Posts tagged with algebraic geometry

roots of `x²⁶•y + x•z⁶ + y¹³•z + x⁹•y¹³ + z²⁶     =   0`

$\dpi{200} \bg_white \large x^{27} \cdot y+x \cdot z^6+y^{13} \cdot z+x^9 \cdot y^{13}+z^{26}$

(Source: imaginary.org)

Witches Kitchen, 1971: To give an approximate sense to the statement … I had to abuse the listeners’ patience for almost two hours. [As a book] it probably takes about 400, 500 pages. …[O]ur thirst for knowledge & discovery indulges itself … in a logical delirium far removed from life, while life itself is going to Hell in a thousand ways…. —Alexander Grothendieck

hi-res

## Geometry of the Nielsen PRIZM

The Nielsen PRIZM groups people into 66 “demographic and geographic market segments” for the purpose of advertising to them.

Each of the segments has a nice description to go along with it. It’s the kind of story you want to hear as a marketer: it uses relatively in-depth knowledge of Americans, plus stereotypes or shallow summaries, to draw a character with enough roundness that you could pitch to him/her. That is, you could write copy or film a creative spot that you believe could speak to members of this cohesive segment.

As I read more deeply into the Nielsen-Claritas PRIZM, however, the 66 segments started to sound like perhaps they were generated by a simple formula. From their slideshow I learned that they divide the US population by:

• affluence
• population density
• kids/no kids + age

Rather than use continuous on the implied cube (3 dimensions above), they lump various ranges together. They also lump the interaction terms unevenly—for example, (suburban & income) is lumped more finely and (urban & income) is lumped more coarsely. Specifically,

• 4 totally -ordered levels of urbanity (measured by population density per zip code) urban  suburban  second city  town & rural
• 14 levels of Affluence Groups (so they consider finer gradations of wealth & income within suburban and low-density zip codes and coarser income gradations in cities and second-cities)

• Three life-stage categories, accommodating both those who do and don’t raise children at some point. {youngish && no kids, kids, oldish && no kids at home}.

Younger folks (this is under-35’s or under-45 DINKs) are less graduated by affluence than families or older folks (over-55’s or over-45 DINKs).

By the way, over-65’s are outside PRIZM’s marketing groups. I guess it’s assumed that they won’t buy big-ticket items or change their ways much unless the Monday lima-bean special becomes 25cents cheaper at Lida’s Diner than Bill’s Diner. Then you’ll see the entire community switch to Lida’s.

Like the MBTI, it assumes that: People fit in rectangles.

Unlike the MBTI, rather than using four sliding scales `[0,1]⁴`, the PRIZM uses discrete, totally ordered sets—something you could build with the `letters` and `combn` functions in `R`.

I started to wonder: is it really true that members of segment 26 are “urbane” and “love the nightlife” — even the empty-nesters and older homeowners of the segment? Is there really a “laid-back atmosphere” to segment 25? Or are these merely colourful papier-mâché rudely draped over a box?

Mostly, of course, I’m concerned with segment 31, the well-known Urban Achievers:

HOW I SEE IT

When I look at a painting, I’m tempted to glance quickly and pass on. In order to appreciate a piece, I imagine the strokes and colour choices that make up the painting. I imagine myself painting the same thing. What would it have felt like to be inside Cy Twombly's hand while he painted Apollo 17? That gives me a better feeling of the art.

When I look at the Nielsen Prizm the same way — try to get inside the heads of its creators — I sense that they adopted the `[0,1]⁸` rectangular structure simply because they’re not aware of alternatives. MBA’s do plenty of mathematics, but I’ve never seen any business mathematics cross over into CW-complexes, 3-tori, arborescences, or Lobachefskyan geometries. It could be that the people who designed the Prizm simply didn’t have anyone on their team who had heard of this stuff. All the quants were working on Wall Street rather than Madison Avenue. (Wacker Drive rather than Michigan Ave.)

The ribbon-farm guy (Venkatesh Rao) is a rocket scientist who crossed over into marketing, but so far I haven’t read enough of his stuff to say if he dove into algebraic geometry—it seems he did more functional analysis, optimisation / control theory, and differential geometry. Which is what I would expect rocket science consists of.

I will admit that the PRIZM’s use of two “matrix” presentations with colour-coding, pictures, defined ranges, and toss-away combinations is quite clear. Probably works better than when I tell clients “Just picture a 5-dimensional manifold, I won’t say the norm because I think it’s spaced differently in the center than the edges—and let’s not get into the interaction terms yet”. But—the bones of their model are really just `[0,1]³`. They’ve dressed it up and they’ve done more than that (segmenting and dropping). But a cube is the underlying architecture.

Is the Prizm simple or oversimplified? I feel it’s the latter. Not that I object to mathematical models of behaviour, emotions, or any human thing—but the hypercube metaphor just doesn’t fit my presumption of the shape of the space.

• Does consumer space have 8 corners to it?
• What’s the best interpretation of “distance” in the consumer space?
• Do all of the lines really cross at right angles, in a hyper-grid? Was that supposed to be implied?

WHEREUNTO

I don’t want to carp about somebody else’s work without at least offering constructive criticism. What are some potentially better ways to think about the space of all consumers—potential buyers of houses, cars, vacations, DVD’s, washers, ‘n’all that?

Mathworld’s picture of a few topological objects gives one starting point:

One thing I noticed pretty quickly: you remember playing Star Fox battle mode? Or any video game where there is a lower-right thumbnail of you on a limited square map—such that when you go leftwards off the map you appear on the right, and when you go upwards off the map you appear on the bottom? As a kid I thought I was flying on the surface of a planet, but in fact it was the surface of a torus. (Why? If you go up to the top of the North Pole you don’t come out again at the South Pole. See the picture of the sphere with B ≠ C, i.e. N ≠ S.)

In other words, a torus (donut) is the product of `a_loop × a_loop`. Whereas a sphere (ball) is the product of `a_loop` (east/west) `× a_line_segment` (north/south).

GEOMETRY

Following from this short lesson in topology, one alternative to multiplying only “linear” dimensions of characteristic attributes would be to multiply lines with loops. For example `a_loop × a_loop × a_line_segment`. I’m not sure what the name for that shape is, but you can imagine it — like a cylindrical torus. And it’s logically possible that there are two circle-like dimensions in marketing. Something like, as politics goes further and further left, it starts to resemble the far right more than the middle. But relevant to marketing.

A second alternative then might be to consider, like in the image above, the endpoints of some line segments from the 3 dimensions of Nielsen. What if some of them were identified rather than left distinct? What kind of shapes could you create with that and would that resemble the consumer space more than a rectangle?

Some other ideas of things to question:

• How do angles meet up? (inner product)
• How do distances work? (norms)
• Look through an algebraic geometry book, or Solid Shape. Are there any shapes—umbilics, furrows, biflecnodes, dimples, trumpets—that have an analogue in the space of all consumers?
• Is backwards just the opposite of forwards? Or does that wrongly assume commutativity?

I don’t know if that would result in a better model. I don’t know if thinking about things this way would reduce wasteful ad spending. I don’t have data to test these ideas on. I just wanted to share this thought.