影像資料在現今社會中相當常見，然而影像資料的高維度性質使其在分析和處理上遇到很大的挑戰，如何降低資料維度將是一個關鍵問題。為了解決這些問題，降維方法被廣泛應用。近年來保留影像資料之原始張量結構的Kronecker包絡主成分分析(KEPCA)在理論與應用上都受到高度重視。在本論文中我們將建立適用於KEPCA的模型選擇方法。在尖峰模型假設下，我們分別推導了KEPCA在樣本數大於或小於等於參數個數情境下之AIC與BIC。模擬實驗與實際資料分析的結果說明了當資料符合或無嚴重偏離Kronecker乘積結構時，KEPCA在兩種準則上的表現都優於PCA；當資料結構偏離KEPCA時，根據不同的偏離程度最終兩種準則皆會選擇PCA。;Image data is ubiquitous in today′s society. However, the high dimensionality of image data poses significant challenges in analysis. Dimension reduction techniques have been widely employed to address these issues. In recent years, Kronecker envelope Principal Component Analysis (KEPCA), which preserves the original tensor structure of image data, has gained attention in both theory and applications. In this thesis, we propose model selection methods for KEPCA. Under the assumption of spiked models, we derive the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for KEPCA in scenarios where the number of samples is greater than, or less than or equal to, the number of parameters, respectively. Simulation studies and empirical data studies confirm that when the data structure is not far from Kronecker product structure, KEPCA outperforms PCA under both criteria. However, when the data deviates from the Kronecker product structure, PCA is preferred instead of KEPCA.
