Trajectory Clustering: Which Clustering Method?

As a newbie in Machine Learning, I have a set of trajectories that may be of different lengths. I wish to cluster them, because some of them are actually the same path and they just SEEM different due to the noise.

In addition, not all of them are of the same lengths. So maybe although Trajectory A is not the same as Trajectory B, yet it is part of Trajectory B. I wish to present this property after the clustering as well.

I have only a bit knowledge of K-means Clustering and Fuzzy N-means Clustering. How may I choose between them two? Or should I adopt other methods?

Any method that takes the "belongness" into consideration? (e.g. After the clustering, I have 3 clusters A, B and C. One particular trajectory X belongs to cluster A. And a shorter trajectory Y, although is not clustered in A, is identified as part of trajectory B.)

=================== UPDATE ======================

The aforementioned trajectories are the pedestrians' trajectories. They can be either presented as a series of (x, y) points or a series of step vectors (length, direction). The presentation form is under my control.

It might be a little late but I am also working on the same problem. I suggest you take a look at TRACLUS, an algorithm created by Jae-Gil Lee, Jiawei Han and Kyu-Young Wang, published on SIGMOD’07. https://hanj.cs.illinois.edu/pdf/sigmod07_jglee.pdf

This is so far the best approach I have seen for clustering trajectories because:

• Can discover common sub-trajectories.
• Focuses on Segments instead of points (so it filters out noise-outliers).
• It works over trajectories of different length.

Basically is a 2 phase approach:

1. Phase one - Partition: Divide trajectories into segments, this is done using MDL Optimization with complexity of O(n) where n is the numbers of points in a given trajectory. Here the input is a set of trajectories and output is a set of segments.

• Complexity: O(n) where n is number of points on a trajectory
• Input: Set of trajectories.
• Output: Set D of segments

2. Phase two - Group: This phase discovers the clusters using some version of density-based clustering like in DBSCAN. Input in this phase is the set of segments obtained from phase one and some parameters of what constitutes a neighborhood and the minimum amount of lines that can constitute a cluster. Output is a set of clusters. Clustering is done over segments. They define their own distance measure made of 3 components: Parallel distance, perpendicular distance and angular distance. This phase has a complexity of O(n log n) where n is the number of segments.

• Complexity: O(n log n) where n is number of segments on set D
• Input: Set D of segments, parameter E that sets neighborhood treshold and parameter MinLns that is the minimun number of lines.
• Output: Set C of Cluster, that is a Cluster of segments (trajectories clustered).

Finally they calculate a for each cluster a representative trajectory, which is nothing else that a discovered common sub-trajectory in each cluster.

They have pretty cool examples and the paper is very well explained. Once again this is not my algorithm, so don't forget to cite them if you are doing research.

PS: I made some slides based on their work, just for educational purposes: http://www.slideshare.net/ivansanchez1988/trajectory-clustering-traclus-algorithm