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摘要: 数据流分类是数据流挖掘领域一项重要研究任务, 目标是从不断变化的海量数据中捕获变化的类结构. 目前, 几乎没有框架可以同时处理数据流中常见的多类非平衡、概念漂移、异常点和标记样本成本高昂问题. 基于此, 提出一种非平衡数据流在线主动学习方法(Online active learning method for imbalanced data stream, OALM-IDS). AdaBoost是一种将多个弱分类器经过迭代生成强分类器的集成分类方法, AdaBoost.M2引入了弱分类器的置信度, 此类方法常用于静态数据. 定义了基于非平衡比率和自适应遗忘因子的训练样本重要性度量, 从而使AdaBoost.M2方法适用于非平衡数据流, 提升了非平衡数据流集成分类器的性能. 提出了边际阈值矩阵的自适应调整方法, 优化了标签请求策略. 将概念漂移程度融入模型构建过程中, 定义了基于概念漂移指数的自适应遗忘因子, 实现了漂移后的模型重构. 在6个人工数据流和4个真实数据流上的对比实验表明, 提出的非平衡数据流在线主动学习方法的分类性能优于其他5种非平衡数据流学习方法.Abstract: Data stream classification is an important research task in the field of data stream mining, which aims to capture changing class structures from the ever-changing massive data. At present, almost no frameworks can simultaneously address the common problems in data stream, such as multi-class imbalance, concept drift, outlier and the exorbitant costs associated with labeling the unlabeled samples. In this paper, we propose an online active learning method for imbalanced data stream (OALM-IDS). AdaBoost is an ensemble classification method that iteratively generates a strong classifier from multiple weak classifiers. AdaBoost.M2 further introduces the confidence degree of weak classifiers, which is suitable for static data. In the method, we firstly define an importance measure of training sample based on imbalanced ratio and adaptive forgetting factor, which makes the AdaBoost.M2 method applying for imbalanced data stream and improves the performance of ensemble classifier. Then, we propose an adaptive adjustment method of marginal threshold matrix, which optimizes the label request strategy. Finally, we define an adaptive forgetting factor based on the concept drift index by bringing the degree of concept drift into the construction process of model, which realizes the model reconstruction after drift. Comparative experiments on six artificial data streams and four real data streams show that the classification performance of the online active learning method is better than those of the existing five learning methods for imbalance data stream.
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Key words:
- Active learning /
- data stream classification /
- multi-class imbalance /
- concept drift
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表 1 数据流的特征
Table 1 Data stream feature
数据流 样本数 特征数 类别数 类分布 漂移次数 异常点 ${\rm{DS} }_{1}$ 200000 21 5 (0.2, 0.2, 0.2, 0.2, 0.2) 0 0 ${\rm{DS}}_{2}$ 200000 21 5 (0.2, 0.2, 0.2, 0.2, 0.2) 3 10 ${\rm{DS}}_{3}$ 200000 21 5 (0.1, 0.3, 0.4, 0.2, 0.1) 0 0 ${\rm{DS}}_{4}$ 200000 21 5 (0.1, 0.3, 0.4, 0.2, 0.1) 3 10 ${\rm{DS}}_{5}$ 200000 21 5 (0.1, 0.3, 0.4, 0.2, 0.1), (0.4, 0.2, 0.1, 0.1, 0.2) 0 0 ${\rm{DS}}_{6}$ 200000 21 5 (0.1, 0.3, 0.4, 0.2, 0.1), (0.4, 0.2, 0.1, 0.1, 0.2) 3 10 Kddcup$99\_10\%$ 494000 42 23 — — — Statlog 570000 10 7 — — — IoT 663000 115 11 — — — HAR 10299 561 6 — — — 
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表 2 6种算法的准确率
Table 2 Precision values of six algorithms
