Feature selection using a multi-strategy improved parrot optimization algorithm in software defect prediction

Related work on software defect prediction

Supervised and unsupervised machine learning techniques have been employed for software defect prediction, including: SVM (Singh, Kaur & Malhotra, 2009), decision trees (DT) (Khoshgoftaar & Seliya, 2003), Bayesian networks (BN) (Yuan et al., 2000), naive Bayes (NB) (Menzies, Greenwald & Frank, 2006), KNN (Kumar Dwivedi & Singh, 2016), multilayer perceptron (MLP) (Carrozza et al., 2013), artificial neural networks (ANN) (Goyal & Bisi, 2015), logistic regression (LR) (Caglayan et al., 2015), multinomial logistic regression (MLR) (Carrozza et al., 2013), random forests (RF) (Bowes, Hall & Petrić, 2018), and unconstrained MLP (Thaher & Khamayseh, 2021).

Khoshgoftaar, Bullard & Gao (2009) are among the pioneers in developing feature selection (FS) techniques for imbalanced data sets. In their research, they applied various wrapper-based feature-ranking and feature subset selection techniques to generate candidate feature sets and combined them with different classification algorithms for prediction. The results showed that applying superior feature selection methods can enhance the performance of software defect prediction (SDP) models and make them more applicable to real-world scenarios. Arar & Ayan (2015) proposed a cost-sensitive ANN based ABC algorithm for software defect prediction (SDP). In this study, feature selection was achieved using correlation-based FS techniques in the Waikato Environment for Knowledge Analysis (WEKA) tool. The results indicated that reducing the number of features did not significantly impact prediction accuracy. Jayanthi & Florence (2019) employed a feature reduction method called principal component analysis (PCA). They addressed errors through maximum likelihood estimation, reducing the potential for errors during the construction of PCA features. The selected features were then fed into a neural network-based classification technique for error prediction. Manjula & Florence (2019) proposed a hybrid method for error prediction that combined the genetic algorithm (GA) for feature optimization and deep neural networks (DNN) for classification. Experiments on the Promise dataset for software defect prediction demonstrated improved classification accuracy with the proposed hybrid method. Xu et al. (2019) introduced a new SDP framework called KPWE, which combines kernel principal component analysis (KPCA) and weighted extreme learning machine (WELM), focusing on feature extraction. Cai et al. (2020) proposed a hybrid multi-objective cuckoo search subsampling software defect prediction model based on SVM (HMOCS-US-SVM). This model uses three undersampling methods to determine the decision domain range and employs a hybrid multi-objective cuckoo search with dynamic local search (HMOCS) to optimize SVM parameters, addressing the class imbalance in defect datasets and parameter selection issues in support vector machines.

The aforementioned methods indicate that the application of preprocessing techniques, such as feature selection, can significantly impact the performance of software defect prediction models. In conjunction with the No Free Lunch (NFL) program theorem in optimization (Wolpert & Macready, 1997), which states that no single classifier is universally optimal for all possible classification problems (Bhaskar, Hoyle & Singh, 2006), this has motivated us to propose an advanced method for predicting software defects. This method treats the stacking ensemble learning classifier as a machine learning technique while employing advanced feature selection methods as wrappers. Specifically, we utilize the novel multi-strategy enhanced parrot optimizer as a search strategy to select the optimal features.

Parrot optimization algorithm

The Parrot Optimization algorithm (POA) (Lian et al., 2024) is a swarm intelligence optimization algorithm inspired by the behaviors of foraging, roosting, communication and fear of strangers observed in domesticated parrots. In each iteration, each individual in POA randomly exhibits one of these four behaviors, accurately simulating the behavioral randomness observed in domesticated parrots. This significantly increases the diversity of the population. Furthermore, by eliminating the traditional two-phase exploration-exploitation structure, POA effectively reduces the risk of getting trapped in local optima.

In the foraging behavior, parrots estimate the approximate location of food by observing its position or considering the location of their owner, then fly towards their respective positions. The positional movement follows the following equations:

(1) $$X_i^{t+1}=(X_i^t-X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}})\times \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{y}(\dim )+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times {\left(1-\frac{t}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}\right)}^{\frac{2t}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}}\times X_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^t$$where $X_i^{t+1}$ represents the position to be updated, $X_i^t$ denotes the current position, $X_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^t$ signifies the mean value within the current population, $X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ represents the best position found from initialization to the current search, $\mathrm{L}\mathrm{e}\mathrm{v}\mathrm{y}(\dim )$ indicates the Levy distribution, $t$ stands for the current iteration number, and $\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}$ denotes the maximum number of iterations.

