A couple of weeks ago, I attended my department's lecture series which featured a professor from the CSM math department giving a talk on what he refers to as "active subspaces". The concept refers a way to reduce the complexity in high dimensional problems by exploiting lower dimensional structure. In simple terms, find what dimensions and variables are not impacting the higher order results and align your axis in a way that eliminates these variables. From a mathematical standpoint, the concept intrigued me.
Having been immersed in the concepts of space and time this week, I was reminded of the talk by certain sections in Einstein's dreams. Given the sensitive relationship often assigned to time and space, this idea got me thinking of time as an exploitable variable. Being a research oriented geophysicist, keenly interested in problems having significant spatio-tenporal elements, I've been pondering whether time is an element or dimension that can ever truly be reduced out of a problem.
Now, it certainly is an element that is reduced out of many data-sets. Some data sets exist in a "time-series" -- intimately linked to time -- and others exist without any sense of time. To give concrete examples, I've worked recently on modeling asteroids, a dataset that is purely spatial as far as my analysis is concerned. An asteroid has a defined shape and I've eliminated the element of rotation with time, as it is considered irrelevant for my goals. I've also worked with tsunami data, analyzing the number of events over a period of time. A third, more subtle example might be my work this past summer with microseismic data from fraccing, a datatset that has definite spatio-temporal elements, but also one that can be analyzed as a function of the entirety of events without any recognition of time.
In thinking about these distinctly unique sets of data, I've returned to the question on whether time can actually ever be completely removed from a set of data. Or rather, whether it should ever be completely removed. For the sake of some analysis, it is probably realistic to assume that it can be removed, but I can't help but feel after the readings this week that both the frame of reference for the timing of data and unique aspects of time itself are crucial to all problems, just in different ways. For example, in my asteroid dataset, the asteroids themselves do not change shape in short times (as far as I know...), however, the directionality of orbiting instruments with respect to the rotation of the asteroid likely has implications on the recoverable data collected. In this case, the element of time now becomes crucial, as the orbiting parameters are not only a function of space, but also time.
The other more "out-there" idea I'll leave you with is the concept of data in non-linear time space. As Einstein's dreams has made evident, the concepts of time vary greatly with a little bit of imagination. What would it look like to analyze data as a function of some type of non-linear time? Would this ever be useful? Obviously slight modifications like log-time or things of that nature are already done, but are more radical time constructs useful in a mathematical sense? If anyone knows of anything that is being done in this nature already, I'd love to hear or read about it.
Complex networks (https://en.wikipedia.org/wiki/Complex_network), at their simplest level, do not have any concept of space or time, but rather rely solely on connectivity. So I think the answer to your first question is yes, time can be irrelevant in some cases, and one can simply disregard it.
ReplyDeleteIn answer to your second question, time is already related nonlinearly to space via the conserved interval s^2 = x^2 - c^2 t^2 in special relativity. This interval is conserved under relativistic frame changes, i.e., Poincare transforms. So just by changing velocity of the viewer, we find nonlinear time.
What did you mean by log time? I thought that was interesting.
Thanks for the links, Lincoln.
ReplyDeleteI grasp conceptually how time and space can be non linearly related, especially after last week's discussion, but am still a bit perplexed by how this is practically applied in data. Perhaps its hidden in layers of problems I have not dealt with yet. Adjusting for relativistic effects from a moving reference frame seems useful, but actually doing data analysis in significantly non-linear time still seems a bit-far more convoluted, as far as the type of problems I deal with are concerned.
As for log time, I was simply referring to viewing data in log space (usually log-log plots). A plotting convention used in a number of problems in classes I've had.