基于最大安全近邻与局部密度的自适应过采样方法

赵小强; 何嘉琦

doi:10.11999/JEIT240441

基于最大安全近邻与局部密度的自适应过采样方法

doi: 10.11999/JEIT240441

赵小强^,,
何嘉琦

兰州理工大学电气工程与信息工程学院兰州 730000

基金项目: 国家自然科学基金(62263021)，甘肃省高校产业支撑计划(2023CYZC-24)

详细信息

作者简介:
赵小强：男，教授，研究方向为故障诊断、数据挖掘和图像处理等

何嘉琦：男，硕士生，研究方向为数据挖掘、不平衡数据分类

通讯作者:
赵小强　xqzhao@lut.edu.cn

中图分类号: TN911; TP274; TP181
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- 被引次数: 0
出版历程
- 收稿日期: 2024-06-03
- 修回日期: 2025-03-30
- 网络出版日期: 2025-04-11
- 刊出日期: 2025-04-01

Adaptive Oversampling Method Based on Maximum Safe Nearest Neighbor and Local Density

ZHAO Xiaoqiang^,,
HE Jiaqi

College of Electrical Engineering and Information Engineering, Lanzhou University of Technology, Lanzhou 730000, China

Funds: The National Natural Science Foundation of China (62263021), The College Industrial Support Project of Gansu Province (2023CYZC-24)

摘要

摘要: 针对不平衡数据过采样的过程中如何合成有效新样本的问题，该文提出一种基于最大安全近邻与局部密度的自适应过采样方法。该方法利用最大安全近邻和局部密度将少数类样本划分为安全样本、边界样本和离群点；在此基础上，通过组合加权设置样本的采样概率，使得靠近边界的“次边界样本”更容易被选择为根样本，并且自适应地调整K近邻的参数K，选择最优合成区域；针对离群点，采用超球面内的随机过采样策略，进一步增加少数类样本的多样性。最后，将所提方法与合成少数类过采样技术(SMOTE)、自适应合成采样方法(ADASYN)等6种过采样方法在13个公开数据集上进行实验分析，结果表明，所提方法相对于对比方法在F1分数(F1-score)指标上分别平均提高了6.9%, 8.8%, 8.2%, 5.8%, 7.2%和12.5%，在几何平均值(G-mean)指标上分别平均提高了3.0%, 2.5%, 3.0%, 3.2%, 5.3%和8.6%，证明所提方法可以有效解决不平衡数据分类问题。
- 不平衡数据 /
- 过采样技术 /
- 最大安全近邻 /
- 次边界样本
Abstract: Objective Traditional classifiers tend to optimize overall accuracy when dealing with imbalanced data sets, often resulting in poor classification performance for minority class samples. Among the available strategies, oversampling methods are widely used due to their strong generalization ability. However, conventional oversampling techniques frequently generate new samples with high overlap rates and limited validity, particularly near decision boundaries. To address this issue, this study proposes an adaptive oversampling approach that selects sub-boundary samples—those located near the boundary samples—for sample generation. In addition, the nearest-neighbor parameter space is constrained to refine the synthetic sample region. This method improves the classifier’s performance when learning from imbalanced data sets. Methods This study first identifies the maximum safe like-neighbors of positive class samples and classifies these samples as either hazardous or safe. The local density of each sample is then calculated, and hazardous samples—those more difficult to classify—are further categorized as either boundary samples or outliers. To provide the classifier with more informative positive class samples, “sub-boundary points” are preferentially selected as root samples using a weighted composite factor. The K-value in the K-nearest neighbor algorithm is adaptively adjusted based on the maximum safe nearest neighbor of each sample to improve neighbor selection. Outliers are oversampled randomly within a hypersphere to generate new samples while minimizing increases in spatial complexity. Results and Discussions To evaluate the feasibility and generalization of the proposed method, Logistic Regression (LR) and Support Vector Machine (SVM) classifiers are employed as base classifiers. The range of the distance adjustment coefficient is first determined by comparing results across selected datasets (Table 3). Once the range is established, the effect of different weight adjustment coefficients on performance is assessed (Table 4). The proposed method is then compared with six existing oversampling techniques across 13 datasets. For most datasets, the proposed method achieves higher values in more than half of the five evaluation metrics considered (Tables 5 and 6). These results demonstrate that the proposed approach effectively improves classifier performance on imbalanced data sets. Conclusions This study introduces the maximum safe nearest neighbor number and local density to classify minority class samples into safe samples, boundary samples, and outliers. A weighted sampling probability, based on both local density and the maximum safe nearest neighbor number, is used to guide adaptive K-nearest neighbor oversampling of safe and boundary samples. Random oversampling within a hypersphere is applied to outliers to preserve informative but rare samples. Comparative experiments confirm that the proposed method performs well across datasets with varying imbalance ratios and remains competitive under highly imbalanced conditions.
- Unbalanced data /
- Over-sampling technique /
- Maximum safe nearest neighbors /
- Sub-boundary points

