在因子實驗中，可以透過數據決定出最具有解釋力的編碼函數，其稱為
本質編碼。Huang (2020) 在全因子設計下，提出了定義以及估計每個因子的本質編碼之準則和方法。而 Huang (2020) 亦把本質編碼的方法，應用於解構配適好的複雜無母數模型上。但當解構時所使用的全因子設計有較多因子或水準時，估計本質編碼所需使用的矩陣維度會變高，使得計算量過大。在本文中，我們針對反應變數配適高斯過程模型後所得的預測函數，將 Huang (2020) 的方法應用於解構此預測函數上，並提出兩種方法來簡化本質編碼估計的計算量。第一個方法可簡化將全因子設計代入高斯過程預測函數所得的配適值向量之計算量。第二個方法為透過高斯過程預測函數，推導出 Huang (2020) 提出的本質編碼估計準則中，特徵分解所作用的矩陣之函數形式。透過模擬計算，我們發現使用第一個方法所估得的本質編碼，其結果與使用 Huang (2020) 的結果會是一致的，但其可有效地降低計算量。另外，我們透過將全因子設計代入第二個方法所推導出的函數套用 Huang (2020) 的方法所估得的本質編碼，歸納出一些特殊的性質。

Essence coding identifies the most explanatory coding functions based on data. Huang (2020) proposed criteria and methods for defining and estimating essence codings for each factor under a full factorial design, and applied the essence coding approach to decompose complex non-parametric fitted models. However, when the full factorial design used in the decomposition involves many factors or levels, the dimensionality of the matrices required for estimating essence codings significantly increases, leading to an excessive computational burden. In this paper, we apply Huang's (2020) method to decompose the prediction function derived from fitting a Gaussian process model to response variable. We introduce two methods to reduce the computational complexity of estimating essence codings. The first method simplifies the computation of the fitted-value vector obtained by substituting all runs of the full factorial design into the Gaussian process prediction function. The second method derives the functional form of the matrix used in eigen-decomposition for estimating essence coding under Gaussian process prediction function. Simulation studies demonstrate that the essence codings estimated using the first method are identical to those obtained by Huang (2020), but with significantly reduced computational effort. Additionally, by substituting all runs of some full factorial designs into the second method's derived function and using Huang's (2020) method to estimate its essence coding, we deduce specific properties of the essence coding in the context of Gaussian processes.

1 緒論 1
2 文獻回顧 4
2.1 本質編碼的定義及估計準則 4
2.2 高斯過程迴歸與核嶺迴歸 6
3 研究方法 9
3.1 全因子設計下 GP 預測模型之運算簡化 9
3.2 利用 GP 預測函數之運算簡化 14
3.2.1 一個生成樣本 15
3.2.2 多個生成樣本 17
4 數據分析 20
4.1 全因子設計下 GP 預測模型簡化運算驗證 20
4.2 函數 f_i(s,t) 之模擬研究 21
5 結論與討論 31
參考文獻 32

1. Huang, J.-M. (2020). Identifying essence codings and effects for multiple factors under full factorial designs. Master’s thesis, National Tsing Hua University, Hsinchu, Taiwan.

2. Kanagawa, M., Hennig, P., Sejdinovic, D., and Sriperumbudur, B. K. (2018). Gaussian processes and kernel methods: A review on connections and equivalences. arXiv, abs/1807.02582v1 [stat.ML].

3. Liu, S. (1999). Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and its Applications, 289(1):267–277.

4. Plumlee, M. and Joseph, V. R. (2018). Orthogonal Gaussian process models. Statistica Sinica, 28(2):601–619.

5. Wu, C. F. J. and Hamada, M. S. (2021). Experiments: Planning, Analysis, and Optimization, 3rd edition. Wiley.