# Resource Mapping Allocation Scheme in 6G Satellite Twin Network
## Appendix A  
There is a set of performance lower bounds \(\mathbf{I^{G}}=\left[I_{1}^{G},\cdots I_{J}^{G}\right]^{T}\) for the domain G with the mean \(\mu\) and the variance \(\sigma^{2}\). The correlation of the domain G can be expressed as  
\[\begin{array}{ll}Ccvs(G^{1},G^{2})&=\sum\limits_{j=1}^{J}\alpha(G^{1})\times \log\left(|\mathbf{r}_{j}^{1}|\right)/\log\biggl{\{}-\frac{\max\left\{R^{2}\right\} }{|\mathbf{r}_{j}^{1}|}\biggr{\}}\\ &=\sum\limits_{j=1}^{J}\alpha(G^{1})\times\log(\sqrt{\left(\text{cap}f_{j}^{1 }\right)^{2}+\left(\text{mem}f_{j}^{1}\right)^{2}+\left(\text{bar}f_{j}^{1} \right)^{2}})\\ &/\log\Biggl{\{}-\frac{\max\left\{R^{2}\right\}}{\sqrt{\left(\text{cap}f_{j}^{ 1}\right)^{2}+\left(\text{mem}f_{j}^{1}\right)^{2}+\left(\text{bar}f_{j}^{1} \right)^{2}}}\Biggr{\}} \tag{11}\]  
The maximum likelihood value is  
\[\frac{1}{\left(2\pi\right)^{\frac{J}{2}}(\sigma^{2})^{\frac{J}{2}}|Ccvs|^{ \frac{J}{2}}}\exp\left[\frac{-\left(\mathbf{I^{G}}-\mathbf{1}\mu\right)^{T}Ccvs^{-1}( \mathbf{I^{G}}-\mathbf{1}\mu)}{2\sigma^{2}}\right]\text{,} \tag{12}\]  
Taking the logarithm of the above equation,  
\[-\frac{J}{2}\log\left(\sigma^{2}\right)-\frac{1}{2}\log\left(|Ccvs|\right)- \frac{-\left(\mathbf{I^{G}}-\mathbf{1}\mu\right)^{T}Ccvs^{-1}(\mathbf{I^{G}}-\mathbf{1}\mu)}{2 \sigma^{2}}\text{,} \tag{13}\]  
The mean \(\hat{\mu}\) and variance \(\hat{\sigma}^{2}\) of the maximum likelihood function can be calculated by the partial derivatives of the mean \(\mu\) and variance \(\hat{\sigma}^{2}\) of the above equation:  
\[\hat{\mu}=\frac{\mathbf{1}^{T}Ccvs^{-1}\mathbf{I^{G}}}{\mathbf{1}^{T}Ccvs^{-1}\mathbf{1}}\text {,} \tag{14}\]  
\[\hat{\sigma}^{2}=\frac{\left(\mathbf{I^{G}}-\mathbf{1}\mu\right)^{T}Ccvs^{-1}(\mathbf{I^{ G}}-\mathbf{1}\hat{\mu})}{n}\text{,} \tag{15}\]  
The logarithmic maximum likelihood value is  
\[-\frac{J}{2}\log\left(\hat{\sigma}^{2}\right)-\frac{1}{2}\log\left(|Ccvs| \right)\text{,} \tag{16}\]  
The covariance matrix of the domain G can be expressed as  
\[\widetilde{Ccvs}=\bigl{(}\begin{matrix}Ccvs&Ccvs^{-1}\\ Ccvs^{-1}&\mathbf{1}\end{matrix}\bigr{)}\text{,} \tag{17}\]  
\(\mathbf{I^{G^{*}}}\) is denoted as the target value. The logarithmic maximum likelihood value is  
\[-\frac{J}{2}\log\left(\sigma^{2}\right)-\frac{1}{2}\log\left(|Ccvs|\right)- \frac{-\left(\mathbf{I^{G^{*}}}-1\hat{\mu}\right)^{T}\widetilde{Ccvs}^{-1}(\mathbf{I^ {G}}-\mathbf{1}\mu)}{2\hat{\sigma}^{2}}\text{,} \tag{18}\]  
Make C = \(\left(1-Ccvs^{-1T}CcvsCvs^{-1}\right)^{T}\). Then, the inverse of the covariance matrix can be expressed as  
\[\widetilde{Ccvs}^{-1}=\bigl{(}\begin{matrix}Ccvs+CCcvs^{-1}&-CcvsC\\ -CCcvs^{-1}&C\end{matrix}\bigr{)}\text{,} \tag{19}\]  
The expanded likelihood function is  
\[\left[\frac{-1}{2\hat{\sigma}^{2}(1-Ccvs^{-1T}Ccvs^{-1})}\right](\hat{I^{G^{* }}}-\hat{\mu})^{2}+\biggl{[}\frac{Ccvs^{-1}Ccvs(\mathbf{I^{G}}-\mathbf{1}\hat{\mu})}{ \hat{\sigma}^{2}(1-Ccvs^{-1T}CcvsCvs^{-1})}\biggr{]}(\hat{I^{G^{*}}}-\hat{\mu}) \text{,} \tag{20}\]The above equation is the quadratic function of \(l^{G^{*}}\). The derivative of the above equation, and equal to 0, can be obtained:  
\[\left[\frac{-1}{2\vartheta^{2}(1-Ccvs^{-1T}CcvsCvs^{-1})}\right](I^{G^{*}}-\hat{ \mu})+\left[\frac{Ccvs^{-1}Ccvs(I^{G}-1\hat{\mu})}{\vartheta^{2}(1-Ccvs^{-1T}CcvsCvs^ {-1})}\right]=0, \tag{11}\]  
The solution is  
\[l^{G^{*}}=\hat{\mu}+Ccvs^{-1T}Ccvs(I^{G}-1\hat{\mu}), \tag{12}\]