在現今大數據的世代中為了因應各種不同的資料型態，新的資料處理方法不斷推陳出新。而統計、電腦科學，以及資訊等各領域專家對於研發新方法的背景與對於其他領域既有方法的理解不完全，因此可能導致引用上的錯誤並使得結果不可靠。在現代研究結果高度流通的環境中，如此的錯誤結果可能在未經驗證的情況下再度被誤用。Sigg & Buhmann (2008) 提出應用EM演算法 (Expectation – Maximization Algorithm)的非負稀疏主成份分析，能夠將高維度的資料進行快速的降維處理，在網路上被廣泛引用。但作者可能不熟悉統計領域的多變量分析理論，故混淆因子分析模型與主成份分析的觀念導致文獻數學公式的推導上有誤。本研究將對於作者錯誤的數學推導進行修正，並應用多變量統計分析(Johnson & Wichern，2007; Applied Multivariate Statistical Analysis)書中的傳統因子分析方法，提出使平均重建誤差平方和最小化的方法。接著使用修改Sigg & Buhmann (2008)文獻的方法後，衍伸出四種不同的因子分析組合，並使用模擬數據做比較，評估何種組合下的誤差平方和較小。統計模擬中，假設資料來自於多變量常態分配。在固定母體期望值下，改變未觀測到的共同因子個數與維度大小，來觀察四種組合所得出的平均誤差平方和何者較小。在實例分析中，我們針對臉部影像的高維度資料進行操作，以了解我們修改過後的方法在高維度資料降維的表現。

In the big data era nowadays, new data processing methods continuously emerge to address various data types. However, experts from fields like statistics, computer science, and information may have an incomplete understanding of the background behind developing new methods and existing methods in other fields. This knowledge gap can lead to misleading usage and unreliable results. In this study, we address an error in the mathematical derivation of the widely cited Expectation-Maximization for Sparse and Non-Negative Principal Component Analysis (emPCA) method proposed by Sigg & Buhmann (2008). We correct these errors by applying traditional factor analysis methods from the book "Applied Multivariate Statistical Analysis" by Johnson & Wichern (2007) to minimize the Mean Squared Error (MSE) and Mean Reconstruction Error (MRE). We extend the modified approach to derive four different factor analysis combinations and evaluate their performance using simulated data. By examining the MSEs resulting from these combinations, we determine the most effective approach. Additionally, we apply our modified method to the high-dimensional Olivetti face dataset to assess its performance in dimensionality reduction.

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