The perching behavior is used to describe when a parrot suddenly flies to any part of its owner’s body and remains stationary there for some time. The perching movement follows the following equation:

(2) $$X_i^{t+1}=X_i^t+X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\times \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{y}(\mathrm{d}\mathrm{i}\mathrm{m})+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \mathrm{o}\mathrm{n}\mathrm{e}\mathrm{s}(1,\mathrm{d}\mathrm{i}\mathrm{m})$$where ones(1, dim) represents a vector of all ones with dimension dim.

In the communication behavior, the parrot either flies towards the flock or does not fly towards the flock. In the Parrot Optimization Algorithm, the probabilities of these two behaviors are equal. The communication behavior follows the following equation:

(3) $$X_i^{t+1}=\{\begin{array}{cc}0.2\times \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \left(1-\frac{t}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}\right)\times (X_i^t-X_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^t),\hfill & \mathrm{i}\mathrm{f}P\le 0.5\hfill \\ 0.2\times \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \exp \left(-\frac{t}{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}\right),\hfill & \mathrm{i}\mathrm{f}P>0.5\hfill \end{array}$$where $P\le 0.5$ represents flying towards the flock, $P>0.5$ represents flying away from the flock.

Parrots exhibit an inherent fear of strangers, maintaining a distance from unfamiliar individuals. They engage in behaviors aimed at seeking a safe environment together with their owners. The fear behavior towards strangers can be represented by the following equation:

(4) $$\begin{array}{rl}{X}_i^{t+1}& =X_i^t+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \cos \left(\frac{1}{2}\pi \times \frac{t}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}\right)\times (X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-X_i^t)\\ & -\cos (\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(0,1)\times \pi )\times {\left(\frac{t}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}\right)}^{\frac{2}{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}}}\times (X_i^t-X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}).\end{array}$$

Stacked ensemble learning

Stacked ensemble learning (Agarwal & Chowdary, 2020) integrates the predictions of multiple models to achieve the final prediction. Its core idea is to use the outputs of various models as new features input into a meta-model, which is then used for final prediction. Stacked ensembles typically consist of two main stages: first, training multiple different machine learning models on the training data, which can be classifiers, regressors, or other types of models; second, utilizing the predictions generated by the base models on the validation set as new features and inputting them into a meta-model. The meta-model is then trained to obtain the final prediction. The framework diagram of stacked ensemble learning is illustrated in Fig. 1.

Dataset acquisition and preprocessing

Traditional static code quality metrics have been proven to be related to software defects (Koru & Liu, 2007; El Emam et al., 2001; Nagappan, Ball & Zeller, 2006; Fenton & Ohlsson, 2000). In this study, 21 defect datasets were selected from the Promise repository, each containing 20 software measurement metrics. The measurement metrics are shown in Table 1, and the number of files and defect file probability in each dataset are shown in Table 2. The Promise dataset is widely used in software defect prediction research because of its origin in real-world software projects, providing significant practical relevance and broad applicability. The datasets not only reflect project characteristics under various development environments but also consider multiple software metrics, including code complexity, comment density, and functional defects, all of which are closely related to the occurrence of software defects. Due to the different dimensions of each software measurement metric, to avoid bias towards certain key features, standardization processing is performed on the metadata of each measurement. The standardization formula is as follows:

(5) $$\widetilde{x_i}=\frac{x_i-\overline{x}}{\delta }$$where $\overline{x}$ is the mean of the dataset, $\delta$ is the standard deviation of the dataset.

By analyzing the defect dataset in Table 2, it was found that the number of defective files in most datasets is relatively small, resulting in a highly imbalanced state of data categories. To balance the dataset, this study employs the SMOTE oversampling technique.

Multi-strategy improved parrot optimizer

In response to the dependency on initial population, premature convergence, and poor global search capability inherent in the Parrot Optimization Algorithm, an improved Parrot Optimization algorithm is proposed. This algorithm first utilizes the Tent chaotic mapping to generate initial positions of parrot populations, laying the foundation for population diversity during the global search process. Secondly, an adaptive t-distribution mutation method is employed to perturb individual positions, enhancing the algorithm’s ability to escape local optima. Finally, a winner-take-all biological competition elimination strategy is introduced, where the parrot populations are mutated by the elite reverse learning strategy, and the original populations are eliminated based on fitness values, thereby strengthening the algorithm’s development capability.

