A feature selection algorithm based on multi-modal multi-objective

Abstract

The application provides a feature selection algorithm based on multi-modal multi-target, which comprises the following steps: initializing population individuals of training data sets, and performing non-dominated sorting on the generated population individuals; calculating fitness value and SDE index value according to the prediction accuracy of KNN and the number of selected features; calculating FER value according to the SDE index value; selecting parent individuals by adopting a roulette strategy according to the FER value, generating new individuals through mutation and crossover operation, and performing non-dominated sorting and mutation operation on the new individuals again; forming a new population, calculating fitness value, putting the new population and the initialized population into an archive set, and performing non-dominated sorting and calculating corresponding SDE index value; and reserving the first N individuals in the archive set as criteria of the least Pareto front number and the lowest SDE index value. The application searches for multiple optimal or suboptimal feature subsets, and the solution set can be used for designing a learning device with less selected features and high classification accuracy for selection by a decision maker.

Description

Technical Field

[0001] This invention relates to a feature selection algorithm based on multimodal and multi-objective modes. Background Technology

[0002] Many evolutionary algorithms have been used for feature selection, such as PSO, GA, DE, ACO, and GP. These early methods modeled the feature selection problem as a single-objective optimization problem and solved it using a single-objective evolutionary algorithm. However, feature selection requires optimization of more than one objective, such as classification accuracy, the number of selected features, computational complexity, and computation time. Subsequently, multi-objective optimization algorithms were also used to solve the feature selection problem. The purpose of feature selection is to select relevant features from the initial dataset to distinguish different categories, and these feature subsets may not be unique. The literature (Kamyab S, Eftekhari M. Feature selection using multimodal optimization techniques[J]. Neurocomputing,2016,171:586-597) uses a multimodal evolutionary algorithm to solve the feature selection problem, hoping to obtain a feature subset with good diversity. This study weights and sums the number of selected features and the classification error rate, thus transforming it into a multimodal single-objective optimization problem for solution. Although the final feature subset can achieve high classification accuracy, the number of selected features may be large.

[0003] Feature selection has two main optimization objectives: classification accuracy and the number of selected features. Ideally, fewer features should be selected, while higher classification accuracy is preferable. The ultimate goal is to find the optimal compromise between these two objectives. However, this optimal compromise may contain different feature combinations with the same number of features that achieve the same classification accuracy. This situation constitutes the multimodal, multi-objective optimization problem in feature selection. Solving this multimodal, multi-objective optimization problem in feature selection is a pressing issue that requires our consideration.

[0004] Multimodal multi-objective optimization problems have multiple Pareto sets (PS), and it is possible for both local and global Pareto sets to exist simultaneously. Therefore, to solve multimodal multi-objective optimization problems, the solution algorithm needs to be able to escape local optima and provide multiple satisfactory solutions at once. The FERDE algorithm, a differential evolution algorithm based on fitness Euclidean distance ratio, can adaptively generate several habitats based on its own information interaction and learning mechanisms, and find multiple global optimal or suboptimal solutions without pre-setting the radius of the habitat. This type of algorithm drives individuals in the population to learn from individuals that are better and closer to them, and it does not require pre-defining parameters such as the number and radius of habitats, thus possessing the advantages of high stability and simple structure. Summary of the Invention

[0005] This invention proposes a feature selection based on fitness Euclidean-distance Ratio Differential Evolution Multimodal Multiobjective Optimization (FS_FERDE_MMO) algorithm. This algorithm uses the prediction accuracy of KNN and the number of selected features as fitness values, and employs the FERDE_MMO algorithm to search for multiple optimal or suboptimal feature subsets in the search space. These solution sets can be used to design learners with a small number of selected features and high classification accuracy for decision-makers to choose from.