数据流 LB BOLE ${\rm{ARF}}_{RE}$ OALE CALMID OALM-IDS ${\rm {DS} }_{1}$ $94.56\pm0.12$ $\boldsymbol{95.61} \;\pm \boldsymbol {0.11} $ $93.54\pm0.13$ $89.78\pm0.21$ $94.76\pm0.16$ $95.48\pm0.15$ ${\rm {DS}}_{2}$ $92.27\pm0.17$ $92.44\pm0.14$ $91.04\pm0.19$ $88.31\pm0.23$ $92.81\pm0.13$ ${\boldsymbol{93.94} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12} }$ ${\rm {DS}}_{3}$ $88.39\pm0.22$ $89.52\pm0.14$ $90.95\pm0.13$ $88.83\pm0.16$ $92.57\pm0.13$ ${\boldsymbol{93.72} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13} }$ ${\rm {DS}}_{4}$ $86.55\pm0.31$ $88.68\pm0.26$ $89.89\pm0.23$ $86.29\pm0.29$ $91.31\pm0.18$ ${\boldsymbol{92.18} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21} }$ ${\rm {DS}}_{5}$ $85.64\pm0.29$ $87.04\pm0.34$ $89.61\pm0.51$ $88.83\pm0.21$ $91.13\pm0.21$ ${\boldsymbol{92.92} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.16} }$ ${\rm {DS}}_{6}$ $82.10\pm0.69$ $83.15\pm0.73$ $86.54\pm0.72$ $83.42\pm0.55$ $90.64\pm0.42$ ${\boldsymbol{92.41} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21} }$ Kddcup$99\_10\%$ $83.87\pm0.43$ $81.09\pm0.56$ $85.48\pm0.65$ $81.01\pm0.36$ $92.06\pm0.19$ ${\boldsymbol{92.07} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18} }$ Statlog $64.55\pm0.31$ $63.78\pm0.61$ $79.97\pm0.39$ $73.78\pm0.43$ $85.40\pm0.34$ ${\boldsymbol{85.68} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.33} }$ IoT $64.03\pm0.48$ $61.54\pm0.43$ $66.66\pm0.53$ $55.81\pm0.51$ $70.85\pm0.54$ ${\boldsymbol{73.12} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.38} }$ HAR $61.63\pm0.53$ $59.76\pm0.46$ $63.22\pm0.49$ $55.16\pm0.69$ $68.64\pm0.71$ ${\boldsymbol{69.98} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.51} }$
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表 3 6种算法的召回率
Table 3 Recall values of six algorithms
数据流 LB BOLE ${\rm{ARF}}_{RE}$ OALE CALMID OALM-IDS ${\rm{DS}}_{1}$ $95.37\pm0.18$ $95.96\pm0.13$ $93.39\pm0.11$ $90.13\pm0.13$ $95.91\pm0.11$ ${\boldsymbol{96.14} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12} }$ ${\rm{DS}}_{2}$ $92.39\pm0.21$ $92.28\pm0.35$ $91.35\pm0.26$ $89.45\pm0.18$ $92.51\pm0.15$ ${\boldsymbol{94.08} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.14} }$ ${\rm{DS}}_{3}$ $87.55\pm0.19$ $88.19\pm0.22$ $86.14\pm0.21$ $88.52\pm0.22$ $90.55\pm0.13$ ${\boldsymbol{92.52} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13} }$ ${\rm{DS}}_{4}$ $84.57\pm0.36$ $86.73\pm0.29$ $87.47\pm0.28$ $83.05\pm0.31$ $89.89\pm0.21$ ${\boldsymbol{92.44} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18} }$ ${\rm{DS}}_{5}$ $84.14\pm0.43$ $86.44\pm0.49$ $87.26\pm0.69$ $83.26\pm0.36$ $90.25\pm0.18$ ${\boldsymbol{91.16} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13} }$ ${\rm{DS}}_{6}$ $83.98\pm1.13$ $81.87\pm0.91$ $84.56\pm1.31$ $78.87\pm0.69$ $90.46\pm0.13$ ${\boldsymbol{90.71} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21} }$ Kddcup$99\_10\%$ $60.82\pm0.71$ $62.75\pm0.64$ $58.17\pm1.32$ $58.44\pm1.63$ $61.88\pm0.43$ ${\boldsymbol{63.71} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.37} }$ Statlog $61.39\pm0.91$ $50.92\pm1.32$ $54.36\pm1.11$ $51.20\pm1.34$ $59.52\pm0.63$ ${\boldsymbol{63.12} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.39} }$ IoT $40.73\pm2.14$ $42.29\pm1.58$ $39.35\pm1.89$ $40.42\pm2.15$ $48.04\pm1.04$ ${\boldsymbol{51.26} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.81} }$ HAR $61.64\pm1.18$ $60.57\pm0.97$ $57.91\pm1.43$ $54.11\pm1.36$ $65.53\pm0.76$ ${\boldsymbol{66.57} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.46} }$
