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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1 Comment so far

  1. David Whitesel on June 9, 2008 9:48 am

    Are you familiar with the contruct diversely Identical?

    This borrowed incite might help;

    “You must already have the notion of distinction to entertain the idea of distinguishing between truth and falsity; and you must already have the notion of identity to entertain the idea of drawing a distinction between identities. ”
    Then further “When psychological functions are not consciously available they are automatically projected, externalized. Undoubtedly, this principle of the transference is mirrored in the extrojection of “externalities” by market-oriented economic theorists. The perspective argued here is that externalities can only be introjected to the market by those functions previously not available to conscious awareness: the Karpenko functors of m-valued Lukasiewicz logics.”

    which seems captured to the this underlying descriptor;

    (MOON, Vol. 2, pp. 597-8):

    “The superposed imagery of Analytical Cubism pictures the perceptual process of phenomenological constitution: how the “socialization of cognition” is projected onto the hyletic data (studied by the Impressionist) to constitute the constant object. Object constancy may be characterized by the perceptual gestalt which allows the object to be recognized as selfsame and enduring, despite the infinite variety of percept-patterns (“profiles”: Abschattungen) through which we apprehend it. In movement around the object, we can apprehend it from as many perspectives as we wish — potentially infinite. There must be a subliminal gestalt-of-superposition allowing us to recognize all these images-of-differing-perspective as signifying the constant object…”

    me thinks these 3 outakes fit into your task.


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