GPSO-ICA: Independent Component Analysis based on Gravitational Particle Swarm Optimization for blind source separation

Abstract

Blind source separation (BSS) is an advanced method of signal processing. Essentially, the problem in BSS is to separate and estimate the original signal from the observed mixed signal source without knowing the characteristics of the original signal. Independent component analysis (ICA) is a popular approach for blind source separation, and because its traditional search scheme is based on a gradient algorithm, a convergence problem will arise. In order to overcome the defect, this paper proposed to apply Particle Swarm Optimization (PSO) and Gravitational Search Algorithm (GSA) to conduct accelerated computing of the rate of convergence of a demixing matrix in ICA. However, the PSO converges prematurely, and the population diversity is reduced rapidly, so that the optimal solution falls into the local optimum. In order to increase the diversity of PSO, GPSO-based ICA algorithm (GPSO-ICA) is proposed that has the exploring ability of GSA, so that the ICA algorithm has a higher convergence rate and better ability to escape local optimization. A series of comparisons is implemented for the ICA algorithms based on PSO, GSA, and GPSO. The results show that GPSO-ICA has better performance than the other methods.

Keywords

Blind source separation Independent Component Analysis Particle Swarm Optimization Gravitation Searching Algorithm

1 Introduction

ICA [1, 2] is a special case of BSS and is a statistical method for exploring the non-Gaussian mutually independent and distributed latencies between random variables. In recent years, ICA has been extensively used in biological signal processing, the capturing of features, data mining, financial time series analysis [3], speech signal processing [4], and digital images [5]. Its medical applications include electroencephalography (EEG), electromyography (EMG), and functional magnetic resonance imaging (fMRI) [6]. In addition, it is used in the image signal of a concurrent multiple-channel single photon emission computed tomography (SPECT) to separate independent signals. All of these applications can be classified as the domain of BSS.

In general, the observed signal is mixed with many unknown signals, and these unknown signals are statistically reorganized into the original signal. If the unknown signals are mutually independent and have non-Gaussian probability distribution, then the measurement criteria of ICA include kurtosis [7], mutual information [8], and maximum likelihood [1, 2]. In the information theory, mutual information [8] is the most frequently used method for measuring the independence between random variables. Under the basic assumption of ICA, when the demixing matrix in the mixed signals is obtained, the independence between demixing signals is higher, and the minimization of mutual information between demixing signals represents the maximization of non-Gaussian characteristic of demixing signals.

This paper therefore uses the above characteristic as the criterion for defining fitness functions. Added to this, the key concept of Infomax [9] is the maximization of the entropy of the demixing signals. The maximization of entropy represents higher independence between signals. According to the definition of Infomax [9], after the conversion of any invertible function, e.g. cumulative distribution function (CDF), the independent signals are distributed uniformly. A Borel Regular Measure (BRM) [10] was proposed to implement a uniformly distributed non-parametric measurement, which could increase computational efficiency greatly in ICA. This paper uses the BRM non-parametric measurement as the criterion for defining fitness functions.

In the traditional ICA algorithm, computing the demixing matrix is a complex calculation. The parametric algorithms use a natural gradient method to gradually compute the demixing matrix. By considering the step size (or learning rate), this method influences the convergence rate directly. If the step size is too small, then the convergence of solving it will be very slow and easily fall into the local optimum. However, if the step size is too large, then the solution obtained is not ideal.

This paper therefore uses the concept of Particle Swarm Optimization (PSO) to increase the effective learning rate [11, 17]. PSO [12] is a swarm-based global optimum search algorithm and has been extensively used in many practical problems, such as the traveling salesman [13], job-shop scheduling [14], feature selection [15], and clustering [16]. It has the most familiar characteristic of the concept of Evolutionary Computation, e.g. the evaluation of fitness. Compared with genetic algorithms, many search solutions are regarded as a biomimetic population, and the overall population is initialized by random numbers. The particles move towards an individual optimal experience or population optimal experience, which is similar to the "crossover" operation of genetic algorithms. The particle in PSO, i.e. each probable potential solution, has a velocity and is computed in the overall search space; at the same time, the particle and particle swarm have memory capacity, which are behaviors and abilities that are completely different from genetic algorithms. Another outstanding characteristic of PSO is that it only needs a simple mathematical calculation, and thus it can work in a machine with only a few program codes.

