Can I use Principal Curves Analysis to fit a Vector Cloud instead of a Point Cloud? I have recently discovered Principal Curves while trying to solve the problem I will describe below.
The principle of Principal Curves is to fit a cloud point to find the "path" running along that points.
My goal is to use something like this to fit a set of GPS tracklogs running along a road, so that the result would be a statistical aproximation of the "real" road (or, more precisely, the region of the road where people - bikers in this case - ACTUALLY ride, which most probably would be side lanes and road shoulders).
The problem is: while classical Principal Curves Analysis consider the point set to have independent points, I believe it would be more appropriate to consider each track to be composed of dependent vectors.
When using points, for example, a track segment with high sampling rate might cause a bias when compared to another with lower sampling rate, although each of them describe only ONE trajectory each, and I am more interested in weighting the trajectories with one another instead of the individual points. (by the way, does that rationale of mine make sense?)
I posted a question on the same topic, but with different formulation, in the Gis.StackExchange site, with some representative pictures:
https://gis.stackexchange.com/questions/70623/how-can-i-statistically-calculate-the-real-road-from-a-set-of-gps-tracks
Thanks for reading!
 A: This sounds like a functional data problem; the individual paths could be see as realisations from some underlying smooth curve (function) that you wish to estimate.
Jim Ramsay has a nice website on FDA with various examples, which may give you a flavour of what FDA is and what it can do.
Looking at @whuber's comment (in the gis.se Q&A) regarding the autocorrelation, that reminds me of some work of Rob Hyndman (@RobHyndman, also of this parish) on functional time series. This is not the same thing as @whuber mentions (the functional timeseries would view the individual realisations as being observed sequentially in time, i.e. the series of tracks represent a time series), but might give you ideas for how to consider handling the autocorrelation in the data.
As for principal curves, the smoothers that are fitted to each variable and that describe the location of the curve along that dimension are essentially plugin components of the technique, at least in the way Hastie & Steutzle originally described the method. You could quite easily adapt the principal curve code in the R package princurve to use a plugin smoother that fitted curves allowing for residual correlation, for example by basing the smoother plugin on the gamm() function from the mgcv package, which can apply certain correlation structures for the covariance matrix of the fitted model. Doing this is not too hard, technically; I've done this myself recently to use Poisson GAMs as the smoothers instead of simple smoothing splines.
If this looks like some half-baked suggestions, it is; I offer them as some observations seeing as the Question hadn't been Answered yet.
