CN117390531B - Automatic depression identification method based on IEEMD electroencephalogram signal decomposition - Google Patents
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Abstract
The invention provides an automatic depression recognition method based on IEEMD electroencephalogram (EEG) signal decomposition, relates to the technical field of EEG signal processing and analysis, and provides an EEG signal decomposition method based on IEEMD. EEMD is improved by adopting additional self-adaptive random white noise to solve the problem of mode aliasing, a mathematical model between an original EEG signal and the additional random white noise is established based on a signal-to-noise ratio theory, and the additional white noise amplitude of each sampling point is self-adaptively determined according to the original signal. IEEMD decomposes the EEG signal to more accurately derive several natural mode functions (IMFs) for modeling the individual neuron signals. In turn, binarized BFN is constructed on different IMFs based on Phase Lag Index (PLI) and proportional threshold strategy. And analyzing the change of the BFN topological structure and the attribute by using a complex network method, exploring potential markers for identifying the depression, and completing high-precision automatic identification of the depression.
Description
Technical Field
The invention relates to the technical field of electroencephalogram signal processing and analysis, in particular to an automatic depression identification method based on IEEMD electroencephalogram signal decomposition.
Background
Depression is the most common mental disorder disease, with low mood, slow thinking, and impaired brain cognitive function as its typical clinical symptoms. Major depressive disorder recurrence rate is about 75% to 80% with very little probability of complete cure. Aiming at the characteristics of high prevalence rate, high mortality rate, high recurrence rate and the like of depression, the irreversible brain function damage caused by time is prevented, and the currently recognized most effective method is early recognition of early intervention.
Traditional face-to-face clinical interview analysis of depression diagnosis methods based on structured scales such as patient health questionnaire 9 (PHQ-9), beck depression self-assessment (BDI) and the like are time consuming and labor intensive and are severely dependent on subjective clinical experience of doctors and authenticity of patient provided information. At present, the proportion of doctors and patients in China is still unbalanced, particularly mental diseases are more prominent, the difference between regional medical resources and economic development level is remarkable, and most depressive patients in some areas cannot be diagnosed accurately in time, even misdiagnosis leads to missing the best treatment opportunity. Therefore, the method has important social value and scientific significance for timely, accurate and effective analysis, diagnosis and identification of the depression.
EEG signals are formed by the sum of the electrical discharges of a large number of pyramidal neurons and are an integral reflection of brain cortex or scalp brain nerve cell activity. Compared with other physiological signals, EEG has the advantages of high time resolution and low cost. Over the past several decades, with the development of interdisciplinary technology, EEG has been widely used for the assisted diagnosis of mental diseases, characterised by its non-falsifiable, non-maskable nature. The nonlinearity and non-stationarity of EEG increases its interpretation complexity, and the Empirical Mode Decomposition (EMD) based time-frequency analysis can effectively reflect its physical meaning. The EMD decomposes the signal according to the time-scale characteristics of the data without any preset basis functions. Thus, EMD has inherent advantages in processing non-linear and non-stationary EEG signals. EMD is directly applied to EEG signal decomposition and is susceptible to modal aliasing interference. Integrated empirical mode decomposition (EEMD) is a noise-assisted data analysis method proposed for the disadvantage of EMD, which solves the problem of mode aliasing to some extent, but still has room for further improvement, such as assisting in the selection and optimization of white noise amplitude and quantity. These parameters tend to significantly affect the extent of improvement of the EEMD over the modal aliasing problem. In one aspect, if the amplitude of the added auxiliary random white noise is too large relative to the original signal, a redundant IMF component may be generated. On the other hand, if the amplitude of the added auxiliary random white noise is too small relative to the original signal, the modal aliasing cannot be sufficiently improved. The predefined constant amplitude method employed by EEMD, while white noise can affect extreme points, such noise cannot completely change or affect all extreme points. Thus, the present invention proposes and implements an improved EEMD (IEEMD) method based on additional adaptive white noise by exploring the magnitude and number of auxiliary white noise.
Disclosure of Invention
In order to solve the technical problems, the invention provides an automatic depression identification method based on IEEMD electroencephalogram signal decomposition, which comprises the following algorithm processing steps:
step one: input of the original EEG signal x (t) and the corresponding random white noise w j (t)；
Step two: the determination of the amplitude of the random white noise comprehensively considers the original signal and the corresponding random white noise, and establishes a mathematical model between the original EEG signal and the additional random white noise according to a signal-to-noise ratio (SNR) theory:
SNR j (i)＝10*log(x(t)/αmp j (i)*w j (t)) (1)
step three: assume an initial amp j (t) is a constant 1, or a constant is multiplied by the standard deviation of the original signal to obtain SNR j (i) Is used for the average signal to noise ratio of (c).