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图 1 参数K对SMOTE的影响

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图 2 边界样本识别示意图

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图 3 过采样过程示意图

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表 1 不平衡数据集信息

数据集	样本数	特征数	特征属性(R/I/N)	不平衡比
wisconsin	683	9	(0/9/0)	1.86
pima	768	8	(8/0/0)	1.87
yeast1	1484	8	(8/0/0)	2.46
haberman	306	3	(0/3/0)	2.78
vehicle0	846	18	(0/18/0)	3.25
new-thyroid2	215	5	(4/1/0)	5.14
glass6	214	9	(9/0/0)	6.38
yeast3	1484	8	(8/0/0)	8.10
ecoli3	336	7	(7/0/0)	8.60
abalone9-18	731	8	(7/0/1)	16.4
glass5	214	9	(9/0/0)	22.78
yeast4	1484	8	(8/0/0)	28.10
yeast5	1848	8	(8/0/0)	32.73

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表 2 二分类混淆矩阵

类别	预测为正样本	预测为负样本
实际正样本	TP	FN
实际负样本	FP	TN

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表 3 指标随调节系数$\alpha $的变化情况

数据集	评价指标	0.01	0.03	0.05	0.07	0.09	0.11	0.13	0.15
yeast1	F1-score	0.572	0.580	0.584	0.591	0.589	0.586	0.579	0.576
yeast1	G-mean	0.662	0.698	0.708	0.711	0.709	0.706	0.700	0.697
glass6	F1-score	0.754	0.754	0.766	0.779	0.779	0.733	0.779	0.757
glass6	G-mean	0.905	0.905	0.908	0.911	0.911	0.884	0.911	0.889
ecoli3	F1-score	0.614	0.628	0.633	0.639	0.613	0.619	0.626	0.619
ecoli3	G-mean	0.893	0.896	0.898	0.900	0.893	0.894	0.886	0.873
abalone9-18	F1-score	0.307	0.372	0.380	0.399	0.414	0.403	0.425	0.419
abalone9-18	G-mean	0.714	0.794	0.787	0.795	0.785	0.773	0.778	0.777
glass5	F1-score	0.395	0.395	0.411	0.434	0.425	0.411	0.356	0.370
glass5	G-mean	0.882	0.882	0.887	0.892	0.934	0.886	0.827	0.830
vehicle0	F1-score	0.879	0.882	0.888	0.885	0.891	0.894	0.899	0.893
vehicle0	G-mean	0.953	0.953	0.953	0.953	0.955	0.958	0.961	0.956

