Isomorphismes

I learned about Zadeh’s fuzzy logic when I was a graduate student…despite the intrinsic interest of the idea, there didn’t seem to be any really impressive results….

When I first heard about “fuzzy logic” control systems (…about 20 years ago — before Google or Wikipedia), I was puzzled. What exactly does the degree of truth of statements have to do with algorithms for controlling trains or elevators? When I asked this question after a dog-and-pony show at a Japanese research lab in the mid-1980s, I got answers … repeating what I already knew about fuzzy logic, without adding anything convincing about the application to control theory.

It sounded to me like technological double-talk. I was sure that the engineers were doing something relevant to control in complicated situations, but the “fuzzy logic” label seemed like a flack’s evocative slogan for a variety of different technologies that didn’t seem to have anything much to do with logic, fuzzy or otherwise.

A friend with a background in chemical engineering set me straight. His explanation went something like this: Standard control systems are linear. That means that controllable outputs (heating, accelerating, braking, whatever) are calculated as a linear function of available inputs (time series of temperature, velocity, and so on).

Linearity makes it easy to design such systems with specified performance characteristics, to guarantee that the system is stable and won’t go off into wild oscillations, and so on. However, the underlying mechanisms may be highly non-linear, and therefore the optimal coefficient choices for a linear control system may be quite different in different regions of a system’s space of operating parameters.

One possible solution is to use different sets of control coefficients for different ranges of input parameters. However, the transition from one control regime to another may not be a smooth one, and a system might even hover at the boundary for a while, switching back and forth.

So the “fuzzy control” idea is to interpolate among the recipes for action given by different linear control systems. If the measured input variables put us halfway between the center of state A and the center of state B, then we should use output parameters that are halfway between state A’s recipe and state B’s recipe. If we’re 2/3 of the way from A to B, then we mix 1/3 of A’s recipe with 2/3 of B’s; and so on.

In the case of the four stages of rice cooking, I suppose that a fuzzy logic controller is able to treat the process as a series of fuzzy or gradient transitions rather than a series of hard, stepwise transitions. … a vaguely analogous method to fit a smoothed piecewise linear model to data about oil recovery as a function of various independent variables, including oil field “age”.

In both cases, the fuzzy approach might well be appropriate, under whatever name (though here’s an alternative story about heating control…).

… And indeed even plain fuzzy is by no means an entirely positive word. When George Bush famously accused Al Gore of “disparaging my [tax] plan with all this Washington fuzzy math”, it was not a warm fuzzy moment.

• …“neuro fuzzy”…
• …a whole fuzzy world out there…
• … semi-skeptical review…

[Update: Fernando Pereira emailed

Petroleum geologists have been pioneers on pretty sophisticated spatiotemporal estimation and smoothing techniques, for instance kriging (aka Gaussian process regression for statisticians). There are tight connections between GP regression and spline smoothing (via the theory of reproducing kernel Hilbert spaces). Either the Saudis are not hiring the best petroleum geologists, or they are being deliberately obfuscating with marketroid talk. I can’t think of any situation in which fuzzy ideas (pun intended) would be preferable to Bayesian statistics for inference.

…]

[Update 2: A review article by David Abramowitch, with slides.

Briefly: the linear regression model. We suppose we can explain or predict y using a vector of variables x. As in Gauß’ estimation theory, y is supposed to be unobservable, and thus has to be estimated. The assumption that y depends on x is expressed this way: the posterior distribution Prob{ Y | X } is different from the prior distribution Prob{ Y }.

The minimization of variance of the difference between [our estimation of Y given X] and [Y] leads to a unique solution: the conditional expectation.

The linear hypothesis says that the estimated value should be an affine expression of X. Moreover, the affine parameters which minimise the variance of the error are given by:



The above linear model coincides with the optimal conditional expectation model when X,Y are Gaussian.