Tent chaotic mapping

During the initialization stage, the POA uses a random number generation method to create the initial population. However, this method leads to uneven distribution across the solution space, which hinders population diversity. To address this, the algorithm employs chaotic mapping, known for its randomness and ability to cover the entire search space globally. This can help speed up convergence and prevent the algorithm from getting stuck in local optima (Arora, Sharma & Anand, 2020). In this study, the Parrot population is initialized using the Tent chaotic mapping method, expressed mathematically as follows:

(6) $$x_{n+1}=\{\begin{array}{cc}\frac{x_n}{u},\hfill & \mathrm{i}\mathrm{f}0\le x_n\le u\hfill \\ \frac{1-x_n}{1-u},\hfill & \mathrm{i}\mathrm{f}\mathrm{u}\le x_n\le 1\hfill \end{array}$$where $x_n$ represents the current state of the chaotic system, $x_{n+1}$ represents the next state calculated based on the current state $x_n$, $u$ is the chaotic state parameter, and this article adopts a value of 0.5 for $u$. The parameter value 0.5 typically lies within the stable region or at the transition point of chaotic systems, ensuring that the generated sequence exhibits chaotic characteristics without being entirely random. In numerical simulations, this value neither leads to complete system instability nor does it restrict the diversity of the generated sequence, providing an adequate balance between chaos and controllability. The population distribution and histogram generated by the Tent map, as derived from Eq. (6), are shown in Figs. 3 and 4, respectively. As shown in Figs. 3 and 4, the initial population distribution generated by the Tent map exhibits good randomness and is relatively uniform.

T-distribution mutation

After completing the update step of the parrot individuals, to further enhance the optimization capability of the POA, this article introduces a mutation strategy based on the t-distribution to generate new individual positions. The t-distribution mutation operator combines the advantages of the Gaussian operator and the Cauchy operator, which helps the algorithm search for better solutions in a wider search space (Li et al., 2020; Mostafa et al., 2020). The individual position update equation for t-distribution mutation is as follows:

(7) $$x_{\mathrm{n}\mathrm{e}\mathrm{w}}=x_i+t(\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r})\times x_i$$where $x_i$ represents the position of the $i$-th individual, $x_{\mathrm{n}\mathrm{e}\mathrm{w}}$ represents the new individual generated based on the $i$-th individual’s position, and $t(\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r})$ follows a t-distribution with degrees of freedom equal to the number of iterations.

When the algorithm starts, the number of iterations is small, the mutation effect of the t-distribution is similar to that of the Cauchy distribution. Figure 5 illustrates the scatter plot of the Cauchy distribution within the range of [−100, 100]. As observed, the Cauchy distribution tends to generate random numbers that are farther from the origin, causing larger disturbances in the mutation term. This results in a stronger global search capability, which helps the algorithm escape from local optima. During the mid-stage of the algorithm, the t-distribution mutation lies between the Cauchy and Gaussian distributions. In the later stages, with a larger number of iterations, the mutation behavior of the t-distribution becomes equivalent to that of the Gaussian distribution. Figure 6 shows the scatter plot of the Gaussian distribution. The random numbers generated by the Gaussian distribution are closer to the origin, resulting in smaller perturbations from the mutation term. This leads to a more detailed local search, improving the algorithm’s convergence accuracy and enhancing its local exploitation ability.

After generating new individual positions, this article follows a greedy strategy to comprehensively evaluate the newly generated individuals with the existing individuals, selecting the optimal individual as the solution for the current step. This strategy ensures that the individuals are updated in the direction of better solutions, thereby improving the optimization capability of the algorithm.

Multi-strategy enhanced parrot optimization algorithm procedure

The specific procedure of the multi-strategy enhanced Parrot Optimization algorithm is as follows:

• Step 1: Initialize the parrot population using Tent chaotic mapping as described in Eq. (6) and then compute the fitness.

• Step 2: Employ the reverse learning strategy to generate reverse solutions according to Eq. (8). Compare the original population with the reverse population, and select populations with higher fitness to join the next generation of parrot populations.

