The development of existing facial coding systems, such as the Facial Action Coding System (FACS), relied on manual examination of facial expression videos for defining Action Units (AUs). To overcome the labor-intensive nature of this process, we propose the unsupervised learning of an automated facial coding system by leveraging computer-vision-based facial keypoint tracking. In this novel facial coding system called the Data-driven Facial Expression Coding System (DFECS), the AUs are estimated by applying dimensionality reduction to facial keypoint movements from a neutral frame through a proposed Full Face Model (FFM). FFM employs a two-level decomposition using advanced dimensionality reduction techniques such as dictionary learning (DL) and non-negative matrix factorization (NMF). These techniques enhance the interpretability of AUs by introducing constraints such as sparsity and positivity to the encoding matrix. Results show that DFECS AUs estimated from the DISFA dataset can account for an average variance of up to 91.29 percent in test datasets (CK+ and BP4D-Spontaneous) and also surpass the variance explained by keypoint-based equivalents of FACS AUs in these datasets. Additionally, 87.5 percent of DFECS AUs are interpretable, i.e., align with the direction of facial muscle movements. In summary, advancements in automated facial coding systems can accelerate facial expression analysis across diverse fields such as security, healthcare, and entertainment. These advancements offer numerous benefits, including enhanced detection of abnormal behavior, improved pain analysis in healthcare settings, and enriched emotion-driven interactions. To facilitate further research, the code repository of DFECS has been made publicly accessible.