SNR＝SNR j (i)/(J/N) (2)
Wherein i e [1,2, …, N ], J e [1,2, …, J ];
step four: the amplitude of each sample point is calculated by adaptively determining the random white noise amplitude of each sample point and applying to the improvement of EEMD by:
αmp j (i)＝10 -(SNR/20) *(x(t)/w j (t)) (3)；
step five: the modified EEMD (IEEMD) algorithm adaptively adds random white noise of the above-described magnitude to the original EEG signal sequence:
x j (t)＝x(t)+amp j (i)*w j (t) (4)；
step six: will x j (t) decomposing into a finite number of IMF combinations, outputting the IMF ij
Step seven: IEEMD decomposes the EEG signals to obtain several more accurate IMF analog independent neuron signals;
step eight: reducing volume conductor effect, calculating connection weight between nodes based on PLI, wherein the calculation process is as follows:
l is the time sequence length, (. Cndot.) is the sign function, ψ x (t) is the instantaneous phase of the signal x (t)The method comprises the steps of performing x (t) Hilbert transformation, calculating all adjacent matrixes of an EEG time window of each IMF according to the number of the obtained IMFs, drawing corresponding BFN, and averaging the adjacent matrixes of a depression group and a healthy control group of each IMF to obtain two groups of adjacent matrixes respectively;
step nine: a large number of weak connections exist in the weighted BFN, and the weighted BFN binarization is adopted to avoid the interference of the weighted BFN on the core connection topology;
step ten: for inter-group Characteristic Path Length (CPL), global efficiency (E glob ) Aggregation coefficient (CC), local efficiency (E loc ) Equal network node degree<K>) Complex network metrics such as small world index (sigma) are subjected to one-factor analysis of variance, and potential markers for identifying depression are explored;
step eleven: the effectiveness of potential markers for automatic identification of depression was evaluated using an SVM classifier.
Further, the specific process in the seventh step is as follows:
s1: according to the sequence of the channels and the EEG time window, the signal-to-noise ratio of the data of the depression group and the healthy control group and the adaptive amplitude of the corresponding random white noise are calculated.
S2: random white noise with adaptive amplitude is added to the corresponding EEG time window signal.
S3: the processed EEG time window signal is decomposed into several IMFs by IEEMD.
S4: the difference in the different EEG time windows results in a slight difference in the number of IMFs generated by IEEMD. The feasibility of subsequent BFN construction is ensured. The minimum number of IEEMD generated IMFs is selected as a threshold, and the same number of IMFs is intercepted after all EEG time windows are decomposed.
S5: and outputting IMF decomposition results of the depression group and the healthy control group in sequence.
Further, a proportional threshold method is adopted in the step nine for BFN binarization, and the threshold calculation process is as follows:
step one: establishing a mathematical model between the average network node degree and the number of network nodes and the number of network edges:
2M＝N*<K> (7)
n is the number of nodes in the network, M is the number of edges in the network, < K > is the average network node degree;
step two: network density is defined as the ratio of the actual number of edges to the maximum number of edges,
step three: the average network node degree and the network density may establish the following relationship:
ρ＝<K>/(N-1) (9)；
step four: studies have shown that the average network node degree is greater than the natural logarithm of the number of network nodes, while the network density is less than 50%, the structure BFN is more efficient,
<K>>2 lnN (10)；
step five: based on equations (9) and (10), and the specific number of BFN network nodes, a binary scale threshold may be calculated, and to minimize the amount of computation, a lower limit of the network density is selected as the scale threshold.
Compared with the prior art, the invention has the beneficial effects that the technical scheme is adopted:
(1) A method for further reducing modal aliasing is explored, and the problem that all extreme points cannot be completely influenced by EEMD preset constant amplitude is solved. An IEEMD method based on self-adaptive random white noise is provided, a mathematical model between an original EEG signal and corresponding random white noise is established through a signal-to-noise ratio theory, and the amplitude of the random white noise added to each sampling point is self-adaptively determined according to the original signal. The EEG signals are decomposed by IEEMD method, and a plurality of IMFs are obtained more accurately to simulate independent neuron signals. IEEMD can completely influence all extreme points, and further effectively solves the problem of EEMD modal aliasing.
