一個好的實驗能有效地收集資料，並提供有用的資訊，使得我們在統計建模時，能提供精確估計及適當的推論。在本文的第一部分，我們主要探討具有空間誤差的反應曲面模型（RSM）之設計及相關應用問題。在具有固定因子效應和空間誤差的反應曲面模型下，我們尋找為估計模型中未知參數的 D 型最適設計。在假設變數實驗設計區域為一個圓，且兩個觀測值之間的相關性，取決於兩個設計點的距離狀況下， 我們給出了最小支持的 $D$ 最適設計的一些性質，以及依據一些可用來尋找正合D 最適設計的數值演算方法，找到數值最適設計結果，並檢視相關性質。此外，反應曲面的另一個主要思想是用於在實驗中獲得最佳反應值。我們希望用這種方法，找到基於電容耦合的堆疊芯片封裝的差分焊盤佈局最適設計。在本論文的第二部分，我們提供了不同焊盤尺寸下的非線性迴歸模型，利用不同頻率與芯片放置間距距離和重疊百分比展示SNR值，給出有關選取頻率的最適實驗設計。最後並使用反應曲面方法，針對非線性迴歸模型中之參數估計。進一步，對於可控制的參數找到最佳設計配置。

Experiments performed with good designs on the settings of controllable factors are essential
collecting useful information efficiently and may provide the precise estimation of statistical
modeling and corresponding inferences. In the first part of this dissertation, our main interest
is to investigate the design problems for first- and second- order response surface models
(RSMs) with correlated errors and related applications. Under the first- and second- order
RSMs with fixed effects and correlated errors, we investigate the D-optimal designs for
estimation of the unknown parameters in the RSMs, for two explanatory variables. The
design region is considered to be within a circle and the correlation between two observations
depends on the distance of the two design points. We present analytical minimally supported
D-optimal design results, and numerical D-optimal designs for circular design region with
spatially correlated error structure.
On the other hand, the main idea of RSM is to use certain experimental procedures to
search for control variables of an optimal response sequentially in an experiment. Based
on such an idea, we would like to find designs optimal for estimation and prediction on the
nonlinear models, used to describe the behaviors of signals obtained from various differential
pad placement designs of a capacitive coupling based stacked die package. In the second
part of this dissertation, we provide parametric models under different pad sizes to exhibit
the patterns of the SNR values versus the frequencies for various die placement designs with
given pitch distance and overlapping percentage. In the end, we use an approach analogous
to the response surface methodology to find the optimal design configuration for each pad
size with an optimal response.

論文審定書 i
誌謝ii
摘要 iii
Abstract iv
1 Introduction 1
2 Preliminaries 4
2.1 Response surface models and information matrices . . . . . . . . . . . . . 4
2.1.1 Regression models with uncorrelated errors . . . . . . . . . . . . . 5
2.1.2 Non-linear regression models with uncorrelated errors . . . . . . . 7
2.1.3 Response surface models with correlated errors . . . . . . . . . . . 9
2.2 The circulant matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Algorithms for optimal designs . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Generalized simulated annealing algorithm . . . . . . . . . . . . . 18
2.3.2 Particle swarm optimization algorithm . . . . . . . . . . . . . . . 22
3 Optimal designs for response surface models with correlated errors 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 D-optimal designs in linear regression model with two dimensional design
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Minimally supported D-optimal designs . . . . . . . . . . . . . . . 31
3.2.2 ExactD-optimal designs for second-order Zernekie polynomial model
on a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 Algorithms for exact D-optimal designs . . . . . . . . . . . . . . . 40
3.2.4 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Case study of wafer measurements . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Corner differential pad placement design for a wireless connection stacked die
package 58
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Stacked corner differential pair analysis . . . . . . . . . . . . . . . . . . . 61
4.3 Parametric model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.1 Estimation results with the ED and GED model . . . . . . . . . . . 64
4.3.2 Validation through the interpolation and extrapolation points . . . . 67
4.4 Characteristics of the response curves . . . . . . . . . . . . . . . . . . . . 68
4.5 Design performance configuration . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Designing experiments for nonlinear model on differential pad placement design
of a capacitive coupling based stacked die package 80
5.1 The D- and I-optimal designs for the GED model . . . . . . . . . . . . . . 81
5.2 Optimal design configuration . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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