The standard PSO algorithm also tends to converge prematurely and gets into a local optimum; its convergence rate becomes very slow later; and it is difficult to escape the local optimum [18, 19]. Therefore, keeping the particle diversity high enough to avoid premature convergence is an important task. Researchers have proposed several algorithms to improve the exploration and development of PSO, such as CLPSO [20], OLPSO [21], and DNSPSO [22]. A recently-proposed swarm-based intelligent algorithm is Gravitational Search Algorithm (GSA) [23]. GSA is based on Newton’s law of gravity and motion; the search space is calculated according to the total force obtained by all individuals. Added to this, researchers have proposed several improved hybrid algorithms, such as A New Hybrid PSOGSA Algorithm [24] and PSGO (Particle Swarm Gravitation Optimization) Algorithm [25].

This paper is inspired by the search strategy of GSA and rapid convergence of PSO. The advantages of the two algorithms are used in the blind source separation problem to implement the diversity of GPSO-ICA. According to [26], who proposed several hybridization methods for heuristic algorithms, the two algorithms can implement crossover at a high level or low level with a co-evolutionary method, e.g. homogeneous or heterogeneous. The GPSO-ICA algorithm proposed in this paper executes PSO and GSA simultaneously, and three criteria are designed for fitness functions: mutual information, kurtosis, and Borel Regular Measure (BRM). The individual and group thoughts of PSO(P_best and P_gbest) are combined with the local search capability of GSA. In experimental evaluation, GPSO-ICA is significantly better than PSO-ICA and GSA-ICA. The BRM-based non-parametric measurement criteria also have significant performance in computingtime.

The remainder of this paper is organized as follows. Section 2 presents the background knowledge and related work. Sections 3 and 4 respectively present the proposed algorithm and experimental evaluations. Section 5 concludes the paper.

2 Related Work

This section reviews related works. The overview focuses on the topics as follows:

Independent Component Analysis (ICA)

Particle Swarm Optimization (PSO)

Gravitation Searching Algorithm (GSA)

Particle Swarm Optimization based ICA (PSOICA)

2.1 Independent Component Analysis

Independent component analysis (ICA) has attracted the attention of researchers of signal processing and neural networks for years. Because ICA is of crucial importance to handling basic problems, such as blind source separation, blind deconvolution, feature extraction, unsupervised learning, data analysis, and compression, it can be used to capture independent components. The independent components must conform to the assumption and limit that the distribution of random variable must be non-Gaussian distributed and mutually independent. The observed signal was mixed with an unknown matrix A and original signal. BSS aims to estimate a demixing matrix W and to approximate the original source signal by W.

The mathematical expressions are Equations. (1) to (3). Here, x = [x₁, x₂,…, x_N] ^T is the mixing signal, and s = [s₁, s₂,…, s_M] ^T is the original signal. Moreover, A is a matrix of dimension N × M, and W is a matrix of dimension M × N. Generally, N = M, and y is obtained from the estimated demixing matrix W to approximate the original signal.

$x = As$ (1)

$y = Wx$ (2)

$y ≅ s$ (3)

The computation of demixing matrix W consists of the following three steps. (1) Establish the objective function for measuring signal independence. (2) Import the demixing matrix and observed signal into the objective function. (3) Optimize the objective function. Optimizing the objective function is equivalent to optimizing the demixing matrix. The gradient descent algorithm can be applied for iterative learning of demixing matrix W, expressed as Equation (4).

