Trajectory Clustering: Which Clustering Method?

Question

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.

Answer 1

I do not know what you mean exactly by 'trajectory'. But what I understand is a one-dimensional vector with numeric values.

My suggestion here is to use Dynamic Time Warping (henceforth, DTW), a method able to 'align' two one-dimensional signals. Then, if you want to measure belongness, DTW provides you with several metrics to state to what extent two signals are similar.

In addition, you can perform clustering based on the metrics provided by DTW instead of using the standard metrics involved whether in Fuzzy C-means or $k$-means.