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表 4 6种算法的F1值
Table 4 F1 values of six algorithms
数据流 LB BOLE ${\rm{ARF}}_{RE}$ OALE CALMID OALM-IDS ${\rm{DS}}_{1}$ $94.96\pm0.11$ ${\boldsymbol{95.80} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.10} }$ $93.42\pm0.13$ $89.93\pm0.15$ $95.33\pm0.11$ ${\boldsymbol{95.80} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.10} }$ ${\rm{DS}}_{2}$ $92.32\pm0.16$ $92.34\pm0.13$ $91.18\pm0.15$ $88.85\pm0.21$ $92.65\pm0.13$ ${\boldsymbol{94.01} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12} }$ ${\rm{DS}}_{3}$ $87.91\pm0.20$ $88.81\pm0.24$ $88.11\pm0.36$ $88.67\pm0.20$ $91.50\pm0.16$ ${\boldsymbol{93.07} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.14} }$ ${\rm{DS}}_{4}$ $85.35\pm0.42$ $87.38\pm0.36$ $88.42\pm0.51$ $84.50\pm0.33$ $90.51\pm0.21$ ${\boldsymbol{92.29} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.20} }$ ${\rm{DS}}_{5}$ $84.85\pm0.41$ $86.67\pm0.43$ $88.30\pm0.46$ $85.36\pm0.48$ $90.62\pm0.21$ ${\boldsymbol{91.93} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18} }$ ${\rm{DS}}_{6}$ $82.97\pm0.87$ $82.43\pm0.71$ $85.35\pm0.91$ $80.59\pm0.63$ $90.46\pm0.39$ ${\boldsymbol{91.53} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.31} }$ Kddcup$99\_10\%$ $73.12\pm0.55$ $72.47\pm0.63$ $72.01\pm0.46$ $72.81\pm0.51$ $73.56\pm0.33$ ${\boldsymbol{74.65} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.20} }$ Statlog $66.18\pm0.83$ $54.32\pm1.91$ $63.85\pm1.03$ $63.42\pm0.98$ $74.42\pm0.36$ ${\boldsymbol{75.19} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.31} }$ IoT $47.01\pm1.24$ $48.40\pm0.96$ $47.34\pm1.89$ $44.94\pm1.36$ $54.26\pm0.65$ ${\boldsymbol{56.73} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.67} }$ HAR $59.93\pm0.91$ $58.81\pm1.21$ $58.52\pm0.79$ $54.43\pm1.13$ $65.43\pm0.63$ ${\boldsymbol{67.76} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.58} }$
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表 5 6种算法的Kappa系数值
Table 5 Kappa coefficient values of six algorithms
数据流 LB BOLE ${\rm{ARF}}_{RE}$ OALE CALMID OALM-IDS ${\rm{DS}}_{1}$ $90.17\pm0.12$ $91.18\pm0.14$ $90.59\pm0.16$ $85.47\pm0.21$ $90.48\pm0.19$ $\boldsymbol{91.31\pm0.12}$ ${\rm{DS}}_{2}$ $88.85\pm0.19$ $88.14\pm0.23$ $87.91\pm0.39$ $83.18\pm0.56$ $89.97\pm0.31$ ${\boldsymbol{90.66} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.23} }$ ${\rm{DS}}_{3}$ $85.25\pm0.22$ $85.86\pm0.38$ $86.68\pm0.29$ $83.91\pm0.39$ $88.91\pm0.26$ ${\boldsymbol{89.93} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21} }$ ${\rm{DS}}_{4}$ $84.15\pm0.55$ $86.04\pm0.63$ $87.14\pm0.66$ $83.42\pm0.71$ $88.92\pm0.33$ ${\boldsymbol{89.33} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.36} }$ ${\rm{DS} }_{5}$ $83.85\pm0.77$ $85.83\pm0.69$ $86.45\pm0.81$ $86.67\pm0.70$ $88.57\pm0.31$ ${\boldsymbol{89.12} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.29} }$ ${\rm{DS} }_{6} $ $81.49\pm1.12$ $82.98\pm1.69$ $84.15\pm1.87$ $79.92\pm1.48$ $89.01\pm0.41$ ${\boldsymbol{89.73} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.28} }$ Kddcup$99\_10\% $ $80.93\pm0.67$ $75.62\pm1.13$ $79.32\pm1.32$ $78.31\pm0.91$ $83.32\pm0.26$ ${\boldsymbol{85.83} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18} }$ Statlog $58.71\pm1.42$ $61.43\pm1.18$ $73.72\pm0.93$ $71.21\pm1.24$ $79.39\pm0.46$ ${\boldsymbol{80.11} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.19} }$ IoT $67.53\pm1.54$ $65.02\pm1.89$ $68.99\pm2.14$ $59.53\pm2.12$ $71.65\pm0.71$ ${\boldsymbol{73.29} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.68} }$ HAR $60.49\pm1.12$ $60.01\pm1.38$ $61.86\pm1.13$ $56.75\pm2.03$ $68.52\pm0.76$ ${\boldsymbol{69.64} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.71} }$
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表 6 参数$\theta $对OALM-IDS的影响
Table 6 Effect of parameter $\theta $ to OALM-IDS
数据流 $\theta $ $b$ 准确率 召回率 F1值 Kappa系数值 0.4 0.17143 $94.21\pm0.16 $ $93.18\pm0.12$ $94.13\pm0.11$ $90.11\pm0.12$ $\rm{DS}_{1}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.180\,26} }$ ${\boldsymbol{95.48} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.15} }$ ${\boldsymbol{96.14}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12}}$ ${\boldsymbol{95.80}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.10}}$ ${\boldsymbol{91.31}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12}}$ 0.6 0.19782 $95.03\pm0.15 $ $93.19\pm0.12$ $95.16\pm0.10$ $91.01\pm0.12$ 0.4 0.17136 $93.01\pm0.12 $ $92.81\pm0.16$ $93.04\pm0.13$ $89.09\pm0.26$ $\rm{DS}_{2}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.191\,78} }$ ${\boldsymbol{93.94}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12}}$ ${\boldsymbol{94.08}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.14}}$ ${\boldsymbol{94.01}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.12}}$ ${\boldsymbol{90.66}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.23}}$ 0.6 0.20000 $93.18\pm0.13 $ $93.16\pm0.14$ $93.75\pm0.12$ $90.07\pm0.23$ 0.4 0.17821 $93.24\pm0.13 $ $92.05\pm0.13$ $92.54\pm0.16$ $88.56\pm0.22$ $\rm{DS}_{3}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.195\,12} }$ ${\boldsymbol{93.72}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13}}$ ${\boldsymbol{92.52}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13}}$ ${\boldsymbol{93.07}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.14}}$ ${\boldsymbol{89.93}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21}}$ 0.6 0.20000 $93.43\pm0.13 $ $92.24\pm0.13$ $92.10\pm0.14$ $88.71\pm0.21$ 0.4 0.18423 $91.63\pm0.21 $ $91.34\pm0.18$ $91.76\pm0.20$ $88.54\pm0.38$ $\rm{DS}_{4}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.198\,77} }$ ${\boldsymbol{92.18}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21}}$ ${\boldsymbol{92.44}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18}}$ ${\boldsymbol{92.29}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.20}}$ ${\boldsymbol{89.33}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.36}}$ 0.6 0.20000 $91.06\pm0.21 $ $91.56\pm0.19$ $91.80\pm0.21$ $88.63\pm0.36$ 0.4 0.18002 $92.01\pm0.16 $ $90.46\pm0.13$ $90.76\pm0.18$ $88.42\pm0.29$ $\rm{DS}_{5}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.197\,22} }$ ${\boldsymbol{92.92} }\;{\boldsymbol{\pm}}\;{\boldsymbol{0.16} }$ ${\boldsymbol{91.16}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.13}}$ ${\boldsymbol{91.93}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18}}$ ${\boldsymbol{89.12}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.29}}$ 0.6 0.20000 $92.50\pm0.16 $ $90.76\pm0.13$ $91.21\pm0.19$ $88.56\pm0.30$ 0.4 0.18331 $91.02\pm0.21 $ $89.03\pm0.22$ $90.32\pm0.31$ $88.12\pm0.28$ $\rm{DS}_{6}$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.199\,23} }$ ${\boldsymbol{92.41}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21}}$ ${\boldsymbol{90.71}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.21}}$ ${\boldsymbol{91.53}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.31}}$ ${\boldsymbol{89.73}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.28}}$ 0.6 0.20000 $91.01\pm0.21 $ $89.92\pm0.22$ $90.12\pm0.31$ $89.13\pm0.28$ 0.4 0.18188 $90.59\pm0.18 $ $63.51\pm0.37$ $73.35\pm0.20$ $83.14\pm0.18$ Kddcup$99\_10\%$ ${\boldsymbol{0.5}}$ ${\boldsymbol{0.199\,61} }$ ${\boldsymbol{92.07}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18}}$ ${\boldsymbol{63.71}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.37}}$ ${\boldsymbol{74.65}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.20}}$ ${\boldsymbol{85.83}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.18}}$ 0.6 0.20000 $91.63\pm0.18 $ $63.63\pm0.37$ $74.43\pm0.21$ $85.61\pm0.18$ 0.4 0.19022 $84.75\pm0.33 $ $62.19\pm0.39$ $74.85\pm0.31$ $78.86\pm0.19$ Statlog ${\boldsymbol{0.5}}$ ${\boldsymbol{0.199\,94} }$ ${\boldsymbol{85.68}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.33}}$ ${\boldsymbol{63.12}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.39}}$ ${\boldsymbol{75.19}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.31}}$ ${\boldsymbol{80.11}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.19}}$ 0.6 0.20000 $85.66\pm0.33 $ $63.01\pm0.39$ $75.19\pm0.31$ $79.89\pm0.19$ 0.4 0.19113 $71.21\pm0.38 $ $49.86\pm0.81$ $51.21\pm0.67$ $71.61\pm0.68$ IoT ${\boldsymbol{0.5}}$ ${\boldsymbol{0.196\,84} }$ ${\boldsymbol{73.12}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.38}}$ ${\boldsymbol{51.26}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.81}}$ ${\boldsymbol{56.73}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.67}}$ ${\boldsymbol{73.29}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.68}}$ 0.6 0.20000 $72.11\pm0.39 $ $50.06\pm0.81$ $54.33\pm0.67$ $71.34\pm0.68$ 0.4 0.18634 $66.54\pm0.52 $ $64.32\pm0.48$ $65.05\pm0.59$ $66.81\pm0.72$ HAR ${\boldsymbol{0.5}}$ ${\boldsymbol{0.195\,47} }$ ${\boldsymbol{69.98}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.51}}$ ${\boldsymbol{66.57}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.46}}$ ${\boldsymbol{67.76}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.58}}$ ${\boldsymbol{69.64}}\;{\boldsymbol{\pm}}\;{\boldsymbol{0.71}}$ 0.6 0.20000 $64.32\pm0.52 $ $65.14\pm0.46$ $66.11\pm0.58$ $64.32\pm0.71$
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