Posts tagged with metric

A road map of mathematical objects by Max Tegmark, via intothecontinuum:

The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures.

For instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.

hi-res

[T]he point of introducing L^p spaces in the first place is … to exploit … Banach space. For instance, if one has |ƒ − g| = 0, one would like to conclude that ƒ = g. But because of the equivalence class in the way, one can only conclude that ƒ is equal to g almost everywhere.

The Lebesgue philosophy is analogous to the “noise-tolerant” philosophy in modern signal progressing. If one is receiving a signal (e.g. a television signal) from a noisy source (e.g. a television station in the presence of electrical interference), then any individual component of that signal (e.g. a pixel of the television image) may be corrupted. But as long as the total number of corrupted data points is negligible, one can still get a good enough idea of the image to do things like distinguish foreground from background, compute the area of an object, or the mean intensity, etc.

Terence Tao

If you’re thinking about points in Euclidean space, then yes — if the distance between them is nil, they are in the exact same spot and therefore the same point.

But abstract mathematics opens up more possibilities.

• Like TV signals. Like 2-D images or 2-D × time video clips.
• Like crime patterns, dinosaur paw prints, neuronal spike-trains, forged signatures, songs (1-D × time), trajectories, landscapes.
• Like, any completenormedvector space. (= it’s thick + distance exists + addition exists + everything’s included = it’s a Banach space)

(Source: terrytao.wordpress.com)

B*tchin’ six dimensional 6-cube. The rainbow colours and glass panes really help this visualisation.

## Examples of 6-dimensional things

If it’s hard to envision 6 dimensions, consider this: the possible tunings of a guitar constitute a 6-dimensional space. You can tune to EADGBE (standard), DADGAB, drop-D, DADGAD, GCCGCC, BEBEBE, CGCFGE, and many others.

(If you consider notes an octave apart to be equivalent, then we’re talking about a quotient space, each distance being topologically on a loop. But that’s just one system of musical valuation — and like the winding number of a complex number, it’s totally apparent that high octaves do not sound exactly the same as low sounds. And doing a 720° is more impressive than a 360°. If the abstract “loop” is unwound, there is a highest note (“1”) and a lowest note (“0”) that can effectively be played on each string (dimension).)

` `

You can also think about 6-D as being the six columns in a table or array. For example the { RBI, on-base percentage, fielding errors, stolen bases, sacrifice flies, and home-runs } for a number of baseball players.

Or you can think about six security prices moving in parallel, from bell to bell at the NYSE.

Again the lowest price is called “0” and the highest is called “1”. This renaming places the jumping Brownian motions inside a secteract. So instead of six 1-D paths it’s one 6-D path:

` `

Enough examples of 6-dimensional things. Back to the 6-cube itself.

Let’s make one.

The bounds of the secteract (its “corners”? Or should I say its 6-corners.) come from filling in each of six slots with either 0 or 1.

There are 64 ways to do this. (two options for each of six slots = 2^6.) For example (0,0,0,0,0,1) is one, (0,0,0,0,1,0) is another, and (0,1,1,0,1,0) is a third out of the 64.

The R programming language was nice enough to write out all of the vertices for me without my having to type much. Here they are:

`> booty=c(0,1) > expand.grid(booty,booty,booty,booty,booty,booty) #rockin everywhere`

```   Var1 Var2 Var3 Var4 Var5 Var6
1     0    0    0    0    0    0
2     1    0    0    0    0    0
3     0    1    0    0    0    0
4     1    1    0    0    0    0
5     0    0    1    0    0    0
6     1    0    1    0    0    0
7     0    1    1    0    0    0
8     1    1    1    0    0    0
9     0    0    0    1    0    0
10    1    0    0    1    0    0
11    0    1    0    1    0    0
12    1    1    0    1    0    0
13    0    0    1    1    0    0
14    1    0    1    1    0    0
15    0    1    1    1    0    0
16    1    1    1    1    0    0
17    0    0    0    0    1    0
18    1    0    0    0    1    0
19    0    1    0    0    1    0
20    1    1    0    0    1    0
21    0    0    1    0    1    0
22    1    0    1    0    1    0
23    0    1    1    0    1    0
24    1    1    1    0    1    0
25    0    0    0    1    1    0
26    1    0    0    1    1    0
27    0    1    0    1    1    0
28    1    1    0    1    1    0
29    0    0    1    1    1    0
30    1    0    1    1    1    0
31    0    1    1    1    1    0
32    1    1    1    1    1    0
33    0    0    0    0    0    1
34    1    0    0    0    0    1
35    0    1    0    0    0    1
36    1    1    0    0    0    1
37    0    0    1    0    0    1
38    1    0    1    0    0    1
39    0    1    1    0    0    1
40    1    1    1    0    0    1
41    0    0    0    1    0    1
42    1    0    0    1    0    1
43    0    1    0    1    0    1
44    1    1    0    1    0    1
45    0    0    1    1    0    1
46    1    0    1    1    0    1
47    0    1    1    1    0    1
48    1    1    1    1    0    1
49    0    0    0    0    1    1
50    1    0    0    0    1    1
51    0    1    0    0    1    1
52    1    1    0    0    1    1
53    0    0    1    0    1    1
54    1    0    1    0    1    1
55    0    1    1    0    1    1
56    1    1    1    0    1    1
57    0    0    0    1    1    1
58    1    0    0    1    1    1
59    0    1    0    1    1    1
60    1    1    0    1    1    1
61    0    0    1    1    1    1
62    1    0    1    1    1    1
63    0    1    1    1    1    1
64    1    1    1    1    1    1

```

And there you have it: an electronic realisation of a secteract. Just as real as a Polyworld life-form.

## 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.

## Unmeasurable Distances

I wrote earlier about the many different ways to measure distance. One way I didn’t include is unmeasurable distance.

Sometimes A is

• tastier,
• sexier,
• cooler,
• more interesting,
• or otherwise better endowed

than B … but it’s impossible to quantify by how much. No problem; just say that A≻B but that |A−B| is undefined.

It’s still the case that if A is sexier than B and B is sexier than C, it must follow that A is sexier than C.

Symbolically: A≻B & B≻C A≻C.

This concept opens up many parts of human experience to the mathematical imagination.

I will also express my view on moral rates of income tax using orderings ≻.

Oh, and if you’re into this kind of thing: using orders instead of measurable quantities kind of saved the economic concept of “utility”. Kind of saved it. At least instead of talking about 174.27819 hedons, nowadays you can just say X is lexicographically preferred to Y. Ordinal utility instead of cardinal utility.

## Circles

A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:

p = ⅔

The subadditive “triangle inequality” A→B→C > A→C no longer holds when p<1.

p = 4

p
= ½
. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
p = 3/2

The moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.

## The Jesus Norm

They say unto him, Rabbi, this woman was taken in adultery, in the very act. Now Moses in the law commanded us, that such should be stoned: but what sayest thou? This they said, tempting him, … they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her.

$\dpi{300} \bg_white \mathrm{Define } \ \|x\|: \begin{cases} \|x\|=\infty & \mathrm{ if \ } \ x \neq 0; \\ \|x\| = 0 & \rm{ otherwise.} \end{cases}$

In other words, unless completely perfect, one is imperfect. This is a different sort of measure than is used in assigning blame in a car crash, or in torts.

Those would take the total sum of damage and fractionally assign responsibility to each party. Allow me the licence to define the “Jesus norm” as above.

It would say that one is either responsible or not responsible.

So if I hurt you at all, I’m equally as responsible as everybody else who hurt you for making you whole again.

## 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.