistent with the distribution of the time-series RSS observation, \(\mathrm{MED}_{\mathbf{rss}_{i}}\) is the median on the RSS observation in Equation (1) and \(MAD(\mathit{rss}(n))\) is the MAD scale estimator mathematically obtained using Equation (6) [28; 36].  
\[MAD(\mathit{rss}(n))=\mathit{median}\Big{\{}\big{|}\mathit{rss}(n)-\mathrm{ MED}_{\mathbf{rss}_{i},i}\Big{|}\Big{\}} \tag{6}\]  
An RSS observation is considered an outlier if its Z-score value obtained using Equation (5) is above a predefined threshold value \"\(\gamma\)\". The values of \(\gamma\) range from \(\pm 0.5\) to \(\pm 5\). That is, for a given RSS observation, in Equation (1), it is considered an RSS outlier if:  
\[\left|z_{MAD}(\mathit{rss}(n))\right|>\left|\gamma\right| \tag{7}\]The MAD scale estimator based on Equation (6) has several drawbacks, including the following: (1) It has only 37% Gaussian efficiency; and (2) the approach to symmetric dispersion, which is based on giving equal weight to positive and negative deviations from the median value, contradicts the general theory of M-estimators [29]. Furthermore, MAD has a high gross error sensitivity, which means that it is more likely to mistake valid observations for outliers. Other well-known scale estimators, such as the least median square (LMS), \(S_{n}\), and \(Q_{n}\) scale estimators, are also used to identify outliers in univariant time-series observations and share the same 50% breakdown point as the MAD scale estimator [29]. Although the LMS is less computationally complex, it has the same gaussian efficiency as the MAD, which is lower than that of the \(S_{n}\) and \(Q_{n}\) scale estimators. The \(Q_{n}\) scale estimator is computationally complex but has the lowest gross error sensitivity. The \(S_{n}\) scale estimator is thought to have moderate computational complexity (lower than \(Q_{n}\) but higher than LMS and MAD) and moderate efficiency (higher than LMS and MAD but lower than \(Q_{n}\)) among the four scale estimators [29]. Thus, as an alternative to MAD, the \(S_{n}\) scale estimator is proposed to be used in determining the Z-score value of an RSS observation. Outlier detection using the Z-score based on the \(S_{n}\) scale estimator is presented in the next subsection.