• Step 3: For each individual, randomly select an updating strategy. If it is foraging behavior, update the individual’s position according to Eq. (1). If it is resting behavior, update the individual’s position according to Eq. (2). If it is social behavior, update the individual’s position according to Eq. (3). If it is fear behavior, update the individual’s position according to Eq. (4).

• Step 4: Optimize the parrot’s positions through t-distribution mutation according to Eq. (7). Compare the fitness values before and after optimization, and select the optimal individuals from the original and new positions using a greedy strategy.

• Step 5: Determine whether the maximum number of iterations has been reached. If the maximum number of iterations has been reached, output the optimal parrot positions and fitness values. Otherwise, return to Step 2 and continue iterating.

The flowchart of the proposed MEPO is illustrated in Fig. 7. The pseudo-code of MEPO is shown in Table 3.

Table 3:

Pseudo-code of MEPO.

Algorithm 1 MEPO algorithm
1: Initialize the MEPO parameters
2: Initialization of the Parrot Population Using the Tent Chaotic Map
3: for $i=1$ to Max_iter do
4:    Generate the elite population based on Eq. (8) and the greedy algorithm.
5:    Calculate the fitness function
6:    Find the best position
7:    for $j=1$ to N do
8:       $St=\mathrm{n}\mathrm{p}.\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}.\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{t}(1,5)$
9:      if $St=1$ then
10:        Behavior 1: The foraging behavior
11:        Update position by Eq. (1)
12:      else if $St=2$ then
13:        Behavior 2: The perching behavior
14:        Update position by Eq. (2)
15:      else if $St=3$ then
16:        Behavior 3: The communicating behavior
17:        Update position by Eq. (3)
18:      else if $St=4$ then
19:        Behavior 4: The fear of strangers’ behavior
20:        Update position by Eq. (4)
21:      end if
22:    end for
23:    Update position based on Eq. (6) and the greedy algorithm.
24: end for
25: Return the best position

DOI: 10.7717/peerj-cs.2815/table-3

Feature selection of software metrics using binary multi-strategy enhanced parrot optimization algorithm

The process of selecting the most relevant metrics is a complex task that involves optimizing multiple objectives. In this particular study, we are focusing on classifying software defects, and we are using the AUC metric to evaluate the performance of our classification models. The AUC (area under the ROC curve) is a crucial measure, and a higher value indicates better classification performance. For this study, we have chosen AUC as the fitness function. When developing solutions, we need to consider how these solutions are represented. In our case, solutions are represented as binary vectors with N elements, where N is the total number of metrics in the original dataset. Each element in the vector is a binary value: 0 means the corresponding metric is not selected, and 1 means it is selected. Figure 8 illustrates this feature selection scheme. The Parrot Optimization (PO) method has shown significant advantages in solving continuous optimization problems. In this study, we have integrated the MEPO algorithm, which uses binary encoding, to develop the BMEPO algorithm. While MEPO is designed for continuous optimization problems, BMEPO is specifically developed to handle binary optimization problems. This approach effectively resolves the optimization problem of metric feature selection.

Heterogeneous data stack ensemble learning framework

Traditional stacking methods often assume that all base models should be trained on the same dataset, ignoring the differences in data feature selection among different models. This study proposes a heterogeneous data stack ensemble framework for software defect prediction. Firstly, for the defect dataset, optimal features adapted to each model are extracted using BMEPO. Secondly, primary learners are trained on feature-filtered data based on different feature selectors, and the outputs serve as inputs to a secondary learner, the logistic regression model. After training the logistic regression model, final predictions are made. The overall framework is illustrated in Fig. 9.

Result analysis of MEPO and experimental comparison with other algorithms

Algorithm comparison on test functions

To demonstrate the superiority of the multi-strategy enhanced parrot optimization algorithm, this study selected several test functions with different characteristics for optimization and compared them with the Parrot Optimization algorithm. To ensure the fairness of the experiments, the number of parrots in all experiments was set to 30, and the maximum number of iterations was set to 500. To avoid result bias caused by randomness, the algorithm was independently run 50 times for each function. Table 4 lists the basic information of the test functions, and Table 5 presents the experimental results of MEPO and PO after 50 independent runs on multiple standard test functions.

Table 4:

Test functions and specific information.

