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Modern industrial processes are continuously monitored by a large number of sensors. Despite having access to large volumes of historical and online sensor data, industrial practitioners still face challenges in the era of Industry 4.0 in effectively utilizing them to perform online process monitoring and fast fault detection and diagnosis. To target these challenges, in this work, we present a novel framework named "FARM" for Fast, Accurate, and Robust online process Monitoring. FARM is a holistic monitoring framework that integrates (a) advanced multivariate statistical process control (SPC) for fast anomaly detection of nonparametric, heterogeneous data streams, and (b) modified support vector machine (SVM) for accurate and robust fault classification. Unlike existing general-purpose process monitoring frameworks, FARM's unique hierarchical architecture decomposes process monitoring into two fault detection and diagnosis, each of which is conducted by targeted algorithms. Here, we test and validate the performance of our FARM monitoring framework on Tennessee Eastman Process (TEP) benchmark dataset. We show that SPC achieves faster fault detection speed at a lower false alarm rate compared to state-of-the-art benchmark fault detection methods. In terms of fault classification diagnosis, we show that our modified SVM algorithm successfully classifies 17 out of 20 of the fault scenarios present in the TEP dataset. Compared with the results of standard SVM trained directly on the original dataset, our modified SVM improves the fault classification accuracy significantly.