$W^{i + 1} = W^{i} + η \nabla_{W^{i}} Obj (x, W^{i})$ (4)

Where i and η are iteration and learning rate, respectively. The gradient term is ∇_{W
ⁱ}Obj (x, Wⁱ) with respect Wⁱ. Obj (x, Wⁱ) is the objective function.

The most fundamental solution of the ICA model is to maximize non-Gaussianity to evaluate the independence of the demixing signal [27], so as to obtain the demixing matrix W. Therefore, the objective function is expressed as Equation (5). The kurtosis value of the signal is estimated by using a statistical fourth-order moment function. When the value is 0, the signal presents Gaussian distribution. When the value is not zero, the signal is non-Gaussian distribution. This method is simple and fast, but sensitive to outliers. The fitness function is defined as Equation (6).

$kurt (y) = E {y^{4}} - 3 (E {y^{2}})^{2}$ (5) $= E {(Wx)^{4}} - 3 (E {(Wx)^{2}})^{2}$

$fitness_function (y) = \frac{1}{abs (kurt (y))}$ (6)

The measured values of negentropy, likelihood function, and mutual information are popular in building the ICA model. The mutual information between the transformed sources (demixing signal) was minimized to find the demixing matrix. Under some assumptions [27], the minimization of mutual information optimization was shown to be equivalent to maximum likelihood and maximum negentropy principles. Therefore, the mutual information is used to measure the independence of the demixing signal, so as to obtain the demixing matrix W. The objective function refers to [28], expressed as Equation (7).

$MI (y_{i}, y_{j}) = \sum_{i \neq j} p (y_{i}, y_{j}) \log \frac{p (y_{i}, y_{j})}{p (y_{i}) p (y_{j})}$ (7) $= \sum_{i \neq j} p ({Wx}_{i}, {Wx}_{j}) \log \frac{p ({Wx}_{i}, {Wx}_{j})}{p ({Wx}_{i}) p ({Wx}_{j})}$

If MI (y_i, y_j) =0, then y_i and y_j are independent of each other. The fitness function is defined as Equation (8).

$fitness_function (y) = MI (y_{i}, y_{j}), y_{i}, y_{j} \in y, i \neq j$ (8)

According to the definition of Infomax [9], the independent signals present uniform distribution by the conversion of any invertible function, e.g. cumulative distribution function (CDF). [10] which is our previous work proposed the Borel Regular Measure (BRM) to implement a uniformly distributed non-parametric measurement for ICA, thus increasing computational efficiency greatly. This paper uses the BRM non-parametric measurement as the criterion of defining fitness functions. The objective function is expressed as Equation (9). The fitness function is defined as Equation (10).

$\begin{matrix} BRM (y_{i}, y_{j}) \\ = \sum_{i \neq j} [μ (B_{r} (g (y_{i}))) - μ (B_{r} (g (y_{j})))] \\ = \sum_{i \neq j} [μ (B_{r} (g ({Wx}_{i}))) - μ (B_{r} (g ({Wx}_{j})))] \end{matrix}$ (9)

μ (B_r (•)) is a borel regular measure. g is the inverse function.

$\begin{matrix} fitness_function (y) & = & BRM (y_{i}, y_{j}), y_{i}, y_{j} \in y, \\ i \neq j \end{matrix}$ (10)

2.2 Particle Swarm Optimization

PSO is a swarm-based intelligent algorithm proposed by Kennedy and Eberhard [12]. There are N individuals (potential solution) in PSO; each individual in iteration t is regarded as a particle in d-dimensional search space, expressed as Equation (11). It flies at a velocity in the search space, expressed as Equation (12). This velocity is adjusted dynamically according to its flight experience and the flight experience of the group. In other words, each particle updates velocity and position according to global and personal best solutions, expressed as Eqs. (13) and (14).