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表 4 不同权重调节系数$ \lambda $下的指标

数据集	评价指标	(0.2,0.8)	(0.4,0.6)	(0.5,0.5)	(0.6,0.4)	(0.8,0.2)
ecoli3	AUC	0.936	0.940	0.936	0.938	0.936
	F1-score	0.616	0.630	0.610	0.610	0.630
	G-mean	0.895	0.898	0.893	0.893	0.895
yeast1	AUC	0.793	0.795	0.794	0.793	0.792
	F1-score	0.592	0.598	0.592	0.600	0.597
	G-mean	0.715	0.721	0.714	0.720	0.718
yeast3	AUC	0.972	0.973	0.972	0.973	0.973
	F1-score	0.732	0.766	0.756	0.765	0.768
	G-mean	0.905	0.909	0.905	0.901	0.899
glass6	AUC	0.961	0.963	0.963	0.965	0.963
	F1-score	0.887	0.871	0.856	0.905	0.886
	G-mean	0.920	0.917	0.914	0.937	0.919
wisconsin	AUC	0.994	0.994	0.995	0.994	0.963
	F1-score	0.922	0.931	0.930	0.928	0.927
	G-mean	0.970	0.976	0.973	0.972	0.967

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表 5 其他采样算法的指标对比

数据集	采样方法	LR					SVM
数据集	采样方法	Acc	AUC	F1-score	G-mean	MCC	Acc	AUC	F1-score	G-mean	MCC
wisconsin	SMOTE	0.971	0.996	0.959	0.970	0.936	0.971	0.988	0.959	0.974	0.938
	ADASYN	0.972	0.996	0.961	0.975	0.941	0.969	0.985	0.958	0.975	0.936
	BSMOTE	0.972	0.996	0.962	0.976	0.939	0.971	0.984	0.960	0.974	0.939
	NaN-SMOTE	0.974	0.996	0.963	0.974	0.943	0.969	0.986	0.958	0.972	0.935
	SMOTE-NaN-DE	0.969	0.996	0.956	0.966	0.933	0.972	0.989	0.961	0.971	0.939
	SADCO	0.972	0.996	0.961	0.972	0.940	0.971	0.986	0.960	0.974	0.938
	本文方法	0.974	0.996	0.963	0.974	0.943	0.972	0.989	0.962	0.976	0.941
pima	SMOTE	0.744	0.829	0.663	0.737	0.465	0.759	0.839	0.692	0.761	0.507
	ADASYN	0.742	0.831	0.668	0.742	0.469	0.753	0.840	0.691	0.760	0.503
	BSMOTE	0.739	0.829	0.670	0.743	0.469	0.738	0.836	0.682	0.751	0.485
	NaN-SMOTE	0.719	0.832	0.680	0.742	0.479	0.723	0.835	0.685	0.746	0.489
	SMOTE-NaN-DE	0.733	0.818	0.622	0.703	0.419	0.754	0.823	0.629	0.706	0.450
	SADCO	0.767	0.830	0.623	0.696	0.470	0.771	0.834	0.636	0.709	0.480
	本文方法	0.767	0.833	0.690	0.760	0.499	0.738	0.839	0.694	0.759	0.504
yeast1	SMOTE	0.708	0.787	0.581	0.703	0.380	0.687	0.792	0.589	0.701	0.388
	ADASYN	0.681	0.786	0.584	0.705	0.379	0.650	0.786	0.581	0.692	0.375
	BSMOTE	0.685	0.785	0.581	0.705	0.376	0.650	0.784	0.578	0.690	0.371
	NaN-SMOTE	0.679	0.788	0.584	0.705	0.380	0.666	0.786	0.587	0.703	0.385
	SMOTE-NaN-DE	0.728	0.781	0.558	0.681	0.364	0.730	0.788	0.575	0.692	0.395
	SADCO	0.726	0.752	0.521	0.644	0.333	0.739	0.736	0.440	0.563	0.298
	本文方法	0.720	0.790	0.593	0.715	0.400	0.722	0.795	0.597	0.718	0.405
haberman	SMOTE	0.676	0.653	0.437	0.594	0.217	0.688	0.688	0.431	0.586	0.270
	ADASYN	0.663	0.650	0.440	0.592	0.215	0.693	0.670	0.467	0.581	0.250
	BSMOTE	0.660	0.612	0.436	0.592	0.202	0.716	0.687	0.434	0.590	0.260
	NaN-SMOTE	0.667	0.658	0.437	0.595	0.209	0.703	0.670	0.410	0.561	0.219
