Parallel Binary Rafflesia Optimization Algorithm and Its Application in Feature Selection Problem

Abstract

The Rafflesia Optimization Algorithm (ROA) is a new swarm intelligence optimization algorithm inspired by Rafflesia’s biological laws. It has the advantages of high efficiency and fast convergence speed, and it effectively avoids falling into local optimum. It has been used in logistics distribution center location problems, and its superiority has been demonstrated. It is applied to solve the problem of continuity, but there are many binary problems to be solved in the actual situation. Thus, we designed a binary version of ROA. We used transfer functions to change continuous values into binary values, and binary values are used to symmetrically represent the meaning of physical problems. In this paper, four transfer functions are implemented to binarize ROA so as to improve the original transfer function for the overall performance of the algorithm. In addition, on the basis of the algorithm, we further improve the algorithm by adopting a parallel strategy, which improves the convergence speed and global exploration ability of the algorithm. The algorithm is verified on 23 benchmark functions, and the parallel binary ROA has a better performance than some other existing algorithms. In the aspect of the application, this paper adopts the datasets on UCI for feature selection. The improved algorithm has higher accuracy and selects fewer features.

1. Introduction

With the rise of the Internet industry, the scale of data available to us is getting larger and larger, so this requires more and more of our ability to perform information filtering. Data mining has received increasing attention in recent years [1,2]. The acquisition of data requires not only comprehensive but also efficient access [3]. Before formally processing data, we need to perform data pre-processing. Feature selection is a good way of data pre-processing [4]. In a large dataset, there are many features that may be unnecessary or redundant and not relevant to our ultimate goal of classification, so it is necessary to reduce the dimensionality of the data and focus on exploring those features that are more representative to achieve a more efficient classification and processing problem [5,6]. However, in our inexperience, it is difficult to estimate which features are useful and which ones are redundant.

Davies has shown that the feature selection problem is an NP problem, which means that we can only find the optimal subset of features by exhaustive enumeration. However, the time cost of using the exhaustive method is excessively high. This cost is unacceptable in most cases. Therefore it is expected that a more optimal method can be used to solve this problem. A heuristic algorithm is a method that expects an optimal or suboptimal solution at an acceptable cost and has become a mainstream approach to solve complex problems [7,8,9,10,11,12]. Particle swarm optimization is a very classical heuristic algorithm developed based on the behavior of biological swarms in nature [13,14,15,16]. Whale optimization algorithm is inspired by the social behavior of humpback whales, which is more competitive than traditional algorithms and has been applied to various fields [17,18,19,20]. In recent years, heuristic algorithms have received more and more attention from scholars, and many excellent algorithms have been proposed [21,22,23,24,25,26,27]. According to the shortcomings of algorithms, many heuristic algorithms for further optimization have also been proposed by scholars [28,29,30,31]. ROA is an algorithm based on the living habits of Rafflesia, and has the advantages of fast convergence and not easily falling into local optimum [32].

Traditional optimization algorithms are generally applied to solve continuous problems. Many optimization algorithms are applied to various problems [33,34,35,36,37]. However, in reality, there are many discrete problems that need to be solved, such as feature selection. To date, many scholars have proposed many binary algorithms, and all of them have achieved good results [38,39,40,41]. The binary PSO algorithm has been applied to the problem of feature selection with good results, and this is the first time that a binary algorithm has been proposed [42]. The improved binary Grey wolf optimization algorithm is proposed by Hu [43]. In the binary algorithm, we will use 1 or 0 to symmetrically indicate whether the corresponding feature in the dataset is to be used for classification. In the paper, a new parallel strategy is proposed and new transfer functions are introduced to achieve better results in the application of the feature selection problem.

Parallel strategies have been widely used in recent years to improve heuristic algorithms and have obtained good results [44,45]. The parallel strategy is to group the original populations, explore them independently of each other, and communicate with them every certain number of iterations through a certain communication strategy to achieve faster convergence and prevent premature maturity [46,47,48]. Schutte has demonstrated the effectiveness of parallel strategies through numerous experiments [49].

The binarization of algorithms is often implemented with transfer functions [50,51]. The S-type transfer function is applied in the binary PSO algorithm to complete the binarization of the continuity values [42]. Zhuang applied a four-family transfer function for the binarization of the quasi-affine transformation evolution algorithm and demonstrated the superiority of the algorithm by 23 benchmark test functions [52]. This paper focuses on the binarization of the ROA algorithm using transfer functions and the optimization of the algorithm by a parallel strategy. The improved algorithm was tested on the UCI dataset for feature selection and compared with the binary GWO and binary PSO algorithms, and the experimental results showed that the algorithm outperformed the above two algorithms in some cases.

The general organization of this paper is as follows. Section 2 describes related works. Section 3 proposes new transfer functions based on mathematical analysis and proposes a parallel strategy to optimize the algorithm. Section 4 demonstrates the performance of the algorithm by testing it on benchmark functions. Section 5 presents the application of the algorithm to feature selection. Section 6 summarizes and discusses this paper.

2. Related Works

2.1. ROA

Rafflesia is a saprophytic organism. Its main axis is extremely short, and there are no leaves and underground stems. When it blooms, it emits a peculiar smell, thus attracting certain insects to pollinate it. In this process, because of its unique structure, some attracted insects may be trapped during pollination and die. After the flowering period has passed, the petals will wither, and the fruit will be ripe at this time. There will be many tiny seeds in the fruit. After the fruit falls to the ground, the seeds will be randomly taken to various places in various ways to find a suitable germination place. Based on the above characteristics, the ROA algorithm is divided into three stages to implement.

