正則化空間型態估計

Abstract

全球暖化和聖嬰現象對世界各地的氣候造成許多變異現象,除了海水溫度 異常升降外,各區域降雨量也產生極大的落差,導致近年來熱浪、水災、 乾旱等天災不斷。研究者們透過大氣動態的發展,觀察並且預測氣候變 遷的脈絡,藉此研究找出減少傷害、增加安全的有效方法。 於大氣科學 中,研究者通常透過某些重要的「空間型態」來分析大氣動態發展的隨機 過程,其中涵蓋了主成份分析及最大變異分析等統計方法。然而在訊噪 比過低時,分析結果多為雜亂難解的估計型態,相對不具任何物理意義。 因此,本論文提出正則化估計方法,可使分析後的估計型態具合理的空間 相關性,同時突顯重要特徵,並能增加其解釋性和精準度。此外,也利用 交替方向乘子法(alternating direction method of multipliers) 建構出新的演算 法,用以加速高維度資料之分析。在實際應用上,透過對印度洋海平面溫 度的資料分析,建立溫度的空間變異結構,進而研究溫度變化對非洲東部 降雨量之影響。

Climate changes are associated with atmospheric dynamics and inevitably affect human life. Recent developments in atmospheric and oceanographic sciences have shown that these dynamics can be studied through spatial patterns of variables of interest. Although principal component analysis (PCA) has been applied to seek spatial patterns, and maximum covariance analysis (MCA) has been applied to discover coupled spatial patterns between two processes, the patterns obtained from PCA or MCA may sometimes be too noisy to be physically mean- ingful when the signal-to-noise ratio is low. Therefore, to obtain more precise estimates of spatial patterns, we propose two regularization approaches, called SpatPCA and SpatMCA, to promote spatial features in dominant spatial patterns by introducing smoothness and sparseness penalties while accounting for their orthogonalities. Our methods allow to analyze data taken at irregularly spaced or sparse locations. Further, the resulting optimization problems can be solved by the alternating direction method of multipliers, which is computationally efficient and easy to implement. Finally, the effectiveness of the proposed methods is demonstrated by several numerical examples that include identifying dominant patterns of sea surface tempera- tures in the Indian Ocean and an application of how precipitations in East Africa are affected by sea surface temperatures in the Indian Ocean