## 2 Strain Gauge Location Optimization Based on a Minimum Correlation Criterion
### The Proposed Improved Correlation Method Based on a Minimum Correlation Criterion  
The aim of this paper is to find the fewest number of strain gauge locations to satisfy the requirements of the accuracy of the inversion loads. The selection principle is that the first strain gauge location is primarily defined, and then the second strain gauge location is determined based on the minimum correlation criterion. But for the third or later strain gauge locations, traditional Pearson correlation cannot address this. The reason is that a third strain location is hard to be simultaneously uncorrelated with the first two strain gauge locations. Therefore, a new strain monitoring location should be selected, which has little correlation with all of the previously determined strain monitoring points and has almost identical Pearson correlation coefficients with all of them. In order to obtain the optimal third strain monitoring point and subsequent strain monitoring points, an improved correlation coefficient method is proposed in this study based on the minimum correlation criterion. The proposed method should be able to evaluate the comprehensive correlation between several variables, which is defined as  
\[{r_{k}}^{\prime}=\sqrt{\frac{r_{1k}^{2}+r_{2k}^{2}+\cdots+r_{lk}^{2}}{l}}\geq \sqrt{\frac{(r_{1k}+r_{2k}+\cdots+r_{lk})^{2}}{l^{2}}}, \tag{2}\]  
where \({r_{k}}^{\prime}\) is the mean square root of the correlation coefficient between the \(k\)-th monitoring point and all previous selected monitoring points, and the strain monitoring point with the smallest \({r_{k}}^{\prime}\) will be adopted as the new monitoring point; \(r_{1k}\), \(r_{2k}\), \(\ldots\), \(r_{lk}\) are the Pearson correlation coefficients between the added strain monitoring point \(k\) with the pre-selected \(l\) monitoring points. According to Equation (2), the equal sign is valid when and only when \(r_{1k}=r_{2k}=\cdots=r_{lk}\). When the strain monitoring point with the smallest \({r_{k}}^{\prime}\) is selected, it indicates that this monitoring point possesses nearly identical Pearson correlation degrees with each ofthe preselected monitoring points. In other words, when the sum of the Pearson correlation coefficient in Equation (2) is smaller, the Pearson correlation coefficients \(r_{1k},r_{2k},\ldots,r_{lk}\) are small and tend to be close with each other. In this case, the proposed improved correlation coefficient method is capable of selecting a new monitoring point which is less correlated with the previous \(l\) characteristic strain monitoring points. The objective of the optimized selection method via the improved correlation coefficient method is to select a set of least correlated strain monitoring locations, and the optimized strain monitoring points will be used as the final strain monitoring locations. At the selected strain monitoring point, the corresponding strain sensor is arranged to form the optimized strain sensor network. The specific flow of the optimal selection method of characteristic strain monitoring points using an enhanced correlation coefficient method is shown in Figure 2.  
Figure 2: Improved correlation method for strain monitoring point optimization. As demonstrated in Figure 2, the four procedures of the proposed two-stage strain gauge location optimization method are described as follows:  
Step 1: Select the original monitoring points  
For the different structures to be monitored on the aircraft, the strain distribution of the aircraft structure under various load conditions is obtained by using the finite element model. Based on engineering experience or force transfer path analysis, \(m\) (\(m\) is usually much larger than the actual monitoring requirement) monitoring points are adopted as the original monitoring points. The locations of the original monitoring strain gauges are usually the positions where the strain values are relatively large and the structural geometry is flat (the stress gradient cannot be too large).