Today, in modern industrial processes, thousands of correlated process variables are measured and stored. Dimension reduction techniques are often employed to construct informative features by discarding redundant information. Slow feature analysis is one such technique that extracts the slowly varying patterns from measured data. Oscillatory behaviour is prevalent in process data due to inadequate control loop tuning and external disturbances such as diurnal temperature variation. Extracting these oscillatory patterns is vital in applications such as control loop monitoring, fault diagnosis. Slow feature analysis may not extract oscillating patterns when the signal to noise ratio is low in process data. This paper proposes the complex probabilistic formulation that extracts slow oscillatory features. We also present the Expectation-Maximization algorithm to obtain the optimal parameter estimates. Finally, three case studies are presented to illustrate the efficacy of the proposed formulation in soft sensing and fault detection applications.

今天，在现代工业过程中，测量和存储了数以千计的相关过程变量。降维技术通常用于通过丢弃冗余信息来构建信息特征。慢特征分析就是这样一种技术，它从测量数据中提取缓慢变化的模式。由于控制回路调谐不足和外部干扰（例如昼夜温度变化），过程数据中普遍存在振荡行为。提取这些振荡模式在控制回路监控、故障诊断等应用中至关重要。当过程数据的信噪比低时，慢速特征分析可能无法提取振荡模式。本文提出了提取缓慢振荡特征的复杂概率公式。我们还提出了期望最大化算法来获得最佳参数估计。最后，提出了三个案例研究来说明所提出的公式在软传感和故障检测应用中的有效性。