$x_{i}^{t} = {x_{i, 1}^{t}, x_{i, 2}^{t}, . . ., x_{i, j}^{t}, . . ., x_{i, d}^{t}}, fori = 1, . . ., N$ (11)

$v_{i}^{t} = {v_{i, 1}^{t}, v_{i, 2}^{t}, . . ., v_{i, j}^{t}, . . ., v_{i, d}^{t}}, fori = 1, . . ., N$ (12)

$v_{i, j}^{t + 1} = ω v_{i, j}^{t} + c_{1} r_{1} (P_{{best}_{i, j}} - x_{i, j}^{t})$ (13) $+ c_{2} r_{2} (P_{{gbest}_{j}} - x_{i, j}^{t})$

$x_{i, j}^{t + 1} = x_{i, j}^{t} + v_{i, j}^{t + 1}$ (14)

where $x_{i, j}^{t}$ and $v_{i, j}^{t}$ denote the position and velocity of particle i in dimension j at iteration t, respectively. ω is an inertia weight that controls the convergence speed. P_{best
_i,j} and P_{gbest
_j} represent the personal best fitness and global best solution among all particles in iteration t, respectively. c1 and c2 denote the constant of social parameter and cognitive element, respectively. r1 and r2 are uniformly random numbers in the interval [0, 1].

The PSO algorithm is detailed in Fig. 1.

Fig.1

A PSO algorithm.

2.3 Gravitation Searching Algorithm

Many heuristic optimization methods are inspired by swarm intelligence. Rashedi et al. [23] proposed an intelligent gravitation search algorithm based on the law of gravity and mass interactions. The searcher agents make up a collection of masses, with each one based on Newton’s law of gravity and motion. N agents (masses) are given; the position of the ith agent in the dth dimension in iteration t is defined as $x_{i}^{d} (t)$ , expressed as Equation (15); and the fitness function is defined and evaluated as follows.

$x_{i} (t) = {x_{i}^{1} (t), x_{i}^{2} (t), . . ., x_{i}^{d} (t)}, fori = 1, . . ., N$ (15)

First, fit_i (t) represent the fitness value of the value of agent i in iteration t, and worst (t) and best (t) are defined as follows (for the minimization problem): $best (t) = min ({fit}_{i} (t)), fori = 1, . . ., N$ (16)

$worst (t) = max ({fit}_{i} (t)), fori = 1, . . ., N$ (17)

The mass m_i (t) and average mass M_i (t) of the agent i in iteration t are given by the following equations:

$m_{i} (t) = \frac{{fit}_{i} (t) - worst (t)}{best (t) - worst (t)}$ (18)

$M_{i} (t) = \frac{m_{i} (t)}{\sum_{j = 1}^{N} m_{j} (t)}$ (19)

The definition of the force acting $F_{i}^{d} (t)$ on agent i in dimension d during iteration t and the acceleration $a_{i}^{d} (t)$ of the agent i in direction d at iteration t are given by Equation (20) to Equation (23).

$F_{i}^{d} (t) = \sum_{j = 1, j \neq i}^{N} {rand}_{j} F_{ij}^{d} (t)$ (20)

$F_{ij}^{d} (t)$ is the force acting on mass i from mass j in dimension d at iteration t.

$F_{ij}^{d} (t) = G (t) \frac{M_{i} (t) M_{j} (t)}{R_{ij} (t) + ∊} (x_{j}^{d} (t) - x_{i}^{d} (t))$ (21)

The gravitational constant G (t) in iteration t is the function of the initial value G₀ and time t. T is the total number of iterations. R_ij (t) is the Euclidian distance between two agents i and j. ∊ is a small constant.