Function name	Function	Dimension	Range	Best solution
Rosenbrock’s function	$f_1={\sum }_{i=1}^{n-1}[100(x_{i+1}-x_i^2{)}^2+(x_i-1{)}^2]$	30	$[-30,30]$	0
Penalized function	$\begin{array}{rl}& f_2=\frac{\pi }{n}\{10\sin (\pi y_1)+\sum _{i=1}^{n-1}[(y_i-1{)}^2(1+10{\sin }^2(\pi y_{i+1}))+(y_n-1{)}^2]\}\\ & +u(x_i,10,100,4)\\ & y_i=1+\frac{x_i+1}{4}\end{array}$ $u(x_i,a,k,m)=\{\begin{array}{cc}k(x_i-a{)}^m,\hfill & \text{if }{x}_i>a\hfill \\ 0,\hfill & \text{if }-a<x_i<a\hfill \\ k(-x_i-a{)}^m,& \text{if }{x}_i<-a\hfill \end{array}$	30	$[-50,50]$	0
Penalized function	$\begin{array}{rl}& f_3=\frac{\pi }{n}\left\{10\sin (\pi y_1)+\sum _{i=1}^{n-1}\left[{(y_i-1)}^2\left(1+10{\sin }^2(\pi y_{i+1})\right)+{(y_n-1)}^2\right]\right\}\\ & +u(x_i,5,100,4)\end{array}$	30	$[-50,50]$	0
Schwefel’Problem 2.26	$f_4=\sum _{i=1}^n-x_i\sin (\sqrt{|x_i|})$	30	$[-500,500]$	$418.98\times n$
Shekel’s foxholes function	$f_5={\left(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2{\left(x_i-a_{ij}\right)}^6}\right)}^{-1}$	2	$[-65,65]$	1
Branin function	$f_6={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)\cos x_1+10$	2	$[-5,15]$	0.3978
Hartmann	$f_7={\sum }_{i=1}^4{c}_i\exp \left(-{\sum }_{j=1}^3{a}_{ij}{(x_j-p_{ij})}^2\right)$	3	$[0,1]$	−3.8627
Hartmann	$f_8={\sum }_{i=1}^4{c}_i\exp \left(-{\sum }_{j=1}^3{a}_{ij}{(x_j-p_{ij})}^2\right)$	3	$[0,1]$	−3.3223

DOI: 10.7717/peerj-cs.2815/table-4

Table 5:

Results of test functions (different algorithms).

Statistic	Algorithms	${\mathit{f}}_{\mathbf{1}}$	${\mathit{f}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{3}}$	${\mathit{f}}_{\mathbf{4}}$	${\mathit{f}}_{\mathbf{5}}$	${\mathit{f}}_{\mathbf{6}}$	${\mathit{f}}_{\mathbf{7}}$	${\mathit{f}}_{\mathbf{8}}$
Best	MEPO	3.4E−15	4.31E−20	1.7E−26	12,569.486	0.998	0.3978	−3.863	−3.322
	PO	4.19E−13	2.79E−9	6.25E−14	−11,483.239	0.998	0.398	−3.861	−3.169
Mean	MEPO	0.006	0.016	2.75E−4	−9,772.386	4.629	0.398	−3.862	−3.276
	PO	0.015	0.023	3.93E−4	−7,453.684	7.974	0.404	−3.833	−2.814
Worst	MEPO	0.149	0.093	0.012	−6,764.510	12.670	0.402	−3.861	−3.086
	PO	0.162	0.105	0.016	−3,453.645	12.670	0.5111	−3.739	−2.281
STD	MEPO	0.022	0.023	0.002	1,634.980	4.838	6.74E−4	4.09E-4	0.068
	PO	0.031	0.093	0.012	1,971.884	5.272	0.017	0.027	0.229
Time (s)	MEPO	1.684	3.656	3.124	1.565	8.066	1.477	3.306	3.189
	PO	1.204	2.401	2.023	1.112	7.411	1.025	1.905	1.982

DOI: 10.7717/peerj-cs.2815/table-5

Upon analyzing Table 5, it is evident that the MEPO and PO algorithms both reach the optimal value for the $f_5$ and $f_6$ test functions. However, for the other test functions, the MEPO algorithm consistently obtains lower optimal values than the PO algorithm. Additionally, when comparing the mean values, worst values, and variances, it is clear that the MEPO algorithm consistently outperforms the PO algorithm. This indicates that the MEPO algorithm possesses superior optimization capabilities compared to the PO algorithm.

