Abstract To facilitate the understanding and analysis of process conditions, a novel wavelet functional principal component analysis is proposed for monitoring batch processes from the functional perspective. In the proposed method, the variables’ trajectories are taken as smooth functions instead of discrete vectors. To this end, the original discrete variables are transferred into continuous functions using wavelet basis functions in an active way. This can not only highlight the subtle shape differences between the normal and faulty variables trajectories but also easily address the uneven-length issue in practical batch processes. Additionally, without unfolding the operation, the 3D matrix is transferred into the functional matrix directly. The functional principal component analysis method is then performed on the functional space to establish monitoring models. Thanks to the compact-support characteristics of the wavelet functions, the proposed method can be directly applied to within-batch detection without data pre-treatment. A numerical case, a case of the simulated penicillin fermentation process, and a case of the laboratorial injection molding process are given to demonstrate the effectiveness of the proposed method.

中文翻译：

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用于批处理过程监控的小波函数主成分分析

摘要 为了便于对过程条件的理解和分析，提出了一种新的小波函数主成分分析方法，从函数的角度对批处理过程进行监控。在所提出的方法中，变量的轨迹被视为平滑函数而不是离散向量。为此，使用小波基函数以主动的方式将原始离散变量转换为连续函数。这不仅可以突出正常和错误变量轨迹之间的细微形状差异，还可以轻松解决实际批处理过程中的长度不均匀问题。此外，无需展开运算，直接将 3D 矩阵转换为函数矩阵。然后对函数空间进行函数主成分分析，建立监测模型。由于小波函数的紧凑支持特性，所提出的方法可以直接应用于批内检测而无需数据预处理。给出了一个数值案例、一个模拟青霉素发酵过程的案例和一个实验室注塑过程的案例，以证明所提出方法的有效性。