## 2 The Principle of Uniform Accuracy Lifetime Design
### Definition and Determination of the Accuracy Loss Weight  
To achieve the optimal design of the accuracy lifetimes of a measurement system, the accuracy loss functions of components should be rationally changed. However, there are two problems that should first be solved: (1) What are the main components that affect the accuracy loss of the system? (2) When the accuracy loss of the main components changes, what are the impacts on the whole system?To solve the above two problems, the accuracy loss weight is proposed. In our opinion, the accuracy loss weight should express the influence coefficient of each component's accuracy loss characteristic on that of the whole system. The greater the accuracy loss weight, the greater the effect of the corresponding component on the whole system. This component should be considered in advance to improve its accuracy loss law. When the accuracy loss weight is small enough, the accuracy loss of the corresponding component can be ignored, and the design model can be simplified. In order to determine the main parameters to be designed and to establish the optimization model, the accuracy loss weight of each component should be calculated.  
For a measurement system with \(n\) components, the accuracy loss \(\delta_{s}(t)\) of the whole system can be expressed as follows:  
\[\delta_{s}(t)=p_{1}\delta_{1}(t)+p_{2}\delta_{2}(t)+\dots+p_{n}\delta_{n}(t), \tag{4}\]  
where \(p_{i}\) is the accuracy loss weight of the \(i\)-th component. Suppose that the accuracy loss data at different times \(t_{i}\) (\(i=1,\,2,\,\dots,\,m\)) are obtained through dynamic accuracy loss experiments: \(\{\delta_{s}(t),\,\delta_{1}(t),\,\delta_{2}(t),\,\dots,\,\delta_{n}(t)\}\). According to the definition of the accuracy loss weight, the multiple linear regression method can be adopted to estimate the accuracy loss weight \(p_{i}\).  
Defining  
\[Y=\left[\begin{array}{c}\delta_{s}(t_{1})\\ \delta_{s}(t_{2})\\ \vdots\\ \delta_{s}(t_{m})\end{array}\right],\,X=\left[\begin{array}{cccc}\delta_{1} (t_{1})&\delta_{2}(t_{1})&\cdots&\delta_{n}(t_{1})\\ \delta_{1}(t_{2})&\delta_{2}(t_{2})&\cdots&\delta_{n}(t_{2})\\ \vdots&\vdots&\cdots&\vdots\\ \delta_{1}(t_{m})&\delta_{2}(t_{m})&\cdots&\delta_{n}(t_{m})\end{array}\right],\,P=\left[\begin{array}{c}p_{1}\\ p_{2}\\ \vdots\\ p_{n}\end{array}\right],\]  
the best estimated value of the accuracy loss weight \(p_{i}\) in matrix form can be obtained in terms of the least-squares principle:  
\[\hat{P}=\left[\begin{array}{c}\hat{p}_{1}\\ \hat{p}_{2}\\ \vdots\\ \hat{p}_{n}\end{array}\right]=\left(X^{T}X\right)^{-1}X^{T}Y. \tag{5}\]  
In accordance with the principle of uniformity design of accuracy lifetimes, the re-designed accuracy loss sequences \(\{\delta_{s}(t)^{\prime},\,\delta_{1}(t)^{\prime},\,\delta_{2}(t)^{\prime},\, \dots,\,\delta_{n}(t)^{\prime}\}\) should satisfy Formula (6):  
\[\left\{\begin{array}{c}\delta(t)^{\prime}|_{t=T_{1}^{\prime}}=p_{1}(\cdot) \delta_{1}(t)^{\prime}+p_{2}(\cdot)\delta_{2}(t)^{\prime}+\dots+p_{n}(\cdot) \delta_{n}(t)^{\prime}|_{t=T_{s}^{\prime}}=\delta_{\text{As}}\\ \delta_{1}(t)^{\prime}|_{t=T_{1}^{\prime}}=\delta_{\Delta 1}\\ \vdots\\ \delta_{n}(t)^{\prime}|_{t=T_{n}^{\prime}}=\delta_{\Delta n}\end{array}\right.. \tag{6}\]  
In addition, the updated uniformity \(\rho^{\prime}\) can be calculated in terms of Formula (3). Meanwhile, \(\rho^{\prime}\) should meet the inequation \(\rho^{\prime}\geq\varepsilon\). After optimization, the uniformity design of accuracy lifetimes can be realized in the system.