## 3 Methods for Contribution Rate Assessment
### Feature Matrix Construction  
The judgment matrix between the object layer and the feature layer is referred to as the feature matrix, which is determined based on the features of the submarine's ELF \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Importance Levels** & \(a_{ij}\) \\ \hline
1 & Elements \(i\) and \(j\) are equally important. & 1 \\
2 & Element \(i\) is slightly more important than element \(j\). & 3 \\
3 & Element \(i\) is clearly more important than element \(j\). & 5 \\
4 & Element \(i\) is strongly more important than element \(j\). & 7 \\
5 & Element \(i\) is extremely more important than element \(j\). & 9 \\
6 & Element \(i\) is slightly less important than element \(j\). & 1/3 \\
7 & Element \(i\) is clearly less important than element \(j\). & 1/5 \\
8 & Element \(i\) is strongly less important than element \(j\). & 1/7 \\
9 & Element \(i\) is extremely less important than element \(j\). & 1/9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Scale for pairwise comparison of importance levels and their assignments. magnetic field signals. Submarines, under different frequency bands, distances, environments, operational conditions, and detection modes, exhibit dynamic changes in the main components of their magnetic fields. Additionally, different types of ELF magnetic fields display multidimensional and multimodal characteristics, necessitating the use of multimodal feature extraction methods to process the signals. To ensure that the magnetic field contribution rate indicators objectively reflect the contribution rate of magnetic fields, the construction of the feature matrix is divided into two steps: signal feature extraction and feature matrix construction. Firstly, the feature extraction of the extremely-low-frequency magnetic field of the submarine is conducted. The preliminary extracted features include the line spectrum and its amplitude, centroid frequency, half-energy bandwidth, higher-order statistics, information entropy, and extrema, as detailed below. Line spectrum and its amplitude. This term usually refers to the phenomenon in spectral analyses where energy is concentrated on discrete spectral lines. The ELF magnetic field signals of submarines often exhibit distinct line spectrum features. For the wake magnetic field and the magnetic field effects of the propeller shaft rotation, the line spectrum features are generally related to the propeller frequency and its harmonic components. As for the magnetic field radiation from internal equipment, the line spectrum features are typically associated with the operational frequencies of the electrical equipment and the power supply frequency. Centroid frequency. The centroid frequency refers to the weighted average frequency of a signal spectrum, which is the inverse of the weighted average of the signal energy (or power) with respect to frequency. It can be used to describe the \"center of mass\" position of the signal spectrum and provides an indicator for measuring the distribution of the signal spectrum. The calculation of the centroid frequency employs a weighted average using the amplitude of the power spectrum as the weight, with the following formula:  
\[f_{\mathrm{c}}\;=\;\frac{\int_{0}^{+\infty}f\cdot P(f)\mathrm{d}f}{\int_{0}^{ +\infty}P(f)\mathrm{d}f} \tag{2}\]  
Here, \(P(f)\) is the power spectrum of the magnetic field signal; thus, the centroid frequency will shift towards the positions with greater amplitude in the power spectrum. Half-energy bandwidth. The half-energy bandwidth refers to the frequency range at which the signal power drops to half of its maximum value, serving as an indicator for measuring the selectivity or resolution of a filter. The spectrum of the submarine's ELF magnetic field often includes low-frequency band spectra. Although not as prominent as the line spectrum features, for signals with low signal-to-noise ratios, the line spectrum features are often submerged in noise due to their low intensity.