How to compute the average path from scattered data (and its variance)?

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Alberto Bacchin on 27 Sep 2021
Answered: William Rose on 29 Sep 2021
I have a set of 2-dimensional vectors which contain each one the positions of a robot during experiments. Each vector has different size. I would like to compute a path that corresponds to the average of all paths (and possibly the variance). The idea is to represents in a compact way the result of all the experiments.
I found a function, but I don't understand how it works. Are there any solution to my problem? I tried to pad all the vectors such that they have the same size and then use mean() command, but results are poor.
Here a plot of some paths I want to average:

William Rose on 27 Sep 2021
This is a good question which arises in slightly different forms in a wide array of problems. One example: how to average an ensemble of GPS position recordings, each of which corresponds to the same course. Another (from my work): find th average hip or knee angle (in 3D), over one stride, when you have a recording of a person walking for many strides on a treadmill, and the stride lengths and durations vary somewhat.
I assume the recordngs do not all have the same number of points.
Resample each vector to have 1000 elements, using interp1(), then average them with mean().
This is probably the best approach in the absence of ther information.
The most obvious "other information" would be a time stamp for each position. If your data is sampled at uniform intervals, then the array index is a time stamp. If you want the average position at each time, and if the recordings are of varying length, then pad at the beginning with the initial position, or pad at the end with the final position, to get vectors of uniform length, then average them. You said you did something like this and the results were bad. But I don't know how you padded. A different padding choice could help.
Alberto Bacchin on 28 Sep 2021
@William Rose Yeah! The fact tha the robot follow a trajectory that is not a math function is really challenging. By the way, I have time stamps. But I would like to use only spatial information because I don't care about time in my analysis.
I tried to pad the end of the vector by repeating their last element to set the same size for all. But this lead to some wrong estimations, espacially at the end. I think it is due to the fact that sometimes the robot was faster, sometimes it was slower (see picture).
This is why I would not use time in synchronization. But it seems the only way...
In the comment to the answer below I described my second approach.

Adam Danz on 27 Sep 2021
Edited: Adam Danz on 27 Sep 2021
It looks like the following assumptions can be made:
• The robot starts at the same location
• There is a uniform temporal sampling interval.
If this is true, why not just average the (x,y) coordinates across all trials?
Alternatively, you could compute the 2D density of (x,y) values using histcounts2 and then use those data to compute the path of highest density within a 2D grid. This approach would require lots of repetitions (more than what is shown in your sample image).
Adam Danz on 28 Sep 2021
I have analyzed similar data created by monkeys steering through a virtual environment but I was working with hundreds of trials. Analyzing 8 trials will produce weak results given the noise within each trial no matter what method you use. I would still recommend using histcounts2 or histogram2 if you want to know the spatial density of trajectory data within a grid.

William Rose on 28 Sep 2021
Since you said you do not care about time, it is OK to normalize all the paths to have the same number of steps. Here is a script that
1. Generates 8 random paths with a different numbers of steps for each
2. Computes 8 paths with the same number of steps, by interpolation
3. Computes the mean+-SD path
4. Plots the mean path and plots 1-SD ellipse at 20 points along the path
The generates the plots below. Each run will have different semi-random walks. Good luck.
William Rose on 29 Sep 2021
I really like the the visual impact of the colored 2D histogram overlaid on te actual paths. It is a very nice example of good graphical communication. The code to make it is very efficient. The 2D histogram approach handles short paths and paths that turn back on themselves better than my approach does. The mean-of-the-normalized paths approach has some attractive features, but as Adam correctly notes, it doesn;t work well if the paths are not reasonably similar. Alberto, you have a couple of potentially good options, or use both!

William Rose on 29 Sep 2021
@Alberto Bacchin, here's another plot you can make, if you use the mean-of-the-normalized paths approach (pathSimulateAndAverage.m). The plot shows the raw paths, mean path, and 90% and 95% confidence regions.
The script pathSimulateAndAverage.m makes the paths and the plot above. It calls plotFilledEllipse.m (attached). Thanks to @Star Strider for providing the basis of plotFilledEllipse().
The script calls chi2inv() in the Stats and Machine Learning Toolbox. If you do not have that toolbox, then, if desired confidence interval (ci)=.90, .95, .99, replace chi2inv(ci,2) with 4.605, 5.991, 9.210.

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