[0006] The technical solution of this invention is implemented as follows: a feature selection algorithm based on multimodal and multi-objective modes, comprising the following steps:

[0007] (1) The initial dataset is divided into a training dataset and a test dataset using the ten-fold cross-validation method. The training dataset is used to calculate the fitness of each individual, and the test dataset is used to validate the classification results.

[0008] (2) Initialize the population individuals in the training dataset, randomly encode each individual, "1" represents that the corresponding feature is selected, and "0" represents that it is not selected, and perform non-dominated sorting on the generated population individuals;

[0009] (3) Calculate the fitness value of each individual based on the prediction accuracy of KNN and the number of selected features, and calculate the SDE (Shift-Based Density Estimation) index value.

[0010] (4) Calculate the fitness Euclidean distance ratio (FER value) of each individual based on the SDE index value of the fitness value; select the parent individual using the roulette wheel strategy based on the FER value; generate new individuals through mutation and crossover operations;

[0011] (5) The new individuals obtained in step (4) are sorted again using non-dominated sorting. The individuals in the top 50% of the population are selected for mutation using the DE / rand / 2 strategy, and the individuals that exceed the boundary during the mutation process are processed.

[0012] (6) Merge the individuals obtained in steps (4) and (5) to form a new population and calculate the fitness value;

[0013] (7) Place the new population from step (6) and the initial population from step (2) into the archive set, and perform non-dominated sorting on the individuals in the archive set and calculate the corresponding SDE index values.

[0014] (8) Select the top N individuals in the archive set as the retained individuals based on the criteria of having the lowest Pareto front number and the lowest SDE index value.

[0015] Furthermore, in step (3), the SDE index value is calculated using formula (1):

[0016]

[0017] in, Let P represent the i-th target value, M be the number of targets, and the value of an individual P is defined as the minimum offset density estimate distance between that individual and other individuals in the population.

[0018] Furthermore, in step (5), the expression for the DE / rand / 2 strategy is as follows:

[0019]

[0020] Among them, r1 i r2 i r3 i ,r4 i r5 i F is a distinct individual index number within the range [1, NP], and is different from the index number i. F ∈ [0, 1], where F is the scaling factor, NP is the population size, and G is the number of iterations.

[0021] Furthermore, in step (5), boundary processing is performed on individuals that exceed the boundary during the mutation process, and the calculation formula is shown in (3):

[0022]

[0023] Where F is 0.9.

[0024] Furthermore, in step (4), the formula for calculating the FER value is as follows:

[0025]

[0026] Among them, FER (j,i) This refers to the ratio of the fitness Euclidean distance between individuals i and j within a population; p i and p j These are the historical best positions of individuals i and j, respectively; p w f(p) represents the worst position of an individual in the current population. j ) and f(p w ) are p respectively j and p w Corresponding fitness;

[0027] Using a roulette wheel strategy, individuals with high FER values ​​are selected as parent individuals. Offspring individuals are then generated through mutation and crossover operations. The mutation formula is as follows:

[0028] v p =x r1 +rand(1,D)·(x r2 -x r3 (5)

[0029] Among them, v p It is a mutated individual, x r1 ,x r2 and x r3 These are three different parent individuals generated by the roulette wheel selection method, where D is the length (i.e., data dimension) of the parent individual;

[0030] A crossover operation is performed on each generated mutated individual vector and its corresponding parent individual vector, as shown in formula (6):

[0031]

[0032] Among them, u p,d It is the offspring individual, p is the historical best position, CR is the crossover probability, which is a constant in the range [0,1], and ν is the crossover probability. p,d It is a mutant individual, χ p,d For the target individual, rand d d is a uniformly distributed random number generated in the range [0,1]. rand It is an integer randomly selected from the set {1,2,...D} to ensure that the mutation vector ν p,d At least one dimension of information is preserved.

