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.

Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions
on Automatic Control, 19, 716–723.
Babii, A., Ghysels, E., & Pan, J. (2022). Tensor principal component analysis. arXiv
preprint arXiv:2212.12981.
Bai, Z., Choi, K. P., & Fujikoshi, Y. (2018). Consistency of AIC and BIC in estimating
the number of significant components in high-dimensional principal component
analysis. The Annals of Statistics, 46, 1050–1076.
Chen, C.-M., Zhang, S.-Q., & Chen, Y.-F. (2010). Face recognition based on MPCA. 2010
The 2nd International Conference on Industrial Mechatronics and Automation, 1,
322–325.
Chen, T.-L., Hsieh, D.-N., Hung, H., Tu, I.-P., Wu, P.-S., Wu, Y.-M., Chang, W.-H., &
Huang, S.-Y. (2014a). γ-SUP: A clustering algorithm for cryo-electron microscopy
images of asymmetric particles. The Annals of Applied Statistics, 8, 259–285.
Chen, T.-L., Huang, S.-Y., Hung, H., & Tu, I.-P. (2014b). An introduction to multilinear
principal component analysis. 中國統計學報, 52, 24–43.
Chung, S.-C., Wang, S.-H., Niu, P.-Y., Huang, S.-Y., Chang, W.-H., & Tu, I.-P. (2020).
Two-stage dimension reduction for noisy high-dimensional images and application
to cryogenic electron microscopy. Annals of Mathematical Sciences and Applications,
5, 283–316.
De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). A multilinear singular value
decomposition. SIAM Journal on Matrix Analysis and Applications, 21, 1253–1278.
Huang, S.-H., & Huang, S.-Y. (2021). On the asymptotic normality and efficiency of
kronecker envelope principal component analysis. Journal of Multivariate Analysis,
184, 104761.
Hung, H., Huang, S.-Y., & Ing, C.-K. (2022). A generalized information criterion for
high-dimensional PCA rank selection. Statistical Papers, 63, 1295–1321.
Jain, A. K., Duin, R. P. W., & Mao, J. (2000). Statistical pattern recognition: A review.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 4–37.
Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components
analysis. The Annals of Statistics, 29, 295–327.
Jolliffe, I. (2002). Principal Component Analysis. Springer Verlag. New York.
Khan, A., & Zubair, S. (2020). A machine learning-based robust approach to identify
dementia progression employing dimensionality reduction in cross-sectional MRI
data. 2020 First International Conference of Smart Systems and Emerging Technologies
(SMARTTECH), 237–242.
Samaria, F. S., & Harter, A. C. (1994). Parameterisation of a stochastic model for human
face identification. Proceedings of 1994 IEEE Workshop on Applications of
Computer Vision, 138–142.
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6,
461–464.
Yang, J., Zhang, D., Frangi, A. F., & Yang, J.-Y. (2004). Two-dimensional PCA: A new
approach to appearance-based face representation and recognition. IEEE Transactions
on Pattern Analysis and Machine Intelligence, 26, 131–137.
Ye, J. (2004). Generalized low rank approximations of matrices. Proceedings of the Twenty-
First International Conference on Machine Learning, 112.
Zhao, L., & Yang, Y.-H. (1999). Theoretical analysis of illumination in PCA-based vision
systems. Pattern Recognition, 32, 547–564.
Zhu, W., Ma, X., Zhu, X.-H., Ugurbil, K., Chen, W., & Wu, X. (2022). Denoise functional
magnetic resonance imaging with random matrix theory based principal component
analysis. IEEE Transactions on Biomedical Engineering, 69, 3377–3388.