In my own research, I have discovered the usefulness of a tool that was developed by mathematicians at least as far back as Sophus Lie, the use of invariants to characterize geometric objects. The idea is pretty simple, given some sort of geometric objects and some notion of equivalence of such objects (e.g., two surfaces are equivalent if you can translate and rotate them so that they coincide with each other), find a finite set of intrinsically defined quantities such that two objects are equivalent exactly when those quantities are equal. This idea has recently found its way into computer vision. If you have a particular person's face in mind, it will appear different depending on the angle from which it is recorded. How does a computer recognize the face? One approach is to compute some invariant quantities and compare those to the ones on file.

My question is, are there any invariants associated to various markets or to various "phases" of markets? I would readily accept that there aren't in any meaningful way, and that I am just infatuated with this idea, but I am interested to know what others might think.

Vinh Tu says:

I order to count, one often needs first to classify. In some cases the classification is trivial. But it can also be pretty complex. For instance, how do you define a trend, or a break-out, or a reversal? You have to discard some part of the data, which you call noise, and fit the "relevant" parts into categories based on invariants. Interesting that you mention computer vision. I've been thinking about computer vision algorithms as well, and how they could be used to classify the features of market movement. There are myriad ways of presenting facets of market data as surfaces of varying dimensions, and I suspect perhaps there may be useful computer vision algorithms to classify areas by flatness, roughness, stability and slope. And, as always, after one has classified and hypothesized, one needs to count. I remind myself that, before jumping to conclusions based on some measurements, one should always check to see how likely it is those measurements may be due to chance. And, as often demonstrated on this web site, monte carlo and bootstrapping techniques can be very useful, both for solving analytically hairy problems as well as double checking ones math. 


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