由於科技的與日俱增，城市透過物聯網技術也變得更加智慧。而智慧化運輸更是對減少交通流量
，以及預測車輛的常用路徑起了重要作用。由內建GPS的車輛所產生的空間資料可用於找出未知
的訊息。這些重要的空間資料常用於許多研究領域。其中一項便是尋找不尋常的路徑
。本論文的研究便是尋找不尋常的路徑，並將其從正常路徑中區別出來。在第一階段，GPS資料
將被匯入、清理、進行預處理。在第二階段，將採取不同的異常值偵測方法來找尋最佳的相似度
測量與群集方法
。在第三階段，不同的群集方法對所有不同的相似度測量將被檢驗與分析，並以相互之間的穩定
性作為標準進行比較。在第四階段，最佳方法的結果將在數位地圖上視覺化以展示不尋常軌跡。
We evaluated our implementation using different similarity measures like Frechet distance, Dynamic Time Warping distance, Longest Common Subsequence distance, Hausdorff distance, Edit distance and lustering methods used are K Means, CLARA (Clustering Large Applications), PAM (Partitioning Around Medoids), Hierarchical clustering, Model Based Clustering and FANNY
(Fuzzy Analysis Clustering).
我們的評估包含了不同的相似度測量，如Frechet distance、Dynamic Time Warping
distance、Longest Common Subsequence distance、Hausdorff distance、Edit
distance；不同的群集方法則包含K Means、CLARA (Clustering Large Applications)、PAM (Partitioning Around Medoids)、Hierarchical clustering、Model Based Clustering 與FANNY (Fuzzy Analysis lustering)。

The smart transportation is of great importance to reduce the traffic and predict the most frequent path taken by the vehicles. One of the key technologies is trajectory analysis, which can be used for the study of ride sharing, traffic flow, abnormal detection, and more. In this thesis, we provide an empirical survey on the trajectory analysis for abnormal detection. The analysis has four stages. In the first stage, importing, cleaning and pre-processing of the GPS data will be initiated. In the second stage, different outliers detection methods will be used to find the best similarity measure and clustering method. In the third stage the validation and analysis of the different clustering methods with all similarity measures will be compared according to the internal and stability measure criteria. In the fourth stage, the method with the best output will be visualized on a digital map to see the abnormal behavior. We implemented different similarity measures, including Frechet distance, Dynamic Time Warping distance, Longest Common Subsequence distance, Hausdorff distance, Edit distance, and different clustering methods: K Means, CLARA (Clustering Large Applications), PAM (Partitioning Around Medoids), Hierarchical clustering, Model Based Clustering and FANNY (Fuzzy Analysis Clustering). We evaluated those methods using the HsinChu Bus trajectories,GeoLife dataset, Rio De Janeiro bus dataset and Android app Go Track dataset. Experimental results show that Frechet distance and Hausdorff distance with Hierarchical clustering provides the best result in terms of internal and stability measure criteria. The higher value of Silhouette width and Dunn Index clearly signifies that Frechet and Hausdorff distance when used with Hierarchical clustering give clusters with strong bonds and hence are able to find the abnormality in the trajectory data.

Contents
Chinese Abstract i
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vii
List of Algorithms ix
1 Introduction 1
2 Methods and Algorithms 4
2.1 Data Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Map matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Path Compression . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Similarity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Dynamic Time Warping (DTW) . . . . . . . . . . . . . . . . . 5
2.2.2 Frechet Distance . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Longest Common Sub-Sequence . . . . . . . . . . . . . . . . . 7
2.2.4 Edit distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.5 Hausdor distance . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Clustering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 K-Means clustering . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.2 Partitioning Around Medoids clustering . . . . . . . . . . . . . 10
2.3.3 Clustering Large Applications . . . . . . . . . . . . . . . . . . 12
2.3.4 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . 13
2.3.5 Fuzzy clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.6 Model Based clustering . . . . . . . . . . . . . . . . . . . . . . 16
3 Cluster Analysis and Validation 18
3.1 Optimal number of clusters . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Elbow method . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Average Silhouette method . . . . . . . . . . . . . . . . . . . . 19
3.2 Comparison of Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Internal measures . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Stability measures . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Cluster Validation Methods . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Silhouette Width . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Dunn Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Compute p-value for Hierarchical clustering . . . . . . . . . . . . . . 22
4 Implementation and Experiments 23
4.1 Importing and cleaning data . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Preprocessing of data . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Conversion to spatial lines . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Data matrix using similarity measures . . . . . . . . . . . . . . . . . 27
4.5 Cluster analysis and validation using 5 similarity measures and 6
clustering techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5.1 DTW distance and 6 clustering techniques . . . . . . . . . . . 28
4.5.2 Frechet distance and 6 clustering techniques . . . . . . . . . . 32
4.5.3 Hausdor distance and 6 clustering techniques . . . . . . . . . 34
4.5.4 Edit distance and 6 clustering techniques . . . . . . . . . . . . 36
4.5.5 LCSS distance and 6 clustering techniques . . . . . . . . . . . 38
4.6 Finding the best clustering technique and similarity measure . . . . . 41
4.7 Visualization of clustering using Frechet distance on two dimensional
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.8 Visualization of the clusters using Frechet distance on the digital map
of city . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.9 Visualization of clustering using Hausdor distance on two dimensional
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.10 Visualization of the clusters using Hausdor distance on the digital
map of city . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.11 Complete table of observations . . . . . . . . . . . . . . . . . . . . . . 46
4.12 p-value calculation of hierarchical clusters . . . . . . . . . . . . . . . 47
4.13 List of the R packages used in coding . . . . . . . . . . . . . . . . . . 48
4.14 Implementation on Go Track dataset . . . . . . . . . . . . . . . . . . 49
4.15 Implementation on Geolife dataset . . . . . . . . . . . . . . . . . . . 49
4.16 Implementation on Rio De Janeiro dataset . . . . . . . . . . . . . . . 50
5 Conclusion 52

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