# An Adaptive Data Rate Algorithm for Power-Constrained End Devices in Long Range Networks
## 3 Link State Evaluation
### Combined Weighting Theory
#### 3.3.2 Improved CRITIC Method  
The CRITIC method is used for objective weight assignment in multiattribute decision making, helping to avoid uncertainties caused by subjective weighting [35]. The traditional CRITIC method reflects data variability through standard deviation but can only capture linear relationships, potentially leading to weighting errors. The improved D-CRITIC method [36] introduces distance correlation, enabling a more accurate measurement of nonlinear relationships between criteria. This results in more stable weight generation, enhancing the accuracy and reliability of decision analysis, especially for complex or large-scale decision matrices.  
Step 1: Data standardization. Let the original decision matrix be \(X=\big{(}x_{ij}\big{)}_{m\times n}\). By standardizing the data, the differences in units or magnitudes between the indicators are eliminated, resulting in the standardized matrix \(X^{\prime}=\big{(}x^{\prime}_{ij}\big{)}_{m\times n}\). The standardization formula is given as Equation (11).  
\[x^{\prime}_{ij}=\frac{x_{ij}-x^{\text{min}}_{j}}{x^{\text{max}}_{j}-x^{\text{ min}}_{j}} \tag{11}\]  
where \(x^{\text{min}}_{j}\) and \(x^{\text{max}}_{j}\) represent the minimum and maximum values of the \(j\)-th indicator, respectively.  
Step 2: Calculate the standard deviation \(s_{j}\) for each indicator. The formula is given as Equation (12).  
\[s_{j}=\sqrt{\frac{1}{m-1}\sum_{i=1}^{m}\Big{(}x^{\prime}_{ij}-\overline{x^{ \prime}_{j}}\Big{)}^{2}} \tag{12}\]  
In Equation (12), \(\overline{x^{\prime}_{j}}\) represents the mean value of the \(j\)-th indicator, and \(m\) is the number of samples.  
Step 3: Calculate the distance correlation coefficient between indicators. Distance correlation (instead of Pearson correlation \(r_{jk}\)) is used, and \(dCor\big{(}c_{j},c_{k}\big{)}\); the calculation formula is given as Equation (13).  
\[dCor(c_{j},c_{k})=\frac{dCov(c_{j},c_{k})}{\sqrt{dVar(c_{j})\cdot dVar(c_{k})}} \tag{13}\]  
In Equation (13), \(dCov(c_{j},c_{k})\) represents the distance covariance between \(c_{j}\) and \(c_{k}\), and \(dVar(c_{j})\) and \(dVar(c_{k})\) represent the distance variances of \(c_{j}\) and \(c_{k}\).  
Step 4: Calculate the information content for each indicator. The information content \(I_{j}\) depends not only on the standard deviation \(s_{j}\) but also on the distance correlation coefficient \(dCov\big{(}c_{j},c_{k}\big{)}\). The information content can be calculated using the following formula, as shown in Equation (14).  
\[I_{j}=s_{j}\cdot\sum_{k\
eq j}\big{(}1-dCov(c_{j},c_{k})\big{)} \tag{14}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Scale** & **Fuzzy Score** & **Meaning** \\ \hline
1 & (1,1,1) & Indicates equal importance between two factors. \\
3 & (\(3-\alpha\), \(3\), \(3+\alpha\)) & The former factor is slightly more important than the latter. \\
5 & (\(5-\alpha\), \(5\), \(5+\alpha\)) & The former is significantly more important than the latter. \\
7 & (\(7-\alpha\), \(7\), \(7+\alpha\)) & The former is strongly more important than the latter. \\
9 & (\(9-\alpha\), \(9\), \(9+\alpha\)) & The former is extremely more important than the latter. \\
2, 4, 6, 8 & (\(x-\alpha\), \(x\), \(x+\alpha\)) & Intermediate values between two adjacent judgments. \\ \hline \end{tabular}
\end{table}
Table 2: Definition of the fuzzy relative importance scale.  
Step 5: Determine the target weights. Based on the information content \(I_{j}\), calculate the objective weight \(\omega_{j}\) for each evaluation indicator. The formula is given as Equation (15).  
\[\omega_{j}=\frac{I_{j}}{\sum\limits_{q=1}^{n}I_{q}} \tag{15}\]  
where \(I_{j}\) represents the information content of the \(j\) indicator.