最大變異斜交轉軸本質編碼

Varimax Obliquely Rotated Essence Codings

林偉捷(Wei-Chieh Lin)

指導教授 ： 鄭少為

國立清華大學/理學院/統計學研究所/碩士(2019年)

摘要

在Peng (2018) 中，提出本質編碼與本質效應的觀念。其所討論的數據型態是反
應變數為函數型變數，而解釋變數則為純量型變數。在其所考慮的函數型線性
模型中，函數型效應是由數個未知的本質編碼與本質效應之線性組合所形成，
其中本質編碼必須在斜交調控矩陣下直交。由於本質編碼與本質效應皆假設未
知，故其會有無窮多組解，為了解決這個問題，Peng (2018) 提出一些準則以唯
一定義本質編碼和效應。但在其提出的定義準則中，斜交調控矩陣必須事先人
為給定，而選取不同的斜交調控矩陣則會導致分析結果亦有所不同。本文將設
定斜交調控矩陣為未知矩陣，並開發方法以利用數據估計之。本文採用因素分
析中最大變異旋轉的概念，對本質編碼引入最大變異準則，以使估得的本質編
碼具有較易於解釋的性質。而這些本質編碼的估計法，則是透過比較不同的斜
交調控矩陣所對應的本質編碼，在最大變異準則上的表現，以選出最合適的斜
交調控矩陣。我們將此法應用於晶圓厚度實驗數據上，並比較其所發現之本質
編碼與之前方法的異同之處。

關鍵字

最大變異旋轉 ； 斜交轉軸 ； 本質編碼 ； 因素分析 ； 函數型線性模型

並列摘要

Peng (2018) proposed the use of essence codings and essence effects for functional linear models with functional response and scalar explanatory variables. In his work, the coefficient functions are assumed to be linear combinations formed by some unknown essence codings and essence effects with the constraints that the essence codings are orthogonal with respect to a known obliqueness control matrix. Under this setup, the essence codings and essence effects have infinitely many solutions because both of them are assumed unknown. To tackle this problem, Peng (2018) further proposed some reasonable optimization criteria to uniquely
define a best solution. In this thesis, we allow the obliqueness control matrix to be unknown, and use data to estimate it. Our estimation procedure adopts the concept of varimax rotation in factor analysis. By varying the obliqueness control matrix and imposing varimax criterion on the corresponding essence codings, we develop the estimators of essence codings with better interpretability. Our estimators are the obliqueness control matrix and its corresponding essence codings that maximize the varimax criterion. We illustrate this method using both simulated
data and a real wafer-thickness data, and compare our results with the ones obtained from the method in Peng (2018).

並列關鍵字

varimax ； rotation ； oblique ； essence ； codings ； factor ； analysis ； functional ； linear ； model

參考文獻

[1] Hsu, Y.-S. (2018). Smoothing functional essence codings, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.

[2] Johnson, R. A. and Wichern, D. W. (2007). Applied Multivariate Statistical Analysis, 6th edition, Pearson Prentice Hall, New Jersey.

[3] 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.

[4] Nelder, J. A. and Mead, R. (1965). A Simplex Method for Function Minimization. Computer Journal, 7, 308-313.

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