I think a force that causes all the data points to move on one direction would be akin to a “translation” or a shift in linear geometric transformations. There are many other linear transformations as far as I’m aware, like mirroring about an axis, scaling, etc.
But what about non linear transformations? We really want to be able to describe any change, that is and combinations of changes.
As I’ve done more thinking on the question I’ve realized at least for the boundary denoting the inside of a cluster of points, the nonlinear transformation of space is there same as every possible group of points linear transformations of space. That means every 3 point combinations for a 2d space has to be computed and stitched together. Anyway that would probably suffice.
Anyway, to attempt a better answer to your question, I will share an example. Let’s imagine each datapoint is my heart rate every minute for the past year. Take every 2 minutes as an observation. What you have is an x, and y. Now you have thousands of observations plotted on a 2d space. You can then look at the observation that followed for each and you will have a new set of coordinates you can plot. Deriving the transformation from one space to another is my goal. You can do this in any number of dimensions you want too, why not all of them? The number of dimensions corresponds to the number of datapointa in your observation.
You may say well what’s the point of all this? Imagine I isolate the 6:58-7:00 observation for each day and it’s future observation 7:00-7:02. My alarm goes off at 7. The observations while I’m sleeping will be every different than the observations after I awake. So we will have a very particular mapping between the two spaces. That mapping is the signature of a real world event. I can predict what will happen today if I have the mapping of that time and extrapolate my 6:58-7:00 observation into the future space.
It seems like nn or any pattern recognition algorithm is essentially deriving this mapping. Why not infer it and use it to predict directly?
Thanks for you’re thoughts on the topic 