The stage of attracting insects is divided into two strategies. The first strategy seeks to replace the inferior individual. In this strategy, 1/3 of the individual with poor fitness is replaced by a new one. Each dimension of the newly added individual is abstracted into a three-dimensional space for calculation, and the dimension model is illustrated in Figure 1. The equations for calculating the individual $X_i$ position are (1)–(3) to update its position.

$$X_{i_k}=X_{bes{t}_k}+d·sin{\beta }_{k}cos{\gamma }_k$$

(1)

$$d=\sqrt{\sum _{k=1}^D{(X_{R_k}-X_{bes{t}_k})}^2}$$

(2)

$$X_{wors{t}_i}=X_i$$

(3)

where d is the distance between $X_i$ and $X_{best}$. $X_R$ is a random individual in the population. $X_{best}$ is the best individual in the population. $X_{worst}$ represents the poor individual in the population. ${\beta }_k$ is a random value between [0, $\pi$/2]; ${\gamma }_k$ is random value between [0, $\pi$].

The second strategy is to update 2/3 of individuals with better fitness. The individual velocity update equation is derived from the insect flapping flight model. The individual speed update equations are as shown in Equations (4)–(7). The individual update Equation is (8):

$$\overrightarrow{v_1}=\frac{{\omega }_0}{2}\sqrt{A^{2}si{n}^2({\omega }_{0}t+\theta )+B^{2}co{s}^2({\omega }_{1}t+\theta )}$$

(4)

$$\overrightarrow{v_2}=\overrightarrow{v_2}{\omega }_{0}cos({\omega }_{0}t+\theta +\varphi )$$

(5)

$$\overrightarrow{v}=\overrightarrow{v_1}+\overrightarrow{v_2}$$

(6)

$$\overrightarrow{le}=C·\overrightarrow{v}·t+(X_{best}-X_{\left(t\right)})·(1-C)·rand$$

(7)

$$X_{\left(t\right)}=X_{\left(t\right)}+\overrightarrow{le}$$

(8)

where $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ represent the translation speed and rotation speed, respectively, and ${\omega }_0$ and ${\omega }_1$ represent the frequency periods of flapping and lateral flapping wings, respectively, both with values of 0.025. A is the amplitude of the wing during movement, with the value of 2.5. B is the lateral offset, with a value of 0.1. $\varphi$ representing the phase difference between translation and rotation, with the value of −0.78545. The value range of $\theta$ is (0, 2pi). The initial value of $v_2$ is within the range (0, 2pi). C is a random number in (−1, 1).

According to the principle of “natural selection by nature, survival of the fittest”, the stage of swallowing insects will eliminate the individual with the worst fitness every certain number of iterations, thus ensuring the quality of the solution and speeding up the efficiency of the algorithm.

At the stage of spreading seeds, the position of the Rafflesia is represented by the initial optimal individual at this stage. At the same time, other individuals will randomly search around for an environment suitable for growth. At this stage, individuals update the equation as follows:

$$X_{{\left(t\right)}_k}=X_{bes{t}_k}+rd·exp(\frac{item}{Max\_iter}-1)·sign(rand-0.5)$$

(9)

where $k(k=1,\cdots ,D)$ is the population dimension. $iter$ and $Max\_iter$ represent the current number of iterations and the maximum number of iterations, respectively. Where $sign(rand-0.5)$ is a symbolic function with a value of 1 or −1. $rd$ represents the range of individual values in the population, and its expression equation is as follows:

$$rd=rand·(ub-lb)+lb$$

(10)

$ub$ is the upper limit of the search space and $lb$ is the lower limit of the search space. Algorithm 1 is the pseudo code for ROA.

Algorithm 1:ROA

Input: f(x): fitness function; N: individual number; d: function dimension; lb: maximum boundary; ub: minimum boundary; $Max\_iter$: maximum number of iteration Output: global optimal value. Initialize the related parameters of ROA Randomly generate the positions of the insects while iter<$Max\_iter$do

2.2. Transfer Function

The transfer function is a common method used in metaheuristic algorithms to convert from continuous variables into binary variables. We can use transfer functions to map a continuous variable to [0, 1] and then convert them into binary values by subsequent processing. It is reasonable and convenient to use transfer functions for binary operations.

In [43], Hu uses transfer functions to binarize the algorithm. As shown in Equation (11), where A denotes a vector of coefficients and D denotes a vector of distances between the current individual and the superior individual.

$$F\left(AD\right)=\frac{1}{1+e^{-10(AD-0.5)}}$$

(11)

After mapping through the transfer function in the above equation, we can use Equation (12) to iterate over the positions of individuals. $rand$ denotes a random number that is uniformly distributed between 0 and 1.

$$X{(t+1)}_i=\left\{\begin{array}{c}0,F\left(AD\right)\le rand\hfill \\ 1,F\left(AD\right)>rand\hfill \end{array}\right.$$

(12)

In this paper, we use four transfer function families to map continuous values. Herein, there are S-shaped, U-shaped, and V-shaped families of transfer functions are proposed by Mirjalili [53,54]. The expressions are given in the following

Table 1. S-shaped families of transfer functions.

Function Name	Formula
$S_1\left(x\right)$	$\frac{1}{1+e^{-2x}}$
$S_2\left(x\right)$	$\frac{1}{1+e^{-x}}$
$S_3\left(x\right)$	$\frac{1}{1+e^{-x/2}}$
$S_4\left(x\right)$	$\frac{1}{1+e^{-x/3}}$

,

Table 2. V-shaped families of transfer functions.

