A Blind Source Separation Method Based on Bounded Component Analysis Optimized by the Improved Beetle Antennae Search
Abstract
:1. Introduction
 (1)
 An improved BCA algorithm based on the Beetle Antennae Search (BAS) algorithm is proposed to address the shortcomings of conventional blind source separation algorithms.
 (2)
 A step decay factor is introduced to the BAS algorithm, which avoids entering local optima during the iteration process.
 (3)
 Simulation results show that the BASBCA algorithm successfully separates the dependent and independent source signals. This algorithm not only has stronger universality but also has faster convergence speed and more stable precision than traditional blind source separation algorithms and BCA algorithms.
2. Theories of BSS Based on BCA
2.1. BSS
2.2. BSS Based on BCA
 (1)
 Assume that the mixing matrix A has rank n, i.e., column full rank, i.e., the number of sensors should be greater than or equal to the number of source signals.
 (2)
 Assume that the distribution of source signals is bounded.
 (3)
 Assume that the branch set S, consisting of the joint distribution of all source signals, can be expressed by the Cartesian product of the branch set Si of each source signal:$$S={S}_{1}\otimes {S}_{2}\otimes \cdots \otimes {S}_{n}$$
3. Blind Source Separation Algorithm Based on BASBCA
3.1. BAS
 An initial position defined as X_{0} is given before beetle foraging.
 Determine the location of the beetle antennas. To ensure the randomness of the aspen search direction, a random factor is defined as P with the following expression.$$P=\frac{rand(n,1)}{\Vert rand(n,1)\Vert}$$
 3.
 The left antenna X_{L} and the right antenna X_{R} are brought into the adaptation function equation to obtain the magnitude of the detected food taste concentration, which is used to update the position of the beetle.$${X}_{k+1}=\left\{\right)separators="">\begin{array}{c}{X}_{k}+E\times \rho \times \frac{{X}_{L}{X}_{R}}{\Vert {X}_{L}{X}_{R}\Vert}f\left({X}_{L}\right)f({X}_{R})\\ {X}_{k}E\times \rho \times \frac{{X}_{L}{X}_{R}}{\Vert {X}_{L}{X}_{R}\Vert}f({X}_{L})\ge f({X}_{R})\end{array}$$$$E=\mathit{cos}\left(\frac{\pi \times k}{2\times M}+\frac{\pi}{2}\right)+1$$
 4.
 Enter the loop process, and when the set maximum number of iterations M is reached, or the adaptation value reaches the set requirement, stop the iteration and output the optimal result.
Algorithm 1 BAS 
Inputs: maximum number of iterations M, fitness function $f\left(\xb7\right)$, step decay factor E, line progress length $\rho $ of the beetle, distance D between the left antenna and the right antenna of the beetle

3.2. BASBCA
 Preprocessing of the received mixed signal, including deaveraging and prewhitening.
 Parameter initialization. The observed mixed signals (in the case of three signals) are used as the position information of the individual beetle X = [X1, X2, X3], the maximum number of iterations M, the step decay factor E, the travel length of the aspen $\rho $, the distance between the left antenna and the right antenna of the beetle D.
 Determine the left antenna X_{L} and right antenna X_{R} of the beetle according to Equations (16) and (17).
 Calculate the objective function J(W) as the fitness of the beetle by Equation (15).
 Update the position of the aspen ${X}_{k+1}$, according to Equation (19).
 Update the step decay factor using Equation (20).
 If the algorithm satisfies the termination condition, go to step 8; otherwise, repeat step 3 until it is satisfied.
 Output separation matrix W and estimate the source signal.
4. Simulation and Experimentation
4.1. Separation of Independent Source Signals
 $\left\rho \right\le 1$, represents the similarity of s and y, and the closer to 1, the more similar,
 $\left\rho \right=1$, which means that s and y are completely similar,
 $\left\rho \right=0$, which means s and y are not similar at all.
y1  y2  y3  

S1  0.97990255  0.01395991  0.00630897 
S2  0.00404252  0.96984297  0.00778653 
S3  0.00741474  0.00730825  0.94597251 
4.2. Separation of Dependent Source Signals
4.3. Blind Source Separation of Images
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Number  Function  DIM  scope  f_{min} 