[0033] Furthermore, in step (2), the maximum value of an individual in each dimension is given before initializing the population. and minimum value The result of initializing the population from the training data is to generate a population of NP and D-dimensional vectors as the target vector, where G is the iteration number, and the vector of the i-th individual is represented as:

[0034]

[0035] When the iteration number G = 0, the i-th individual is generated according to formula (2):

[0036]

[0037] Here, rand[0,1] is a uniformly distributed random number generated in [0,1], and D is the data dimension.

[0038] Furthermore, in step (2), the specific method for randomly encoding each individual is as follows: the position vector value x of the i-th individual id Compare with the set threshold θ, if x id If the value is greater than θ, then feature d is selected; otherwise, feature d is not selected.

[0039] The beneficial effects of this invention are:

[0040] The multimodal, multi-objective integrated feature selection algorithm of this invention introduces non-dominated solution sorting into the FERDE multimodal optimization algorithm and uses the improved FERDE_MMO algorithm as a feature subset search method to find multiple Pareto optimal solution sets.

[0041] The challenge of solving multimodal multi-objective optimization problems lies in finding as many Pareto optimal solutions as possible during the search process, while preserving individuals with small objective space distances but large decision space distances during environment selection. This invention improves the FERDE multimodal optimization algorithm, proposing a feature selection algorithm based on multimodal multi-objectives. This algorithm uses the prediction accuracy of KNN and the number of selected features as fitness values ​​and calculates the SDE index. Then, it employs the FERDE_MMO algorithm to search for multiple optimal or suboptimal feature subsets. Compared to the MO_PSO and MO_Ring_PSO_SCD algorithms, the FS_FERDE_MMO algorithm can quickly find more multimodal feature subsets and obtain a better Pareto front. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 Comparison of the number of equivalent feature subsets obtained from different datasets;

[0044] Figure 2 Comparison of Pareto fronts obtained from different datasets;

[0045] Figure 3 Comparison of runtime results across different datasets;

[0046] Figure 4 This is a schematic diagram illustrating the method for calculating fitness values. Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0048] A feature selection algorithm based on multimodal and multi-objective methods includes the following steps:

[0049] (1) The initial dataset is divided into a training dataset and a test dataset using the ten-fold cross-validation method. The training dataset is used to calculate the fitness of each individual, and the test dataset is used to validate the classification results.

[0050] (2) Initialize the population individuals in the training dataset, randomly encode each individual, "1" represents that the corresponding feature is selected, and "0" represents that it is not selected, and perform non-dominated sorting on the generated population individuals;

[0051] Before initializing the population, the maximum values ​​of individuals in each dimension are given. and minimum value The result of initializing the population from the training data is to generate a population of NP and D-dimensional vectors as the target vector, where G is the iteration number, and the vector of the i-th individual is represented as:

[0052]

[0053] When the iteration number G = 0, the i-th individual is generated according to formula (2):

[0054]

[0055] Where rand[0,1] is a uniformly distributed random number generated in [0,1], and D is the data dimension;

[0056] The specific method for randomly encoding each individual is as follows: the position vector value x of the i-th individual id Compare with the set threshold θ, if x id If the value is greater than θ, then feature d is selected; otherwise, feature d is not selected.

[0057] (3) Calculate the fitness value of each individual based on the prediction accuracy of KNN and the number of selected features, and calculate the SDE (Shift-Based Density Estimation) index value.

[0058] The method for calculating fitness values ​​is as follows: Figure 4 As shown, the topmost rectangular squares represent the probability of each feature being selected. These are decimals between 0 and 1. If the corresponding value is greater than or equal to 0.5 (assuming the threshold θ is 0.5), the corresponding feature is selected; if the corresponding value is less than 0.5, the corresponding feature is discarded. In the diagram, the values ​​corresponding to the 1st, 3rd, 6th, and 8th features are greater than or equal to 0.5, so the selected features are {1, 3, 6, 8}, and the number of selected features is 4. The selected feature combination is fed into the KNN classifier to obtain the corresponding classification accuracy. This yields the values ​​of two targets, completing the evaluation of an individual. Before the algorithm evaluation, the individual might be a series of consecutive decimals, but after evaluation, it becomes a discrete binary number. Each binary individual is represented by the binary strings "0" and "1", where "1" indicates that the individual's corresponding feature is selected, and "0" indicates that it is not selected.

