Online Prediction of Chaotic Time Series Based on Correntropy Kernel Learning Evolving Fuzzy Systems
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摘要: 演化模糊系统(EFSs)是在线学习领域中广泛应用的方法. 然而, 大部分EFSs往往隐含高斯噪声假设, 在重尾、偏态及强相关等非高斯扰动下易出现性能退化. 为此, 提出一种相关熵核学习演化模糊系统(CKL-EFS). CKL-EFS融合结构自组织与鲁棒递归学习两种机制, 以提升非高斯噪声环境下的在线建模能力. 在结构演化方面, 模型基于数据密度与激活度实现数据云的按需生成与效用驱动移除, 并通过在线归属更新维护数据云统计量, 从而抑制规则库无界膨胀并保持紧凑结构; 在参数更新方面, 采用核学习策略将输入映射至高维特征空间, 并引入改进相关熵准则对大误差样本自适应降权, 以增强对离群冲击与非高斯噪声的鲁棒性. 此外, 给出模型的计算复杂度分析. 在多个公开数据集上的实验结果表明, 所提CKL-EFS在预测精度与抗非高斯扰动稳定性方面均优于现有代表性方法.Abstract: Evolving fuzzy systems (EFSs) have been widely used in online learning. However, most existing EFSs implicitly rely on a Gaussian-noise assumption and may suffer performance degradation under non-Gaussian disturbances such as heavy-tailed, skewed, and strongly correlated noise. To address this issue, this paper proposes a correntropy kernel learning EFSs (CKL-EFS). CKL-EFS integrates two mechanisms—structural self-organization and robust recursive learning—to enhance online modeling capability in non-Gaussian noise environments. For structural evolution, the model performs on-demand data-cloud generation and utility-driven pruning based on data density and activation, and maintains data-cloud statistics via online assignment updates, thereby preventing unbounded rule-base growth and preserving a compact structure; For parameter updating, a kernel learning strategy is employed to map inputs into a high-dimensional feature space, and an improved correntropy criterion is introduced to adaptively down-weight large-error samples, improving robustness against outlier perturbations and non-Gaussian noise. In addition, the computational complexity of the proposed model is analyzed. Experimental results on multiple public datasets demonstrate that CKL-EFS consistently outperforms representative existing methods in both predictive accuracy and stability against non-Gaussian disturbances.1)
1 ①https://biendata.com/competition/kdd_20182)2 ②https://www.kaggle.com/datasets/thomaswrightanderson/river-aire-discharge-time-series3)3 https://www.xjwind.com/ -
图 1 CKL-EFS模型结构及内部驱动原理
Fig. 1 CKL-EFS model architecture and internal driving principles
图 2 Lorenz-96数据集: 不同模型的规则数量和运行时间
Fig. 2 Lorenz-96 dataset: Number of rules and runtime for different models
图 3 伦敦气象数据集: 不同预测因子的波动趋势和非高斯噪声分析
Fig. 3 London meteorological dataset: Fluctuation trend and non-Gaussian noise analysis of different predictors
图 4 伦敦气象数据集: CKL-EFS的1步超前预测结果
Fig. 4 London meteorological dataset: 1-step-ahead prediction results of CKL-EFS
图 5 利兹水文数据集: 非高斯噪声强度分析和特征可视化分析
Fig. 5 Leeds hydrological dataset: non-Gaussian noise intensity analysis and feature visualization analysis
图 6 利兹水文数据集: 不同模型的1步预测散点图
Fig. 6 Leeds hydrological dataset: 1-step-ahead prediction scatter plot by different models
图 7 利兹水文数据集: 多步超前预测的NRMSE、SMAPE和MAE
Fig. 7 Leeds hydrological dataset: NRMSE, SMAPE and MAE of multi-step-ahead prediction
图 8 新疆风电数据集: 风功率波动及非高斯噪声强度分析
Fig. 8 Xinjiang wind power dataset: Analysis of power fluctuations and non-Gaussian noise intensity
表 1 数据集的参数设置
Table 1 Parameter settings of datasets
数据集 维度 样本数 训练集 测试集 Lorenz-96混沌系统 40 2 400 1 800 600 伦敦气象数据集 15 4 000 3 000 1 000 利兹水文数据集 24 1 200 1 000 200 新疆风电数据集 15 3 000 2 500 500 
