應用等距特徵映射法分析太平洋島嶼氣候資料

Abstract

在處理複雜或高維度資料時，降維是一個相當重要的課題。傳統的降維方法，如主成分分析，通常為建立在歐式空間中。然而當資料具有曲面的結構時，在歐式距離看似相近的兩點，本質上可能距離非常遠，因而在降維時造成誤差。基於流形學習的非線性降維方法，如等距特徵映射或局部線性嵌入，即是針對上述問題所發展。本研究應用等距特徵映射法分析太平洋島嶼氣候資料，以瞭解其數據的結構。在等距特徵映射中，首先根據原始距離構建鄰接圖，接著利用最短距離演算法所計算出的最短路徑來近似測地線距離。最後，將測地線距離矩陣輸入到多元尺度法中以建立能夠維持高維度裡相似性的二維座標。結果顯示第一個等距特徵映射維度與緯度有關係。另外從資料類別的敏感度分析中，我們可以推斷，對於一些國家，它們可能在空間距離上較近，但是，溫度或降雨距離可能要大得多。此外，藉由本研究我們也可以觀察資料結構隨時間的變化。

Dimensionality reduction is an important issue while dealing with complex datasets. Traditional methods, such as principal component analysis (PCA) or multidimensional scaling (MDS), are usually based on Euclidian distance. However, this may underestimate the true distance between data points while there exists a curved structure of data. In order to solve the problem above, nonlinear dimensionality reduction methods based on manifold learning, such as isometric feature mapping (Isomap) or locally linear embedding (LLE), are proposed. In this study, Isomap is applied to help us understand the structure of climate data in the Pacific Islands. In Isomap, the neighborhood graph is constructed based on the original distance. Next, geodesic distances is approximated by the shortest path calculated by using Dijkstra’s algorithm. Finally, the geodesic distance matrix is imported into multidimensional scaling (MDS) to create a two-dimension map representing their dissimilarities in a higher dimension. Results have shown that the first Isomap dimension has a relationship with latitude. The sensitive analysis of variables has shown that though some countries are spatially close, their temperature or rainfall distance are much farther. Also, we can observe the change of data structure along time through this research.