# New Quality Control Algorithm Based on GNSS Sensing Data for a Bridge Health Monitoring System
## 2 Quality Control Algorithm and Quality Measures
### Quality Measures
#### 2.3.1 Redundancy Number  
Equation (5) illustrated that both random errors and outliers affect the residuals of measurements contained in the outliers. Unfortunately, these two errors cannot be differentiated by the least squares method, which is limited to random error. Therefore, it is difficult to detect outliers by using residuals estimated by the least squares method. However, J = \(Q_{v}w\) has a singular and idempotent matrix structure. With this matrix, it is possible to check for the redundancy number if the sum of the diagonal elements of the matrix (known as the trace of the matrix) is the same as the degrees of freedom:  
\[Trace(Q_{v}w)=rank(Q_{v}P)=n-u=r \tag{44}\]  
where r is the total redundancy. If \(r_{i}\) denotes the diagonal elements of J = \(Q_{v}P\), from Equation (6), \(r_{i}\) is redefined as follows:  
\[\sum_{i=1}^{n}r_{i}=rr_{i}=(Q_{v}P)_{ii}=Q_{v_{i}}P_{i}=e_{i}^{T}Q_{v_{i}}P_{i}e_ {i} \tag{45}\]  
Figure 3: Quality control flowchart.  
Because of the nature of the idempotent matrix, the elements of this matrix are distributed in the range of \(0\leq r_{i}\leq 1\). The closer the redundancy is to 1, the closer the variance of residuals is to the variance of the'measurements. Thus, in conclusion, this relation is the same with random errors of residual and measurements; therefore, the estimated coordinates (GNSS positioning solution) with a high level of precision can be obtained. However, if the redundancy is closer to 0, the variance of residuals has a smaller value close to 0; therefore, this value cannot be a correct comparison with the value of the measurements. In other words, a low level of redundancy is insufficient for separating outliers. Furthermore, the measurements have a higher possibility of containing outliers; moreover, a high level of redundancy means that the outliers can be easily detected because the outliers have an impact on the residual. Nevertheless, if the redundancy becomes 0, the outliers cannot be detected. Therefore, estimated coordinates may not be accurate. In general, when the level of redundancy is higher than 0.5, it can be considered well measured [6]. If \(\
abla v\) is reformed through Equation (46), it can be expressed as \(\
abla\overline{v}=Q_{v}w\
abla l\), and the individual \(\
abla v_{i}\) can be defined as:  
\[\
abla v_{i}=r_{i}\cdot\
abla l_{i} \tag{46}\]  
This can be seen as the contribution of a single measurement \(l_{i}\) against the whole redundancy \(r\). Relative redundancy can then be defined as the average of the diagonal elements of \(M\):  
\[\
abla v_{i}=r_{i}\cdot\
abla l_{i} \tag{47}\]  
\[M=Q_{v}w=I-Q_{\overline{l}}w=I-U \tag{48}\]  
in which:  
\[\mathrm{U}=Q_{\overline{l}}w=AQ_{\overline{r}}A^{T}w \tag{49}\]  
The U matrix is also an idempotent matrix, which can be described as:  
\[Trace(U)=Rank(U)=Rank(Q_{x})=U \tag{50}\]  
\[U_{i}\cdot\
abla l_{i}=\
abla l_{i}-r_{i}\
abla l_{i}, \tag{51}\]  
If an outlier (\(\
abla l_{i}\)) occurs in only one measurement (\(l_{i}\)), it will be reflected in the corresponding residual (\(v_{i}\)), as much as \(\
abla v_{i}=r_{i}\
abla l_{i}\). Hence, it is now clear that an outlier (\(\
abla l_{i}\)), coupled with a large number \(r_{i}\), will more greatly affect the corresponding \(v_{i}\).