Global vs. Local in Complex Systems.

The type of models I build fall under the very broad category of Complex Systems. I’ve talked a bit about them here, but I’ve found that I don’t get many hits when I actually write about what I do. But I thought I’d take another stab at the apple and write a tiny bit about the nature of my models, because I think it’s relevant to many types of systems modeling that you might actually be interested in, like climate science, ecology, or economics.

The difference between the kind of modeling I do and those, is they are generally very broad, large-scale, continuous-domain models. My models are smaller, restricted, and hybrid-domain. Continuous domain models are used for things where most model variables and infinitely sub-divisible. Like time, and distance, and temperature. Hybrid models are used for when certain variables can only vary discretely, like number of people, exam rooms, etc..

Climate and economic models are usually concerned with long-term, global effects. And it should be noted here that “global” does not necessarily mean “earth”. In the case of climate and economy it often does. But “global” is a technical term in engineering meaning that the effects we’re concerned with represent values ranging over the entire model, rather than a “local” portion of it. I’m not going to get in to the strict definitions of “global” and “local” using mathematics, because they get technical and boring and pedantic, if you’re not a mathematician.

To be a little less handwavy about it (but still pretty handwavy), imagine a continuous function on the real numbers. Let’s let f(x)=sin(x) + 2* cos (x+1) – 2* sin^2(x). The domain of the function is the entire real numbers (Meaning, I’m allowed to put any number I care to in for “x”.) . And when I let x vary over the real numbers, I get a strange undulating curve:

I’ve circled two areas of interest, one in red, one in blue. The red circle surrounds a local minimum. When I’m confined to a small area, this looks like it’s as low as I can get. However, the blue circle surrounds a (non-unique) global minimum. If I look at the entire global space, I will never find any place lower than this one. Now, that’s pretty easy to do with well-behaved functions like this. But when your function is “height of the tide” and your global space is “planet earth for the next 100 years”, it can be impossible to know if you’re in a local minimum or a global minimum. The space is too hard to search. A local minimum might masquerade as a global one.

But in many ways, it’s much easier to predict global effects than local ones. This is because local effects are subject to randomness in often alarming ways that global effects are not. This is why the weather is hard to predict, but we’re all pretty sure it’ll be warmer in July than February. In general, simulations are good at predicting broad global trends. They’re not as good at predicting local effects, and they’re not good at distinguishing local phenomena from global phenomena.

So, in my models, the “global” effects I’m looking to predict are things like, “How much can I reduce average  inpatient boarding censuses by reorganizing the types of surgeries performed on various days of the week?” In climate science, a global effect is along the lines of, “How will monsoon weather patterns shift given a 40% reduction in summer time Arctic sea-ice?” And we’re decent at that kind of thing (Though I just made that climate question up, and have no idea if anyone is looking at such a thing, or what the answer might be.). In my world, a “local” effect would be, “How crowded is the emergency department going to be on Tuesday, August 3rd?” In climate, it might be, “What will the rain gauge look like in Florida in July of 2015?” We can’t answer these kind of questions, and we will never be able to. They’re too dependent on too many factors that are influenced by randomness.

It is really tempting to try to predict local phenomena. That’s what the media wants. That’s what politicians want. People ask if tornadoes are the result of global warming. Or if a particular bond issue led to revitalization in a midwestern city. But we can never truly know those things. It’s easy to look at situations in hindsight and see causes and effects. But for local phenomena, small-scale events in a large system, the truth is, causality is obscure. Generally, the best we can do is say that local phenomena are made more or less likely by global trends.

So when the media and politicians make sensational claims about complex systems, feel free to ignore them. The actual papers done by competent scientists are always carefully qualified with detailed lists of assumptions and alternative scenarios. Predictions are tempered with confidence intervals and probabilities. Those things rarely get reported. But, when good, well-validated models are used to predict broad trends? Pay attention. Because we know how to do that pretty well, these days. And the results can be enlightening.