	SMOTE-NaN-DE	0.722	0.609	0.409	0.554	0.236	0.729	0.676	0.422	0.561	0.263
	SADCO	0.546	0.633	0.436	0.576	0.151	0.490	0.461	0.438	0.540	0.139
	本文方法	0.677	0.680	0.518	0.667	0.306	0.630	0.703	0.505	0.640	0.284
vehicle0	SMOTE	0.930	0.984	0.872	0.952	0.839	0.962	0.995	0.926	0.973	0.905
	ADASYN	0.923	0.978	0.860	0.947	0.824	0.955	0.994	0.914	0.970	0.890
	BSMOTE	0.913	0.976	0.851	0.951	0.812	0.959	0.993	0.920	0.973	0.898
	NaN-SMOTE	0.936	0.985	0.881	0.954	0.849	0.963	0.995	0.928	0.972	0.908
	SMOTE-NaN-DE	0.918	0.979	0.853	0.942	0.814	0.952	0.992	0.907	0.966	0.881
	SADCO	0.922	0.968	0.855	0.936	0.813	0.950	0.990	0.905	0.967	0.880
	本文方法	0.944	0.986	0.895	0.960	0.866	0.967	0.995	0.935	0.976	0.916
new-thyroid2	SMOTE	0.977	0.998	0.927	0.961	0.915	0.981	0.998	0.948	0.989	0.940
	ADASYN	0.967	0.998	0.910	0.969	0.895	0.967	0.998	0.913	0.980	0.901
	BSMOTE	0.963	0.998	0.898	0.966	0.882	0.963	0.998	0.904	0.977	0.956
	NaN-SMOTE	0.963	0.997	0.863	0.874	0.857	0.977	0.997	0.925	0.949	0.912
	SMOTE-NaN-DE	0.977	0.998	0.923	0.937	0.912	0.986	0.997	0.962	0.992	0.956
	SADCO	0.953	0.989	0.824	0.840	0.820	0.977	0.997	0.921	0.925	0.912
	本文方法	0.981	0.998	0.939	0.952	0.929	0.991	0.998	0.973	0.994	0.989
glass6	SMOTE	0.925	0.954	0.757	0.890	0.727	0.972	0.948	0.886	0.934	0.876
	ADASYN	0.916	0.952	0.743	0.902	0.714	0.967	0.913	0.873	0.931	0.861
	BSMOTE	0.935	0.949	0.772	0.883	0.741	0.967	0.904	0.862	0.900	0.852
	NaN-SMOTE	0.935	0.959	0.790	0.895	0.766	0.967	0.967	0.848	0.875	0.840
	SMOTE-NaN-DE	0.935	0.952	0.786	0.895	0.758	0.949	0.958	0.820	0.908	0.793
	SADCO	0.898	0.957	0.716	0.890	0.687	0.953	0.960	0.825	0.906	0.802
	本文方法	0.935	0.951	0.792	0.913	0.767	0.981	0.968	0.920	0.940	0.913
yeast3	SMOTE	0.908	0.967	0.679	0.894	0.651	0.931	0.971	0.736	0.904	0.710
	ADASYN	0.887	0.968	0.651	0.917	0.638	0.906	0.971	0.694	0.925	0.679
	BSMOTE	0.876	0.966	0.623	0.910	0.618	0.902	0.970	0.682	0.920	0.666
	NaN-SMOTE	0.910	0.967	0.678	0.884	0.648	0.933	0.971	0.747	0.913	0.723
	SMOTE-NaN-DE	0.919	0.967	0.701	0.892	0.671	0.939	0.972	0.759	0.908	0.734
	SADCO	0.909	0.965	0.674	0.881	0.643	0.919	0.970	0.708	0.892	0.680
	本文方法	0.919	0.968	0.699	0.892	0.670	0.940	0.973	0.761	0.903	0.795
ecoli3	SMOTE	0.869	0.937	0.595	0.888	0.579	0.872	0.938	0.598	0.890	0.583
	ADASYN	0.851	0.933	0.572	0.889	0.562	0.860	0.935	0.576	0.883	0.561
	BSMOTE	0.860	0.930	0.585	0.895	0.576	0.881	0.943	0.622	0.907	0.611
	NaN-SMOTE	0.863	0.943	0.577	0.873	0.556	0.881	0.944	0.608	0.883	0.586
	SMOTE-NaN-DE	0.860	0.938	0.579	0.883	0.564	0.884	0.942	0.621	0.896	0.605
	SADCO	0.857	0.932	0.576	0.881	0.560	0.863	0.938	0.581	0.884	0.566
	本文方法	0.890	0.938	0.640	0.900	0.622	0.902	0.940	0.665	0.907	0.647