To further verify the convergence of the MEPO algorithm, iterative convergence curves for test functions were plotted, with the horizontal axis representing the number of iterations and the vertical axis representing the average value over 50 independent runs. As shown in the figure below, on the f₆ test function, the MEPO and PO algorithms converge almost simultaneously. In the f₂, f₅ and f₈ test functions, the MEPO algorithm converges later than the PO algorithm, but the average value at convergence is lower than that of the PO algorithm. In the f₁, f₃, f₄ and f₇ test functions, the MEPO algorithm converges earlier than the PO algorithm, and the average value at convergence is lower than that of the PO algorithm. The iterative convergence curves for the test functions are shown in Fig. 10.

In conclusion, the MEPO algorithm, due to the incorporation of mechanisms such as chaotic mapping, reverse solution generation, and t-distribution mutation, incurs a slight disadvantage in terms of computation time. However, when applied to optimization problems, it demonstrates better performance and convergence compared to the PO algorithm.

Result analysis of BMEPO and experimental comparison with other algorithms

In the software defect datasets, we use the Whale Optimization Algorithm (Nasiri & Khiyabani, 2018) Feature Selector (BWO), Parrot Optimization Algorithm Feature Selector (BPO), and Crowned Porcupine Optimization Algorithm (Abdel-Basset, Mohamed & Abouhawwash, 2024) Feature Selector (BCPO) as baseline algorithms to compare with the Binary Multi-strategy Enhanced Parrot Optimization Algorithm Feature Selector (BMEPO). The population size is set to 10, the maximum number of iterations to 50, and the dimensionality of the population of individuals to 20. These four feature selection algorithms are applied to 20 defect datasets from the Promise repository. The classification models selected are decision tree, SVM, and KNN, with default parameters for each machine learning model. The Area Under the Curve (AUC) metric is used to evaluate the performance of the feature selection algorithms. Each algorithm is executed 30 times to ensure fairness, and we report the average and variance of the AUC values over these 30 runs. The results of the comparison experiments are presented in Tables 7–9, and Fig. 11. To further evaluate the computational efficiency of the algorithms, we conduct a comparative experiment on the average runtime of the BMEPO, BPO, BWO, and BCPO algorithms across 16 datasets using DT, SVM, and KNN models. The experimental results are shown in Table 10.

As shown in Fig. 11, the feature selection performance of the BMEPO algorithm is superior to that of the BPO, BWO, and BCPO algorithms. By averaging the AUC metrics of all data in Table 7, it can be concluded that the average AUC obtained by the BMEPO algorithm for feature selection based on the Decision Tree model is 2.3 percent higher than that of the BPO algorithm, 6.8 percent higher than that of the BWO algorithm, and 1.8 percent higher than that of the BCPO algorithm. By averaging the AUC metrics of all data in Table 8, it can be concluded that the average AUC obtained by the BMEPO algorithm for feature selection based on the SVM model is 1.7 percentage points higher than that of the BPO algorithm, 5.2 percentage points higher than that of the BWO algorithm, and 1.4 percentage points higher than that of the BCPO algorithm. By averaging the AUC metrics of all data in Table 9, it can be concluded that the average AUC obtained by the BMEPO algorithm for feature selection based on the KNN model is 2 percentage points higher than that of the BPO algorithm, 5.5 percentage points higher than that of the BWO algorithm, and 1.4 percentage points higher than that of the BCPO algorithm.

As shown in Table 10, the execution time of the BMEPO algorithm is higher than that of the BPO and BWO algorithms but lower than that of the BCPO algorithm across different classification models. This difference in execution time is primarily because the BMEPO algorithm incorporates a hybrid strategy to enhance its global search capability, which increases the computational cost per iteration.

In summary, although the execution time of the BMEPO algorithm is higher compared to the BPO and BWO algorithms, its significant improvement in feature selection performance demonstrates its superiority in balancing search efficiency and feature selection quality. Moreover, unlike other algorithms that have limitations in optimization performance on specific classification models, the BMEPO algorithm outperforms the BPO, BWO, and BCPO algorithms across all classification models (decision tree, SVM, and KNN). This further validates the excellent generalization capability and stability of the BMEPO algorithm, indicating its broad applicability in feature selection tasks.

Result analysis of HEDSE and experimental comparison with other algorithms

In this section, we will discuss the impact of heterogeneous data stacking ensembles on the performance of prediction models.