Function Name	Formula
$V_1\left(x\right)$	$\left|erf\left(\frac{\sqrt{\pi }}{2}x\right)\right|$
$V_2\left(x\right)$	$\left|tanh\right(x\left)\right|$
$V_3\left(x\right)$	$\left|\frac{x}{\sqrt{1+x^2}}\right|$
$V_4\left(x\right)$	$\left|\frac{2}{\pi }arctan\left(\frac{2}{\pi }x\right)\right|$

and

Table 3. U-shaped families of transfer functions.

Function Name	Formula
$U_1\left(x\right)$	$min(\left|x^{1.5}\right|,1)$
$U_2\left(x\right)$	$min(\left|x^2\right|,1)$
$U_3\left(x\right)$	$min(\left|x^3\right|,1)$
$U_4\left(x\right)$	$min(\left|x^4\right|,1)$

. Guo proposed a family of Z-shaped transfer functions in [55] and used them in binarization. The expressions are shown in

Table 4. Z-shaped families of transfer functions.

Function Name	Formula
$Z_1\left(x\right)$	$\sqrt{1-2^x}$
$Z_2\left(x\right)$	$\sqrt{1-5^x}$
$Z_3\left(x\right)$	$\sqrt{1-8^x}$
$Z_4\left(x\right)$	$\sqrt{1-{20}^x}$

.

The position update equation of the S-shaped transfer functions is (12), which means that the particle is more likely to become 0 when its velocity is small and 1 when its velocity is large. However, because the nature of the U-shaped, V-shaped, and Z-shaped transfer functions is different from that of S-shaped transfer functions, it is not possible to use the same position update method as the S-shaped transfer functions. Zhuang proposed the location update Equation (13). In this updated formula, when the individual’s speed is higher, there is a higher possibility of changing to a complementary position.

$$X(t+1)=\left\{\begin{array}{c}1-X\left(t\right),F\left(AD\right)>rand\hfill \\ X\left(t\right),F\left(AD\right)\le rand\hfill \end{array}\right.$$

(13)

3. Analysis and Proposed Parallel Binary ROA

In this section, a mathematical analysis of the search space of ROA is carried out. The initial transfer function is changed on the basis of mathematical analysis, and the algorithm is binarized. The algorithm is optimized using a parallel strategy, and two inter-group communication methods are proposed to improve the convergence speed of the algorithm and reduce the risk of premature convergence.

3.1. Mathematical Analysis

In the ROA algorithm, the location of the insect can be at any point in the search space, but in the binary ROA, its location can only be chosen between 0 and 1. Therefore, the search space of the original algorithm needs to be analyzed to obtain its specific range so that the corresponding transfer function can be better constructed to binarize the algorithm.

In order to make it easier to understand, we take one dimension of the individual to analyze. From (7) and (8), we know that $X_i=X_i+cvt+(X_{best}-X_i)·(1-c)·rand$, where $v=v_1+v_2$, $v_1=\frac{{\omega }_0}{2}\sqrt{A^{2}si{n}^2({\omega }_{0}t+\theta )+B^{2}co{s}^2({\omega }_{1}t+\theta )}$, $v_2=v_2{\omega }_{0}cos({\omega }_{0}t+\theta +\varphi )$. Furthermore, it is known that the values of $X_i$ and $X_{best}$ can only be chosen between 0 or 1, so the final value of X has the four following cases.

(1)

if $X_i=0$ and $X_{best}=0$

$X_i=0+cvt+(0-0)·(1-c)·rand=cv$

(2)

if $X_i=0$ and $X_{best}=1$

$X_i=0+cvt+(1-0)·(1-c)·rand=cv+(1-c)·rand$

(3)

if $X_i=1$ and $X_{best}=0$

$X_i=1+cvt+(0-1)·(1-c)·rand=1+cv-(1-c)·rand$

(4)

if $X_i=1$ and $X_{best}=1$

$X_i=1+cvt+(1-1)·(1-c)·rand=1+cv$

We can know that c is a random number in (−1, 1) and $rand$ is a random number between (0, 1). After the above analysis, we derive $vs.\in (-0.156,0.188)$, and conclude that the range of values of $X_i\in (-1.188,2.156)$. The conclusions we analyze here can be used for the improvement of the transfer function later.

3.2. New Transfer Functions

The transfer function is extremely important for the binary of the algorithm. In Section 2.2, we refer to four families of transfer functions to binarize the algorithm. This section will improve the transfer functions based on the mathematical analysis of the algorithm in Section 3.1.

The original transfer function is not fully applicable in this algorithm’s binary process because the search space to which the original transfer function is adapted does not match this algorithm. An acceptable transfer function allows individuals to converge to the optimum more quickly, while an unsuitable transfer function may cause the algorithm to converge slowly. We take S-shaped transfer functions as an example, and we hope that the greater the value of the individual in the process of position update, the greater the probability that the individual position will take the value of 1, and vice versa for the value of 0. From Section 3.1, we know that the search space of the ROA is [−1.188, 2.156], the center of the search space is 0.484, and the range of values of the $S_1$ transfer function in the search space is [0.0850, 0.9868]. Therefore, we deform the transfer functions so that we can achieve better results, and the deformed S-shaped transfer functions are shown in

Table 5. Improved S-shaped families of transfer functions.

Function Name	Formula
$S_1\left(x\right)$	$\frac{1}{1+e^{-2(x-0.484)}}$
$S_2\left(x\right)$	$\frac{1}{1+e^{-5(x-0.484)}}$
$S_3\left(x\right)$	$\frac{1}{1+e^{-10(x-0.484)}}$
$S_3\left(x\right)$	$\frac{1}{1+e^{-15(x-0.484)}}$

.