$G (t) = G (G_{0}, t) = G_{0} * \exp (- α \frac{t}{T})$ (22)

$\begin{matrix} a_{i}^{d} (t) = \frac{F_{i}^{d} (t)}{M_{i} (t)} = \\ \sum_{j \in Kbest, j \neq i} G (t) {rand}_{j} \frac{M_{j} (t)}{R_{ij} (t) + ∊} (x_{j}^{d} (t) - x_{i}^{d} (t)) \end{matrix}$ (23)

The Kbest in Equation (23) is the set of first K agents with the best fitness value and biggest mass. Kbest is a function of time, at the beginning, all agents apply the force and as time passes, Kbest is decreased linearly and at the end there will be just one agent applying force to the others. In this study, the Equation (20) would be change to:

$F_{i}^{d} (t) = \sum_{j \in Kbest, j \neq i} {rand}_{j} F_{ij}^{d} (t)$ (24)

Therefore, the position and velocity in solution space of each agent could be calculated as follows:

$v_{i}^{d} (t + 1) = {rand}_{i} * v_{i}^{d} (t) + a_{i}^{d} (t)$ (25)

$x_{i}^{d} (t + 1) = x_{i}^{d} (t) + v_{i}^{d} (t + 1)$ (26)

The GSA algorithm is detailed in Fig. 2.

Fig.2

A GSA algorithm.

2.4 Particle Swarm Optimization based ICA

Lee [10], Gao [17], and Nian et al. [29] proposed using the PSO optimization algorithm in ICA, thus enhancing the solving precision and convergence rate. PSOICA uses the matrix element of the demixing matrix to assume the position and velocity of a particle, defined as follows. K is the size of a swarm. $W_{i} = [W_{i 1}, W_{i 2}, . . ., W_{iL}], 1 \leq i \leq K, L = M \times N$ (27)

$v_{i} = [v_{i 1}, v_{i 2}, . . ., v_{iL}], 1 \leq i \leq K, L = M \times N$ (28)

The local best position P_i and global best position G are defined as follows.

$P_{i} = [P_{i 1}, P_{i 2}, . . ., P_{iL}], 1 \leq i \leq K, L = M \times N$ (29)

$\begin{matrix} G = [G_{1}, G_{2}, . . ., G_{L}] \\ = arg min_{i} {{Fitness}_{i}}, \\ 1 \leq i \leq K, L = M \times N \end{matrix}$ (30)

The fitness function is defined as Equations (6), (8) and (10), corresponding to PSO-kurtosis-ICA, PSO-infomax-ICA, and PSO-BRM-ICA. The optimal solution is obtained according to Eqs. (13) and (14).

The PSO-ICA algorithm is detailed in Fig. 3.

Fig.3

A PSO-ICA algorithm.

GSA is used in ICA algorithm according to the concept of PSO-ICA algorithm. The algorithm is shown in Fig. 4.

Fig.4

A GSA-ICA algorithm.

3 Proposed Algorithm

The standard PSO converges prematurely, and the population diversity is reduced rapidly, so that the optimal solution turns into a local optimum. In order to enhance the diversity of PSO, Gravitational Particle Swarm Optimization (GPSO) is proposed with the exploring ability of GSA, so that the ICA algorithm has a higher convergence rate and better ability to escape local optimization. This paper calls it GPSO-ICA. PSO is combined with GSA, and the PSO and GSA algorithms are executed simultaneously; in each iterative operation, the individual and swarm thoughts (P_best and P_gbest) are combined with the local search capability of GSA by PSO and GSA systems, and the diversity helps obtain a better solution. The particle velocity is updated based on Equation (13) and Equation (25):

$\begin{matrix} v_{i}^{d} (t + 1) = ω v_{i}^{d} (t) & + c_{1} {rand}_{i} (P_{{best}_{i}^{d} (t)} - x_{i}^{d} (t)) \\ + c_{2} {rand}_{i} (P_{gbest}^{d} (t) - x_{i}^{d} (t)) \\ + \frac{1}{2} [c_{3} {rand}_{i} a_{i}^{d} (t) + v_{{GSA}_{i}}^{d} (t)] \end{matrix}$ (31)

We hybridize PSO with GSA to ICA using a low-level coevolution heterogeneous hybrid [26]. The hybrid is low-level, because both algorithms are concurrently operative and isomeric. The two algorithms influence the final result and evolve together. The GPSO-ICA process is shown in Fig. 5.

Fig.5

A GPSO-ICA algorithm.

