## 3 Case Study
### Comparison and Analysis of the Gross Error Removal Effect  
In this paper, the LMS-IGG-ELM gross errors identification method is utilized to compare with the popular DBSCAN clustering algorithm, the traditional Romanovsky criterion and the 3\(\sigma\) method to verify and analyze the effects of different methods, which are exhibited in Tables 4 and 5:  
Where \(N_{d}\) is the number of gross errors identified, and \(r_{d}\)(%) is expressed as:  
\[r_{d}(\%)=\frac{N_{d}}{total\quad gross\quad errors}\times 100\% \tag{23}\]  
From Tables 4 and 5, using the traditional 3\(\sigma\) method and Romanovsky criterion can only identify extremely obvious gross errors, so the identification ability of the gross errors is poor; using the DBSCAN clustering algorithm can identify 16 gross errors, the identification rate is 72.73%; using the LMS-IGG-ELM method can identify 22 gross errors with an identification rate of 100%. We adjusted the size values of these gross errors, conducted five sets of experiments, and the resulting gross errors identification effect was basically consistent, which verified the generalization and reliability of the method. Through comparison, the LMS-IGG-ELM method has better resistance to gross errors, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multirow{2}{*}{**Raw**} & \multirow{2}{*}{**a*/mm**} & \multirow{2}{*}{**b*/mm**} & \multicolumn{4}{c}{**The Identified Situation of Gross Errors**} \\ \cline{5-8}  & & & & & **LMS-IGG-ELM** &
\begin{tabular}{c} **DBSCAN** \\ **Clustering** \\ **Algorithm** \\ \end{tabular} & \multirow{2}{*}{**Romanovsky Criterion**} & \multirow{2}{*}{**3\(\sigma\) Method**} \\  & & & & **Method** & **Algorithm** & **Criterion** \\ \hline
10 November 2016 & 38.85 & \(-\)5.85 & 33.00 & \(\surd\) & \(\surd\) & \\
11 November 2016 & 38.75 & 4.25 & 43.00 & \(\surd\) & \(\surd\) & \\
13 November 2016 & 38.82 & \(-\)5.82 & 33.00 & \(\surd\) & \\
14 November 2016 & 38.81 & 9.19 & 48.00 & \(\surd\) & \(\surd\) & \\
15 November 2016 & 38.78 & \(-\)6.78 & 32.00 & \(\surd\) & \\
16 November 2016 & 38.70 & 7.30 & 46.00 & \(\surd\) & \(\surd\) & \\
18 November 2016 & 38.67 & \(-\)6.67 & 32.00 & \(\surd\) & \\
19 November 2016 & 38.60 & 8.40 & 47.00 & \(\surd\) & \(\surd\) \\
21 November 2016 & 38.52 & \(-\)6.02 & 32.50 & \(\surd\) & \(\surd\) \\
22 November 2016 & 38.48 & \(-\)6.48 & 32.00 & \(\surd\) & \(\surd\) \\
24 November 2016 & 38.47 & 4.53 & 43.00 & \(\surd\) & \(\surd\) \\
25 November 2016 & 38.48 & 3.52 & 42.00 & \(\surd\) & \(\surd\) \\
5 March 2017 & 26.05 & 6.95 & 33.00 & \(\surd\) & \(\surd\) \\
6 March 2017 & 25.72 & \(-\)12.22 & 13.50 & \(\surd\) & \\
7 March 2017 & 25.36 & 3.64 & 29.00 & \(\surd\) & \\
8 March 2017 & 25.00 & 3.50 & 28.50 & \(\surd\) & \\
1 August 2017 & 28.24 & 29.76 & 58.00 & \(\surd\) & \(\surd\) & \\
15 September 2017 & 36.56 & \(-\)5.56 & 31.00 & \(\surd\) & \(\surd\) & \\
16 September 2017 & 36.48 & 26.52 & 63.00 & \(\surd\) & \(\surd\) & \(\surd\) \\
17 September 2017 & 36.73 & 26.27 & 63.00 & \(\surd\) & \(\surd\) & \(\surd\) \\
18 September 2017 & 36.74 & \(-\)6.74 & 30.00 & \(\surd\) & \(\surd\) & \\
19 September 2017 & 36.63 & \(-\)6.63 & 30.00 & \(\surd\) & \(\surd\) & \\ \hline \hline \end{tabular}  
*: Gross errors of the displacement measurement value. b*: Displacement measurement after inserting gross error. \(\surd\): Gross error is identified.  
\end{table} 
Table 4: Gross errors insertion positions and gross errors identification under the four methods. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Gross Errors Identification Method** & \(N_{d}\) & \(r_{d}\) (\%) \\ \hline LMS-IGG-ELM method & 22 & 100 \\ DBSCAN clustering algorithm & 16 & 72.73 \\ Romanovsky criterion & 3 & 13.64 \\ 3\(\sigma\) method & 2 & 9.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison table of gross errors identification rate of measuring point PL13-3.