[0059] The SDE index value is calculated using formula (1):

[0060]

[0061] in, Let P represent the i-th target value, M be the number of targets, and the value of an individual P is defined as the minimum offset density estimate distance between that individual and other individuals in the population.

[0062] (4) Calculate the fitness Euclidean distance ratio (FER value) of each individual based on the SDE index value of the fitness value; select the parent individual using the roulette wheel strategy based on the FER value; generate new individuals through mutation and crossover operations;

[0063] The formula for calculating the FER value is as follows:

[0064]

[0065] Among them, FER (j,i) This refers to the ratio of the fitness Euclidean distance between individuals i and j within a population; p i and p j These are the historical best positions of individuals i and j, respectively; p w f(p) represents the worst position of an individual in the current population. j ) and f(p w ) are p respectively j and p w Corresponding fitness;

[0066] Using a roulette wheel strategy, individuals with high FER values ​​are selected as parent individuals. Offspring individuals are then generated through mutation and crossover operations. The mutation formula is as follows:

[0067] v p =x r1 +rand(1,D)·(x r2 -x r3 (5)

[0068] Among them, v p It is a mutated individual, x r1 ,x r2 and x r3 These are three different parent individuals generated by the roulette wheel selection method, where D is the length (i.e., data dimension) of the parent individual;

[0069] A crossover operation is performed on each generated mutated individual vector and its corresponding parent individual vector, as shown in formula (6):

[0070]

[0071] Among them, u p,d It is the offspring individual, p is the historical best position, CR is the crossover probability, which is a constant in the range [0,1], and ν is the crossover probability. p,d It is a mutant individual, χ p,d For the target individual, rand d d is a uniformly distributed random number generated in the range [0,1]. rand It is an integer randomly selected from the set {1,2,...D} to ensure that the mutation vector ν p,d At least one dimension of information is preserved.

[0072] (5) The new individuals obtained in step (4) are sorted again using non-dominated sorting. The individuals in the top 50% of the population are selected for mutation using the DE / rand / 2 strategy, and the individuals that exceed the boundary during the mutation process are processed.

[0073] The expression for the DE / rand / 2 strategy is as follows:

[0074]

[0075] Among them, r1 i r2 i r3 i ,r4 i r5 i F is a distinct individual index number within the range [1, NP], and is different from the index number i, F ∈ [0, 1], F is the scaling factor, NP is the population size, and G is the number of iterations;

[0076] For individuals that exceed the boundary during the mutation process, boundary processing is performed, and the calculation formula is shown in (3):

[0077]

[0078] Where F is 0.9.

[0079] (6) Merge the individuals obtained in steps (4) and (5) to form a new population and calculate the fitness value;

[0080] (7) Place the new population from step (6) and the initial population from step (2) into the archive set, and perform non-dominated sorting on the individuals in the archive set and calculate the corresponding SDE index values.

[0081] (8) Select the top N individuals in the archive set as the retained individuals based on the criteria of having the lowest Pareto front number and the lowest SDE index value.

[0082] To verify the effectiveness of the improved algorithm, datasets were selected from the UCI Machine Learning Repository for comparative experiments: WBCO, Glass, Wine, Zoo, SPECT, Sonar, SRBCT, and DLBCT. These datasets are widely used to evaluate learning algorithms and can provide a comprehensive test of performance for feature selection problems. Table 1 presents information about these datasets. These datasets come from different domains and contain varying numbers of samples, features, and class attributes. The number of samples ranges from 83 to 699, and the feature dimensions range from 9 to 5469. The number of classes corresponds to the number of class attributes, including both binary and multi-class problems. The diversity of these datasets indicates that they can, to a certain extent, verify the performance of feature selection algorithms under different conditions.