Similar deformation operations are performed for other transfer function families, which enable the values that can be obtained in the original search space to be more evenly distributed between [0, 1], resulting in better results. The deformed transfer functions are shown in

Table 6. Improved V-shaped families of transfer functions.

Function Name	Formula
$V_1\left(x\right)$	$\left|erf\left(\frac{\sqrt{\pi }}{2}(4·(x-0.484))\right)\right|$
$V_2\left(x\right)$	$\left|tanh\right(3·(x-0.484)\left)\right|$
$V_3\left(x\right)$	$|\frac{\sqrt{1+1.{672}^2}}{1.672}·\frac{(x-0.484)}{\sqrt{1+{(x-0.484)}^2}}|$
$V_4\left(x\right)$	$|\frac{1}{\frac{2}{\pi }\ast arctan(\frac{\pi }{2}·5.016)}·\frac{2}{\pi }arctan\left(\frac{6}{\pi }(x-0.484)\right)|$

,

Table 7. Improved U-shaped families of transfer functions.

Function Name	Formula
$U_1\left(x\right)$	$min(\left|\frac{{(x-0.484)}^{1.5}}{0.{516}^{1.5}}\right|,1)$
$U_2\left(x\right)$	$min(\left|\frac{{(x-0.484)}^2}{0.{516}^2}\right|,1)$
$U_3\left(x\right)$	$min(\left|\frac{{(x-0.484)}^3}{0.{516}^3}\right|,1)$
$U_4\left(x\right)$	$min(\left|\frac{{(x-0.484)}^4}{0.{516}^4}\right|,1)$

and

Table 8. Improved Z-shaped families of transfer functions.

Function Name	Formula
$Z_1\left(x\right)$	$\frac{\sqrt{1-2^{x-0.484}}}{\sqrt{1-2^{-1.602}}}$
$Z_2\left(x\right)$	$\frac{\sqrt{1-5^{x-0.484}}}{\sqrt{1-5^{-1.602}}}$
$Z_3\left(x\right)$	$\frac{\sqrt{1-8^{x-0.484}}}{\sqrt{1-8^{-1.602}}}$
$Z_4\left(x\right)$	$\frac{\sqrt{1-{20}^{x-0.484}}}{\sqrt{1-{20}^{-1.602}}}$

. A comparison of the original and improved transfer function series is shown in Figure 2.

3.3. Parallel Strategy

Parallel strategy is often used in the optimization of algorithms. Through parallels, we can obtain better quality solutions and faster convergence. The concrete practice of a parallel strategy is to explore the optimal solution through grouping, and each group is independent of the other. Then, when certain conditions are met, communication between groups is carried out, thus completing the information exchange between groups.

In the parallel strategy used in this algorithm, we group the original population into a main population and two subpopulations. Subpopulation 1 selects the better individuals of the main population to search because optimal individual may appear in the vicinity of the better ones. Subpopulation 2 randomly selects individuals in the main population to search so as to prevent the population from falling into a local optimum.

The communication strategy is extremely important for the algorithm. A good communication method can make the algorithm obtain faster convergence and make the algorithm less likely to fall into the local optimum. In order to achieve faster convergence and improve the quality of the solution, two intergroup communication strategies are proposed. Strategy 1 compares the best individuals in the subpopulation with the best in the main population after every certain number of iterations, and if the individuals in the subpopulation are better, the worst individuals in the main population are replaced. This strategy can make the algorithm converge faster. Strategy 2 is to monitor the optimal individuals in the main population, and if no update is made for a long time, the algorithm is considered to be possibly stuck in a local optimum, and then, some individuals that are optimal in the main population are updated. The update is performed by combining the different dimensions of previously good individuals to form new individuals, and then replacing some of the best individuals in the main population with new ones. Figure 3 depicts the communication strategy for a parallel strategy.

4. Experiments, Results, and Analysis

In this section, we will use 23 benchmark functions to examine the exploration and development capabilities of the algorithm. The benchmark test function used, although specifically for evaluating continuous optimization algorithms, still has the original properties of the function when each dimension can only take binary values, so the benchmark test function can be used to detect binary optimization. In

Table 9. Unimodal benchmark functions.

Name	Function	Space	${\mathit{D}}_{\mathbf{im}}$	${\mathit{f}}_{\mathbf{min}}$
Sphere	$f_1\left(x\right)={\sum }_{i=1}^n{x}_i^2$	[−100, 100]	30	0
Schwefel’s function 2.21	$f_2\left(x\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	[−10, 10]	30	0
Schwefel’s function 1.2	$f_3\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	[−100, 100]	30	0
Schwefel’s function 2.22	$f_4\left(x\right)=ma{x}_i\{\left|x_i\right|,1\le i\le n\}$	[−100, 100]	30	0
Rosenbrock	$f_5\left(x\right)={\sum }_{i=1}^{n-1}[100(x_{i+1}-x_i^2)+{(x_i-1)}^2]$	[−30, 30]	30	0
Step	$f_6\left(x\right)={\sum }_{i=1}^n{\left([x_i+0.5]\right)}^2$	[−100, 100]	30	0
Dejong’s noisy	$f_7\left(x\right)={\sum }_{i=1}^{n}i{x}_i^4+random(0,1)$	[−1.28, 1.28]	30	0

, $f_1-f_7$ represents unimodal benchmark functions; in

Table 10. Common multimodal benchmark functions.