(2) The invention provides a BFN binarization method based on the relationship between network density and average node degree, which ensures that the constructed BFN is more effective and ensures the comparability of BFN among groups.
(3) IMF and its difference matrix analysis based on IEEMD EEG signal decomposition showed that the inter-group differences were mainly present in the three brain areas LF, LT and LC. Based on the statistical analysis of the BFN connection numbers corresponding to the difference matrix, the BFN corresponding to the IMF1 and the IMF4 is selected as a further research object. Based on the method, BFN of the depression patients is studied based on complex network measurement indexes, the complex network measurement indexes with obvious differences among groups are used as potential biomarkers for depression identification, SVM is used as a classifier, and the identification accuracy rate of 92.38% is the highest.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a diagram of an original EEG signal and corresponding random white noise according to an embodiment of the present invention;
FIG. 3 is an adaptive random white noise amplitude added in accordance with an embodiment of the present invention;
FIG. 4 is a graph showing the comparison of the effect of EEMD and IEEMD addition of random white noise on the polarization effect;
FIG. 5 is a comparison of EEG signal decomposition results based on EMD, EEMD, IEEMD, respectively, according to an embodiment of the present invention;
FIG. 6 is a graph showing the adjacency matrix between the depression group and the healthy control group, and the difference matrix between the two groups according to the embodiment of the present invention;
FIG. 7 is a 3D graph of the distribution of electrodes corresponding to the difference matrix on each IMF component, and statistics of connection numbers according to an embodiment of the present invention;
FIG. 8 is a graph of the differential analysis results between the complex network metrics sets of the three brain regions and the whole brain region of IMF1 in accordance with an embodiment of the present invention;
FIG. 9 is a graph of the differential analysis results between the complex network metrics sets of the three brain regions and the whole brain region of IMF4 in accordance with an embodiment of the present invention;
fig. 10 shows the result of identifying depression using mixed potential markers as SVM inputs in an embodiment of the present invention.
Detailed Description
The following describes in further detail the embodiments of the present invention with reference to the drawings and examples. The following examples are illustrative of the invention and are not intended to limit the scope of the invention.
The invention provides an IEEMD-based EEG signal decomposition method, and the decomposed EEG signal is used for BFN construction so as to realize automatic identification of depression. EEMD is improved by adopting additional self-adaptive random white noise to solve the problem of mode aliasing, a mathematical model of an original signal and the added random white noise is established based on a signal-to-noise ratio theory, and the amplitude of the added white noise of each sampling point is self-adaptively determined according to the original signal. IEEMD decomposes the EEG signal to more accurately derive several IMFs that simulate individual neuron signals. In turn, a binarized BFN is constructed on different IMFs based on the Phase Lag Index (PLI) and the proportional threshold. Then, the change of the BFN topological structure and the attribute is analyzed by using a complex network method, potential biomarkers for detecting the depression are explored, and the high-precision automatic identification of the depression is completed.
Referring to fig. 1, the present invention provides a technical solution: an EEG signal decomposition and BFN construction-based automatic depression identification method comprises the following steps:
step one: input of the original EEG signal x (t) and the corresponding random white noise w j (t), please refer to fig. 2;
step two: the determination of the amplitude of the random white noise comprehensively considers the original signal and the corresponding random white noise, and establishes a mathematical model between the original EEG signal and the additional random white noise according to a signal-to-noise ratio (SNR) theory:
SNR j (i)＝10*log(x(t)/αmp j (i)*w j (t)) (1)
step three: assume an initial amp j (t) is a constant 1, or a constant is multiplied by the standard deviation of the original signal to obtain SNR j (i) Is used for the average signal to noise ratio of (c).
SNR＝SNR j (i)/(J*N) (2)
Where i e [1,2, …, N ], J e [1,2, …, J ].
