## 2 Theory and Analysis
### The Classification and Identification Method
#### 2.2.1 Similarity Comparison of Radiation Emission Components  
(1) Similarity comparison based on peak components  
Peak components are the amplitude and frequency corresponding to a series of peaks in the test data. From a large number of test results, the amplitude is relatively unstable. Therefore, in the similarity analysis of peak components, the main considerations are the peak frequencies corresponding to each peak in the test data. This means that, for the spectrum of an arbitrary piece of equipment, the matrix form of its peak components is:  
\[Fre=[f_{1},f_{2},\ldots,f_{n}] \tag{2}\]  
In order to calculate this similarity between the equipment to be identified and the standard equipment, a novel concept based on the Jaccard index is proposed in this work. The Jaccard index [43], also known as the Jaccard similarity coefficient, is one of the most frequently used evaluation measurements in the comparison of sequences. In this paper, in order to be consistent with the other two distances discussed below, the modified Jaccard correlation coefficient is introduced as follows:  
\[D_{accard}(Fre_{1},Fre_{0})=1-jaccard(Fre_{1},Fre_{0})=1-\frac{|Fre_{1} \cap Fre_{0}|}{|Fre_{1}\cup Fre_{0}|} \tag{3}\]  
Among this formulation, \(Fre_{1}\) and \(Fre_{0}\) are the peak component matrices of the spectrum data to be identified and of the standard spectrum data. The modified Jaccard similarity coefficient indicates the proportion of the intersection of sets in the union of sets. The greater its value (closer to 1), the lower the similarity between the two sets. On the contrary, the higher the similarity, the smaller the calculated value (closer to 0). Thus, the similarity comparison by peak components \(S_{1}\) can be viewed as  
\[S_{1}=D_{jaccard}(Fre_{1},Fre_{0}) \tag{4}\]  
Figure 4: The classification and identification method of the equipment spectrum using radiation emission components. Similarity comparison based on broadband components  
Broadband components represent waveforms with a bandwidth formed by signals such as damped oscillation and spike waves. Broadband components have relatively wide bandwidths. At the same time, the spectrum of a piece of equipment can often include 3-5 broadband components. Therefore, two sets of data are used to characterize a broadband component. On the one hand, \(F^{i}\) is used to represent all frequency points occupied by the \(i\)th broadband component in the measured spectrum. On the other hand, \(Amp^{i}\) is used to represent the amplitude information of the frequency points corresponding to the \(i\)th broadband component. The details of \(F^{i}\) and \(Amp^{i}\) are shown in Equations (5) and (6), where 1-n indicates the sampling points of the \(i\)th broadband component. \[F^{i}=\left[f_{1}^{i},f_{2}^{i},\ldots,f_{n}^{i}\right] \tag{5}\]  
\[Amp^{i}=\left[amp_{1}^{i},amp_{2}^{i},\ldots,amp_{n}^{i}\right] \tag{6}\]  
To calculate the similarity between the equipment to be identified and the standard equipment, the frequency \(F_{0}^{i}\) and amplitude \(Amp_{0}^{i}\) information of the \(i\)th broadband components of standard equipment are first extracted. The details of \(F_{0}^{i}\) and \(Amp_{0}^{i}\) are shown in Equations (7) and (8):  
\[F_{0}^{i}=\left[f_{10}^{i},f_{20}^{i},\ldots,f_{n0}^{i}\right] \tag{7}\]  
\[Amp_{0}^{i}=\left[amp_{10}^{i},amp_{20}^{i},\ldots,amp_{n0}^{i}\right] \tag{8}\]  
Then, the amplitude information \(Amp_{1}^{i}\) of the corresponding frequency of the equipment to be identified is extracted according to the frequency information \(F_{0}^{i}\) of standard equipment. \[Amp_{1}^{i}=\left[amp_{11}^{i},amp_{21}^{i},\ldots,amp_{n1}^{i}\right] \tag{9}\]  
After extracting the amplitude information of the equipment to be identified, the similarity problem of comparing the \(i\)th wideband components is transformed into the problem of describing the similarity between the vectors \(Amp_{0}^{i}\) and \(Amp_{1}^{i}\). Of course, there are many methods to describe the degree of similarity between curves, such as that in [44].