Name	Function	Space	${\mathit{D}}_{\mathbf{im}}$	${\mathit{f}}_{\mathbf{min}}$
Schwefel	$f_8\left(x\right)={\sum }_{i=1}^n-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)$	[−500, 500]	30	−12,569
Rastringin	$f_9\left(x\right)={\sum }_{i=1}^n[x_i^2-10cos\left(2\pi x_i\right)+10]$	[−5.12, 5.12]	30	0
Ackley	$f_{10}\left(x\right)=-20exp(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2})-$	[−32, 32]	30	0
	$exp\left(\frac{1}{n}{\sum }_{i=1}^{n}cos\left(2\pi x_i\right)\right)+20+e$
Griewank	$f_{11}\left(x\right)=\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	[−600, 600]	30	0
Generalized penalized 1	$f_{12}\left(x\right)=\frac{\pi }{n}\{10sin\left(\pi y_1\right)+{\sum }_{i=1}^{n-1}{(y_i-1)}^2[1+$	[−50, 50]	30	0
	$10si{n}^2\left(\pi y_{i+1}\right)]+{(y_n-1)}^2\}+{\sum }_{i=1}^{n}u(x_i,10,100,4)$
	$y_i=1+\frac{x_i+1}{4}u(x_i,a,k,m)=$
	$\left\{\begin{array}{c}k{(x_i-a)}^m{x}_i>a\hfill \\ 0,-a<x_i<a\hfill \\ k{(-x_i-a)}^m{x}_i<-a\hfill \end{array}\right.$
Generalized penalized 2	$f_{13}\left(x\right)=0.1\{si{n}^2\left(3\pi x_1\right)+$	[−50, 50]	30	0
	${\sum }_{i=1}^n{(x_i-1)}^2[1+si{n}^2(3\pi x_i+1)]+$
	${(x_n-1)}^2[1+si{n}^2\left(2\pi x_n\right)]\}+$
	${\sum }_{i=1}^{n}u(x_i,10,100,4)$

, $f_8-f_{13}$ represents common multimodal benchmark functions; and in

Table 11. Multimodal benchmark functions in low dimension.

Name	Function	Space	${\mathit{D}}_{\mathbf{im}}$	${\mathit{f}}_{\mathbf{min}}$
Fifth of Dejong	$f_{14}\left(x\right)=\left(\frac{1}{500}{\sum }_{j=1}^{25}\frac{1}{j+{\sum }_{i=1}^2{(x_i-a_{ij})}^6}\right)$	[−65, 65]	2	1
Kowalik	$f_{15}\left(x\right)={\sum }_{i=1}^{1}1{[a_i-\frac{x_1(b_i^2+b_i{x}_2)}{b_i^2+b_i{x}_3+x_4}]}^2$	[−5, 5]	4	0.00030
Six-hump camel back	$f_{16}\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	[−5, 5]	2	−1.0316
Branins	$f_{17}\left(x\right)={(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6)}^2+10(1-\frac{1}{8\pi })cos{x}_1+10$	[-5, 5]	2	0.398
Goldstein–Price	$f_{18}\left(x\right)=[1+{(x_1+x_2+1)}^2(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2)]$	[−2, 2]	2	3
	$\times [30+{(2x_1-3x_2)}^2\times (18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2)]$
Hartman 1	$f_{19}\left(x\right)=-{\sum }_{i=1}^4{c}_{i}exp(-{\sum }_{j=1}^3{a}_{ij}{(x_j-p_{ij})}^2)$	[1, 3]	3	−3.86
Hartman 2	$f_{20}\left(x\right)=-{\sum }_{i=1}^4{c}_{i}exp(-{\sum }_{j=1}^6{a}_{ij}{(x_j-p_{ij})}^2)$	[0, 1]	6	−3.32
Shekel 1	$f_{21}\left(x\right)=-{\sum }_{i=1}^5{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	[0, 10]	4	−10.1532
Shekel 2	$f_{22}\left(x\right)=-{\sum }_{i=1}^7{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	[0, 10]	4	−10.4028
Shekel 3	$f_{23}\left(x\right)=-{\sum }_{i=1}^{10}{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	[0, 10]	4	−10.5363

, $f_{14}-f_{23}$ represents multimodal benchmark functions in low dimension. $D_{im}$ in the table is the dimension of the function, $f_{min}$ denotes the minimum value that the function can obtain, and $space$ denotes the search space of the function.

There is no optimal local solution in the unimodal function, but only one global optimal solution and the convergence speed of the algorithm can be measured by the single-peaked function. The common multimodal function is relatively complex, with multiple locally optimal solutions, so it can test whether the algorithm can dispose of the local optimal and thus reach the optimal global solution. Besides having multiple locally optimal solutions, the multimodal functions in low dimension with its too low dimensionality makes the algorithm more prone to prematureness, so it can strictly verify the convergence results of the algorithm.

Experimental Results

In experiments, we examine the impact of the improved transfer functions and the addition of the parallel strategy on the algorithm. Due to space limitation, we chose four transfer functions, namely $S_3,V_2,U_3,Z_3$, in four families of transfer functions as examples for our experiments. We also compared the improved algorithm with BPSO and BGWO to perform the experiments [42,43].

In our experiments, we will judge the algorithm’s merit by the quality of the solution and its stability. In the experiments, the algorithms were run 20 times on each benchmark function, and 150 iterations were passed during each run, with a total of 30 individuals in the population.

Table 12. The statistical results of the original transfer function.