Step four: the amplitude value of each sampling point is calculated by the following formula, namely, the random white noise amplitude of each sampling point is adaptively determined, as shown in fig. 3, and fig. 3 is the adaptive random white noise amplitude calculated by the invention according to fig. 2, and is applied to the improvement of EEMD:
αmp j (i)＝10 -(SNR/20) *(x(t)/w j (t)) (3)
step five: the modified EEMD (IEEMD) algorithm adaptively adds the random white noise described above to the original EEG signal sequence:
x j (t)＝x(t)+amp j (i)*w j (t) (4)
fig. 4 shows a comparison of the result of the extreme value influence by the EEMD added constant amplitude and the IEEMD added adaptive amplitude random white noise in the embodiment of the present invention, where fig. 4 (b) is an enlarged result of the red dotted line area in fig. 4 (a), and it can be seen that the IEEMD can sufficiently influence all the extreme value points compared with the EEMD.
Step six: will x j (t) decomposing into a finite number of IMF combinations, outputting the IMF ij 。
Step seven: IEEMD decomposes the EEG signals to obtain several more accurate IMF analog independent neuron signals. The procedure is summarized as follows:
(1) According to the sequence of the channels and the EEG time window, the signal-to-noise ratio of the data of the depression group and the healthy control group and the adaptive amplitude of the corresponding random white noise are calculated.
(2) Random white noise with adaptive amplitude is added to the corresponding EEG time window signal.
(3) The EEG time window signal is decomposed into several IMFs by IEEMD.
(4) The difference in the different EEG time windows results in a slight difference in the number of IMFs generated by IEEMD. The feasibility of subsequent BFN construction is ensured. The minimum number of IEEMD decomposed IMFs is selected as a threshold, and the same number of IMFs is truncated after decomposing all EEG time windows.
(5) And outputting IMF decomposition results of the depression group and the healthy control group in sequence.
Fig. 5 shows a comparison of the results of the decomposition of the EEG signal based on EMD, EEMD, IEEMD, which decomposes the EEG signal into 5 IMFs, 7 IMFs and 10 and IMFs, respectively, from which it can be seen that the EMD and EEMD do not completely decompose the EEG signal compared to the IEEMD, and that there is still relatively more pattern aliasing. Similar characteristic time scales of IMF2 and IMF3, such as in fig. 5 (a), are distributed in different components, resulting in overlapping and interacting adjacent IMF waveforms. The minimum number of IMFs generated by EEMD decomposed EEG signals of each time window is 7, and the number of IMFs of the EEG signals of each time window generated by the IEEMD through statistical analysis is distributed between 7 and 11, and the average value and standard deviation are 8.21+/-2.79 and p is 0.05. There was no statistically significant difference, indicating that the minimum threshold policy screening did not result in information loss. Thus, the first 7 IMF components are selected in the embodiment of the invention.
Step eight: reducing volume conductor effect, calculating connection weight between nodes based on PLI, wherein the calculation process is as follows:
l is the time sequence length, (. Cndot.) is the sign function, ψ x (t) is the instantaneous phase of the signal x (t),is an x (t) hilbert transform. And calculating all adjacency matrixes of the EEG time window of each IMF according to the obtained number of the IMFs, and further drawing the corresponding BFN. The adjacency matrix of the depression group and the healthy control group of each IMF was averaged to obtain two adjacency matrices, respectively.
Step nine: there are a large number of weak connections in the weighted BFN, avoiding their interference to the core connection topology, weighted BFN binarization is an effective solution. Studies have been carried out (l.dutan, et al., "Quantitative comparison of resting-state functional connectivity derived from fNIRS and fMRI: a simultaneous recording study," neuroimage., "2012,60 (4): 2008-2018.) to demonstrate that the average network node degree is greater than the natural logarithm of the number of network nodes, while the network density is less than 50% and that the BFN is more efficient to construct. Based on the theory, the invention designs and realizes a proportional threshold method for BFN binarization, and the threshold calculation process is as follows:
(1) Establishing a mathematical relationship between the average network node degree and the number of network nodes and the number of network edges:
2M＝N*<K> (7)
n is the number of nodes in the network, M is the number of edges in the network, < K > is the average network node degree,
(2) Network density is defined as the ratio of the actual number of edges to the maximum number of edges.
(3) The average network node degree and the network density may establish the following relationship:
ρ＝<K>/(N-1) (9)
(4) In the present invention n=106, i.e. 106 EEG electrodes are used to verify the validity of the inventive method, its average network node degree.
<K>>2 lnN＝9.33 (10)
Finally, a ratio threshold of 8.89% is obtained based on equations (9) and (10).