4 Experimental Result

This section evaluates the performance of the proposed algorithm.

4.1 Experimental environment

All experiments were performed on a computer with a core(TM) i5-5200U 2.20 GHz Intel CPU with 8 GB RAM running on Microsoft Windows 10.

4.2 Parameter settings and dataset

The basic parameter settings of all experiments are listed in Table 1. All experimental results were collected from 10 independent runs, each of 100 iterations, and the average computation time was expressed in seconds. In the experiments, the speech and music signals were sampled from ICA’99 Test Sets which are available at: http://sound.media.mit.edu/ica-bench/. The music signal from the Beethoven Symphony(source signal 1) and the speech signal from a male speaker1(source signal 2) are shown in Fig. 6. The sample rate of signals is 22050 and the duration of signals is 7 seconds. A 2 × 2 mixing matrix was specified to mix the source signals according to Equation (1) as given in Fig. 7. No delayed and convolved sources were considered.

Table 1
Parameter Settings for algorithms

Algorithm: Parameter: Value:

PSO-ICA Number of particles 4

Inertia ω 0.4 to 0.9

c₁, c₂ 2

GSA-ICA Number of agents 10

Initial value of G 100

α 20

∊ 1.0e-6

GPSO-ICA Number of particles 4

Number of agents 10

Inertia ω 0.4 to 0.9

c₁, c₂, c₃ 2

Initial value of G 100

α 20

∊ 1.0e-6

Algorithm:	Parameter:	Value:
PSO-ICA	Number of particles	4
	Inertia ω	0.4 to 0.9
	c₁, c₂	2
GSA-ICA	Number of agents	10
	Initial value of G	100
	α	20
	∊	1.0e-6
GPSO-ICA	Number of particles	4
	Number of agents	10
	Inertia ω	0.4 to 0.9
	c₁, c₂, c₃	2
	Initial value of G	100
	α	20
	∊	1.0e-6

Fig.6

Source signals.

Fig.7

Mixed signals.

4.3 Evaluation of signal-to-interference ratios

The waveforms of the demixing signals using PSO-infomax-ICA, PSO-kurtosis-ICA, and PSO-BRM-ICA algorithms are respectively displayed in Figs. 8, 9 and 10. The waveforms of the demixing signals using GSA-infomax-ICA, GSA-kurtosis-ICA, and GSA-BRM-ICA algorithms are respectively displayed in Figs. 11, 12 and 13. The waveforms of the demixing signals using GPSO-infomax-ICA, GPSO-kurtosis-ICA, and GPSO-BRM-ICA algorithms are respectively displayed in Figs. 14, 15 and 16. GPSO-ICA has a significant effect according to demixing signals. Furthermore, a quantitative comparison in different algorithms is conducted by measuring the signal-to-interference ratios (SIRs), defined as follows:

Fig.8

The demixing signals on PSO-infomax-ICA.

Fig.9

The demixing signals on PSO-kurtosis-ICA.

Fig.10

The demixing signals on PSO-BRM-ICA.

Fig.11

The demixing signals on GSA-infomax-ICA.

Fig.12

The demixing signals on GSA-kurtosis-ICA.

Fig.13

The demixing signals on GSA-BRM-ICA.

Fig.14

The demixing signals on GPSO-infomax-ICA.

Fig.15

The demixing signals on GPSO-kurtosis-ICA.

Fig.16

The demixing signals on GPSO-BRM-ICA.

$SIR (db) = 10 \log_{10} \frac{\sum_{t} ∥ s_{m}^{t} ∥^{2}}{\sum_{t} ∥ y_{m}^{t} - s_{m}^{t} ∥^{2}}$ (32)

s_m means the source signals, and y_m are the demixed signals. The experimental results are shown in Fig. 17. The kurtosis-based measurement on the different algorithm has poor performances, especially in signal 1. The kurtosis measurement is simple and fast, however, it is sensitive to outliers. In thePSO-based algorithm, we can see from the measurement of BRM is better than the measurement of infomax. However, in the GSA-based algorithm, it is opposite. The GPSO-ICA proposed in this paper has a significant effect. Using the measurement of BRM has the outstanding performance than others.

