基於空間動態結構的正規化主成分分析於時空資料的建模與應用

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研究生：	李漢岳 Lee, Han-Yueh
論文名稱：	基於空間動態結構的正規化主成分分析於時空資料的建模與應用 Regularized PCA on spatial dynamic patterns with applications to spatial-temporal modeling
指導教授：	徐南蓉 Hsu, Nan-Jung
口試委員:	黃信誠 Huang, Hsin-Cheng 陳春樹 Chen, Chun-Shu
學位類別：	碩士 Master
系所名稱：	理學院 - 統計學研究所 Institute of Statistics
論文出版年：	2019
畢業學年度：	107
語文別：	英文
論文頁數：	50
中文關鍵詞：	經驗正交函數、投影梯度下降演算法、懲罰最小平方法、卡爾曼濾波、空間預測、時間預測
外文關鍵詞：	Empirical orthogonal functions, projected gradient descent, penalized least squares, Kalman filter, spatial prediction, temporal forecast
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本研究提出一個尋找時空資料中動態空間結構的正規化方法。模型的建立以Spatial PCA (Wang與Huang, 2016)為基礎，但進一步的將其尋找重要的空間型態(spatial dominant pattern)，延伸至尋找具有時間預測能力的空間型態。本研究所設計損失函數同時考量空間預測損失、時間預測損失、以及基底函數的平滑度，在空間隨機效果模型(spatial random effect model)的架構下，透過投影梯度下降演算法(manifold projected gradient descent)搭配最大期望演算法(Expectation-maximization algorithm)，可有效率的進行參數估計，並取得隨時間演進下可解釋最大空間變異的正交基底函數。此外，本研究亦透過模擬實驗以及以及實際資料分析(西太平洋風速資料集)驗證此方法的效果。結果顯示，在時間預測力以及描繪動態空間結構的表現均較Spatial PCA為佳。

This thesis suggests a regularized method on finding the dynamic spatial patterns for spatial temporal data analysis. The proposed method adopts the idea of spatial PCA (Wang and Huang, 2017) on extracting the spatial dominate patterns, but further enhances on the spatial patterns with better temporal forecast abilities. By minimizing a well-designed objective function which simultaneously considers spatial prediction loss, temporal forecast loss and smoothness penalty, a set of orthogonal basis functions are solved to account for major spatial variations across time in a spatial random effect model framework. The resulting optimization problem for finding the basis functions and other model parameters can be solved efficiently using the projected gradient descent algorithm with orthogonality constraint embedded in an EM algorithm. The effectiveness of the proposed method is demonstrated by simulation experiments and applied to wind speed data observed at the western Pacific Ocean. As a result, the spatial basis functions and the corresponding model derived in this thesis outperform the spatial PCA in terms of representing the dynamic spatial patterns as well as on temporal forecasting in low-dim settings.

Abstract vii
Introduction 1
1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Purpose of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Overview of SpatPCA 5
1 Spatial PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Estimation for Φ . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Estimation for Λ and σ2 . . . . . . . . . . . . . . . . . . . . . 7
3 Prediction for ξt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
The Proposed Method 9
1 Temporal-Spatial PCA . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Filtering and Forecasting for fξtgT t=1 . . . . . . . . . . . . . . . . . . . 10
3 Estimation for Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Estimation for the Rest of Parameters . . . . . . . . . . . . . . . . . . 15
5 Spatial Prediction and Temporal Forecast for Yt . . . . . . . . . . . . . 17
6 Tuning Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . 19
Simulation Study 23
1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Performance Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Compare Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Spatial PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Case I: K^ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Case II: K^ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Case III: Auto-selected K^ . . . . . . . . . . . . . . . . . . . . 31
Application 39
1 Data Description and Preprocessing . . . . . . . . . . . . . . . . . . . 39
2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 Tuning Parameters Selection . . . . . . . . . . . . . . . . . . . 39
2.2 Proposed model application . . . . . . . . . . . . . . . . . . . 41
2.3 Performance comparison . . . . . . . . . . . . . . . . . . . . . 44
2.4 Eigen-image Comparison . . . . . . . . . . . . . . . . . . . . . 45
Conclusion 47
References 49
                                

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