Fig. 6 shows an example of the adjacency matrix of the depression group and the healthy control group and the difference matrix between the two groups generated according to the methods of the seventh step of 7 IMF components, the eighth step of PLI-based BFN construction, and the ninth step of weighted BFN binarization. As shown in fig. 6 (a) and 6 (b), the yellow cross point indicates a high correlation, and the green cross point indicates a low correlation. Two sets of binarized adjacency matrices generated over 7 IMFs show complex but rather simple patterns. However, it is difficult to quickly and clearly demonstrate the topology differences between the depressed group and the healthy control group from fig. 6 (a) and 6 (b). For this reason, the embodiment of the invention provides an inter-group difference matrix, and as shown in fig. 6 (c), yellow crossing points at the upper left corner of the difference matrix all show aggregation phenomenon, which indicates that the depression patient has abnormal functions in the corresponding brain region.
Furthermore, in order to find more valuable information from the difference matrix, the present invention draws a 3D map of the difference matrix from the EEG electrode position distribution, as shown in fig. 7 (a) -7 (g). Although the connection distribution of the 7 sub-graphs is different, the overall distribution exhibits a certain regularity. Most of the differential connections are mainly distributed in the three areas Left Forehead (LF), left Temporal (LT) and Left Center (LC), except for a few connections in the Right Forehead (RF) and Right Center (RC) areas. According to the general computing theory (M.Satyananayanan, "Pervasive computing: vision and challenges," IEEE. Personal. Communications.,2001,8 (4): 10-17.), not all IMFs are equally affected by depression. If all IMFs are treated equally, the computational complexity will be severely increased. Therefore, the invention calculates the connection number of the corresponding BFN of the difference matrix, and the statistical result is shown in fig. 7 (h). It can be seen that IMF1 and IMF4 have significantly more connections than the other five components. The invention comprehensively considers the position distribution of EEG electrodes in the 7 difference matrix 3D graphs, and the brain region with high connection density difference can better reflect the actual brain function change, so the embodiment further provides the difference analysis of complex network metrics of LF, LT, LC and other regions of IMF1 and IMF 4.
Step ten: for inter-group Characteristic Path Length (CPL), global efficiency (E glob ) Aggregation coefficient (CC), local efficiency (E loc ) Equal network node degree<K>) Complex network metrics such as small world index (sigma) are analyzed by one-way variance to explore potential markers for depression.
FIGS. 8 and 9 are single-factor analysis of variance of complex network metrics between groups for IMF1 (FIG. 8), CC for LF and LC regions, E for LT region for the depressed group compared to healthy control group loc The measurement indexes such as sigma and the like of the whole brain region are obviously reduced, and CPL of the LF region is obviously increased; for IMF4 (fig. 9), the depressed group LF and CC of LT regions, the whole brain region<K>Metrics like sigma and CLP of LC region are significantly reduced, while CLP of LC region is significantly increased. There was no significant difference in complex network metrics between other regional groups of IMF1 and IMF 4. Studies have shown that indices of significant differences between groups can be used as classification features or biomarkers in general (S.Sun, et al, "Graph theory analysis of functional connectivity in major depression disorder with high-density resting state EEG data," IEEE.T.Neur.Sys.Reh.,2019,27 (3): 429-439.). The results obtained from figures 8 and 9 show that there are significant differences in the 10 metrics between the depressive group and the healthy control group in the examples of the present invention.
Step eleven: the effectiveness of potential markers for automatic identification of depression was evaluated using an SVM classifier. FIG. 10 shows the use of LF-CLP, LF-CC, LT-E in IMF1 in accordance with an embodiment of the present invention loc LC-CC, FB-sigma, LF-CC, LT-CC, LC-CLP, FB-in IMF4<K>and FB-sigma as a 10-fold cross-validation result of the SVM common input, whereinThe highest detection accuracy was 92.07% (see 10: fold-6).
Further analyzing the effectiveness of each complex network metric as a potential marker, table 1 provides 10-fold cross-validation results for each complex network metric alone as an input to the SVM, as can be seen from table 1, the ability of the 10 complex network metrics as potential markers to identify depression, in order from strong to weak: FB-sigma of IMF4, LF-CC of IMF1, FB-sigma of IMF1, FB-17 of IMF4<K>LC-CLP of IMF4, LC-CC of IMF1, LT-CC of IMF4, LT-E of IMF1 loc LF-CLP of IMF1, with the highest recognition accuracy of 92.38%.
