Posts tagged with quasimetrics

## Manifolds, Star Fox, and Self-versus-Other

Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

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Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods" that Newton & Descartes supposed make up the fabric of space.

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved" metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around "0", one’s own perspective — and spread out where the “others” stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.

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The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

## Noncommutative distances between industries

The distance from your house to the grocery must be the same as the distance back, but 20th-century mathematicians speculated about circumstances where this might not be the case.

Very small-scale physics is non-commutative in some ways and so is distance in finance.

But non-commutative logic isn’t really that exotic or abstract.

• Imagine you’re hiring. You could hire someone from the private sector, charity sector, or public sector. It’s easier for v managers to cross over into b | c than for c | b managers to cross over into v.

So private is close to public, but not the other way around. Or rather, v is closer to b than b is to v.  δv, | < δb| . (same for δ| vc |)

• Perhaps something similar is true of management consulting, or i-banking? Such is the belief, at least, of recent Ivy grads who don’t know what to do but want to “keep their options open”.

This might be more of a statement about average distance to other industries ∑ᵢ δ| consulting, xᵢ | being low, rather than a comparison between δ| consulting, x |   and   δ| x, consulting | . Can you cross over from energy consulting to actual energy companies just as easily as the reverse?

• Imagine you’re want a marketing consultant. Maybe some “verticals” are more respected than others? So that a firm from vertical 1 could cross over into vertical 2 but not vice versa.
• Is it easier for sprinters to cross over into distance running, or vice versa? I think distance runners have a more difficult time getting fast. If it’s easier for one type to cross over, then δ| sprinter, longdist |    δ| longdist, sprinter |.
• It’s easier to roll things downhill than uphill. So the energy distance δ | top, bottom |  <  δ | bottom, top |.
• It’s usually cheaper to ship one direction than the other. Protip: if you’re shipping PACA (donated clothes) from the USA to Central America, crate your donation on a Chiquita vessel returning to point of export.

Noncommutative distance, homies. (quasimetric) And I didn’t invoke quantum field theory or Alain Connes. Just business as usual.

Another quasimetric! Actually a pseudo-quasi-metric. Facebook’s news feed decides what status updates to show you based on three factors:

1. affinity u — how much you like somebody (FB’s guess)
2. type of post w — videos and photos are more interesting than “Chris liked Lili’s status update”
3. time since post d — show newer stuff

$\dpi{200} \bg_white \sum_\mathrm{graph} \scriptstyle \text{affinity}_e \; \cdot \scriptstyle \text{ type}_e \; \cdot \scriptstyle \text{ time decay}_e$

The affinity score is a pseudoquasimetric — it’s a one-way measure. If I’m always looking at your profile because you’re a fascinating girl that I met at a party and have been stalking ever since, your updates will show up more in my feed.

But, say you’re the girl and you didn’t think I was interesting at all and never looked at my profile after you Accepted my offer of internet-ual befriendship, my stories won’t be interesting to you and are unlikely to be displayed.

In other words, stalkers don’t show up in your news feed.

The properties of a quasimetric, again, are:

1. never below zero
2. the only “distance of zero” is the distance of something to itself
3. dist( ①→③ )  ≤   dist( ①→② )   +   dist( ②→③ )

Had to drop the second one because there might be lots of FB friends Ⓑ,Ⓒ,Ⓓ,Ⓔ who are not important to me Ⓐ and thus dist( Ⓐ→Ⓑ ) = 0 | Ⓐ≠Ⓑ. (But maybe not?)

Anyway, I threw a "pseudo" on the "quasimetric" just in case lots of different people are of zero interest.

Math news out.

## Temperature Preference

Here’s another example of a quasimetric.  My girlfriend was arguing that winter is worse than summer.  Her reasoning was this:  if the ideal temperature is 72 Fahrenheit, plus or minus, then winter deviates much further from ideal than does summer.  In Indiana, temperatures often get down to daily highs in the 10’s, 20’s, 30’s in the winter — but they don’t get up to 110’s, 120’s, 130’s in the summer.  (And that doesn’t even take nighttime / lows into account.)

But since people do choose to live in cold places, their preferences mustn’t be symmetrical.  It must be that colder-than-ideal is not as bad as hotter-than-ideal.  Probably because you can wear a coat, but not an anti-coat!  Well, I hate wearing coats, she said.

So her preferences are more or less symmetric. But other people’s climate preferences are a quasi-metric.

## quasimetrics

A quasi-metric is just an asymmetrical measure of distance.

Physical distance is measured symmetrically. The distance from Bloomington to Madrid is the same as the distance from Madrid to Bloomington. Obviously! But if you measured investments symmetrically, you would err.  In finance, gains are good and losses are bad.  Also obvious!  But what that means, mathematically, is that there is only one direction: up (OK, there’s also negative up). Asymmetrical distance.

I think of this like the down on a duck’s back, or the hair on your arm.  All the hairs are pointing the same way.  ”Hairs” = a 1-form.  Which assigns to any point in the theoretical financial space a number: your portfolio returns.

$\large \dpi{200} \bg_white \quad \{ \mathrm{what\ could\ happen\} \ space} \longrightarrow \ \quad \$

OK, that’s a less obvious way to think about it.  But the modeling point is robust:  any time you’re implementing a financial model, don’t penalize gains!  This idea has only recently been incorporated at my brokerage: they now report the forecast likelihood of 10%, 20%, 30% losses to my portfolio — rather than so-called “risk %” which is really standard deviation, squared.