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表 6 高不平衡比数据集上采样算法的指标对比

数据集	采样方法	LR					SVM
数据集	采样方法	Acc	AUC	F1-score	G-mean	MCC	Acc	AUC	F1-score	G-mean	MCC
abalone9-18	SMOTE	0.835	0.890	0.338	0.774	0.336	0.852	0.892	0.384	0.820	0.393
	ADASYN	0.827	0.894	0.328	0.770	0.327	0.847	0.894	0.378	0.829	0.393
	BSMOTE	0.841	0.857	0.313	0.734	0.304	0.859	0.853	0.318	0.706	0.295
	NaN-SMOTE	0.858	0.890	0.361	0.779	0.355	0.866	0.886	0.376	0.775	0.367
	SMOTE-NaN-DE	0.891	0.854	0.330	0.652	0.295	0.885	0.849	0.341	0.688	0.323
	SADCO	0.824	0.804	0.284	0.706	0.263	0.854	0.872	0.303	0.689	0.277
	本文方法	0.870	0.883	0.411	0.817	0.413	0.885	0.892	0.424	0.811	0.421
glass5	SMOTE	0.860	0.937	0.324	0.821	0.356	0.949	0.963	0.573	0.863	0.597
	ADASYN	0.851	0.924	0.309	0.816	0.343	0.949	0.968	0.573	0.863	0.597
	BSMOTE	0.855	0.937	0.316	0.818	0.350	0.944	0.968	0.547	0.860	0.574
	NaN-SMOTE	0.869	0.929	0.378	0.878	0.424	0.944	0.912	0.520	0.805	0.535
	SMOTE-NaN-DE	0.865	0.936	0.333	0.823	0.366	0.930	0.961	0.480	0.854	0.497
	SADCO	0.860	0.905	0.242	0.583	0.228	0.935	0.978	0.535	0.911	0.563
	本文方法	0.879	0.937	0.425	0.934	0.486	0.949	0.960	0.573	0.862	0.597
yeast4	SMOTE	0.858	0.876	0.260	0.786	0.294	0.870	0.896	0.277	0.794	0.310
	ADASYN	0.840	0.874	0.247	0.800	0.289	0.852	0.893	0.247	0.774	0.278
	BSMOTE	0.885	0.868	0.305	0.799	0.335	0.904	0.885	0.338	0.798	0.360
	NaN-SMOTE	0.861	0.879	0.266	0.788	0.299	0.873	0.892	0.276	0.785	0.305
	SMOTE-NaN-DE	0.892	0.865	0.315	0.803	0.344	0.900	0.886	0.334	0.805	0.361
	SADCO	0.836	0.859	0.246	0.796	0.287	0.860	0.873	0.267	0.787	0.299
	本文方法	0.894	0.870	0.324	0.804	0.351	0.908	0.888	0.354	0.809	0.378
yeast5	SMOTE	0.934	0.985	0.478	0.965	0.541	0.946	0.985	0.530	0.972	0.584
	ADASYN	0.931	0.985	0.457	0.964	0.532	0.945	0.986	0.527	0.971	0.581
	BSMOTE	0.930	0.984	0.463	0.963	0.529	0.944	0.986	0.520	0.971	0.576
	NaN-SMOTE	0.939	0.985	0.491	0.967	0.552	0.945	0.981	0.521	0.961	0.572
	SMOTE-NaN-DE	0.935	0.986	0.483	0.966	0.545	0.945	0.986	0.524	0.971	0.579
	SADCO	0.944	0.986	0.520	0.971	0.576	0.937	0.986	0.593	0.967	0.554
	本文方法	0.945	0.985	0.524	0.971	0.579	0.949	0.985	0.546	0.973	0.597

