Solved – Similarity measures between curves

Question

I would like to compute the measure of similarity between two ordered sets of points—the ones under User compared with the ones under Teacher:

The points are curves in 3D space, but I was thinking that the problem is simplified if I plotted them in 2 dimensions like in the picture. If the points overlap, similarity should be 100%.

Answer 1

You are comparing trajectories, or curves. This is a studied topic. Procrustes analysis and dynamic time warping, as EMS says, are tools of the trade. Once you've aligned the curves you'll want to measure the distance, say the Fréchet distance. If you want to share some of your data we could take a crack at it ourselves.

Relevant reading:

• Curve Matching, Time Warping, and Light Fields: New Algorithms for Computing Similarity between Curves
• Discovering Similar Multidimensional Trajectories (uses the LCSS for robustness)

If you disregard the temporal dimension:

You could fit the User and Teacher to multivariate Gaussian densities and find the volume of their product--that's pretty easy. If you want more accuracy, you could use a nonparametric density estimate instead.