[0083] Table 1. Feature Selection Dataset Information Table

[0084] Serial Number Data Name Number of features Number of samples Number of categories 1 WBCO 9 699 2 2 Glass 9 214 7 3 Wine 13 178 3 4 Zoo 17 101 7 5 SPECT 22 267 2 6 Sonar 60 208 2 7 SRBCT 2308 83 4 8 DLBCL 5469 77 2

[0085] To verify the effectiveness of the improved algorithm, the dataset was first randomly divided into a training set (70%) and a test set (30%). During training, each individual represented a feature subset. Classification accuracy is often used as a performance evaluation metric for feature subsets; this paper employs the KNN classifier to obtain the classification performance of features selected by various feature selection algorithms. KNN is a simple learning algorithm, yet it performs well on many practical problems. In the proposed method, the population size is set to 100, and the stopping condition is set to 100 generations.

[0086] Experiments were conducted to analyze the performance of the FS_FERDE_MMO algorithm in solving multimodal multiobjective optimization problems in feature selection, and compared it with the single-modal multiobjective particle swarm optimization algorithm (MO_PSO) (Xue B, Zhang M, Browne W N. Particleswarm optimization for feature selection in classification: A multi-objective approach[J]. IEEE Transactions on Cybernetics, 2013, 43(6): 1656-1671.) and the multimodal multiobjective particle swarm optimization algorithm (MO_Ring_PSO_SCD) (Yue CT, Qu BY, Liang J JA multiobjective particle swarm optimizer using ring topology for solving multimodal multiobjective problems[J]. IEEE Transactions on Evolutionary Computation, 2018, 22(5): 805-817.). Tables 2, 3 and 4 show the results of the three algorithms. The third column in the table represents the classification accuracy, the fourth column provides the list of selected features, and the fifth column is the number of equivalent feature subsets. "-" indicates that the dataset has a large number of selected features, which are not listed here. In Tables 3 and 4, rows with multiple equivalent feature subsets are marked with a gray background.

[0087] Table 2 Feature combinations selected by MO_PSO

[0088]

[0089]

[0090]

[0091] Table 3 Feature combinations selected by MO_Ring_PSO_SCD

[0092]

[0093]

[0094]

[0095] Table 4 Feature combinations selected by FS_FERDE_MMO

[0096]

[0097]

[0098]

[0099] Table 2 shows that the classification accuracy is 0.9781 when the three features {F2, F3, F7} of the WBCO dataset are selected. Similarly, in Table 4, the classification accuracy is also 0.9781 when either {F1, F2, F7} or {F2, F3, F7} is selected. This indicates that the FS_FERDE_MMO algorithm found two equivalent feature subsets, while the MO_PSO algorithm found only one feature combination. According to the UCI dataset (http: / / archive.ics.uci.edu / ml / index.php), the first feature in WBCO is slice thickness, and the third feature is cell shape uniformity. Clearly, the first feature is easier to extract than the third. If only three features can be selected, researchers tend to choose {F1, F2, F7} over these two equivalent feature subsets. This result demonstrates that multimodal multi-objective optimization algorithms can not only provide decision-makers with more choices but also potentially reduce the difficulty of feature extraction. For other datasets, the FS_FERDE_MMO algorithm also found several equivalent feature subsets. Among these, SRBCT and DLBCL are two high-dimensional datasets. Experimental results show that FS_FERDE_MMO can find more feature subset combinations in the SRBCT dataset. Although the MO_PSO, MO_Ring_PSO_SCD, and FS_FERDE_MMO algorithms can only find one feature subset in DLBCL, FS_FERDE_MMO finds the combination with the fewest features while maintaining the same accuracy. Therefore, FS_FERDE_MMO can also achieve good experimental results when processing high-dimensional data.

