Category: Aside

I like e day better, but I think I’m in the minority. Pi still makes many appearances in the analysis of dynamic systems. Anyway, here’s a cool video that links the two, avoiding the usual infinite series of Euler’s formula.

An older, shorter version is here.

Notice that the definition of e^x as a function satisfying f(x+y)=f(x)f(y) is much like reasoning from Reality Checks. The same logic gives rise to the Boltzmann distribution in the concept of temperature in partitions of thermodynamic systems.

It’s hard to work under these conditions.

Nelson Repenning & colleagues have a nice new paper on problem formulation. It’s set in a manufacturing context, but the advice is as relevant for building models as for building motorcycles:

Anatomy of a Good Problem Statement
A good problem statement has five basic elements:
• it references something that the organization cares about and connects that element to a clear and specific goal or target;
• it contains a clear articulation of the gap between the current state and the goal;
• the key variables—the target, the current state and the gap—are quantifiable,if not immediately measurable;
• it is neutral as possible concerning possible diagnoses or solutions;
• it is sufficiently small in scope that you can tackle it quickly.

I haven’t had time to write much lately. I spent several weeks in arcane code purgatory, discovering the fun of macros containing uninitialized thread pointers that only fail in 64 bit environments, and for different reasons on Windows, Mac and Linux. That’s a dark place that I hope never again to visit.

Now I’m working fun things again, but they’re secret, so I can’t discuss details. Instead, I’ll just share a little observation that came up in the process.

Frequently, we do calibration or policy optimization on models with a lot of parameters. “A lot” is actually a pretty small number – like 10 – when you have to do things by brute force. This works more often than we have a right to expect, given the potential combinatorial explosion this entails.

However, I suspect that we (at least I) don’t fully appreciate what’s going on. Here are two provable facts that make sense upon reflection, but weren’t part of my intuition about such problems:

• An n-dimensional sphere inscribed in an n-dimensional hypercube occupies approximately none of the volume for large n.
• For random points distributed in n-dimensional space, the distance to one’s nearest neighbor approaches the distance to one’s farthest neighbor for large n.

In other words, R^n gets big really fast, and it’s all corners. The saving grace is probably that sensible parameters are frequently distributed on low-dimensional manifolds embedded in high dimensional spaces. But we should probably be more afraid than we typically are.