## 1 Introduction
For example, [PERSON] et al. [18] proposed a two-dimensional deep correlated representation learning (2D-DCRL) approach for nonlinear batch process monitoring. Another example is [PERSON] et al. [19], who introduced a sparsity-constrained optimization framework within the DCCA model to improve fault detection performance in nonlinear industrial processes. These approaches eliminate the need for computationally intensive kernel operations and the dependency on predefined kernel functions and parameters. However, their reliance on static training datasets and the use of feedforward neural networks limit their ability to capture temporal features in sequential data. Their static assumption and deficient temporal modeling restrict their effectiveness in monitoring processes where new fault types emerge or system characteristics change dynamically over time. To address these challenges, this study attempts to enhance the DCCA framework by incorporating a novel network structure that combines the Broad Learning System (BLS) with the Pyraformer model to enhance fault detection in dynamic and complex nonlinear processes. The former is known for its efficiency in handling high-dimensional data with minimal computational cost, while the latter excels in capturing long-range temporal dependencies through its attention mechanism. This integration enhances DCCA frameworks by enabling real-time adaptation to dynamic industrial data streams. It provides a robust solution to capture spatial and temporal correlations and addresses challenges in monitoring fault patterns that evolve in high-dimensional industrial environments. Figure 1 displays the overall architecture of this integrated framework and illustrates the flow of data through the BLS and Pyraformer components and their interaction within the DCCA structure. To validate the effectiveness of the proposed method, we performed an ablation study and applied the proposed method to the TE process, CSTR process, and injection molding process. To demonstrate our method's superiority in detection accuracy, we compared our method with other temporal models (e.g., the LSTM-DCCA and Autoformer-DCCA) in terms of the Fault Detection Rate (FDR) and False Alarm Rate (FAR). The contributions of this study rest on the following aspects: (1) We extend the DCCA framework by integrating the Pyraformer model, which employs its Pyramidal Attention Mechanism to capture long-range temporal dependencies and multi-scale temporal patterns. This integration improves the modeling of dynamic industrial processes while ensuring computational efficiency and suitability for high-dimensional time-series data. (2) We propose an incremental learning mechanism within the DCCA framework based on the BLS. This mechanism allows the model to adapt to evolving industrial data streams without requiring retraining. It enhances fault detection by updating feature mappings and effectively captures spatial correlations. (3) We validate the proposed Incremental Pyraformer-DCCA framework through extensive experiments on the Tennessee Eastman (TE) process, the Continuous Stirred-Tank Reactor (CSTR) system, and the injection molding process. The results demonstrate its adaptability and reliability for complex real-time industrial tasks that require substantial processing power. The rest of this paper is organized as follows: Section 2 introduces the background of Canonical Correlation Analysis (CCA), including the principles of linear CCA and the foundations of DCCA. Section 3 presents the Pyraformer model's structure and the incremental learning methods employed within the Broad Learning System (BLS). Section 4 details the application of the proposed Incremental Pyraformer-DCCA framework to the Tennessee Eastman (TE) process, Continuous Stirred-Tank Reactor (CSTR) system, and injection molding process, followed by the Conclusion in Section 5. Figure 1: Outline of the present study.