Table 1 depression identification performance of complex network metrics as potential markers
The foregoing description is merely illustrative of the preferred embodiments of the present disclosure and the technical principles applied thereto, and it should be understood by those skilled in the art that the scope of the invention in the embodiments of the present disclosure is not limited to the specific combination of the features described above, but encompasses other technical solutions formed by any combination of the features described above or their equivalents, such as the features described above and the features disclosed in the embodiments of the present disclosure (but not limited to) having similar functions, being interchanged.
Claims (3)
1. An automatic depression identification method based on IEEMD electroencephalogram signal decomposition is characterized in that: the method comprises the following algorithm processing steps:
step one: input of raw EEG signalsx(t) And corresponding random white noisew j (t)；
Step two: the determination of the amplitude of the random white noise comprehensively considers the original signal and the corresponding random white noise, and establishes a mathematical model between the original EEG signal and the additional random white noise according to the SNR theory:
(1)
step three: assume an initial stateamp j (t) Is a constant 1, or a constant multiplied by the standard deviation of the original signal to obtain SNR j (i) Average signal to noise ratio of (a);
(2)
wherein the method comprises the steps ofi ∈ [1,2,…,N]，j ∈[1,2,…,J]；
Step four: the amplitude of each sample point is calculated by adaptively determining the random white noise amplitude of each sample point and applying to the improvement of EEMD by:
(3)；
step five: the improved EEMD, IEEMD algorithm, adaptively adds random white noise of the above-described magnitude to the original EEG signal sequence:
(4)；
step six: will beDecomposing into a limited number of IMF combinations, outputting +.>；
(5)；
Step seven: IEEMD decomposes the EEG signals to obtain several more accurate IMF analog independent neuron signals;
step eight: reducing volume conductor effect, calculating connection weight between nodes based on PLI, wherein the calculation process is as follows:
(6)；
Lis the time sequence length, (-) is a sign function,is a signalx(t) Instantaneous phase of>Is thatx(t) Hilbert transform, according to the number of the obtained IMFs, calculating all adjacent matrixes of an EEG time window of each IMF, further drawing corresponding BFN, and averaging adjacent matrixes of a depression group and a healthy control group of each IMF to obtain two groups of adjacent matrixes respectively;
step nine: a large number of weak connections exist in the weighted BFN, and the weighted BFN binarization is adopted to avoid the interference of the weighted BFN on the core connection topology;
step ten: for inter-group characteristic path length CPL, global efficiency E glob Aggregation coefficient CC, local efficiencyE loc Equal network node degree<K>Performing single-factor analysis of variance on the small world index sigma complex network metric, and exploring a potential marker for identifying depression;
step eleven: the effectiveness of potential markers for automatic identification of depression was evaluated using an SVM classifier.
2. The automatic depression identification method based on IEEMD brain electrical signal decomposition according to claim 1, wherein: the specific process in the seventh step is as follows:
s1: calculating the signal-to-noise ratio of the data of the depression group and the healthy control group and the self-adaptive amplitude of the corresponding random white noise according to the sequence of the channel and the EEG time window;
s2: adding random white noise with adaptive amplitude to the corresponding EEG time window signal;
s3: decomposing the processed EEG time window signal into a plurality of IMFs through IEEMD;
s4: the difference in the different EEG time windows resulted in a slight difference in the number of IMFs generated by IEEMD;
the feasibility of the subsequent BFN construction is ensured;
selecting the minimum number of IMFs generated by IEEMD as a threshold value, decomposing all EEG time windows, and then intercepting the same number of IMFs;
s5: and outputting IMF decomposition results of the depression group and the healthy control group in sequence.
3. The automatic depression identification method based on IEEMD brain electrical signal decomposition according to claim 1, wherein: in the step nine, a proportional threshold method is adopted for BFN binarization, and the threshold calculation process is as follows:
step one: establishing a mathematical relationship model between the average network node degree and the number of network nodes and the number of network edges:
(7)
Nis the number of nodes in the network,Mis the number of edges in the network,<K>is the average network node degree;
step two: network density is defined as the ratio of the actual number of edges to the maximum number of edges,
(8)；
step three: the average network node degree and the network density may establish the following relationship:
(9)；
step four: studies have shown that the average network node degree is greater than the natural logarithm of the number of network nodes, while the network density is less than 50%, the structure BFN is more efficient,
(10)；
step five: based on equations (9) and (10), and the specific number of BFN network nodes, a binary scale threshold may be calculated, and to minimize the amount of computation, a lower limit of the network density is selected as the scale threshold.
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