# New Quality Control Algorithm Based on GNSS Sensing Data for a Bridge Health Monitoring System
## 2 Quality Control Algorithm and Quality Measures
### Quality Control Algorithm Procedure
#### 2.2.2 Identification  
An outlier, detected through the previously described process, is identified using the w-test data snooping technique [4]. To identify which satellite measurements are responsible, this stage is implemented with new alternative hypothesis testing, which has a standard normal distribution under the null hypothesis. However, under the alternative hypothesis, the distribution of the statistic will have non-centrality. In other words, the w-test for outlier \(\
abla l_{i}\) relies on the null hypothesis, which shows that the measurements are outlier-free, as well as on the alternative hypothesis which, when proved true, indicates the existence of an outlier of magnitude \(\
abla l\):  
\[\text{Null hypothesis }H_{0}:E[\
abla\hat{l}_{i}]=0 \tag{34}\]  
\[\text{Alternative hypothesis }H_{a}:E[\
abla\hat{l}_{i}]=e\
abla l_{i}\
eq 0 \tag{35}\]  
To consider a critical value of \(\lambda_{0}\) by the determined \(\alpha\) and \(\beta\), \(\lambda_{0}\) is rewritten as follows:  
\[\lambda_{0}=\frac{1}{\sigma_{0}^{2}}\
abla_{0}l^{T}PQ_{v}P\
abla_{0}l=\frac{ \
abla_{0}l^{2}}{\sigma_{0}^{2}}e^{T}PQ_{v}Pe \tag{36}\]  
\[|\
abla_{0}l|=\sigma_{0}\sqrt{\frac{\lambda_{0}}{e^{T}PQ_{v}Pe}} \tag{37}\]  
It is then easy to derive a statistic, which tests the alternative hypothesis \(H_{a}\):  
\[H_{a}:\
abla_{0}l=e_{i}\
abla_{0}l_{i} \tag{38}\]  
\[|\
abla_{0}l|=\sigma_{0}\sqrt{\frac{\lambda_{0}}{P_{i}(Q_{v}P)_{ii}}} \tag{39}\]  
On the basis of the previous information, the one-dimensional (one-tailed) test statistic can test the alternative hypothesis, whereby the w-test is defined by the ratio between the size of the outlier and its variance.  
Calculation of the test statistic is as follows:  
\[w_{i}=\delta_{i}=\frac{v_{i}}{\sigma_{v_{i}}}=\frac{v_{i}}{\sigma_{0}\sqrt{q_ {v_{i}}}} \tag{40}\]  
\[w_{i}=\frac{e_{i}^{T}Pv}{\sigma_{0}\sqrt{e_{i}PQ_{v}P\epsilon_{i}}}=\frac{(Pv )_{ii}}{\sigma_{0}\sqrt{(PQ_{v}P)_{ii}}} \tag{41}\]  
Fixing the significance level \(\alpha\) and the power of test \(\beta\), the non-centrality parameter can then be estimated.  
The test statistic for \(w_{i}\) is \(N_{1-\alpha/2}(0,1)\) for the situation where:  
\[(|w_{i}|=\frac{v_{i}}{\sigma_{v_{i}}})>(N_{1-\frac{\alpha}{2}}(0,1)=\sqrt{F_ {1-\alpha_{0}:1,\infty(0,1)}}) \tag{42}\]  
\[\sqrt{\lambda_{0}}=\sqrt{F_{1-\alpha,1,\infty}}+\sqrt{F_{\beta,1,\infty}} \tag{43}\]  
If the \(i\)th measurement is a supposed outlier, the test is carried out with respect to each measurement; the greatest value that exceeds the critical value is deemed an outlier and is removed from the model. The test statistic is iterated from the first element of the vector; each element has a value of 1. This method can screen for the existence of any potential outliers in individual measurements. The process is repeated until an absolute value or the greatest value is detected. The w-test is then repeated to determine if any further outliers exist. If another outlier is found, it is removed fromthe model; moreover, the measurement that was first deemed an outlier is reinstated and the model is retested.  
From Figures 1 and 2 it is evident that, from the significance level and the power of test that the non-centrality parameter can be estimated, \(\sqrt{\lambda_{0}}\). This procedure can then be repeated until no further outliers are detected. This type of procedure for screening each individual measurement for the presence of an outlier is known as data snooping.