## 3 Fast Channel Information Prediction Based Random Forest
### RB and Power Allocation Algorithm for Energy-Efficient Optimization  
In order to facilitate the analysis and solution of the energy-efficiency optimization problem P2, the RB and power allocation are performed when the predicted SINR and CC allocation (\(\chi_{\mathrm{n}}^{\mathrm{kj}}(\mathrm{t})\) or \(a\) is known) are known. The set of component carriers allocated to user k with MCSj is defined as \(\mathbb{C}_{\mathrm{k}}=\left\{\mathrm{n}|\chi_{\mathrm{n}}^{\mathrm{kj}}=1, \forall\mathrm{n},\forall\mathrm{j}\in\mathbb{J}_{\mathrm{m}}\right\}\) and the set of users with MCSj allocated carrier n is defined as \(\mathbb{U}_{\mathrm{n}}=\left\{\mathrm{k}|\chi_{\mathrm{n}}^{\mathrm{kj}}=1, \forall\mathrm{k}  
\[\sum_{k\in U_{n}}\sum_{j\in J_{m}}\eta^{kj}_{n,m}\ \leq 1,\ \forall m, \forall n\in\mathbb{C}_{k} \tag{14a}\] \[\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{k\in U_{n}}\sum_{j\in J_{m}}S^{kj}_{n,m} \leq P_{T}\] (14b) \[0\leq\eta^{kj}_{n,m}\leq 1,\forall n,\forall m,\forall k,\forall j\] (14c) \[S^{kj}_{n,m}\geq 0,\forall n,\forall m,\forall k,\forall j \tag{14d}\]  
P\({}_{3}\) is a fractional programming problem whose numerator \(\hat{R}(b,\hat{p})\) is a concave function and its denominator \(\hat{P}(b,\hat{p})\) is a positive affine function. According to the nonlinear fractional programming theory, Formula (14) is a quasi-concave function, which can be solved by converting P\({}_{3}\) into a non-fractional subtraction form using Dinkelbach theory. Therefore, the optimization problem P\({}_{3}\) is transformed into a non-fractional subtraction form P\({}_{4}\) and the \(\rho_{i}\) series is constructed to approximate the optimal energy efficiency \(\rho^{*}\). When the component carrier allocation is known, it is assumed that the energy efficiency \(\rho=\frac{\hat{R}(b,\hat{p})}{\hat{P}(b,\hat{p})}\), the objective function, and constraints of P\({}_{4}\) are as follows:  
\[P_{4}\ :f(\rho)=\max_{b,\hat{p}}[\hat{R}(b,\hat{p})-\rho\hat{P}(b,\hat{p})] \tag{15}\]  
s.t. Formulas (14a), (14b), (14c), (14d)  
**Theorem 1**. : _The necessary and sufficient condition for \(f(\rho)\) to obtain the maximum energy efficiency \(\rho^{*}\) is to satisfy: \(\max_{b,\hat{p}}\ \left[\hat{R}(b,\hat{p})-\rho^{*}\hat{P}(b,\hat{p})\right]= \hat{R}(b^{*},\hat{p}^{*})-\rho^{*}\hat{P}(b^{*},\hat{p}^{*})=0\) and when the maximum energy efficiency is \(\rho^{*}\), the optimal value \(\{b^{*},\hat{p}^{*}\}\) of the independent variable corresponding to the problem \(P_{4}\) is also the optimal solution of problem \(P_{3}\)._  
Proof. : With the CC allocation known, let S be the set of feasible solutions for the RB and power allocation scheme \(\{b,\hat{p}\}\) of the P\({}_{3}\), so it satisfies the constraints Formulas (14a), (14b), (14c), and (14d); definition \(x=\{b,\hat{p}\}\) is any feasible solution in the set S, \(\forall x\in\mathbb{S}\). Sufficient condition: In the case of known CC allocation, if \(x^{*}=\{b^{*},\hat{p}^{*}\}\) is a feasible solution for optimal RB and power allocation, then \(\rho^{*}=\frac{\hat{R}(x^{*})}{\hat{P}(x^{*})}\geq\frac{\hat{R}(x^{*})}{\hat{ P}(x^{*})}\), so we obtain the following conclusion: (a): \(\hat{R}(x)-\rho^{*}\hat{P}(x)\leq 0\); and (b): \(\hat{R}(x^{*})-\rho^{*}\hat{P}(x^{*})=0\). From (a) we can obtain \(\max_{x}\{\hat{R}(x)-\rho^{*}\hat{P}(x)|\forall x\in\mathbb{S}\}=0\); from (b) the maximum value is obtained when \(x=x^{*}\). Necessary condition: if \(x^{*}=\{b^{*},\hat{p}^{*}\}\) is a solution of \(\max_{x}\{\hat{R}(x)-\rho^{*}\hat{P}(x)|\forall x\in\mathbb{S}\}\), namely \(\hat{R}(x^{*})-\rho^{*}\hat{P}(x)=0\), then \(\hat{R}(x)-\rho^{*}\hat{P}(x)\leq\hat{R}(x^{*})-\rho^{*}\hat{P}(x^{*})=0\), therefore, we can obtain (c): \(\{\hat{R}(x)-\rho^{*}\hat{P}(x)|\forall x\in\mathbb{S}\}\leq 0\); and (d): \(\hat{R}(x^{*})-\rho^{*}\hat{P}(x^{*})=0\).