[0100] Figure 1A bar chart is provided to show the number of equivalent feature subsets obtained by the MO_PSO, MO_Ring_PSO_SCD, and FS_FERDE_MMO algorithms on the WBCO, Glass, Wine, Zoo, SPETC, and SRBCT datasets. The horizontal axis represents the number of selected features, and the vertical axis represents the number of equivalent feature subsets. Different colored bars represent the three different algorithms. Figure 1 It is known that regardless of the number of features selected, MO_PSO can only find one feature combination because it is a single-modal multi-objective optimization algorithm. Even if an equivalent feature combination is found, it will be deleted during environment selection. However, in Figure 1 In (a), when the number of selected features is 4, FS_FERDE_MMO found two sets of equivalent feature subsets, while MO_PSO and MO_Ring_PSO_SCD each found only one set of equivalent feature subsets. Figure 1 In (b), when the number of selected features is 7, FS_FERDE_MMO finds two equivalent feature subsets, while MO_PSO and MO_Ring_PSO_SCD can only find one equivalent feature subset. Figure 1 In (e), when the number of features is 3, 4, or 8, FS_FERDE_MMO can find multiple feature subsets, while MO_PSO and MO_Ring_PSO_SCD can only find one equivalent feature subset. Figure 1 (f) shows that in SRBCT, FS_FERDE_MMO finds more feature subsets than MO_PSO and MO_Ring_PSO_SCD. This leads to the conclusion that MO_PSO, regardless of the number of selected features, can only find one feature combination because it is a unimodal multi-objective optimization algorithm; even if an equivalent feature combination is found, it will be deleted during environment selection. However, the FS_FERDE_MMO multimodal multi-objective optimization algorithm can provide decision-makers with more equivalent feature subsets and may also reduce the difficulty of feature extraction. Furthermore, as a multimodal multi-objective optimization algorithm, FS_FERDE_MMO finds more feature subsets than MO_Ring_PSO_SCD.

[0101] Figure 2 The Pareto fronts of the MO_PSO, MO_Ring_PSO_SCD, and FS_FERDE_MMO algorithms are presented on different datasets. The horizontal axis represents the number of selected features, and the vertical axis represents the classification error rate. Since both objective values ​​are better when smaller, a Pareto front further to the lower left corner indicates better algorithm performance. Figure 2As can be seen, while maintaining good multimodal solution search capabilities, the FS_FERDE_MMO algorithm can obtain Pareto fronts that are the same as or better than those of MO_PSO and MO_Ring_PSO_SCD. Although FS_FERDE_MMO did not obtain the optimal Pareto front in SRBCT, it can find more feature subsets, and in this dataset, the Pareto front of FS_FERDE_MMO is inferior to that of MO_PSO.

[0102] Figure 3 This graph compares the running times of three multimodal algorithms—MO_PSO, MO_Ring_PSO_SCD, and FS_FERDE_MMO—on different datasets. The vertical axis represents a log10 distribution. As shown in the graph, while maintaining good multimodal solution search and Pareto front acquisition capabilities, FS_FERDE_MMO outperforms MO_Ring_PSO_SCD on all datasets (except Wine). Figure 3 The results demonstrate the efficiency of FS_FERDE_MMO.

[0103] In summary, the multimodal multi-objective FS_FERDE_MMO algorithm finds multiple equivalent feature subsets without affecting classification accuracy. This not only provides decision-makers with more choices but also reduces the computational resources consumed by feature extraction.