F1  ${f}_{1}(x)={\displaystyle \sum _{i=1}^{D}}{x}_{i}^{2}$  30  [−100, 100]  0 
F2  ${f}_{2}(x)={\displaystyle \sum _{i=1}^{D}}{x}_{i}+{\displaystyle \prod _{i=1}^{D}}{x}_{i}$  30  [−10, 10]  0 
F3  ${f}_{3}(x)={\displaystyle \sum _{i=1}^{D1}}[100{({x}_{i}^{2}{x}_{i+1})}^{2}+{({x}_{i}1)}^{2}]$  30  [−30, 30]  0 
F4  ${f}_{4}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{D}}{x}_{i}^{2}{\displaystyle \prod _{i=1}^{D}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{{x}_{i}}{\sqrt{i}})+1$  30  [−600, 600]  0 
F5  ${F}_{5}(x)={\displaystyle \sum _{i=1}^{n}}[{x}_{i}^{2}10\mathrm{c}\mathrm{o}\mathrm{s}(2\pi {x}_{i}+10)]$  30  [−5.12, 5.12]  0 
F6  ${f}_{12}(x)=20+e20\mathrm{e}\mathrm{x}\mathrm{p}(0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^{D}}{x}_{i}^{2}})\mathrm{e}\mathrm{x}\mathrm{p}(\frac{1}{D}{\displaystyle \sum _{i=1}^{D}}\mathrm{c}\mathrm{o}\mathrm{s}(2\pi {x}_{i}))$  30  [−32, 32]  0 
Number  BAS  GWO  WOA  

BEST  STD  TIME  BEST  STD  TIME  BEST  STD  TIME  
F1  8.43 × 10^{−9}  1.32 × 10^{−9}  0.124  1.28 × 10^{−19}  3.39 × 10^{−19}  0.176  0.279  0.13  0.085 
F2  4.47 × 10^{−14}  0.348  0.092  2.01 × 10^{−9}  3.41 × 10^{−9}  0.182  1.24 × 10^{−16}  4.82  0.098 
F3  22.63  7.06 × 10^{−7}  0.096  0.269 × 10^{2}  0.601  0.213  4.14 × 10^{2}  5.36 × 10^{2}  0.103 
F4  1.01 × 10^{−5}  1.11 × 10^{−4}  0.085  1.84 × 10^{−3}  4.12 × 10^{−3}  0.219  0.038  0.016  0.089 
F5  −1.92 × 10^{−3}  0.136  0.081  0.398  0.545 × 10^{2}  0.198  1.64 × 10^{−2}  0.462 × 10^{2}  0.079 
F6  2.35 × 10^{−3}  0.542  0.076  2.19 × 10^{−4}  3.54 × 10^{−4}  0.192  5.09 × 10^{−3}  1.61  0.073 
y1  y2  y3  

S1  0.95674313  0.39900721  0.04960461 
S2  0.47858537  0.98699327  0.47763681 
S3  0.35747411  0.50449069  0.97789969 
FastICA  SCA  BCA  BASBCA  

similarity coefficient  0.61478832 0.79113256 0.86792361  0.75382446 0.83249712 0.84562887  0.94112682 0.89978596 0.96120685  0.99913226 0.97777913 0.98947889 
PI  0.37858537  0.33259634  0.18382564  0.09763681 
SIR  17.78995818  18.58967436  23.98567492  27.52189291 
Time  19.36  20.21  29.68  25.11 
y1  y2  y3  

S1  0.99705728  0.86008656  0.66929815 
S2  0.87889169  0.99727447  0.82707136 
S3  0.75027519  0.89507997  0.98287344 
FastICA  SCA  BCA  BASBCA  

similarity coefficient  0.52367416 0.71235987 0.79534238  0.76545382 0.63541287 0.77852536  0.93564258 0.90312354 0.95238421  0.99705728 0.99727447 0.98287344 
PI  0.49896372  0.39986341  0.21368547  0.15236846 
SIR  11.6689403  12.9658332  22.6854931  24.6235874 
Time  20.68  21.93  31.42  26.57 
s1  s2  s3  

s1  1  0.27858537  0.34960461 
s2  0.27858537  1  0.12389069 
s3  0.34960461  0.12389069  1 
FastICA  SCA  BCA  BASBCA  

similarity coefficient  0.81583695 0.80563229 0.83652718  0.85426875 0.82156371 0.84685932  0.96189537 0.92678932 0.95345874  0.98256749 0.96585374 0.99136457 
PI  0.29356417  0.27998659  0.14567789  0.11268894 
SIR  18.1235845  20.2314567  23.2589671  28.9358791 
Time  21.56  24.13  33.15  28.35 
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Tang, M.; Wu, Y. A Blind Source Separation Method Based on Bounded Component Analysis Optimized by the Improved Beetle Antennae Search. Sensors 2023, 23, 8325. https://doi.org/10.3390/s23198325
Tang M, Wu Y. A Blind Source Separation Method Based on Bounded Component Analysis Optimized by the Improved Beetle Antennae Search. Sensors. 2023; 23(19):8325. https://doi.org/10.3390/s23198325
Chicago/Turabian StyleTang, Mingyang, and Yafeng Wu. 2023. "A Blind Source Separation Method Based on Bounded Component Analysis Optimized by the Improved Beetle Antennae Search" Sensors 23, no. 19: 8325. https://doi.org/10.3390/s23198325