Function	BPSO		BGWO		BROA_S3		BROA_V2		BROA_U3		BROA_Z3
	adv	std	adv	std	adv	std	adv	std	adv	std	adv	std
f1	0.9333	0.6397	3.1000	1.2959	0.0667	0.2537	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f2	0.7333	0.5833	2.8000	1.2149	0.1667	0.3790	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f3	8.0333	6.6978	117.4667	103.4004	0.4667	1.2243	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f4	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	0.8000	0.4068	0.0000	0.0000	1.0000	0.0000
f5	259.8333	112.4192	16.9667	38.5920	31.0333	48.3432	30.0333	18.1668	29.0000	0.0000	34.7667	41.6270
f6	8.7000	1.1265	13.2333	2.1485	8.1667	1.0933	7.5000	0.0000	7.5000	0.0000	7.5000	0.0000
f7	4.7210	3.0065	35.4002	19.3117	0.7211	1.5594	0.0006	0.0006	0.0005	0.0004	0.0010	0.0011
f8	−24.5149	0.3653	−25.2441	0.0000	−25.1880	0.2135	−25.1319	0.2909	−25.0758	0.3423	−25.2441	0.0000
f9	1.0000	0.6433	2.5667	1.5241	0.2333	0.4302	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f10	0.5548	0.3511	1.3355	0.2614	0.0239	0.1309	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f11	0.0223	0.0130	0.1358	0.0601	0.0025	0.0066	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f12	1.7746	0.0771	2.0588	0.1769	1.6790	0.0382	1.6690	0.0000	1.6690	0.0000	1.6690	0.0000
f13	0.1000	0.0643	0.0000	0.0000	0.0067	0.0254	0.0200	0.0407	0.0067	0.0254	0.0000	0.0000
f14	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000
f15	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000
f16	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f17	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000
f18	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000
f19	−0.3348	0.0000	−0.3337	0.0063	−0.3348	0.0000	−0.3348	0.0000	−0.3348	0.0000	−0.3348	0.0000
f20	−0.1657	0.0000	−0.1334	0.0554	−0.1657	0.0000	−0.1657	0.0000	−0.1657	0.0000	−0.1657	0.0000
f21	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000
f22	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000
f23	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000

shows the experimental results when the algorithm uses the original transfer function, and

Table 13. Statistical results of applying improvement strategies.

Function	BPSO		BGWO		BROA_S3		BROA_V2		BROA_U3		BROA_Z3
	adv	std	adv	std	adv	std	adv	std	adv	std	adv	std
f1	0.9333	0.6397	3.1000	1.2959	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f2	0.7333	0.5833	2.8000	1.2149	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f3	8.0333	6.6978	117.4667	103.4004	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f4	1.0000	0.0000	1.0000	0.0000	1.0000	0.0000	0.8667	0.3457	0.0000	0.0000	1.0000	0.0000
f5	259.8333	112.4192	16.9667	38.5920	3.5000	19.1703	28.0333	5.2947	29.0000	0.0000	29.4667	24.1400
f6	8.7000	1.1265	13.2333	2.1485	7.5000	0.0000	7.5000	0.0000	7.5000	0.0000	7.5000	0.0000
f7	4.7210	3.0065	35.4002	19.3117	0.0004	0.0003	0.0005	0.0005	0.0003	0.0004	0.0008	0.0008
f8	−24.5149	0.3653	−25.2441	0.0000	−25.2441	0.0000	−25.2441	0.0000	−25.2161	0.1536	−25.2441	0.0000
f9	1.0000	0.6433	2.5667	1.5241	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f10	0.5548	0.3511	1.3355	0.2614	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f11	0.0223	0.0130	0.1358	0.0601	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f12	1.7746	0.0771	2.0588	0.1769	1.6690	0.0000	1.6690	0.0000	1.6690	0.0000	1.6690	0.0000
f13	0.1000	0.0643	0.0000	0.0000	0.0000	0.0000	0.0033	0.0183	0.0000	0.0000	0.0000	0.0000
f14	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000	12.6705	0.0000
f15	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000	0.1484	0.0000
f16	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
f17	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000	27.7029	0.0000
f18	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000	600.0000	0.0000
f19	−0.3348	0.0000	−0.3337	0.0063	−0.3348	0.0000	−0.3348	0.0000	−0.3348	0.0000	−0.3348	0.0000
f20	−0.1657	0.0000	−0.1334	0.0554	−0.1657	0.0000	−0.1657	0.0000	−0.1657	0.0000	−0.1657	0.0000
f21	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000	−5.0552	0.0000
f22	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000	−5.0877	0.0000
f23	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000	−5.1285	0.0000

shows the experimental results when the algorithm uses the improved strategy. For the convenience of reading, the experimental data are retained to four decimal places using rounding.

As can be seen from Table 12, BROA_U3, BROA_V2, and BROA_Z3 have better results on the unimodal benchmark functions. This proves the good convergence performance of the ROA algorithm using the original V-shaped, U-shaped, and Z-shaped transfer function binarization. BGWO and BROA_Z3 achieved the best results on f8 and f13. BROA_U3, BROA_V2, and BROA_Z3 achieved the best performance on f9–f12, which indicates their good global exploration capability. In contrast, the ROA algorithm was optimized with the S-shaped transfer function and performs poorly. In the multimodal benchmark functions in low dimensions, all algorithms achieve better results.

Table 13 shows the experimental results of the algorithm after applying the improved strategy. For ease of reading, the improved algorithm is indicated in blue font if it is better than the original algorithm, and in red font if the improved algorithm is inferior to the original algorithm. The improved BROA_S3 achieves better results on both f1–f3 and f5–f13, indicating that the improved strategy is very effective on it, allowing it to obtain better convergence performance and more easily escape the local optimum. The improved BROA_V2 has better results on f5, f7, f8, f13; the improved BROA_U3 has better results on f7, f8, f13; and the improved BROA_V2 has better results on f5, f7. The differences between the fitness values of the improved algorithm and the original algorithm on the test functions are shown in Figure 4. On the figure, we can more clearly see the performance improvement due to the algorithm improvement. The above results show that the performance of the algorithm is significantly improved after applying the improved strategy. The parallel strategy of the algorithm allows it to converge to the optimum faster and effectively prevents the algorithm from falling into a local optimum. Among the algorithms that use different transfer functions for binarization, the improved strategy is the most effective in optimizing the algorithm for the binarization of the S-shaped transfer function.

