## 3 Stress Measurements and Data Updating Based on Grey Theory
### Data Updating
#### 3.3.1 Theoretical Background of Grey Theory  
Grey systems are those with incomplete known information. A basic viewpoint of grey theory is that all random variables are grey variables within certain variation ranges, and related predictive models can be established based on limited data without exploring the uncertainties influencing the system behaviors or separating the system behaviors from the noise behaviors. Grey variables should be processed following some innovative rules to reveal the inherent laws among them, rather than traditional statistical laws. Accumulated generating operation (AGO) is a fundamental grey-theory-based data processing method, which can transform disordered random sequences into well-ordered regular arrays.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Factor** & **Sum of Squares** & **DF** & **Mean Square** & **F Value** & **Prob \textgreater{} F** & **R2** & **Adequate Precision** \\ \hline Regression model (factor) & \(2.758\times 10^{10}\) & 1 & \(2.758\times 10^{10}\) & \(1.780\times 10^{8}\) & \(<\)0.0001 & & \\ \hline Residual & 774.60 & 5 & 154.92 & & & 1.0000 & 23,532.462 \\ \hline Lack of fit & 774.60 & 3 & 258.20 & & & & \\ \hline \end{tabular}
\end{table}
Table 2: Significant test results for the established regression model.  
Figure 7: Temperature discrepancy–stress plot.  
The original sequence and the generated sequence are, respectively, assumed as \(\left\{x^{(0)}\right\}=\left(x^{(0)}{}_{(1)},x^{(0)}{}_{(2)},\ldots,x^{(0)}{}_{(n )}\right)\) and \(\left\{x^{(1)}\right\}=\left(x^{(1)}{}_{(1)},x^{(1)}{}_{(2)},\ldots,x^{(1)}{}_{( n)}\right)\). If  
\[x^{(1)}{}_{(k)}=\sum_{m=1}^{k}x^{(0)}{}_{(m)} \tag{2}\]  
\(\left\{x^{(1)}\right\}\) is the first-order accumulative sequence of \(\left\{x^{(0)}\right\}\), written as 1-AGO sequence.  
Assuming the system characteristic data are \(\left\{x^{(0)}\right\}=\left(x^{(0)}{}_{(1)},x^{(0)}{}_{(2)},\ldots,x^{(0)}{}_ {(n)}\right)^{T}\), \(\left\{z^{(1)}\right\}\) is the neighboring averagely generated sequence of \(\left\{x^{(1)}\right\}\), which can be written as:  
\[z^{(1)}{}_{(k)}=0.5x^{(1)}{}_{(k)}+0.5x^{(1)}{}_{(k-1)},\,(k=2,3,\ldots,n) \tag{3}\]  
The following grey differential equation can be regarded as GM (1,1) model:  
\[x^{(0)}{}_  
(2) compare \(x^{(0)}{}_{(k+1)}-x^{(0)}{}_{(k)}\) with \(x^{(0)}{}_{(k+1)}-x^{(0)}{}_{(k)}\); (3) when \(x^{(0)}{}_{(k+1)}-x^{(0)}{}_{(k)}\
otin\left(0.5\times\left(\overset{\wedge}{x^{ (0)}{}_{(k+1)}-x^{(0)}{}_{(k)}}\right),1.5\times\left(\overset{\wedge}{x^{(0)}{ }_{(k+1)}-x^{(0)}{}_{(k)}}\right)\right)\), \(x^{(0)}{}_{(k+1)}-x^{(0)}{}_{(k)}\) is replaced by \(x^{(0)}{}_{(k+1)}-x^{(0)}{}_{(k)}\) to form the updated stress sequence; otherwise, the original value of the measured stress is kept in the updated stress sequence. The updated stress sequences obtained in this way are also shown in Figure 8. As can be seen, the updated stress sequences are extremely close to the values predicted by GM (1,1) models, demonstrating their fidelities. The updated stress sequences are regarded as the reference data in the subsequent model updating.