[0104] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A feature selection algorithm based on multimodal and multi-objective modes, characterized in that: For breast cancer feature selection, the following steps are included: (1) The initial dataset is divided into a training dataset and a test dataset using the ten-fold cross-validation method. The training dataset is used to calculate the fitness of each individual, and the test dataset is used to validate the classification results. (2) Initialize the population individuals in the training dataset, randomly encode each individual, "1" represents that the corresponding feature is selected, "0" represents that it is not selected, and perform non-dominated sorting on the generated population individuals; (3) Calculate the fitness value of each individual based on the prediction accuracy of KNN and the number of selected features, and calculate the SDE index value; the features are variable information of breast cancer characteristics, specifically the thickness of the slice and the uniformity of cell shape; wherein, the SDE index value is calculated using formula (1): （1） in, Let P represent the i-th target value of P, M be the number of targets, and the value of an individual P is defined as the minimum offset density estimate distance between that individual and other individuals in the population. (4) Calculate the fitness Euclidean distance ratio (FER) of each individual based on the SDE index value of the fitness value; select the parent individual using the roulette wheel strategy based on the FER value; generate new individuals through mutation and crossover operations; (5) Perform non-dominated sorting on the new individuals obtained in step (4), select the top 50% of the population individuals and perform mutation operation using the DE / rand / 2 strategy, and perform boundary processing on individuals that exceed the boundary during the mutation process; wherein, the expression of the DE / rand / 2 strategy is as follows: （2） Among them, r1 i r2 i r3 i r4 i r5 i Let i be a distinct individual index of an integer in the range [1, NP], and different from index i. F is the scaling factor, NP is the population size, and G is the number of iterations; for individuals that exceed the boundary during the mutation process, boundary processing is performed, and its calculation formula is shown in (3): （3） Where F is 0.9; This refers to the target vector; (6) Merge the individuals obtained in steps (4) and (5) to form a new population and calculate the fitness value; (7) Place the new population from step (6) and the initial population from step (2) into the archive set, and perform non-dominated sorting on the individuals in the archive set and calculate the corresponding SDE index values; (8) Select the top N individuals in the archive set as the individuals to be retained, based on the criteria of the lowest Pareto front number and the lowest SDE index value of the individuals in the archive set in step (7).

2. The feature selection algorithm based on multimodal and multi-objective modes according to claim 1, characterized in that: In step (4), the formula for calculating the FER value is as follows: （4） Among them, FER (j,i) This refers to the ratio of fitness Euclidean distance between individuals i and j within a population; and These are the historical best positions of individuals i and j, respectively; This represents the worst position of an individual in the current population. and They are respectively and Corresponding fitness; Using a roulette wheel strategy, individuals with high FER values ​​are selected as parent individuals. Offspring individuals are then generated through mutation and crossover operations. The mutation formula is as follows: （5） Among them, v p It is a mutated individual, x r1 , x r2 and x r3 These are three different parent individuals generated by the roulette wheel selection method, where D is the length of the parent individual; A crossover operation is performed on each generated mutated individual vector and its corresponding parent individual vector, as shown in formula (6): （6） in, is the offspring individual, p is the historical best position, and CR is the crossover probability, which is a constant in the range [0, 1]. It is a mutated individual. For the target individual, These are uniformly distributed random numbers generated within the range [0, 1]. It is an integer randomly selected from the set {1,2,...D} to ensure the mutation vector At least one dimension of information is preserved.

3. The feature selection algorithm based on multimodal and multi-objective modes according to claim 1, characterized in that: In step (2), the maximum value of an individual in each dimension is given before initializing the population. and minimum value The result of initializing the population from the training data is to generate a population of NP-dimensional, D-dimensional individuals as the target vector, where G is the iteration number, and the vector of the i-th individual is represented as: , i=1,2,…,NP (1) When the iteration number G=0, the i-th individual is generated according to formula (2): （2） Here, rand[0, 1] is a uniformly distributed random number generated in [0, 1], and D is the data dimension.

4. The feature selection algorithm based on multimodal and multi-objective modes according to claim 3, characterized in that: In step (2), the specific method for randomly encoding each individual is as follows: the position vector value x of the i-th individual id Compare with the set threshold θ, if x id If the value is greater than θ, then feature d is selected; otherwise, feature d is not selected.

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