# An Effective Surrogate Ensemble Modeling Method for Satellite Coverage Traffic Volume Prediction
## 3 Satellite Coverage Traffic Volume Modeling and Prediction Approximation
### Surrogate Ensemble Modeling for SCTV Prediction
Then the sequence of the surrogates according to GMSE in the ascending order can be obtained, and denote the candidate model set with the ranking sequence as \(\left\{M_{(i)}^{k}\right\}_{i=1,2,\ldots,n_{M}}\) with the corresponding GMSE and predictor sets denoted as \(\left\{\text{GMSE}_{(i)}^{k}\right\}_{i=1,2,\ldots,n_{M}}\) and \(\left\{\hat{y}_{(i)}^{k}\right\}_{i=1,2,\ldots,n_{M}}\). Here, \(n_{M}\) is the total number of the candidate surrogates. To choose the relatively more accurate surrogates from the candidate set \(\left\{M_{(i)}^{k}\right\}_{i=1,2,\ldots,n_{M}}\) so as to compose a more accurate ensemble, the first issue is to define the number of contributing surrogates \(n_{M}^{k*}\leq n_{M}\) to be selected. There are two important points that should be taken into consideration during setting the threshold for the \"more accurate\" candidate selection. On one hand, because each domain has distinct SCTV features and different surrogates have different performance, the threshold is preferred to be decided adaptively in each sub-domain rather than simply fixed by a specific number or ratio. On the other hand, considering that GMSE values of the inaccurate surrogates might greatly deviate from those of the accurate surrogates, the threshold could be determined by borrowing the idea of identifying the outliers [28] so as to rationally screen out the surrogates with low accuracy (or comparatively large GMSE values). According to these considerations, the number of contributing surrogates \(n_{M}^{k}\) to be selected in the \(k\)th sub-domain is set as  
\[ \begin{array}{l}\text{GMSE}_{(n_{M}^{k})}^{k}\leq t^{k}+\delta C^{k}\\ \text{GMSE}_{(n_{M}^{k}+1)}^{k}>t^{k}+\delta C^{k},\end{array}  \tag{19}\]  
where \(\delta\in\mathbb{N}^{+}\) is a user-defined control parameter. From our numerical experience, it is appropriate to take 1 to 3 for \(\delta\). When the value of \(\delta\) is small, there may be less candidate surrogates chosen as positively contributing models. With the enlarged \(\delta\) value, it is likely to select more candidate surrogates into the ultimate ensemble. \(t^{k}\) and \(C^{k}\) are the mean and standard deviation of the GMSE set in the \(k\)th sub-domain. To eliminate the negative effect of the inaccurate models GMSE so as to obtain a robust estimation, \(t^{k}\) and \(C^{k}\) are solved using the first \(M=\left[3n_{M}/4\right]\) (\(\left[\cdot\right]\) denotes the rounding operation) elements of the set \(\left\{\text{GMSE}_{(i)}^{k}\right\}_{i=1,2,\ldots,n_{M}}\)[29]  
\[ \begin{array}{l}t^{k}=\frac{1}{M}\sum\limits_{i=1}^{M}\text{GMSE}_{(i)}^{k }\\ C^{k}=\sqrt{\frac{1}{M}\sum\limits_{i=1}^{M}\left(\text{GMSE}_{(i)}^{k}-t^{k} \right)^{2}}.\end{array}  \tag{20}\]  
From Equations (19) and (20) it can be observed that for different sub-domains, there may be different types as well as different numbers of the contributing surrogates. To combine the independent contributing models into the ultimate ensemble for each sub-domain SCTV prediction, the associated weights are calculated for the \(k\)th sub-domain by  
\[w_{(i)}^{k}=\frac{\sum\limits_{j=1,j\
eq i}^{n_{M}^{k}}\sqrt{\text{GMSE}_{(j)} ^{k}}}{\left(n_{M}^{k*}-1\right)\sum\limits_{j=1}^{n_{M}^{k}}\sqrt{\text{GMSE }_{(j)}^{k}}},i=1,\ldots,n_{M}^{k*}, \tag{21}\]  
and the prediction value of the ensemble in the \(k\)th sub-domain is \[\hat{y}_{PWTA}^{k}=\sum\limits_{i=1}^{n_{M}^{k*}}w_{(i)}^{k}\hat{y}_{(i)}^{k}. \tag{22}\]  
After the surrogate ensemble modeling procedure, eight ensembles are obtained for the eight sub-domains. Notice that for points at the boundary of the sub-domains which belong to two or more sub-domains at the same time, their SCTV values are determined by averaging the predictions of the ensembles from the sub-domains sharing these boundaries.