函數型線性模型中的最大變異直交轉軸本質編碼

Varimax Orthogonally Rotated Essence Codings in Functional Linear Models

呂長祐(Chang-Yu Lu)

指導教授 ： 鄭少為

國立清華大學/理學院/統計學研究所/碩士(2021年)

摘要

本文考慮函數型線性模型 Y (t) = β0(t) + Xβ(t) + ε(t)，其中 Y (t) 為函數型反應變數，X 為純量型解釋變數，β(t) 為係數函數，ε(t) 為具同質性和獨立性之誤差。對此模型，我們進一步假設模型中的 β(t) 為某些未知的本質編碼Φ(t) 之線性組合，亦即 β(t) = ΓΦ(t)，其中 Γ 稱為本質效應。我們將 Γ 和 Φ(t)視為類似於因素分析中的潛變數與負荷矩陣的結構，在本文中探討對 Φ(t) 做直交旋轉的問題。在考慮 Peng (2018) 既有的本質編碼序列式定義準則，以及 Lin (2019) 所提出之最大變異旋轉準則下，我們透過子空間的觀點，重新對本質編碼定義準則，以使得所估得的本質編碼能在相同的子空間內同時極大化這兩種準則。並討論極大化這兩種準則的求解方法，和本質效應的估計與檢定。最後，我們將這些方法分別應用於模擬數據以及真實晶圓數據上，並討論其分析結果。

關鍵字

最大解釋變異 ； 因素分析 ； 最大變異直交旋轉 ； 梯度投影法

並列摘要

This paper considers the functional linear model Y (t) = β0(t) + Xβ(t) + ε(t), where Y (t) is a functional response variable, X is the model matrix formed by scalar explanatory variables, β(t) contains all coefficient functions, ε(t) are er-rors with homogeneity and independence assumptions. We further assume that every coefficient function in β(t) is a linear combination of some unknown essence codings Φ(t), i.e., β(t) = ΓΦ(t), where Γ is a matrix of the coefficients of Φ(t). The elements in Γ are called essence effects. Because both Γ and Φ(t) are assumed unknown, the essence effects and codings possess a structure similar to the latent variables and loading matrices in factor analysis. In this paper, we study the prob-lem of orthogonal rotation of Φ(t), which functions on Φ(t) in a similar manner as rotating the loading matrix in a factor analysis. Considering the existing cri-terion in Peng (2018), which sequentially defines essence codings with maximum explained variation, and the varimax rotation criterion in Lin (2019), we propose a criterion for defining and estimating essence codings from the perspective of sub-space. This new criterion allows the estimated essence codings to simultaneously maximize explained variation and varimax. We also offer the estimation method for essence codings under the new criterion, and the estimation and testing pro-cedures for essence effects. Finally, we apply the new method on simulated data and real wafer data, and discuss their analysis results.

並列關鍵字

maximum explained variation ； factor analysis ； varimax orthogonal rotation ； gradient projection algorithm

參考文獻

[1] Bernaards, C. A. and Jennrich, R. I. (2005). Gradient Projection Algorithms and Software for Arbitrary Rotation Criteria in Factor Analysis. Educational and Psychological Measurement, 65(5), 676-696.

[2] Liao, Y.-F. (2018). Identifying essence codings and effects in functional linear models with homogeneous and independent errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.

[3] Lin, W.-C. (2019). Varimax Obliquely Rotated Essence Codings, Master the-sis, National Tsing Hua University, Hsinchu, Taiwan.

[4] Peng, P.-R. (2018). Identifying essence codings and effects in functional lin-ear models with heterogeneous and correlated errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.

[5] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd edition. Springer, New York.