In order to demonstrate that the improved algorithm is significantly different from the original algorithm, the Wilcoxon rank-sum test was performed on the results of the algorithm before and after improvement at the 5% significance level. We assume that there is no significant difference between the improved algorithm and the original algorithm. The obtained results are shown in

Table 14. p-value of the Wilcoxon rank-sum test.

Function	PBROA_S3	PBROA_V2	PBROA_U3	PBROA_Z3
f1	0.0156	1	1	1
f2	0.0425	1	1	1
f3	0.2500	1	1	1
f4	0.0001	0.2500	0.0000	1
f5	0.0080	0.8659	0.7465	0.0021
f6	0.0625	1	1	1
f7	0.0013	0.0023	0.0034	0.0033
f8	1	0.2500	0.0425	1
f9	1	1	1	1
f10	0.2500	1	1	1
f11	0.1250	1	1	0.2500
f12	0.1250	1	1	1
f13	1	0.1250	0.0335	1
f14	1	1	1	1
f15	1	1	1	1
f16	1	1	1	1
f17	1	1	1	1
f18	1	1	1	1
f19	1	1	1	1
f20	1	1	1	1
f21	1	1	1	1
f22	1	1	1	1
f23	1	1	1	1

. The original hypothesis is rejected if the value p in the table is less than 0.05, proving that the improved algorithm is significantly different from the one before improvement. From the table, we can see that PBROA_S3 has p-values less than 0.05 on f1, f2, f4, f5, f7; PBROA_V2 has p-values less than 0.05 on f7; PBROA_U3 has p-values less than 0.05 on f4, f7, f8; and PBROA_Z3 has p-values lower than 0.05 on f5, f7. From the results analyzed above, we know that there is a significant difference between the improved algorithm and the original algorithm. This further proves that the improved strategy enhances the performance of the algorithm.

5. Application of Feature Selection

Because the original data have a lot of redundancy, it makes the data processing difficult and time consuming. Feature selection is a very important data pre-processing approach, where the raw data are initially processed by feature selection, and the processed data become more accurate and streamlined. In this section, we apply the improved algorithm to feature selection.

5.1. Dataset

The experiments test the performance of the algorithm using 13 datasets. These datasets are all from the UCI machine learning repository and vary in number and dimensionality [56]. The specific parameters of the datasets are shown in

Table 15. Information about the testing datasets.

Dataset	Instances	Number of Features	Number of Categories	Attribute Types
Breast Cancer	284	9	2	Categorical
Breast Cancer Wisconsin	699	9	6	Integer
Glass	214	9	6	Real
Iris	150	4	3	Real
South German Credit	1000	20	4	Integer, real
Wine	178	13	3	Integer, real
Flags	194	28	8	Categorical, integer
Dermatology	366	33	6	Categorical, integer
Credit Approval	690	15	2	Categorical, integer, real
Chess(kr-vs-kp)	3196	36	2	Categorical
Hepatitis	155	19	2	Categorical, integer, real
Image Segmentation	210	19	7	Real
Statlog (Australian Credit Approval)	690	14	2	Categorical, integer, real
Wall-Following Robot Navigation Data	5456	25	4	Real

.

5.2. Experimental Results and Analysis

KNN is a simple method for data classification in data mining, which is based on the principle that each individual type can be represented by its K-nearest neighbors [57]. The calculation of distance in KNN generally uses the Manhattan distance or the Euclidean distance. The specific calculation is as in Equation (14)

$$D_p(x,x_{test})={\left(\sum _{i=1}^n{|x_{{}_i}-x_{tes{t}_i}|}^p\right)}^{\frac{1}{p}}$$

(14)

where p can be taken as 1 or 2; when $p=1$, this equation indicates the calculation of Manhattan distance, and when $p=2$, this equation indicates the calculation of Euclidean distance. Where x denotes training data, $x_{test}$ denotes the test data, and n denotes the dimensionality of the data.

In this experiment, the K-fold cross-validation is adopted, the original dataset is divided into K parts, and K-1 parts are used as training data each time, leaving one datum as the test datum. After repeating K experiments, the final result is taken as the average of K experiments. In the experiment, we should balance the classification accuracy and the number of features selected. The purpose of feature selection is to streamline the experimental data, so we want to obtain a lower classification error and, at the same time, select as few features as possible. The experimental error is calculated as shown in Equation (15)

$$fitness=a·error+(1-a)\frac{flag}{dim}$$

(15)

where $error$ denotes the classification error, a denotes the proportion of the classification error in the fitness, and in this experiment, a takes the value of 0.99. $flag$ denotes the number of selected features, and $dim$ denotes the total number of features in this dataset.

Euclidean and 5 are the values of the parameters of Distance and NumNeighbors in KNN, and the value of K-fold parameter in cross validation is set to 2 in this experiment. In the experiments, the number of all populations was set to 30, the number of iterations was set to 150, and it was run 20 times on each dataset. In the algorithm used in the experiments, when the individual takes the value of 0, it means that the current top feature is not selected, and when the individual takes the value of 1, this means that the current top feature is selected.

The data in

Table 16. The result of fitness value.

