Exploring Stationarity From the Frequency-domain Perspective: Frequency Stationary Subspace Analysis and Its Application to BFIP
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摘要: 非平稳信号的平稳分量提取是信号处理与过程监测的核心问题. 现有平稳子空间分析方法局限于时域二阶平稳性, 难以刻画频域非平稳特征. 为此, 提出频域平稳子空间分析方法(FreqSSA). 该方法以功率谱密度时不变性定义频域平稳性, 揭示时域与频域平稳性的等价及包含关系; 进而构建谱幅度、相位与结构三类频域非平稳统计矩阵, 建立时频联合非平稳统计框架, 并以最小化非对角元素平方和为目标函数实现子空间分离; 最后采用联合对角化策略, 实现正交解混矩阵的高效迭代优化. 真实高炉炼铁数据的实验结果表明, FreqSSA在平稳分量提取精度与过程监测性能上均优于现有主流方法.Abstract: Stationary component extraction underpins signal processing and process monitoring, yet conventional stationary subspace analysis remains confined to time-domain second-order constraints, rendering it inadequate for frequency-dependent nonstationarities. We introduce Frequency-Domain Stationary Subspace Analysis (FreqSSA), which establishes frequency-domain stationarity via power spectral density time-invariance and formalizes its structural relations with time-domain stationarity. By constructing spectral magnitude, phase, and structure matrices within a unified time–frequency statistical framework, FreqSSA achieves subspace separation through off-diagonal energy minimization, with efficient optimization enabled by joint diagonalization employing analytically derived rotation angles. Validation on industrial blast furnace data confirms FreqSSA＇s superior extraction accuracy and monitoring performance over existing approaches.
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图 2 各种方法(ICA、SSA和SWNMF)的频域投影与平稳性检验表现
Fig. 2 Frequency-domain projections and stationarity test performances across different methods (ICA, SSA and SWNMF)
图 3 各种方法(SSAE-ISFA、AD-TFM-AT和FreqSSA)的频域投影与平稳性检验表现
Fig. 3 Frequency-domain projections and stationarity test performances across different methods (SSAE-ISFA, AD-TFM-AT and FreqSSA)
表 1 高炉炼铁过程监测变量说明
Table 1 Description of monitored variables in blast furnace ironmaking process
编号 描述 单位 传感器类型 物理意义 $ {\cal{V}}_1 $ 冷风流量 $ 10^4 \text{m}^3/\text{s} $ 孔板流量计 反映鼓风系统送入高炉的冷风总量, 直接影响炉内燃烧强度与温度分布 $ {\cal{V}}_2 $ 冷风压力 MPa 压阻式压力变送器 表征鼓风系统克服炉内阻力所需的送风压力, 与炉况透气性密切相关 $ {\cal{V}}_3 $ 总压降 kPa 差压变送器 反映气体从高炉下部上升至上部过程中的总体阻力损失, 直接关联炉内料柱透气性 $ {\cal{V}}_4 $ 热风压力 MPa 压阻式压力变送器 表征经热风炉预热后的送风压力, 是衡量热风系统工作状态的核心指标 $ {\cal{V}}_5 $ 实际风速 m/s 皮托管风速仪 反映风口处鼓风射流对炉缸内物料的搅拌强度, 影响炉缸活跃程度 $ {\cal{V}}_6 $ 透气性指数 $ - $ 计算量 综合反映炉内料柱允许气体通过的难易程度, 定义为冷风流量与总压降之比 $ {\cal{V}}_7 $ 阻力指数 $ - $ 计算量 表征气体通过料柱时单位流量所遇到的阻力, 定义为总压降与冷风流量平方之比 注: 所有变量的采样间隔均为60s. 
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表 2 各数据集描述
Table 2 Description of each dataset
数据集 样本数 时长 时间段 模型训练集$ T_1 $ 7000 4.86 d 1月3日00:00 $ \sim $ 1月7日20:40 正常测试集$ N_1 $ 1000 16.67 h 1月15日08:00 $ \sim $ 1月16日00:40 故障测试集$ F_1 $ 680 11.33 h 1月25日14:00 $ \sim $ 1月26日01:20 故障测试集$ F_2 $ 550 9.17 h 2月10日09:00 $ \sim $ 2月10日18:10 故障测试集$ F_3 $ 470 7.83 h 3月5日13:00 $ \sim $ 3月5日20:50
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表 3 测试数据集(N1、F1、F2和F3)的误报率和漏报率监测性能表现
Table 3 Monitoring performance of the false alarm rate and missed detection rate for the test data sets (N1, F1, F2 and F3)
方法 指标 N1‡ F1 F2 F3 ‡ ICA 误报率 7.60% − − − 漏报率 − 36.61% 14.89% 61.25% SSA 误报率 2.50% − − − 漏报率 − 21.12% 7.08% 30.74% SWNMF 误报率 5.10% − − − 漏报率 − 34.64% 25.74% 51.30% SSAE-ISFA 误报率 4.70% − − − 漏报率 − 66.57% 25.28% 68.86% AD-TFM-AT 误报率 4.40% − − − 漏报率 − 40.33% 20.28% 70.29% FreqSSA 误报率 0% − − − 漏报率 − 16.87% 5.40% 22.00% 注: 对于正常测试集N1, 仅评估误报率指标, 该值越低表示性能越好; 对于故障验证数据集, 共评估3项指标, 这些指标取值越低表示性能越好. ‡表示各数据集中的正常样本与故障样本均由现场专家验证. “−”表示对应指标无需计算.
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表 4 各FreqSSA变体的消融实验表现
Table 4 Ablation experiment performance of each FreqSSA variant
方法 指标 N1 F1 F2 F3 FreqSSA-A 误报率 1.60% −† − − 漏报率 − 20.50% 8.54% 28.74% FreqSSA-AF 误报率 0.70% − − − 漏报率 − 18.46% 7.57% 24.83% FreqSSA-AS 误报率 0.40% − − − 漏报率 − 17.05% 6.83% 22.81% FreqSSA-GD 误报率 0.20% − − − 漏报率 − 17.77% 6.60% 22.53% FreqSSA 误报率 0% − − − 漏报率 − 16.87% 5.40% 22.00%
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表 5 模型参数取值
Table 5 Model parameter values
参数 取值范围 最优值 窗长$ L $ (64, 96, 128, 160) 128 滑动步长$ S $ (32, 48, 64, 80) 64 FFT点数$ F $ (128, 192, 256, 320) 256 SC维度$ k $ (1, 2, 3, 4) 2
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表 6 所提出方法在测试数据集中的参数敏感性表现
Table 6 Parameter sensitivity performance of the proposed method in the test dataset
参数 取值 N1|误报率 F1|漏报率 F2|漏报率 F3|漏报率 窗长 64 0.30% 18.27% 6.94% 23.12% 96 0.15% 17.33% 6.28% 22.33% 128 0% 16.87% 5.40% 22.00% 160 0.20% 17.52% 6.32% 22.56% 滑动步长 32 0.15% 16.92% 6.43% 21.57% 48 0.15% 17.33% 6.13% 22.00% 64 0% 16.87% 5.40% 22.00% 80 0.20% 17.00% 5.93% 21.75% FFT数 128 0.15% 17.44% 6.13% 22.48% 192 0.05% 17.12% 5.93% 21.88% 256 0% 16.87% 5.40% 22.00% 320 0.10% 16.87% 5.85% 21.68% SC维度 1 0% 26.64% 24.92% 39.25% 2 0% 16.87% 5.40% 22.00% 3 3.20% 14.93% 4.64% 20.65% 4 8.60% 12.59% 3.92% 19.80%
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