Suppose those random variables are represented as scalar time series with nonlinear relationship (thus applying linear correlation or covariance would be futile) that may change over time, then is there a machine learning algorithm that would approximate or "learn" such relationship?

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    $\begingroup$ If a machine learned linear relationship between $x^2$ and $y$, does that means the machine learned non-linear relationship between $x$ and $y$? $\endgroup$ May 28 '17 at 14:55
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    $\begingroup$ What do you know about their relationship? What kinds of relationships are more likely than others? The "no free lunch" theorem (roughly) says there's no hope if you don't have any priors on the kinds of relationships that are plausible; but in practice we usually can narrow down the types of relationships that are plausible. So to make this answerable, I suspect you're going to need to provide more context about your specific situation. And welcome to the site, by the way! $\endgroup$
    – D.W.
    May 28 '17 at 16:11
  • $\begingroup$ fuzzy logic is the science of adaptive learning of the non-linear transfer function by initially applying random noise and comparing the response with predicted value, Sony uses this method for low lightn camera servo focus to get the fastest response time. But it is used in every field from Engineering to the Stock Market. There are also algorithms for optimizing PID (proportional, integral and derivative ) response in motor servo control systems with dynamic load and power control to establish the k gain factors for each variable. $\endgroup$ May 29 '17 at 3:25
  • $\begingroup$ All modems have an adaptive EQ learning algorithm for matching frequency dependant attenuation, prop delay , phase shift with f and echo cancellation. The algorithms are diverse from impulse response, swept stimulus/response to cross correlation noise reduction methods. $\endgroup$ May 29 '17 at 3:29

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