# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 2 Tolerance Analysis Considering Form Errors
### Variance Separation Method  
Considering a model in n-dimensional space \(R^{n}\).  
\[y=f(x)=f(x_{1},x_{2},\cdots,x_{n}) \tag{1}\]  
Figure 3: The process of the tolerance analysis. The left part shows the traditional tolerance analysis process, and the right part shows the novel tolerance analysis process.  
All the input variables \(x_{1},x_{2},\cdots,x_{n}\) are independent random variables, and \(y\) is the random output. The joint probability density function (PDF) of \(x\) can be described by \(f_{X}(x)\) with the given distribution parameters \(\theta\).  
Now considering a special case, the distribution parameter \(\theta\) contains uncertainty, and the uncertain of \(\theta\) can be described by the PDF \(f_{\theta}(\theta)\), where \(\theta=\left(\theta_{1},\theta_{2},\cdots\theta_{p}\right)\) are p-dimensional independent variables. The uncertainty of the input variable \(x\) results from two aspects, the randomness of \(x\) itself and the uncertainty of parameter \(\theta\). Once the distribution parameter was determined, the PDF of \(x\) can be described by the conditional PDF \(f_{X}(x|\theta)\). The family of PDFs (probability density functions) was introduced to represent the total uncertainty of variable \(x\), as shown in Figure 4.  
To analyze the uncertainty of \(y\), a double-loop procedure is implemented: the sample of parameters must be generated first, then the sample of variable x can be generated. To simplify the procedure, [PERSON] [20] proposed the separation method based on the probability integral transformation (PIT) to separate the uncertainties of variables and their distribution parameters.  
Denoting \(F_{X}(x|\theta)\) as the cumulative distribution functions (CDF) of variable \(x\), and supposing \(u\) as a standard uniformly distributed random variable, \(u\sim U(0,1)\), then based on the PIT theory, Equation (2) can be proved be true.  
\[u=F_{X}(x|\theta) \tag{2}\]  
By applying the inverse probability integral transformation, the original variable \(x\) can be represented as follow.  
\[x=F_{X}^{-1}(u|\theta) \tag{3}\]  
Then the original function Equation (1) can be transformed to Equation (4), which leads to a single-loop procedure.  
\[y=f(x)=f\big{(}F_{X}^{-1}(u|\theta)\big{)} \tag{4}\]  
[PERSON] [21] improved the above procedure to a generalized separation method. The corresponding PIT can be given as:  
\[F_{\omega}(\omega)=F_{X}(x|\theta),\quad x=F_{X}^{-1}(F_{\omega}(\omega,\theta)) \tag{5}\]  
where the auxiliary variable \(\omega\) is a standard random variable obeying to the same type of distribution with the original variable \(x\). For example, if \(x\) is distributed as normal, then the auxiliary variable \(\omega\) is distributed as standard normal.  
Based on Equation (5), the original function Equation (1) can be described as follow.  
\[y=f(x)=f\big{(}F_{X}^{-1}(F_{\omega}(\omega,\theta))\big{)} \tag{6}\]  
Figure 4: Family of PDFs for a single variable.  
Equations (5) and (6) are the generalized separation model from the auxiliary variable and the distribution parameter to the output, and it is more efficient than Equation (4).