In quantitative finance, factor models are essential for portfolio construction and risk management. Suppose you're a quant researcher tasked with building a statistical factor-based equity risk model. First, walk me through how you would leverage Principal Component Analysis (PCA) to identify and extract these statistical factors. What are the key advantages and potential limitations of using PCA for this application? Then, looking beyond the model's theoretical construction, what critical engineering and financial business considerations would you take into account when deploying such a model in a production environment?

在量化投资领域，因子模型是构建投资组合和管理风险的核心工具。你作为一名量化研究员，被要求构建一个基于统计因子的股票风险模型。请你阐述如何利用主成分分析（PCA）来识别和提取这些统计因子，并详细说明PCA在此应用中的优势和潜在局限性。在实际部署模型时，你还会考虑哪些关键的工程和金融业务方面的问题？