First, we use the BMEPO algorithm to perform feature selection on the defect test datasets for the DT model (BMEPO-FS-DT), SVM model (BMEPO-FS-SVM), and KNN model (BMEPO-FS-KNN), respectively. We select the feature indices that yield the best values of fitness function. The selection results are shown in Table 11.

Second, we constructed a defect prediction model based on stacked ensemble learning, where DT, SVM, KNN, and linear regression models were employed as base learners, with a linear regression model serving as the meta-learner. The defect data underwent feature selection using the DT model feature filter, SVM model feature filter, KNN model feature filter, and without feature filter, respectively, and were then input into the stacked ensemble model for classification. For the sake of consistency in expression, we collectively refer to the aforementioned methods as the homogeneous data-stacked ensemble learning model (HODSE).

Subsequently, we utilized feature filters based on DT, SVM, and KNN models to perform feature selection on defect data, and the selected features were input into the corresponding base learners of the stacked ensemble model for classification. The results were compared with those of the homogeneous data stacked ensemble model to validate the effectiveness of the heterogeneous data stacked ensemble model. Furthermore, to comprehensively evaluate the impact of different feature selection combinations on classification performance, we designed and conducted ablation experiments to systematically analyze the performance of the DT+SVM, DT+KNN, and SVM+KNN feature filters. The experimental results are shown in Fig. 12 and Table 12.

From Table 12 and Fig. 12, it can be observed that the proposed method outperforms all the comparison methods in terms of the average AUC across 16 projects, demonstrating its significant performance advantage. Compared to the homogeneous stacked ensemble models, the proposed method achieves an average AUC value higher by 22.79%, 5.35%, 16.8%, and 15.6% than the models with no feature selection, DT-based feature selection, SVM-based feature selection, and KNN-based feature selection, respectively. In the ablation study, the proposed method improves the AUC mean by 15.3%, 5.6%, and 1.9% compared to heterogeneous stacked ensemble models with DT+SVM feature selection, DT+KNN feature selection, and SVM+KNN feature selection. Furthermore, according to the Win/Draw/Lost statistics, our method achieved seven wins, one draw, and eight losses out of the 16 projects, with a significantly higher win rate than the other methods. Meanwhile, the Cliff’s Delta values indicate a substantial effect advantage of our method over the comparison models, especially when compared to the model without feature selection and the SVM-based feature selection model, with Cliff’s Delta values reaching 0.9531 and 0.8593, respectively. Cliff’s Delta is a non-parametric effect size measure used to quantify the amount of difference between two methods. The range of values is [−1, 1], with values closer to 1 indicating a large effect size. Typically, Cliff’s Delta is categorized into four effect levels, as shown in Table 13.

Finally, to validate the advancement of the proposed framework, we perform a comparative analysis with currently advanced machine learning models (Random Forest (Breiman, 2001), LightGBM (Ke et al., 2017), Natural Gradient Boosting (NGBoost) (Duan et al., 2020)) and two state-of-the-art defect prediction methods. The state-of-the-art methods include: Iterative Ensemble Genetic Classifier Algorithm (IEGCA), which combines genetic algorithms for feature selection and uses multiple binary classifiers along with an ensemble voting strategy for defect prediction (Ali et al., 2024); Nested-Stacking method, which improves the defect prediction model’s fitting ability and generalization ability by combining various boosting algorithms and custom deep learning algorithms based on the stacking criterion (Chen, Wang & Song, 2022). The experimental results are shown in Fig. 13 and Table 14.

From Fig. 13, it is evident that the proposed method achieves overall superior performance compared to RF, LightGBM, NGBoost, and two state-of-the-art algorithms, IECGA and Nested-Stacking. As shown in Table 14, the proposed method achieves an average AUC across 16 projects that is 18.98% higher than RF, 19.89% higher than LightGBM, 16.16% higher than NGBoost, 1.75% higher than IECGA, and 5.62% higher than Nested-Stacking. Furthermore, the Win/Draw/Loss statistics indicate that the proposed method achieved 11 wins, 0 draws, and five losses across the 16 projects, demonstrating a significantly higher win rate compared to other methods. Cliff’s Delta analysis further reveals that the proposed method provides significant improvements over RF, LightGBM, NGBoost, and Nested-Stacking, and a slight improvement over IECGA.