I started reading something on topology and statistical distributions. This is one of my research topics. My intuition tells me there is something to the notion that linear analyses fail when the space is hyperbolic, and succeed despite themselves when the space is spherical. In this reading the author said that the distribution sits on a manifold. Why a manifold? I’ll take that under advisement and wait for it to become more meaningful.

I’ve sketched tons of graphics. Some of those I’ve used in this series of posts on the normal distribution. We’ll look at one more, a normal distribution on a small sphere. The tails get longer, but like a squeezed water balloon, the noise that fills that additional length had to come from somewhere. We usually think of the distribution as sitting on a flat Euclidean plane. When the distribution is sitting on a sphere, flat goes out the window. The noise comes from the bottom of the distribution moving outward from the mean.

The figure shows how the shape of the distribution would change and how the tails reach around the sphere. In the figure, the increase in tail length is projected back a Euclidean plane. Lost in this is the height loss that happens as the number of standard deviations, or sigmas increase, as the firm grows larger.

Notice that in this particular figure, I’m taking the Frequentist point of view that of a distribution containing random noise, rather than knowledge.

Back in my radar mechanic days, the story about using a Styrofoam cup and a microwave transmitter as a bug was making it’s rounds. Microwaves are fun stuff. If you change the shape of the container, you change the frequencies emitted. That container is typically a metal waveguide. These waveguides are firm. They don’t get deformed in typical usage. But in this application as a bug, the lack of firmness is essential. A room vibrates, so the cup vibrates. Sound waves bounce around the room, so they eventually deform the Styrofoam cup. Those deformations clip frequencies from the square microwave pulse filling the cup.

The figure shows the opening of a Styrofoam cup in blue. The red sphere fits inside the cup. When you pick the cup up, the cup is deformed. The deformation makes the opening thinner, but longer. The frequencies that were the size of the red sphere are now clipped as only frequencies the size of the brown sphere can fit into the cup. You could use that Styrofoam cup to squeeze out some Morris code.

In the last two posts I wrote about how black swans clip the tail of the distribution. Black swans are typically big price/valuation losses on in the financial markets, aka missed quarters and such. But, these black swans are like setting an epsilon in calculus when you are trying to find the convergence of a function, so these tail movements happen all the time by tiny price fluctuations that happen every day once you’re a public company. Your normal distribution is like that Styrofoam cup. Your price constantly vibrates. This also means that your outliers might be under the distribution one day and not the next. Your real option tracking portfolio would act similarly, so the strategic decisions driven by those real options would oscillate as well.

Given that a statistical distribution has a surface defined by a function, and functions can be analyzed via a Fourier analysis, the shape of the distribution is doing the Styrofoam cup thing inside the distribution, not just relative to the outliers at the base of the distribution.

In this figure, I’ve drawn the largest sphere, the largest frequency, that could fit under the distribution. I also drew a much smaller sphere. Notice that with an unlimited budget, there is no physical limit to how small that smaller sphere could be. Alas, at some point you end up with a laser, aka another way to pick up the vibrations from a Styrofoam cup. But, budget and significant use keep us from getting smaller frequencies into our Fourier analysis. This is much like doing a factor analysis. There is always another factor, but the smaller those factors get, the more expensive it is to capture the underlying data. Beyond budget, you might want to ask just how much company cognition you can dedicate to ever finer factors.

I did not show the spheres between the largest and smallest that would fit or pack under or inside the distribution. Also not shown are the lifecycle of the distribution. The largest sphere gets smaller as the company gets larger. The spheres start out small in a Poisson distribution and get larger as those Poisson distributions tend to the normal.

The above figure roughly lays out the Poisson distributions under the normal while those Poisson distributions tend to the normal. Frequentist probabilities use the law of large numbers to find macro-level behavior, and the law of small numbers to find micro-level behavior. Poisson distribution provide the basis for Markov chains. Markov chains begin chipping away at the notion of that a distribution only contains random noise. Markov chains begin to structure the contents of a normal distribution, the normal distribution being large. Poisson distributions, aka small distributions constitute traversals of the area under the distribution, aka vectors, like a vector of differentiation.

The Poisson distribution here starts as an outlier. It follows a chain of vectors until it gets to the mean. It could be a random walk under the normal. It need not pass through the mean. When we seek out our next discontinuous technology, our random walk would be a Levy flight. Imposing structure happens as we gain knowledge of the systems under the normal. As this structure is imposed, the probabilities become less Frequentist and more Bayesian.

The next few figures illustrate how the technology adoption lifecycle imposes structure on the contents of the normal distribution, on the once random variables.

Here I wanted to show how the B2B early adopter was an outlier. Yes, I called that early adopter a weird person. They don’t make a good reference case for other prospects in their vertical. The technology adoption lifecycle is organized by pragmatism. Marketing would set the width of each slice of pragmatism. Those widths can change. But, business cases and other reference data needs to be generated for each slice. Market to a few, since traversal across the market takes time, but try selling to just one at a time. Requirements collection needs to be bound by the width of the pragmatism slices as well.

In the above figure I’ve put the B2B early adopter, who is an outlier in our category’s normal distribution, is also an outlier in the normal distribution of the vertical that that B2B early adopter does business in. In the bowling ally, the seats and dollars you will take from that vertical is the only workable stage gating you’d want to do before deciding to take on building that B2B early adopter’s product visualization.

Structure comes out of nowhere, well, out of the daily operations, experimentation, and other efforts to reduce the uncertainty of the innovating organization. Geometries change, the shapes of the noise change, distributions change, but they change in organized ways. Frequentist probabilities are not the only ones out there. Some people have found that they have topological problems. Know who to call.

Enjoy? Comments?

April 29, 2015 at 6:39 am |

[…] Here I’m using circles as a Fourier analysis of the value chain. I’ve followed the Styrofoam cup as microphone notion of saying the circles fit the largest area between the constraining elevations of both the taxonomy […]