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表 2 Lorenz-96数据集: 在不同非高斯扰动下的预测性能(度量标准: MAE)
Table 2 Lorenz-96 dataset: Prediction performance under various non-Gaussian disturbance (metric: MAE)
噪声类型 模型 值($ \times \; 10^{-1} $) 理想观测
(无噪声)CEFNS[29] 5.88 PSO-ALMMo*[40] 5.01 RMCEFS[25] 5.45 MEEFIS[21] 3.01 ePL-KRLS-DISCO[22] 3.15 EPL-KGLA[30] 4.36 MIIPSO-EFS[18] 3.75 CKL-EFS 3.59 重尾脉冲型非高斯扰动
(加$\alpha $-稳定噪声)CEFNS[29] 5.92 PSO-ALMMo*[40] 7.48 RMCEFS[25] 5.73 MEEFIS[21] 6.61 ePL-KRLS-DISCO[22] 6.47 EPL-KGLA[30] 5.29 MIIPSO-EFS[18] 6.41 CKL-EFS 3.72 强相关低频非高斯扰动
(加粉红噪声)CEFNS[29] 6.13 PSO-ALMMo*[40] 7.98 RMCEFS[25] 5.77 MEEFIS[21] 6.85 ePL-KRLS-DISCO[22] 7.32 EPL-KGLA[30] 5.41 MIIPSO-EFS[18] 7.13 CKL-EFS 3.97
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表 3 Lorenz-96数据集: 非高斯性增强时的性能变化规律(以$ \alpha $-稳定噪声为例, 度量标准: MAE)
Table 3 Lorenz-96 dataset: Performance trend as non-Gaussianity increases (via $ \alpha $-stable noise, metric: MAE)
模型 稳定性指数$ \alpha_0 $ 2.0 1.8 1.6 1.4 1.2 CEFNS[29] 5.57 5.92 6.44 7.11 8.15 PSO-ALMMo*[40] 5.35 7.48 8.35 9.22 10.31 RMCEFS[25] 5.60 5.73 6.10 6.72 7.54 MEEFIS[21] 4.15 6.61 7.25 8.12 9.05 ePL-KRLS-DISCO[22] 4.02 6.47 7.05 7.88 8.96 EPL-KGLA[30] 4.95 5.29 5.70 6.22 6.95 MIIPSO-EFS[18] 4.40 6.41 7.20 8.05 9.28 CKL-EFS 3.62 3.72 3.85 4.10 4.45
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表 4 伦敦气象数据集: 不同模型的1步超前预测结果
Table 4 London meteorological dataset: 1-step-ahead prediction results by different models
性能指标 模型 温度 风速 NRMSE ($ \times \; 10^{-1}$) CEFNS[29] 2.53 3.11 PSO-ALMMo*[40] 1.71 8.03 RMCEFS[25] 1.86 2.91 MEEFIS[21] 0.75 3.47 ePL-KRLS-DISCO[22] 1.68 4.35 EPL-KGLA[30] 1.13 2.80 MIIPSO-EFS[18] 1.39 4.52 CKL-EFS 0.54 2.13 SMAPE ($\times \; 10^{-1} $) CEFNS[29] 4.32 2.19 PSO-ALMMo*[40] 5.03 5.13 RMCEFS[25] 3.93 2.21 MEEFIS[21] 1.84 3.02 ePL-KRLS-DISCO[22] 2.28 2.87 EPL-KGLA[30] 2.12 1.43 MIIPSO-EFS[18] 2.75 2.52 CKL-EFS 0.87 0.74 规则数量 CEFNS[29] 3 5 PSO-ALMMo*[40] 9 9 RMCEFS[25] 6 10 MEEFIS[21] 379 329 ePL-KRLS-DISCO[22] 2 4 EPL-KGLA[30] 2 4 MIIPSO-EFS[18] 12 8 CKL-EFS 15 10 运行时间(s) CEFNS[29] 28.77 35.12 PSO-ALMMo*[40] 527.69 172.41 RMCEFS[25] 8.61 14.03 MEEFIS[21] 11.26 6.59 ePL-KRLS-DISCO[22] 50.73 64.32 EPL-KGLA[30] 13.36 19.65 MIIPSO-EFS[18] 363.21 81.05 CKL-EFS 30.42 22.97
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表 5 利兹水文数据集: 不同模型的1步超前预测结果
Table 5 Leeds hydrological dataset: 1-step-ahead prediction results by different models
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表 6 新疆风电数据集: 不同模型的1步超前预测结果
Table 6 Xinjiang wind power dataset: 1-step-ahead prediction results by different models
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表 7 新疆风电数据集: CKL-EFS在不同超参数扰动下的SMAPE变化
Table 7 Xinjiang wind power dataset: SMAPE variation of CKL-EFS under different hyperparameter perturbations
设置 SMAPE ($ \times \; 10^{-1} $) SMAPE差值 相对变化 最优参数 2.11 – – 核带宽$ \sigma \uparrow 20 \% $ 2.17 0.06 +2.8% 正则化因子$ \gamma \uparrow 20\% $ 2.14 0.03 +1.4% 形状参数$ \alpha \uparrow 20\% $ 2.18 0.07 +3.3% 尺度参数$ \beta \uparrow 20\% $ 2.16 0.05 +2.4% 核带宽$ \sigma \downarrow 20\% $ 2.15 0.04 +1.9% 正则化因子$ \gamma \downarrow 20\% $ 2.13 0.02 +0.9% 形状参数$ \alpha \downarrow 20\% $ 2.18 0.07 +3.3% 尺度参数$ \beta \downarrow 20\% $ 2.16 0.05 +2.4%
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