Dataset	BPSO	BGWO	PBROA_S3	PBROA_V2	PBROA_U3	PBROA_Z3
Breast Cancer	0.1094	0.1030	0.1020	0.1039	0.1079	0.1010
Breast Cancer Wisconsin	0.0042	0.0046	0.0037	0.0040	0.0041	0.0036
Glass	0.1273	0.1258	0.1279	0.1214	0.1283	0.1208
Iris	0.0025	0.0029	0.0025	0.0025	0.0025	0.0025
South German Credit	0.4647	0.4421	0.4544	0.4569	0.4580	0.4594
Wine	0.0020	0.0034	0.0018	0.0019	0.0017	0.0018
Flags	0.2239	0.1877	0.2068	0.2197	0.2201	0.2120
Dermatology	0.0035	0.0046	0.0031	0.0032	0.0032	0.0035
Credit Approval	0.0817	0.0817	0.0704	0.0784	0.0780	0.0748
Chess(kr-vs-kp)	0.0287	0.0323	0.0225	0.0314	0.0382	0.0257
Hepatitis	0.0271	0.0226	0.0098	0.0179	0.0176	0.0195
Image Segmentation	0.0068	0.0046	0.0067	0.0098	0.0066	0.0087
Statlog (Australian Credit Approval)	0.0668	0.0764	0.0600	0.0660	0.0663	0.0649
Wall-Following Robot Navigation Data	0.0447	0.0500	0.0359	0.0395	0.0406	0.0408

are the fitness values, where the best data are indicated in bold for ease of reading. In addition, we also use Figure 5 to show the experimental results, which makes the comparison of the experimental results clearer. From the data in the table and figure, we can see that BROA_Z3 achieves the best results for the datasets Breast Cancer, Breast Cancer Wisconsin, and Glass. For the dataset Iris, several algorithms achieve better results due to its low dimensionality. For BGWO, it performs well in the datasets South German Credit, Flags, and Image Segmentation. BROA_S3 has good performance in the datasets Dermatology, Credit Approval, Chess(kr-vs-kp), Hepatitis, Statlog (Australian Credit Approval), and Wall-Following Robot Navigation datasets. These results show the superiority of the parallel binary ROA algorithm, which can outperform BGWO and BPSO algorithms in most cases, and BROA_S3 has higher accuracy among several improved algorithms. Compared with the traditional algorithm, the new algorithm proposed in this paper has a better convergence performance, it does not easily fall into local optimum, and tends to find better solutions. The above results also confirm the superiority of the improved parallel optimization algorithm in practical applications, where the parallel strategy can explore more cases and obtain better results.

The data in

Table 17. The number of selected features.

Dataset	BPSO	BGWO	PBROA_S3	PBROA_V2	PBROA_U3	PBROA_Z3
Breast Cancer	3.1	3.6	3.5	3.65	3.3	3.4
Breast Cancer Wisconsin	3.75	4.15	3.3	3.6	3.65	3.25
Glass	5.35	5.05	4.8	6.4	6.2	4.75
Iris	1	1.15	1	1	1	1
South German Credit	9.65	9	8.45	8.4	9.2	9
Wine	2.65	4.45	2.4	2.45	2.25	2.3
Flags	10.6	11.25	10	9.7	10.75	9.9
Dermatology	11.45	15.15	10.15	10.45	10.45	11.4
Credit Approval	5.2	6.35	5.5	4.55	5.15	5.15
Chess (kr-vs-kp)	19.5	19.95	19.45	22.6	29.15	18.95
Hepatitis	6.05	6.05	6.55	6.65	6.1	6.7
Image Segmentation	8.5	8.65	8.2	7.45	8	7.6
Statlog (Australian Credit Approval)	4.2	4.55	4.15	3.55	3.9	4.45
Wall-Following Robot Navigation	4.85	6.65	4.6	4.7	4.45	4.6

are the number of selected features, where the best data are indicated in bold for the ease of reading. To make the experimental effect more understandable, we also show the experimental results with Figure 6. From the data in the table and figure, we can see that, for the dataset Iris, several algorithms give the best results. BPSO gives the best results for the dataset Breast Cancer, Hepatitis. BROA_S3 gives the best results for the dataset Dermatology. BROA_V2 selected fewer features on the datasets South German Credit, Flags, Credit Approval, Image Segmentation, and Statlog (Australian Credit Approval). BROA_U3 has better results on the datasets Wine and Wall-Following Robot Navigation. BROA_Z3 selected fewer features on the dataset Breast Cancer Wisconsin, Glass, Chess (kr-vs-kp). From the analysis of the above results, it is known that BROA_S3 is more accurate but biased in the selection of more features, and BROA_V2 will be more biased to select fewer features to achieve classification.

6. Conclusions

The parallel binary Rafflesia Optimization Algorithm is able to solve discrete problems, and in this paper, it is applied to the problem of feature selection with good results. In this paper, the transfer function is crucial in the binary algorithm. We analyze the search space of the original algorithm and propose new transfer functions based on it, thus binarizing the original algorithm. To further improve the performance of the algorithm, a parallel strategy is used to optimize the algorithm. The remarkable performance of the parallel binary Rafflesia Optimization Algorithm is demonstrated by running on the benchmark functions. Finally, we successfully applied the algorithm to the problem of feature selection and achieved the classification of features by KNN and cross-validation methods. Parallel binary ROA has been shown to possess acceptable classification accuracy in experiments. In this paper, we only used KNN to implement feature selection, which can be combined with a neural network for classification in the future and other ways to reduce the classification error. The binary conversion of the algorithm also increases the computation time, and future work will also include reducing the running time of the algorithm.