Posts tagged with nonlinear

## Nonlinear Superdimensions?

I’m not the first person to say "ceteris paribus is a lie". What this aphorism means is that if you make a c.p. assumption in order to think something through, then the conclusion you reach may be irrelevant to the real world.

Worse, because people don’t understand models, someone might take your careful “A implies B” statement to mean “Both A and B are the case”. For example rather than Edgeworth boxes implying that trade be always mutually beneficial, people might take you to mean that

1. exchange is characterised by Edgeworth boxes
2. private transactions are always Pareto optimal

which is not at all what the theory’s saying. The theory is just connecting assumptions to conclusion: yes, if this were true, then that would surely follow. Which is great because some people don’t actually think such things through.

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Anyway. Ceteris paribus assumptions make thinking easier, but they hamstring whatever you find out—so that it may be useless, or (hopefully not) worse than useless: misleading.

But maybe it’s possible to keep the crutch of c.p. and make it less foolish.

There are some situations where it’s impossible to do what I’m going to suggest—like where space overlaps itself. But in Euclidean spaces it is possible.

Econometricians are already familiar with principal components analysis. You make one “composite dimension” which is composed of a fixed combination of existing dimensions.

`composite = .4 × X₁   +   .2 × X₂   +   1.7 × X₃`

This is what I’m calling a “super-dimension”.

You hold all other things constant so you can think logically about a situation that has the geometry of a single straight line. By creating a composite dimension maybe one could still use the handy ceteris-paribus assumption but roll more of real-life into the model too.

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For example let’s say as wealth ↑, trips to the emergency room ↓. Then you could form a composite dimension with a positive coefficient on wealth, negative on emergency room visits, and talk about both at once with everything else held constant. One step forward relative to talking only about only wealth ↑↓.

But wait — maybe these are only linearly related around a small neighbourhood of some point. Well, we could still create a composite “super-dimension” by varying the coefficients. This could either come in the form of pre-transforming wealth to be log of wealth, or something else — like a threshold effect where we use two or three linear pieces (eg, rich enough with slope=0, way too poor with slope=0, and middle with a linear decrease). In general, whereas linear means `+k+k+k+k+k+k+…`, nonlinear can be interpreted as `+1.2k+k+.9k+.8k+k+1.1k+1.3k+1.2k+1.4k+…`. So instead of constructing a composite dimension with fixed coefficients before ignoring everything else, perhaps one could vary the coefficients along with the space.

That’s all. This may not be a new idea.

George Soros: “I would have bought even more Italian bonds.

• At 6% or 7%, it’s a very attractive speculation. It won’t stay up there forever. If things go wrong it could go up to 10% and you would lose a lot of your money.
• But at 5%, it would be a very nice, stable, long-term investment”.

Filed under #nonlinear and #nonmonotonic.

thank you for the introduction to Newcomb’s paradox! Could you do a post on your favorite paradoxes?
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The decision theory paradoxes I’m familiar with are:

• Ellsberg Paradox— Theorists encode both
1. situations with unknown probabilities, such as the chance of extraterrestrial intelligence in the Drake Equation or the chance of someone randomly coming up and killing you, and
2. situations that are known to have a “completely random” outcome, like fair dice or the `runif` function in `R`,
the same way. However the two differ materially and so do behavioural responses to the types of situations.
• Allais Paradox — The difference between 100% chance and 99% chance in people’s minds is not the same as the difference between 56% chance and 55% chance in people’s minds. (In other words, the difference is nonlinear.) At least when those numbers are written on paper.

Prospect theory proposes the following [0,1]→[0,1] function describing how "we" perceive probabilities
(Remember that it shouldn’t be taken for granted that everybody thinks the same, or that it’s possible to simnply re-map a person’s probability judgment onto another probability. Perhaps the codomain needs to change to something other than [0,1], for example a poset or a von Neumann algebra.)
• Newcomb’s Paradox — This one has a self-referential feel to it. At least as of today, the story is well told on Wikipedia. The Newcomb paradox seems to undercut the notion that “more is always preferred to less” — a central tenet of microeconomics. However, I believe it’s really undercutting the way we reason about counterfactuals. I actually don’t like this one as much as the Ellsberg and Allais paradoxes, which teach an unambiguous lesson.
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Despite the name, they’re notreally paradoxes. They are just evidence that probability + utility theory ≠ what’s going on inside our 10^10 neurons. I don’t think Herb Simon would be surprised at that. (Simon is famous for arguing to economists that “economic agents” — both people and firms — have a finite computational capacity, so we shouldn’t put too much faith in the optimisation paradigm.)

You can find out a lot more about each of these paradoxes by googling. As is my way, I’ve tried to provide the shortest-possible intro on the subject. Twenty-two slides opening the door for you.

I also think it’s interesting how the calculus disproves Zeno’s paradox and how a proper measure-theory-conscious theory of martingales disproves the St. Petersburg paradox. I also think Vitali sets and the Banach-Tarski paradox are compelling arguments against the real numbers. Particularly since everything practical is accomplished with (finite) `float`s, I’m not sure why people hold on to  in the face of those results.

But personally, I’m more interested in decision theory / choice theory than those pure-maths clarifications.

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I know I am forgetting several interesting paradoxes which have revolutionised the way people think. (Zeno thought his reasoning was so revolutionary that he concluded, via modus tollens, that the world didn’t actually exist. One of many religions that has come to such a belief, not to mention Neo and Morpheus thought so.)

hi-res

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.
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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.
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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.

[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.

Mark Liberman, in When “Fuzzy” Means “Smoothed Piecewise Linear”

One cool thing to imagine: the multi-dimensional space of parameters of the control system, the space of all possible tunings of the knobs — and how a few multi-dimensional charts — how do they meet up in this high-dimensional space? — link together.

## arXiv

Hola Nerds,

Have you ever found yourself browsing tumblr and thinking to yourself: “Instead of looking at pictures of dyed hair, I should really be thinking about chirped dissipative solitons" ?

Well, now you can. I ported the following arXiv feeds to tumblr:

If you’d like me to add another, tweet @isomorphisms. Or if you don’t tweet, use http://isomorphismes.tumblr.com/ask.

Gracious thanks to codecogs, perl monks, p.t. campbell, tumblr, and of course the arXiv & supporting institutions.