Fig.17

Comparison of SIR.

4.4 Convergence Rates

This subsection evaluates the convergence rates of PSO-ICA, GSA-ICA, and GPSO-ICA based on three different fitness functions (Infomax, Kurtosis, and BRM), as shown in Figs. 18, 19, and 20. In the fitness function evaluation based on Infomax, GPSO-ICA converges slower than GSA-ICA, but it has a better fitness value according to Equation (8) seen in Fig. 18. In the fitness function evaluation based on kurtosis, GPSO-ICA and GSA-ICA have a better convergence rate and fitness value according to Equation (6) seen in Fig. 19. However, the kurtosis measurement is sensitive to outliers. The performance is not good enough. Finally, in the fitness function evaluation based on BRM according to Equation (10), GPSO-ICA has preferable performance in convergence rate and fitness value seen in Fig. 19. Add to this, the measurement of BRM has outstanding convergence steps than infomax and kurtosis. In summary, the highest performing variants were GPSO-ICA.

Fig.18

The convergence rate on infomax measurement.

Fig.19

The convergence rate on kurtosis measurement.

Fig.20

The convergence rate on BRM measurement.

4.5 Comparison of Time Complexity

GPSO-ICA is the concurrent operation of PSO and GSA. Its execution time is longer than PSO or GSA separately, as shown in Fig. 21. This paper uses the non-parametric measurement criterion BRM (Borel Regular Measure) to increase the overall computational efficiency. The measurement of BRM in computation cost is lower than the measurements of infomax and kurtosis. Moreover, according to the experiments on the SIR value and convergence rate, GPSO-BRM-ICA has a significant effect. To sum up the aforesaid experiments, the GPSO-ICA based on the BRM algorithm enhances the SIR value and execution efficiency effectively in the blind source separation problem. The detailed experimental data are shown in Table 2.

Fig.21

Comparison of time complexity.

Table 2

Summary of experiments (*: outstanding result)

Algorithm:	SIR of demixing signal 1:	SIR of demixing signal 2:	Convergence steps:	Elapsed time(sec)
Fast-ICA [7]	2.3461	5.6373	68	42.37
DPSO-ICA [29]	3.6887	6.2774	41	48.15
PSO-infomax-ICA	3.7877	4.3924	37	47.94
PSO-kurtosis-ICA	1.1637	7.5414	24	40.45
PSO-BRM-ICA	7.5194	7.1711	13	*23.18
GSA-infomax-ICA	7.8718	8.9714	8	43.13
GSA-kurtosis-ICA	1.1709	11.367	10	38.57
GSA-BRM-ICA	7.4625	7.2418	13	*22.37
GPSO-infomax-ICA	10.1276	12.3339	13	94.87
GPSO-kurtosis-ICA	1.2307	13.0887	13	81.25
GPSO-BRM-ICA	*11.3915	*17.2432	*7	*47.43

5 Conclusions and Future work

In this work the advantages of PSO and GSA are used to enhance the diversity of particle and local search strategies to avoid the local optimum trap. The convergence rate is increased in ICA, and the blind source separation problem is solved effectively. The Borel Regular Measure (BRM) is imported to propose a simple and rapid evaluation algorithm, thus increasing overall computational efficiency. The experimental results show that the GPSO-ICA based on the BRM algorithm has significant effectiveness. In a future work, the PSO and GSA integration mechanism and a more appropriate fitness function will be further designed to upgrade the quality of GPSO-ICA, which will be used in other fields, such as image enhancement, image segmentation, and noisereduction.

Footnotes

Acknowledgement

This research is financially supported by the Ministry of Science and Technology of Taiwan (under grants Nos. 104-2221-E-006-119-MY3 and 104-3115-E-194-001).

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