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参考文献(23)

[1]	李艳霞, 柴毅, 胡友强, 等. 不平衡数据分类方法综述[J]. 控制与决策, 2019, 34(4): 673–688. doi: 10.13195/j.kzyjc.2018.0865. LI Yanxia, CHAI Yi, HU Youqiang, et al. Review of imbalanced data classification methods[J]. Control and Decision, 2019, 34(4): 673–688. doi: 10.13195/j.kzyjc.2018.0865.
[2]	GUO Haixiang, LI Yijing, SHANG J, et al. Learning from class-imbalanced data: Review of methods and applications[J]. Expert Systems with Applications, 2017, 73: 220–239. doi: 10.1016/j.eswa.2016.12.035.
[3]	SHIN K, HAN J, and KANG S. MI-MOTE: Multiple imputation-based minority oversampling technique for imbalanced and incomplete data classification[J]. Information Sciences, 2021, 575: 80–89. doi: 10.1016/j.ins.2021.06.043.
[4]	苏逸, 李晓军, 姚俊萍, 等. 不平衡数据分类数据层面方法: 现状及研究进展[J]. 计算机应用研究, 2023, 40(1): 11–19. doi: 10.19734/j.issn.1001-3695.2022.05.0250. SU Yi, LI Xiaojun, YAO Junping, et al. Data-level methods of imbalanced data classification: Status and research development[J]. Application Research of Computers, 2023, 40(1): 11–19. doi: 10.19734/j.issn.1001-3695.2022.05.0250.
[5]	THABTAH F, HAMMOUD S, KAMALOV F, et al. Data imbalance in classification: Experimental evaluation[J]. Information Sciences, 2020, 513: 429–441. doi: 10.1016/j.ins.2019.11.004.
[6]	CHAWLA N V, BOWYER K W, HALL L O, et al. SMOTE: Synthetic minority over-sampling technique[J]. Journal of Artificial Intelligence Research, 2002, 16: 321–357. doi: 10.1613/jair.953.
[7]	ABDI L and HASHEMI S. To combat multi-class imbalanced problems by means of over-sampling techniques[J]. IEEE Transactions on Knowledge and Data Engineering, 2016, 28(1): 238–251. doi: 10.1109/TKDE.2015.2458858.
[8]	CHEN Baiyun, XIA Shuyin, CHEN Zizhong, et al. RSMOTE: A self-adaptive robust SMOTE for imbalanced problems with label noise[J]. Information Sciences, 2021, 553: 397–428. doi: 10.1016/j.ins.2020.10.013.
[9]	HAN Hui, WANG Wenyuan, and MAO Binghuan. Borderline-SMOTE: A new over-sampling method in imbalanced data sets learning[C]. The International Conference on Intelligent Computing Advances in Intelligent Computing, Hefei, China, 2005: 878–887. doi: 10.1007/11538059_91.
[10]	HE Haibo, BAI Yang, GARCIA E A, et al. ADASYN: Adaptive synthetic sampling approach for imbalanced learning[C]. 2008 IEEE International Joint Conference on Neural Networks, Hong Kong, China, 2008: 1322–1328. doi: 10.1109/IJCNN.2008.4633969.
[11]	SOLTANZADEH P and HASHEMZADEH M. RCSMOTE: Range-Controlled synthetic minority over-sampling technique for handling the class imbalance problem[J]. Information Sciences, 2021, 542: 92–111. doi: 10.1016/j.ins.2020.07.014.
[12]	XU Zhaozhao, SHEN Derong, NIE Tiezheng, et al. A cluster-based oversampling algorithm combining SMOTE and k-means for imbalanced medical data[J]. Information Sciences, 2021, 572: 574–589. doi: 10.1016/j.ins.2021.02.056.
