誤差最小降維模型於流形之分析

Abstract

本研究主要目的是要呈現一個新的流形分析降維方法，透過分析得到原始地圖的每一點之間的距離矩陣後，誤差最小法是透過最小化變形地圖的與原始地圖的距離矩陣誤差，在低維空間去建構新的變形座標矩陣，若是誤差趨近於零，新的座標矩陣就能完整保留原始座標矩陣的距離特性。 本研究使用了台北市做案例分析，由於台北市本身豐富的多樣化地理環境與密集的城市建築，點與點之間路徑的距離不可能皆為直線距離，透過原始座標系統與交通路網建構出一個台北市的距離矩陣，透過MDS與誤差最小化方法分別作出新的變形變量圖，而這個變量圖就是以距離為主要變數將台北市的每個座標點重新分配位置，希望能在維持台北市地圖結構與維持距離屬性之間取得平衡。

The research presents the new method for manifold analysis. Error Minimizing method is a dimensional reduction method that find the new node coordinate by seeking minimal error. the error is equal to the square of new distance over the square of original distance. The new distance will be accordant with original distance when error is close to 0, therefore the new node coordinate in new 2-D space maintain the characteristic. As a research case, Taipei city has diverse geographical environment and the great dense of buildings, therefore the real path between two nodes can not be a straight line. The research is committed to develop a cartogram which the distance between two nodes will change according to the real distance constructed by traffic route for automobile. obtain the balance of that maintain the structure of the map and maintain the characteristic of distance.