Kronecker包絡主成分分析模型選擇方法及其應用

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：93

、訪客IP：18.217.104.170

姓名

徐若瑄(JO-HSUAN HSU) 查詢紙本館藏

畢業系所

數學系

論文名稱

Kronecker包絡主成分分析模型選擇方法及其應用
(Model Selection Methods for Kronecker Envelope Principal Component Analysis and their Applications)

相關論文

★ New insights on ′′A semi-parametric model for wearable sensor-based physical activity monitoring data with informative device wear"	★ A parametric model for wearable sensor-based physical activity monitoring data with informative device wear
★ 透過隨機投影降維的函數型資料變異數分析—以穿戴式裝置資料為例	★ 在PU類型資料之下比較三種邏輯斯迴歸模型
★ 邏輯斯迴歸的子取樣方法之比較	★ 用於函數型資料之兩步驟共變異數分析在穿戴裝置資料之應用
★ 兩個具時空效應之隨機場的獨立性檢定

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 赤池訊息準則
★ 貝氏訊息準則
★ 維度縮減
★ Kronecker包絡
★ 主成分分析
★ 維度估計

關鍵字(英)

論文目次

摘要 i
Abstract ii
目錄 iii
一、緒論 1
二、維度縮減方法介紹 4
2.1 主成分分析 4
2.1.1 PCA 情況一：n > p 6
2.1.2 PCA 情況二：n ≤ p 8
2.2 Kronecker 包絡主成分分析 11
2.2.1 TPCA 估計方法 13
2.2.2 KEPCA 情況一：n > p 16
2.2.3 KEPCA 情況二：n ≤ p 19
三、模擬資料分析 22
3.1 模擬實驗一 22
3.1.1 情境1：n > p 22
3.1.2 情境2：n ≤ p 26
3.1.3 總結 28
3.2 模擬實驗二 29
3.2.1 實驗設定 29
3.2.2 實驗結果 29
3.2.3 總結 31
四、實際資料分析 33
4.1 Olivetti 人臉資料 33
4.1.1 資料來源 33
4.1.2 分析結果 34
4.2 MRI 資料 36
4.2.1 資料來源 36
4.2.2 分析結果 37
五、結論 39
參考文獻 41
附錄 42
1 MRI 資料篩選 43
2 MRI 圖像預處理 43

參考文獻

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.

指導教授

黃世豪

審核日期

2023-8-16

推文