[13]	高雷阜, 张梦瑶, 赵世杰. 融合簇边界移动与自适应合成的混合采样算法[J]. 电子学报, 2022, 50(10): 2517–2529. doi: 10.12263/DZXB.20210265. GAO Leifu, ZHANG Mengyao, and ZHAO Shijie. Mixed-sampling algorithm combining cluster boundary movement and adaptive synthesis[J]. Acta Electronica Sinica, 2022, 50(10): 2517–2529. doi: 10.12263/DZXB.20210265.
[14]	黄海松, 魏建安, 康佩栋. 基于不平衡数据样本特性的新型过采样SVM分类算法[J]. 控制与决策, 2018, 33(9): 1549–1558. doi: 10.13195/j.kzyjc.2017.0649. HUANG Haisong, WEI Jian’an, and KANG Peidong. New over-sampling SVM classification algorithm based on unbalanced data sample characteristics[J]. Control and Decision, 2018, 33(9): 1549–1558. doi: 10.13195/j.kzyjc.2017.0649.
[15]	SHI Shengnan, LI Jie, ZHU Dan, et al. A hybrid imbalanced classification model based on data density[J]. Information Sciences, 2023, 624: 50–67. doi: 10.1016/j.ins.2022.12.046.
[16]	TAO Xinmin, GUO Xinyue, ZHENG Yujia, et al. Self-adaptive oversampling method based on the complexity of minority data in imbalanced datasets classification[J]. Knowledge-Based Systems, 2023, 277: 110795. doi: 10.1016/j.knosys.2023.110795.
[17]	周玉, 岳学震, 刘星, 等. 不平衡数据集的自然邻域超球面过采样方法[J]. 哈尔滨工业大学学报, 2024, 56(12): 81–95. doi: 10.11918/202311030. ZHOU Yu, YUE Xuezhen, LIU Xing, et al. A natural neighborhood hypersphere oversampling method for imbalanced data sets[J]. Journal of Harbin Institute of Technology, 2024, 56(12): 81–95. doi: 10.11918/202311030.
[18]	LENG Qiangkui, GUO Jiamei, JIAO Erjie, et al. NanBDOS: Adaptive and parameter-free borderline oversampling via natural neighbor search for class-imbalance learning[J]. Knowledge-Based Systems, 2023, 274: 110665. doi: 10.1016/j.knosys.2023.110665.
[19]	THEJAS G S, HARIPRASAD Y, IYENGAR S S, et al. An extension of Synthetic Minority Oversampling Technique based on Kalman filter for imbalanced datasets[J]. Machine Learning with Applications, 2022, 8: 100267. doi: 10.1016/j.mlwa.2022.100267.
[20]	胡峰, 王蕾, 周耀. 基于三支决策的不平衡数据过采样方法[J]. 电子学报, 2018, 46(1): 135–144. doi: 10.3969/j.issn.0372-2112.2018.01.019. HU Feng, WANG Lei, and ZHOU Yao. An oversampling method for imbalance data based on three-way decision model[J]. Acta Electronica Sinica, 2018, 46(1): 135–144. doi: 10.3969/j.issn.0372-2112.2018.01.019.
[21]	ALCALÁ-FDEZ J, SÁNCHEZ L, GARCÍA S, et al. KEEL: A software tool to assess evolutionary algorithms for data mining problems[J]. Soft Computing, 2009, 13(3): 307–318. doi: 10.1007/s00500-008-0323-y.
[22]	LI Junnan, ZHU Qingsheng, WU Quanwang, et al. A novel oversampling technique for class-imbalanced learning based on SMOTE and natural neighbors[J]. Information Sciences, 2021, 565: 438–455. doi: 10.1016/j.ins.2021.03.041.
[23]	LI Junnan, ZHU Qingsheng, WU Quanwang, et al. SMOTE-NaN-DE: Addressing the noisy and borderline examples problem in imbalanced classification by natural neighbors and differential evolution[J]. Knowledge-Based Systems, 2021, 223: 107056. doi: 10.1016/j.knosys.2021.107056.