Posts tagged with quotient space

## 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.

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.

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.

## The Geometry of Musical Chords

by Dmitri Tymoczko

A musical chord can be represented as a point in an orbifold. (An orbifold is a quotient manifold.)

Line segments join notes of one chord to those of another. Composers in a wide range of styles have exploited the (non-Euclidean) geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation.

(via Artemy Kolchinsky)

book